enVision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers

enVision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers

Go through the enVision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers regularly and improve your accuracy in solving questions.

enVision Math Common Core 5th Grade Answers Key Topic 5 Use Models and Strategies to Divide Whole Numbers

Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 5.1

Envision STEM Project: Average Temperature

Does Research Use a weather site from the Internet or another source of daily weather reports to find the average daily temperature for your city or town for every day of one month? The average daily temperature is the average temperature for a whole 24-hour period.
Journal: Write a Report that Includes what you found about daily temperatures. Also in your report:
• Find the average daily high temperature for the month. Which day had the greatest high temperature?
• Find the average daily low temperature for the month. Which day had the least low temperature?
• Makeup and solve division problems based on your data.

Review What You Know

VOC

• dividend
• quotient
• divisor
• remainder

Choose the best term from the box. Write it in the blank.

Question 1.
In the equation 80 ÷ 10 = 8, the number 80 is the ___.
Answer:

80 is called dividend

Question 2.
The number used to divide another number is the ___.
Answer:

The number used to divide another number is the Divisor

Question 3.
The result of dividing two numbers is the ___.
Answer:

The result of dividing two numbers is the quotient.

Multiplication and Division

Multiply or divide.

Question 4.
630 ÷ 9
Answer:

70

Question 5.
480 ÷ 6
Answer:

80

Use Models and Strategies to Divide Whole Numbers

Question 6.
755 ÷ 5
Answer:

151

Question 7.
657 ÷ 9
Answer:

73

Question 8.
57 × 13
Answer:

741

Question 9.
71 × 109
Answer:

7739

Question 10.
For the state fair next month, 132 people volunteered to plan the fair’s activities. The volunteers formed 12 equal groups. How many volunteers were in each group?
Answer:

Number of volunteers = 132

Number of groups formed = 12

Now, the number of volunteers in each group =

132/12

= 11

Therefore, the number of volunteers in each group = 11

Question 11.
A town is holding a competition for various athletic games. Each community has 14 players. 112 communities are competing in the games. How many players are competing?
A. 1,676
B. 1,568
C. 126
D. 98
Answer:

1568

Estimate

Question 12.
A county has a goal to build 12,000 bus stop shelters in 48 months. If the county builds 215 bus shelters each month, will it reach its goal? Explain one way to estimate the answer.
Answer:

No

Explanation:

Because, if 215 bus shelters were built each month, in 48 months only 10320 bus shelters were built. 1680 bus shelters will remain.

215×48 = 10320.

12000-10320 = 1680.

Pick a Project

PROJECT 5A
How much does a field trip cost?
Project: Plan an Educational Field Trip
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 5.2

PROJECT 5B
How does an assembly line work?
Project: Design an Assembly Line for Toy Vehicles
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 6.2

PROJECT 5C
How do marathon runners get enough water?
Project: Position Water Stations
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 6.3

3-ACT MATH PREVIEW

Math Modeling

Flapjack Stack

Before watching the video, think:
Pancakes have been around for thousands of years and are popular all over the world. Some pancakes are made from potatoes and served with applesauce and sour cream. Other pancakes are sweet and served with blueberries and maple syrup.
Maybe I should plant a maple tree.
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 6.33

Lesson 5.1 Use Patterns and Mental Math to Divide

Solve & Share
A bakery sells muffins to local grocery stores in boxes that hold 20 muffins each. How many boxes are used if 60 muffins are sold? 600 muffins? 6,000 muffins? Solve this problem any way you choose.

Find the answer for 60 muffins. Then you can look for relationships to help find the answers for 600 and 6,000 muffins. Show your work!
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 6.5

Look Back! How can you use multiplication to help you divide 6,000 by 202?

Visual Learning Bridge

Essential Question How Can Patterns Help You Question Divide Multiples of 10?

A.
A jet carries 18,000 passengers in 90 trips. The plane is full for each trip. How many passengers does the plane hold?
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 6.6
Find 18,000 ÷ 90, the number of passengers on each trip.
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 6.7

B.
Think of a basic fact to help you. 18 ÷ 9 = 2
Think about patterns in place value and multiples of 10:
180 ÷ 90 = 18 tens ÷ 9 tens = 2
1,800 ÷ 90 = 180 tens ÷ 9 tens = 20
18,000 ÷ 90 = 1,800 tens ÷ 9 tens = 200
So, the jet can hold 200 passengers.

C.
Or, use multiplication.
90 × 2 = 180
90 × 20 = 1,800
90 × 200 = 18,000
So 18,000 = 90 = 200
The jet can hold 200 passengers.

Think of multiplication to help you divide.
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 7.1

Convince Me! Look for Relationships If the jet above carried 10,000 people in 50 trips, how many people did it carry each trip? The jet carried the same number of people each trip.

What basic fact helped you find the answer?

Guided Practice

Do You Understand?

Question 1.
Why is 210 ÷ 30 the same as 21 tens ÷ 3 tens?
Answer:

Because 21 tens are 210

3 tens are 30

so, 210 ÷ 30 is the same as 21 tens ÷ 3 tens.

Use Models and Strategies to Divide Whole Numbers 2

Question 2.
A jet carried 12,000 people in 40 trips. If the jet was full each trip, how many people did it carry for each trip?
Use a basic fact to help you.
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 7.5

Do You Know How?

Answer:

Given,

The jet carried 12,000 people in 40 trips.

12000 ÷ 40 = 300

Therefore, 300 people carried in each trip.

In 3-9, find each quotient. Use mental math.

Question 3.
210 ÷ 30 = 21 tens ÷ 3 tens = ____
Answer:

210 ÷ 30 = 21 tens ÷ 3 tens =

21 ÷ 3  = 7

Question 4.
480 ÷ 60 = 48 tens ÷ 6 tens = ___
Answer:

480 ÷ 60 = 48 tens ÷ 6 tens =

48 ÷ 6 = 8.

Question 5.
15,000 ÷ 30 = 1,500 tens ÷ 3 tens = ___
Answer:

15,000 ÷ 30 = 1,500 tens ÷ 3 tens =

1500 ÷ 3 = 500.

Question 6.
___ = 8,100 ÷ 90
Answer:

8,100 ÷ 90 =

810 ÷ 9 = 90.

Question 7.
2,800 ÷ 70 = ___
Answer:

2,800 ÷ 70 = 40

Question 8.
30,000 ÷ 50 = ___
Answer:

30,000 ÷ 50 =

3000 ÷ 5 = 600.

Question 9.
__= 1,800 ÷ 60
Answer:

1,800 ÷ 60 =

180 ÷ 6 = 30.

Independent Practice

Leveled Practice In 10-25, use mental math to find the missing numbers.

Question 10.
560 ÷ 70 = 56 tens ÷ 7 tens = ___
Answer:

560 ÷ 70 = 56 tens ÷ 7 tens =

56 ÷ 7 =

8.

Question 11.
360 ÷ 60 = 36 tens ÷ 6 tens = ___
Answer:

360 ÷ 60 = 36 tens ÷ 6 tens =

36 ÷ 6

= 6.

Question 12.
6,000 ÷ 50 = 600 tens ÷ 5 tens = ___
Answer:

6,000 ÷ 50 = 600 tens ÷ 5 tens =

600 ÷ 5

= 120.

Question 13.
24,000 ÷ 60 = 2,400 tens ÷ 6 tens = ___
Answer:

24,000 ÷ 60 = 2,400 tens ÷ 6 tens =

2400 ÷ 6 =

400.

Question 14.
___= 2,000 ÷ 20
Answer:

2,000 ÷ 20 =

200 ÷ 2 =

100.

Question 15.
6,300 ÷ 90 = ____
Answer:

630 ÷ 9 =

70.

Question 16.
___ ÷ 10 = 24
Answer:

___ ÷ 10 = 24

Let the number be x

x ÷ 10 = 24

x= 10×24 = 240

Therefore, number is 240.

Question 17.
21,000 ÷ ___ = 700
Answer:

21,000 ÷ ___ = 700

Let the number be x

21000 ÷ x = 700

x = 21000 ÷ 700

x= 30

Therefore, number is 30.

Question 18.
2,500 ÷ 50 = ___
Answer:

2,500 ÷ 50 =

250 ÷ 5 = 50.

Use Models and Strategies to Divide Whole Numbers 3

Question 19.
72,000 ÷ ____ = 800
Answer:

72,000 ÷ ____ = 800

Let the number be x

72000 ÷ x = 800

x = 72000 ÷ 800

x = 90

Therefore, number is 90.

Question 20.
56,000 ÷ ___ = 800
Answer:

56,000 ÷ ___ = 800

Let the number be x

56,000 ÷ x = 800

x = 56,000 ÷ 800

x = 560 ÷ 8 = 70

Therefore, number is 70.

Question 21.
___ ÷ 10 = 100
Answer:

___ ÷ 10 = 100

Let the number be x

x ÷ 10 = 100

x = 10 x 100

x = 1000

Therefore, number is 1000.

Question 22.
45,000 ÷ 90 = ____
Answer:

45,000 ÷ 90 = ____

4500 ÷ 9 = 500.

Question 23.
___ = 42,000 ÷ 70
Answer:

___ = 42,000 ÷ 70

4200 ÷ 7 = 600.

Question 24.
64,000 ÷ ___ = 800
Answer:

64,000 ÷ ___ = 800

Let the number be x

64,000 ÷ x = 800

x = 64000 ÷ 800

x = 640 ÷ 8 = 80

Therefore, number is 80.

Question 25.
32,000 ÷ ___ = 400
Answer:

32,000 ÷ ___ = 400

Let the number be x

32,000 ÷ x = 400

x = 32000 ÷ 400

x = 320 ÷ 4 = 80

Therefore, the number is 80.

Problem Solving

Question 26.
The table shows the number of passengers who flew on airplane flights in or out of one airport. Each flight had the same number of passengers. How many passengers were on each flight?
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 8.1
Answer:

Given,

Total passengers = 27000

Number of flights = 90

Let the number of passengers were on each flight be x

Then,

x = 27000 ÷ 90

x = 2700 ÷ 9

x = 300

Therefore, the number of passengers were on each flight are 300.

Question 27.
Algebra A truck delivers 478 dozen eggs to stores in one day. Write and solve an equation to find n, the number of eggs the truck delivers in one day.
Answer:

1 dozen = 12 eggs

Number of dozens = 478

Equation = 478 x 12

n = 478 x 12

n = 5736 eggs

Therefore, the truck driver delivers 5736 eggs in one day.

Question 28.
Paula wants to divide 480 tomatoes equally among 80 baskets. How many tomatoes will Paula put in each basket?
Answer:

Number of Tomatoes = 480

Number of baskets = 80

Now,

= 480/80

= 6

Therefore, paul keeps 6 tomatoes in each basket.

Question 29.
Be Precise Ernesto measured the width of each of the three coins shown below.
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 8.3
What is the difference in width between the widest coin and the least wide coin?
Answer:

The width of the widest coin = 0.84 inch

The width of the least wide coin = 0.7 inch

Difference between them

= 0.84 – 0.7

= 0.14 inch

Therefore, the difference in width between the widest coin and the least wide coin = 0.14 inches.

Question 30.
Higher-Order Thinking A baker uses 30 grams of sea salt for each batch of bread. Sea salt comes in an 18-kilogram package or an 800-gram package. Which size package should the baker buy so that no sea salt is left after all of the batches are made? Explain.
1 kilogram equals 1,000 grams
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 8.4
Answer:

Salt for one batch of bread = 30 grams

For 800 grams package,

Now,

= 800/30

= 26 batches.

For 18 kg salt,

1 kg = 1000 grams

18 kg = 18 x 1000 = 18000

Now, 18000/30

= 600 batches.

For 18 kg divided by 30 then, 600 batches no salt will be left.

But in, 800 grams pack 26 batches are left

Therefore, The baker should use 18 kg so that no salt will be left.

Assessment Practice

Question 31.
Which is 2,400 divided by 80?
A. 3
B. 4
C. 30
D. 40
Answer:

2400 ÷ 80

240 ÷ 8 = 30

Therefore, C is the correct option.

Question 32.
Which expression has a quotient of 70?
A. 420 ÷ 60
B. 4,200 ÷ 6
C. 4,200 ÷ 60
D. 4,200 ÷ 600
Answer:

4,200 ÷ 60 = 420 ÷ 6 = 70

Therefore, C is correct option.

Lesson 5.2 Estimate Quotients with 2-Digit Divisors

Solve & Share
Kyle’s school needs to buy posters for a fundraiser. The school has a budget of $147. Each poster costs $13. About how many posters can his school buy? Solve this problem any way you choose.

You can find compatible numbers to estimate quotients. Show your work!
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 9.1

Look Back! Make Sense and Persevere What numbers are close to 147 and 13 that would be easy to divide using mental math?

Visual Learning Bridge.

Essential Question
How Can You Use Compatible Numbers to Estimate Quotients?

A.
Ella earned $159 by selling bracelets. Each bracelet was the same price. About how much did each bracelet cost?

Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 9.22

You can use division to find the price.
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 9.2
You know the total amount earned and the number of bracelets.

B.
The question asks, “About how much?” So, an estimate is enough.
Use compatible numbers to estimate 159 ÷ 75.
Think: 159 is close to 160. Is there a number close to 75 that divides 160 evenly? Try 80.
160 ÷ 80 = 2
So, 160 and 80 are compatible numbers.
16 can be divided evenly by 8.
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 9.4

C.
Since 160 ÷ 80 = 2, 159 ÷ 75 is about 2.
Ella charged about $2 for each bracelet.
Use multiplication to check for reasonableness:
2 × 80 = 160.

Convince Me! Make Sense and Persevere Suppose Ella earned $230 selling the 75 bracelets. Estimate the price of each bracelet. What compatible numbers did you use?

Guided Practice

Do You Understand?

Question 1.
Ella has 425 more bracelets to sell. She wants to store these in bags that hold 20 bracelets each. She estimates she will need about 25 bags. Do you agree? Why or why not?

Answer:

Given,

Number of bracelets = 425

Number of bracelets in each = 20

Therefore, the Number of bags = 425 ÷ 20 = 21.25 which is approximately equal to 22.

Hence I agree with  Ella because the number of bags is approximately equal to 25.

Do You Know How?

In 2-7, estimate using compatible numbers.

Question 2.
287 ÷ 42
Answer:

287 is approximately equal to 280

42 is approximately equal to 40

280 ÷ 40 = 7

Therefore, the estimation of 287 ÷ 42 is 7.

Question 3.
320 ÷ 11
Answer:

11 approximately equals 10

320 ÷ 10 = 32

Therefore, the estimation of 320 ÷ 11 is 32

Question 4.
208 ÷ 72
Answer:

208 approximately equals 210

72 approximately equals 70

210 ÷ 70 = 3

Therefore, the estimation of 208 ÷ 72 is 3

Question 5.
554 ÷ 62
Answer:

554 approximately equals 540

62 approximately equals 60

540 ÷ 60 = 9

Therefore, the estimation of 554 ÷ 62 is 9.

Question 6.
815 ÷ 23
Answer:

815 approximately equals 820

22 approximately eq820uals to 20

820 ÷ 20 = 41

Therefore, the estimation of 815 ÷ 23 is 41

Question 7.
2,491 ÷ 48
Answer:

2491 approximately equals 2500

48 approximately equals 50

2500 ÷ 50 = 50

Therefore, the estimation of 2,491 ÷ 48 is 50.

Independent Practice

Leveled Practice In 8-10, fill in the blanks to find each estimate.

Question 8.
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 10
Answer:

80

and the estimate is 5

Question 9.
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 10.1
Answer:

40

and the estimate is 7

Question 10.
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 11
Answer:

70

and the estimate is 40

In 11-22, estimate using compatible numbers.

Question 11.
228 ÷ 19
Answer:

220 approximately equals 228

20 approximately equals 19

220 ÷ 20 = 11

Therefore, the estimation of 228 ÷ 19 is 11.

Question 12.
1,784 ÷ 64
Answer:

1800 approximately equals 1784

60 approximately equals 64

1800 ÷ 60 = 30

Therefore, the estimation of 1784 ÷ 64 is 30.

Question 13.
7,260 ÷ 83
Answer:

7200 approximately equals 7260

80 approximately equals 83

7200 ÷ 80 = 90

Therefore, the estimation of 7260 ÷ 83 is 90.

Question 14.
2,280 ÷ 12
Answer:

2400 approximately equals 2280

2400 ÷ 12 = 200

Therefore, the estimation of 2280 ÷ 12 is 20.

Question 15.
485 ÷ 92
Answer:

450 approximately equals 485

90 approximately equals 92

450 ÷ 90 = 5

Therefore, the estimation of 485 ÷ 92 is 5.

Question 16.
540 ÷ 61
Answer:

60 approximately equals 61

540 ÷ 60 = 9

Therefore, the estimation of 540 ÷ 61 is 9.

Question 17.
1,710 ÷ 32
Answer:

1800 approximately equals 1710

30 approximately equals 32

1800 ÷ 30 = 60

Therefore, the estimation of 1710 ÷ 32 is 60.

Question 18.
2,740 ÷ 67
Answer:

2800 approximately equals 2740

70 approximately equals 67

2800 ÷ 70 = 40

Therefore, the estimation of 2740 ÷ 67 is 40.

Question 19.
4,322 ÷ 81
Answer:

4400 approximately equals 4322

80 approximately equals 81

4400 ÷ 80 = 55

Therefore, the estimation of 4322 ÷ 81 is 55.

Question 20.
5,700 ÷ 58
Answer:

60 approximately equals 58

5700 ÷ 60 = 95

Therefore, the estimation of 5700 ÷ 60 is 95.

Question 21.
7,810 ÷ 44
Answer:

7875 approximately equals 7810

45 approximately equals 44

7875 ÷ 45 = 175

Therefore, the estimation of 7810 ÷ 44 is 175.

Question 22.
6,395 ÷ 84
Answer:

6400 approximately equals 6395

80 approximately equals 84

6400 ÷ 80 = 80

Therefore, the estimation of 6395 ÷ 84 is 80.

Problem Solving

Question 23.
A model with Math The sign shows the price of baseball caps for different pack sizes. Coach Lewis will buy the medium-size pack of caps. About how much will each cap cost? Write an equation to model the problem.
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 11.1
Answer:

Given,

Number of medium size caps = 32

Total cost = $270

30 approximately equals 32

Cost of each cap = $270 ÷ 30 = $9.

The cost of each cap approximately equals $9.

Question 24.
There are 91 days until the craft sale. Autumn needs to make 817 rings before the sale. She wants to make about the same number of rings each day. About how many rings should she make each day? Explain how Autumn can use compatible numbers to estimate.
Answer:

Number of days left = 91 which approximately equals 90

Number of rings to be made = 817 which approximately equals 810

Therefore, rings should she make each day are 810 ÷ 90 = 9

Approximately, she should make 9 rings each day.

Question 25.
Higher-Order Thinking A company purchased 3,128 bottles of water. Each department needs 55 bottles. Find compatible numbers to estimate the number of departments that can get the bottles they need. Explain.
Answer:

Given,

Number of bottles = 3128 which approximately equals 3080

Number of bottles each department need = 55

The number of departments that can get the bottles they need is 3080 ÷ 55 = 56

Therefore, the number of departments that can get the bottles they need is approximately equaled to 56.

Question 26.
Rita had $20. Then, she saved $5.85 each week for 8 weeks. How much money does she have now? Use the bar diagram to solve the problem. Show your work.
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 13.5
Answer:

Given, The amount Rita have = $20

The amount she saved = $5.85

Total number of weeks = 8

Now,

$5.85 x 8

= 44.64

Now, $44.64 + $20

= $66.80.

Assessment Practice

Question 27.
Lea bought 225 flowers and 12 vases. She put about the same number of flowers in each vase. Which is the best estimate for the number of flowers in each vase?
A. 40 flowers
B. 30 flowers
C. 20 flowers
D. 10 flowers
Answer:

C. 20 flowers

Question 28.
A school has 617 students. Each class has between 28 and 32 students. Which is the best estimate of the number of classes in the school?
A. 14 classes
B. 20 classes
C. 30 classes
D. 60 classes
Answer:

B. 20 classes

Lesson 5.3 Use Models and Properties to Divide with 2-Digit Divisors

Solve & Share
A parking lot has 270 parking spaces. Each row has 18 parking spaces. How many rows are in this parking lot? Solve this problem any way you choose.

You can use appropriate tools, such as grid paper, to solve the problem. Show your work!
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 15.1

Look Back! How can you use estimation to check that your answer to the problem above is reasonable?

Answer:

Total number of parking spaces = 270

Number of parking spaces in each row = 18

Therefore, the total number of parking rows = 270 ÷ 18 =15.

Visual Learning Bridge

Essential Question How Can You Use Area Models and Properties to Find Quotients?

A.
Emily has a rectangular garden with an area of 360 square feet. The length of her garden measures 20 feet. How many feet wide is her garden?

Think 20 × w = 360 or 360 ÷ 20 = w.
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 15.2
You can have w stand for the unknown side. Use place value and the Distributive Property to find the unknown side length.

Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 15.3

B.
20 × 10 = 200 and 20 × 20 = 400, so, w is between 10 and 20. That is, W= 10 + ?
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 15.4
20 × ? = 160

C.
20 × 8 = 160
So w = 10 + 8 = 18.
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 15.6
Multiply to check:
20 × 18 = 20 (10 + 8) = 200 + 160 = 360
So 360 ÷ 20 = 18.
The garden is 18 feet wide.

Convince Me! Make Sense and Persevere Use the diagram, place value, and the Distributive Property to find the quotient
408 ÷ 12. Hint: Find the value of x and solve.
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 15.8

Answer:

12(30 + x) = 360+48

12(30 + x) = 408

30 + x = 408 ÷ 12

30 + x = 34

x = 34 – 30 = 4

Therefore value of x is 4.

Guided Practice

Do You Understand?

Question 1.
Write the missing numbers to find 154 ÷ 11.
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 16.1
154 = 11 × __ + 11 × ___
= 11 × (___ + ___)
= 11 × ___
So, 154 ÷ 11 = ___
Answer:

154 = 11 × 10 + 11 × 4
= 11 × (10 + 4)
= 11 × 14
So, 154 ÷ 11 = 14

Do You Know How?

Question 2.
Use the diagram to find 156 ÷ 12.
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 16.2
So, 156 ÷ 12 = ___

Answer:

156 = 12 × 10 + 12 × 3
= 12 × (10 + 3)
= 12 × 13
So, 156 ÷ 12 = 13

In 3 and 4, use grid paper or draw a picture to find each quotient.

Question 3.
682 ÷ 22
Answer:

682 = 22 × 30 + 22 × 1
= 22 × (30 + 1)
= 22 × 31
So, 682 ÷ 22 = 31

Question 4.
143 ÷ 11
Answer:

143 = 11 × 10 + 11 × 3
= 11 × (10 + 3)
= 11 × 13
So, 143 ÷ 11 = 13

Start by estimating how many tens will be in the quotient.
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 16.5

Independent Practice

Leveled Practice In 5-11, use grid paper or draw a picture to find each quotient.

Question 5.
Use the diagram to find 182 ÷ 13.
So, 182 ÷ 13 = ___
Envision Math Common Core Grade 5 Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 16.6

Answer:

Question 6.
342 ÷ 38
Answer:

Question 7.
720 ÷ 16
Answer:

Question 8.
608 ÷ 19
Answer:

Question 9.
752 ÷ 47
Answer:

Question 10.
375 ÷ 25
Answer:

Question 11.
576 ÷ 24
Answer:

Problem Solving

Question 12.
Angelo is training for a long-distance bicycle ride. He travels 15 miles each hour. How many hours will it take him to ride 210 miles?
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 30.1
Answer:

Total Number of miles  = 210

Number of miles each hour = 15

The number of hours it will take to ride 210 miles = 210 ÷ 15 = 14.

Question 13.
Higher-Order Thinking A rectangular doormat is 21 inches long and has an area of 714 square inches. Find its width. Will the doormat fit in an entryway that is 36 inches wide? Show your work.
Answer:

Given,

Total area = 714 square inches

Length  = 21 inches

Then, length x width = area

21 x width = 714

width = 714 ÷ 21 = 34

Therefore, the Width of the doormat is 34 inches.

Yes, the doormat fits in an entryway that is 36 inches wide.

Question 14.
Use the map. How much longer is the distance from the library to the park to the train station than the distance from the library straight to the train station?
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 30.2
Answer:

Distance from the library to the park to the train station is 2.14  + 2.96 = 5.1 mi

Distance from the library straight to the train station is 3.82 mi

Difference = 5.1 – 3.82 = 1.28.

Question 15.
Algebra If you walk from the train station to the library, then to the park, and then back to the train station, how many miles would you walk in all? Write an equation to model your work.
Answer:

Distance from the library straight to the train station is 3.82 mi

Distance from the library to the park to the train station is 2.14  + 2.96 = 5.1 mi

Total distance = 3.82 + 5.1 = 8.92 mi

Question 16.
Make Sense and Persevere Explain how you can use the picture to show that 391 ÷ 23 = 17.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 30.3
Answer:

391 = 23 × 10 + 23 × 7
= 23 × (10 + 7)
= 23 × 17
So, 391 ÷ 23 = 17

Assessment Practice

Question 17.
There are 16 rows of chairs in the auditorium. Each row has the same number of chairs. There are 512 chairs in all. How many chairs are in each row?
A. 22 chairs
B. 30 chairs
C. 32 chairs
D. 33 chairs
Answer:

C. 32 chairs

Question 18.
A patio has an area of 286 square feet. If the length of the patio is 22 feet, what is the width?
A. 10 feet
B. 13 feet
C. 14 feet
D. 144 feet
Answer:

B. 13 feet

Lesson 5.4 Use Partial Quotients to Divide

Activity

Solve & Share
A hotel sets up tables for a conference for 156 people. If each table seats 12 people, how many tables will be needed? Solve this problem any way you choose.

You can use estimation to help solve this problem. Think about how many groups of 12 you can take away from 156. Show your work!
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 35.1

Look Back! Make Sense and Persevere How can you check that the answer to a division problem is correct?

Visual Learning Bridge

Essential Question How Can You Use Partial Quotients to Solve Division Problems?

A.
A theater has 375 seats arranged in rows with 15 seats in each row. How many rows are in this theater? Let r equal the number of rows. Think: 15 × r = 375 or 375 ÷ 15 = r.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 35.2
An area model can help you find how many 15s are in 375.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 35.3

B.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 35.4
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 35.5
375 ÷ 15 = 25 because 15 × 25 = 375.
So, there are 25 rows and 0 additional seats in the theater.

Convince Me! Critique Reasoning Yimil’s solution to the problem above is shown on the right. Is his solution correct? Explain.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 35.6

Guided Practice

Do You Understand?

Question 1.
Show one way of using partial quotients to find 233 ÷ 11.
Answer:

Partial quotients are 20,1

Question 2.
How can you use estimation to check that your answer to Problem 1 is correct?
Answer:

240 approximately equals 233

12 approximately equals 11

240 ÷ 12 = 20. It approximately equals 21.

Do You Know How?

In 3-6, use partial quotients to divide. Show your work.

Question 3.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 200
Answer:

partial quotients are 10,4

Question 4.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 201
Answer:

partial quotients are 20,2

Question 5.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 202
Answer:

Question 6.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 203
Answer:

partial quotients are 30.

Independent Practice

Leveled Practice In 7-16, use partial quotients to divide. Show your work.

Question 7.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 35.7
Answer:

Question 8.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 35.8
Answer:

Question 9.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 204
Answer:

partial quotients are 10,2

Question 10.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 205
Answer:

partial quotients are 40, and 5  leftover

Question 11.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 206
Answer:

partial quotients are 10,9, and 0 leftover

Question 12.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 207
Answer:

partial quotients are 10,5, and 0  leftover

Question 13.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 208
Answer:

partial quotients are 30,1, and 2  leftover

Question 14.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 209
Answer:

partial quotients are 20,2, and o left over

Question 15.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 210
Answer:

partial quotients are 10,7, and 0 leftovers.

Question 16.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 211
Answer:

partial quotients are 30,8, and 2 leftovers.

Problem Solving

Question 17.
A 969-acre wildlife preserve has 19 cheetahs. About how many acres does each cheetah have to itself, if each cheetah roams the same number of acres?
Answer:

Number of cheetahs = 19

Number of acres = 969

Number of acres does each cheetah have to itself = 969 ÷ 19 = 51

Question 18.
A factory produces 272 chairs in an 8-hour shift. If the factory produces the same number of chairs each hour, how many chairs does it produce in 30 minutes?
Answer:

Number of chairs = 272

Number of hours per shift = 8

Number of chairs in 1 hour = 272 ÷ 8 = 34

Therefore, the number of chairs it produces in 30 minutes = 34/ 2 = 17.

Question 19.
A cafeteria can seat 5 × 102 students. Each table has 2 × 101 seats. How many tables are in the cafeteria?
Answer:

Number of students in cafeteria = 500

Number of seats in a table = 20

Number of tables = 500 ÷ 20 = 25.

Question 20.
A model with Math Peter is driving 992 miles from Chicago to Dallas. His sister Anna is driving 1,068 miles from Phoenix to Dallas. Write and solve an equation to find how much farther Anna drives than Peter drives.
Answer:

Peter is driving 992 miles from Chicago to Dallas

His sister Anna is driving 1,068 miles from Phoenix to Dallas

Difference in Distance = 1068 – 992 = 78 miles.

Question 21.
Write a multiplication equation and a division equation that represent the model shown below.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 40.1
Answer:

Question 22.
Higher-Order Thinking How can you use partial quotients to find 325 ÷ 13? Explain.
Answer:

Assessment Practice

Question 23.
Which expressions are equivalent to 35?
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 51 1,400 ÷ 4
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 51 420 ÷ 12
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 51 875 ÷ 25
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 51 7,700 ÷ 22
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 51 14,000 ÷ 40
Answer:

420 ÷ 12 = 35

875 ÷ 25 = 35

Question 24.
Which expressions are equivalent to 22?
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 51 704 ÷ 32
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 51 1,078 ÷ 49
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 51 1,890 ÷ 30
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 51 1,430 ÷ 65
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 51 4,500 ÷ 50
Answer:

704 ÷ 32
1,078 ÷ 49
1,430 ÷ 65

 

Lesson 5.5 Use Sharing to Divide: Two-Digit Divisors

Solve & Share
The Recycling Club has $294 to purchase one set of recycling bins for each of the 14 members. Each of the 14 sets of bins will be identical to the others and cost the same amount. What is the greatest amount they can spend on one set of bins? Use objects or draw pictures to help solve this problem. Explain how you found your answer.

Using appropriate tools like play money or place-value blocks can help you divide.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 51.1

Look Back! Why can you use division to answer this question?

Visual Learning Bridge

Essential Question How Can You Record Division with a Two-Digit Divisor?

A.
Orchard workers have 258 grapefruit seedlings to plant in 12 equal rows. How many seedlings will be in each row?
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 51.2
Estimate: 258 ÷ 12 is close to 250 ÷ 10 = 25.

You can think about place-value and area models to solve the problem.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 51.3

B.
Regroup the blocks to fill the 12 rows.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 51.4
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 51.5
258 ÷ 12 = 21 R6 because 12 × 21 + 6 = 258.
There will be 21 seedlings in each row with 6 seedlings leftover.

Convince Me! Reasoning What does the remainder mean in the problem above?

Guided Practice

Do You Understand?

Question 1.
If the orchard has 200 seedlings and 12 are planted in each row, how many rows will be filled? Draw place-value blocks to show your answer.
Answer:

Question 2.
In Problem 1, what does the remainder represent?
Answer:

The remainder represents the remaining orchards after planting 12 in a row.

Do You Know How?

In 3 and 4, divide. Write the missing numbers.

Question 3.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 51.9
Answer:

Question 4.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 51.10
Answer:

An estimate can help you decide if your answer is reasonable.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 51.11

Independent Practice

Leveled Practice In 5-13, divide. Write the missing numbers.

Question 5.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 51.12
Answer:

Question 6.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 51.13
Answer:

Question 7.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 51.14
Answer:

Question 8.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 220
Answer:

Question 9.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 221
Answer:

Question 10.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 222
Answer:

Question 11.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 223
Answer:

Question 12.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 224
Answer:

Question 13.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 225
Answer:

Problem Solving

Question 14.
Rita’s family is moving from Grand Junction to Dallas. The moving van averages 60 miles each hour. About how many hours does the van take to reach Dallas? Explain your work.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 52.1
Answer:

Total miles = 980

Average speed = 60 miles each hour

Time it will take = 980 ÷ 60 = 16.33 hours

Question 15.
Due to construction delays on the trip from Little Rock to Chicago, a van driver averaged 48 miles each hour. About how long did that trip take?
Answer:

Total miles = 660

Number of miles each hour = 48

Time taken = 660 ÷ 48 = 13.75 hours

Question 16.
Higher-Order Thinking A scientist needs 72 milliliters of distilled water for each of 15 experiments. She has a bottle that contains 975 milliliters of distilled water. Is there enough water in the bottle for all 15 experiments? Explain.
Answer:

Total distilled water = 975 milliliters

Milliliters needed for  experiments =72

Number of experiments can be done = 972 ÷ 72 = 13.5

Therefore, there is no enough water in the bottle for 15 experiments.

Question 17.
A model with Math The Port Lavaca fishing pier is 3,200 feet long. One person is fishing for every ten feet of length. Write and solve an equation to find how many people are fishing from the pier.
Answer:

Total length = 3200

One person fishing for every ten feet of length

Therefore, number of people = 3200 ÷ 10 = 320.

Question 18.
Todd made a table to show different plans he can use to save $500. Complete the table. Which plan can Todd use to save $500 in less than 16 weeks and have $20 extra? Explain how you found your answer.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 52.6
Answer:

Assessment Practice

Question 19.
Find an expression that produces a quotient of 9 R15. Write the expression in the box.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 52.7
Answer:

375 ÷ 40

Lesson 5.6 Use Sharing to Divide: Greater Dividends

Activity

Solve & Share
The city built a skate park that cost $3,240 and will be paid for over two years in equal monthly payments. How much is each monthly payment? Use objects or draw pictures to help solve this problem. Explain how you found your answer.

To make sense of the problem, you need to read carefully to find all the important information.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 52.8

Look Back! Use Structure How did you use what you know about place value to find the answer to the problem?

Visual Learning Bridge

Essential Question
How Can You Record Division with a Two-Digit Divisor and a Four-Digit Dividend?

A.
Jake works at a flower shop. The shop just received a delivery of 1,830 roses. If the roses are distributed evenly among 15 coolers, how many roses should Jake put in each cooler?

You can use place-value and area models to solve the problem.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 52.9

Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 52.19

B.
There are not enough thousands to put one thousand in each group, so regroup the thousands into hundreds.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 52.10
1,500 + 300 + 30 = 1,830
So 1,850 ÷ 15 = 122.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 52.11
1,830 ÷ 15 = 122 because 15 × 122 = 1,830.
There should be 122 roses in each cooler with no roses left over.

Convince Me! Reasoning Why is 122 a reasonable answer for the problem?

Another Example
Divide 4,108 ÷ 85
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 53.1

Think 41 hundred divided into 82 equal groups.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 53.2

4,108 ÷ 82 = 50 R8 because 82 × 50 + 8 = 4,108.

Guided Practice

Do You Understand?

Question 1.
Use place-value blocks to model
3,710 ÷ 18.
Answer:

Do You Know How?

In 2-5, divide. Use place-value blocks to help.

Question 2.
4,632 ÷ 15
Answer:

Question 3.
3,332 ÷ 30
Answer:

Question 4.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 230
Answer:

Question 5.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 231
Answer:

Independent Practice

In 6, draw an area model for the division problem.

Question 6.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 232

Answer:

In 7-12, divide. Use place-value blocks or area models to help.

Question 7.
7,905 ÷ 35
Answer:

Question 8.
5,500 ÷ 90
Answer:

Question 9.
2,838 ÷ 11
Answer:

Question 10.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 233
Answer:

Question 11.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 234
Answer:

Question 12.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 235
Answer:

Problem Solving

Question 13.
Number Sense The booster club picked 1,370 apples. They plan to sell bags of apples with 15 apples in each bag. How many bags can they make? Explain.
Answer:

Number of apples = 1370

Number of apples in each bag = 15

Number of bags= 1370 ÷ 15 =91.33 approximately equals to 91.

Question 14.
Mason teaches ice skating. He earns $24.50 per lesson. How much does he earn in 5 days if he gives 6 lessons per day?
Answer:

Money earned per lesson=  $24.50

Number of lessons per day = 6

Total earned in a day = $24.50 x 6 = $147

In 5 days, she will earn = 147 x 5 = $735

Question 15.
Reasoning A delivery to the flower shop is recorded at the right. The shop makes centerpiece arrangements using 36 flowers that are all the same type. Will they be able to make at least 10 arrangements using each type of flower? At least 100 arrangements? Explain.
Envision Math Common Core 5th Grade Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 54.1
Answer:

Question 16.
Amelia and Ben have two different answers for 1,955 ÷ 85. Without dividing, how can you tell who might be correct? Amelia: 1,955 ÷ 85 = 23
Ben: 1,955 ÷ 85 = 203
Answer:

I can tell by estimating the values.

1955 approximately equals 2000

85 approximately equals 80

2000 ÷ 80 = 25 which is close to 23. so, amelia is correct

Question 17.
Higher-Order Thinking Estimate the quotient of 4,839 ÷ 15 to the nearest hundred. Explain how you found the estimate.
Answer:

4839 is close to 4800

Now, 4800 ÷15 = 320.

so, the estimation of the quotient of 4,839 ÷ 15 is close to 300.

Assessment Practice

Question 18.
Find 5,092 ÷ 38. How can you check the reasonableness of your answer?
Answer:

5,092 ÷ 38 = 134

I can check reasonableness  by estimating the values

5000 close to 5092

40 is close to 38

5000 ÷ 40 = 125 which is close to 134.

Lesson 5.7 Choose a Strategy to Divide

Activity

Solve & Share

Choose a strategy to solve each problem. Explain your solutions.
Problem 1: Bob’s Citrus and Nursery sells cartons of citrus fruits. There are 24 oranges in each carton. They have 5,643 oranges to pack into cartons. How many cartons can they fill?
Problem 2: This year, 4,338 grapefruits have been harvested so far. Bob’s has 18 storage bins for grapefruits. If the grapefruits are distributed evenly among the 18 bins, how many grapefruits are in each bin?

Envision Math Common Core Grade 5 Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 58.12

The numbers and the situations in the problems can help you choose strategies. Show your work!
Envision Math Common Core Grade 5 Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 58.13

Look Back! Make Sense and Persevere How are your two strategies alike? How are they different?

Visual Learning Bridge

Essential Question
What Are Some Different Strategies I Can Use to Solve a Division Problem?

A.
A company has three printers. The printer in Room 102 is for all of their 15 employees to use. If the available pages are distributed evenly among the employees, how many pages can each employee use?
Envision Math Common Core Grade 5 Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 58.14

Answer:

Number of pages available to print = 3720

Number of employees = 15

Number of pages can each employee use = 3720 ÷ 15 = 248

B.
You can first estimate.
3,720 ÷ 15 is between 3,000 ÷ 15 = 200 and 4,500 ÷ 15 = 300.
Envision Math Common Core Grade 5 Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 58.15
Each employee can print 248 pages.

Convince Me! Reasoning How can you check your answer?

Another Example
How many 32-page brochures can the 40 144 printers in Room 101 print?
Envision Math Common Core Grade 5 Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 58.16
The printer in Room 101 can print 144 brochures.

You need to find how many 32s are in 4,618. You can use division and show partial quotients. You can estimate 4,618 ÷ 32 using 4,500 ÷ 30 = 150.
Envision Math Common Core Grade 5 Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 58.17

Guided Practice

Do You Understand?

Question 1.
Can the remainder be greater than the divisor? Why or why not?
Answer:

No. the Remainder can not be greater than the divisor. If the remainder is greater than the divisor, the division is incomplete.

Do You Know How?

Question 2.
Estimate 452 ÷ 21.
Answer:

452 is close to 460

21 is close to 20

460 ÷ 20 = 23.

Question 3.
Divide.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 240

Answer:

Yes, the answer is reasonable.

Remember to check that your answer is reasonable.
Envision Math Common Core Grade 5 Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 60.1

Independent Practice

In 4-11, estimate and then find the quotient. Use your estimate to check for reasonableness.

Question 4.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 241
Answer:

Estimation is 400 ÷ 50 = 8

Yes, the estimation is reasonableness.

Question 5.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 242
Answer:

Estimation is 640 ÷ 80 = 8

Yes, the estimation is reasonableness.

Question 6.
761 ÷ 5
Answer:

Estimation is 750 ÷ 5 = 150

Yes, the estimation is reasonableness.

Question 7.
510 ÷ 30
Answer:

Estimation is 510 ÷ 30 = 17

Yes, the estimation is reasonableness.

Question 8.
7,704 ÷ 24
Answer:

Estimation is 7,680 ÷ 24 = 320

Yes, the estimation is reasonableness.

Question 9.
7,830 ÷ 33
Answer:

Estimation is 7821 ÷ 33 = 237

Yes, the estimation is reasonableness.

Question 10.
3,136 ÷ 64
Answer:

Estimation is 3,200 ÷ 64 = 50

Yes, the estimation is reasonableness.

Question 11.
6,253 ÷ 71
Answer:

Estimation is 6,300 ÷ 70 = 90

Yes, the estimation is reasonableness.

Problem Solving

For 12, use the table at right.

Question 12.
Make Sense and Persevere Bob’s sells tangelo gift cartons each December. Last year, they shipped a total of 3,300 tangelos. If each carton sells for $28, how much money did Bob’s earn from the tangelo gift cartons sold?
Envision Math Common Core Grade 5 Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 62.1
Answer:

Number of tangelos = 3300

Number of tangelos per carton = 12

Number of cartons = 3300 ÷ 12 = 275

Money earned = $28 x 275 = $7700.

Question 13.
Higher-Order Thinking A score is a group of 20 things. For example, a period of 20 years is called a score of years. The Statue of Liberty was dedicated in 1886. About how many scores of years ago was that?
Answer:

The score for every 20 years

The Statue of Liberty was dedicated in 1886.

1886 ÷ 20 = 94.3 approximately equals to 94.

Question 14.
At an automobile plant, each car is inspected by 34 different workers before it is shipped to a dealer. One day, workers performed 9,690 inspections. How many cars were shipped? Explain.
Envision Math Common Core Grade 5 Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 62.2
Answer:

workers performed 9,690 inspections

each car is inspected by 34 different workers

Therefore, number of cars were inspected = 9,690 ÷ 34 = 285

Assessment Practice

Question 15.
For which division problems is 46 the quotient? Write those division problems in the box.
Envision Math Common Core Grade 5 Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 62.3
Answer:

Lesson 5.8 Make Sense and Persevere

Problem Solving

Activity

Solve & Share

Write a word problem for the equation: 2,530 ÷ 23 = q

Solve your problem any way you choose.

Thinking Habits
Be a good thinker! These questions can help you.
• What information is provided?
• How are the quantities related to each other?
• What strategies do I know for solving this type of problem?
• What tools might help me?
• How can I check that my solution makes sense?
Envision Math Common Core Grade 5 Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 62.4

Look Back! Make Sense and Persevere Does your word problem ask you to find the equal number in each group, or the number of equal groups?

Visual Learning Bridge

Essential Question How Can You Make Sense of Problems and Persevere in Solving Them?

A.
For 3 months, a fifth-grade class raised money for charities. If the class divides the money equally among 32 different organizations, how many dollars will each organization receive, and how many dollars will be leftover?
Envision Math Common Core Grade 5 Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 62.9

You can make sense of the problem by answering these questions. How much money was raised in all?
How much should each organization get?
Envision Math Common Core Grade 5 Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 62.10

Here’s my thinking.
Envision Math Common Core Grade 5 Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 62.11

B.
How can I make sense of and solve the problem?
I can
• identify the quantities given.
• understand how the quantities are related.
• choose and implement an appropriate strategy.
• check to be sure my work and answer make sense.

C.
First, I can write an addition equation to find the total amount raised:
1,104 + 2,117 + 3,275 = 6,496
Next, I can write a division equation to model equal sharing:
6,496 ÷ 32 =
(6400 + 96) ÷ 32 =
(6400 ÷ 32) + (96 ÷ 32) =
200 + 3 = 203
So, each organization will receive $203.

Convince Me! Critique Reasoning Julio says that you can solve this problem by dividing each month’s total by 32 and then adding the three quotients together. Do you agree? Do you think his approach is easier or harder? Justify your answer.

Stuck? Try solving a simpler problem.
Envision Math Common Core Grade 5 Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 71.5

Guided Practice

Make Sense and Persevere
Dana starts with 875 stamps in her stamp collection. Her grandparents give her 332 stamps. Then, she buys 72 more. How many pages in her scrapbook can she fill?
Envision Math Common Core Grade 5 Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 72.5

Question 1.
What do you know?
Answer:

Dana starts with 875 stamps in her stamp collection.

Her grandparents give her 332 stamps.

Then, she buys 72 more

Question 2.
What are you trying to find?
Answer:

How many pages in her scrapbook can she fill?

Question 3.
How are the quantities related? What is the answer to the problem? Write equations to model your work.
Answer:

875 + 332 + 72 = 1279

Number of stamps in a page = 24

The number of pages in her scrapbook she can fill = 1279 ÷ 24 = 53.2 approximately equals 53.

Independent Practice

Make Sense and Persevere
Tanya is saving for a vacation. She wants to have at least $75 for each of the 12 days of her trip. If she saves $85 each month for 10 months, will she save enough money?

Question 4.
Use the strategy of mental math to find the total amount she will save. Then, write a division equation to see if she will save enough.
Answer:

Amount for 12 days = $75 x 12 = $900

Amount for 10 months = $85 x 10 = $850

She can not save enough money.

Question 5.
Jorge says he can solve this problem differently. He says that he can compare 85 × 10 and 75 × 12. Do you agree? Explain your thinking.
Answer:

Yes. I agree with Jorge. Because it is easy to estimate and compare both values of amounts.

Problem Solving
Performance Task

Pumpkin Patch Farms
The table shows the number of seeds the owners of Pumpkin Patch Farms received from different seed suppliers. Each of the pumpkins they harvest usually weighs between 10 and 12 pounds. There are 60 rows, and the farmers will plant the same number of seeds in each row. How many seeds will they plant in each row?
Envision Math Common Core Grade 5 Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 75.1

Question 6.
Make Sense and Persevere What do you know? What are you trying to find?
Answer:

Question 7.
Reasoning How are the quantities in the problem-related? What steps are needed to solve the problem?
Answer:

Question 8.
A model with Math Writes equations with variables to represent the steps needed to solve the problem.

Think about the problem-solving strategies to help you!
Envision Math Common Core Grade 5 Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 75.2

Question 9.
Be Precise Solve the equations and answer the question.
Answer:

Question 10.
Reasoning What strategy can you use to check that your answer makes sense?
Answer:

Topic 5 Fluency Practice Activity

Point & Tally

Work with a partner. Get paper and a pencil. Each partner chooses light blue or dark blue.
At the same time, Partner 1 and Partner 2 each point to one of their black numbers. Both partners find the product of the two numbers.
The partner who chose the color where the product appears gets a tally mark. Work until one partner has seven tally marks.
Envision Math Common Core Grade 5 Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 76.1
Envision Math Common Core Grade 5 Answers Topic 5 Use Models and Strategies to Divide Whole Numbers 76.2

Topic 5 Vocabulary Review

Glossary

Understand Vocabulary

Word List
• compatible numbers
• dividend
• divisor
• estimate
• multiple
• product
• quotient
• remainder

Choose the best term from the Word List. Write it on the blank.

Question 1.
One way to estimate the answer to a division problem is to replace the divisor and dividend with ___
Answer:

Estimate and multiple

Question 2.
The part that is left when you divide into equal groups is called the ____
Answer:

remainder

Question 3.
To decide where to place the first digit of a quotient, ____ is the number of digits in the answer.
Answer:

The first digit of the quotient will be in the Hundreds Place, If the first digit is less than the divisor, If the first digit is greater than or equal to the divisor,  the first digit of the quotient will be in the thousand’s place.

Question 4.
The answer to a division problem is the ______
Answer:

Remainder.

For each of these terms, give an example and a non-example.

Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 75.1

Use Vocabulary in Writing

Question 8.
Write a division problem with a 3-digit dividend, a divisor of 20, and the remainder of 10. Use at least three of the terms in the Word List to explain how you chose the numbers for your example.
Answer:

Topic 5 Reteaching

Set A
pages 181-184
Find 32,000 ÷ 80 using mental math.
Use basic facts and place-value patterns to help.
32 ÷ 8 = 4
320 ÷ 80 = 4
3,200 ÷ 80 = 40
32,000 ÷ 80 = 400

Remember to look for a basic division fact in the numbers. Check your answer by multiplying. Find each quotient. Use mental math.

Question 1.
360 ÷ 40
Answer:

360 ÷ 40

36 ÷ 4 = 9.

Question 2.
270 ÷ 90
Answer:

270 ÷ 90

27 ÷ 9 = 3

Question 3.
2,100 ÷ 30
Answer:

2,100 ÷ 30

210 ÷ 3 = 70

Question 4.
4,800 ÷ 80
Answer:

4,800 ÷ 80

480 ÷ 8 = 60

Question 5.
72,000 ÷ 80
Answer:

72,000 ÷ 80

7200 ÷ 8 = 900

Question 6.
81,000 ÷ 90
Answer:

81,000 ÷ 90

8100 ÷ 9

900

Set B
pages 185-188
Estimate 364 ÷ 57.
Use compatible numbers and patterns to divide.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 75.2
So, 364 ÷ 57 is about 6.

Remember that compatible numbers are numbers that are easy to compute mentally.

Estimate using compatible numbers.

Question 1.
168 ÷ 45
Answer:

168 ÷ 45

160 ÷ 40

4

Question 2.
525 ÷ 96
Answer:

525 ÷ 96

500 ÷ 100

5

Question 3.
379 ÷ 63
Answer:

379 ÷ 63

360 ÷ 60

6

Question 4.
234 ÷ 72
Answer:

234 ÷ 72

210 ÷ 70

3

Question 5.
$613 ÷ 93
Answer:

$613 ÷ 93

$630 ÷ 90

$7

Question 6.
$748 ÷ 92
Answer:

$748 ÷ 92

$720 ÷ 90

$8

Set C
pages 189-192

Find 195 ÷ 13.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 75.3
Draw a model to help you find 131 the number of tens and ones in the quotient.
1 ten + 5 ones = 15.
So, 195 ÷ 13 = 15

Remember to find the number of tens first, then find the number of ones.

Use a model to find each quotient.

Question 1.
180 ÷ 15
Answer:

Question 2.
154 ÷ 14
Answer:

Question 3.
351 ÷ 27
Answer:

Question 4.
192 ÷ 16
Answer:

Question 5.
143 ÷ 11
Answer:

Question 6.
217 ÷ 31
Answer:

Question 7.
130 ÷ 26
Answer:

Question 8.
270 ÷ 18
Answer:

Set D
pages 193–196

Find 336 ÷ 21 using partial quotients.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 78.1
Add the partial quotients: 10 + 6 = 16. So, 336 = 21 = 16.

Remember to add the partial quotients to find the actual quotient.

Use partial quotients to divide.

Question 1.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 250
Answer:

Partial quotients are 10, 9 and sum is 19

Question 2.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 251
Answer:

Partial quotients are 40,2 and sum is 42

Question 3.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 252
Answer:

Partial quotients are 40,1 and sum is 41

Question 4.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 253
Answer:

partial quotients are 20,3 and sum is 23.

Question 5.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 254
Answer:

PArtial quotients are 60,6 and sum is 66.

Question 6.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 255
Answer:

partial quotients are 30,4 and sum is 34.

Question 7.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 256
Answer:

partial quotients are 20,8 and sum is 28

Question 8.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 257
Answer:

partial quotients are 10,1 and sum is 11.

Set E
pages 197-200

Find 461 ÷ 50.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 80.1

Remember that you can check your answer by multiplying the quotient by the divisor, and then adding any remainder.

Question 1.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 260
Answer:

Question 2.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 261
Answer:

Question 3.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 262
Answer:

Question 4.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 263
Answer:

Question 5.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 264
Answer:

Question 6.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 265
Answer:

Question 7.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 266
Answer:

Question 8.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 267
Answer:

Question 9.
Ivan uses 30 craft sticks to make each toy cabin. He has a box of 342 craft sticks. How many toy cabins can Ivan make? How many sticks will be left?
Answer:

Set F
pages 201-204

Find 3,657 ÷ 23.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 81.1

Remember you can multiply the divisor by powers of 10 to estimate the quotient.

Divide. Use place-value blocks to help.

Question 1.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 270
Answer:

Question 2.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 271
Answer:

Question 3.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 272
Answer:

Question 4.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 273
Answer:

Question 5.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 274
Answer:

Question 6.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 275
Answer:

Question 7.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 276
Answer:

Question 8.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 277
Answer:

Set G
pages 205-208

Find 789 ÷ 19.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 85.2

Remember that you can check your answer by multiplying the quotient by the divisor, and then adding any remainder.

Question 1.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 282
Answer:

Question 2.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 281
Answer:

Question 3.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 290

Answer:

Question 4.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 283
Answer:

Question 5.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 284
Answer:

Question 6.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 285
Answer:

Question 7.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 286
Answer:

Question 8.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 287
Answer:

Question 9.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 288
Answer:

Question 10.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 289
Answer:

Set H
pages 209-212

Think about these questions to help you make sense and persevere in solving problems.

Thinking Habits

• What do I know?
• What do I need to find?
• What’s my plan for solving the problem?
• What else can I try if I get stuck?
• How can I check that my solution makes sense?
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 86.1

Selena is planning to visit her aunt in 5 weeks. She has saved $365 but thinks the trip will cost $500. She plans to save the same amount each week so she has $500 for the trip. How much does she need to save each week?
I can write an equation to find how much more money Selena needs:
500 – 365 = 135
Then divide the amount she needs by 5 weeks: 135 ÷ 5 = 27
Selena needs to save $27 each week.
My answer is reasonable because 365 + 27 + 27 + 27 + 27 + 27 = 500.

Remember to think about what steps are needed to solve each problem. Solve. Show your work.

Question 1.
The football coach spent a total of $890 including $50 in tax for 35 shirts for the team. Each shirt cost the same amount. What was the price of one shirt before tax was added?
Answer:

Total cost = $890 including tax

Total cost before tax added = $890 – $50 = $840

Total number of shirts = 35

Therefore, Price of one shirt = $840 ÷ 35 = $24

Question 2.
A gymnast practices 6 days each week. She practices the same number of hours each day. If she practices a total of 120 hours in a 4-week period, how many hours each day does she practice?
Answer:

Number of days in the 4-week period = 4×7 = 28

But, she practices only 6 days, Therefore, 6 days in a 4-week period = 6×4 = 24

Total number of hours in 4-week period = 120

Therefore, the number of hours each day she practices = 120 ÷ 24 =5.

Question 3.
Nathan works the same number of hours each day, 5 days each week. He earns $12 per hour. Last week he earned $420. How many hours did he work each day last week? Write equations to model your work.
Answer:

Total earned = $420

Number of days in each week = 5

Total earned in each day = $420 ÷ 5 = $84

Total earned per hour = $12

Therefore, the number of hours he works each day = $84 ÷$12 = 7.

 

Question 4.
A high-rise apartment building has 15 floors with 26 apartments on each floor. There are 3 kinds of apartments in the building: 1-, 2-, and 3-bedroom. The building has the same number of each kind of apartment. How many of each kind of apartment is in the building? Show your work.

Answer:

The  number of floors = 15

The number of Apartments = 26

Now, 15 x 26 = 390 Apartments.

Given that, There are three kinds of apartments

= 390/3

= 130

Therefore, there are 130 apartments in the building.

 

Topic 5 Assessment Practice

Question 1.
Select all of the following equations the number 60 will make true.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 90 420 ÷ Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 91 = 70
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 90 1,800 ÷ Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 91 = 300
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 90 5,400 ÷ Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 91 = 90
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 90 2,400 ÷ Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 91 = 40
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 90 500 ÷ Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 91 = 10
Answer:

Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 90 5,400 ÷ Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 91 = 90
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 90 2,400 ÷ Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 91 = 40

Question 2.
Which of the following is the best estimate of 487 ÷ 67?
A. 80
B. 70
C. 10
D. 7
Answer:

B. 70

Question 3.
The carnival committee has purchased 985 small prizes. The prizes are to be divided equally among the 20 game booths. A. In what place will the first digit of the quotient be?
B. How many prizes will each booth have?
C. How many prizes will be left?
Answer:

A. The first digit of the quotient be in Ten’s place.

B. 985 /20 = 49 as Quotient

Therefore, each booth have 49 prizes in each booth

C. 985 /20 = 5 as remainder

Therefore, 5 prizes will be left.

 

Question 4.
A rectangular living room has an area of 425 square feet. The width of the room is 17 feet.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 91.8
Write a number in the box to show the missing dimension. What is the length of the room?
____ feet
Answer:

Given, The Area of the living room = 425 square feet.

Now, 85/17 = 5

Therefore, the length of the room = 20 + 5 = 25 feet.

Question 5.
A. Divide.
2,700 ÷ 30 = ____
B. Select all the expressions that are equal to 2,700 ÷ 30.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 93 270 ÷ 3
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 93 270 tens ÷ 3 tens
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 93 2,700 ÷ 3 tens
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 93 2,700 ÷ 3
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 93 2,700 tens ÷ 30
Answer:

A. 90

B.Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 93 270 ÷ 3
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 93 270 tens ÷ 3 tens
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 93 2,700 ÷ 3 tens

Question 6.
Select the quotient for each expression.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 93.1
Answer:

Question 7.
Use the table.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 93.2
A. Using Plan B, how many weeks will it take Althea to reach her savings goal? Write the missing number in the table.
B. Show how you found your answer to A.
Answer:

Althea’s plans for savings = $384

It approximately equals $390

Amount to save each week = $30 according to plan B

Therefore, Number of weeks needed = $390 ÷ $30  = 13

Question 8.
Five Star Farm purchased 2,400 apple trees. If 80 trees can be planted on each acre of land, how many acres will be needed to plant all the trees?
A. Identify which expression represents the problem.
A. 2,400 × 80
B. 80 × 2,400
C. 80 ÷ 2,400
D. 2,400 ÷ 80
B. How many acres will be needed to plant all the trees?
Answer:

A.

D. 2,400 ÷ 80

B.

Total number of apple trees = 2400

Number of trees planted on each acre land = 80

Number of acres needed to plant all the trees = 2400 ÷ 80 = 30

 

Question 9.
Mrs. Reiss has 264 crayons for her art class of 22 students. How many crayons will each student get if the crayons are divided equally? Use the model.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 94.1
Answer:

Number of students = 22

Number of Crayons = 264

Now,

264/22

= 12

Therefore, Each Student gets 12 crayons.

Question 10.
Select all the expressions that have the value of 9.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 96 270 ÷ 3
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 96 250 ÷ 25
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 96 270 ÷ 30
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 96 207 ÷ 23
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 96 189 ÷ 21
Answer:

  1. 270 ÷ 3
  2. 270 ÷ 30
  3. 207 ÷ 23
  4. 189 ÷ 21

Question 11.
Kari wants to find 3,277 ÷ 29.
A. Without doing the division, which number will the quotient be closest to?
A. 1
B. 10
C. 100
D. 1,000
B. What is the exact quotient?
Answer:

A.

The quotient be closest to 100

B. The exact quotient =

3,277 ÷ 29.

= 113

Question 12.
The cost to rent a lodge for a family reunion is $975. If 65 people attend and pay the same price, how much does each person pay?
A. Which of the following expressions represents the problem?
A. 975 + 65
B. 975 ÷ 65
C. 975 × 65
D. 975 ÷ 2
B. How much does each person pay?
A. $16
B. $15
C. $14
D. $13
Answer:

$975 ÷ 65

= $15

Therefore, Each person pays $15.

Question 13.
Shady Rivers summer camp has 188 campers this week. If there are 22 campers to each cabin, what is the least number of cabins needed?
A. 7 cabins
B. 8 cabins
C. 9 cabins
D. 10 cabins
Answer:

188/22

= 8 cabins.

Question 14.
The area of a rectangular banquet hall is 7,400 square feet. The length of one side of the hall is 82 feet. Explain how you can use compatible numbers to estimate the width of the hall.
Answer:

Given,

The area of the rectangle = 7,400 square feet.

The length of one side = 82 feet.

We know that, The area of the rectangle =

Area = length x width

7400 = 82 x n

n = 7400/82

= 90.2 feet

Therefore, the width of the rectangular banquet hall = 90.2 feet.

Question 15.
The cost of renting a bus is $1,344. Tony wants to find how much each person will pay if 32 people ride the bus and share the cost equally. Fill in the partial quotients that are missing from Tony’s work below.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 96.2
Answer:

Question 16.
Jessie made 312 mini energy bars. She puts 24 bars in each bag. She plans to sell each bag for $6.
A. Write two equations with variables that Jessie can use to find the amount of money she will earn if she sells all of the bags.
B. How much will she earn if she sells all of the bags?
Answer:

Question 17.
Select all of the following equations the number 40 will make true.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 97 280 ÷ Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 98 = 7
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 97 800 ÷ Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 98 = 20
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 97 4,000 ÷ Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 98 = 10
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 97 3,200 ÷ Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 98 = 80
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 97 800 ÷ Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 98 = 200
Answer:

  1. 280 ÷ 40  =  7
  2. 800 ÷ 40 = 20
  3. 3200 ÷ 40 = 80

Question 18.
Select the quotient for each expression.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 98.1
Answer:

  1. 2700 ÷ 30 = 90
  2. 270 ÷ 30 = 9
  3. 2400 ÷ 30 = 80
  4. 240 ÷ 30 = 8

Question 19.
Charles burns 4,350 calories hiking 15 miles of the Appalachian Trail. How many calories does he burn each mile?
A. Identify which expression represents the problem.
A. 4,350 ÷ 15
B. 4,350 × 15
C. 4,350 – 15
D. 4,350 ÷ 10
B. How many calories does he burn each mile?
Answer:

A. 4,350 ÷ 15

B. In each mile he burns 290 calories.

Question 20.
Find the quotient 432 ÷ 48.
Answer:

432 ÷ 48 = 9

Question 21.
Which partial quotients could be added to find 465 ÷ 15?
A. 20 and 1
B. 30 and 1
C. 30 and 9
D. 30 and 10
Answer:

B. 30 + 1

465 ÷ 15 = 31

30 + 1 = 31

Question 22.
The table shows the number of students going on field trips. For each trip, one adult is needed for every 15 students.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 99.6
How many adults are needed to go on the seventh-grade trip?
Answer:

Given,

Number of students in seventh grade = 225

Condition is one adult is needed for every 15 students.

Number of adults are needed to go on the seventh-grade trip = 225 ÷ 15 = 15

Therefore, the number of adults are needed to go on the seventh-grade trip is 15.

Topic 5 Performance Task

School Supplies
A store had a sale on school supplies in August. The store manager recorded how many of several types of items were sold. Each of the same types of item costs the same amount. Use the information in the table to answer the questions.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 99.7

Question 1.
Backpack sales totaled $1,200. How much did each backpack cost? Write an equation to model your work.
Answer:

Given,

Total sales = $1200

Total number of backpacks = 60

cost of each backpack = $1200 ÷ 60 =  $120 ÷ 6 = $20.

Therefore, the cost of each backpack is $20.

Question 2.
The store sold 71 packages of pens. Use compatible numbers to estimate how many pens were in each package. Show your work.
Answer:

Given,

Number of packages = 71

Number of pens sold = 568

number of pens in each package is 568 ÷ 71 = 8

Therefore, the number of pens in each package is 8

Question 3.
There were 16 pencils in each box. Olivia wants to find how many boxes of pencils were sold.
Part A
When Olivia divides 784 by 16, in which place should she write the first digit of the quotient? Tell how you know without dividing.
Part B
How many boxes of pencils were sold?
Answer:

Number of pencils sold = 784

Number of pencils in each box = 16

Number of boxes of pencils sold = 784 ÷ 16 = 49

Therefore, the number of boxes of pencils sold is 49.

 

Question 4.
The store manager has ordered the calculators shown, but the shipment has been delayed.
Part A
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 99.8
If all the calculators ordered are sold, the total sales would be $2,014. Was the number of calculators ordered less than or greater than 100? How do you know without dividing?

Part B
How many calculators were ordered? Write an equation to model your work.
Answer:

Given,

Total sales = $2,014

Cost for each calculator = $19

Therefore number of calculators = $2,014 ÷ $19 = 106

So, the number of calculators ordered was 106 and it is greater than 100.

we can say it not by diving. 19 x 100 = 1900 and the total sales given is $2014. So it must be greater than 100.

Question 5.
The manager wants to order 408 more notebooks. The notebooks are shipped in packages of 12. He used partial quotients to find the number of packages to order. His work is shown on the right. Is his solution correct? Explain.
Envision Math Common Core 5th Grade Answer Key Topic 5 Use Models and Strategies to Divide Whole Numbers 99.9
Answer:

No. The model to find the number of packages to order is not correct.

Explanation:

408 ÷ 12 = 34

But his work shown at right is showing the value is 40

So, it is not correct and the correct value is 34.

Question 6.
An additional 40 packages of paper were ordered at a total cost of $520. How much did each package of paper cost? Write an equation to model your work.
Answer:

Each package of paper cost $13

Because,

$520 ÷ 40

$52 ÷ 4 = $13.

enVision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers

Go through the enVision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers regularly and improve your accuracy in solving questions.

enVision Math Common Core 4th Grade Answers Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers

Essential Questions:
How can you multiply by multiples of 10, 100, and 1,000? How can you multiply whole numbers?
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 1

enVision STEM Project: Maps and Math
Do Research Use the Internet or other sources to find information about three of Earth’s features on a topographic map, such as mountains or oceans. Write two facts about each of the features you researched.
Journal: Write a Report Include what you found. Also in your report:

  • Write the height or depth of each feature you researched.
  • Estimate to find 10 times the heights or depths of the features you researched.

Review What You Know

Vocabulary
Choose the best term from the box. Write it on the blank.

  • breaking apart
  • compensation
  • inverse operations
  • partial products

Question 1.
Multiplication and division are _________.
Answer:
Multiplication and division are ____inverse operations____.

 

Use Strategies and Properties to Multiply by 1-Digit Numbers 1

Question 2.
A mental math method used to rewrite a number as the sum of numbers to form an easier problem is called __________.
Answer:
A mental math method used to rewrite a number as the sum of numbers to form an easier problem is called __breaking apart________.

 

Question 3.
Choosing numbers close to the numbers in a problem to make the computation easier, and then adjusting the answer is called ________.
Answer:
Choosing numbers close to the numbers in a problem to make the computation easier, and then adjusting the answer is called ___partial products_____.

 

 

Multiplication

Find each product.
Question 4.
6 × 2
Answer:
6 × 2 = 12.

Explanation:
6 × 2 = (3 × 2) + (3 × 2)
6 × 2 = 6 + 6
6 × 2 =12.

 

Question 5.
8 × 9
Answer:
8 × 9 = 72.

Explanation:
8 × 9 = (4 × 9) + (4 × 9)
8 × 9 = 36 + 36
8 × 9 = 72.

Question 6.
6 × 5
Answer:
6 × 5 = 30.

Explanation:
6 × 5 = (2 × 5) + (4 × 5)
6 × 5 = 10 + 20
6 × 5 = 30.

 

 

Question 7.
7 × 8
Answer:
7 × 8 = 56.

Explanation:
7 × 8 = (3 × 8) + (4 × 8)
7 × 8 = 24 + 32
7 × 8 = 56.

 

 

Question 8.
4 × 8
Answer:
4 × 8 = 32.

Explanation:
4 × 8 = (1 × 8) + (3 × 8)
4 × 8 = 8 + 24
4 × 8 = 32.

Use Strategies and Properties to Multiply by 1-Digit Numbers 2

 

Question 9.
3 × 7
Answer:
3 × 7 = 21.

Explanation:
3 × 7 = (2 × 7) + (1 × 7)
3 × 7 = 14 + 7
3 × 7 = 21.

 

 

Rounding

Round each number to the nearest ten.
Question 10.
16
Answer:
Nearest ten of 16 is 20.

 

Question 11.
82
Answer:
Nearest ten of 82 is 80.

 

 

Question 12.
35
Answer:
Nearest ten of 35 is 40.

 

 

Question 13.
53
Answer:
Nearest ten of 53 is 50.

 

 

Question 14.
24
Answer:
Nearest ten of 24 is 20.

 

 

Question 15.
49
Answer:
Nearest ten of 49 is 50.

 

 

Round each number to the nearest hundred.
Question 16.
868
Answer:
Nearest hundred of 868 is 900.

 

 

Question 17.
499
Answer:
Nearest hundred of 499 is 500.

 

 

Question 18.
625
Answer:
Nearest hundred of 625 is 600.

 

 

Question 19.
167
Answer:
Nearest hundred of 167 is 200.

 

 

Question 20.
341
Answer:
Nearest hundred of 341 is 300.

 

 

Question 21.
772
Answer:
Nearest hundred of 772 is 800.

 

 

Question 22.
919
Answer:
Nearest hundred of 919 is 900.

 

 

Question 23.
552
Answer:
Nearest hundred of 552 is 600.

 

 

Question 24.
321
Answer:
Nearest hundred of 321 is 300.

 

 

Problem Solving
Question 25.
Critique Reasoning Tyler says, 9 × 7 is greater than 7 × 9 because the greater number is first.” Explain Tyler’s error.
Answer:
Tyler’s is incorrect because 9 × 7 and 7 × 9 are same value numbers, just numbers are interchanged.

Explanation:
9 × 7 = (5 × 7) + (4 × 7)
9 × 7 = 35 + 28
9 × 7 = 63.

7 × 9 = (3 × 9) + (4 × 9)
7 × 9 = 27 + 36
7 × 9 =  63.

 

 

Pick a Project

PROJECT ЗА
How zesty is Key lime pie?
Project: Create Data for Key Lime Pie Ingredients
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 2

Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Digit-Numbers-Pick a Project-PROJECT ЗА

 

 

 

PROJECT ЗВ
What do a dozen Florida panthers weigh?
Project: Draw a Captioned Picture
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 3

Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Digit-Numbers-Pick a Project-PROJECT ЗВ

 

 

PROJECT 3C
What’s the mass of a giraffe?
Project: Create a Song
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 4

Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Digit-Numbers-Pick a Project-PROJECT 3C

 

3-ACT MATH PREVIEW
Math Modeling
Covered Up
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 5
I can…model with math to solve a problem that involves estimating and computing with area models.

Lesson 3.1 Multiply by Multiples of 10, 100, and 1,000

Solve & Share
Find the products for 3 × 4, 3 × 40, 3 × 400, and 3 × 4,000. Solve these problems using any strategy you choose.
I can… find the products of multiples of 10, 100, and 1,000 using mental math and place-value strategies.
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 6

Look Back! What pattern do you notice in the products?
Answer:
The patern noticed in the products is that for every product extra zero is added after the number.

Explanation:
3 × 4 = (3 × 1) + ( 3 × 3) = 3 + 9 = 12.
3 × 40 = (3 × 10) + (3 × 30) = 30 + 90 = 120.
3 × 400 = (3 × 100) + ( 3 × 300) = 300 + 900 = 1200.
3 × 4,000 = (3 × 1000) + ( 3 × 3000) = 3000 + 9000 = 12000.

 

 

Essential Question
How Can You Multiply by Multiples of 10, 100, and 1,000?
Answer:
We multiply the multiplicand by the non-zero digit of the multiplier. Then we add the same number of zero to the extreme right of the product as the multiplier has.

Explanation:
We use Multiply by Multiples of 10, 100, and 1,000 as follow:
We can understand this method as follows.
1. First we count the zeros.
2. We multiply the non – zero numbers.
3. We place the same number of zero to the extreme right of the product

 

 

Visual Learning Bridge
Calculate 3 × 50, 3 × 500, and 3 × 5,000 using basic multiplication facts and properties of operations.

The Associative Property of Multiplication states that you can change the grouping of the factors and the product stays the same.
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 7

One Way
Find 3 × 50, 3 × 500, and 3 × 5,000.
Use basic facts and place value.
3 × 50 = 3 × 5 tens
= 15 tens
= 150
3 × 500 = 3 × 5 hundreds
= 15 hundreds
= 1,500
3 × 5,000 = 3 × 5 thousands
= 15 thousands
= 15,000

Another Way
Find 3 × 50, 3 × 500, and 3 × 5,000.
Break apart numbers. Use the Associative Property of Multiplication.
3 × 50 = 3 × (5 × 10)
= (3 × 5) × 10
= 15 × 10
= 150
3 × 500 = 3 × (5 × 100)
= (3 × 5) × 100
= 15 × 100
= 1,500
3 × 5,000 = 3 × (5 × 1,000)
= (3 × 5) × 1,000
= 15 × 1,000
= 15,000

Convince Me! Reasoning What patterns do you see in the number of zeros in the products above?
Answer:
The pattern noticed in the zero’s in the products above is that for every number zero is added more from one to last number products.

Explanation:
Extra zero is added for every product from one to last number.

 

Use Strategies and Properties to Multiply by 1-Digit Numbers 4

Another Example!
Use place value to calculate 5 × 400 and 6 × 5,000.
5 × 400 = 5 × 4 hundreds
= 20 hundreds
= 2,000

6 × 5,000 = 6 × 5 thousands
= 30 thousands
= 30,000

If the product of the basic fact ends in zero, the product has one more zero than you see in the factors.

 

 

Guided Practice
Do You Understand?
Question 1.
Show how you can use the basic fact 5 × 8 = 40 to determine the product of 5 × 800.
Answer:
5 × 800 = (5 × 8 × 100) = 4000.

Explanation:
5 × 8 = 40.
5 × 800 = (5 × 8 × 100)
5 × 800 = 40  × 100
5 × 800 = 4000.

 

 

Question 2.
Bob said 4 × 500 = 200. Explain his error using place value.
Answer:
His error is he missed zero to write in place value of the number.
4 × 500 = 2000 not 4 × 500 = 200.

Explanation:
4 × 500 = 200.
4 × 500 = 4 × 5 × 100
4 × 500 = 20 × 100
4 × 500 = 2000.

 

 

Do You Know How?
For 3-5, use strategies you learned to help multiply.
Question 3.
8 × 7 = ________
8 × 70 = ________
8 × 700 = ________
8 × 7,000 = ________
Answer:
8 × 7 = __56______.
8 × 70 = __560______.
8 × 700 = __5600______.
8 × 7,000 = ___56000_____.

Explanation:
8 × 7 = 56.
8 × 70 = 8 × 7 tens
= 56 tens
= 560.
8 × 700 = 8 × 7 hundreds
= 56 hundreds
= 5600.
8 × 7,000 = 8 × 7 thousands
= 56 thousands
= 56000.

 

 

Question 4.
7 × 70
Answer:
7 × 70 = 490.

Explanation:
7 × 70 = 7 × 7 tens
= 49 tens
= 490.

 

 

Question 5.
2 × 700
Answer:
2 × 700 = 1400.

Explanation:
2 × 700 = 2 × 7 hundreds
= 14 hundreds
= 1400.

 

 

Independent Practice
Leveled Practice For 6-11, use basic facts, place value, and properties to help multiply.

You can use place-value strategies to calculate each product.
Question 6.
3 × 70 = ________
3 × 700 = ________
3 × 7,000 = ________
Answer:
3 × 70 = __210______.
3 × 700 = ___2100_____.
3 × 7,000 = ___21000_____.

Explanation:
3 × 70 = 3 × 7 tens
= 21 tens
= 210.
3 × 700 = 3 × 7 hundreds
= 21 hundreds
= 2100.
3 × 7,000 = 3 × 7 thousands
= 21 thousands
= 21000.

 

 

Question 7.
________ = 6 × 40
________ = 6 × 400
________ = 6 × 4,000
Answer:
___240_____ = 6 × 40.
____2400____ = 6 × 400.
__24000______ = 6 × 4,000.

Explanation:
6 × 40 = 6 × 4 tens
= 24 tens
= 240.
6 × 400 = 6 × 4 hundreds
= 24 hundreds
= 2400.
6 × 4,000 = 6 × 4 thousands
= 24 thousands
= 24000.

 

 

Question 8.
8 × 50 = ________
8 × 500 = ________
8 × 5,000 = ________
Answer:
8 × 50 = ___400_____.
8 × 500 = ___4000_____.
8 × 5,000 = __40000______.

Explanation:
8 × 50 = 8 × 5 tens
= 4 hundreds
= 400.
8 × 500 = 8 × 5 hundreds
= 4 thousands
= 4000.
8 × 5,000 = 8 × 5 thousands
= .4 ten thousands
= 40000.

 

 

Question 9.
4 × 2,000
Answer:
4 × 2,000 = 8000.

Explanation:
4 × 2,000 = 4 × 2 thousands
= 8 thousands
= 8000.

 

 

Question 10.
700 × 4
Answer:
700 × 4 = 2800.

Explanation:
700 × 4 = 7 hundreds × 4
= 28 hundreds
= 2800.

 

 

Question 11.
6 × 60
Answer:
6 × 60 = 360.

Explanation:
6 × 60 = 6 × 6 tens
= 36 tens
= 360.

 

 

Problem Solving
Question 12.
enVision® STEM The Mississippi River is about 8 times the length of the Hudson River. If the Hudson River is about 300 miles long, about how many miles long is the Mississippi River? Write and solve an equation.
Answer:
Length of Mississippi River = 2400 miles.

Explanation:
Length of the Hudson River = 300 miles long.
The Mississippi River is about 8 times the length of the Hudson River.
=> Length of Mississippi River = 8 × Length of the Hudson River
= 8 × 300
= 8 × 3 hundreds
= 24 hundreds
= 2400 miles.

 

 

Question 13.
Ted, Jason, and Angelina are trying to raise $200 for a local shelter. Ted raised $30. Jason raised $90. How much money m does Angelina need to raise to reach their goal?
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 8
Answer:
Amount to be raised by m, Angelina = $80.

Explanation:
Amount raised by Ted = $30.
Amount raised by Jason = $90.
Total amount Ted, Jason, and Angelina are trying to raise = $200.
Amount to be raised by m, Angelina = Total amount Ted, Jason, and Angelina are trying to raise  – (Amount raised by Ted + Amount raised by Jason )
= $200 – ($30 + $90)
= $200 – $120
= $80.
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Digit-Numbers-Lesson 3.1 Multiply by Multiples of 10, 100, and 1,000-Problem Solving-13

 

 

For 14-15, use the table at the right.
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 9
Question 14.
Make Sense and Persevere There are 9 girls and 4 adults in Aimee’s scout troop. How much did the troop pay for tickets to the amusement park?
Answer:
Total amount of fee to 9 girls and 4 adults troop paying for tickets to the amusement park = $430.

Explanation:
Number of girls to the amusement park = 9.
Amount of fee to each girl to the amusement park = $30.
Number of adults to the amusement park = 4.
Amount of fee to each adult to the amusement park = $40.
Total amount of fee to 9 girls and 4 adults troop paying for tickets to the amusement park = ( Number of girls to the amusement park × Amount of fee to each girl to the amusement park =) + ( Number of adults to the amusement park × Amount of fee to each adult to the amusement park )
= (9 × $30) + ( 4 × $40)
= $270 + $160
= $430.

 

 

Question 15.
Higher Order Thinking Tina visited Funland with her mom and a friend They bought tickets for Plan C. How much money did they save on the two children’s tickets for Plan C instead of buying separate tickets for Plan A and Plan B?
Answer:
Amount of money they saved on the two children’s tickets for Plan C instead of buying separate tickets for Plan A and Plan B = $20.

Explanation:
Amount of fee to each girl to the amusement park in Plan A = $20.
Amount of fee to each girl to the amusement park in Plan B = $30.
Amount of fee to each adult to the amusement park = $40.
Amount of tickets for Plan C (combined A and B) for adult = $60.
Amount of tickets for Plan C (combined A and B) for kids = $40.
Amount for two children in Plan A = 2 × $20 = $40.
Amount for two children in Plan B = 2 × $30 = $60.
Amount for two children in Plan C (combined A and B) = 2 × $40 = $80.
Difference:
Amount of money they saved on the two children’s tickets for Plan C instead of buying separate tickets for Plan A and Plan B = (Amount for two children in Plan A  + Amount for two children in Plan B ) – Amount for two children in Plan C (combined A and B)
= ($40 + $60) – $80.
= $100 – $80
= $20.

 

 

Assessment Practice
Question 16.
Brandon says 4 × 800 is greater than 8 × 4,000.
Renee says 4 × 800 is less than 8 × 4,000.
A. Without calculating the answer, explain ” how to use place-value strategies or the Associative Property to find which is greater.
Answer:
Brandon says 4 × 800 is greater than 8 × 4,000 and Renee says 4 × 800 is less than 8 × 4,000 both are correct.

Explanation:
4 × 800 is greater than 8 × 4,000.
=> place value of 4000 is thousand greater than 800.
4 × 800 is less than 8 × 4,000.
=> place value of 800 is hundred less than thousands.

 

 

B. Without calculating the answer, explain how to use relationships or basic facts to find which is less.
Answer:
Without calculating the answer, we can use relationships or basic facts to find which is less by the place value of the numbers given.

Explanation:
We can use relationships or basic facts to find which is less by the place value of the numbers given without calculating the answer. Which is having less place value that is considered as less and vice versa.

 

 

Lesson 3.2 Estimate Products

Solve & Share
Sarah earns $48 a week babysitting. She saves all of her earnings for 6 weeks. Estimate to determine about how much money Sarah saves. Solve this problem using any strategy you choose.
I can …use rounding to estimate products and check if my answer is reasonable
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 10

Look Back! Is your estimate more or less than the amount Sarah actually earned? Explain.
Answer:
My estimation for Sarah earnings is around $290. My estimation saving is more than the actual savings of her.

Explanation:
Amount of money Sarah earns for babysitting for a week = $48.
Number of weeks She saves all of her earnings = 6.
Amount of money Sarah saves = Amount of money Sarah earns for babysitting for a week × Number of weeks She saves all of her earnings
= $48 × 6
= $288.

 

 

Essential Question
How Can You Estimate When You Multiply?
Answer:
We can estimate when we multiply by using multiplication facts to get the solution and solve the problem.

Explanation:
We can estimate when we multiply by checking the numbers given and doing their multiplication or by using multiplication facts to get the solution and solve the problem.

 

 

Visual Learning Bridge
Mr. Hector’s class sold calendars and note pads for 3 weeks as a class fundraiser. About how much did Mr. Hector’s class make selling calendars? selling note pads?
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 11

Use compatible numbers to estimate how many calendars C, were sold.
28 + 73 + 63 = C
164 = C
Estimate $4 × 164.
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 12
Estimate by using compatible numbers. 164 is close to 150.
4 × 150 = 600
Mr. Hector’s class made about $600 selling calendars.

Use rounding to estimate how many note pads n, were sold.
272 + 475 + 232 = n
979 = n
Estimate $3 × 979.
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 13
To the nearest thousand, 979 rounds to 1,000.
3 × 1,000 = 3,000
Mr. Hector’s class made about $3,000 selling note pads.

Convince Me! Construct Arguments Could you solve the problem above by rounding to estimate the total number of calendars and the total number of note pads? Explain.
Answer:
We can estimate by rounding to estimate the total number of calendars and the total number of note pads and solve the problem.

Explanation:
We can estimate by rounding to estimate the total number of calendars and the total number of note pads and solve the problem. By rounding the total there cannot be any difference much its just rounding the number to the nearest tens, hundreds or thousands.

 

 

Another Example!
Mrs. Li’s class made $2,089 selling neon pencil packs. A student calculated that Mr. Hector’s class and Mrs. Li’s class together made $8,865 selling the school supplies. Is the student’s calculation reasonable?

You can estimate to check if an answer is reasonable.

Round Mrs. Li’s total to the nearest thousand: $2,000. Then, add the rounded amounts. $2,000 + $600 + $3,000 = $5,600

The student’s calculation is not reasonable because $8,865 is not close to $5,600.

 

 

Guided Practice
Do You Understand?
Question 1.
Mr. Harm’s class sold 1,275 items at $5 per item. One of his students calculates the class raised $2,375. Is the student’s calculation reasonable? Explain.
Answer:
No, student’s calculation is not reasonable because class raised $6375 not $2375.

Explanation:
Number of items Mr. Harm’s class sold  = 1275.
Cost of each item Mr. Harm’s class sold = $5
Total cost of amount of items Mr. Harm’s class sold  = Number of items Mr. Harm’s class sold × Cost of each item Mr. Harm’s class sold
= 1275 × $5
= $6375.
One of his students calculates the class raised $2,375.

 

 

Do You Know How?
For 2-5, estimate the product.
Question 2.
6 × 125
Answer:
6 × 125 = 15000.

Explanation:
6 × 125 = 6 × (100 + 25)
6 × 125 = 600 + 150
6 × 125 = 750.

 

 

Question 3.
4 × 2,610
Answer:
4 × 2,610 = 10440.

Explanation:
4 × 2,610 = 4 × (2000 + 600 + 10)
4 × 2,610 = 8000 + 2400 + 40
4 × 2,610 = 10400 + 40
4 × 2,610 = 10440.

 

 

Question 4.
538 × 3
Answer:
538 × 3 = 1614.

Explanation:
538 × 3 = (500 + 30 + 8) × 3
538 × 3 = 1500 + 90 +  24
538 × 3 = 1590 + 24
538 × 3 = 1614.

 

Question 5.
314 × 7
Answer:
314 × 7 = 2198.

Explanation:
314 × 7 = (300 + 10 + 4) × 7
314 × 7 = 2100 + 70 + 28
314 × 7 = 2170 + 28
314 × 7 = 2198.

 

 

Independent Practice
Leveled Practice For 6-8, estimate the product.
Question 6.
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 14

Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.2 Estimate Products-Guided Practice-Independent Practice-6

Explanation:
287 Rounded to 290.
=> 3 × 290=270.

 

 

Question 7.
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 15

Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.2 Estimate Products-Guided Practice-Independent Practice-7

Explanation:
1310 Rounded to 1300.
=> 6 × 1300 = 7800.

 

 

Question 8.
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 16

Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.2 Estimate Products-Guided Practice-Independent Practice-8

Explanation:
62 Rounded to 60.
=> 9 × 60 = 540.

 

 

For 9-11, estimate to check if the answer is reasonable.
Question 9.
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 17

Answer:
Its reasonable because 3430 is the rounded number of 3402 and there is not much difference.

Explanation:
7 × 486 = 3402.
486 is rounded to 490.
=> 7 × 490 = 3430.

 

 

 

Question 10.
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 18

Answer:
Its not reasonable because 5 × 1240 = 6200 not 9200.

Explanation:
5 × 1240 = 6200.
1240 Rounded to 1240.
=> 5 × 1240 = 6200

 

 

Question 11.
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 19

Answer:
Its reasonable because 2610 is the rounded figure is 2583 and there is not much difference.

Explanation:
9 × 287 = 2583.
287 Rounded to 290.
9 × 290 = 2610.

 

 

Problem Solving
For 12-13, use the graph at the right.
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 20
Question 12.
The students voted on a school mascot. Which mascot had 4 times as many of votes as the flamingo?
Answer:
Mascot had 4 times as many of votes as the flamingo = Manatee = 158 votes.

Explanation:
Number of votes on Flamingo = 38.
Number of votes on Bobcat = 78.
Number of votes on Manatee = 158.
Number of votes on Alligator = 22.
Number of votes on Black Bear = 50.
Comparison:
Mascot had 4 times as many of votes as the flamingo = 4 × Number of votes on Flamingo
= 4 × 38
= 152.
=> Number of votes on Manatee = 158.

 

 

Question 13.
Construct Arguments Explain how you could estimate the number of students who voted on a school mascot. Then give your estimate.
Answer:
We can estimate by by adding the number of votes, they voted in school.
Total number of students = 346.

Explanation:
Number of votes on Flamingo = 38.
Number of votes on Bobcat = 78.
Number of votes on Manatee = 158.
Number of votes on Alligator = 22.
Number of votes on Black Bear = 50.
Total number of students = Number of votes on Flamingo + Number of votes on Bobcat + Number of votes on Manatee + Number of votes on Alligator + Number of votes on Black Bear
= 38 + 78 + 158 + 22 + 50
= 116 + 158 + 22 + 50
= 274 + 22 + 50
= 296 + 50
= 346.

 

 

Question 14.
Number Sense Ellie says the product of 211 and 6 is 1,866. Is this calculation reasonable? Explain.
Answer:
This calculation is not reasonable because 211 × 6 = 1266 not 1866.

Explanation:
211 × 6 = (200 + 11) × 6
= 1200 + 66
= 1266.

 

 

Question 15.
Higher Order Thinking An adult sleeps about 480 minutes per day. An infant sleeps about 820 minutes per day. About how many more minutes does an infant sleep than an adult in one week? Solve the problem two different ways.
Answer:
1Way:
Time taken more by infant than an adult to sleep per one week = Time an infant sleeps per day – Time an adult sleeps per day
= 820 – 480
= 340 minutes.
2Way:
Time an adult sleeps per day  = Time taken more by infant than an adult to sleep per one week + Time an infant sleeps per day
=> 820 = ??? + 480
=> 820 – 480 = ???
= 340 minutes = ???

Explanation:
1Way:
Time an adult sleeps per day = 480 minutes.
Time an infant sleeps per day = 820 minutes.
Time taken more by infant than an adult to sleep per one week = Time an infant sleeps per day – Time an adult sleeps per day
= 820 – 480
= 340 minutes.
2Way:
Time an adult sleeps per day = 480 minutes.
Time an infant sleeps per day = 820 minutes.
Time an adult sleeps per day  = Time taken more by infant than an adult to sleep per one week + Time an infant sleeps per day
=> 820 = ??? + 480
=> 820 – 480 = ???
= 340 minutes = ???

 

 

Assessment Practice
Question 16.
Gul can run 800 meters in 139 seconds. About how long would it take her to run 6 times as far if she runs at the same speed? Choose the best estimate.
A. About 140 seconds
B. About 600 seconds
C. About 1,200 seconds
D. About 6,000 seconds
Answer:
Time taken by Gul to run 6 times as far if she runs at the same speed = 834 seconds.
C. About 1,200 seconds

Explanation:
Time taken by Gul to run 800 meters = 139 seconds.
Time taken by Gul to run 6 times as far if she runs at the same speed = 6 × Time taken by Gul to run 800 meters
= 6 × 139
= 834 seconds.

 

 

Question 17.
A new two-seater car weighs 1,785 pounds. About how much would 8 of the two-seater cars weigh? Choose the best estimate.
A. About 160 pounds
B. About 1,600 pounds
C. About 16,000 pounds
D. About 60,000 pounds
Answer:
Weight of 8 of the two-seater cars weigh = 14280 pounds.
C. About 16,000 pounds

Explanation:
Weight of new two-seater car = 1,785 pounds.
Weight of 8 of the two-seater cars weigh = 8 × Weight of new two-seater car
= 8 × 1785
= 14280 pounds.

 

 

 

Lesson 3.3 Use Arrays and Partial Products to Multiply

Solve & Share
A music room has chairs arranged in rows. There are 6 rows. Each row has 18 chairs. How many chairs are in the music room? Solve this problem any way you choose. Explain your answer.
I can … use arrays and partial products to multiply.
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 21

Look Back! Would the answer be different if there were 18 rows with 6 chairs in each row? Explain.
Answer:
No, there won’t be any change in the answer because 18 × 6 or 6 × 18 are both the same just numbers are interchanged, product = 108.

Explanation:
Number of rows chairs arranged = 18.
Number of chairs in each row = 6.
Total number of chairs used = Number of rows chairs arranged × Number of chairs in each row
= 18 × 6
= 108.
Total number of chairs used = Number of chairs in each row × Number of rows chairs arranged
= 6 × 18
= 108.

 

 

Essential Question
How Can You Use an Array and Partial Products to Multiply?
Answer:
We can use an array and Partial Products to Multiply by multiplying each digit of a number in turn with each digit of another where each digit maintains its place.

Explanation:
The partial product method involves multiplying each digit of a number in turn with each digit of another where each digit maintains its place.

 

 

Visual Learning Bridge
The windows of a building on International Drive are in an array. There are 5 rows with 13 windows in each row. How many windows are in this array?
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 22

What You Show
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 23

What You Think
Find 5 × 13.
Estimate: 5 × 13 is about 5 × 10 = 50.
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 24
The numbers 15 and 50 are called partial products. 65 is the product.

What You Record
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 25
There are 65 windows in the array.
The product, 65, is close to the estimate of 50. The answer is reasonable.
You can use place value to break factors apart and the Distributive Property to find partial products.

Convince Me! Use Structure How are the partial products represented with the place-value blocks?
Answer:
The partial product method involves multiplying each digit of a number in turn with each digit of another where each digit maintains its place.

Explanation:
The partial product method involves multiplying each digit of a number in turn with each digit of another where each digit maintains its place.
For Example:
2 in 23 would actually be 20.
For instance, 23 x 42 would become (20 x 40) + (20 x 2) + (3 x 40) + (3 x 2).

 

 

Another Example!
Find 2 × 126.
Estimate: 2 × 126 is about 2 × 100 = 200.
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 26

 

 

 

Guided Practice
Do You Understand?
Question 1.
Explain how an array shows multiplication.
Answer:
An array is a way to represent multiplication and division using rows and columns.  The word problem will determine which way to draw the array.

Explanation:
Rows represent the number of groups. Columns represent the number in each group or the size of each group. The word problem will determine which way to draw the array.

 

 

Do You Know How?
For 2-3, complete each calculation.
Question 2.
2 × 24
Answer:
2 × 24 = 48.

Explanation:
2 × 24 = ??
=> 2 × 2 tens = 2 × 20 = 40.
=> 2 × 4 ones = 2 × 4 = 8.
2 × 24 = 40 + 8 = 48.

 

 

Question 3.
3 × 218
Answer:
3 × 218 = 654.

Explanation:
3 × 218 = ??
=> 3 × 2 hundreds = 3 × 200 = 600.
=> 3 × 1 tens = 3 × 10 = 30.
=> 3 × 8 ones = 3 × 8 = 24.
3 × 218 = 600 + 30 + 24 = 654.

 

Independent Practice
Leveled Practice For 4-5, multiply. Complete each equation.
Question 4.
Find 3 × 13.
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 27

Answer:
3 × 13 = 39.
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.3 Use Arrays and Partial Products to Multiply-Independent Practice-4

Explanation:
3 × 13 = ??
3 rows, 1 ten in each.
=> 3 × 10 = 30.
3 rows, 3 ones in each.
=> 3 × 3= 9.
=> 30 + 9 = 39..

 

 

Question 5.
Find 2 × 105
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 28

Answer:
2 × 105 = 210.
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.3 Use Arrays and Partial Products to Multiply-Independent Practice-5

Explanation:
2 × 105 = ??
2 rows, 1 hundred in each.
=> 2 × 100 = 200.
2 rows, 5 ones in each.
=> 2 × 5 = 10.
=> 200 + 10 = 210.

 

 

Problem Solving
Question 6.
Model with Math What multiplication equation do the place-value blocks show? Find the product. Then write a problem that could be solved using this model.
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 29
Answer:
300 + 20 + 4 = 324 blocks 3 rows.
Multiplication equation:
3 × 324 = 972.

Explanation:
3 rows, 3 hundred in each.
=> 3 × 300 = 900.
3 rows, 2 tens in each.
=> 3 × 20 = 60.
3 rows, 4 ones in each.
=> 3 × 4 = 12.
3 × 324 = 900 + 60+ 12 = 972.

 

 

Question 7.
How many marbles are in 3 large bags and 4 small bags?
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 30
Answer:
Total number of marbles in 4 small bags = 60.
Total number of marbles in 3 large bags = 240.

Explanation:
Number of marbles in small bag = 15.
Total number of marbles in 4 small bags = 4 × Number of marbles in small bag
= 4 × 15
= 60.
Number of marbles in large bag = 80.
Total number of marbles in 3 large bags = 3 × Number of marbles in large bag
= 3 × 80
= 240.

 

 

Question 8.
Higher Order Thinking How can the Distributive Property be used to find 4 × 875? Draw an array.
Answer:
4 × 875 = 3500.
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.3 Use Arrays and Partial Products to Multiply-Higher Order Thinking-8

Explanation:
4 × 875 =???
= 4 × 8hundreds = 4 × 800 = 3200.
= 4 × 7tens = 4 × 70 = 280.
= 4 × 5ones = 4 × 5 = 20.
4 × 875 = 3200 + 280 + 5 = 3500.

 

 

Question 9.
Select all the expressions that have a value of 312.
☐ (3 × 100) + (3 × 4)
☐ 3 × 12
☐ 3 × 112
☐ (30 × 10) + (3 × 12)
☐ 3 × 104
Answer:
(3 × 100) + (3 × 4) and 3 × 104 expressions have 312.

Explanation:
312 = ???
(3 × 100) + (3 × 4) = 300 + 12 = 312.
3 × 12 = 36.
3 × 112 = 336.
(30 × 10) + (3 × 12) = 300 + 36 = 336.
3 × 104 = 312.

 

 

Question 10.
Which are correct partial products for 54 × 9?
A. 36, 45
B. 36, 450
C. 540, 36
D. 32, 450
Answer:
54 × 9 = 450 + 36 = 486.
B. 36, 450 is the correct partial products for 54 × 9.

Explanation:
54 × 9 = ??
= 5tens × 9 = 9 × 50 = 450.
= 4ones × 9 = 4 × 9 = 36.
54 × 9 = 450 + 36 = 486.

 

 

Lesson 3.4 Use Area Models and Partial Products to Multiply

Solve & Share
Use only the numbers shown on the diagram and the operation symbols (+, -, ×, +) to determine the area of the unshaded rectangle below. Solve this problem using any strategy you choose.
I can use area models and partial products to multiply.
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 31

Look Back! Use Structure Are these equations equal? Explain. 8 × (10 – 4) =  (8 × 10) – (8 × 4)
Answer:
Yes, these equations are equal 8 × (10 – 4) =  (8 × 10) – (8 × 4)  = 48.

Explanation:
8 × (10 – 4) = (8 × 10) – (8 × 4) = ??
8 × (10 – 4) = 80 – 32 = 48.
(8 × 10) – (8 × 4) = 80 – 32 = 48.

 

 

Essential Question
How Can You Use an Area Model and Partial Products to Multiply?
Answer:
The partial product method involves multiplying each digit of a number in turn with each digit of another where each digit maintains its place. Write each factor in expanded form. Then, draw your model. Next, multiply to find the area of each smaller rectangle. Finally, add those products to find the total area.

Explanation:
First, write each factor in expanded form. Then, draw your model. Next, multiply to find the area of each smaller rectangle. Finally, add those products to find the total area.
The partial product method involves multiplying each digit of a number in turn with each digit of another where each digit maintains its place.

 

 

Visual Learning Bridge
A garden is in the shape of a rectangle. It is 8 feet wide and 25 feet long. What is the area of the garden?
A numerical expression contains numbers and at least one operation. 8 × 25 is a numerical expression.

What You Show
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 32

What You Think
Estimate: 8 × 25 is about 8 × 30 = 240. 20
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 33

What You Record
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 34
The area of the garden is 200 square feet.
The product, 200, is close to the estimate of 240. The answer is reasonable.

Convince Me! Use Structure Does 12 – (4 × 2) = (12 – 4) (12 – 2)? Explain.
Answer:
No, this equation is not correct because 12 – (4 × 2) = 24 where as (12 – 4) (12 – 2) =80.

Explanation:
12 – (4 × 2) = (12 × 4) – (12 × 2) = 48 – 24 = 24.
(12 – 4) (12 – 2) = 8 × 10 = 80.

 

 

 

Another Example!
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 35

 

Guided Practice
Do You Understand?
Question 1.
What numerical expression is shown by the area model below?
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 36

Answer:
265 × 7 = 1855.

Explanation:
(200 + 60 + 5) 7 = 265 × 7
=> 7 × 200 = 1400.
=> 7 × 60 = 420.
=> 7 × 5 = 35.
1400 + 420 + 35 = 1855.

 

 

Do You Know How?
Question 2.
Multiply. Use the area model and partial products.
Envision Math Common Core 4th Grade Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 37

Answer:
560 × 7 = 3920.

Explanation:
(500 + 60 ) 7 = 560 × 7
=> 7 × 500 = 3500
=> 7 × 60 = 420.
3500 + 420 = 3920.

 

 

Independent Practice
Leveled Practice For 3-6, multiply. Use the area model and partial products.
Question 3.
Envision Math Common Core 4th Grade Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 38
Answer:
34 × 6 = 204.

Explanation:
(30+ 4) 6 = 34 × 6
=> 6 × 30 = 180.
=> 6 × 4 = 24.
180 + 24 = 204.

 

 

Question 4.
Envision Math Common Core 4th Grade Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 39
Answer:
99 × 2 = 198.

Explanation:
(90 + 9) 2 = 99 × 2
=> 2 × 90 = 180.
=> 2 × 9 = 18.
180 + 18 = 198.

 

 

Question 5.
Envision Math Common Core 4th Grade Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 40
Answer:
3 × 185 = 555.

Explanation:
3 (100 + 80 + 5) = 3 × 185
=> 3 × 100 = 300.
=> 3 × 80 = 240.
=> 3 × 5 = 15.
300 + 240 + 15 = 555.

 

 

Question 6.
Envision Math Common Core 4th Grade Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 41
Answer:
8 × 440 = 3520.

Explanation:
8 ( 400 + 40) = 8 × 440
=> 8 × 400 = 3200.
=> 8 × 40 = 320.
3200 + 320 = 3520.

 

 

Problem Solving
Question 7.
Model with Math Last year, Anthony’s grandmother gave him 33 silver coins and 16 gold coins to start a coin collection. Now Anthony has six times as many coins in his collection. How many coins does Anthony have in his collection? Complete the bar diagram to show your work.
Envision Math Common Core 4th Grade Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 42

Answer:
Number of coins Anthony have in his collection now = 294.

Explanation:
Number of silver coins Anthony’s grandmother gave him = 33.
Number of gold coins Anthony’s grandmother gave him = 16.
Now Anthony has six times as many coins in his collection.
Number of coins Anthony have in his collection now = 6 × (Number of silver coins Anthony’s grandmother gave him + Number of gold coins Anthony’s grandmother gave him)
= 6 × (33 + 16)
= (6 × 33) + (6 × 16)
= 198 + 96
= 294.
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.4 Use Area Models and Partial Products to Multiply-Problem Solving-7

 

 

Question 8.
Wyatt said he used partial products to write 7 × 870 = 5,600 + 49. Explain Wyatt’s error and use math to justify your explanation.
Answer:
7 × 870 = 5600 + 490 not 5,600 + 49  because Wyatt’s error is that he took partial products value wrong and got the result wrong answer.

Explanation:
7 × 870 = (7 × 800) + (7 × 70)
7 × 870 = 5600 + 490
7 × 870 = 6090.
5,600 + 49 = 5649.

 

 

Question 9.
Higher Order Thinking Tod Mountain is a mountain peak near Tyler, Texas. A ranger hiked 607 feet to and from the peak, each way. The ranger hiked 3 times in the past four weeks. How far did the ranger hike on Tod Mountain over the past four weeks?
Answer:
Length of the ranger hiked on Tod Mountain over the past four weeks = 1821 feet.

Explanation:
Length of the ranger hiked to and from the peak, each way of Tod Mountain = 607 feet.
The ranger hiked 3 times in the past four weeks.
Length of the ranger hiked on Tod Mountain over the past four weeks = 3 × Length of the ranger hiked to and from the peak, each way of Tod Mountain
= 3 × 607
= (3 × 600) + (3 × 7)
= 1800 + 21
= 1821 feet.

 

Question 10.
Wendy plans to bring beverages for the school picnic. She has 5 gallons of iced tea. Also, she will bring 2 gallons of lemonade for every 10 people. How many total gallons of lemonade and iced tea does Wendy need to bring for 40 people? Complete the table.
Envision Math Common Core 4th Grade Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 43
Answer:
Total gallons of lemonade and iced tea Wendy needs to bring for 40 people = 28.

Explanation:
Number of gallons of iced tea Wendy has for every 10 people = 5.
Number of gallons of lemonade Wendy has for every 10 people = 2.
Total gallons of lemonade and iced tea Wendy needs to bring for 40 people = 4 × (Number of gallons of iced tea Wendy has for every 10 people + Number of gallons of lemonade Wendy has for every 10 people)
= 4 × (5 + 2)
= (4 × 5) + (4 × 2)
= 20 + 8
= 28.

 

 

Assessment Practice
Question 11.
What is the missing factor?
6 × ? = 264
Envision Math Common Core 4th Grade Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 44
A. 50
B. 44
C. 40
D. 30
Answer:
6 × 44 = 264.
B. 44

Explanation:
6 × ? = 264
=> ? = 264 ÷ 6
=> ? = 44.
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.4 Use Area Models and Partial Products to Multiply-Assessment Practice-11

 

Question 12.
Which are the partial products for this area model?
7 × 228
Envision Math Common Core 4th Grade Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 45
A. 1,400, 144, 56
B. 14,000, 140, 56
C. 56, 14, 1,400
D. 56, 140, 1,400
Answer:
7 × 228 = 1400 + 140 + 56 = 1596.
D. 56, 140, 1,400

Explanation:
7 × 228 = (7 × 200) + (7 × 20) + (7 × 8)
7 × 228 = 1400 + 140 + 56
7 × 228 =1540 + 56
7 × 228 = 1596.

 

 

Lesson 3.5 More Use Area Models and Partial Products to Multiply

Solve & Share
The horseshoe-pit model below has an area of 228 square feet. The length of one part of the pit was erased by mistake. What is the length of the missing section, x? Solve any way you choose. Explain how you found the answer.
I can … use area models and partial products to multiply.
Envision Math Common Core 4th Grade Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 46

Look Back! What multiplication equation can be used to represent the horseshoe-pit model above?
Answer:
Length of the second section missing horseshoe-pit model = x = 8 feet.

Explanation:
The horseshoe-pit model below has an area of 228 square feet. The length of one part of the pit was erased by mistake. What is the length of the missing section, x? Solve any way you choose.
Area of the horseshoe-pit model = 228 square feet.
Length of the first section horseshoe-pit model = 30 feet.
Length of the second section missing horseshoe-pit model = x feet.
Breadth of the horseshoe-pit model  = 6 feet.
Area of the horseshoe-pit model = length × Breadth
=> 228 = (Length of the first section horseshoe-pit model + Length of the second section missing horseshoe-pit model) × 6
=> 228 = (30 + x) × 6
=> 228 ÷ 6 = 30 + x
=> 38 = 30 + x
=> 38 – 30 = x
=> 8 feet = x.

 

 

Essential Question
How Do You Multiply with Greater Numbers?
Answer:
We multiply greater number Multiplication by using the place values of each number given and multiply and get the result to that problem.

Explanation:
When multiplying by larger numbers with two digits or more, use one placeholding zero when multiplying by the tens digit, two placeholding zeros when multiplying by the hundreds digit, three zeros when multiplying by the thousands digit, and so forth.

 

Visual Learning Bridge
The Rails-to-Trails Preservation opened a new section of a biking trail. The section is 6 yards wide. What is its area? Calculate 6 × 1,842.
The process for multiplying is the same regardless of the number of digits.
Envision Math Common Core 4th Grade Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 47

Find 6 × 1,842 using an area model and partial products.
Estimate: 6 × 1,842 is about 6 × 2,000 = 12,000.
Envision Math Common Core 4th Grade Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 48

Remember you can find partial products in any order.
Envision Math Common Core 4th Grade Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 49
The new section is 11,052 square yards.

Convince Me! Use Structure How is the process of using partial products to find the final product the same for each of these calculations? How is it different? Explain.
Envision Math Common Core 4th Grade Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 50
Answer:
The process of using partial products to find the final product for each of these calculations is not same, its completely different because the numbers given for multiplication having different place values and result of their products also different.

Explanation:
5 × 34 = (5 × 30) + (5 × 4)
5 × 34 = 150 + 20
5 × 34 = 170.

5 × 234 = (5 × 200) + (5 × 30) + (5 × 4)
5 × 234 = 1000 + 150 + 20
5 × 234 = 1150 + 20
5 × 234 = 1170.

5 × 1234 = (5 × 1000) + (5 × 200) + (5 × 30) + (5 × 4)
5 × 1234 = 5000 + 1000 + 150 + 20
5 × 1234 = 6000 + 150 + 20
5 × 1234 = 6150 + 20
5 × 1234 = 6170.

 

Guided Practice
Do You Understand?
Question 1.
What multiplication expression is shown by the area model below?
Envision Math Common Core 4th Grade Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 51
Answer:
4 × (1000 + 400 + 5) = 4 × 1405 = 5620.

Explanation:
4 × (1000 + 400 + 5) = ????
4 × (1000 + 400 + 5) = (4 × 1000) + (4 × 400) + ( 4 × 5)
4 × (1000 + 400 + 5) = 4000 + 1600 + 20
4 × (1000 + 400 + 5) = 5600 + 20
4 × (1000 + 400 + 5) = 5620.
=> 4 × 1405 = 5620. (1000 + 400 + 5 = 1405)

 

 

Do You Know How?
For 2, multiply. Use the area model and partial products.
Question 2.
5 × 1,117
Envision Math Common Core 4th Grade Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 52
Answer:
5 × 1,117 = 5585.

Explanation:
Length of the given rectangle figure = 1000 + 100 + 10 + 7 = 1117.
Breadth of the given rectangle figure = 5.
Area of the given rectangle figure = Length of the given rectangle figure × Breadth of the given rectangle figure
= 1117 × 5
= (1000 × 5) + (100 × 5) + (10 × 5) + (7 × 5)
= 5000 + 500 + 50 + 35
= 5500 + 50 + 35
= 5550 + 35
= 5585.

 

 

Independent Practice
For 3-8, multiply. Use the area model and partial products.
Envision Math Common Core 4th Grade Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 53

Question 3.
Envision Math Common Core 4th Grade Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 54
Answer:
1092 × 8 = (1000 × 8) + (90 × 8) + (2 × 8) = 8736.

Explanation:
Length of the given rectangle figure = 1000 + 90+ 2 = 1092.
Breadth of the given rectangle figure = 8.
Area of the given rectangle figure = Length of the given rectangle figure × Breadth of the given rectangle figure
= 1092× 8
= (1000 × 8) + (90 × 8) + (2 × 8)
= 8000 + 720 + 16
= 8720 + 16
= 8736.

 

Question 4.
Envision Math Common Core 4th Grade Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 55

Answer:
4261 × 3 = (4000 × 3) + (200× 3) + (60 × 3) + (1 × 3) = 12783.

Explanation:
Length of the given rectangle figure = 4000 + 200 + 60 + 1= 4261.
Breadth of the given rectangle figure = 3.
Area of the given rectangle figure = Length of the given rectangle figure × Breadth of the given rectangle figure
= 4261 × 3
= (4000 × 3) + (200× 3) + (60 × 3) + (1 × 3)
= 12000 + 600 + 180 + 3
= 12600 + 180 + 3
= 12780 + 3
= 12783.

 

 

Question 5.
Envision Math Common Core 4th Grade Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 56
Answer:
1990× 2 = (1000 × 2) + (900× 2) + (90 × 2) = 3980.

Explanation:
Length of the given rectangle figure = 1000 + 900 + 90 = 1990
Breadth of the given rectangle figure = 2.
Area of the given rectangle figure = Length of the given rectangle figure × Breadth of the given rectangle figure
= 1990× 2
= (1000 × 2) + (900× 2) + (90 × 2)
= 2000 + 1800 + 180
= 3800 + 180
= 3980.

 

 

Question 6.
Envision Math Common Core 4th Grade Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 57

Answer:
2347 × 6 = (2000 × 6) + (300× 6) + (40 × 6) + (7 × 6) = 14082.

Explanation:
Length of the given rectangle figure = 2000 + 300 + 40 + 7 = 2347
Breadth of the given rectangle figure = 6.
Area of the given rectangle figure = Length of the given rectangle figure × Breadth of the given rectangle figure
= 2347 × 6
= (2000 × 6) + (300× 6) + (40 × 6) + (7 × 6)
= 12000 + 1800 + 240 + 42
= 13800 + 240 + 42
= 14040 + 42
= 14082.

 

Question 7.
Envision Math Common Core 4th Grade Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 58

Answer:
3022 × 5 = (3000 × 5) + (20 × 5) + (2 × 5)  = 15110.

Explanation:
Length of the given rectangle figure = 3000 + 20 + 2 = 3022.
Breadth of the given rectangle figure = 5.
Area of the given rectangle figure = Length of the given rectangle figure × Breadth of the given rectangle figure
= 3022 × 5
= (3000 × 5) + (20 × 5) + (2 × 5)
= 15000 + 100 + 10
= 15100 + 10
= 15110.

 

 

Question 8.
Envision Math Common Core 4th Grade Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 59
Answer:
1999 × 7 = (1000 × 7) + (900 × 7) + (90 × 7) + (9 × 7) = 13993.

Explanation:
Length of the given rectangle figure = 1000 + 900 + 90 + 9 = 1999.
Breadth of the given rectangle figure = 7.
Area of the given rectangle figure = Length of the given rectangle figure × Breadth of the given rectangle figure
= 1999 × 7
= (1000 × 7) + (900 × 7) + (90 × 7) + (9 × 7)
= 7000 + 6300 + 630 + 63
= 13300 + 630 + 63
= 13930 + 63
= 13993.

 

 

Problem Solving
Question 9.
Be Precise There are usually 365 days in each year. Every fourth year is called a leap year and has one extra day in February. How many days are there in 8 years if 2 of the years are leap years?
Answer:
Number of days in 8 years = 2922.

Explanation:
Number of days in each year = 365.
Every fourth year is called a leap year and has one extra day in February.
=> Number of days in leap year = 366.
Number of days in 8 years = 6 ( Number of days in each year) + 2 (Number of days in leap year)
= (6 × 365) + (2 × 366)
= 2190 + 732
= 2922.

 

 

Question 10.
There are 1,250 seeds in each package. There are 5 packages. How many seeds are there in all?
Answer:
1250 × 5 = (1000 × 5) + (200 × 5) + ( 50 × 5) = 6250.

Explanation:
Number of seeds in each package = 1250.
Number of packages = 5.
Total number of seeds in all = Number of seeds in each package × Number of packages
= 1250 × 5
= (1000 × 5) + (200 × 5) + ( 50 × 5)
= 5000 + 1000 + 250
= 6000 + 250
= 6250.

 

 

Question 11.
A cat breeder has 6 Sphynx kittens and 7 Persian kittens for sale. If all 13 kittens sell, how much money will the breeder earn? Write and solve equations. Tell what your variables represent.
Envision Math Common Core 4th Grade Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 60

Answer:
Total money the breeder earns = $9613.

Explanation:
Number of Sphynx kittens a cat breeder has for sale = 6.
Number of Persian kittens a cat breeder has for sale = 7.
Cost of each Sphynx kitten = $698.
Cost of each Persian kitten = $775.
Total money the breeder earns = ( Number of Sphynx kittens a cat breeder has for sale × Cost of each Sphynx kitten) + (Number of Persian kittens a cat breeder has for sale × Cost of each Persian kitten )
= (6 × $698) + (7 × $775)
= $4188 + $5425
= $9613.

 

 

Question 12.
Higher Order Thinking Patricia creates a design using 1,025 tiles. She doubles the number of tiles to make a second design. Her third design uses 3 times as many tiles as the second design. How many tiles does Patricia use in her third design? Explain.
Answer:
Number of tiles Patricia uses for creating third design = 6150.

Explanation:
Number of tiles Patricia uses for creating first design = 1025.
She doubles the number of tiles to make a second design.
=> Number of tiles Patricia uses for creating second design = 2×1025 = 2050.
Her third design uses 3 times as many tiles as the second design.
=> Number of tiles Patricia uses for creating third design = 3 × Number of tiles Patricia uses for creating second design
=> 3 × 2050
=> 6150.

 

 

Assessment Practice
Question 13.
Which are the correct partial products of 3 × 3,672?
A. 6; 21; 1,800; 9,000
B. 9,000; 1,800; 210; 6
C. 210; 1,800; 9,000
D. 6; 210; 180; 9,000
Answer:
The correct partial products of 3 × 3,672:
B. 9,000; 1,800; 210; 6

Explanation:
3 × 3,672 = (3 × 3000) + (3 × 600) + (3 × 70) + (3 × 2)
3 × 3,672 = 9000 + 1800 + 210 + 6
3 × 3,672 = 10800 + 210 + 6
3 × 3,672 = 11010 + 6
3 × 3,672 = 11016.

 

 

Question 14.
Select all the expressions that have 1,600 as a partial product.
☐ 4 × 4,381
☐ 8 × 3,240
☐ 4 × 1,408
☐ 2 × 7,881
☐ 8 × 2,021
Answer:
Expressions that have 1,600 as a partial product:
8 × 3,240
4 × 1,408
2 × 7,881.

Explanation:
4 × 4,381 = (4 × 4000) + (4 ×300) + (4 × 80) + (4 × 1)
4 × 4,381 = 16000 + 1200 + 320 + 4
4 × 4,381 = 17200 + 320 + 4
4 × 4,381 = 17520 + 4
4 × 4,381 = 17524.

8 × 3,240 = (8 × 3000) + (8 ×200) + (8 × 40) + (8 × 1)
8 × 3,240 = 24000 + 1600 + 320 + 8
8 × 3,240 = 25600 + 320 + 8
8 × 3,240 = 25920 + 8
8 × 3,240 = 25928.

4 × 1,408 = (4 × 1000) + (4 ×400) + (4 × 8)
4 × 1,408 = 4000 + 1600 + 32
4 × 1,408 = 5600 + 32
4 × 1,408 = 5632.

2 × 7,881 = (2 × 7000) + (2 ×800) + (2 × 80) + (2 × 1)
2 × 7,881 = 14000 + 1600 + 160 + 2
2 × 7,881 = 15600 + 160 + 2
2 × 7,881 = 15760 + 2
2 × 7,881 = 15762.

8 × 2,021 = (8 × 2000) + (8 × 20) + (8 × 1)
8 × 2,021 = 16000 + 160 + 8
8 × 2,021 = 16160 + 8
8 × 2,021 = 16168.

 

 

Lesson 3.6 Mental Math Strategies for Multiplication

Solve & Share
Determine the products for the expressions given below. Use mental math to solve. Explain your thinking. Solve these problems using any strategy you choose.
I can … use mental math strategies based on place value and properties of operations to multiply.
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 61

Look Back! Describe two different ways to find 4 × 97.
Answer:
Way:1:
4 × 97 = (4 × 100) – (4 × 3) = 400 – 12 = 388.
Way:2:
4 × 97 = 4 × (90 + 7) = (4 × 90) + (4 × 7) = 360 + 28 = 388.

Explanation:
Way:1:
4 × 97 = 4 × (100 – 3)
4 × 97 = (4 × 100) – (4 × 3)
4 × 97 = 400 – 12
4 × 97 = 388.
Way:2:
4 × 97 = 4 × (90 + 7)
4 × 97 = (4 × 90) + (4 × 7)
4 × 97 = 360 + 28
4 × 97 = 388.

 

 

Essential Question
How Can You Multiply Mentally?
Answer:
We can Multiply Mentally by rounding the numbers to nearest place and doing the required operation necessary to solve the problem.

Explanation:
For example: Any time you square a two-digit number that ends in 5, the last digits of the answer will be 25 and the digits before that are given by multiplying the first digit of the number by the number that’s one greater.

 

Visual Learning Bridge
Three cyclists rode their bikes the distances shown in the table. Use mental math to calculate the total distance Pam and Anna each rode.
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 62

Multiply 4 × 325 to find the distance Pam rode.
To multiply mentally, you can break apart numbers using place value and the Distributive Property.
Think:
4 × 325 = 4 × 300 + 4 × 25
= 1,200 + 100
= 1,300
Pam rode 1,300 miles.

Multiply 8 × 25 to find the distance Anna rode.
To multiply mentally, you can break apart and rearrange numbers using the Commutative and Associative Properties.
Think: 8 × 25 = (4 × 2) × 25
= 2 × (4 × 25)
= 2 × 100
= 200
Anna rode 200 miles.

Convince Me! Use Structure in the work for 4 × 325 above, why was 25 not broken into 20 and 5? Explain.
Answer:
25 was not broken into 20 and 5 because 325 already was broken into two parts 300 and 25 , so its not broken again.

Explanation:
To multiply mentally, you can break apart numbers using place value and the Distributive Property.
4 × 325 = 4 × 300 + 4 × 25
4 × 325 = 1,200 + 100
4 × 325 = 1,300.
Pam rode 1,300 miles.

 

 

Another Example!
Determine the distance George rode.
Find 3 × 398. 400 is close to 398.
Find 3 × 400 and adjust the answer.
3 × 400 = 1,200
398 + 2 = 400 3 × 2 = 6
Adjust the answer by subtracting 6. 1,200 – 6 = 1,194
George rode 1,194 miles in 3 months.
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 63

Guided Practice
Do You Understand?
Question 1.
Why was 6 subtracted from 1,200 in the problem above?
Answer:
6 was subtracted from 1,200 in the problem above, because 398 was rounded nearest to hundreds number 400 and it was rearranged to get its actual figure.

Explanation:
George rode ???? miles in 3 months.
3 × 398. 400 is close to 398.
3 × 400 = 1,200
398 + 2 = 400.
3 × 2 = 6.
Adjust the answer by subtracting 6.
1,200 – 6 = 1,194.
George rode 1,194 miles in 3 months.

 

Do You Know How?
For 2, multiply mentally to find the product. Explain which strategy you used.
Question 2.
8 × 903?
Answer:
8 × 903 = 7224.

Explanation:
8 × 903 = (8 × 900) + (8 × 3)
8 × 903 = 7200 + 24
8 × 903 = 7224.

 

 

Independent Practice
Leveled Practice For 3-10, multiply mentally to find each product. Explain which strategy you used.
Question 3.
6 × 250 = (3 + _____) × 250
Answer:
6 × 250 = (3 + __3___) × 250 = 1500.

Explanation:
6 × 250 = (3 + 3) × 250
6 × 250 = (3 × 250) + (3 × 250)
6 × 250 = 750 + 750
6 × 250 = 1500.

 

Question 4.
4 × 506 = 4 × (_____+ _____)
Answer:
4 × 506 = 4 × (__500___+ ___6__) = 2024.

Explanation:
4 × 506 = 4 × (__500___+ ___6__)
4 × 506 = (4 × 500) + (4 × 6)
4 × 506 = 2000 + 24
4 × 506 = 2024.

 

Question 5.
4 × 1,995
Answer:
4 × 1,995 = 4 (2000 – 5) = 7980.

Explanation:
4 × 1,995 = 4 (2000 – 5)
4 × 1,995 = (4 × 2000) – (4 × 5)
4 × 1,995 = 8000 – 20
4 × 1,995 = 7980.

 

 

Question 6.
22 × 5
Answer:
22 × 5 = (20 + 2) 5 = 100 + 10 = 110.

Explanation:
22 × 5 = (20 + 2) 5
22 × 5 = (20 × 5) + (2 × 5)
22 × 5 = 100 + 10
22 × 5 = 110.

 

 

Question 7.
404 × 6
Answer:
404 × 6 = (400 + 4) 6 = (400 × 6) + ( 4 × 6) = 2424.

Explanation:
404 × 6 = (400 + 4) 6
404 × 6 = (400 × 6) + ( 4 × 6)
404 × 6 = 2400 + 24
404 × 6 = 2424.

 

Question 8.
7 × 250
Answer:
7 × 250 = 7 ( 200 + 50) = 1400 + 350 = 1750.

Explanation:
7 × 250 = 7 ( 200 + 50)
7 × 250 = (7 × 200) + (7 × 50)
7 × 250 = 1400 + 350
7 × 250 = 1750.

 

 

Question 9.
2 × 395
Answer:
2 × 395 = 2 (400 – 5) = 800 – 10 = 790.

Explanation:
2 × 395 = 2 (400 – 5)
2 × 395 = (2 × 400) – (2 × 5)
2 × 395 = 800 – 10
2 × 395 = 790.

 

 

Question 10.
9 × 56
Answer:
9 × 56 = 9 (50 + 6) = (9 × 50) + ( 9 × 6) = 450 + 54 = 504.

Explanation:
9 × 56 = 9 (50 + 6)
9 × 56 = (9 × 50) + ( 9 × 6)
9 × 56 = 450 + 54
9 × 56 = 504.

 

 

Problem Solving
Question 11.
Each elephant at a zoo eats 100 pounds of hay and 5 pounds of fruits and vegetables every day. How many pounds of food does the zoo need to feed one elephant for one week? Use mental math to solve.
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 64
Answer:
Number of pounds of food the zoo needs to feed one elephant for one week = 735.

Explanation:
Number of pounds of hay every day each elephant at a zoo eats = 100.
Number of pounds of fruits and vegetables every day each elephant at a zoo eats = 5.
1 Week = 7 days.
Number of pounds of food the zoo needs to feed one elephant for one week = 7 (Number of pounds of hay every day each elephant at a zoo eats + Number of pounds of fruits and vegetables every day each elephant at a zoo eats)
= 7 (100 + 5)
= (7 × 100) + (7 × 5)
= 700 + 35
= 735.

 

 

Question 12.
Ashley and 3 friends are planning a trip. The cost of the trip is $599 per person. How much will the trip cost Ashley and her friends? Explain the mental math strategy you used to find the answer.
Answer:
Total trip cost  of Ashley and her friends = $2396.
The mental math strategy you used to find the answer is break apart numbers using place value and the Distributive Property.

Explanation:
Number of people planning for a trip =(Ashley and 3 friends) = 1 + 3 = 4.
Cost of the trip per person = $599.
Total trip cost  of Ashley and her friends = Number of people planning for a trip × Cost of the trip per person
= 4 × $599
= (4 × 500) + (4 × 99)
= 2000 + 396
= $2396.

 

 

Question 13.
Kyle has a rock collection. On Monday, he found 16 new rocks. On Tuesday, he gave 9 rocks to his friends. After giving away the rocks, Kyle had 122 rocks left in his collection. How many rocks did Kyle have to start with?
Answer:
Number of rocks Kyle have to start with = 129.

Explanation:
Number of new rocks on Monday, Kyle found = 16.
Number of rocks on Tuesday, Kyle gave to his friends = 9.
Number of rocks left in his collection after giving = 122.
Number of rocks Kyle have to start with = Number of new rocks on Monday, Kyle found – Number of rocks on Tuesday, Kyle gave to his friends + Number of rocks left in his collection after giving
= 16 – 9 + 122
= 7 + 122
= 129.

 

 

Question 14.
Critique Reasoning Quinn used compensation to find the product of 4 × 307. First, she found 4 × 300 = 1,200. Then she adjusted the product by subtracting 4 groups of 7 to get her final answer of 1,172. Explain Quinn’s mistake and find the correct answer.
Answer:
4 × 307 = 1228 not 1172. Quinn’s mistake is that 4 × 300 = 1200 is correct yet subtracting  did  4 groups of 7  is wrong, they needed to be added to get the correct answer.

Explanation:
4 × 307 = 4 (300 + 7)
4 × 307 = (4 × 300) + (4 × 7)
4 × 307 = 1200 + 28
4 × 307 = 1228.

 

 

Question 15.
Higher Order Thinking Do you think it would be better to use breaking apart and the Distributive Property or compensation to find the product of 5 × 328? Explain why and show how to find the product.
Answer:
It would be better to use breaking apart and the Distributive Property or compensation to find the product of 5 × 328 because its a big number to do multiplication at once, so breaking them into parts and doing the process helps to get correct answer without any mistakes.

Explanation:
5 × 328 = 5 (300 + 28)
5 × 328 = (5 × 300) + (5 × 28)
5 × 328 = 1500 + 140
5 × 328 =1640.

 

 

Assessment Practice
Question 16.
Select all of the expressions that show how to use mental math to find the product of 4 × 27.
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 65
☐ (4 × 20) + (4 × 7)
☐ 4 × (20 × 7)
☐ (4 × 30) – (4 × 3)
☐ (4 × 25) + (4 × 2)
☐ 4 × 2 × 7
Answer:
The expressions that show how to use mental math to find the product of 4 × 27:
(4 × 20) + (4 × 7)
4 × (20 × 7)
(4 × 30) – (4 × 3)
(4 × 25) + (4 × 2)

Explanation:
4 × 27 = 4 (20 + 7)
4 × 27 = (4 × 20) + (4 × 7)
4 × 27 = 80 + 28
4 × 27 = 108.

4 × 27 = 4 (30 – 3)
4 × 27 = (4 × 30) – (4 × 3)
4 × 27 = 120 – 12
4 × 27 = 108.

4 × 27 = 4 (25 + 2)
4 × 27 = (4 × 25) + (4 × 2)
4 × 27 = 100 + 8
4 × 27 = 108.

 

 

 

Lesson 3.7 Choose a Strategy to Multiply

Solve & Share
A cineplex has 4 movie theaters. Each theater has 342 floor seats and 85 mezzanine seats. How many people can the cineplex seat? Solve this problem using any strategy you choose. Explain your solution.
I can… choose an appropriate strategy to multiply.
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 66
Look Back! What strategy or strategies did you use to solve this problem? Explain why.
Answer:
Manipulation tools strategy the use of blocks, fruits, balls, or other manipulation tools help students learn the basics of place value, addition, subtraction, and other areas of basic math. It is used to solve this problem.
Number of people in the cineplex seat = 1708.

Explanation:
A cineplex has 4 movie theaters. Each theater has 342 floor seats and 85 mezzanine seats. How many people can the cineplex seat.
Number of floor seats each theater has = 342.
Number of mezzanine seats each theater has = 85.
Number of movie theaters a cineplex has = 4.
Number of people in the cineplex seat = 4 (342 + 85)
= (4 × 342) + (4 × 85)
= 1368 + 340
= 1708.

 

 

Essential Question
What Strategy Will You Use to Multiply?
Answer:
The Strategy Used to do Multiply  is using the place value of the numbers given and do the required operation to solve the problem.

Explanation:
The place value of the numbers given and do the required operation to solve the problem can be followed to multiply.

 

 

Visual Learning Bridge
Paying for pothole damage to cars can be costly. The table shows the cost of some car repairs. What is the total cost of the repairs?
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 67

Find the cost for the tires, c.
4 × $125 = 0
Estimate: 4 × 125 is about 4 × 120 = 480.
Break apart 125 using place value and the Distributive Property.
4 × 125 = 4 × (100 + 25)
= 4 × 100 + 4 × 25
= 400 + 100
= 500
c = 500
The tire repairs cost $500.

Find the cost of the paint, y.
2 × $1,450 = y
Estimate: 2 × 1,450 is about 2 x× 1,500 = 3,000.
Use an area model and partial products.
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 68
To paint the car costs $2,900.

Find the total cost of the repairs, z.
$500 + $2,900 = 2
Estimate: 500 + 2,900 is about 500 + 3,000 = 3,500.
Use compensation to add.
500 + 3,000 = 3,500
3,000 is 100 more than 2,900.
Subtract 100.
3,500 – 100 = 3,400
z = 3,400
The total cost of the repairs was $3,400.

Convince Me! Reasoning Explain why the answers to each part of the problem above are reasonable.
Answer:
The total cost of the repairs was $3,400. The answers to each part of the problem above are reasonable because they are correctly solved using all required strategies.

Explanation:
The cost for the tires, c.
4 × $125 = c.
4 × 125 = 4 × (100 + 25)
4 × 125 = 4 × 100 + 4 × 25
4 × 125 = 400 + 100
4 × 125 = 500.
c = 500
The tire repairs cost $500.

The cost of the paint, y.
2 × $1,450 = y
2 × 1,450 = 2 (1000 + 450)
2 × 1,450 = (2 × 1000) + (2 × 450)
2 × 1,450 = 2000 + 900
2 × 1,450 = $2900.

The total cost of the repairs, z.
x + y = z.
=> $500 + $2,900
=> $3400.

 

Guided Practice
Do You Understand?
Question 1.
A tire shop sells 3 tires that cost $175 each, which includes a fourth tire for free. Is this more or less expensive than buying 4 tires that cost $135 each?
Answer:
Cost of all tires a tire shop sells = $700  is more than Cost of all tires = $540.

Explanation:
Number of tires a tire shop sells = 3.
Cost of each tire a tire shop sells = $175.
Number of tires includes a fourth tire for free a tire shop sells = 1.
Cost of all tires a tire shop sells = (Number of tires a tire shop sells + Number of tires includes a fourth tire for free a tire shop sells) × Cost of each tire a tire shop sells
= (3 + 1) 175
= (3 × 175) + (1 × 175)
= 525 + 175
= $700.

Number of tires = 4.
Cost of each tire = $135.
Cost of all tires = Number of tires × Cost of each tire
= 4 × 135
= 4 (100 + 35)
= (4 × 100) + (4 × 35)
= 400 + 140
= $540.

 

 

Do You Know How?
For 2-5, find each product by choosing an appropriate strategy.
Question 2.
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 69
Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.7 Choose a Strategy to Multiply-Do You Know How-2

Explanation:
74 × 6 = (70 × 6) + (4 × 6)
74 × 6 = 420 + 24
74 × 6 = 444.

 

Question 3.
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 70
Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.7 Choose a Strategy to Multiply-Do You Know How-3

Explanation:
819 × 5 = (800 × 5) + (19 × 5)
819 × 5 = 4000 + 95
819 × 5 = 4095.

 

Question 4.
4 × 309
Answer:
4 × 309 = 1236.

Explanation:
4 × 309 = (4 × 300) + (4 × 9)
4 × 309 = 1200 + 36
4 × 309 = 1236.

 

Question 5.
3 × 354
Answer:
3 × 354 = 1062.

Explanation:
3 × 354 = (3 × 300) + (3 × 50) + (3 × 4)
3 × 354 = 900 + 150 + 12
3 × 354 = 1050 + 12
3 × 354 = 1062.

 

Independent Practice
For 6-21, find each product by choosing an appropriate strategy.
Question 6.
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 71

Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.7 Choose a Strategy to Multiply-Do You Know How-Independent Practice-6

Explanation:
25 × 6 = (20 × 6 ) + ( 5 × 6)
25 × 6 = 120 + 30
25 × 6 = 150.

 

Question 7.
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 72

Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.7 Choose a Strategy to Multiply-Do You Know How-Independent Practice-7

Explanation:
83 × 5 = (80 × 5) + ( 3 × 5)
83 × 5 = 400 + 15
83 × 5 = 415.

 

 

Question 8.
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 73

Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.7 Choose a Strategy to Multiply-Do You Know How-Independent Practice-8

Explanation:
62 × 4 = (60 × 4) + (2 × 4)
62 × 4 = 240 + 8
62 × 4 = 248.

 

 

Question 9.
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 74

Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.7 Choose a Strategy to Multiply-Do You Know How-Independent Practice-9

Explanation:
89 × 7 = (100 × 7) – (11 × 7)
89 × 7 = 700 – 77
89 × 7 = 623.

 

 

Question 10.
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 75

Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.7 Choose a Strategy to Multiply-Do You Know How-Independent Practice-10

Explanation:
245 × 3 = (250 × 3) – (5 × 3)
245 × 3 = 750 – 15
245 × 3 = 735.

 

 

Question 11.
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 76

Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.7 Choose a Strategy to Multiply-Do You Know How-Independent Practice-11

Explanation:
318 × 9 = (300 × 9) + (18 × 9)
318 × 9 = 2700 + 162
318 × 9 = 2862.

 

 

Question 12.
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 77

Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.7 Choose a Strategy to Multiply-Do You Know How-Independent Practice-12

Explanation:
103 × 5 = (100 × 5) + (3 × 5)
103 × 5 = 500 + 15
103 × 5 = 515.

 

 

Question 13.
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 78

Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.7 Choose a Strategy to Multiply-Do You Know How-Independent Practice-13

Explanation:
314 × 8 = (300 × 8) + (14 × 8)
314 × 8 = 2400 + 112
314 × 8 = 2512.

 

 

Question 14.
4 × 4,347
Answer:
4 × 4,347 = 17388.

Explanation:
4 × 4,347 = (4 × 4000) + (4 × 300) + (4 × 40) + (4 × 7)
4 × 4,347 = 16000 + 1200 + 160  + 28
4 × 4,347 = 17200 + 160  + 28
4 × 4,347 = 17360 + 28
4 × 4,347 =17388.

 

 

Question 15.
6 × 2,716
Answer:
6 × 2,716 = 16296.

Explanation:
6 × 2,716 = (6 × 2000) + ( 6 × 700) + ( 6 × 10) + (6 × 6)
6 × 2,716 = 12000 + 4200 + 60 + 36
6 × 2,716 = 16200 + 60 + 36
6 × 2,716 = 16260 + 36
6 × 2,716 = 16296.

 

 

Question 16.
7 × 1,287
Answer:
7 × 1,287 = 9009.

Explanation:
7 × 1,287 = (7 × 1000) + (7 × 200) + (7 × 80) + (7 × 7)
7 × 1,287 = 7000 + 1400 + 560 + 49
7 × 1,287 = 8400 + 560 + 49
7 × 1,287 = 8960 + 49
7 × 1,287 = 9009.

 

Question 17.
3 × 1,942
Answer:
3 × 1,942 = 5826.

Explanation:
3 × 1,942 = (3 × 1000) + ( 3 × 900) + (3 × 40) + (3 × 2)
3 × 1,942 = 3000 + 2700 + 120 + 6
3 × 1,942 = 5700 + 120 + 6
3 × 1,942 = 5820 + 6
3 × 1,942 = 5826.

 

 

Question 18.
2,319 × 5
Answer:
2,319 × 5 = 11595.

Explanation:
2,319 × 5 = (2000 × 5) + (300 × 5) + (10 × 5) + (9 × 5)
2,319 × 5 = 10000 + 1500 + 50 + 45
2,319 × 5 =  11500 + 50 + 45
2,319 × 5 = 11550 + 45
2,319 × 5 = 11595.

 

 

Question 19.
1,467 × 5
Answer:
1,467 × 5 = 7335.

Explanation:
1,467 × 5 = (1000 × 5) + (400 × 5) + (60 × 5) + (7 × 5)
1,467 × 5 = 5000 + 2000 + 300 + 35
1,467 × 5 = 7000 + 300 + 35
1,467 × 5 = 7300 + 35
1,467 × 5 = 7335.

 

 

Question 20.
6,006 × 9
Answer:
6,006 × 9 = 54054.

Explanation:
6,006 × 9 = (6000 × 9) + (6 × 9)
6,006 × 9 = 54000 + 54
6,006 × 9 = 54054.

 

 

Question 21.
9,749 × 5
Answer:
9,749 × 5 = 48745.

Explanation:
9,749 × 5 = (9000 × 5) + (700 × 5) + (40 × 5) + (9 × 5)
9,749 × 5 = 45000 + 3500 + 200 + 45
9,749 × 5 = 48500 + 200 + 45
9,749 × 5 = 48700 + 45
9,749 × 5 = 48745.

 

 

Problem Solving
Question 22.
Make Sense and Persevere Maura’s dance team wants to buy costumes that cost $56 each. They have $523 saved in a fund. How much money will they have left in the fund after they buy 9 costumes?
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 79

Answer:
Amount of money left in the fund after they buy 9 costumes = $19.

Explanation:
Cost of each costumes Maura’s dance team wants to buy = $56.
Total amount saved in fund by Maura’s dance team = $523.
Cost of 9 costumes Maura’s dance team wants to buy = 9 × Cost of each costumes Maura’s dance team wants to buy
= 9 × $56
= (9 × 50) + (9 × 6)
= 450 + 54
= $504.
Amount of money left in the fund after they buy 9 costumes = Total amount saved in fund by Maura’s dance team – Cost of 9 costumes Maura’s dance team wants to buy
= $523 – $504
= $19.

 

Question 23.
Elaine rents a car for 5 days. It costs $44 each day to rent the car and $7 each day for insurance. At the end of the trip she spends $35 to fill the car with gas. What is the total cost for Elaine to use the car?
Answer:
Total cost for Elaine to use the car = $86.

Explanation:
Number of days Elaine rents a car = 5.
Cost of renting car each day = $44.
Cost of each day for insurance = $7.
Cost for filling car with gas = $35.
Total cost for Elaine to use the car = Cost of renting car each day + Cost of each day for insurance + Cost for filling car with gas
= $44 + $7 + $35
= $51 + $35
= $86.

 

 

Question 24.
At the Math Club ceremony, there were 17 tables with 8 people seated at each table. Each guest received 2 certificates. How many certificates were distributed at the ceremony?
Answer:
Total number of certificates were distributed at the ceremony = 272.

Explanation:
Number of tables at the Math Club ceremony = 17.
Number of people seated at each table = 8.
Number of certificates each guest received = 2.
Total number of people at the Math Club ceremony = Number of tables at the Math Club ceremony × Number of people seated at each table
= 17 × 8
= 136.
Total number of certificates were distributed at the ceremony = Total number of people at the Math Club ceremony × Number of certificates each guest received
= 136 × 2
= (100 × 2) + (36 × 2)
= 200 + 72
= 272.

 

 

Question 25.
Water erupting from geysers can reach a temperature of 244°F. The average temperature in Yellowstone National Park is 35°F. Calculate the difference between these two temperatures.
Answer:
Difference between these two temperatures = 87°F.

Explanation:
Temperature of Water erupting from geysers = 244°F
Average temperature in Yellowstone National Park = 35°F
Average temperature of Water erupting from geysers = 244°F ÷ 2
= 122°F
Difference between these two temperatures = Average temperature of Water erupting from geysers  – Average temperature in Yellowstone National Park
= 122°F – 35°F
= 87°F.

 

Question 26.
Higher Order Thinking On Monday, Paolo sold 21 tickets to the dance. On Tuesday, he sold three times as many tickets as he sold on Monday. On Wednesday, he sold twice as many tickets as he sold on Tuesday. How many total tickets did Paolo sell in the three days?
Answer:
Total number of tickets Paolo sells in the three days = 210.

Explanation:
Number of tickets to the dance Paolo sold on Monday = 21.
On Tuesday, he sold three times as many tickets as he sold on Monday.
=> Number of tickets to the dance Paolo sold on Tuesday = 3 × Number of tickets to the dance Paolo sold on Monday
= 3 × 21
= 63.
On Wednesday, he sold twice as many tickets as he sold on Tuesday.
=> Number of tickets to the dance Paolo sold on Wednesday = 3 × Number of tickets to the dance Paolo sold on Tuesday
=> 2 × 63
=> 126.
Total number of tickets Paolo sells in the three days = Number of tickets to the dance Paolo sold on Monday + Number of tickets to the dance Paolo sold on Tuesday + Number of tickets to the dance Paolo sold on Wednesday
= 21 + 63 + 126
= 84 + 126
= 210.

 

 

Assessment Practice
Question 27.
Mr. Tran would like to buy a new sofa that costs $934. He can pay the total all at once, or he can make a $125 payment each month for 8 months. How much less does it cost to pay all at once? Write equations to show how you solve. Tell what your variables represent.
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 80
Answer:
$66 less it costs to pay all at once for Mr. Tran.

Explanation:
Cost of new sofa Mr. Tran wants to buy = $934.
Cost of each month payment = $125.
Number of months for each month payment = 8.
Total amount for each month payment = Cost of each month payment × Number of months for each month payment
= $125 × 8
= $1000.
Difference:
Total amount for each month payment  – Cost of new sofa Mr. Tran wants to buy
= $1000 – $934
= $66.

 

 

Lesson 3.8 Problem Solving

Model with Math
Solve & Share
Kevin took 120 color photos and 128 black and white photos on a field trip. Marco took 2 times as many photos as Kevin. How many photos did Marco take? Solve this problem using any strategy you choose. Use the bar diagram to help.
I can … apply the math I know to solve problems.
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 81

Answer:
Number of photos Marco took = 496.

Explanation:
Number of color photos Kevin took on a field trip = 120.
Number of white photos  Kevin took on a field trip = 128.
Marco took 2 times as many photos as Kevin.
=> Number of photos Marco took = 2 (Number of color photos Kevin took on a field trip + Number of white photos  Kevin took on a field trip)
=> 2  (120 + 128)
=> 2 × 248
=> 496.
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Model with Math-Solve & Share

 

 

 

Thinking Habits
Be a good thinker! These questions can help you.

  • How can I use math I know to solve this problem?
  • How can I use pictures, objects, or an equation to represent the problem?
  • How can I use numbers, words, and symbols to solve the problem?

Look Back! Model with Math What representation did you use to solve the problem and show relationships?
Answer:
Number of photos Marco took = 2 (Number of color photos Kevin took on a field trip + Number of white photos  Kevin took on a field trip)
=> 2  (120 + 128)
=> 2 × 248
=> 496.

Explanation:
Number of color photos Kevin took on a field trip = 120.
Number of white photos  Kevin took on a field trip = 128.
Marco took 2 times as many photos as Kevin.
=> Number of photos Marco took = 2 (Number of color photos Kevin took on a field trip + Number of white photos  Kevin took on a field trip)
=> 2  (120 + 128)
=> 2 × 248
=> 496.

 

Essential Question
How Can You Represent a Situation with a Math Model?
Answer:
We can represent a Situation with a Math Model using the given data, knowing what is asked and then solving the problem to find the solution asked for.

Explanation:
A mathematical model is a description of a system using mathematical  concepts and language . The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences and engineering  disciplines, as well as in non-physical systems such as the social sciences.

 

 

Visual Learning Bridge
An art show uses 9 teams of art judges. If each team judges the work of 13 painters and 14 sculptors, how many artists attend the show?
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 82
What do you need to find?
I need to find how many artists each team judges.
I need to find the total number of artists.

How can I model with math?
I can
use bar diagrams and equations to represent and solve this problem.
decide if my results make sense.

Here’s my thinking.
Find 9 × 27.
Use a bar diagram. Write and solve an equation.
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 83
a = 9 × 27
a = 243
There are 243 artists at the show.

Convince Me! Model with Math How could you decide if your answer makes sense?
Answer:
Number of artists attended the show = 243.

Explanation:
An art show uses 9 teams of art judges. If each team judges the work of 13 painters and 14 sculptors, how many artists attend the show?
Number of teams of art judges an art show uses = 9.
Number of painters each team judges the work = 13.
Number of sculptors each team judges the work = 14.
Number of artists attended the show = Number of teams of art judges an art show uses (Number of painters each team judges the work + Number of sculptors each team judges the work)
= 9 (13 + 14)
= 9 × 27
= (9 × 20) + (9 × 7)
= 180 + 63
= 243.

 

 

Guided Practice
Model with Math
When you model with Sharon’s Stationery Store contains 1,219 boxes of cards. May’s Market contains 3 times as many boxes of cards. How many boxes, b, does May’s Market contain?
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 84
Question 1.
Explain how to use a picture to represent the problem and show relationships.
Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Model with Math-Guided Practice-1

Explanation:
Number of boxes of cards Sharon’s Stationery Store contains = 1,219.
May’s Market contains 3 times as many boxes of cards.
Number of boxes May’s Market contains = ???

 

Question 2.
What equation can you write to represent the problem?
Answer:
Number of boxes May’s Market contains = 3 × Number of boxes of cards Sharon’s Stationery Store contains
=> 3 × 1219 = ???

Explanation:
Number of boxes of cards Sharon’s Stationery Store contains = 1,219.
May’s Market contains 3 times as many boxes of cards.
=> Number of boxes May’s Market contains = 3 × Number of boxes of cards Sharon’s Stationery Store contains
=> 3 × 1219

 

Question 3.
What is the solution to the problem?
Answer:
Number of boxes May’s Market contains = 3 × 1219 = 3657.

Explanation:
Number of boxes of cards Sharon’s Stationery Store contains = 1,219.
May’s Market contains 3 times as many boxes of cards.
=> Number of boxes May’s Market contains = 3 × Number of boxes of cards Sharon’s Stationery Store contains
=> 3 × 1219
=> (3 × 1000) + (3 × 200) + (3 × 10) + (3 × 9)
=> 3000 + 600 + 30 + 27
=> 3600 + 30 + 27
=> 3630 + 27
=> 3657.

 

 

Independent Practice
Model with Math Annie has 6 albums of stamps in her stamp collection. Each album contains 440 stamps. How many stamps, does Annie have in her collection? Use Exercises 4-6 to answer the question.
Question 4.
Draw a picture and write an equation to represent the problem.
Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Model with Math-Guided Practice-Independent Practice-4

Explanation:
Number of albums of stamps in her stamp collection Annie has = 6.
Number of stamps each album contains = 440.
Total number of stamps Annie have in her collection = ???

 

 

Question 5.
What previously learned math can you use to solve the problem?
Answer:
Number of albums of stamps in her stamp collection Annie has = 6.
Number of stamps each album contains = 440.
Total number of stamps Annie have in her collection = Number of albums of stamps in her stamp collection Annie has × Number of stamps each album contains
= 6 × 440

Explanation:
Number of albums of stamps in her stamp collection Annie has = 6.
Number of stamps each album contains = 440.
Total number of stamps Annie have in her collection = Number of albums of stamps in her stamp collection Annie has × Number of stamps each album contains
= 6 × 440

 

 

Question 6.
What is the solution to the problem? Explain why your solution makes sense.
Answer:
Total number of stamps Annie have in her collection = 2640.

Explanation:
Number of albums of stamps in her stamp collection Annie has = 6.
Number of stamps each album contains = 440.
Total number of stamps Annie have in her collection = Number of albums of stamps in her stamp collection Annie has × Number of stamps each album contains
= 6 × 440
= (6 × 400) + (6 × 40)
= 2400 + 240
= 2640.

 

 

Problem Solving
Performance Task
Hauling Fuel A truck like the one shown delivers a load of gasoline to a gas station 3 times a week. The storage tank at the gas station holds 9 loads of fuel. How much more gas does the storage tank hold than the truck?
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 85
Question 7.
Make Sense and Persevere What do you know and what do you need to determine?
Answer:
Number of times a truck delivers a load of gasoline to a gas station a week = 3.
Quantity of loads of fuel a storage tank at the gas station holds = 9.
Quantity of loads of fuel a storage tank holds = 2700 gallons.
Quantity of gas the storage tank holds = ???
Quantity of gas the truck holds = ???

Explanation:
Known information:
Number of times a truck delivers a load of gasoline to a gas station a week = 3.
Quantity of loads of fuel a storage tank at the gas station holds = 9.
Quantity of loads of fuel a storage tank holds = 2700 gallons.
To determine:
Quantity of gas the storage tank holds = ???
Quantity of gas the truck holds = ???

 

Question 8.
Make Sense and Persevere What do you need to know to determine how much more the tank holds than the truck?
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 86

Answer:
16200 gallons more gas the storage tank hold than the truck.

Explanation:
Number of times a truck delivers a load of gasoline to a gas station a week = 3.
Quantity of loads of fuel a storage tank at the gas station holds = 9.
Quantity of loads of fuel a storage tank holds = 2700 gallons.
Quantity of gas the storage tank holds = ???
Quantity of gas the truck holds = ???
Quantity of gas the storage tank holds = Quantity of loads of fuel a storage tank at the gas station holds  × Quantity of loads of fuel a storage tank holds
= 9 × 2700
= 9 ( 2000 + 700)
= (9 × 2000) + (9 × 700)
= 18000 + 6300
= 24300 gallons.
Quantity of gas the truck holds = Number of times a truck delivers a load of gasoline to a gas station a week  × Quantity of loads of fuel a storage tank holds
= 3 × 2700
= 3 ( 2000 + 700)
= (3 × 2000) + (3 × 700)
= 6000 + 2100
= 8100 gallons.
Difference:
= Quantity of gas the storage tank holds – Quantity of gas the truck holds
= 24300 – 8100
= 16200 gallons.

Question 9.
Model with Math Explain how to use bar diagrams and equations to represent the problem and solve.
Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Problem Solving-9-

Explanation:
Quantity of gas the storage tank holds = Quantity of loads of fuel a storage tank at the gas station holds  × Quantity of loads of fuel a storage tank holds
= 9 × 2700
= 9 ( 2000 + 700)
= (9 × 2000) + (9 × 700)
= 18000 + 6300
= 24300 gallons.
Quantity of gas the truck holds = Number of times a truck delivers a load of gasoline to a gas station a week  × Quantity of loads of fuel a storage tank holds
= 3 × 2700
= 3 ( 2000 + 700)
= (3 × 2000) + (3 × 700)
= 6000 + 2100
= 8100 gallons.
Difference:
= Quantity of gas the storage tank holds – Quantity of gas the truck holds
= 24300 – 8100
= 16200 gallons.
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Problem Solving-9

 

 

Topic 3 Fluency Practice Activity

Follow the Path
Shade a path from START to FINISH. Follow the differences that are correct. You can only move up, down, right, or left.
I can … subtract multi-digit whole numbers.
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 87
Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Topic 3 Fluency Practice Activity-Follow the Path

Explanation:
812 – 44 = 768. (Correct)
621 – 85 = 536. (Correct)
543 – 97 = 446. (Correct)
111 – 87 = 76. (Incorrect)
684 – 485 = 299. (Incorrect)
929 – 879 = 150. (Incorrect)
341 – 299 =  142. (Incorrect)
836 – 788 = 48. (Correct)
876 – 55 = 72. (Incorrect)
922 – 87 = 865. (Incorrect)
511 – 423 = 112. (Incorrect)
486 – 230 =256. (Correct)
178 – 98 =80. (Correct)
912 – 842 = 170. (Incorrect)
312 – 219 = 193. (Incorrect)
767 – 31 = 636. (Incorrect)
825 – 789 = 36. (Correct)
123 – 53 = 30. (Incorrect)
282 – 32 = 150. (Incorrect)
986 – 887 = 199. (Incorrect)
698 – 12 = 586. (Incorrect)
333 – 111 = 222. (Correct)
342 – 88 = 254. (Correct)
293 – 95 = 198. (Correct)
876 – 543 = 333. (Correct)

 

Topic 3 Vocabulary Review

Understand Vocabulary
Word List

  •  array
  • Associative Property of Multiplication
  • Commutative Property of Multiplication
  • compensation
  • Distributive Property
  • estimate
  • numerical expression
  • partial product

Question 1.
Circle the property shown by 4 × (6 + 2) = (4 × 6) + (4 × 2).
Associative
Commutative
Distributive
Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Topic 3 Vocabulary Review-1

Explanation:
The property shown by 4 × (6 + 2) = (4 × 6) + (4 × 2):
Associative

 

Question 2.
Circle the property shown by 2 × 134 = 134 × 2.
Associative
Commutative
Distributive
Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Topic 3 Vocabulary Review-2

Explanation:
The property shown by 2 × 134 = 134 × 2:
Commutative

 

 

Question 3.
Circle the property shown by (1 × 3) × 7 = 1 × (3 × 7).
Associative
Commutative
Distributive
Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Topic 3 Vocabulary Review-3

Explanation:
The property shown by (1 × 3) × 7 = 1 × (3 × 7):
Distributive

 

Question 4.
Draw a line from each vocabulary word to its example.
Envision Math Common Core Grade 4 Answer Key Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 88
Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Topic 3 Vocabulary Review-4

Explanation:
Each vocabulary word is matched to the examples:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Topic 3 Vocabulary Review-4

 

 

Use Vocabulary in Writing
Question 5.
Find 4 × 114. Use at least three terms from the Word List to describe how to find the product.
Answer:
4 × 114 = 456.

Explanation:
4 × 114 = ??? { Estimate}
4 × 114 = (4 × 100) + (4 × 10) + (4 × 4) { numerical expression}
4 × 114 = 400 + 40+ 16
4 × 114 = 440 + 16 {partial products}
4 × 114 = 456.

 

 

Topic 3 Reteaching

Set A pages 81-84

Use basic facts, place value, and multiplication properties to multiply by multiples of 10 and 100.
Find 4 × 60.
4 × 60 = 4 × 6 tens
4 × 60 = (4 × 6) tens
4 × 60 = 24 × 10
4 × 60 = 240

Find 4 × 600.
4 × 600 = 4 × 6 hundreds
4 × 600 = (4 × 6) hundreds
4 × 600 = 24 × 100
4 × 600 = 2,400

Find 4 × 6,000.
4 × 6,000 = 4 × 6 thousands
4 × 6,000 = (4 × 6) thousands
4 × 6,000 = 24 × 1,000
4 × 6,000 = 24,000

Remember when the product of a basic fact ends in zero, the answer will have an extra zero.

Question 1.
8 × 60
Answer:
8 × 60 = 480.

Explanation:
8 × 60 = (8 × 6) tens
8 × 60 = 48 tens
8 × 60 = 48 × 10
8 × 60 = 480.

 

 

Question 2.
3 × 10
Answer:
3 × 10 = 30.

Explanation:
3 × 10 = (3 × 1) tens
3 × 10 = 3 tens
3 × 10 = 3 × 10
3 × 10 = 30.

 

Question 3.
6 × 50
Answer:
6 × 50 = 300.

Explanation:
6 × 50 = (6 × 5) tens
6 × 50 = 30 tens
6 × 50 = 30 × 10
6 × 50 = 300.

 

 

Question 4.
5 × 300
Answer:
5 × 300 = 1500.

Explanation:
5 × 300 = (5 × 3) hundreds
5 × 300 = 15 × hundreds
5 × 300 = 15 × 100
5 × 300 = 1500.

 

Question 5.
7,000 × 4
Answer:
7,000 × 4 = 28000.

Explanation:
7,000 × 4 = thousands (7 × 4)
7,000 × 4 = thousands  × 28
7,000 × 4 = 1000 × 28
7,000 × 4 = 28000.

 

 

Question 6.
2 × 900
Answer:
2 × 900 = 1800.

Explanation:
2 × 900 = (2 × 9) hundreds
2 × 900 = 18 × hundreds
2 × 900 = 18 × 100
2 × 900 = 1800.

 

Question 7.
80 × 8
Answer:
80 × 8 = 640.

Explanation:
80 × 8 = tens (8 × 8)
80 × 8 = tens × 64
80 × 8 = 10 × 64
80 × 8 = 640.

 

 

Question 8.
400 × 5
Answer:
400 × 5 = 2000.

Explanation:
400 × 5 = hundreds (4 × 5)
400 × 5 = hundreds × 20
400 × 5 = 100 × 20
400 × 5 = 2000.

 

 

Question 9.
30 × 9
Answer:
30 × 9 = 270.

Explanation:
30 × 9 = tens ( 3 × 9)
30 × 9 = tens × 27
30 × 9 = 10 × 27
30 × 9 = 270.

 

 

Question 10.
5 × 8,000
Answer:
5 × 8,000 = 40000.

Explanation:
5 × 8,000 = (5 × 8) thousands
5 × 8,000 = 40 × thousands
5 × 8,000 = 40 × 1000
5 × 8,000 = 40000.

 

 

Question 11.
700 × 8
Answer:
700 × 8 = 5600.

Explanation:
700 × 8 = hundreds (7 × 8)
700 × 8 = hundreds × 56
700 × 8 = 100 × 56
700 × 8 = 5600.

 

 

Question 12.
9,000 × 6
Answer:
9,000 × 6 = 54000.

Explanation:
9,000 × 6 = thousands (9 × 6)
9,000 × 6 = thousands × 54
9,000 × 6 = 1000 × 54
9,000 × 6 = 54000.

 

 

Question 13.
7 × 9,000
Answer:
7 × 9,000 = 63000.

Explanation:
7 × 9,000 = (7 × 9) thousands
7 × 9,000 = 63 × thousands
7 × 9,000 = 63 × 1000
7 × 9,000 = 63000.

 

Question 14.
5 × 100
Answer:
5 × 100 = 500.

Explanation:
5 × 100 = (5 × 1) hundreds
5 × 100 = 5 × hundreds
5 × 100 = 5 × 100
5 × 100 = 500.

 

 

Question 15.
20 × 5
Answer:
20 × 5 = 100.

Explanation:
20 × 5 = tens (2 × 5)
20 × 5 = tens × 10
20 × 5 = 10 × 10
20 × 5 = 100.

 

Question 16.
5 × 4,000
Answer:
5 × 4,000 = 20000.

Explanation:
5 × 4,000 = (5 × 4) thousands
5 × 4,000 = 20 × thousands
5 × 4,000 = 20 × 1000
5 × 4,000 = 20000.

 

 

Question 17.
5 × 500
Answer:
5 × 500 = 2500.

Explanation:
5 × 500 = (5 × 5) hundreds
5 × 500 = 25 × hundreds
5 × 500 = 25 × 100
5 × 500 = 2500.

 

Question 18.
3 × 2,000
Answer:
3 × 2,000 = 6000.

Explanation:
3 × 2,000 = (3 × 2) thousands
3 × 2,000 = 6 × thousands
3 × 2,000 = 6 × 1000
3 × 2,000 = 6000.

 

 

Set B pages 85-88
Use rounding to estimate 9 × 1,993.
Round 1,993 to 2,000.
Envision Math Common Core Grade 4 Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 89
So, 9 × 1,993 is about 18,000.

Remember to round a three-digit number to the nearest hundred and a four-digit number to the nearest thousand.

Estimate each product.
Question 1.
8 × 7,632
Answer:
8 × 7,632 is about 64000.

Explanation:
8 × 7,632 = ?????
Rounding 7632 to nearest thousand.
=> 8 × 7,632 = 8 × 8000
=> 8 × 7,632 =  64000.

 

Question 2.
493 × 3
Answer:
493 × 3 is about 1500.

Explanation:
493 × 3 = ????
Rounding 493 to nearest hundred.
=> 493 × 3 = 500 × 3
=> 493 × 3 = 1500.

 

 

Question 3.
9,379 × 5
Answer:
9,379 × 5 is about 45000.

Explanation:
9,379 × 5 = ?????
Rounding 9379 to nearest thousand.
=> 9,379 × 5 = 9000 × 5
=> 9,379 × 5 = 45000.

 

 

Question 4.
678 × 6
Answer:
678 × 6 is about 4200.

Explanation:
678 × 6 = ????
Rounding 678 to nearest hundred.
=> 678 × 6 = 700 × 6
=> 678 × 6 = 4200.

 

 

Question 5.
707 × 4
Answer:
707 × 4 is about 2800.

Explanation:
707 × 4 = ????
Rounding 707 to nearest hundred.
=> 707 × 4 = 700 × 4
=> 707 × 4 = 2800.

 

 

Question 6.
5,703 × 3
Answer:
5,703 × 3 is about 18000.

Explanation:
5,703 × 3 = ?????
Rounding 5703 to nearest thousand.
=> 5,703 × 3 = 6000 × 3
=> 5,703 × 3 = 18000.

 

Question 7.
483 × 6
Answer:
483 × 6 is about 3000.

Explanation:
483 × 6 = ????
Rounding 483 to nearest hundred.
=> 483 × 6 = 500 × 6
=> 483 × 6 = 3000.

 

 

Question 8.
6 × 8,166
Answer:
6 × 8,166 is about 48000.

Explanation:
6 × 8,166 = ?????
Rounding 9379 to nearest thousand.
=> 6 × 8,166 = 6 × 8000
=> 6 × 8,166 = 48000.

 

 

Set C pages 89-92
Use an array and partial products to find 3 × 121.
Envision Math Common Core Grade 4 Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 90

Remember to line up the partial products carefully.

Question 1.
Envision Math Common Core Grade 4 Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 91
Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Set C pages 89-92-1

Explanation:
75 × 5 = (70 × 5) + ( 5 × 5)
75 × 5 = 350 + 25
75 × 5 = 375.

 

 

Question 2.
Envision Math Common Core Grade 4 Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 92
Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Set C pages 89-92-2

Explanation:
253 × 4 = (200 × 4) + ( 50 × 4) + (3 × 4)
253 × 4 = 800 + 200 + 12
253 × 4 = 1000 + 12
253 × 4 = 1012.

 

 

Question 3.
Envision Math Common Core Grade 4 Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 93
Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Set C pages 89-92-3

Explanation:
214 × 7 = (200 × 7) + (10 × 7) + (4 × 7)
214 × 7 = 1400 + 70 + 28
214 × 7 = 1470 + 28
214 × 7 = 1498.

 

 

Question 4.
Envision Math Common Core Grade 4 Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 94
Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Set C pages 89-92-4

Explanation:
1341 × 7 = (1000 × 7) + (300 × 7) + (40 × 7) + (1 × 7)
1341 × 7 = 7000 + 2100 + 280 + 7
1341 × 7 = 9100 + 280 + 7
1341 × 7 = 9380 + 7
1341 × 7 = 9387.

 

 

 

Set D pages 93-96
Use the Distributive Property to find 5 × 45.
Use place value to break apart 45 as 40 + 5.
Envision Math Common Core Grade 4 Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 95
Use the Distributive Property:
5 × 45 = 5 × (40 + 5)
= (5 × 40) + (5 × 5)
= 200 + 25
= 225
So, 5 × 45 = 225.

Remember that you can use the Distributive Property to help multiply larger numbers.

Question 1.
7 × 45
Answer:
7 × 45 = 315.

Explanation:
7 × 45 = 7 × (40 + 5)
= (7 × 40) + (7 × 5)
= 280+ 35
= 315.

 

 

Question 2.
326 × 9
Answer:
326 × 9 = 2934.

Explanation:
326 × 9 = (300 + 26)  9
= (300 × 9) + (26 × 9)
= 2700 + 234
= 2934.

 

 

Question 3.
720 × 6
Answer:
720 × 6 = 4320.

Explanation:
720 × 6 = (700 + 20) 6
= (700 × 6) + (20 × 6)
= 4200 + 120
= 4320.

 

 

Question 4.
3 × 46
Answer:
3 × 46 = 138.

Explanation:
3 × 46 = 3 ( 40 + 6)
= (3 × 40) + (3 × 6)
= 120 + 18
= 138.

 

 

Question 5.
371 × 8
Answer:
371 × 8 = 2968.

Explanation:
371 × 8 = (300 + 71) 8
= (300 × 8) + (71 × 8)
= 2400 + 568
= 2968.

 

 

Question 6.
5 × 95
Answer:
5 × 95 = 475.

Explanation:
5 × 95 = 5 ( 90 + 5)
= (5 × 90) + (5 × 5)
= 450 + 25
= 475.

 

 

Question 7.
88 × 3
Answer:
88 × 3 = 264.

Explanation:
88 × 3 = (80 + 8) 3
= (80 × 3) + ( 8 × 3)
= 240 + 24
= 264.

 

 

Question 8.
4 × 1,865
Answer:
4 × 1,865 = 7460.

Explanation:
4 × 1,865 = 4 ( 1000 + 800 + 60 + 5)
= (4 × 1000) + ( 4 × 800) + (4 × 60) + (4 × 5)
= 4000 + 3200 + 240 + 20
= 7200 + 240 + 20
= 7440 + 20
= 7460.

 

 

Question 9.
57 × 3
Answer:
57 × 3 = 171.

Explanation:
57 × 3 = (50 + 7) 3
= (50 × 3) + ( 7 × 3)
= 150 + 21
= 171.

 

Question 10.
209 × 7
Answer:
209 × 7 = 1463.

Explanation:
209 × 7 = (200 + 9) 7
= (200 × 7) + (9 × 7)
= 1400 + 63
= 1463.

 

 

Set E pages 97-100
All your strategies may be used to find larger products.
Use the Distributive Property to find 3 × 1,275.
Use expanded notation and place value to construct an area model. 1,000
Envision Math Common Core Grade 4 Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 96

Remember to estimate to check that your answer is reasonable.

Question 1.
1,468 × 4
Answer:
1,468 × 4 = 5872.

Explanation:
1,468 × 4 = (1000 + 400 + 60 + 8) 4
1,468 × 4 = (1000 × 4) + (400 × 4) + ( 60 × 4) + (8 × 4)
1,468 × 4 = 4000 + 1600 + 240 + 32
1,468 × 4 = 5600 + 240 + 32
1,468 × 4 = 5840 + 32
1,468 × 4 = 5872.
Check:
??? × 4 = 5872
= ??? = 5872 ÷ 4
= ??? = 1468. (Answer is reasonable)

 

 

Question 2.
361 × 3
Answer:
361 × 3 = 1083.

Explanation:
361 × 3 = (300 + 60 + 1) 3
361 × 3 = (300 × 3) + (60 × 3) + (1 × 3)
361 × 3 = 900 + 180 + 3
361 × 3 = 1080 + 3
361 × 3 = 1083.
Check:
??? × 3 = 1083.
??? = 1083 ÷ 3
??? = 361. (Answer is reasonable)

 

 

 

Question 3.
25 × 7
Answer:
25 × 7 = 175.

Explanation:
25 × 7 = (20 + 5) 7
25 × 7 = (20 × 7) + (5 × 7)
25 × 7 = 140 + 35
25 × 7 = 175.
Check:
?? × 7 = 175.
=> ??= 175 ÷ 7
=> ?? = 25. (Answer is reasonable)

 

 

Question 4.
2,189 × 7
Answer:
2,189 × 7 = 15323.

Explanation:
2,189 × 7 = (2000 + 100 + 80 + 9) 7
2,189 × 7 = (2000 × 7) + ( 100× 7) + (80 × 7) + ( 9 × 7)
2,189 × 7 = 14000 + 700 + 560 + 63
2,189 × 7 = 14700 + 560 + 63
2,189 × 7 =  15260 + 63
2,189 × 7 = 15323.
Check:
???? × 7 = 15323.
???? = 15323 ÷ 7
???? = 2189. (Answer is reasonable)

 

 

Question 5.
6 × 987
Answer:
6 × 987 = 5922.

Explanation:
6 × 987 = 6 ( 900 + 80 + 7)
6 × 987 = (6 × 900) + (6 × 80) + (6 × 7)
6 × 987 = 5400 + 480 + 42
6 × 987 = 5880 + 42
6 × 987 = 5922.
Check:
6 × ??? = 5922.
??? = 5922 ÷ 6
??? = 987.(Answer is reasonable)

 

 

Question 6.
8 × 22
Answer:
8 × 22 = 176.

Explanation:
8 × 22 =  8 (20 + 2)
8 × 22 = (8 × 20) + (8 × 2)
8 × 22 = 160 + 16
8 × 22 = 176. (Answer is reasonable)

 

 

Question 7.
763 × 5
Answer:
763 × 5 = 3815.

Explanation:
763 × 5 = (700 + 60 + 3) 5
763 × 5 = (700 × 5) + (60 × 5) + (3 × 5)
763 × 5 = 3500 + 300 + 15
763 × 5 = 3800 + 15
763 × 5 = 3815. (Answer is reasonable)

 

 

Question 8.
14 × 9
Answer:
14 × 9 = 126.

Explanation:
14 × 9 = (10 + 4) 9
14 × 9 = (10 × 9) + (4 × 9)
14 × 9 = 90 + 36
14 × 9 = 126. (Answer is reasonable)

 

 

Question 9.
171 × 8
Answer:
171 × 8 = 1368.

Explanation:
171 × 8 = (100 + 70 + 1) 8
171 × 8 = (100 × 8) + (70 × 8) + (1 × 8)
171 × 8 = 800 + 560 + 8
171 × 8 = 1360 + 8
171 × 8 = 1368. (Answer is reasonable)

 

 

Question 10.
22 × 9
Answer:
22 × 9 = 198.

Explanation:
22 × 9 = (20 + 2) 9
22 × 9 = (20 × 9) + (2 × 9)
22 × 9 = 180 + 18
22 × 9 = 198. (Answer is reasonable)

 

 

Question 11.
1,409 × 5
Answer:
1,409 × 5 = 7045.

Explanation:
1,409 × 5 = (1000 + 400 + 9) 5
1,409 × 5 = (1000 × 5) + (400 × 5) + (9 × 5)
1,409 × 5 = 5000 + 2000 + 45
1,409 × 5 = 7000 + 45
1,409 × 5 =  7045. (Answer is reasonable)

 

 

Question 12.
17 × 6
Answer:
17 × 6 = 102.

Explanation:
17 × 6 = (10 + 7) 6
17 × 6 = (10 × 6) + (7 × 6)
17 × 6 = 60 + 42
17 × 6 = 102. (Answer is reasonable)

 

 

 

 

Set F pages 101-104
You can use properties of operations or compensation to multiply mentally.
Use the Commutative and Associative Properties to find 8 × 50.
8 × 50 = (2 × 4) 50
= (4 × 2) × 50
= 4 × (2 × 50)
= 4 × 100
= 400
Use compensation to find 9 × 67.
Think: 67 = 70 – 3
Multiply mentally 9 × 70.
9 × 70 = 630
Adjust by subtracting
9 × 3, or 27.
Subtract: 630 – 27 = 603
So, 9 × 67 = 603.

Remember that you can use diagrams or properties of operations to help multiply mentally

Question 1.
18 × 2
Answer:
18 × 2 = 36.

Explanation:
18 × 2 = (10 + 8) 2
18 × 2 = (10 × 2) + (8 × 2)
18 × 2 = 20 + 16
18 × 2 = 36.

 

 

Question 2.
48 × 5
Answer:
48 × 5 = 240.

Explanation:
48 × 5 = (50 – 2) 5
48 × 5 = (50 × 5) – (2 × 5)
48 × 5 = 250 – 10
48 × 5 = 240.

 

 

Question 3.
52 × 7
Answer:
52 × 7 = 364.

Explanation:
52 × 7 = (50 + 2) 7
52 × 7 = (50 × 7) + (2 × 7)
52 × 7 = 350 + 14
52 × 7 = 364.

 

 

Question 4.
33 × 6
Answer:
33 × 6 = 198.

Explanation:
33 × 6 = (30 + 3) 6
33 × 6 = (30 × 6) + (3 × 6)
33 × 6 = 180 + 18
33 × 6 = 198.

 

 

Question 5.
97 × 7
Answer:
97 × 7 = 679.

Explanation:
97 × 7 = (100 – 3) 7
97 × 7 = (100 × 7) – (3 × 7)
97 × 7 = 700 – 21
97 × 7 = 679.

 

 

 

Question 6.
88 × 4
Answer:
88 × 4 = 352.

Explanation:
88 × 4 = (90 – 2) 4
88 × 4 = (90 × 4) – (2 × 4)
88 × 4 = 360 -8
88 × 4 = 352.

 

 

Question 7.
239 × 4
Answer:
239 × 4 = 956.

Explanation:
239 × 4 = (200 + 30 + 9) 4
239 × 4 = (200 × 4) + (30 × 4) + (9 × 4)
239 × 4 = 800 + 120 + 36
239 × 4 = 920 + 36
239 × 4 = 956.

 

 

Question 8.
148 × 5
Answer:
148 × 5 = 740.

Explanation:
148 × 5 = (100 + 40 + 8) 5
148 × 5 = (100 × 5) + (40 × 5) + (8 × 5)
148 × 5 = 500 + 200 + 40
148 × 5 = 700 + 40
148 × 5 = 740.

 

 

Question 9.
233 × 6
Answer:
233 × 6 = 1398.

Explanation:
233 × 6 = (200 + 30 + 3) 6
233 × 6 = (200 × 6) + (30 × 6) + (3 × 6)
233 × 6 = 1200 + 180 + 18
233 × 6 = 1380 + 18
233 × 6 = 1398.

 

 

Question 10.
937 × 7
Answer:
937 × 7 = 6559.

Explanation:
937 × 7 = (900 + 30 + 7) 7
937 × 7 = (900 × 7) + (30 × 7) + (7 × 7)
937 × 7 = 6300 + 210 + 49
937 × 7 = 6510 + 49
937 × 7 = 6559.

 

 

 

Set G pages 105-108
You can use the strategy that fits the problem best. You know place-value strategies, the Distributive Property, area models, and compensation.
Place value: 8 × 359 = 8 × 3 hundreds + 8 × 5 tens + 8 × 9 ones

The Distributive Property: 7 × 4,056 = 7 × (4,000 + 50 + 6)
= (7 × 4,000) + (7 × 50) + (7 × 6)

Compensation: 6 × 495
(6 × 500) – (6 × 5)

Remember to use an estimate to check if your answer is reasonable.

Question 1.
Envision Math Common Core Grade 4 Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 97
Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Set G pages 105-108-1

Explanation:
43 × 8 = (40 + 3) 8
43 × 8 = (40 × 8) + (3 × 8)
43 × 8 = 320 + 24
43 × 8 = 344.
Check:
?? × 8 = 344.
=> ?? = 344 ÷ 8
=> ?? = 43.

 

Question 2.
Envision Math Common Core Grade 4 Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 98
Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Set G pages 105-108-2

Explanation:
57 × 9 = (60 – 3) 9
57 × 9 = (60 × 9) – (3 × 9)
57 × 9 = 540 – 27
57 × 9 = 513.
Check:
?? × 9 = 513.
=> ?? = 513 ÷ 9
=> ??  = 57.

 

 

 

Question 3.
Envision Math Common Core Grade 4 Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 99
Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Set G pages 105-108-3

Explanation:
215 × 7 = (200 + 10 + 5) 7
215 × 7 = (200 × 7) + (10 × 7) + (5 × 7)
215 × 7 = 1400 + 70 + 35
215 × 7 = 1470 + 35
215 × 7 = 1505.
Check:
?? × 7 = 1505.
=> ?? = 1505 ÷ 7
=> ??  = 215.

 

Question 4.
Envision Math Common Core Grade 4 Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 100
Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Set G pages 105-108-4

Explanation:
869 × 2 = (800 + 60 + 9)  2
869 × 2 = (800 × 2) + (60 × 2) + (9 × 2)
869 × 2 = 1600 + 120 + 18
869 × 2 = 1720 + 18
869 × 2 = 1738.
Check:
??? × 2 = 1738.
=> ??? = 1738 ÷ 2
=> ??? = 869.

 

 

Question 5.
Envision Math Common Core Grade 4 Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 101
Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Set G pages 105-108-5

Explanation:
4233 × 7 = (4000 + 200 + 30 + 3) 7
4233 × 7 = (4000 × 7) + (200 × 7) + (30 × 7) + (3 × 7)
4233 × 7 = 28000 + 1400 + 210 + 21
4233 × 7 = 29400 + 210 + 21
4233 × 7 = 29610 + 21
4233 × 7 = 29631.
Check:
??? × 7 = 29631.
=> ??? = 29631 ÷ 7
=>???  = 4233.

Question 6.
Envision Math Common Core Grade 4 Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 102
Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Set G pages 105-108-6

Explanation:
3261 × 4 = (3000 + 200 + 60 + 1) 4
3261 × 4 = (3000 × 4) + ( 200 × 4) + (60 × 4) + (1 × 4)
3261 × 4 = 12000 + 800 + 240 + 4
3261 × 4 = 12800 + 240 + 4
3261 × 4 = 13040 + 4
3261 × 4 =  13044.
Check:
??? × 4 =  13044.
=> ??? =  13044 ÷ 4
=> ??? =  3261.

 

 

 

Question 7.
Envision Math Common Core Grade 4 Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 103
Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Set G pages 105-108-7

Explanation:
1250 × 8 = (1000 + 200 + 50) 8
1250 × 8 = (1000 × 8) + (200 × 8) + (50 × 8)
1250 × 8 = 8000 + 1600 + 400
1250 × 8 = 9600 + 400
1250 × 8 = 10000.
Check:
??? × 8 = 10000.
=> ??? = 10000 ÷ 8
=> ??? = 1250.

 

 

 

Question 8.
Envision Math Common Core Grade 4 Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 104
Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Set G pages 105-108-8

Explanation:
2239 × 5 = (2000 + 200 + 30 + 9) 5
2239 × 5 = (2000 × 5) + (200 × 5) + (30 × 5) + (9 × 5)
2239 × 5 = 10000 + 1000 + 150 + 45
2239 × 5 = 11000 + 150 + 45
2239 × 5 = 11150 + 45
2239 × 5 = 11195.
Check:
??? × 5 = 11195.
=> ??? = 11195 ÷ 5
=> ???  = 2239.

 

 

 

Set H pages 109-112
Think about these questions to help you model with math.
Thinking Habits

  • How can I use math I know to help solve this problem?
  • How can I use pictures, objects, or an equation to represent the problem?
  • How can I use numbers, words, and symbols to solve the problem?
    Envision Math Common Core Grade 4 Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 105

Remember that a bar diagram can help you write an equation.

Mia has a collection of 34 dolls. A toy store’s warehouse has 5 times as many dolls.
Question 1.
Use pictures, a bar diagram, or an equation to find the number of dolls in the warehouse.
Answer:
Number of dolls in the warehouse = 5 × Number of dolls collection Mia has
= 5 × 34
= 170.

Explanation:
Number of dolls collection Mia has = 34.
A toy store’s warehouse has 5 times as many dolls.
=> Number of dolls in the warehouse = 5 × Number of dolls collection Mia has
= 5 × 34
= 170.

 

Question 2.
How can you decide if your answer is reasonable?
Answer:
To be reasonable means to be as much as is appropriate or fair.

Explanation:
In math, reasonableness can be defined as checking to verify that the result of the solution or the calculation of the problem is correct or not, be either estimating or by plugging in your result to check it.

 

Topic 3 Assessment Practice

Question 1.
Use numbers from the box to show how to multiply 126 by 3.
Envision Math Common Core Grade 4 Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 106
Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Topic 3 Assessment Practice-1

Explanation:
126 × 3 = (100 + 20 + 6) 3
126 × 3 = (100 × 3) + ( 20 × 3) + (6 × 3)
126 × 3 = 300 + 60 + 18
126 × 3 = 360 + 18
126 × 3 = 378.

 

Question 2.
Alberto rode the train 198 miles round trip 4 times last month. Explain how to use mental math to find the total distance Alberto traveled.
Answer:
Total distance Alberto traveled = 792.

Explanation:
Number of miles Alberto rode the train = 198.
Number of trips he did last month = 4.
Total distance Alberto traveled = Number of miles Alberto rode the train × Number of trips he did last month
= 198 × 4
= (100 + 90 + 8) 4
= (100 × 4) + (90 × 4) + (8 × 4)
= 400 + 360 + 32
= 760 + 32
= 792.

 

Question 3.
Mr. Ortiz sells tortillas in bags of 25 and in bags of 50. If he sells 4 bags of 50 and 6 bags of 25, how many tortillas did Mr. Ortiz sell?
A. 10 tortillas
B. 350 tortillas
C. 170 tortillas
D. 3,500 tortillas
Answer:
Total number of tortillas  Mr. Ortiz sells = 350.
B. 350 tortillas

Explanation:
Number of tortillas in bags of 25 Mr. Ortiz sells = 6.
Number of tortillas in bags of 50 Mr. Ortiz sells = 4.
Total number of tortillas  Mr. Ortiz sells = (25 × Number of tortillas in bags of 25 Mr. Ortiz sells) + 50 ×Number of tortillas in bags of 50 Mr. Ortiz sells)
= (25 × 6) + (50 × 4)
= 150 + 100
= 350.

 

 

Question 4.
A. What are the partial products when finding 1,874 × 3? Select all that apply.
☐ 5,400
☐ 2,000
☐ 3,000
☐ 210
☐ 2,400
☐ 12
Answer:
The partial products when finding 1,874 × 3 are:
☐ 5,400
☐ 3,000
☐ 210
☐ 2,400
☐ 12

Explanation:
1,874 × 3 = (1000 + 800 + 70 + 4) 3
1,874 × 3 = (1000 × 3) + (800 × 3) + (70 × 3) + (4 × 3)
1,874 × 3 = 3000 + 2400 + 210 + 12
1,874 × 3 = 5400 + 210 + 12
1,874 × 3 = 5610 + 12
1,874 × 3 = 5622.

 

B. Find the product of 1,874 and 3.
Answer:
1,874 × 3 = 5622.

Explanation:
1,874 × 3 = (1000 + 800 + 70 + 4) 3
1,874 × 3 = (1000 × 3) + (800 × 3) + (70 × 3) + (4 × 3)
1,874 × 3 = 3000 + 2400 + 210 + 12
1,874 × 3 = 5400 + 210 + 12
1,874 × 3 = 5610 + 12
1,874 × 3 = 5622.

 

Question 5.
A science class is growing some fruit and vegetable plants on a plot of land behind their school. Each section is laid out in rows.
Envision Math Common Core Grade 4 Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 107
A. There are 9 strawberry plants planted in each row. Write and solve an equation to find how many strawberry plants were planted.
Answer:
Total number of strawberry plants were planted = 585.

Explanation:
Number of strawberry plants planted in each row = 9.
Number of rows strawberry plants planted = 65.
Total number of strawberry plants were planted = Number of rows strawberry plants planted  × Number of strawberry plants planted in each row
= 65 × 9
= (60 + 5) 9
= (60 × 9) + (5 × 9)
= 540 + 45
= 585.

 

B. There are 9 pepper plants planted in each row. Draw an area model and show the partial products to find how many pepper plants are planted behind the school.
Answer:
Number of pepper plants are planted behind the school = 162.

Explanation:
Number of pepper plants planted in each row = 9.
Number of rows pepper plants planted = 18.
Number of pepper plants are planted behind the school = Number of pepper plants planted in each row × Number of rows pepper plants planted
= 9 × 18
= 9 (10 + 8)
= (9 × 10) + (9 × 8)
= 90 + 72
= 162.
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Topic 3 Assessment Practice-5B

 

 

 

Question 6.
Select all the expressions that could be used to find 1,235 × 9.
☐ 9 × (1,000 + 200 + 20 + 5)
☐ 9 × (1,000 + 200 + 30 + 5)
☐ (9 × 1,000) + (9 × 200) + (9 × 30) + (9 × 5)
☐ 9 × 1,235
☐ 1,235 + 9
Answer:
1,235 × 9 = 11115.
The expressions that are used to find 1,235 × 9:
9 × (1,000 + 200 + 30 + 5)
(9 × 1,000) + (9 × 200) + (9 × 30) + (9 × 5)
9 × 1,235

Explanation:
1,235 × 9 = (1000 + 200 + 30 + 5) 9
1,235 × 9 = (1000 × 9) + (200 × 9) + (30 × 9) + (5 × 9)
1,235 × 9 = 9000 + 1800 + 270 + 45
1,235 × 9 = 10800 + 270 + 45
1,235 × 9 = 11070 + 45
1,235 × 9 = 11115.

 

Question 7.
Draw a model to find 365 × 3.
365 × 3 = _________
Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Topic 3 Assessment Practice-7
365 × 3 = __1095____.

Explanation:
365 × 3 = (300 + 60 + 5) 3
365 × 3 = (300 × 3) + (60 × 3) + (5 × 3)
365 × 3 = 900 + 180 + 15
365 × 3 = 1080 + 15
365 × 3 = 1095.

 

 

Question 8.
Select all of the expressions that have a product of 640.
☐ (6 × 100) + (4 × 10)
☐ 9 × 54
☐ 60 × 4 × 1
☐ 160 × 4
☐ (6 × 100) × (4 × 10)
Answer:
The expressions that have a product of 640:
☐ 160 × 4

Explanation:
640 = (600 + 40)
640 = 160 × 4

 

 

Question 9.
The table shows the number of hot drinks sold in a busy coffee shop in 1 week.
Envision Math Common Core Grade 4 Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 108

A. If the same number of cups of coffee were sold for 6 weeks in a row, how many cups would be sold in all?
Answer:
Number of cups of coffee were sold in all = 5010.

Explanation:
Number of cups of coffee were sold in a week = 835.
Number of cups of coffee were sold in all = 6 × Number of cups of coffee were sold in a week
= 6 × 835
= 6 (800 + 30 + 5)
= (6 × 800) + ( 6 × 30) + ( 5 × 6)
= 4800 + 180 + 30
= 4980 + 30
= 5010.

 

B. The special drink of the month was an iced mocha. The shop sold 5 times as many iced mochas in one week as hot mochas. How many more iced mochas were sold than hot mochas in 4 weeks? Explain.
Answer:
1468 more iced mochas were sold than hot mochas in 4 weeks.

Explanation:
The shop sold 5 times as many iced mochas in one week as hot mochas.
Number of iced mochas were sold in a week = 567.
Number of hot mochas were sold in a week = 200.
Number of more iced mochas were sold than hot mochas in 4 weeks = (4 × Number of iced mochas were sold in a week) – (4 × Number of hot mochas were sold in a week )
= (4 × 567) – (4 × 200)
= 2268 – 800
= 1468.

 

 

Question 10.
Tickets for a resort cost $1,182 each for adults and $459 each for children. Find the cost for 3 adult tickets and 3 child tickets. Explain how you know your answer is reasonable.
Answer:
Cost for 3 adult tickets and 3 child tickets = $4923.
Answer is reasonable because the solution is correct for given problem.

Explanation:
Cost of Tickets for each adult = $1182.
Cost of Tickets for each children = $459.
Cost for 3 adult tickets and 3 child tickets = (3 × Cost of Tickets for each adult ) +  (3 × Cost of Tickets for each children)
= (3 × $1182) + (3 × $459)
= $3546 + $1377
= $4923.

 

 

Question 11.
Which expression has the same value as 5 × 617?
A. (5 × 6) + (5 ×1) + (5 × 7)
B. (5 × 60) + (5 × 10) + (5 × 70)
C. (5 × 600) + (5 × 1) + (5 × 7)
D. (5 × 600) + (5 × 10) + (5 × 7)
Answer:
Expression has the same value as 5 × 617 is:
D. (5 × 600) + (5 × 10) + (5 × 7)

Explanation:
5 × 617 = 5 (600  10 + 7)
5 × 617 = (5 × 600) + (5 × 10) + (5 × 7)
5 × 617 = 3000 + 50 + 35
5 × 617 = 3050 + 35
5 × 617 = 3085.
A. (5 × 6) + (5 ×1) + (5 × 7) = 30 + 5 + 35 = 35 + 335 = 70.
B. (5 × 60) + (5 × 10) + (5 × 70) = 300 + 50 + 350 = 350 + 350 = 700.
C. (5 × 600) + (5 × 1) + (5 × 7) = 3000 + 5 + 35 = 3005 + 35 = 3040.
D. (5 × 600) + (5 × 10) + (5 × 7) = 3000 + 50 + 35 = 3050 + 35 = 3085.

 

 

 

 

 

Question 12.
Which of the following is equivalent to (400 × 3) + (36 × 3)?
A. 436 ÷ 3
B. 400 × 10
C. 436 × 3
D. 400 × 3 + 36
Answer:
Equivalent to (400 × 3) + (36 × 3) is:
C. 436 × 3 = 1308.

Explanation:
(400 × 3) + (36 × 3) = 1200 + 108
(400 × 3) + (36 × 3) = 1308.
A. 436 ÷ 3 = 145.33
B. 400 × 10 = 4000.
C. 436 × 3 = 1308.
D. 400 × 3 + 36 = 1200 + 36 = 1236.

 

 

 

Question 13.
Find 3 × 312. Draw a bar diagram to solve.
Answer:
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Topic 3 Assessment Practice-13

Explanation:
3 × 312 = 3 (300 + 10 + 2)
3 × 312 = (3 × 300) + (3 × 10) + (3 × 2)
3 × 312 = 900 + 30 + 6
3 × 312 = 930 + 6
3 × 312 = 936.

 

 

 

Question 14.
Bea’s Bakery bakes 215 cookies and 45 muffins every hour. How many baked goods are baked in 4 hours?
Answer:
Number of baked goods are baked in 4 hours = 1040.

Explanation:
Number of cookies Bea’s Bakery bakes every hour = 215.
Number of muffins Bea’s Bakery bakes every hour = 45.
Number of baked goods are baked in 4 hours = (4 × Number of cookies Bea’s Bakery bakes every hour) + (4 × Number of muffins Bea’s Bakery bakes every hour)
= (4 × 215)+ (4 × 45)
= 860 + 180
= 1040.

 

 

Question 15.
Write and solve an equation that represents the given bar diagram.
Envision Math Common Core Grade 4 Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 109
Answer:
Total number of biscuits made in 4 hours = b = 192.

Explanation:
Number of hours used to make them = 1.
Number of biscuits made every hour = 48.
Total number of biscuits made in 4 hours = b = 4 × Number of biscuits made every hour
= 4 × 48
= 4 (40 + 8)
= (4 × 40) + (4 × 8)
= 160 + 32
= 192.

 

 

Question 16.
Which of the following is equivalent to 8 × 493? Select all that apply.
☐ (8 × 400) + (8 × 90) + (8 × 3)
☐ (8 × 400) + 93
☐ 493 × 8
☐ (8 × 500) – (8 × 93)
☐ (8 × 500) – (8 × 7)
Answer:
8 × 493 = 3944.
Equivalent to 8 × 493 is:
(8 × 400) + (8 × 90) + (8 × 3)
493 × 8
(8 × 500) – (8 × 7)

Explanation:
8 × 493 = 8 (400 + 90 + 3)
8 × 493 = (8 × 400) + (8 × 90) + (8 × 3)
8 × 493 = 3200 + 720 + 24
8 × 493 = 3920 + 24
8 × 493 = 3944.
Expression given:
(8 × 400) + (8 × 90) + (8 × 3) = 3200 + 720 + 24 = 3920 + 24 = 3944.
(8 × 400) + 93 = 3200 + 93 = 3293.
493 × 8 = (400 × 8) + (90 × 8) + (3 × 8) = 3200 + 720 + 24 = 3920 + 24 = 3944.
(8 × 500) – (8 × 93) = 4000 – 744 = 3256.
(8 × 500) – (8 × 7) = 4000 – 56 = 3944.

 

 

 

Question 17.
Mr. Luca would like to purchase a digital keyboard for each of his 3 nieces and 1 nephew. The keyboard costs $105.
A. Mr. Luca thinks the total cost should be about $200. Is this amount reasonable? Explain.
Answer:
No, the amount is not reasonable because total cost of the keyboard Mr. Luca  wants to buy for each of his 3 nieces and 1 nephew is $420 not $200. His estimation is wrong.

Explanation:
Cost of the keyboard = $105.
Number of nieces Mr. Luca has = 3.
Number of nephew Mr. Luca has = 1.
Total cost of the keyboard Mr. Luca  wants to buy for each of his 3 nieces and 1 nephew = (3 × $105) + (1 × $105)
= $315 + $105
= $420.

 

 

B. Write and solve an equation to find the total cost of the keyboards. Explain why your answer is reasonable.
Answer:
Total cost of the keyboard Mr. Luca  wants to buy for each of his 3 nieces and 1 nephew = (3 × $105) + (1 × $105)
= $315 + $105
= $420.
My answer is reasonable because its calculation is correct with correct solution for the problem.

Explanation:
Cost of the keyboard = $105.
Total cost of the keyboard Mr. Luca  wants to buy for each of his 3 nieces and 1 nephew = (3 × $105) + (1 × $105)
= $315 + $105
= $420.

 

 

Question 18.
What is the product of 5 × 1,903?
A. 9,515
B. 9,505
C. 9,155
D. 965
Answer:
5 × 1,903 = 9515.
A. 9,515
C. 9,155

Explanation:
5 × 1,903 = 5 (1000 + 900 + 3)
5 × 1,903 = (5 × 1000) + (5 × 900) + (5 × 3)
5 × 1,903 = 5000 + 4500 + 15
5 × 1,903 = 9500 + 15
5 × 1,903 = 9515.

 

 

Question 19.
How can you use mental math to find the product of 6 × 295?
Answer:
We can use mental math to find the product of 6 × 295 by using partial products or by distributive property.

Explanation:
6 × 295 = 6 (200 + 95)
6 × 295 = (6 × 200 ) + (6 × 95)
6 × 295 = 1200 + 570
6 × 295 = 1770.

 

 

Question 20.
A. Select all the partial products for 8 × 321.
☐ 8
☐ 80
☐ 160
☐ 1,600
☐ 2,400
Answer:
Partial products for 8 × 321 :
☐ 8
☐ 160
☐ 2,400

Explanation:
8 × 321 = 8 ( 300 + 20 + 1)
8 × 321 = (8 × 300) + (8 × 20) + (8 × 1)
8 × 321 = 2400 + 160 + 8
8 × 321 = 2560 + 8
8 × 321 = 2568.

 

 

B. Find the product for Part A using the partial products.
Answer:
8 × 321 = 2568.

Explanation:
8 × 321 = 8 ( 300 + 20 + 1)
8 × 321 = (8 × 300) + (8 × 20) + (8 × 1)
8 × 321 = 2400 + 160 + 8
8 × 321 = 2560 + 8
8 × 321 = 2568.

 

 

Topic 3 Performance Task

Buying Classroom Computers Jorge’s school can purchase computers and printers for the prices shown. Information about the fourth-grade classes are given in the table.
Envision Math Common Core Grade 4 Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 110

Question 1.
Jorge’s teacher is Ms. Sanchez. His class wants to buy 8 desktop computers and 3 printers.
Envision Math Common Core Grade 4 Answers Topic 3 Use Strategies and Properties to Multiply by 1-Digit Numbers 111
Part A
What is the total cost of 8 desktop computers? Use place-value strategies and properties of operations.
Answer:
Cost of 8 desktop computers = $8400.

Explanation:
Cost of desktop computers = $1050.
Cost of 8 desktop computers = 8 × $1050
= 8 (1000 + 50)
= (8 × 1000) + (8 × 50)
= 8000 + 400
= $8400.

 

 

Part B
What is the total cost of 3 printers? Draw an array and show the partial products to find the cost.
Answer:
Total cost of 3 printers = $1024.
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Topic 3 Performance Task-1Part B

Explanation:
Cost of printers= $128.
Total cost of 3 printers = 8 × $128
= 8(100 + 20 + 8)
= (8 × 100) + (8 × 20) + (8 × 8)
= 800 + 160 + 64
= 960 + 64
= $1024.

 

 

Part C
Did Jorge’s class raise enough money to buy 8 computers and 3 printers? Explain.
Answer:
No, Jorge’s class did not raise enough money to buy 8 computers and 3 printers because cost of them is $9424 and their raise amount $9000 is less to buy them.

Explanation:
Total cost of 3 printers = $1024.
Cost of 8 desktop computers = $8400.
Money raised by Jorge’s class = $9000.
Total amount of money to buy 8 computers and 3 printers = Total cost of 3 printers + Cost of 8 desktop computers
= $1024 + $8400
= $9424.
Difference:
= Total amount of money to buy 8 computers and 3 printers – Money raised by Jorge’s class
= $9424 – $9000
= $424.

Question 2.
Rachel’s teacher is Mr. Jones. Her class wants to buy 7 computers and 2 printers.
Part A
How much more do 7 desktop computers cost than 7 laptop computers? Use bar diagrams and equations to represent and solve the problem.
Answer:
$1904 Cost of 7 desktop computers is more cost than 7 laptop computers.
Envision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Topic 3 Performance Task-2-Part AEnvision-Math-Common-Core-4th-Grade-Answers-Key-Topic-3-Use-Strategies-and-Properties-to-Multiply-by-1-Lesson 3.8 Problem Solving-Topic 3 Performance Task-2-Part A-

Explanation:
Cost of desktop computers = $1050.
Total cost of 7 desktop computers = 7 × Cost of desktop computers
= 7 × $1050
= 7 ( 1000 + 50)
= (7 × 1000) + (7 × 50)
= 7000 + 350
= $7350.
Cost of laptop computers = $798.
Total cost of 7 laptop computers = 7 × Cost of laptop computers
= 7 × $798
= 7 ×( 700+ 90 + 8)
= (7 × 700) + (7 × 90) + (7 × 8)
= 4900 + 490 + 56
= 5390 + 56
= $5446.
Cost of 7 desktop computers more cost than 7 laptop computers = Total cost of 7 desktop computers – Total cost of 7 laptop computers
= $7350 – $5446
= $1904.

 

 

Part B
What is another way to find the difference in the cost of 7 desktop computers and 7 laptop computers? Explain.
Answer:
Difference between 7 desktop computers and 7 laptop computers is $1904.

Explanation:
Another way:
Cost of 7 desktop computers = $7350.
Cost of 7 laptop computers = $5446.
Difference:
= Total cost of 7 desktop computers – Total cost of 7 laptop computers
= $7350 – $5446
= $1904.

 

 

 

Question 3.
Miranda’s teacher is Ms. Katz. Her class wants to buy 9 laptop computers and 4 printers. Miranda said the total cost should be $7,494. Is this amount reasonable? Explain. Does the class have enough money?
Answer:
No, the amount is not reasonable because there is a difference $200 less than the actual amount to that of expected amount.

Explanation:
Cost of laptop computer = $798.
Cost of printer = $128.
Total cost expected to be by  Miranda = $7,494
Cost of 9 laptop computers and 4 printers = (9 × Cost of laptop computer) + ( 4 × Cost of printer)
= (9 × $798) + (4 × $128)
= $7182 + $512
= $7694.
Difference:
= Cost of 9 laptop computers and 4 printers – Total cost expected to be by  Miranda
= $7694 – $7494
= $200.

enVision Math Common Core Grade 4 Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers

Go through the enVision Math Common Core Grade 4 Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers regularly and improve your accuracy in solving questions.

enVision Math Common Core 4th Grade Answers Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers

Essential Questions:
How can mental math be used to divide? How can quotients be estimated? How can the steps for dividing be explained?
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1

enVision STEM Project: Music and Division
Do Research: Use the Internet or other resources to find an example of a woodwind instrument, a brass instrument, a stringed instrument, and a percussion instrument.
Journal: Write a Report Include what you found. Also in your report:

  • Explain how each instrument you researched uses energy to make sounds. Include information about how the sounds are produced.
  • An octave spans 8 white keys on a piano. The last key of an octave begins the next octave. Explain why you can divide by 7 to find the number of octaves on a piano with 52 white keys.

Review What You Know

Vocabulary

Choose the best term from the box. Write it on the blank.

  • compatible numbers
  • equation
  • divisible
  • round
  • division
  • variable

Question 1.
A(n) __________ uses the equal sign (=) to show two expressions have the same value.
Answer:
An equation uses the equal sign (=) to show two expressions have the same value.

Question 2.
One way to estimate a product is to _________ each factor.
Answer:
One way to estimate a product is to round each factor.

Question 3.
You use ________ when you find the number of equal groups.
Answer:
You use division when you find the number of equal groups.

Question 4.
Numbers that are easy to compute mentally are called __________
Answer:
Numbers that are easy to compute mentally are called Compatible numbers.

Use Strategies and Properties to Divide by 1-Digit Numbers 1

Division Facts

Find each quotient.
Question 5.
27 ÷ 9
Answer:
The quotient of 27 ÷ 9 is 3.

Explanation:
Given that 27 ÷ 9, here we need to find the quotient. The quotient is the number obtained by dividing one number by another diving number. So the quotient of 27 ÷ 9 is 3.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers1-1

Question 6.
30 ÷ 5
Answer:
The quotient of 30 ÷ 5 is 6.

Explanation:
Given that 30 ÷ 5, here we need to find the quotient. The quotient is the number obtained by dividing one number by another diving number. So the quotient of 30 ÷ 5 is 6.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-2

Question 7.
32 ÷ 4
Answer:
The quotient of 32 ÷ 4 is 8.

Explanation:
Given that 32 ÷ 4, here we need to find the quotient. The quotient is the number obtained by dividing one number by another diving number. So the quotient of 32 ÷ 4 is 8.

Question 8.
54 ÷ 9
Answer:
The quotient of 54 ÷ 9 is 6.

Explanation:
Given that 54 ÷ 9, here we need to find the quotient. The quotient is the number obtained by dividing one number by another diving number. So the quotient of 54 ÷ 9 is 6.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-3

Question 9.
28 ÷ 7
Answer:
The quotient of 28 ÷ 7 is 4.

Explanation:
Given that 28 ÷ 7, here we need to find the quotient. The quotient is the number obtained by dividing one number by another diving number. So the quotient of 28 ÷ 7 is 4.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-4

Question 10.
72 ÷ 9
Answer:
The quotient of 72 ÷ 9 is 8.

Explanation:
Given that 72 ÷ 9, here we need to find the quotient. The quotient is the number obtained by dividing one number by another diving number. So the quotient of 72 ÷ 9 is 8.

Question 11.
56 ÷ 8
Answer:
The quotient of 56 ÷ 8 is 7.

Explanation:
Given that 56 ÷ 8, here we need to find the quotient. The quotient is the number obtained by dividing one number by another diving number. So the quotient of 56 ÷ 8 is 7.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-5

Question 12.
18 ÷ 3
Answer:
The quotient of 18 ÷ 3 is 6.

Explanation:
Given that 18 ÷ 3, here we need to find the quotient. The quotient is the number obtained by dividing one number by another diving number. So the quotient of 18 ÷ 3 is 6.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-6

Question 13.
15 ÷ 5
Answer:
The quotient of 15 ÷ 5 is 3.

Explanation:
Given that 15 ÷ 5, here we need to find the quotient. The quotient is the number obtained by dividing one number by another diving number. So the quotient of 15 ÷ 5 is 3.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-7

Rounding

Round each number to the nearest hundred.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 2
Question 14.
864
Answer:
900.

Explanation:
The rounding to the nearest 864 is 900.

Question 15.
651
Answer:
700.

Explanation:
The rounding to the nearest 651 is 700.

Question 16.
348
Answer:
400.

Explanation:
The rounding to the nearest 348 is 400.

Question 17.
985
Answer:
1000.

Explanation:
The rounding to the nearest 985 is 1000.

Question 18.
451
Answer:
500.

Explanation:
The rounding to the nearest 451 is 500.

Question 19.
749
Answer:
800.

Explanation:
The rounding to the nearest 749 is 800.

Division as Sharing

Question 20.
Make Sense and Persevere Julio has 47 marbles. He keeps his two favorite marbles, then equally shares the remaining marbles between 5 friends. How many marbles does each friend receive? Explain.
Answer:
The number of marbles does each friend receive is 9 marbles.

Explanation:
Given that Julio has 47 marbles and he keeps his two favorite marbles, so Julio has 47 – 2 =45 marbles. And then equally shares the remaining marbles between 5 friends. So the number of marbles does each friend receive is 45 ÷ 5 which is 9 marbles.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-14

Use Strategies and Properties to Divide by 1-Digit Numbers 2

Pick a Project

PROJECT 5A
How many passengers did trains like those in the Gold Coast Railroad Museum carry? Project: Make a Model of a Train
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 3

PROJECT 5B
How far do sailfish migrate?
Project: Make a Migration Map
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 4

PROJECT 5C
How much food do Portuguese Water Dogs need?
Project: Create a Brochure on Portuguese Water Dogs
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 5

3-ACT MATH PREVIEW

Math Modeling
Snack Attack
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 6
I can … model with math to solve a problem that involves rounding, estimating and computing with whole numbers.

Lesson 5.1 Mental Math: Find Quotients

Solve & Share
José has 270 hockey cards to arrange equally in 9 boxes. Each box can hold the same number of cards. How many cards should José place in each box? Solve this problem using any strategy you choose.
I can … make sense of quantities and use mental math and place-value strategies to divide.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 7

Look Back! Reasoning What multiplication equation could help you find the number of cards José should place in each box?

Answer:
Jose should place 30 cards in each box.

Explanation:
Given that Jose has 270 hockey cards to arrange equally in 9 boxes. As each box can hold the same number of cards. So the number of cards in each box will be 270 ÷ 9 which will be 30 cards.

Essential Question
How Can You Divide Mentally?

Visual Learning Bridge
Mr. Díaz ordered a supply of 1,800 pastels. He wants to divide them equally among his class and 5 other art classes. How many pastels does each class receive?
If Mr. Diaz stores the pastels so each class will receive new pastels 5 times a year, how many pastels are handed out each of the 5 times?
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 8

Find 1,800 ÷ 6.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 9
The basic division fact is 18 ÷ 6 = 3.
18 hundreds ÷ 6= 3 hundreds or 300.
1,800 ÷ 6 = 300
Each class will receive 300 pastels.

Find 300 ÷ 5.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 10
The basic division fact is 30 ÷ 5= 6.
30 tens ÷ 5 = 6 tens or 60.
300 ÷ 5 = 60
Each class will receive 60 pastels 5 times a year.

Convince Me! Use Structure Write the missing dividends for each of the following equations. How did you determine each dividend?
_______ ÷ 7 = 70
______ ÷ 8 = 50
______ ÷ 4 = 800

Answer:
The missing digits will be 490, 400, 3200.

Explanation:
To find the missing dividend, we will perform multiplication. So the missing digit will be
70 × 7 which is 490. So the missing digit will be 490 ÷ 7 = 70.
50 × 8 which is 400. So the missing digit will be 400 ÷ 8 = 50.
800 × 4 which is 3200. So the missing digit will be 3200 ÷ 4 = 800.

Use Strategies and Properties to Divide by 1-Digit Numbers 3

Guided Practice

Do You Understand?
Question 1.
Explain how 32 ÷ 4 can help you solve 320 ÷ 4.
Answer:
32 ÷ 4 = 8.
320 ÷ 4 = 80.

Explanation:
Since 320 is 10 times 32. The answer for 320 divided by 4 is going to be 10 times the answer for 32 divided by 4.
32 ÷ 4 = 8.
320 ÷ 4 = 80.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-8

Question 2.
Mrs. Gall orders 240 folders and divides them equally among 3 classes. How many folders does each class receive? What basic fact did you use?
Answer:
The number of folders does each class receive is 80 folders.

Explanation:
Given that Mrs. Gall orders 240 folders and divides them equally among 3 classes. So the number of folders does each class receive is 240 ÷ 3 which is 80 folders. The basic fact that we have used is 24 ÷ 3 = 8.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-9

Do You Know How?
For 3-4, use basic facts and place value to find each quotient.
Question 3.
28 ÷ 7 = ______
280 ÷ 7 = _______
2,800 ÷ 7 = ________
Answer:
28 ÷ 7 = 4.
280 ÷ 7 = 40.
2,800 ÷ 7 = 400.
The basic fact that we have used is 28 ÷ 7 = 4.

Explanation:
Given that 28 ÷ 7 and we need to find the quotient of the given equation. So the quotient of 28 ÷ 7 is 4.
The quotient of 280 ÷ 7 is 40.
The quotient of 2,800 ÷ 7 is 400. The basic fact that we have used is 28 ÷ 7 = 4.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-4

Question 4.
______ = 64 ÷ 8
______ = 640 ÷ 8
______ = 6,400 ÷ 8
Answer:
8 = 64 ÷ 8
80 = 640 ÷ 8
800 = 6,400 ÷ 8
The basic fact that we have used is 64 ÷ 8 = 8.

Explanation:
Given that 64 ÷ 8 and we need to find the quotient of the given equation. So the quotient of 64 ÷ 8 is 8.
The quotient of 640 ÷ 8 is 80.
The quotient of 6,400 ÷ 8 is 800. The basic fact that we have used is 64 ÷ 8 = 8.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-11

Independent Practice

Leveled Practice For 5-24, use basic facts and place value to divide.
Question 5.
36 ÷ 9 = _______
360 ÷ 9 = _______
3,600 ÷ 9 = _______
Answer:
36 ÷ 9 = 4.
360 ÷ 9 = 40.
3,600 ÷ 9 = 400.
The basic fact that we have used is 36 ÷ 9 = 4.

Explanation:
Given that 36 ÷ 9 and we need to find the quotient of the given equation. So the quotient of 36 ÷ 9 is 4.
The quotient of 360 ÷ 9 is 40.
The quotient of 3,600 ÷ 9 is 400. The basic fact that we have used is 36 ÷ 9 = 4.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-12

Use Strategies and Properties to Divide by 1-Digit Numbers 4

Question 6.
______ = 10 ÷ 2
______ = 100 ÷ 2
______ = 1,000 ÷ 2
Answer:
5 = 10 ÷ 2
50 = 100 ÷ 2
500 = 1,000 ÷ 2
The basic fact that we have used is 10 ÷ 2 = 5.

Explanation:
Given that 10 ÷ 2 and we need to find the quotient of the given equation. So the quotient of 10 ÷ 2 is 5.
The quotient of 100 ÷ 2 is 50.
The quotient of 1,000 ÷ 2 is 500. The basic fact that we have used is 10 ÷ 2 = 5.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-13

Question 7.
45 ÷ 5 = _______
450 ÷ 5 = ________
4,500 ÷ 5 = _________
Answer:
45 ÷ 5 = 9.
450 ÷ 5 = 90.
4,500 ÷ 5 =900.
The basic fact that we have used is 45 ÷ 5 = 9.

Explanation:
Given that 45 ÷ 5 and we need to find the quotient of the given equation. So the quotient of 45 ÷ 5 is 9.
The quotient of 450 ÷ 5 is 90.
The quotient of 4,500 ÷ 5 is 900. The basic fact that we have used is 45 ÷ 5 = 9.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-14

Question 8.
_______ = 24 ÷ 8
_______ = 240 ÷ 8
_______ = 2,400 ÷ 8
Answer:
3 = 24 ÷ 8
30 = 240 ÷ 8
300 = 2,400 ÷ 8
The basic fact that we have used is 24 ÷ 8 = 3.

Explanation:
Given that 24 ÷ 8 and we need to find the quotient of the given equation. So the quotient of 24 ÷ 8 is 3.
The quotient of 240 ÷ 8 is 30.
The quotient of 2,400 ÷ 8 is 300. The basic fact that we have used is 24 ÷ 8 = 3.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-17

Question 9.
2,000 ÷ 5
Answer:
The quotient is 400.

Explanation:
Given that 2,000 ÷ 5 and we need to find the quotient of the given equation. So the quotient of 2,000 ÷ 5 is 400.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-18

Question 10.
360 ÷ 4
Answer:
The quotient is 90.

Explanation:
Given that 360 ÷ 4 and we need to find the quotient of the given equation. So the quotient of 360 ÷ 4 is 900.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-19

Question 11.
540 ÷ 9
Answer:
The quotient is 60.

Explanation:
Given that 540 ÷ 9 and we need to find the quotient of the given equation. So the quotient of 540 ÷ 9 is 60.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-20

Question 12.
160 ÷ 4
Answer:
The quotient is 40.

Explanation:
Given that 160 ÷ 4 and we need to find the quotient of the given equation. So the quotient of 160 ÷ 4 is 40.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-16

Question 13.
900 ÷ 3
Answer:
The quotient is 300.

Explanation:
Given that 900 ÷ 3 and we need to find the quotient of the given equation. So the quotient of 900 ÷ 3 is 300.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-21

Question 14.
3,200 ÷ 8
Answer:
The quotient is 400.

Explanation:
Given that 3,200 ÷ 8 and we need to find the quotient of the given equation. So the quotient of 3,200 ÷ 8 is 400.

Question 15.
360 ÷ 6
Answer:
The quotient is 60.

Explanation:
Given that 360 ÷ 6 and we need to find the quotient of the given equation. So the quotient of 360 ÷ 6 is 60.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-23

Question 16.
1,800 ÷ 3
Answer:
The quotient is 600.

Explanation:
Given that 1,800 ÷ 3 and we need to find the quotient of the given equation. So the quotient of 1,800 ÷ 3 is 600.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-24

Question 17.
7,200 ÷ 8
Answer:
The quotient is 900.

Explanation:
Given that 7,200 ÷ 8 and we need to find the quotient of the given equation. So the quotient of 7,200 ÷ 8 is 900.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-26

Question 18.
500 ÷ 5
Answer:
The quotient is 100.

Explanation:
Given that 500 ÷ 5 and we need to find the quotient of the given equation. So the quotient of 500 ÷ 5 is 100.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-27

Question 19.
350 ÷ 7
Answer:
The quotient is 50.

Explanation:
Given that 350 ÷ 7 and we need to find the quotient of the given equation. So the quotient of 350 ÷ 7 is 50.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-77

Question 20.
6,300 ÷ 9
Answer:
The quotient is 700.

Explanation:
Given that 6,300 ÷ 9 and we need to find the quotient of the given equation. So the quotient of 6,300 ÷ 9 is 700.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-78

Question 21.
1,600 ÷ 2
Answer:
The quotient is 800.

Explanation:
Given that 1,600 ÷ 2 and we need to find the quotient of the given equation. So the quotient of 1,600 ÷ 2 is 800.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-79

Question 22.
210 ÷ 7
Answer:
The quotient is 30.

Explanation:
Given that 210 ÷ 7 and we need to find the quotient of the given equation. So the quotient of 210 ÷ 7 is 30.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-80

Question 23.
4,800 ÷ 6
Answer:
The quotient is 800.

Explanation:
Given that 4,800 ÷ 6 and we need to find the quotient of the given equation. So the quotient of 4,800 ÷ 6 is 800.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-81

Question 24.
600 ÷ 6
Answer:
The quotient is 100.

Explanation:
Given that 600 ÷ 6 and we need to find the quotient of the given equation. So the quotient of 600 ÷ 6 is 100.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-82

Problem Solving

Question 25.
If you know 20 ÷ 5 = 4, how does that help you calculate 200 ÷ 5?
Answer:
200 ÷ 5 = 40.

Explanation:
Here, 20 ÷ 5 = 4 is a basic fact. So the value of 200 ÷ 5 = 40

Question 26.
A bakery produced two batches of bread with 80 loaves in each batch. It sold 30 loaves each hour. How many loaves of bread were sold in 4 hours? How many loaves of bread were left to sell?
Answer:
The number of loaves of bread that were sold in 4 hours is loaves is 120 loaves.

Explanation:
Given that the bakery produced two batches of bread with 80 loaves in each batch and it was sold 30 loaves each hour. So the number of loaves of bread that were sold in 4 hours is loaves.30 × 4 = 120 loaves.

Question 27.
An engineer designed and built a solar race car. If there are 810 solar cells arranged in 9 rows, how many solar cells are in each row?
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 11
Answer:
The number of solar cells is in each row is 90rows.

Explanation:
Given that if there are 810 solar cells arranged in 9 rows. So the number of solar cells are in each row is 810 ÷ 9 which is 90 rows.

Question 28.
Model with Math On Saturday afternoon, 350 people attended a play. The seating was arranged in 7 equal rows. Draw a bar diagram and solve an equation to find p, how many people sat in each row.
Answer:
The number of people sat in each row is 50 people.

Explanation:
Given that 350 people attended a play and seating was arranged in 7 equal rows. So the number of people sat in each row is 350 ÷ 7 which is 50 people.

Question 29.
Higher Order Thinking Molly and five friends picked a total of 300 oranges. If they each picked the same number of oranges, how many oranges did Molly pick? Explain.
Answer:
The number of oranges did Molly picked is 60 oranges.

Explanation:
Given that Molly and five friends picked a total of 300 oranges, so the number of oranges did Molly picked is 300 ÷ 5 which is 60 oranges.

Assessment Practice

Question 30.
Find 240 ÷ 8.
A. 3
B. 10
C. 30
D. 80
Answer:
30.

Explanation:
Given that 240 ÷ 8 will be 30.

Question 31.
What basic fact helps to solve 180 ÷ 6? What is 180 ÷ 6?
A. 18 ÷ 3; 60
B. 18 ÷ 3; 30
C. 18 ÷ 6; 60
D. 18 ÷ 6; 30
Answer:
Option D.

Explanation:
Given that to solve 180 ÷ 6 with the basic fact. So the basic fact will be 18 ÷ 6 which is 3.

Lesson 5.2 Mental Math: Estimate Quotients

Solve & Share
Three friends at a video arcade win a total of 248 tickets. They decide to share the tickets equally. About how many tickets will each friend receive? Solve this problem using any strategy you choose.
I can … use compatible numbers to estimate quotients when dividing with 3-digit dividends.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 12

Look Back! is an exact answer or an estimate needed for the problem above? Explain.

Answer:
Each friend will receive 82.67 tickets.

Explanation:
Given that there are three friends at a video arcade who win a total of 248 tickets and they decide to share the tickets equally. Let the number of tickets each friend would receive be X. So
238 = 3 × X
X = 248 ÷ 3
On solving we will get 82.67 tickets.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-22

Essential Question
How Can You Estimate Quotients to Solve Problems?

Visual Learning Bridge
Max wants to make 9 rubber band balls using about the same number of rubber bands for each ball. He bought a jar of 700 rubber bands. Estimate to find about how many rubber bands Max can use for each ball.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 13

Use Compatible Numbers
Estimate 700 ÷ 9.
What number close to 700 is easily divided by 9?
Try multiples of ten near 700.
710 is not easily divided by 9.
720 is 72 tens and can be divided by 9.
720 ÷ 9= 80
Max can use about 80 rubber bands for each ball.

Use Multiplication
Estimate 700 ÷ 9.
9 times what number is about 700?
9 × 8 = 72, so 9 × 80 = 720.
700 ÷ 9 is about 80.
Max can use about 80 rubber bands for each ball.

Convince Me! Construct Arguments What compatible numbers can you use to estimate 132 ÷ 6? Why is rounding not a good way to estimate 132 ÷ 6?

Guided Practice

Do You Understand?
Question 1.
Max wants to make 9 rubber band balls using 80 bands each from his package of 700. Will Max be able to make more or fewer balls than he wanted?
Answer:

Question 2.
Max decides to use 700 rubber bands to make 8 balls. Is it reasonable to say he would use about 90 rubber bands to make each ball? Explain.
Answer:

Do You Know How?
For 3-10, estimate each quotient. Use multiplication or compatible numbers. Show your work.
Question 3.
48 ÷ 5
Answer:
The estimated quotient will be 10.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 48 ÷ 5 will be 50 ÷ 5 which is 10. So the estimated quotient will be 10.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-28

Question 4.
235 ÷ 8
Answer:
The estimated quotient will be 30.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 235 ÷ 8 will be 240 ÷ 8 which is 30. So the estimated quotient will be 30.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-29

Question 5.
547 ÷ 6
Answer:
The estimated quotient will be 90.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 547 ÷ 6 will be 540 ÷ 6 which is 90. So the estimated quotient will be 90.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-30

Question 6.
192 ÷ 5
Answer:
The estimated quotient will be 38.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 192 ÷ 5 will be 190 ÷ 5 which is 38. So the estimated quotient will be 38.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-34

Question 7.
662 ÷ 8
Answer:
The estimated quotient will be 83.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 662 ÷ 8 will be 664 ÷ 8 which is 83. So the estimated quotient will be 83.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-35

Question 8.
362 ÷ 3
Answer:
The estimated quotient will be 120.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 362 ÷ 3 will be 360 ÷ 3 which is 120. So the estimated quotient will be 120.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-36

Question 9.
41 ÷ 2
Answer:
The estimated quotient will be 20.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 41 ÷ 2 will be 40 ÷ 2 which is 20. So the estimated quotient will be 20.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-37

Question 10.
211 ÷ 4
Answer:
The estimated quotient will be 55.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 211 ÷ 4 will be 220 ÷ 4 which is 55. So the estimated quotient will be 55.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-38

Independent Practice

For 11-26, estimate each quotient.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 14
Question 11.
430 ÷ 9
Answer:
The estimated quotient will be 50.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 430 ÷ 9 will be 450 ÷ 9 which is 50. So the estimated quotient will be 50.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-39

Question 12.
620 ÷ 7
Answer:
The estimated quotient will be 90.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 620 ÷ 7 will be 630 ÷ 7 which is 90. So the estimated quotient will be 90.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-40

Question 13.
138 ÷ 5
Answer:
The estimated quotient will be 28.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 138 ÷ 5 will be 140 ÷ 5 which is 28. So the estimated quotient will be 28.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-41

Question 14.
232 ÷ 6
Answer:
The estimated quotient will be 40.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 232 ÷ 6 will be 240 ÷ 6 which is 40. So the estimated quotient will be 40.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-42

Question 15.
342 ÷ 8
Answer:
The estimated quotient will be 45.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 342 ÷ 8 will be 360 ÷ 8 which is 45. So the estimated quotient will be 45.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-43

Question 16.
652 ÷ 6
Answer:
The estimated quotient will be 110.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 652 ÷ 6 will be 660 ÷ 6 which is 110. So the estimated quotient will be 110.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-44

Question 17.
59 ÷ 9
Answer:
The estimated quotient will be 7.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 59 ÷ 6 will be 63 ÷ 9 which is 7. So the estimated quotient will be 7.

Question 18.
813 ÷ 8
Answer:
The estimated quotient will be 100.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 813 ÷ 8 will be 800 ÷ 8 which is 100. So the estimated quotient will be 100.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-45

Question 19.
637 ÷ 6
Answer:
The estimated quotient will be 105.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 637 ÷ 6 will be 630 ÷ 6 which is 105. So the estimated quotient will be 105.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-46

Question 20.
481 ÷ 4
Answer:
The estimated quotient will be 120.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 481 ÷ 4 will be 480 ÷ 4 which is 120. So the estimated quotient will be 120.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-47

Question 21.
747 ÷ 8
Answer:
The estimated quotient will be 95.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 747 ÷ 8 will be 760 ÷ 8 which is 95. So the estimated quotient will be 95.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-49

Question 22.
232 ÷ 9
Answer:
The estimated quotient will be 25.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 232 ÷ 9 will be 225 ÷ 9 which is 25. So the estimated quotient will be 25.

Question 23.
552 ÷ 7
Answer:
The estimated quotient will be 80.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 552 ÷ 7 will be 560 ÷ 7 which is 80. So the estimated quotient will be 80.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-50

Question 24.
52 ÷ 5
Answer:
The estimated quotient will be 10.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 52 ÷ 5 will be 50 ÷ 5 which is 10. So the estimated quotient will be 10.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-51

Question 25.
392 ÷ 2
Answer:
The quotient will be 196.

Explanation:
Here, the quotient of 392 ÷2 is 196.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-52

Question 26.
625 ÷ 3
Answer:
The estimated quotient will be 210.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 625 ÷ 3 will be 630 ÷ 3 which is 210. So the estimated quotient will be 210.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-53

Problem Solving

For 27-28, use the table at the right.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 15
Question 27.
Ada sold her mugs in 3 weeks. About how many mugs did Ada sell each week?
Answer:
The number of mugs Ada sells each week will be 80 mugs.

Explanation:
Given that Ada sold her mugs in 3 weeks and there are five mugs and each mug is equal to 50 mugs. So there will be 5 × 50 which is 250 mugs. So the number of mugs Ada sells each week will be 250 ÷ 3, so the estimated quotient will be 240 ÷ 3 which will be 80 mugs.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-54

Question 28.
Ben sold his mugs in 6 weeks. About how many mugs did Ben sell each week?
Answer:
The number of mugs Ben sells each week will be 30 mugs.

Explanation:
Given that Ada sold her mugs in 6 weeks and there are five mugs and each mug is equal to 50 mugs. So there will be 4 × 50 which is 200 mugs. So the number of mugs Ada sells each week will be 200 ÷ 6, so the estimated quotient will be 180 ÷ 6 which will be 30 mugs.

Question 29.
enVision® STEM The International Space Station takes 644 minutes to orbit Earth 7 times. About how long does each orbit take?
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 16
Answer:
The number of minutes does each orbit takes is 92 minutes.

Explanation:
Given that the International Space Station takes 644 minutes to orbit Earth 7 times, so the number of minutes does each orbit takes is 644 ÷ 7 which is 92 minutes.

Question 30.
There are 60 minutes in 1 hour and 24 hours in 1 day. About how many times does the International Space Station orbit Earth each day?
Answer:

Question 31.
Construct Arguments Complete by writing > or < in the Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 17. Without dividing, explain how you know which quotient is greater.
930 ÷ 4 Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 17 762 ÷ 4
Answer:
930 ÷ 4 > 762 ÷ 4.

Explanation:
Given that to find out which quotient is greater without dividing. Here, we can say that by seeing the dividend, 930 is greater than 762. So 930 ÷ 4 quotient is greater than 762 ÷ 4.

Question 32.
Higher Order Thinking Explain how to find a better estimate for 260 ÷ 5 than the one shown below.
Round 260 to 300, and then estimate 300 ÷ 5.
300 ÷ 5 = 60, so 260 ÷ 5 is about 60.
Answer:

Assessment Practice

Question 33.
Kaylee wanted to divide 133 pieces of candy equally into 7 boxes. She decides to put 19 pieces in each box. Use estimation to determine if this answer seems reasonable.
Answer:
Kaylee is correct.

Explanation:
Given that Kaylee wanted to divide 133 pieces of candy equally into 7 boxes and she decides to put 19 pieces in each box. So the estimated answer will be 140 ÷ 7 which will be 20 pieces. So Kaylee is correct.

Lesson 5.3 Mental Math: Estimate Quotients for Greater Dividends

Solve & Share
Jimi has 3,000 tickets to sell at the school carnival. Jimi separated the tickets into groups of 8 tickets. About how many groups did Jimi make? Solve this problem using any strategy you choose.
I can … estimate quotients for 4-digit dividends.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 18

Look Back! Reasoning What basic fact did you use to solve the problem above? How does this help you?

Essential Question
How Can You Estimate Quotients Using Patterns and Place Value?

Visual Learning Bridge
On “Clean Up Your Town Day,” 1,320 people volunteered to clean up the Springville parks. The volunteers were divided equally into teams to work in each of the town’s parks. About how many people were on each team?
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 19

Use Multiplication Patterns
Estimate 1,320 ÷ 6.
6 times what number is about 1,320?
You know 6 × 2 = 12, and
6 × 20 = 120, so
6 × 200 = 1,200.
1,200 is close to 1,320.
There are about 200 people on each team.

Use Division Facts and Place-Value Patterns
Find compatible numbers to estimate 1,320 ÷ 6.
You know 12 ÷ 6 = 2, and
120 ÷ 6 = 20, So
1,200 ÷ 6 = 200.
1,320 ÷ 6 is about 200.
There are about 200 people on each team.

Convince Me! Construct Arguments Complete the calculations at the right. Explain how you can use the calculations to estimate 1,296 ÷ 4.
4 × 100 = __________
4 × 200 = __________
4 × 300 = __________
4 × 400 = __________

Another Example!
You can use rounding to estimate quotients.
Estimate 357 ÷ 8 by rounding the dividend.
Round: 400 ÷ 8
4008 ÷ 50
So, 357 ÷ 8 is about 50.

Estimate 5,582 ÷ 7 by rounding the dividend.
Round: 5,600 ÷ 7
5,600 ÷ 7 = 800
So, 5,582 ÷ 7 = is about 800.

Guided Practice

Do You Understand?
Question 1.
When estimating 1,320 ÷ 6, why is rounding not a good strategy?
Answer:
1,320 ÷ 6 is 220.

Explanation:
Given that the equation is 1,320 ÷ 6 which is 220. As the dividend is divisible by 6. So rounding is not a good strategy.

Question 2.
When dividing a 4-digit number by a 1-digit number, how many digits can the quotient have?
Answer:
Here, the quotient can still have 4 digits depending on the 1 digit.

Do You Know How?
For 3-8, estimate each quotient.
Question 3.
3,340 ÷ 8
Answer:
The estimated quotient will be 417.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 3,340 ÷ 8 will be 3,336 ÷ 8 which is 417. So the estimated quotient will be 417.

Question 4.
2,943 ÷ 7
Answer:
The estimated quotient will be 420.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 2,943 ÷ 7 will be 2,940 ÷ 7 which is 420. So the estimated quotient will be 420.

Question 5.
552 ÷ 9
Answer:
The estimated quotient will be 60.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 552 ÷ 9 will be 540 ÷ 9 which is 60. So the estimated quotient will be 60.

Question 6.
776 ÷ 4
Answer:
The quotient will be 194.

Explanation:
Here, the quotient of 776 ÷ 4 is 194.

Question 7.
2,013 ÷ 5
Answer:
The estimated quotient will be 400.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 2,013 ÷ 5 will be 2,000 ÷ 5 which is 400. So the estimated quotient will be 400.

Question 8.
281 ÷ 3
Answer:
The estimated quotient will be 90.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 281 ÷ 3 will be 270 ÷ 3 which is 90. So the estimated quotient will be 90.

Independent Practice

For 9-20, estimate each quotient.
Question 9.
61 ÷ 2
Answer:
The estimated quotient will be 30.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 61 ÷ 2 will be 60 ÷ 2 which is 30. So the estimated quotient will be 30.

Question 10.
7,779 ÷ 7
Answer:
The estimated quotient will be 1,100.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 7,779 ÷ 7 will be 7,770 ÷ 7 which is 1,100. So the estimated quotient will be 1,100.

Question 11.
3,688 ÷ 6
Answer:
The estimated quotient will be 600.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 3,688 ÷ 6 will be 3,600 ÷ 6 which is 600. So the estimated quotient will be 600.

Question 12.
497 ÷ 8
Answer:
The estimated quotient will be 60.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 497 ÷ 8 will be 480 ÷ 8 which is 60. So the estimated quotient will be 60.

Question 13.
5,684 ÷ 9
Answer:
The estimated quotient will be 600.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 5,684 ÷ 9 will be 5,400 ÷ 9 which is 600. So the estimated quotient will be 600.

Question 14.
5,346 ÷ 6
Answer:
The estimated quotient will be 900.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 5,346 ÷ 6 will be 5,400 ÷ 6 which is 900. So the estimated quotient will be 900.

Question 15.
508 ÷ 7
Answer:
The estimated quotient will be 70.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 508 ÷ 7 will be 490 ÷ 7 which is 70. So the estimated quotient will be 70.

Question 16.
92 ÷ 3
Answer:
The estimated quotient will be 30.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 92 ÷ 3 will be 90 ÷ 3 which is 30. So the estimated quotient will be 30.

Question 17.
647 ÷ 3
Answer:
The estimated quotient will be 200.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 647 ÷ 3 will be 600 ÷ 3 which is 200. So the estimated quotient will be 200.

Question 18.
3,958 ÷ 8
Answer:
The estimated quotient will be 500.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 3,958 ÷ 8 will be 4,000 ÷ 8 which is 500. So the estimated quotient will be 500.

Question 19.
224 ÷ 3
Answer:
The estimated quotient will be 70.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 224 ÷ 3 will be 210 ÷ 3 which is 70. So the estimated quotient will be 70.

Question 20.
2,438 ÷ 5
Answer:
The estimated quotient will be 500.

Explanation:
Here, Compatible numbers are numbers that are easy to compute with mental math. We can also use compatible to find two estimates that the quotient is between. So 2,438 ÷ 5 will be 2,500 ÷ 5 which is 500. So the estimated quotient will be 500.

Problem Solving

Question 21.
Laura’s dog eats 1 bag of dog food every 6 days. About how many bags will her dog eat in 1 year? About how many bags will Laura’s dog eat in 10 years? Explain.
Answer:
The number of bags will Laura’s dog eat in 10 years is 600 bags.

Explanation:
Given that Laura’s dog eats 1 bag of dog food every 6 days, as there are 365 days for 1 year. So for 1 year, it will be 365 ÷ 6 which will be approx 360 ÷ 6 = 60 bags. So the number of bags will Laura’s dog eat in 10 years is 60 × 10 which will be 600 bags.

Question 22.
Model with Math During the school year, a bus driver made 7 trips to the museum. The distance from the school to the museum is 36 miles. Write and solve equations to find how many miles the bus driver drove for the 7 trips.
Answer:
The number of miles the bus driver drives for 7 trips is approx 280 miles.

Explanation:
Given that a bus driver made 7 trips to the museum. The distance from the school to the museum is 36 miles. So the equation for 7 trips will be 36 × 7 which will be 252 miles, which will be approx 40 × 7 = 280 miles.

Question 23.
Vocabulary Use a vocabulary word to complete the statement.
Multiplication and division are _________ operations because they undo each other.
Answer:
Multiplication and division are inverse operations because they undo each other.

Question 24.
Ramón’s sister wants to buy a car that costs $7,993. She earns $9 for every hour she works. About how many hours must Ramón’s sister work to earn enough money to buy the car?
Answer:
The number of hours must Ramón’s sister work to earn enough money to buy a car is 900 hours.

Explanation:
Given that Ramón’s sister wants to buy a car that costs $7,993 and she earns $9 for every hour she works. So the number of hours must Ramón’s sister work to earn enough money to buy a car is 7,993 ÷ 9 which will be 8,100 ÷ 9 which is 900 hours.

Question 25.
Number Sense Eight students can sit at one cafeteria table. About how many tables are needed for 231 students? Explain.
Answer:
The number of tables required for 231 students is approx 30 tables.

Explanation:
Given that eight students can sit at one cafeteria table. So the number of tables required for 231 students is 231 ÷ 8 which will be 240 ÷ 8 which is 30 tables.

Question 26.
Higher Order Thinking At Camp Summer Fun, 4 campers share a tent. The camp is expecting 331 campers. About how many tents will they need? Will the actual number of tents needed be more or less than your estimate? Explain.
Answer:
The number of tents will they need is 80 tents.

Explanation:
Given that 4 campers share a tent and the camp is expecting 331 campers. So the number of tents will they need is 320 ÷ 4 which is 80 tents. Here, we need more tents for the extra 11 students.

Assessment Practice

Question 27.
Nadine has 1,424 pictures to put in 7 folders on her computer. She wants the same number in each folder. Which is the best estimate of the number of pictures she should put in each folder?
A. About 200
B. About 300
C. About 2,000
D. About 3,000
Answer:
The number of pictures she should put in each folder is about 200.

Explanation:
Given that Nadine has 1,424 pictures to put in 7 folders on her computer and she wants the same number in each folder. So the number of pictures she should put in each folder is 1424 ÷ 7 which will be 1400 ÷ 7 which is about 200.

Question 28.
Sven needs to save $239 to buy a bike. Which is the best estimate of the amount he needs to save each month to have enough in 3 months?
A. About $60
B. About $70
C. About $80
D. About $90
Answer:
The best estimate of the amount he needs to save each month to have enough in 3 months is about $80.

Explanation:
Given that Sven needs to save $239 to buy a bike, so the best estimate of the amount he needs to save each month to have enough in 3 months is 239 ÷ 3 which is 240 ÷ 3 which is about $80.

Lesson 5.4 Interpret Remainders

Solve & Share
There are 47 students taking a field trip. The students are being driven in cars to a play by adult volunteers. Each driver can take at most 4 students. How many cars are needed for the field trip? Will each car have four students? Use counters or draw pictures to solve this problem. Explain how you found your answer.
I can… apply what I know about dividing items into equal groups to solve problems.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 20

Look Back! Suppose there were only 46 students. Would the number of cars needed for the field trip change? Explain.

Essential Question
After Dividing, What Do You Do Question her with the Remainder?

Visual Learning Bridge
When you divide with whole numbers, any whole number that remains after the division is complete is called the remainder.

Ned has 27 soccer cards in an album. He put 6 cards on each page. He knows 27 ÷ 6 = 4 with 3 left over, because 6 × 4 = 24 and 24 + 3 = 27.
Use an R to represent the remainder: 27 ÷ 6 = 4 R3
How do you use the remainder to answer questions about division?
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 21

How many pages did Ned fill?
To answer this question, find how many groups of 6 there are. The remainder can be ignored.
27 ÷ 6 = 4 R3
Ned filled 4 pages.

How many pages did Ned work on?
To answer this question, find how many groups of 6 are filled or started. Because there is a remainder, add 1 to the quotient.
27 ÷ 6 = 4 R3
Ned worked on 5 pages.

How many cards did Ned put on the fifth page?
The answer to this question is the remainder.
27 ÷ 6= 4 R3
Ned put 3 cards on the fifth page.

Convince Me! Critique Reasoning The calculation to the right is incorrect. What error was made? What is the correct answer?
45 ÷6 = 6 R9

Another Example!
There are 20 apples to arrange in gift baskets, with 6 apples in each basket. How many baskets can be filled? Explain the meaning of the remainder.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 22
3 equal groups of 6 with 2 left over
20 ÷ 3 = 6 R2, because
3 × 6 = 18 and 18 + 2 = 20.
The remainder, 2, represents the number of apples not placed into gift baskets.

Guided Practice

Do You Understand?
Question 1.
When a divisor is 3, can the remainder be 5? Explain.
Answer:
No, the remainder cannot be 5.

Explanation:
No, the remainder cannot be 5. Because in the division the remainder will always be less than the divisor.

Question 2.
Dave is packing 23 sweaters into boxes. Each box will hold 3 sweaters. How many boxes will he need? Explain how the remainder affects your answer.
Answer:
The number of boxes required is about 8 boxes.

Explanation:
Given that Dave is packing 23 sweaters into boxes and each box will hold 3 sweaters. So the number of boxes required is 23 ÷ 3 which is 7.6 approx 8 boxes. Here the remainder will be rounded off.

Do You Know How?
For 3-6, find the number of groups and the number left over. Draw an array if needed.
Question 3.
47 ÷ 3 = ________ with _________ left over
Answer:
47 ÷ 3 = 15 with 2 left over.

Explanation:
Given that 47 ÷ 3, so the remainder will be 15, and the leftover will be 2.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-31

Question 4.
29 ÷ 2 = ________ with _________ left over
Answer:
29 ÷ 2 = 14 with 1 left over.

Explanation:
Given that 29 ÷ 2, so the remainder will be 14 and the leftover will be 1.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-83

Question 5.
62 ÷ 5 = ________ with _________ left over
Answer:
62 ÷ 5 = 12 with 2 left over.

Explanation:
Given that 62 ÷ 5, so the remainder will be 12 and the leftover will be 2.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-84

Question 6.
86 ÷ 6 = ________ with _________ left over
Answer:
86 ÷ 6 = 14 with 2 left over.

Explanation:
Given that 86 ÷ 6, so the remainder will be 14 and the leftover will be 2.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-85

Independent Practice

For 7-10, find the number of groups and the number left over.
Question 7.
18 ÷ 4 = ______ with ______ left over
Answer:
18 ÷ 4 = 4 with 2 left over.

Explanation:
Given that 18 ÷ 4, so the remainder will be 4 and the leftover will be 2.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-86

Question 8.
22 ÷ 6 = _______ with ________ left over
Answer:
22 ÷ 6 = 3 with 4 left over.

Explanation:
Given that 22 ÷ 6, so the remainder will be 3 and the leftover will be 4.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-87

Question 9.
31 ÷ 8 = ________ with _________ left over
Answer:
31 ÷ 8 = 3 with 7 left over.

Explanation:
Given that 31 ÷ 8, so the remainder will be 3 and the leftover will be 7.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-90

Question 10.
32 ÷ 9 = _______ with _________ left over
Answer:
32 ÷ 9 = 3 with 5 left over.

Explanation:
Given that 32 ÷ 9, so the remainder will be 3 and the leftover will be 5.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-91

For 11-13, interpret each remainder.
Question 11.
59 football cards 3 cards on each page. How many pages can Alex complete?
Answer:
The number of pages can Alex complete is 19 pages.

Explanation:
Given that 59 football cards and 3 cards on each page, so the number of pages can Alex complete is 59 ÷ 3 which is about 19 pages.

Question 12.
55 baseball cards 4 cards on each page. How many cards are on the last page?
Answer:
The number of cards is on the last page is about 13 pages.

Explanation:
Given that 55 baseball cards and 4 cards on each page, so the number of cards is on the last page is about 13 pages.

Question 13.
84 stickers 5 stickers on each page. How many pages will have some stickers on them?
Answer:
The number of pages that will have stickers on them is 17 pages.

Explanation:
Given that 84 stickers and 5 stickers on each page, so the number of pages that will have stickers on them are 84 ÷ 5 which is about 17 pages.

Problem Solving

For 14-15, use the table at the right.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 23
Question 14.
Samuel has 85 prize tickets. How many marbles can he get?
Answer:
The number of marbles can he get is about 12 marbles.

Explanation:
Given that Samuel has 85 prize tickets, so the number of marbles can he get is 85 ÷ 7 which is about 12 marbles.

Question 15.
Inez chose 8 rings and 12 stickers. How many tickets did she use?
Answer:
The total number of tickets is 120 tickets.

Explanation:
Given that Inez chose 8 rings which is 8 × 9 = 72 and 12 stickers which is 12 × 4 = 48. So the total number of tickets is 72 + 48 which is 120 tickets.

Question 16.
Keiko makes necklaces like the one in the picture at the right. She has 19 blue beads and 13 red beads. How many necklaces can Keiko make? How many of each color bead will be left over?
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 24
Answer:
Keiko makes 6 necklaces and 1 bead of each color.

Explanation:
Given that Keiko has 19 blue beads and 13 red beads, as she places one blue bead and one red bead. So she can make 6 necklaces and one red bead, the blue bead will be leftover.

Question 17.
Critique Reasoning Amanda calculated 34 ÷ 8= 3 R10. Is Amanda’s answer correct? If not, what is the correct answer? Explain.
Answer:
No, Amanda is not correct.

Explanation:
No, Amanda is not correct. Here, we can see the remainder is greater than the divisor, as remainder should not be greater than the divisor.

Question 18.
Higher Order Thinking Write a problem that requires adding 1 to the quotient when interpreting the remainder.
Answer:
Each table can seat 4 students. How many tables will be needed to seat 26 students?

Assessment Practice

Question 19.
There are 39 children at a park. They want to make teams with 9 children on each team. Two of the children go home. How many complete teams can they make? Explain.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 25
Answer:
The number of complete teams can they make is 4 teams.

Explanation:
Given that there are 39 children at a park and they want to make teams with 9 children on each team and two of the children go home, so there will be 39 – 2 = 37 students. So the number of complete teams can they make is 37 ÷ 9 is about 4 teams.

Lesson 5.5 Use Partial Quotients to Divide

Solve & Share
Sally’s bird feeder holds 6 cups of bird feed. How many times can Sally’s bird feeder be filled using a 72-cup bag of bird feed? Use counters or draw pictures to solve this problem. Explain how you found your answer.
I can … divide by thinking about multiplication, estimation, properties, and place value.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 26

Look Back! Reasoning How can you use multiplication to check your answer?

Essential Question
How Can You Use Partial Quotients to Solve Division Problems?

Visual Learning Bridge
There are 3 seats in each row of an airplane for passengers. If there are 63 people boarding the airplane, how many rows of seats are needed for the passengers?
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 27

One Way
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 28
There are 21 groups of 3 in 63.
21 rows of seats are needed.

Another Way
Sometimes you can use a different first estimate.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 29
There are 21 groups of 3 in 63.
21 rows of seats are needed.

Convince Me! Use Structure How can you use the relationships between multiplication and division to check your answer?

Another Example!
Break apart 69 using place value. Use the Distributive Property to show 69 ÷ 3.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 30
69 ÷ 3 = (60 + 9) ÷ 3
= (60 ÷ 3) + (9 ÷ 3)
= 20 + 3
= 23

Guided Practice

Do You Understand?
Question 1.
Harry used the model below to find 84 ÷ 4. Use the Distributive Property to show why Harry’s strategy works.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 31
Answer:
84 ÷ 4 is 21.

Explanation:
Given that Harry used the model to find 84 ÷ 4, so the Distributive Property shows why Harry’s strategy works is
= (80 ÷ 4) + (4 ÷ 4)
= 20 + 1
= 21.

Do You Know How?
For 2-3, use partial quotients to divide. Use counters or draw pictures as needed.
Question 2.
How many groups of 4 are in 48?
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 32
Answer:
There are 12 groups of 4 in 48.

Explanation:
There are 12 groups of 4 in 48.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-33

Question 3.
How many groups of 6 are in 78?
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 33
Answer:
There are 13 groups of 6 in 78.

Explanation:
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-33-1

Independent Practice

For 4-9, use partial quotients to divide. You may use counters or draw pictures to help.
Question 4.
How many groups of 6 are in 90?
90 ÷ 6
Answer:
There are 15 groups of 6 in 90.

Explanation:
There are 15 groups of 6 in 90.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-33-2

Question 5.
How many groups of 5 are in 85?
85 ÷ 5
Answer:
There are 17 groups of 5 in 85.

Explanation:
There are 17 groups of 5 in 85.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-33-3

Question 6.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 34
Answer:
There are 39 groups of 2 in 78.

Explanation:
There are 39 groups of 2 in 78.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-33-4

Question 7.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 35
Answer:

Question 8.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 36
Answer:
There are 42 groups of 2 in 84.

Explanation:
There are 42 groups of 2 in 84.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-33-5

Question 9.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 37
Answer:
There are 19 groups of 3 in 57.

Explanation:
There are 19 groups of 3 in 57.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-33-6

Problem Solving

Question 10.
Model with Math A collection of 64 stickers is placed into 4 equal piles. How many stickers are placed in each pile? Use the bar diagram to write and solve an equation.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 38
Answer:
The number of stickers is placed in each pile is 16 piles.

Explanation:
Given that 64 stickers are placed into 4 equal piles, so the number of stickers are placed in each pile is
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-33-7

Question 11.
A chef is baking cookies for 3 parties. For each party, the chef uses the same number of eggs. She has 2 dozen eggs. What is the greatest number of eggs the chef can use for each party?
Answer:
The greatest number of eggs the chef can use for each party is 8 eggs.

Explanation:
Given that A chef is baking cookies for 3 parties and for each party, the chef uses the same number of eggs and she has 2 dozen eggs which are 2 × 12 = 24 eggs. So the greatest number of eggs the chef can use for each party is 24 ÷ 3 which is 8 eggs.

Question 12.
Show how to use the Distributive Property to divide 54 by 2 by breaking 54 into 40 + 14.
Answer:

Question 13.
Higher Order Thinking Amanda wants to put some of her books on 4 shelves with 6 books on each shelf and the rest on 6 shelves with 3 books on each shelf. Can Amanda arrange her books this way? Explain.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 39
Answer:
Amanda can arrange books 4 by 6 and 3 by 6.

Explanation:
Given that Amanda put some of her books on 4 shelves with 6 books, so there will be 4 × 6 which is 24. As the total number of books is 42, so the remaining books will be 42 – 24 = 18. And the rest are on 6 shelves with 3 books on each shelf, these can be arranged as 18 ÷ 6 which is 3.

Assessment Practice

Question 14.
Select all correct combinations of partial quotients that can be used to find 96 ÷ 3.
☐ 30, 2
☐ 30,10,2
☐ 10, 10, 10, 2
☐ 10, 10, 10, 6
☐ 20, 10, 2
Answer:
The correct combination of partial quotients is options A, C, and E.

Explanation:
The correct combination of partial quotients that can be used to find 96 ÷ 3 is 30,2 and 10, 10, 10, 2 and 20, 10, 2.

Question 15.
Use the Distributive Property to find 84 ÷ 7. Which is the missing number?
84 ÷ 7 = (70 + _______) = 7
= (70 ÷ 7) + (_____ ÷ 7)
= 10 + 2
= 12
A. 4
B. 14
C. 24
D. 34
Answer:
The missing number is 14.

Explanation:
Here, we are using Distributive Property to find 84 ÷ 7 which is
84 ÷ 7 = (70 + 14) = 7
= (70 ÷ 7) + (14 ÷ 7)
= 10 + 2
= 12.

Lesson 5.6 Use Partial Quotients to Divide: Greater Dividends

Solve & Share
The new reading room in the library is 9 feet long. It is divided into a reading area and a help desk area. It has a total area of 153 square feet. What is the total width of the new reading room? Use the model to help solve this problem. Solve this problem using any strategy you choose.
I can … divide by thinking about multiplication, estimation, and place value.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 40

Look Back! Reasoning If the area of the reading room was 216 square feet and the length was still 9 feet, would the width be more or less than 20? Explain.

Essential Question
How Can You Use Partial Quotients to Divide Greater Dividends?

Visual Learning Bridge
A total of 277 people signed up to audition for a talent show. Five people at a time were brought in for a group interview. How many group interviews were needed to audition all 277 people?
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 41

Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 42

Convince Me! Use Structure How can you use multiplication and addition to check the answer above?

Another Example!
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 43

Guided Practice

Do You Understand?
Question 1.
Hilary has 254 tokens to use for games at Pizza Mania. She would like to use an equal number of tokens for each of 3 visits she has planned. Will Hilary be able to use the same amount of tokens for each visit?
Answer:
She can use 84 tokens with 2 visits.

Explanation:
Given that Hilary has 254 tokens to use for games at Pizza Mania and she would like to use an equal number of tokens for each of the 3 visits she has planned. So, Hilary can use 254 ÷ 3 which is 84 with 2 leftovers. So she can use 84 tokens with 2 visits.

Do You Know How?
For 2-3, use partial quotients to divide.
Question 2.
How many 4s are in 6,787?
6,787 ÷ 4
Answer:
The number of 4’s in 6,787 is 1696 with the remainder 3.

Explanation:
Given that 6,787 ÷ 4, so the number of 4’s in 6,787 is 1696 with the remainder 3.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-55

Question 3.
How many 5s are in 6,209?
6,209 ÷ 5
Answer:
The number of 5’s in 6,209 is 1241 with the remainder 4.

Explanation:
Given that 6,209 ÷ 5, so the number of 5’s in 6,209 is 1241 with the remainder 4.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-56

Independent Practice

For 4-11, use partial quotients to divide.
Question 4.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 44
Answer:
The partial quotient is 17.

Explanation:
Given that 153 ÷ 9, the partial quotient is 17.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-57

Question 5.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 45
Answer:
The partial quotient is 56 with 2 leftovers.

Explanation:
Given that 450 ÷ 8, the partial quotient is 56 with 2 leftovers.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-59

Question 6.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 46
Answer:
The partial quotient is 942.

Explanation:
Given that 2,826 ÷ 3, the partial quotient is 942.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-60

Question 7.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 47
Answer:
The partial quotient is 1346 with 6 leftovers.

Explanation:
Given that 9,428 ÷ 7, the partial quotient is 1346 with 6 leftovers.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-62

Question 8.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 48
Answer:
The partial quotient is 616 with 6 leftovers.

Explanation:
Given that 4,318 ÷ 7, the partial quotient is 616 with 6 leftovers.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-73

Question 9.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 49
Answer:
The partial quotient is 2,114 with 1 leftover.

Explanation:
Given that 8,457 ÷ 4, the partial quotient is 2,114 with 1 leftover.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-74

Question 10.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 50
Answer:
The partial quotient is 712 with 3 leftovers.

Explanation:
Given that 5,699 ÷ 8, the partial quotient is 712 with 3 leftovers.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-75

Question 11.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 51
Answer:
The partial quotient is 1,522 with 1 leftover.

Explanation:
Given that 4,567 ÷ 3, the partial quotient is 1,522 with 1 leftover.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-76

Problem Solving

Question 12.
After a state fair, three 4th-grade classes volunteered to clean up the trash from the fairgrounds. In total, they collected 1,281 pounds of trash. If each class collected the same amount, how many pounds of trash did each class collect?
Answer:
The number of pounds of trash did each class collects is 427 pounds.

Explanation:
Given that after a state fair, three 4th-grade classes volunteered to clean up the trash from the fairgrounds. In total, they collected 1,281 pounds of trash. If each class collected the same amount, the number of pounds of trash did each class collects is 1281 ÷ 3 which is 427 pounds.

Question 13.
enVision® STEM An electric car can travel 4 miles on one-kilowatt hour of electricity. How many kilowatt hours of electricity would it take for Shawn to drive his electric car to his grandmother’s house and back? Shawn lives 56 miles from his grandmother.
Answer:
The number of kilowatt-hours is 28 kilowatt-hours.

Explanation:
Given that an electric car can travel 4 miles on the one-kilowatt hour of electricity and Shawn to drive his electric car to his grandmother’s house and back, so 56 × 2 which is 112 miles Shawn needs to drive. So the number of kilowatt-hours of electricity would it take for Shawn to drive his electric car to his grandmother’s house and back is 112 ÷ 4 which is 28 miles.

For 14-15, use the table at the right.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 52
Question 14.
Number Sense Use estimation to find which material allows the students to make the least number of headbands.
Answer:
Given that the beads are 6,437 ÷ 7 which is around 900, buttons are 3,636 ÷ 3 which is around 1000 and ribbons are 3,870 ÷ 9 which is around 450. So the ribbon allows students to make the least number of headbands.

Question 15.
Higher Order Thinking How many headbands can the students make? Explain.
Answer:

Assessment Practice

Question 16.
Select all correct combinations of partial quotients and a remainder that can be used to find 4,567 ÷ 7.
☐ 600; 50; 2.
☐ 500; 10; 50; R3
☐ 500; 100; 50; 2; R3
☐ 600; 50; R17
☐ 600; 50; 2; R3
Answer:
600; 50; 2; R3.

Explanation:
The correct combinations of partial quotients and a remainder that can be used to find 4,567 ÷ 7 is 600; 50; 2; R3.

Question 17.
Which is the quotient? 3,858 ÷ 8
A. 4,082 R2
B. 472 R2
C. 482 R2
D. 481 R8
Answer:
482 with 2 leftovers.

Explanation:
The quotient of 3,858 ÷ 8 is 482 with 2 leftovers.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-65

Lesson 5.7 Use Sharing to Divide

Solve & Share
A class collected $238 to be distributed equally to two charities. How much money will each charity receive? Use objects or draw pictures to help solve this problem. Explain how you found your answer.
I can … sort objects into equal-sized groups to divide.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 53

Look Back! When might you need to divide something into equal groups in everyday life?
Answer:
The money will each charity receive is $119.

Explanation:
Given that a class collected $238 to be distributed equally to two charities. So the money will each charity receive is $238 ÷ 2 which is $119.

Essential Question
How Can Place Value don Help You Divide?

Visual Learning Bridge
The craft club made 375 key chains. They sold 137 of the key chains at the school fair. The rest need to be packed into 2 boxes with the same number of key chains in each box. How many key chains will go in each box?
First, subtract to find how many key chains need to be packed.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 54

Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 55

Convince Me! Use Appropriate Tools Tell how you would evenly divide the money shown among 4 people using only $10 bills and $1 bills.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 56

Another Example!
Find 55 ÷ 4.
Divide the tens equally into 4 groups. Regroup 1 ten as 10 ones and then divide the ones equally into 4 groups. There are 3 ones left over.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 57

Guided Practice

Do You Understand?
Question 1.
Draw a picture to explain why 423 ÷ 3 = 141.
Answer:

Question 2.
The art teacher displayed 48 paintings on 3 walls. If each wall had the same number of paintings, how many paintings were on each wall?
Answer:
The number of paintings was on each wall is 16 paintings.

Explanation:
Given that the art teacher displayed 48 paintings on 3 walls and if each wall had the same number of paintings, so the number of paintings were on each wall is 48 ÷ 3 which is 16 paintings.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-66

Do You Know How?

For 3-4, tell how many are in each group and how many are left over. Use placevalue blocks or draw pictures as needed.
Question 3.
176 magazines divided equally into 5 boxes
Answer:
35 with 1 leftover.

Explanation:
Given that 176 magazines were divided equally into 5 boxes which is 176 ÷ 5 = 35 with 1 leftover.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-67

Question 4.
56 marbles divided equally into 3 bags
Answer:
18 with 2 leftovers.

Explanation:
Given that 56 marbles were divided equally into 3 bags which is 56 ÷ 3 = 18 with 2 leftovers.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-68

Independent Practice

For 5-8, use place-value blocks or a drawing to divide. Record remainders.
Question 5.
71 ÷ _____ = ______ R2
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 58
Answer:
The equation will be 71 ÷ 3 = 23 with 2 leftovers.

Explanation:
In the above image, we can see that there are three circles with 23 in each circle and 2 ones that are leftover. So the equation will be 71 ÷ 3 = 23 with 2 leftovers.

Question 6.
_____ = 176 ÷ ______
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 59
Answer:
The equation will be 176 ÷ 4 = 44.

Explanation:
In the above image, we can see that there are three circles with 23 in each circle and 2 ones that are leftover. So the equation will be 176 ÷ 4 = 44.

Question 7.
46 ÷ 3
Answer:
The equation will be 46 ÷ 3 = 15 with 1 leftover.

Explanation:
On solving the equation we will get 46 ÷ 3 = 15 with 1 leftover.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-69

Question 8.
65 ÷ 4
Answer:
The equation will be 65 ÷ 4 = 16 with 1 leftover.

Explanation:
On solving the equation we will get 65 ÷ 4 = 16 with 1 leftover.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-70

Problem Solving

Question 9.
Model with Math A company with 65 employees is moving to a new location. All of the employees are divided into groups of 5 for the move. Write an equation and find g, the number of groups used for the move.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 60
Answer:
The equation will be 65 ÷ 5 = 13.

Explanation:
Given that a company with 65 employees is moving to a new location and all of the employees are divided into groups of 5 for the move. So the equation will be 65 ÷ 5 = g, and the value of g is 13.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-71

Question 10.
Maya used a drawing to divide 86. She made groups of 17 with 1 left over. Draw a picture to determine how many groups Maya made.
Answer:
Given that Maya used a drawing to divide 86 she made groups of 17 with 1 leftover, so the picture will be
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-72

Question 11.
Number Sense A science museum has 2,400 gemstones displayed equally in 3 cases. How many gemstones are in each case? What basic fact did you use to determine the quotient?
Answer:

Explanation:
Given that a science museum has 2,400 gemstones displayed equally in 3 cases, so the number of gemstones is in each case is 2400 ÷ 3 which can also be written as 24 ÷ 3 = 8. So the number of gemstones is in each case is 8 gemstones.

Question 12.
Mr. Harold has 268 books on 4 shelves in the classroom library. He has the same number of books on each shelf. To find the number of books on each shelf, he divided 268 by 4. How many books are on each shelf?
Answer:
The number of books on each shelf is 67 books.

Explanation:
Given that Mr. Harnold has 268 books on 4 shelves in the classroom library and he has the same number of books on each shelf, so the number of books on each shelf is 268 ÷ 4 which is 67 books are on each shelf.

Question 13.
Higher Order Thinking Five fourth grade classes from an elementary school took a trip to the United States Capitol. There were 25 students in each class. At the Capitol, a maximum of 40 students were allowed on a tour at one time. What is the least number of tours needed so all the students were able to take a tour?
Answer:
The least number of tours was needed so all the students were able to take a tour is 3 with 1 leftover.

Explanation:
Given that five fourth grade classes from an elementary school took a trip to the United States Capitol and there were 25 students in each class and at the Capitol, so the number of students will be 25 × 5 which is 25 students. And a maximum of 40 students was allowed on a tour at one time, so the least number of tours was needed so all the students were able to take a tour is 125 ÷ 40 which is 3 with 1 leftover.

Assessment Practice

Question 14.
Which division equation is represented by the drawing below?
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 61
A. 72 ÷ 6 = 12
B. 62 ÷ 3 = 24
C. 64 ÷ 3 = 24
D. 72 ÷ 3 = 24
Answer:
72 ÷ 3 = 24.

Explanation:
In the above image, we can see that three circles with 24 in each circle. So the equation will be 72 ÷ 3 = 24.

Question 15.
What is the missing divisor? 2,244 ÷ n = 374
A. 3
B. 4
C. 6
D. 7
Answer:
The missing divisor is 6.

Explanation:
The missing divisor for the equation 2,244 ÷ n =374 is 2,244 ÷ 374 which is 6.

Lesson 5.8 Continue Sharing to Divide

Solve & Share
Sara volunteers at a clothing recycling center. She packs T-shirts into bins for display. She packs the same number of T-shirts into 3 bins. How many T-shirts does Sara pack in each bin? She packs shorts into 2 bins with the same number in each bin. How many pairs of shorts does Sara pack in each bin?
I can … use place value and sharing to divide.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 62

Look Back! Explain how you can estimate answers to the problems above.

Essential Question
How Can You Record Division With a 1-Digit Divisor?

Visual Learning Bridge
Helen has 55 postcards. As an art project, she plans to glue the same number of postcards onto 4 poster boards. How many postcards can Helen put on each poster board?
You can use place-value blocks to solve the problem.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 63

Divide the tens.
Estimate: 55 ÷ 4 is close to 60 ÷ 4 = 15.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 64

Divide the ones.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 65
Helen can put 13 postcards on each of the poster boards. This quotient is reasonable since it is close to the estimate of 15.

Convince Me! Reasoning What does the remainder mean in the problem above?

Another Example!
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 66

Guided Practice

Do You Understand?
Question 1.
Explain how place-value blocks can help you with division.
Answer:
Here, place-value allows us to take the complication addition, subtraction, multiplication, and division problems and reduce and reduce them to a simpler problem.

Do You Know How?
For 2-3, estimate, and then find the quotient. Use place-value blocks or draw pictures as needed.
Question 2.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 67
Answer:
16 with 2 leftovers.

Explanation:
The quotient of 82 ÷ 5 is 16 with 2 leftovers.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-92

Question 3.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 68
Answer:
94 with 1 leftover.

Explanation:
The quotient of 659 ÷ 7 is 94 with 1 leftover.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-93

Independent Practice

For 4-11, find each quotient. Use place-value blocks or draw pictures as needed.
Question 4.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 69
Answer:
26.

Explanation:
The quotient of 78 ÷ 3 is 26.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-95

Question 5.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 70
Answer:
28 with 2 leftovers.

Explanation:
The quotient of 86 ÷ 3 is 28 with 2 leftovers.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-96

Question 6.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 71
Answer:
52 with 1 leftover.

Explanation:
The quotient of 417 ÷ 8 is 52 with 1 leftover.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-98

Question 7.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 72
Answer:
23 with 1 leftover.

Explanation:
The quotient of 93 ÷ 4 is 23 with 1 leftover.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-99

Question 8.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 73
Answer:
65 with 6 leftovers.

Explanation:
The quotient of 526 ÷ 8 is 65 with 6 leftovers.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-100

Question 9.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 74
Answer:
12 with 4 leftovers.

Explanation:
The quotient of 88 ÷ 7 is 12 with 4 leftovers.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-101

Question 10.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 75
Answer:
253 with 2 leftovers.

Explanation:
The quotient of 761 ÷ 3 is 253 with 2 leftovers.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-102

Question 11.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 76
Answer:
16.

Explanation:
The quotient of 96 ÷ 6 is 16.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-103

Problem Solving

Question 12.
Some of the tallest selenite crystals in a cave in Chihuahua, Mexico are 40 feet tall. Nathan is 4 feet tall. About how many times as tall as Nathan are the tallest crystals?
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 77
Answer:

Question 13.
Model with Math The Galveston-Port Bolivar Ferry takes cars across Galveston Bay. One day, the ferry transported a total of 350 cars over a 5-hour period. The ferry took the same number of cars each hour. How many cars did it take each hour? Complete the bar diagram to help.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 78
Answer:
The number of cars did it take each hour is 70 cars.

Explanation:
Given that the Galveston-Port Bolivar Ferry takes cars across Galveston Bay. One day, the ferry transported a total of 350 cars over a 5-hour period. So the number of cars did it take each hour is 350 ÷ 5 which is 70 cars.
Envision-Math-Common-Core-4th-Grade-Answers-Topic-5-Use-Strategies-and-Properties-to-Divide-by-1-Digit-Numbers-78

Question 14.
Zelda has a piece of fabric that is 74 inches long. She wants to divide it into 2 equal pieces. What is the length of each piece?
Answer:
The length of each piece is 37 inches long.

Explanation:
Given that Zelda has a piece of fabric that is 74 inches long and she wants to divide it into 2 equal pieces. So the length of each piece is 74 ÷ 2 which is 37 inches long.

Question 15.
Higher Order Thinking Maggie is making trail mix. She makes 4 batches of the recipe shown. She divides it into 3 equal-sized bags. How many ounces are in each bag?
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 79
Answer:
There will be 24 ounces in each bag.

Explanation:
Given that Maggie is making trail mix. So the total weight of trail mix is 8 + 5 + 2 + 3 which is 18 oz. As she makes 4 batches it will be 18 × 4 which is 72 and she divides it into 3 equal-sized bags, so 72 ÷ 3 which is 24 ounces in each bag.

Assessment Practice

Question 16.
Find the quotient.
Envision Math Common Core 4th Grade Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 80
A. 112
B. 114
C. 121
D. 122
Answer:
112.

Explanation:
The quotient of 784 ÷ 7 is 112.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-104

Question 17.
Find the quotient. 60 ÷ 5
A. 10
B. 12
C. 25
D. 55
Answer:
12.

Explanation:
The quotient of 60 ÷ 5 is 12.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-105

Lesson 5.9 Choose a Strategy to Divide

Solve & Share
Choose a strategy to solve each problem. Explain your solutions. Problem 1 There are 135 fourth-grade students. Each lunch table seats 6 students. How many tables are needed to seat all of the fourth graders? Problem 2 A high school football stadium has 5 sections. Each section seats the same number of people. A total of 1,950 people can be seated in the stadium. How many people can sit in each section?
I can … follow a series of steps that breaks the division into simpler calculations.
Envision Math Common Core Grade 4 Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 81

Look Back! Look for Relationships Are either of the problems above easily solved using mental math? Explain.

Essential Question
How Do You Choose a Strategy to Divide?

Visual Learning Bridge
What strategy should I use to solve the problems below?
Envision Math Common Core Grade 4 Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 82

How many packages of hot dogs were used on Saturday?
Think: I can use partia! quotients.
Envision Math Common Core Grade 4 Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 83
There were 105 packages of hot dogs used on Saturday.

On Sunday, the 3 food stands each sold the same number of hot dogs. How many hot dogs were sold at each stand?
Think: I can break 216 apart and divide with mental math.
216 ÷ 3 = (210 + 6) ÷ 3
= (210 ÷ 3) + (6 ÷ 3)
= 70 + 2
= 72
72 hot dogs were sold at each stand.
You can use the Distributive Property.

Convince Me! Reasoning How do you decide which is the best method to use?

Another Example!
Only 75 cookies were sold on Monday, Tuesday, and Thursday. The same number of cookies were sold each day. How many cookies were sold each day? 75 ÷ 3 = c
Envision Math Common Core Grade 4 Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 84

Guided Practice

Do You Understand?
Question 1.
Vickie’s estimated quotient was 80. The actual quotient she calculated was 48. Is her actual quotient reasonable? Explain.
Answer:

Do You Know How?
For 2 and 3, divide.
Question 2.
Envision Math Common Core Grade 4 Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 85
Answer:
The quotient is 319.

Explanation:
Given that 2,871 ÷ 9 which we will get 319 as quotient.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-106

Question 3.
Envision Math Common Core Grade 4 Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 86
Answer:
The quotient is 117.

Explanation:
Given that 468 ÷ 4 which we will get 117 as quotient.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-107

Independent Practice

For 4-11, divide.
You can use different strategies to divide.
Question 4.
Envision Math Common Core Grade 4 Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 87
Answer:
The quotient is 411.

Explanation:
Given that 3,288 ÷ 8 which we will get 411 as quotient.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-108

Question 5.
Envision Math Common Core Grade 4 Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 88
Answer:
The quotient is 49 with 2 leftovers.

Explanation:
Given that 247 ÷ 5 which we will get 49 as a quotient with 2 leftovers.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-109

Question 6.
Envision Math Common Core Grade 4 Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 89
Answer:
The quotient is 230.

Explanation:
Given that 1,380 ÷ 6 which we will get 230 as a quotient.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-110

Question 7.
Envision Math Common Core Grade 4 Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 90
Answer:
The quotient is 796.

Explanation:
Given that 3,980 ÷ 5 which we will get 796 as a quotient.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-111

Question 8.
Envision Math Common Core Grade 4 Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 91
Answer:
The quotient is 61 as a quotient with 1 leftover.

Explanation:
Given that 367 ÷ 6 which we will get 61 as a quotient with 1 leftover.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-112

Question 9.
Envision Math Common Core Grade 4 Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 92
Answer:
The quotient is 1194 as a quotient.

Explanation:
Given that 3,582 ÷ 3 which we will get 1194 as a quotient.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-113

Question 10.
Envision Math Common Core Grade 4 Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 93
Answer:
The quotient is 189 as a quotient.

Explanation:
Given that 756 ÷ 4 which we will get 189 as a quotient.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-114

Question 11.
Envision Math Common Core Grade 4 Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 94
Answer:
The quotient is 166 as a quotient with 3 leftovers.

Explanation:
Given that 999 ÷ 6 which we will get 166 as a quotient with 3 leftovers.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-115

Problem Solving

Question 12.
A family of four drove from San Francisco to New York. They drove the same number of miles each day for 6 days. How many miles did they drive each day? How can you interpret the remainder?
Envision Math Common Core Grade 4 Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 95
Answer:
The number of miles did they drive each day is about 484 miles and extra 2 miles.

Explanation:
Given that a family of four drove from San Francisco to New York and then they drove the same number of miles each day for 6 days. So the number of miles did they drive each day is 2906 ÷ 6 which is about 484 miles and on 1 day they drove an extra 2 miles.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-116

Question 13.
Without dividing, how can you tell if the quotient for 5,873 ÷ 8 is greater than 700? Explain whether the quotient is less than 800.
Answer:
Yes, the quotient is less than 800.

Explanation:
The quotient for 5,873 ÷ 8 is 734, and here the quotient is less than 800.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-117

Question 14.
Reasoning A square dance set is made up of 4 couples (8 dancers). There are 150 people at a square dance. What is the greatest number of sets possible at the dance? Describe the steps you would take to solve this problem.
Answer:
The greatest number of sets possible at the dance is 18 with 6 leftovers.

Explanation:
Given that a square dance set is made up of 4 couples (8 dancers) and there are 150 people at a square dance. So the greatest number of sets possible at the dance is 150 ÷ 8 which is 18 with 6 leftovers.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-118

Question 15.
Higher Order Thinking Is rounding or using compatible numbers a more useful method when finding quotients in division problems? Explain.
Answer:
Compatible number is better because they always give me groups that we can divide by. For example, if we take 151 ÷ 3 we will round off 151 to 150 and then we will perform division.

Question 16.
Ron’s Tires has 1,767 tires for heavy-duty trucks. Each heavy-duty truck needs 6 tires. How many heavy-duty trucks can get all new tires at Ron’s?
Answer:
The number of heavy-duty trucks can get all new tires at Ron is 294 new tries.

Explanation:
Given that Ron’s Tires has 1,767 tires for heavy-duty trucks and each heavy-duty truck needs 6 tires. So the number of heavy-duty trucks can get all new tires at Ron is 1,767 ÷ 6 which is 294 new tires.
Envision Math Common Core 4th Grade Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 1-119

Assessment Practice

Question 17.
Select all correct equations.
☐ 5658 ÷ 70 R5
☐ 3,613 ÷ 6 = 600 R13
☐ 3,288 ÷ 4 = 822
☐ 218 ÷ 3 = 72 R2
☐ 6,379 ÷ 7 = 911
Answer:
The correct equations are 3,288 ÷ 4 = 822 and 218 ÷ 3 = 72 R2.

Question 18.
Find 6,357 ÷ 8.
A. 814 R1
B. 794 R5
C. 794 R1
D. 784
Answer:
794 R5

Explanation:
The quotient of 6,357 ÷ 8 is 794 with 5 as remainder.

Lesson 5.10 Problem Solving

Model with Math
Solve & Share
Allen set a goal to do at least 120 minutes of outdoor activities a day, Monday through Friday. He made a list of each activity and the amount of time he spends doing it every week. The same amount of time is spent every day doing the activities. Is Allen spending enough time each day on outdoor activities to meet his goal? What math can you use to solve this problem?
I can … use a drawing, diagram, or table to model a problem.
Envision Math Common Core Grade 4 Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 96

Thinking Habits
Be a good thinker! These questions can help you.

  • How can I use math I know to help solve this problem?
  • Can I use pictures, objects, or an equation to represent this problem?
  • How can I use numbers, words, and symbols to solve the problem?

Look Back! Model with Math What hidden question do you have to answer before you can determine if Allen met his goal? What math can you use to find the answer to this hidden question and the original question?

Essential Question
How Can You Apply Math You Know to Solve Problems?

Visual Learning Bridge
A class is making decorations using same-size straws. They use the straws to make triangles, squares, pentagons, and hexagons. One package of straws is used for each group of polygons with the same number of sides. How many decorations can the class make?
Envision Math Common Core Grade 4 Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 97
What math can you use to solve the problem? I need to divide to find how many decorations can be made from one box of straws.

How can I model with math?
I can

  • use previously learned concepts and skills. find and answer any hidden questions.
  • decide if my results make sense.

Here’s my thinking.

Each polygon has a different number of sides.
I will divide 1,500 straws by the number of sides for each polygon:
1,500 ÷ 3 = 500 triangles
1,500 ÷ 4 = 375 squares
1,500 ÷ 5 = 300 pentagons
1,500 ÷ 6 = 250 hexagons
I will add all the decorations together:
500 + 375 + 300 + 250 = 1,425
The class can make 1,425 decorations.

Convince Me! Reasoning Another class made 200 octagon-shaped decorations. How many straws did they use?

Guided Practice

Model with Math Miguel is going camping with 3 friends. He packed sandwiches for everyone to share equally. How many sandwiches did Miguel pack for each camper?
Envision Math Common Core Grade 4 Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 98
Question 1.
What hidden question do you need to solve first? Write and solve an equation to find the answer. Tell what your variable represents.
Answer:

Question 2.
Complete the bar diagram. Write and solve an equation to find the number of sandwiches, s, for each camper.
You can use a bar diagram and write an equation to model with math.
Answer:

Independent Practice

Model with Math
Jodi delivers 54 newspapers on Saturday and 78 newspapers on Sunday. She makes bundles of 6 newspapers. How many bundles does Jodi make on Saturday and Sunday combined?
Question 3.
Explain how you could use a picture to represent the problem and show the relationships. Define variables.
Answer:

Question 4.
Write and solve equations to represent the problem. Explain how you can check that your solution is reasonable.
Answer:

Problem Solving

Assessment Practice
Dog Grooming
Patricia and Antonio own a dog grooming business. To attract new customers, they offered free dog baths with the purchase of a grooming service. During the first 6 days of the promotion, they bathed 26 beagles, 12 boxers, 17 pugs, and 5 golden retrievers. Patricia and Antonio each bathed the same number of dogs each day.
Envision Math Common Core Grade 4 Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 99
Question 5.
Reasoning What are the quantities given in the problem?
Answer:

Question 6.
Make Sense and Persevere What do you need to know to determine how many dogs Patricia bathed each day?
Answer:

Question 7.
Model with Math Draw a bar diagram. Write and solve an equation to find d, how many dogs were bathed in all.
You model with math when you use a picture or equation to represent the problem.
Answer:

Question 8.
Make Sense and Persevere Find how many dogs Patricia bathed each day. Explain how you were able to find the solution.
Answer:

Topic 5 Fluency Practice Activity

Follow the path
Shade a path from START to FINISH. Follow the sums and differences that are between 1,000 and 1,200. You can only move up, down, right, or left.
I can … add and subtract whole numbers with regrouping.
Envision Math Common Core Grade 4 Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 100

Topic 5 Vocabulary Review

Understand Vocabulary
Choose the best term from the box. Write it on the blank.
Word List

  • dividend
  • division
  • divisor
  • equation
  • partial quotients
  • quotient
  • remainder

Question 1.
The answer to a division problem is called the ________
Answer:
The answer to a division problem is called the quotients.

Question 2.
The number to be divided in a division problem is called the ___________
Answer:
The number to be divided in a division problem is called the dividend.

Question 3.
A way to divide that finds quotients in parts until only a remainder, if any, is left is using __________
Answer:
A way to divide that finds quotients in parts until only a remainder, if any, is left is using

Question 4.
The number by which another number is divided is called the ___________
Answer:

Question 5.
The operation that tells how many equal groups there are or how many are in each group is called __________
Answer:

For each of these terms, give an example and a non-example.
Envision Math Common Core Grade 4 Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 101
Answer:

Use Vocabulary in Writing
Question 8.
Megan made 21 loom bracelets to share equally among her 7 friends. How many bracelets will each friend receive? Write and solve an equation. Use at least 3 terms from the Word List to describe your equation.
Answer:

Topic 5 Reteaching

Set A pages 169-172

A school district shares 2,700 chairs equally among 3 school buildings. How many chairs will each school building have?
Envision Math Common Core Grade 4 Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 102
Find 2,700 ÷ 3 = c.
The basic fact is 27 ÷ 3 = 9.
27 hundreds = 3 = 9 hundreds, or 900.
2,700 ÷ 3 = 900
Each school building will have 900 chairs.

Remember you can use basic division facts and place value to divide mentally.

Question 1.
250 ÷ 5
Answer:

Question 2.
810 ÷ 9
Answer:

Question 3.
3,200 ÷ 4
Answer:

Question 4.
4,200 ÷ 7
Answer:

Question 5.
1,000 ÷ 2
Answer:

Question 6.
240 ÷ 4.
Answer:

Question 7.
450 ÷ 5
Answer:

Question 8.
720 ÷ 9
Answer:

Question 9.
3,600 ÷ 4
Answer:

Question 10.
4,900 ÷ 7
Answer:

Question 11.
2,000 ÷ 2
Answer:

Question 12.
280 ÷ 4
Answer:

Question 13.
2,100 ÷ 7
Answer:

Question 14.
560 ÷ 8
Answer:

Set B pages 173-180

Use multiplication to estimate 420 ÷ 8.
8 times what number is about 420?
8 × 5 = 40,
So, 8 × 50 = 400.
So, 420 ÷ 8 is about 50.
Use compatible numbers to estimate 1,519 ÷ 7.
What number close to 1,519 is easily divided by 7?
Try division facts to help find compatible numbers for 1,519
1,519 is close to 1,400.
14 ÷ 7 = 2,
so 1,400 ÷ 7 = 200.
So, 1,519 ÷ 7 is about 200.

Remember basic facts can help you find a number that is easily divided by the divisor.

Estimate each quotient.

Question 1.
718 ÷ 8
Answer:

Question 2.
156 ÷ 4
Answer:

Question 3.
482 ÷ 8
Answer:

Question 4.
174 ÷ 3
Answer:

Question 5.
843 ÷ 7
Answer:

Question 6.
321 ÷ 2
Answer:

Question 7.
428 ÷ 6
Answer:

Question 8.
811 ÷ 9
Answer:

Question 9.
5,616 ÷ 8
Answer:

Question 10.
7,224 ÷ 8
Answer:

Question 11.
6,324 ÷ 9
Answer:

Question 12.
3,627 ÷ 9
Answer:

Question 13.
331 ÷ 4
Answer:

Question 14.
1,222 ÷ 6
Answer:

Question 15.
2,511 ÷ 5
Answer:

Question 16.
362 ÷ 6
Answer:

Question 17.
4,940 ÷ 7
Answer:

Question 18.
9,312 ÷ 3
Answer:

Set C pages 181-184

Tom is putting 14 apples into bags. Each bag holds 4 apples. How many bags can Tom fill? Will any apples be left over?
Envision Math Common Core Grade 4 Answer Key Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 103
Use a model to represent 14 ÷ 4.
Tom can fill 3 bags. There will be 2 apples left over.

Remember to make sure the remainder is less than the divisor.
Question 1.
22 pickles
3 pickles on each plate
22 ÷ 3 = _____ with ______ left over
How many plates have 3 pickles?
Answer:

Question 2.
19 stamps
2 stamps on each envelope
19 ÷ 2 = _____ with _____ left over
How many stamps are not on an envelope?
Answer:

Set D pages 185-192

Find 357 ÷ 7.
Use a model. Divide by finding partial quotients.
Envision Math Common Core Grade 4 Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 104
Divide. Use the Distributive Property.
357 ÷ 7 = (350 + 7) ÷ 7
= (350 ÷ 7) + (7 ÷ 7)
= 50 + 1
= 51

Remember to add the partial quotients to find the actual quotient.

Use partial quotients to solve.
Question 1.
There are 81 chairs in 3 equal groups. How many chairs are in each group?
Answer:

Question 2.
There are 174 games scheduled for 6 different leagues. Each league has the same number of games scheduled. How many games does each league have scheduled?
Answer:

Question 3.
There were 1,278 people at the last basketball game. The stands were divided into 6 sections. The same number of people sat in each section. How many people sat in each section?
Answer:

Set E pages 193–196

Margaret packed 68 books equally into 2 boxes. How many books did Margaret pack in each box?
Find 68 ÷ 2.
Envision Math Common Core Grade 4 Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 105
Divide the tens into two equal groups. Then divide the ones into two equal groups.
Envision Math Common Core Grade 4 Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 106
68 ÷ 2 = 34, because 2 × 34 = 68.
Margaret packed 34 books in each box.

Remember to check if your answer is reasonable.

Tell how many are in each group and how many are left over.
Question 1.
138 books; 5 stacks
Answer:

Question 2.
55 shells; 3 jars
Answer:

Question 3.
217 pens; 7 cases
Answer:

Question 4.
154 shoes; 4 boxes
Answer:

Question 5.
195 seeds; 6 planters
Answer:

Set F pages 197-200

Find 147 ÷ 6.
Estimate 120 ÷ 6 = 20.
Envision Math Common Core Grade 4 Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 107
147 ÷ 6 = 24 R3 is close to 20, so the answer is reasonable.

Remember to estimate the quotient to check if your answer is reasonable.

Question 1.
710 ÷ 9
Answer:

Question 2.
657 ÷ 5
Answer:

Question 3.
398 ÷ 8
Answer:

Question 4.
429 ÷ 2
Answer:

Question 5.
470 ÷ 6
Answer:

Question 6.
255 ÷ 4
Answer:

Set G pages 201-204

Find 8,951 ÷ 8.
Estimate: 8,800 ÷ 8 = 1,100.
One strategy is partial quotients.
Envision Math Common Core Grade 4 Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 108

Remember you can use your estimate to check if your answer is reasonable.

Choose a strategy to divide.
Question 1.
4,649 ÷ 4
Answer:

Question 2.
2,843 ÷ 3
Answer:

Question 3.
8,478 ÷ 6
Answer:

Question 4.
6,399 ÷ 9
Answer:

Question 5.
379 ÷ 2
Answer:

Question 6.
3,812 ÷ 5
Answer:

Question 7.
4,793 ÷ 5
Answer:

Question 8.
5,957 ÷ 7
Answer:

Set H pages 205-208

Think about these questions to help you model with math.
Thinking Habits

  • How can I use math I know to help solve this problem?
  • Can I use pictures, objects, or an equation to represent the problem?
  • How can I use numbers, words, and symbols to solve the problem?
    Envision Math Common Core Grade 4 Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 109

Remember a bar diagram can help you write an equation.

A paint supplier delivered 1,345 cans of paint to 5 different hardware stores. Each store got the same number of paint cans. How many cans of paint were delivered to each store?
Question 1.
How can you use pictures, a bar diagram, and an equation to find the number of paint cans delivered to each store?
Answer:

Question 2.
How can you decide if your answer makes sense?
Answer:

Topic 5 Assessment Practice

Question 1.
Select all the equations that are reasonable estimates for the quotient 184 ÷ 8.
☐ 160 ÷ 8 = 20
☐ 200 ÷ 5 = 40
☐ 180 ÷ 9 = 20
☐ 150 ÷ 5 = 30
☐ 180 ÷ 6 = 30
Answer:

Question 2.
Draw a bar diagram for the equation, and then solve. 2,400 ÷ 6 = m
Answer:

Question 3.
Mrs. Bollis has two pieces of fabric to make costumes. One piece is 11 yards long and the other is 15 yards long. Each costume requires 3 yards of fabric. How many costumes can Mrs. Bollis make? How do the remainders affect the number of costumes she can make?
Answer:

Question 4.
A. Write an equation to show how to divide 453 into 3 equal groups.
Answer:

B. Complete the model to solve the equation in A.
Envision Math Common Core Grade 4 Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 110
Answer:

Question 5.
What is the best estimate for 3,350 ÷ 8?
A. 600
B. 200
C. 400
D. 800
Answer:

Question 6.
Draw an array and solve the equation. 48 ÷ 9 = ?
Answer:

Question 7.
Use compatible numbers to estimate the quotient 530 ÷ 9. Then find the exact answer.
Answer:

Question 8.
Find 4,800 ÷ 6 using a place-value strategy. What basic fact did you use?
Answer:

Question 9.
Select all the equations in which the remainder is 5.
☐ 59 ÷ 9 = 6 R?
☐ 352 ÷ 6 = 58 R?
☐ 788 ÷ 9 = 87 R?
☐ 1,486 ÷ 7 = 212 R?
☐ 2,957 ÷ 8 = 369 R?
Answer:

Question 10.
Which of the following expressions does NOT have a remainder of 3?
A. 52 ÷ 7
B. 123 ÷ 7
C. 451 ÷ 7
D. 794 ÷ 7
Answer:

Question 11.
Which of the following is NOT equivalent to 63 ÷ 3?
A. (60 + 3) ÷ 3
B. (33 + 30) ÷ 3
C. (60 ÷ 3) + 3
D. (60 ÷ 3) + (3 ÷ 3)
Answer:

Question 12.
Select all the quotients that are reasonable estimates for 472 ÷ 6.
☐ 450 ÷ 5
☐ 480 ÷ 6
☐ 500 ÷ 5
☐ 450 ÷ 3
☐ 1,200 ÷ 6 = 200
Answer:

Question 13.
Use partial quotients to find the quotient. Choose numbers from the box to complete the calculations. Use each number once.
Envision Math Common Core Grade 4 Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 111
Answer:

Question 14.
Find 1,600 ÷ 8. What basic fact did you use?
Answer:

Question 15.
The fourth graders are going to the science museum.
Envision Math Common Core Grade 4 Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 112
Groups of 8 students can see a special exhibit on space travel. How many groups will be needed so everyone can see the exhibit?
Answer:

Question 16.
A. Divide.
432 ÷ 8 = _________
Answer:

B. How can you use the answer from Part A to find 4,320 ÷ 8?
Answer:

Question 17.
The Pizza Stand gives patrons a free pizza when they collect 8 coupons. How many free pizzas can Mrs. Fowler get if she has 78 coupons? How does the remainder affect the number of free pizza’s she gets? How many more coupons does she need for the next free pizza? Explain.
Answer:

Question 18.
Estimate 257 ÷ 5. Explain how you can use multiplication to estimate the quotient.
Answer:

Question 19.
Use an equation to show how to separate 128 into 4 equal groups. Explain how to check the answer using multiplication.
Answer:

Question 20.
Write and solve an equation that shows one way to estimate 1,792 ÷ 6.
Answer:

Question 21.
Draw a picture to explain why 675 ÷ 5 = 131 R2.
Answer:

Question 22.
For each equation, mark a check to show the correct missing number.
Envision Math Common Core Grade 4 Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 113
Answer:

Question 23.
Holly uses 7 sheets of tissue paper to make one flower. If she bought a package with 500 sheets of tissue paper, about how many flowers will Holly be able to make? Use compatible numbers to estimate the number of flowers.
Answer:

Topic 5 Performance Task

Saving What You Earn
Trista’s older brother Ryan got a job. Ryan would like to buy the items shown with his earnings. Ryan earns $8 for each hour he works.
Question 1.
Ryan is curious and wants to know the amount of time it will take him to earn enough money to buy the items shown.
Part A
How many hours does Ryan need to work to earn enough money to buy the computer? Explain how to use place value and mental math to solve.
Envision Math Common Core Grade 4 Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 114
Answer:

Part B
How many hours does Ryan need to work to buy the video game? Use a model to show how to find the quotient. Explain how to interpret the remainder.
Answer:

Part C
How many hours will Ryan need to work to earn enough money to buy the smart phone? Use partial quotients to divide.
Answer:

Question 2.
Ryan gets a raise. He now earns $9 an hour. Ryan decides to start saving for a car. He works 9 hours a week.
Envision Math Common Core Grade 4 Answers Topic 5 Use Strategies and Properties to Divide by 1-Digit Numbers 115
Part A
How many hours does Ryan need to work to earn enough money to buy the car, as well as pay for the taxes, title, and plates as shown? Draw bar diagrams to help write and solve equations.
Answer:
Ryan has to work 5,887hrs and 55 min.

Explanation:
Ryan earns $9 an hr. He works 9hrs a week, or 1hr 17 min a day. $11.52 per day. The car costs $52,753, + $235(taxes, titles, and plates)= $52,988. 52,988/9=5,887:55

Part B
About how many weeks will Ryan need to work to buy the car and pay for the taxes, title, and plates? Explain.
Answer:
3,504 weeks

Explanation:
One week= 168 hrs.
5,887:55/168=3,504 weeks

Part C
How many actual weeks does Ryan need to work to buy the car and pay for the taxes, title, and plates? Show your work. Explain why your solution is reasonable.
Answer:

Envision Math Common Core Grade 7 Answer Key Topic 3 Analyze And Solve Percent Problems

enVision Math Common Core Grade 7 Answer Key Topic 3 Analyze And Solve Percent Problems

Go through the enVision Math Common Core Grade 7 Answer Key Topic 3 Analyze And Solve Percent Problems regularly and improve your accuracy in solving questions.

enVision Math Common Core 7th Grade Answers Key Topic 3 Analyze And Solve Percent Problems

Topic 3 Essential Question

How can percents show proportional relationships between quantities and be used to solve problems?

3-ACT MATH

Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 1
The Smart Shopper
Why do stores and manufacturers print coupons? It seems like they lose money every time you use one. Well, some coupons are designed to steer you toward a specific brand and gain your loyalty. Stores also offer coupons to get you into the store, counting on you buying other items while you are there. If you’re clever, you can use multiple coupons. Think about this during the 3-Act Mathematical Modeling lesson.3
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 2

Topic 3 enVision STEM Project

Did You Know?
One of the first popular activity trackers was a pedometer, which measures number of steps taken. Some sources trace the history of the pedometer back to Leonardo da Vinci.
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 3
A pedometer called Manpo-kei (10,000 steps meter) was introduced in Japan in the mid1960s. Research led by Dr. Yoshiro Hatano indicated that 10,000 steps a day is the ideal energy output to maintain health.
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 4
Today, most activity trackers are electronic devices that can sync to a computer or a smartphone.

Many activity trackers are wearable technology. Some common places to wear activity trackers are on the wrist, arm, or chest. There are even collar-mounted activity trackers for dogs.
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 5
Your Task: Analyze Activity Tracker Data
Activity trackers, also called fitness trackers, have become quite popular in recent years. But is the data collected actually helpful to the user? You and your classmates will explore the types of data that an activity tracker collects, and how that data can help users reach their activity and fitness goals.
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 6

Topic 3 GET READY!

Review What You Know!

Vocabulary
Choose the best term from the box. Write it on the blank.

  • percent
  • proportion
  • rate
  • ratio

Question 1.
A ________is a ratio in which the first term is compared to 100.

Answer:
A percent is a ratio in which the first term is compared to 100.

Explanation:
In the above-given question,
given that,
A percent is a ratio in which the first term is compared to 100.
for example:
38/100 = x/100.
x = 38%.
38/100 = 38.

Analyze And Solve Percent Problems 1

Question 2.
A ratio that relates two quantities with different units of measure is a _________.

Answer:
A ratio that relates two quantities with different units of measure is a rate.

Explanation:
In the above-given question,
given that,
A ratio that relates two quantities with different units of measure is a rate.
for example:
34mi/2 gal.
we can convert any rate to a unit rate.

Question 3.
A statement that two ratios are equal is called a _________.

Answer:
A statement that two ratios are equal is called a proportion.

Explanation:
In the above-given question,
given that,
A statement that two ratios are equal is called a proportion.
for example:
5/10 = 1/2.

Question 4.
The relationship “3 students out of 5 students” is an example of a _________.

Answer:
The relationship “3 students out of 5 students” is an example of a ratio.

Explanation:
In the above-given question,
given that,
The relationship “3 students out of 5 students” is an example of a ratio.

Fractions, Decimals, and Percents

Write each number in two equivalent forms as a fraction, decimal, or percent.
Question 5.
0.29

Answer:
The number infraction is 29/100.
decimal = 0.29.
percent = 29%.

Explanation:
In the above-given question,
given that,
the number is 0.29.
decimal = 0.29.
percent = 29%.
fraction = 29/100.

Question 6.
35%

Answer:
The number of fraction is 35/100.
decimal = 0.35.
percent = 35%.

Explanation:
In the above-given question,
given that,
the number is 35%.
decimal = 0.35.
percent = 35%.
fraction = 35/100.

Question 7.
\(\frac{2}{5}\)

Answer:
The number infraction is 2/5.
decimal = 40.0..
percent = 0.4%.

Explanation:
In the above-given question,
given that,
the number is 2/5.
decimal = 40.0.
percent = 0.4%.
fraction = 2/5.

Analyze And Solve Percent Problems 2

Proportions

Find the unknown number in each proportion.
Question 8.
\(\frac{x \text { days }}{4 \text { years }}=\frac{365.25 \text { days }}{1 \text { year }}\)

Answer:
x = 1461.

Explanation:
In the above-given question,
given that,
x/4 = 365.25.
x = 365.25 x 4.
x = 1461.

Question 9.
\(\frac{33,264 \text { feet }}{x \text { miles }}=\frac{5,280 \text { feet }}{1 \text { mile }}\)

Answer:
x = 1.569.

Explanation:
In the above-given question,
given that,
3364/x = 5280/1.
x = 5280/3364.
x = 1.569.

Question 10.
A cooking magazine shows a photo of a main dish on the front cover of 5 out of the 12 issues it publishes each year. Write and solve a proportion to determine how many times a photo of a main dish will be on the front cover during the next 5 years.

Answer:
The number of times a photo of a main dish will be on the front cover during the next 5 years = 3 times.

Explanation:
In the above-given question,
given that,
A cooking magazine shows a photo of a main dish on the front cover of 5 out of the 12 issues it publishes each year.
5/100 x 12.
0.05 x 12.
0.6.
0.6 x 5 = 3.
so the number of times a photo of the main dish will be on the front cover during the next 5 years = 3 times.

Language Development

Fill in the spider map by writing new vocabulary terms on each diagonal and related ideas.
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 7

Answer:
25/100 = 0.25.
50/100 = 0.5.
75/100 = 0.75.
100/100 = 1.

Explanation:
In the above-given question,
given that,
the percent of all the terms in problem.
the four terms are 25, 50, 75, and 100.
25/100 = 0.25.
50/100 = 0.5.
75/100 = 0.75.
100/100 = 1.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-3-Analyze And Solve Percent Problems-1

Pick A Project

PROJECT 3A
How can you make sure you are getting the best deal?
PROJECT: SEARCH FOR BARGAINS
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 8

Answer:
Yes, I am getting the best deal.

Explanation:
In the above-given question,
given that,
they kept the discount of 50% on summer sale.
50/100 = 0.5.
so  I am getting the best deal.

PROJECT 3B
What makes a song’s lyrics easy to remember?
PROJECT: WRITE AND PERFORM A CHEER OR A RAP
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 9

PROJECT 3C
If you were to play basketball, what strategies would you practice?
PROJECT: ESTIMATE BASKETS
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 10

PROJECT 3D
What are some elements of your favorite app?
PROJECT: DESIGN AN APP ICON
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 11

Lesson 3.1 Analyze Percents of Numbers

Solve & Discuss It!
Jaime’s older brother and his three friends want to split the cost of lunch. They also want to leave a 15%-20% tip. How much should each person pay?
I can… understand, find, and analyze percents of numbers.
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 12
Reasoning Which line on the receipt will you use to calculate the tip?

Answer:
The amount should each person pay = 30%

Explanation:
In the above-given question,
given that,
Jaime’s older brother and his three friends want to split the cost of lunch.
They also want to leave a 15%-20% tip.
15/100 – 20/100.
50/100 = 0.5.
20/100 = 0.2.
0.5 – 0.2 = 0.3.
30/100.
so the amount should each person pay = 30%.

Focus on math practices
Reasoning How would the amount each person pays change if the tip is determined before or after the bill is split?

Essential Question
How do percents show the relationship between quantities?

Try It!

Kita’s phone had a fully charged battery. With normal usage, her phone will last 18 hours. How much time is left on Kita’s phone battery with 12% charge remaining?
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 13
Kita’s phone battery has __________ hours remaining.

Answer:
The time left on Kita’s phone battery with 12% charge remaining = 3 hours.

Explanation:
In the above-given question,
given that,
Kita’s phone had a fully charged battery.
With normal usage, her phone will last 18 hours.
21/100 = x/18.
0.21 = x/18.
x/18 x 0.21.
x = 3.78.
so the time left on Kita’s phone battery with 12% charge remaining = 3 hours.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-3-Analyze And Solve Percent Problems-2
Convince Me! Why is 51% of a number more than half of the number?

Analyze And Solve Percent Problems 3

Try It!

a. Find 0.08% of 720.

Answer:
0.576.

Explanation:
In the above-given question,
given that,
0.08% of 720.
0.08/100 x 720.
0.0008 x 720 = 0.576.

b. Find 162.5% of 200.

Answer:
162.5.

Explanation:
In the above-given question,
given that,
162.5% of 200.
162.5/100 x 200.
1.625 x 100 = 162.5.

c. Find 0.3% of 60.

Answer:
0.18.

Explanation:
In the above-given question,
given that,
0.3% of 60.
0.3/100 x 60.
0.003 x 60 = 0.18.

KEY CONCEPT
A percent is one way to represent the relationship between two quantities, generally that of a part to the whole.
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 14

Do You Understand?
Question 1.
Essential Question How do percents show the relationship between quantities?

Answer:
x = 9.

Explanation:
In the above-given question,
given that,
60/100 = x/15.
0.6 = x/15.
0.6 x 15 = x.
x = 9.

Question 2.
Reasoning How does a value that is greater than 100% of the original value or less than 1% of the original value compare to the original value?.

Answer:
The original value is greater than 1%.

Explanation:
In the above-given question,
given that,
the original value is 9%.
9/100 = 0.09.
1/100 = 0.01.
0.01 is greater than 0.09.
so the original value is greater than 1%.

Question 3.
Construct Arguments Gene stated that finding 25% of a number is the same as dividing the number by \(\frac{1}{4}\). Is Gene correct? Explain.

Answer:
Yes, Gene was correct.

Explanation:
In the above-given question,
given that,
Gene stated that finding 25% of a number is the same as dividing the number by \(\frac{1}{4}\).
25/100 = 0.25.
1/4 of 100.
so Gene was correct.

Do You Know How?
Question 4.
An 8-ounce serving of apples contains 8% of your daily vitamin C. How many ounces of apples would you need to get 100% of your daily vitamin C?

Answer:
The number of ounces of apples would you need to get 100% of your daily vitamin C = 100-ounce.

Explanation:
In the above-given question,
given that,
An 8-ounce serving of apples contains 8% of your daily vitamin C.
for 8-ounce it will get 8%.
for 100 ounces it will get 100%.
so 100 ounces of apples would you need to get 100% of your daily vitamin C.

Question 5.
Find the percent of each number.
a. 59% of 640

Answer:
377.6.

Explanation:
In the above-given question,
given that,
59% of 640.
59/100 x 640.
0.59 x 640.
377.6.

b. 0.20% of 3,542

Answer:
7.084.

Explanation:
In the above-given question,
given that,
0.20% of 3542.
0.20/100 x 3542.
0.002 x 3542.
7.084.

c. 195% of 568

Answer:
1107.6.

Explanation:
In the above-given question,
given that,
195% of 568.
195/100 x 568.
1.95 x 568.
1107.6.

d. 74% of 920

Answer:
680.8.

Explanation:
In the above-given question,
given that,
74% of 920.
74/100 x 920.
0.74 x 920.
680.8.

Question 6.
Water is 2 parts hydrogen and 1 part oxygen (H2O). For one molecule of water, each atom has the atomic mass unit, u, shown. What percent of the mass of a water molecule is hydrogen?
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 15

Answer:
The mass of a water molecule in hydrogen = 2.02.

Explanation:
In the above-given question,
given that,
Water is 2 parts hydrogen and 1 part oxygen (H2O).
For one molecule of water, each atom has an atomic mass unit.
the mass of hydrogen is 1.01.
1.01 x 2 = 2.02.
so the mass of a water molecule in hydrogen is 2.02.

Practice & Problem Solving

Leveled Practice In 7-8, fill in the boxes to solve.
Question 7.
A local Little League has a total of 60 players, 80% of whom are right-handed. How many right-handed players are there?
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 16

Answer:
The number of right-handed players is there = 48.

Explanation:
In the above-given question,
given that,
A local Little League has a total of 60 players,
80% of whom are right-handed.
80/100 x 60.
0.8 x 60.
48.
so the number of right-handed players is there = 48.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-3-Analyze And Solve Percent Problems-3

Question 8.
Sandra’s volleyball team has a total of 20 uniforms. 20% are medium-sized uniforms. How many uniforms are medium-sized?
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 17

Answer:
The number of uniforms is medium-sized = 4.

Explanation:
In the above-given question,
given that,
Sandra’s volleyball team has a total of 20 uniforms.
20% are medium-sized uniforms.
20/100 = 0.20.
0.20 x 20 = 4.
so the number of uniforms are medium-sized = 4.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-3-Analyze And Solve Percent Problems-4

Question 9.
Meg is a veterinarian. In a given week, 50% of the 16 dogs she saw were Boxers. Steve is also a veterinarian. In the same week, 7 of the 35 dogs he saw this week were Boxers. Each wants to record the part, the whole, and the percent.
a. Does Meg need to find the part, the whole, or the percent?

Answer:
Meg needs to find the whole = 8.

Explanation:
In the above-given question,
given that,
Meg is a veterinarian. In a given week, 50% of the 16 dogs she saw were boxers.
Steve is also a veterinarian.
In the same week, 7 of the 35 dogs he saw this week were Boxers.
50/100 x 16.
0.5 x 16 = 8.
7/35 x 100.
0.2 x 100 = 20.

b. Does Steve need to find the part, the whole, or the percent?

Answer:
Steve needs to find the part.

Explanation:
In the above-given question,
given that,
Steve is also a veterinarian.
In the same week, 7 of the 35 dogs he saw this week were Boxers.
50/100 x 16.
0.5 x 16 = 8.
7/35 x 100.
0.2 x 100 = 20.
so steve needs to find the part.

Question 10.
Olivia is a stockbroker. She makes 4% of her sales in commission. Last week, she sold $7,200 worth of stocks.
a. How much commission did she make last week?

Answer:
The much commission did she make last week = $288.

Explanation:
In the above-given question,
given that,
Olivia is a stockbroker.
She makes 4% of her sales in commission.
Last week, she sold $7,200 worth of stocks.
4/100 x 7200.
0.04 x 7200 = 288.
so the much commission did she make last week = $288.

b. If she were to average that same commission each week, how much would she make in commissions in a year, treating a year as having exactly 52 weeks?

Answer:
The much would she make in commissions in a year, treating a year as having exactly 52 weeks = $3744.

Explanation:
In the above-given question,
given that,
If she were to average that same commission each week.
52/100 = 0.52.
0.52 x 7200 = 3744.
so the much she make in commission is $3744.

Question 11.
The registration fee for a used car is 0.8% of the sale price of $5,700. How much is the fee?

Answer:
The fee is $45.6.

Explanation:
In the above-given question,
given that,
The registration fee for a used car is 0.8% of the sale price of $5,700.
0.8/100 x 5700.
0.008 x 5700.
45.6.
so the fee is $45.6.

Question 12.
The total cost of an item is the price plus the sales tax.
Find the sales tax to complete the table. Then find the total cost of the item.
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 18

Answer:
The total cost of the item = $1.6.

Explanation:
In the above-given question,
given that,
The total cost of an item is the price plus the sales tax.
4/100 x 40.
0.04 x 40.
1.6.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-3-Analyze And Solve Percent Problems-5

Question 13.
Is 700% of 5 less than 10, greater than 10 but less than 100, or greater than 100? Explain your reasoning.

Answer:
35 is greater than 10 but less than 100.

Explanation:
In the above-given question,
given that,
700/100 x 5.
7 x 5.
35.
35 is greater than 10 but less than 100.

Question 14.
Is 250% of 44 less than 100, greater than 100 but less than 150, or greater than 150? Explain your reasoning.

Answer:
110 is greater than 100 but less than 150.

Explanation:
In the above-given question,
given that,
250% of 44.
250/100 x 44.
2.5 x 44.
110.

Question 15.
The seed and skin of a typical avocado is about 30%-40% of the avocado’s weight. For an 8-ounce avocado, how many ounces of edible fruit does it have?

Answer:
The number of ounces of edible fruit does it have = 5.6 ounces.

Explanation:
In the above-given question,
given that,
The seed and skin of a typical avocado is about 30%-40% of the avocado’s weight.
30/100 – 40/100.
30 + 40 = 70.
70/100 = 0.7.
0.7 x 8 = 5.6.
so the number of ounces of edible fruit does it have = 5.6 ounces.

Question 16.
A new health drink has 130% of the recommended daily allowance (RDA) for a certain vitamin. The RDA for this vitamin is 45 mg. How many milligrams of the vitamin are in the drink?

Answer:
The milligrams of the vitamin is in the drink = 60.75 mg.

Explanation:
In the above-given question,
given that,
A new health drink has 130% of the recommended daily allowance (RDA) for a certain vitamin.
The RDA for this vitamin is 45 mg.
135/100 x 45.
1.35 x 45.
60.75.
so the milligrams of the vitamin is in the drink = 60.75 mg.

Question 17.
Make Sense and Persevere 153 is 0.9% of what number? Tell which equivalent ratios you used to find the solution.

Answer:
The equivalent ratios used to find the solution = 1.377.

Explanation:
In the above-given question,
given that,
153 is 0.9%.
153 x 0.9/100.
153 x 0.009.
1.377.
so the equivalent ratio used to find the solution is 1.377.

Question 18.
Construct Arguments Brad says that if a second number is 125% of the first number, then the first number must be 75% of the second number. Is he correct? Justify your answer.

Answer:
Yes, Brad was correct.

Explanation:
In the above-given question,
given that,
Brad says that if a second number is 125% of the first number.
the first number must be 75%of the second number.
125/100 = 1.25.
75/100 = 0.75.
so he was correct.

Question 19.
Higher-Order Thinking Mark and Joe work as jewelers. Mark has an hourly wage of $24 and gets overtime for every hour he works over 40 hours. The overtime pay rate is 150% of the normal rate. Joe makes 5% commission on all jewelry he sells. Who earns more money in a week if Mark works 60 hours and Joe sells $21,000 worth of jewelry? Explain.

Answer:
Joe earns more money than the mark.

Explanation:
In the above-given question,
given that,
Mark and Joe work as jewelers.
Mark has an hourly wage of $24 and gets overtime for every hour he works over 40 hours.
The overtime pay rate is 150% of the normal rate.
5/100 = 0.05.
40 x 60 = 2400.
so Joe earns more money than the mark.

Assessment Practice

Question 20.
Pamela and John work as tutors at two different test-prep companies. Pamela earns $20 per hour. John earns $65 per pupil. Pamela works 40 hours each week. John has 11 pupils. Who earns more money in a week? Explain.

Answer:
The more money earns in a week = Pamela.

Explanation:
In the above-given question,
given that,
Pamela and John work as tutors at two different test-prep companies.
Pamela earns $20 per hour.
John earns $65 per pupil.
Pamela works 40 hours each week.
John has 11 pupils.
40 x $20 = $800.
$65 x 11 = $715.
so pamela earns more than john.

Question 21.
An Olympic-sized pool, which holds 660,000 gallons of water, is only 63% full. The pool maintenance company adds more water, filling the pool to 90% full. How many gallons of water did they add?
A. 244,200
B. 594,000
C. 178,200
D. 415,800

Answer:
The number of gallons of water they add = 178,200 gallons.

Explanation:
In the above-given question,
given that,
An Olympic-sized pool, which holds 660,000 gallons of water, is only 63% full.
The pool maintenance company adds more water, filling the pool to 90% full.
63 – 90 = 27.
27 x 660,000 = 178,200.

Lesson 3.2 Connect Percent and Proportion

Solve & Discuss It!
A florist is making flower arrangements for a party. He uses purple and white flowers in a ratio of 3 purple flowers to 1 white flower. How many flowers will he need in order to make 30 identical arrangements?
I can… use proportions to solve percent problems.
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 19

Look for Relationships
How are the number of purple flowers related to the number of white flowers?

Answer:
The ratio of purple flowers to the number of white flowers = 1: 3.

Explanation:
In the above-given question,
given that,
A florist is making flower arrangements for a party.
He uses purple and white flowers in a ratio of 3 purple flowers to 1 white flower.
the ratio is 1 : 3.
so the ratio of purple flowers to the number of white flowers = 1: 3.

Focus on math practices
Make Sense and Persevere If the florist can only buy white flowers in groups of flowers that have 3 white flowers and 2 red flowers, how many red flowers will the florist have to purchase? Explain your answer.

Answer:
The ratio of white flowers to the number of red flowers = 12.

Explanation:
In the above-given question,
given that,
If the florist can only buy white flowers in groups of flowers that have 3 white flowers and 2 red flowers.
2 + 3 = 5.
5 x 6 = 30.
2 + 2 + 2 + 2 + 2 = 12.
so the ratio of white flowers to the number of red flowers = 12.

Essential Question
How does proportional reasoning relate to percent?

Try It!

Camila makes 2 of her 5 shots attempted. Is the percent of shots she made more than, less than, or the same as Emily’s percent of shots?
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 20
Camila made _______% of her shots.
Camila’s percent of the shots made is _______ Emily’s.

Answer:
Camila made 2/5% of her shots.
Camila’s percent of the sots made is 40 Emily’s.

Explanation:
In the above-given question,
given that,
Camila makes 2 of her 5 shots attempted.
2/5 = p/100.
p = 2/5 x 100.
p = 0.4 x 100.
p = 40.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-3-Analyze And Solve Percent Problems-6

Convince Me! A hockey goalie stops 37 out of 40 shots. What percent of attempted goals did she stop?

Answer:
The percent of attempted goals did she stop = 14.8%

Explanation:
In the above-given question,
given that,
A hockey goalie stops 37 out of 40 shots.
37/100 x 40.
0.37 x 40.
14.8 %.
so the percent of attempted goals did she stop = 14.8%.

Try It!

a. Megan’s room is expanded so the width is 150% of 3 meters. What is the new width?

Answer:
The new width = 4.5 meters.

Explanation:
In the above-given question,
Megan’s room is expanded so the width is 150% of 3 meters.
150/100 x 3.
15/10 x 3.
1.5 x 3.
4.5 meters.

b. Use the soy milk label in Example 3. What is the recommended amount of iron needed each day? Round your answer to the nearest mg.

Answer:
The recommended amount of iron needed each day =

Explanation:
In the above-given question,
given that,

KEY CONCEPT
Percent problems represent a kind of proportional relationship. You can use proportional reasoning to solve percent problems.
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 21

Do You Understand?
Question 1.
Essential Question How does proportional reasoning relate to percent?

Answer:
part/whole = p/100.

Explanation:
In the above-given question,
given that,
Percent problems represent a kind of proportional relationship.
if the whole is 100%.
part is p.
part/whole = p/100.

Question 2.
Reasoning Why does one of the ratios in a percent proportion always have a denominator of 100?

Answer:
The proportion always has a denominator of 100.

Explanation:
In the above-given question,
given that,
the whole is always 100%.
so part /whole = p/100.
so the proportion always has a denominator of 100.

Question 3.
Construct Arguments The proportion \(\frac{75}{W}=\frac{150}{100}\) can be used to find the whole, w. Use the language of percent to explain whether w is less than or greater than 75.

Answer:
The weight is greater than 75.

Explanation:
In the above-given question,
given that,
The proportion \(\frac{75}{W}=\frac{150}{100}\) can be used to find the whole, w.
75/w = 150/100.
75/w = 1.5.
w = 1.5 x 75.
w = 112.5.
the weight is greater than 75.

Do You Know How?
Question 4.
Write a percent proportion for the bar diagram shown.
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 22

Answer:
The percent proportion for the bar diagram is 11.56

Explanation:
In the above-given question,
given that,
68/100 = 17/p.
0.68 = 17/p.
0.68 x 17 = p.
11.56 = p.

Question 5.
Use a proportion to find each value.
a. 2% of 180
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 23

Answer:
n = 3.6.

Explanation:
In the above-given question,
given that,
2% of 180.
2/100 = n/180.
0.02 = n/180.
n = 0.02 x 180.
n = 3.6.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-3-Analyze And Solve Percent Problems-7

b. What percent is 17 out of 40?
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 24

Answer:
p = 6.8.

Explanation:
In the above-given question,
given that,
17/100 = p/40.
0.17 = p/40.
p = 0.17 x 40.
p = 6.8.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-3-Analyze And Solve Percent Problems-8

Question 6.
Construct Arguments Gia researches online that her car is worth $3,000. She hopes to sell it for 85% of that value, but she wants to get at least 70%. She ends up selling it for $1,800. Did she get what she wanted? Justify your answer.

Answer:
Yes, she gets what she wanted.

Explanation:
In the above-given question,
given that,
Gia researches online that her car is worth $3,000.
she hopes to sell it for 85% of that value, but she wants to get at least 70%.
She ends up selling it for $1,800.
85/100 x 3000.
0.85 x 3000.
70/100 x 1800.
0.7 x 1800.

Practice & Problem Solving

Leveled Practice In 7-8, fill in the boxes to solve.
Question 7.
The rabbit population in a certain area is 200% of last year’s population. There are 1,100 rabbits this year. How many were there last year?
\(\frac{1,100}{w}\) = \(\frac{}{}\)
There were _______ rabbits last year.

Answer:
The number of rabbits last year = 2200.

Explanation:
In the above-given question,
given that,
The rabbit population in a certain area is 200% of last year’s population.
There are 1,100 rabbits this year.
1100/w = 200/100.
1100/w = 2.
w = 2 x 1100.
w = 2200.
so the number of rabbits last year = 2200.

Question 8.
A company that makes hair-care products had 3,000 people try a new shampoo. Of the 3,000 people, 9 had a mild allergic reaction. What percent of the people had a mild allergic reaction?
\(\frac{9}{3,200}\) = \(\frac{p}{}\)
Percent = _________%

Answer:
The percent of the people who had a mild allergic reaction = 8.4%.

Explanation:
In the above-given question,
given that,
A company that makes hair-care products had 3,000 people try a new shampoo.
Of the 3,000 people, 9 had a mild allergic reaction.
9/3200 = p/3000.
0.0028 = p/3000.
p = 3000 x 0.0028.
p = 8.4%.
so the percent of the people had a mild allergic reaction = 8.4%.

Question 9.
A survey was given to people who owned a certain type of car. What percent of the people surveyed were completely satisfied with the car?
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 25

Answer:
The percent of the people surveyed were completely satisfied with the car = 1100.

Explanation:
In the above-given question,
given that,
A survey was given to people who owned a certain type of car.
completely satisfied = 1100.
somewhat satisfied = 740%.
not satisfied = 160%.
so the percent of the people surveyed were completely satisfied with thw car = 1100%.

Question 10.
The Washingtons buy a studio apartment for $240,000. They pay a down payment of $60,000
a. Their down payment is what percent of the purchase price?

Answer:
The down payment is 0.4.

Explanation:
In the above-given question,
given that,
The Washingtons buy a studio apartment for $240,000.
They pay a down payment of $60,000.
240000/60000 = 0.4.

b. What percent of the purchase price would a $12,000 down payment be?

Answer:
The percent of the purchase price would be a $12000 down payment = $20.

Explanation:
In the above-given question,
given that,
The Washingtons buy a studio apartment for $240,000.
240000/12000 = 20.
so the percent of the purchase price would be a $120000 down payment = $20.

Question 11.
A restaurant customer left $3.50 as a tip. The tax on the meal was 7% and the tip was 20% of the cost including tax.
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 26
a. What piece of information is not needed to compute the bill after tax and tip?

Answer:
The piece of information is not needed to compute the bill after tax and tip = $0.455.

Explanation:
In the above-given question,
given that,
A restaurant customer left $3.50 as a tip.
The tax on the meal was 7% and the tip was 20% of the cost including tax.
20% – 7% = 13%.
13/100 = 0.13.
3.50 x 0.13 = 0.455.
so the piece of information is not needed to compute the bill after tax and tip = $0.455.

b. Make Sense and Persevere What was the total bill?

Answer:

Question 12.
Reasoning What is a good estimate for 380% of 60? Explain.

Answer:
The good estimate is 228.

Explanation:
In the above-given question,
given that,
380% of 60.
380/100 x 60.
3.8 x 60 = 228.
so the good estimate is 228.

Question 13.
Critique Reasoning Marna thinks that about 35% of her mail is junk mail. She gets about twice as much regular mail as junk mail. Is she correct? Explain.

Answer:
Yes, she was correct.

Explanation:
In the above-given question,
given that,
Marna thinks that about 35% of her mail is junk mail.
She gets about twice as much regular mail as junk mail.
35% + 35%.
70%.
70/100 = 0.7.
so she was correct.

Question 14.
Hypatia has read 13 chapters of a 22-chapter book. What percent of the chapters has she read?

Answer:
The chapters she has read = 9%.

Explanation:
In the above-given question,
given that,
Hypatia has read 13 chapters of a 22-chapter book.
22 – 13 = 9.
9 percent = 9/100.
so 9% of the chapters has she read.

Question 15.
A school year has 4 quarters. What percent of a school year is 7 quarters?

Answer:
The percent of a school year = 28%.

Explanation:
In the above-given question,
given that,
A school year has 4 quarters.
4/100 = 7/q.
0.04 = 7/q.
q = 0.04 x 7.
q = 0.28.
so the percent of a school year = 28%.

Question 16.
Construct Arguments A survey found that 27% of high school students and 94% of teachers and school employees drive to school. The ratio of students to employees is about 10 to 1. Roger states that the number of students who drive to school is greater than the number of teachers and employees who drive to school. Explain how Roger’s statement could be correct.

Answer:
The ratio of students to employees is about 10 to 1.

Explanation:
In the above-given question,
given that,
A survey found that 27% of high school students and 94% of teachers and school employees.
The ratio of students to employees is about 10 to 1.
27/100 = 0.27.
94/100 = 0.94.
so the ratio of students to employees is about 10 to 1.

Question 17.
Higher Order Thinking Stefan sells Jin a bicycle for $114 and a helmet for $18. The total cost for Jin is 120% of what Stefan spent originally to buy the bike and helmet. How much did Stefan spend originally? How much money did he make by selling the bicycle and helmet to Jin?
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 27

Answer:
The money did he make by selling the bicycle and helmet to Jin = $132.

Explanation:
In the above-given question,
given that,
Stefan sells Jin a bicycle for $114 and a helmet for $18.
The total cost for Jin is 120% of what Stefan spent originally to buy the bike and helmet.
$114 + $18 = $132.
so the money did he make by selling the bicycle and helmet to Jin = $132.

Assessment Practice

Question 18.
Last month Nicole spent $30. This month she spent 140% of what she spent last month. Write a proportional equation to represent the situation. How much did Nicole spend this month?

Answer:
The amount did Nicole spend this month = $42.

Explanation:
In the above-given question,
given that,
Last month Nicole spent $30.
This month she spent 140% of what she spent last month.
140/100 x $30.
1.4 x $30.
$42.
so the amount did Nicole spend this month = $42.

Question 19.
Mr. Jones, the owner of a small store buys kayak paddles for $50.00 each, and sells them for 180% of the purchase price.
PART A
A customer buys a paddle for $97.65, which includes the selling price and sales tax. What is the sales tax rate?
A 7.65%
B. 4.7%
C. 9.75%
D. 8.5%

Answer:
Option C was correct.

Explanation:
In the above-given question,
given that,
Mr. Jones, the owner of a small store buys kayak paddles for $50.00 each and sells them for 180% of the purchase price.
A customer buys a paddle for $97.65, which includes the selling price and sales tax.
180/100 = 1.8.
1.8 x 50.
90.
so option C was correct.

PART B
If Mr. Jones buys paddles for $35 instead of $50, and uses the same percent increase on price, how many paddles must Mr. Jones sell for the total paddle sales, before tax, to be at least $250? Explain your answer.

Answer:
Mr. Jones sells for the total paddle sales = $40.

Explanation:
In the above-given question,
given that,
If Mr. Jones buys paddles for $35 instead of $50 and uses the same percent increase on the price.
$250/$50.
$.5.
$35 + $5 = $40.

Lesson 3.3 Represent and Use the Percent Equation

Solve & Discuss It!
Fran is shopping for a new pair of shoes. She did some research and has narrowed the options to the two pairs she likes the most. Based on buyers’ reviews, which pair do you recommend that she buy? Explain your thinking.

I can.. represent and solve percent problems using equations.
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 28

Model with Math
How can you use what you know about ratios to compare the reviews?

Focus on math practices
Model with Math Describe another situation in which you could use ratios to make a decision.

Essential Question
How are percent problems related to proportional reasoning?

Try It!

An elephant weighs 15,000 pounds on Earth and 2,500 pounds on the Moon. Assuming the weights are proportional, what percent of its weight on Earth is its weight on the Moon?
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 29
The elephant’s weight on the Moon is about __________% of its weight on Earth.

Answer:
The elephant’s weight on the Moon is about 6% of its weight on Earth.

Explanation:
In the above-given question,
given that,
An elephant weighs 15,000 pounds on Earth and 2,500 pounds on the Moon.
Assuming the weights are proportional.
2500 = p x 15000.
p = 15000/2500.
p = 6.
so the elephant’s weight on the Moon is about 6% of its weight on Earth.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-3-Analyze And Solve Percent Problems-9

Convince Me! How does the percent describe how the weights are related?

Try It!

To make a profit, a clothing store sells board shorts at 115% of the amount they paid for them. How much did the store pay for the board shorts shown?
Envision Math Common Core 7th Grade Answer Key Topic 3 Analyze And Solve Percent Problems 30

Answer:
The store pays for the board shorts = $32.2.

Explanation:
In the above-given question,
given that,
To make a profit, a clothing store sells board shorts at 115% of the amount they paid for them.
the cost of the short is $28.
115/100 x $28.
1.15 x  $28 = $32.2.
so the store pay for the board shorts = $32.2.

KEY CONCEPT

The percent equation shows how a percent relates proportional quantities. The percent is a constant of proportionality and the equation has the same form as y = mx.
Envision Math Common Core 7th Grade Answers Topic 3 Analyze And Solve Percent Problems 31

Do You Understand?
Question 1.
Essential Question How are percent problems related to proportional reasoning?

Answer:
The percent problem related to proportional reasoning is y = mx.

Explanation:
In the above-given question,
given that,
The percent equation shows how a percent relates to proportional quantities.
The percent is a constant of proportionality and the equation has the same form as y = mx.
part/whole = percent.
part = percent x whole.

Question 2.
Reasoning A waiter at a restaurant receives $11 as a tip on a $47.20 bill. He usually receives tips that are 20% of the total bill. Is the tip amount what the waiter typically receives? Explain.

Answer:
The tip amount is the waiter typically receives = $11.64.

Explanation:
In the above-given question,
given that,
A waiter at a restaurant receives $11 as a tip on a $47.20 bill.
He usually receives tips that are 20% of the total bill.
$47.20 + $11.
$58.2.
20/100 x $58.2.
0.2 x $58.2 = $11.64.

Question 3.
Construct Arguments Sara used an equation to solve the problem below. Justify each step of her work. About 11% of people are left-handed. How many people would you expect to be left-handed in a class of 30 students?
l = 0.11 • 30
ļ = 3.3
about 3 students

Answer:
The number of people would you accept to be left-handed = 3 students.

Explanation:
In the above-given question,
given that,
11% of people are left-handed.
11/100 x 30.
0.11 x 30 = 3.3.
so the number of people would you accept to be left-handed = 3 students.

Do You Know How?
Question 4.
An auto insurance company pays 12% commission to its agents for each new insurance policy they sell. How much commission does an agent make on a $1,000 policy?

Answer:
The commission does an agent makes on a $1000 policy = $10.

Explanation:
In the above-given question,
given that,
An auto insurance company pays a 12% commission to its agents for each new insurance policy they sell.
12/100 x $1000.
0.12 x $1000.
$10.
so the commission does an agent make on a $1000 policy = $10.

Question 5.
Curt and Melanie are mixing blue and yellow paint to make seafoam green paint. Use the percent equation to find how much yellow paint they should use.
Envision Math Common Core 7th Grade Answers Topic 3 Analyze And Solve Percent Problems 32

Answer:
The quantity of yellow paint they should use = 1.5.

Explanation:
In the above-given question,
given that,
Curt and Melanie are mixing blue and yellow paint to make seafoam green paint.
70% = 70/100.
70/100 = 0.7.
30/100 = 30%.
30% = 0.3.
0.3 + 0.7 = 1.
1 x 1.5 = 1.5.
so the quantity of yellow paint they should use = 1.5.

Question 6.
Bill paid $35.99 in tax on a laptop that cost $449.99. About what percent sales tax did Bill pay?

Answer:
The percent sales tax did Bill pay = 4.8%.

Explanation:
In the above-given question,
given that,
Bill paid $35.99 in tax on a laptop that cost $449.99
$449.99 + $35.99.
$485.98.
so the percent of sales tax did Bill pay = 4.8%.

Practice & Problem Solving

Leveled Practice In 7 and 8, solve each percent problem.
Question 7.
In a survey of 500 voters, 430 said they would vote for the same candidate again. What percent of the voters would vote the same way again?
part = percent. whole
_______ = p% • ________
_______ = p%

Answer:
The percent of the votes would vote the same way again = 0.086.

Explanation:
In the above-given question,
given that,
In a survey of 500 voters, 430 said they would vote for the same candidate again.
500 = p% x 430.
p% = 430/500.
p% = 0.86.
p = 0.086.
so the percent of the votes would vote the same way again = 0.086.

Question 8.
The local newspaper has letters to the editor from 40 people. If this number represents 5% of all of the newspaper’s readers, how many readers, r, does the newspaper have?
part = percent. whole
______ = ______ • r
______ = r

Answer:
The number of readers r does the newspaper have = 0.00125.

Explanation:
In the above-given question,
given that,
The local newspaper has letters to the editor from 40 people.
If this number represents 5% of all of the newspaper’s readers.
5% = 40 x r.
5/100 = 40 x r.
0.05 = 40r.
r = 0.05/40.
r = 0.00125.
so the number of readers r does the newspaper have = 0.00125.

Question 9.
Make Sense and Persevere What percent of the 16-gigabyte hard drive shown is used for photos?
Envision Math Common Core 7th Grade Answers Topic 3 Analyze And Solve Percent Problems 33

Answer:
The percent of the 16-gigabyte hard drive shown is used for photos = 0.002075.

Explanation:
In the above-given question,
given that,
percent = part/whole.
percent = 3.32/16.
percent = 0.2075.
0.2075% = 0.002075.
so the percent of the 16-gigabyte hard drive shown is used for photos = 0.002075.

Question 10.
A shirt that normally costs $30 is on sale for $21.75. What percent of the regular price is the sale price?

Answer:
The percent of the regular price = 0.0825.

Explanation:
In the above-given question,
given that,
A shirt that normally costs $30 is on sale for $21.75.
$30 – $21.75 = $8.25.
$8.25% = 0.0825.
so the percent of the regular price = 0.0825.

Question 11.
Complete the table.
Envision Math Common Core 7th Grade Answers Topic 3 Analyze And Solve Percent Problems 34

Answer:
The commission = $30.72.

Explanation:
In the above-given question,
given that,
the sales of the earning commission = $768.
the commission rate = 4%.
4/100 = 768/c.
0.04 = 768/c.
c = 0.04 x 768.
c = $30.72.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-3-Analyze And Solve Percent Problems-10

Question 12.
Complete the table.
Envision Math Common Core 7th Grade Answers Topic 3 Analyze And Solve Percent Problems 35

Answer:
The sales tax = $1.7991.

Explanation:
In the above-given question,
given that,
the selling price is $39.98.
the tax rate is 4.5%.
4.5 /100 x $39.98.
0.045 x $39.98.
$1.7991.
so the sales tax = $1.7991.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-3-Analyze And Solve Percent Problems-11

Question 13.
A restaurant automatically charges a 20% gratuity if a party has 6 or more people. How much gratuity is added to a party of 6 on a $141 bill?

Answer:
The much gratuity is added to a party of 6 on a $141 bill = $169.2.

Explanation:
In the above-given question,
given that,
A restaurant automatically charges a 20% gratuity if a party has 6 or more people.
20/100 = 0.2.
0.2 x 6 = 1.2.
1.2 x $141 = 169.2.

Question 14.
Make Sense and Persevere A large university accepts 70% of the students who apply. Of the students the university accepts, 25% actually enroll. If 20,000 students apply, how many enroll?

Answer:
The number of students that enroll = 200 students.

Explanation:
In the above-given question,
given that,
A large university accepts 70% of the students who apply.
Of the students the university accepts, 25% actually enroll.
70 + 25 = 95.
95/100 x 20000.
0.95 x 20000.
200.
so the number of students that enroll = 200 students.

Question 15.
Model with Math There are 4,000 books in the town’s library. Of these, 2,600 are fiction. Write a percent equation that you can use to find the percent of the books that are fiction. Then solve your equation.

Answer:
The percent of the books that are fiction = 0.01538.

Explanation:
In the above-given question,
given that,
There are 4,000 books in the town’s library.
Of these, 2,600 are fiction.
4000/2600 = 1.538.
1.538% = 0.01538.

Question 16.
A salesperson earns 4% commission on furnace sales.
a. What is the commission that the salesperson earns on the sale of $33,000 worth of furnaces?

Answer:
The salesperson earns on the sale of $33000 = $1320.

Explanation:
In the above-given question,
given that,
A salesperson earns a 4% commission on furnace sales.
4/100 x $33000.
0.04 x $33000.
$1320.
so the salesperson earns on the sale of $33000 = $1320.

b. Suppose the salesperson doubles his sales of furnaces. What would be true about the commission? Explain without using any calculations.

Answer:
The salesperson doubles his sales of furnaces = $2640.

Explanation:
In the above-given question,
given that,
A salesperson earns a 4% commission on furnace sales.
4/100 x $33000.
0.04 x $33000.
$1320.
1320 x $2 = $2640.
so the salesperson doubles his sales of furnaces = $2640.

Question 17.
Heidi earns 3% commission on the jewelry she sells each week. Last week, she sold the pieces of jewelry shown.
Envision Math Common Core 7th Grade Answers Topic 3 Analyze And Solve Percent Problems 36
a. How much did she make in commission?
Answer:
She makes in commission is 21.9%.

Explanation:
In the above-given question,
given that,
Heidi earns 3% commission on the jewelry she sells each week.
$110 + $275 + $200 + $145.
$730.
3% of $730.
3/100 x 730.
0.03 x 730.
21.9.

b. Reasoning How much did the jewelry store take in from her sales? How do you know?

Answer:
The jewelry store takes in from her sales 29.2%.

Explanation:
In the above-given question,
given that,
Heidi earns 3% commission on the jewelry she sells each week.
$110 + $275 + $200 + $145.
$730.
4% of $730.
4/100 x 730.
0.04 x 730.
29.2.

Question 18.
Higher Order Thinking in a company, 60% of the workers are men. If 1,380 women work for the company, how many workers are there in all? Show two different ways that you can solve this problem.

Answer:
The number of workers in all = 828 workers.

Explanation:
In the above-given question,
given that,
In a company, 60% of the workers are men.
If 1,380 women work for the company.
60% of 1380.
60/100 x 1380.
6/10 x 1380.
0.6 x 1380.
828.
so the number of workers in all = 828 workers.

Assessment Practice

Question 19.
A salesperson starts working 40 hours per week at a job with two options for being paid. Option A is an hourly wage of $19. Option B is a commission rate of 8% on weekly sales. How much does the salesperson need to sell in a given week to earn the same amount with each option?
A. $9,500
B. $4,750
C. $760
D. $320

Answer:
The salesperson needs to sell in a given week to earn the same amount with each option = $320.

Explanation:
In the above-given question,
given that,
A salesperson starts working 40 hours per week at a job with two options for being paid.
Option A is an hourly wage of $19. Option B is a commission rate of 8% on weekly sales.
8/100 x 40.
0.08 x 40.
3.2.
so the salesperson needs to sell in a given week to earn the same amount with each option = $320.

Question 20.
At a real estate agency, an agent sold a house for $382,000. The commission rate is 5.5% for the real estate agency. The commission for the agent is 30% of the amount the real estate agency gets. How much did the agent earn in commission? Explain your answer.

Answer:
The agent earn in commission = $114600.

Explanation:
In the above-given question,
given that,
At a real estate agency, an agent sold a house for $382,000.
The commission rate is 5.5% for the real estate agency.
The commission for the agent is 30% of the amount the real estate agency gets.
30/100 x $382000.
0.3 x $382000.
114600.
so the agent earn in commission = $114600.

Topic 3 Mid-Topic Checkpoint

Question 1.
Vocabulary Explain how the percent equation relates proportional quantities. Lesson 3-3

Answer:
The percent equation shows how a percent relates to proportional quantities.
y = mx.

Explanation:
In the above-given question,
given that,
The percent equation shows how a percent relates proportional quantities.
The percent is a constant of proportionality and the equation has the same form as y = mx.
part/whole = percent.
part = percent x whole.

Question 2.
Colleen buys a movie for $20 and pays 7% sales tax. Her cousin, Brad, lives in another state. Brad buys the same movie for $22 and pays 6% sales tax. Who pays more sales tax? How much more? Lessons 3-1 and 3-3

Answer:
Brad pays more sales tax = $0.1.

Explanation:
In the above-given question,
given that,
Colleen buys a movie for $20 and pays 7% sales tax.
Her cousin, Brad, lives in another state.
Brad buys the same movie for $22 and pays 6% sales tax.
7/100 x $20.
0.07 x 20 = $1.4.
6/100 x $22.
0.06 x 22 = $1.3.
Brad pays more sales tax = $0.1.

Question 3.
Kamesh and Paolo each read 40 books in one year. Kamesh read 12 nonfiction books. Thirty-five percent of the books Paolo read were nonfiction. Who read more nonfiction books? How many more? Lesson 3-2

Answer:
Kamesh read more notifications.

Explanation:
In the above-given question,
given that,
Kamesh and Paolo each read 40 books in one year.
Kamesh read 12 nonfiction books.
Thirty-five percent of the books Paolo read were nonfiction.
35% x 40.
0.35 x 40 = $14.
35% x 12.
0.35 x 12 = $4.2.
so Kamesh read more notifications.

Question 4.
Val buys a computer for $920. If this is 115% of what the store paid for the same computer, how much did the store earn on the sale? Lesson 3-3

Answer:
The much did the store earn on the sale = $1058.

Explanation:
In the above-given question,
given that,
Val buys a computer for $920.
If this is 115% of what the store paid for the same computer.
115/100 x $920.
1.15 x $920 = $1058.
so the much did the store earn on the sale = $1058.

Question 5.
For each situation, select the percent to answer the question. Lessons 3-1, 3-2, and 3-3

Answer:
25%, 5%, and 5%

Explanation:
In the above-given question,
given that,
In an auto repair shop, 14 of the 56 cars received oil changes.
14/56 = 0.25.
0.25 x 100 = 25%.
Harry pays $3.50 sales tax on a $70 item.
In a box of 250 paperclips, 50 are red.
250/50 = 5%.

Envision Math Common Core 7th Grade Answers Topic 3 Analyze And Solve Percent Problems 37

Question 6.
Explain how you can use proportional reasoning to determine the whole if you know that 21 is 60% of the whole. Lesson 3-2

Answer:
12.6%.

Explanation:
In the above-given question,
given that,
21 is 60%.
60/100 x 21.
0.6 x 21.
12.6.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-3-Analyze And Solve Percent Problems-12

Topic 3 Mid-Topic Performance Task

The coach of a women’s basketball team wants each of her starting players to make at least 75% of the free throws attempted during regular season games. The table shows the statistics for the starting players after the first 15 games.
Envision Math Common Core 7th Grade Answers Topic 3 Analyze And Solve Percent Problems 38

PART A
Use the table. Find the percentage of free throws made by each player. Round to the nearest whole percent.

Answer:
The percentage of free throws made by each player = 1.5%, 1.1%, 1.4%, 1.8%, and 0.7%.

Explanation:
In the above-given question,
given that,
Wilson free throws attempted = 36/24 = 1.5.
Bartholdi free throws attempted = 42/37 = 1.1.
Jhonson free throws attempted = 22/15 = 1.4.
Garcia free throws attempted = 29/16 = 1.8.
O’Malley free throws attempted = 14/12 = 0.7.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-3-Analyze And Solve Percent Problems-13

PART B
Choose one of the players with a free-throw percentage less than 75%. Determine a number of free throws the player could attempt and make during the next 10 games to increase her free-throw percentage to at least 75%.
Answer:

PART C
Choose one of the players with a free-throw percentage greater than 75%. Determine the number of free throws that the player could miss during the next 10 games and still maintain an overall percentage of at least 75%.
Answer:

Lesson 3.4 Solve Percent Change and Percent Error Problems

Explain It!
Nadia lives in the town of Bayville. Quinn lives in the town of Portside. Nadia and Quinn each claim that her respective town’s population is growing more rapidly.

I can… solve problems involving percent change and percent error.
Envision Math Common Core 7th Grade Answers Topic 3 Analyze And Solve Percent Problems 39

A. Write an argument to support Nadia. Why might she argue that Bayville’s population is growing more rapidly?

Answer:
The population is growing more rapidly because they have the Historic place, LightHouse, and Beachfront.

Explanation:
In the above-given question,
given that,
Nadia lives in the town of Bayville.
Quinn lives in the town of Portside.
Nadia’s population has last year 3399.
Quinn has population has 1870.
Nadia population has this year 3702.
Quinn’s population has 1926.
Nadia has a Historic place, LightHouse, and Beachfront.
so the population is growing more rapidly.

B. Write an argument to support Quinn. Why might she argue that Portside’s population is growing more rapidly?

Answer:
The population is growing more rapidly because they have the Aquarium.

Explanation:
In the above-given question,
given that,
Nadia lives in the town of Bayville.
Quinn lives in the town of Portside.
Nadia’s population has last year 3399.
Quinn has population has 1870.
Nadia population has this year 3702.
Quinn’s population has 1926.
Quinn has a Historic place, LightHouse, and Beachfront.
so the population is growing more rapidly.

c. Whose reasoning is more logical? Explain why.

Answer:
Nadia’s reasoning is more logical.

Explanation:
In the above-given question,
given that,
Nadia lives in the town of Bayville.
Quinn lives in the town of Portside.
Nadia’s population has last year 3399.
Quinn has population has 1870.
Nadia population has this year 3702.
Quinn’s population has 1926.
the population of Nadia is very high.
so Nadia’s reasoning is more logical.

Focus on math practices
Critique Reasoning Suppose Bayville’s population is expected to grow 3% next year. Nadia says that means the population will increase by 300 people. Is Nadia’s reasoning correct? Explain.

Answer:
Yes, Nadia was correct.

Explanation:
In the above-given question,
given that,
Bayville’s population is expected to grow 3% next year.
3/100 = 0.03.
300.
so Nadia was correct.

Essential Question
How is finding percent error similar to finding percent change?

Try It!

What will be the length of the alligator next year if its length changes by the same percent as it changed this year?
percent change • length this year = change in length
Envision Math Common Core 7th Grade Answers Topic 3 Analyze And Solve Percent Problems 40
0.1875 • ________ = ________
_______ + _______ = ________
Next year, the length of the alligator will be ________ inches long.

Answer:
Next year, the length of the alligator will be 45 inches long.

Explanation:
In the above-given question,
given that,
percent change • length this year = change in length.
0.1875 x 38 = 7.125.
7.125 + 38 = 45.125.
so next year, the length of the alligator will be 45 inches long.

Convince Me! Why is the increase in the alligator’s length different from year to year, even though the percent change stayed the same?

Try It!

The specification for the length of a bolt is 4.75 inches. A machinist makes a bolt that is 4.769 inches long. What is the percent error of the bolt’s length?
Envision Math Common Core 7th Grade Answers Topic 3 Analyze And Solve Percent Problems 41

Answer:
The percent error of the bolt’s = 0.019.

Explanation:
In the above-given question,
given that,
The specification for the length of a bolt is 4.75 inches.
A machinist makes a bolt that is 4.769 inches long.
4.769 – 4.75 = 0.019.
so the percent error of the bolt’s = 0.019.

KEY CONCEPT

Percent change and percent error problems are kinds of percent problems. You can use the percent equation to solve them.
amount of change = percent change • original amount
The amount of change is the difference between original and new values
Envision Math Common Core 7th Grade Answers Topic 3 Analyze And Solve Percent Problems 42

Do You Understand?
Question 1.
Essential Question How is finding percent error similar to finding percent change?

Answer:
Yes, percent error and percent change problems are the same.

Explanation:
In the above-given question,
given that,
Percent change and percent error problems are kinds of percent problems.
amount of change = percent change • original amount.
The amount of change is the difference between original and new values.

Question 2.
Reasoning Give an example of a problem in which the percent error is greater than 20%, but less than 50%. Explain how you determined the percent error.

Answer:
The percent error is less than 20% but less than 50%.

Explanation:
In the above-given question,
given that,
percent error = percent change x original amount.
percent error = 5% x 3%.
percent error = 15%.
so the percent error is less than 20% but less than 50%.

Question 3.
Construct Arguments A store manager marked up a $10 flash drive by 20%. She then marked it down by 20%. Explain why the new price of the flash drive is not $10.

Answer:
The new price of the flash drives is $2.

Explanation:
In the above-given question,
given that,
A store manager marked up a $10 flash drive by 20%.
She then marked it down by 20%.
20/100 x 10.
0.2 x 10 = 2.
so the new price of the flash drives is $2.

Do You Know How?
Question 4.
Lita’s softball team won 8 games last month and 10 this month. What was the percent change in games the team won? Was it an increase or decrease?

Answer:
The percent change in games the team won = 80.

Explanation:
In the above-given question,
given that,
Lita’s softball team won 8 games last month and 10 this month.
percent change = percent change x original amount.
percent change = 8 x 10.
percent change = 80.

Question 5.
What is the percent change in the price of a gallon of gas, to the nearest whole percent? Is it an increase or a decrease?
Envision Math Common Core 7th Grade Answers Topic 3 Analyze And Solve Percent Problems 43

Answer:
The percent change in the price of a gallon of gas = 42.

Explanation:
In the above-given question,
given that,
3.499 last year.
3.079 this year.
3.499 – 3.079 = 0.42.
42/100 = 42.
so the percent change in the price of a gallon of gas = 42.

Question 6.
Several students measured a 25-mm-long nail and wrote the measurements shown in the table below. Whose measurement had the greatest percent error? Round to the nearest percent.
Envision Math Common Core 7th Grade Answers Topic 3 Analyze And Solve Percent Problems 44

Answer:
Tenicia has the greatest percent error.

Explanation:
In the above-given question,
given that,
Several students measured a 25-mm-long nail and wrote the measurements.
26 – 25 = 1.
25 – 23 = 2.
25 – 25 = 0.
so Tenicia had the greatest measurement.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-3-Analyze And Solve Percent Problems-14

Practice & Problem Solving

Leveled Practice In 7-8, use the bar diagram and fill in the boxes to solve.
Question 7.
The original quantity is 10 and the new quantity is 13. What is the percent change? Is it an increase or decrease?
Envision Math Common Core 7th Grade Answers Topic 3 Analyze And Solve Percent Problems 45
The percent decrease is _______ %.

Answer:
The percent decrease is 1.3%.

Explanation:
In the above-given question,
given that,
The original quantity is 10 and the new quantity is 13.
percent change = 10 x 13.
percent change = 130.
130/100 = 1.3.
so the percent decrease is 1.3%.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-3-Analyze And Solve Percent Problems-15

Question 8.
The original quantity is 5 and the new quantity is 3. What is the percent change? Is it an increase or decrease?
Envision Math Common Core 7th Grade Answers Topic 3 Analyze And Solve Percent Problems 46
The percent increase is _______%.

Answer:
The percent increase is 0.006%.

Explanation:
In the above-given question,
given that,
The original quantity is 10 and the new quantity is 13.
3 = p x 5.
percent change = 3/5.
0.6/100 = 0.006..
so the percent increase is 0.006%.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-3-Analyze And Solve Percent Problems-16

Question 9.
At noon, a tank contained 10 cm of water. After several hours, it contained 7 cm of water. What is the percent decrease of water in the tank?

Answer:
The percent decrease of water in the tank = 3%.

Explanation:
In the above-given question,
given that,
At noon, a tank contained 10 cm of water. After several hours, it contained 7 cm of water.
percent change = 10 – 7.
percent change = 3.
so the percent decrease of water in the tank = 3%.

Question 10.
Craig likes to collect vinyl records. Last year he had 10 records in his collection. Now he has 12 records. What is the percent increase of his collection?

Answer:
The percent increase of his collection = 2%.

Explanation:
In the above-given question,
given that,
Craig likes to collect vinyl records. Last year he had 10 records in his collection.
Now he has 12 records.
percent increase = 12 – 10.
percent increase = 2.
so the percent increase of his collection = 2%.

Question 11.
Carl bought an airline ticket. Two weeks ago, the cost of this flight was $300.
Envision Math Common Core 7th Grade Answers Topic 3 Analyze And Solve Percent Problems 47
What is the percent increase?

Answer:
The percent increase = 75%.

Explanation:
In the above-given question,
given that,
Carl bought an airline ticket. Two weeks ago, the cost of this flight was $300.
percent increase = 375 – 300.
percent increase = 75.
so the percent increase = 75%.

Question 12.
On Monday, a museum had 150 visitors. On Tuesday, it had 260 visitors.
a. Estimate the percent change in the number of visitors to the museum.

Answer:
The percent change in the number of visitors to the museum = 39000.

Explanation:
In the above-given question,
given that,
On Monday, a museum had 150 visitors. On Tuesday, it had 260 visitors.
percent change = 150 x 260.
percent change = 39000.
so the percent change in the number of visitors to the museum = 39000.

b. About how many people would have to visit the museum on Wednesday to have the same percent change from Tuesday to Wednesday as from Monday to Tuesday? Explain your answer.

Answer:
The percent change from Tuesday to Wednesday = 110.

Explanation:
In the above-given question,
given that,
On Monday, a museum had 150 visitors. On Tuesday, it had 260 visitors.
percent change = 150 – 260.
percent change = 110.
so the percent change from Tuesday to Wednesday = 110.

Question 13.
Rihanna has a container with a volume of 1.5 liters. She estimates the volume to be 2.1 liters. What is the percent error?

Answer:
The percent error = 0.6.

Explanation:
In the above-given question,
given that,
Rihanna has a container with a volume of 1.5 liters.
She estimates the volume to be 2.1 liters.
percent error is similar to the percent change.
percent change = 2.1 – 1.5.
percent change = 0.6.
so the percent error = 0.6.

Question 14.
The label on a package of bolts says each bolt has a diameter of 0.35 inch. To be in the package, the percent error of the diameter must be less than 5%. One bolt has a diameter of 0.33 inch. Should it go in the package? Why or why not?

Answer:
Yes, it would go in the package.

Explanation:
In the above-given question,
given that,
The label on a package of bolts says each bolt has a diameter of 0.35 inches.
To be in the package, the percent error of the diameter must be less than 5%. One bolt has a diameter of 0.33 inches.
percent error = 0.35 – 0.33.
percent error = 0.02.
5/100 x 0.02 = 0.05 x 0.02.
0.001.
it would go in the package.

Question 15.
A band expects to have 16 songs on their next album. The band writes and records 62.5% more songs than they expect to have in the album. During the editing process, 50% of the songs are removed. How many songs will there be in the final album?

Answer:
The number of songs will there be in the final album = 200.

Explanation:
In the above-given question,
given that,
A band expects to have 16 songs on their next album.
The band writes and records 62.5% more songs than they expect to have in the album.
During the editing process, 50% of the songs are removed.
50% – 62.5% = 12.5%.
12.5% x 16.
200.
so the number of songs will there be in the final album = 200.

Question 16.
Make Sense and Persevere in the first week of July, a record 1,060 people went to the local swimming pool. In the second week, 105 fewer people went to the pool. In the third week, 135 more people went to the pool than in the second week. In the fourth week, 136 fewer people went to the pool than in the third week.
What is the percent change in the number of people who went to the pool between the first and last weeks?

Answer:
The percent change in the number of people who went to the pool between the first and last weeks = 924.

Explanation:
In the above-given question,
given that,
Persevere in the first week of July, a record 1,060 people went to the local swimming pool.
In the second week, 105 fewer people went to the pool. In the third week, 135 more people went to the pool than in the second week.
In the fourth week, 136 fewer people went to the pool than in the third week.
percent change =1060 – 136.
percent change = 924.
so the percent change in the number of people between first and last weeks = 924.

Question 17.
Be Precise You have 20 quarters. You find 40% more quarters in your room. Then you go shopping and spend 50% of the total number of quarters.
a. Write an expression that represents the total number of quarters you take with you when you go shopping.

Answer:
The total number of quarters you take with you when you go shopping = 2 quarters.

Explanation:
In the above-given question,
given that,
You have 20 quarters. You find 40% more quarters in your room.
Then you go shopping and spend 50% of the total number of quarters.
50 – 40 = 10%.
20 x 10/100.
20 x 0.1.
2.
so the total number of quarters you take with you and go shopping = 2 quarters.

b. How much money do you have left?

Answer:
The money I have left = 18 quarters.

Explanation:
In the above-given question,
given that,
You have 20 quarters. You find 40% more quarters in your room.
Then you go shopping and spend 50% of the total number of quarters.
20 – 2 = 18.
so the money I have left = 18 quarters.

Question 18.
Higher Order Thinking The dot plot shows predictions for the winning time in a 200-meter sprint. The winner finished the race in 22.3 seconds. Find the greatest percent error for a prediction to the nearest tenth of a percent. Justify your answer.
Envision Math Common Core 7th Grade Answers Topic 3 Analyze And Solve Percent Problems 48

Answer:
The greatest percent error for a prediction to the nearest tenth of a percent = 22.2 seconds.

Explanation:
In the above-given question,
given that,
The dot plot shows predictions for the winning time in a 200-meter sprint.
The winner finished the race in 22.3 seconds.
200/100 = 0.02%.
22.3 – 0.02 = 22.2 seconds.

Assessment Practice

Question 19.
The amount of money in a savings account increases from $250 to $270 in one month. If the percent increase is the same for every month, how much money will be in the account at the end of the next month?
A. $291.60
B. $295
C. $289.60
D. $300

Answer:
The much money will be in the account at the end of the next month = 0.2.

Explanation:
In the above-given question,
given that,
The amount of money in the savings account increases from $250 to $270 in one month.
270 – 250 = 20.
20% = 20/100.
20/100 = 0.2.

Question 20.
A meteorologist predicted that there would be 1.0 inches of rainfall from a storm. Instead, there were 2.2 inches of rainfall. Which statements are true?
☐ The prediction was off by 35%.
☐ If the percent error should be less than 60%, the prediction was acceptable.
☐ The percent error of the prediction was about 55%.
☐ If the percent error should be less than 20%, the prediction was acceptable.
☐ The difference between the predicted and actual rainfall was 1.2 inches.

Answer:
The statements are true is the difference between the predicted and actual rainfall was 1.2 inches.

Explanation:
In the above-given question,
given that,
A meteorologist predicted that there would be 1.0 inches of rainfall from a storm.
Instead, there were 2.2 inches of rainfall.
2.2 – 1.0 = 1.2.
so the difference between predicted and actual rainfall = 1.2 inches.

3-ACT MATH

3-Act Mathematical Modeling:
The Smart Shopper
Envision Math Common Core 7th Grade Answers Topic 3 Analyze And Solve Percent Problems 49

АСТ 1

Question 1.
After watching the video, what is the first question that comes to mind?
Answer:

Question 2.
Write the Main Question you will answer.
Answer:

Question 3.
Construct Arguments Make a prediction to answer this Main Question. Explain how you arrived at your prediction.
Envision Math Common Core 7th Grade Answers Topic 3 Analyze And Solve Percent Problems 50
Answer:

Question 4.
On the number line below, write a number that is too small to be the answer. Write a number that is too large.
Envision Math Common Core Grade 7 Answer Key Topic 3 Analyze And Solve Percent Problems 51
Answer:

Question 5.
Plot your prediction on the same number line.
Answer:

ACT 2

Question 6.
What information in this situation would be helpful to know? How would you use that information?
Answer:

Question 7.
Use Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.
Answer:

Question 8.
Model with Math Represent the situation using mathematics. Use your representation to answer the Main Question.
Envision Math Common Core Grade 7 Answer Key Topic 3 Analyze And Solve Percent Problems 52
Answer:

Question 9.
What is your answer to the Main Question? Is it higher or lower than your prediction? Explain why.
Envision Math Common Core Grade 7 Answer Key Topic 3 Analyze And Solve Percent Problems 53
Answer:

ACT 3

Question 10.
Write the answer you saw in the video.
Envision Math Common Core Grade 7 Answer Key Topic 3 Analyze And Solve Percent Problems 54
Answer:

Question 11.
Reasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
Answer:

Question 12.
Make Sense and Persevere Would you change your model now that you know the answer? Explain.
Envision Math Common Core Grade 7 Answer Key Topic 3 Analyze And Solve Percent Problems 55
Answer:

ACT 3

Question 13.
Model with Math Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?
Envision Math Common Core Grade 7 Answer Key Topic 3 Analyze And Solve Percent Problems 56
Answer:

Question 14.
Be Precise Describe how you would tell the friends to use their coupons.
Answer:

SEQUEL
Question 15.
Make Sense and Persevere Suppose the $20 coupon requires a purchase of $100 or more. How would that affect your solution?

Answer:
They affect the solution = $80.

Explanation:
In the above-given question,
given that,
Suppose the $20 coupon requires a purchase of $100 or more.
$100 – $20 = $80.
so they affect the solution = $80.

Lesson 3.5 Solve Markup and Markdown Problems

Solve & Discuss It!
Clare subscribes to an online music streaming service for a yearly fee of $96. Starting next month, there will be a 12% increase in the fee.
The ad for another music streaming service is shown below. Should Clare switch? Explain.

I can… solve problems involving percent markup and markdown.
Envision Math Common Core Grade 7 Answer Key Topic 3 Analyze And Solve Percent Problems 57

Model with Math
You can use the percent equation to determine the percent increase.

Focus on math practices
Make Sense and Persevere What is another problem-solving method you could use to check that your solution makes sense?

Essential Question
How are the concepts of percent markup and percent markdown related to the percent equation?

Try It!

What is the percent markup on a $300 phone sold for $465?
Envision Math Common Core Grade 7 Answer Key Topic 3 Analyze And Solve Percent Problems 58
markup = percent markup • cost
________ = p • 300
________ = p
The percent markup on the phone is _________%.

Answer:
The percent markup on the phone is 1.55%.

Explanation:
In the above-given question,
given that,
the percent markup on a $300 phone sold for $465.
markup = percent markup x cost.
markup = p x 300.
465/300 = p.
p = 1.55.
so the percent markup on the phone is 1.55%.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-3-Analyze And Solve Percent Problems-17

Convince Me! How does the percent equation help solve markup problems?

Try It!

What is the selling price for a $45 pair of shoes with a 15% markup?

Answer:
The markup = 0.003.

Explanation:
In the above-given question,
given that,
the selling price = $45.
markup = 15%.
markup = percent markup x cost.
15% = p x $45.
0.15 = p x $45.
p = 0.15/45.
p = 0.003.

Try It!

Find the percent markdown for an $80 jacket that is on sale for $48.

Answer:
The markup = $32.

Explanation:
In the above-given question,
given that,
the selling price = $45.
markup = 15%.
markdown = $80 – $48.
markdown = $32.

KEY CONCEPT
You can solve markup and markdown problems using the percent equation.
Envision Math Common Core Grade 7 Answer Key Topic 3 Analyze And Solve Percent Problems 59

Do You Understand?
Question 1.
Essential Question How are the concepts of markup and markdown related to the percent equation?

Answer:
markup and markdown are proportional to each other.

Explanation:
In the above-given question,
given that,
markup = percent markup x original value.
markdown = percent markdown x original value.
markup = new value is greater than the original value.
markdown = new value is less than the original value.

Question 2.
Reasoning What does the amount of the markup or markdown represent in the percent equation?

Answer:
The amount of the markup or markdown is proportional to each other.

Explanation:
In the above-given question,
given that,
markup = percent markup x original value.
markdown = percent markdown x original value.
so the amount of the markup or markdown is proportional to each other.

Question 3.
Generalize When an item is marked up by a certain percent and then marked down by the same percent, is the sale price equal to the price before the markup and markdown?

Answer:
markup = percent markup x original value.
markdown = percent markdown x original value.

Explanation:
In the above-given question,
given that,
When an item is marked up by a certain percent and then marked down by the same percent.
markup = percent markup x original value.
markdown = percent markdown x original value.

Do You Know How?
Question 4.
An item costs $4 before tax and $4.32 after sales tax. What is the sales tax rate?
Envision Math Common Core Grade 7 Answer Key Topic 3 Analyze And Solve Percent Problems 60

Answer:
The sales tax rate = $8.32.

Explanation:
In the above-given question,
given that,
An item costs $4 before tax and $4.32 after sales tax.
$4 + $4.32.
$8.32.
so the sales tax rate = $8.32.

Question 5.
Sheila buys two concert tickets from her friend. She pays $90 for the two tickets. She looks at the tickets and sees that each ticket has a face value of $52.50.
a. How much of a markdown did her friend give Sheila? Explain how you know.

Answer:
The percent markdown = $1.174.

Explanation:
In the above-given question,
given that,
Sheila buys two concert tickets from her friend. She pays $90 for the two tickets.
She looks at the tickets and sees that each ticket has a face value of $52.50.
markdown = percent markdown x original value.
$52.50 = percent markdown x $90.
percent markdown = $90/$52.50.
percent markdown = $1.714.

b. What was the percent markdown, rounded to the nearest whole percent?

Answer:
The percent markdown rounded to the nearest whole percent = $2.

Explanation:
In the above-given question,
given that,
Sheila buys two concert tickets from her friend. She pays $90 for the two tickets.
She looks at the tickets and sees that each ticket has a face value of $52.50.
markdown = percent markdown x original value.
$52.50 = percent markdown x $90.
percent markdown = $90/$52.50.
percent markdown = $1.714.
$2.

Question 6.
Find the sale prices.
a. $4,200 with a 35% markdown

Answer:
The sale price = $1470.

Explanation:
In the above-given question,
given that,
the sales price = percent markdown x original value.
sales price = 35% x $4200.
sales price = 0.35 x $4200.
sales price = $1470.

b. $5,000 with a 44% markdown

Answer:
The sale price = $2200.

Explanation:
In the above-given question,
given that,
the sales price = percent markdown x original value.
sales price = 44% x $5000.
sales price = 0.44 x $5000.
sales price = $2200.

Practice & Problem Solving

Leveled Practice In 7-8, fill in the boxes to solve.
Question 7.
A $300 suit is marked down by 20%. Find the sale price rounded to the nearest dollar.
markdown = percent markdown • original price
markdown = $_________
original price – markdown = sale price
$ _______ – $ ______
sale price = $ ________

Answer:
The sale price = $300.

Explanation:
In the above-given question,
given that,
A $300 suit is marked down by 20%.
sale price = original price – markdown.
sale price = $300 – 20%.
sale price = $300 – 0.2.
sale price = $299.8.

Question 8.
The selling price of an item is $650 marked up from the wholesale cost of $450. Find the percent markup from wholesale cost to selling price.
selling price – markup = wholesale cost
$ _______ – $ ______ = $ _______
markup = percent markup × wholesale cost
_______ = _______ × _______
The percent markup is about ________.

Answer:
The markup = $250.

Explanation:
In the above-given question,
given that,
The selling price of an item is $650 marked up from the wholesale cost of $450.
wholesale cost = selling price – markup.
$450 = $650 – markup.
markup = $250.

Question 9.
Karen purchased the DVD player shown in the sign on the right. Find the percent markdown rounded to the nearest percent.
Envision Math Common Core Grade 7 Answer Key Topic 3 Analyze And Solve Percent Problems 61

Answer:
The markdown = 27,048.143.

Explanation:
In the above-given question,
given that,
Karen purchased the DVD player shown in the sign on the right.
markdown = percent markdown x original value.
markdown = $175.90 x $153.77.
markdown = 27,048.143.

Question 10.
A store manager instructs his employees to mark up all items by 30%. A store clerk puts a price tag of $30 on an item that the store bought for $27. As an employee, you notice that this selling price is incorrect.
a. Find the correct selling price. Round to the nearest dollar.

Answer:
The correct selling price is $3.

Explanation:
In the above-given question,
given that,
A store manager instructs his employees to mark up all items by 30%.
A store clerk puts a price tag of $30 on an item that the store-bought for $27.
selling price = original price – mark up.
selling price = $30 – $27.
selling price = $3.
so the correct selling price = $3.

b. What was the clerk’s likely error?

Answer:
The clerk’s likely error is he said his employees to mark up all items by 30%.

Explanation:
In the above-given question,
given that,
A store clerk puts a price tag of $30 on an item that the store-bought for $27.
selling price = original price – mark up.
selling price = $30 – $27.
selling price = $3.
so the correct selling price = $3.

Question 11.
Nate has $50 to spend at the grocery store. He fills his shopping cart with items totaling $46. At checkout he will have to pay 6% sales tax on all items in the cart. Does he have enough money to buy everything in his cart? Explain.

Answer:
Enough money to buy everything in his cart = $96.

Explanation:
In the above-given question,
given that,
Nate has $50 to spend at the grocery store.
He fills his shopping cart with items totaling $46.
$50 + $46.
$96.
enough money to buy everything in his cart = $96.

Question 12.
A department store buys 300 shirts at a cost of $1,800 and sells them for $10 each. Find the percent markup rounded to the nearest percent.

Answer:
The percent markup = $180.

Explanation:
In the above-given question,
given that,
A department store buys 300 shirts at a cost of $1,800 and sells them for $10 each.
markup = percent markup x original price.
$10 = p x $1800.
p = $180.
so the percent markup rounded to the nearest percent = $180.

Question 13.
Make Sense and Persevere A computer store buys a computer system at a cost of $465.60. The selling price was first at $776, but then the store advertised a 30% markdown on the system.
a. Find the current sale price. Round to the nearest cent if necessary.

Answer:
The sales price nearest cent = $310.

Explanation:
In the above-given question,
given that,
A computer store buys a computer system at a cost of $465.60.
The selling price was first at $776, but then the store advertised a 30% markdown on the system.
sales price = original price – markdown.
sales price = $465.60 – $776.
sales price = 310.4.
so the sales price nearest cent = $310.

b. Members of the store’s loyalty club get an additional 10% off their computer purchases. How much do club members pay for the computer with their discount?

Answer:

Explanation:
In the above-given question,
given that,
Members of the store’s loyalty club get an additional 10% off their computer purchases.

Question 14.
Higher Order Thinking A sporting goods store manager was selling a kayak set for a certain price. The manager offered the markdowns shown on the right, making the one-day sale price of the kayak set $328. Find the original selling price of the kayak set.
Envision Math Common Core Grade 7 Answer Key Topic 3 Analyze And Solve Percent Problems 62

Answer:
The selling price = $312.2.

Explanation:
In the above-given question,
given that,
A sporting goods store manager was selling a kayak set for a certain price.
The manager offered the markdowns shown on the right, making the one-day sale price of the kayak set $328.
original selling price = markdown x 40%.
selling price = $328 x 0.4.
selling price = $131.2.

Assessment Practice

Question 15.
Eliza cannot decide which of two bicycles to buy. The original price of each is $380. The first is marked down by 50%. The second is marked down by 30% with an additional 20% off. Which bicycle should Eliza buy if the bicycles are the same except for the selling price? Explain your answer.

Answer:
The selling price = 0.16.

Explanation:
In the above-given question,
given that,
Eliza cannot decide which of two bicycles to buy.
The original price of each is $380.
The first is marked down by 50%.
The second is marked down by 30% with an additional 20% off.
original selling price = markdown x 20%.
selling price = 80% x 0.2.
selling price = 0.16.

Question 16.
A shoe store uses a 50% markup for all of the shoes it sells. The store also charges a 10% sales tax on all purchases. What would be the purchase price, including sales tax, of a pair of shoes that has a wholesale cost of $57?

Answer:
The purchase price, including sales tax, of a pair of shoes that has a wholesale cost = $2.28.

Explanation:
In the above-given question,
given that,
A shoe store uses a 50% markup for all of the shoes it sells.
The store also charges a 10% sales tax on all purchases.
wholesale cost = 40% x $57.
wholesale cost = 0.04 x 57.
wholesale cost = $2.28.

Lesson 3.6 Solve Simple Interest Problems

Explore It!
Gerard compares the offers at two different banks to decide where he should open a savings account.
I can… apply percent reasoning to solve simple interest problems.
Envision Math Common Core Grade 7 Answer Key Topic 3 Analyze And Solve Percent Problems 63

A. Draw a representation to show how much would be in the first savings account if Gerard’s initial deposit were d dollars.
Answer:

B. Draw a representation to show how much would be in the second savings account if Gerard’s initial deposit were d dollars.
Answer:

C. Use the two representations you drew to explain how the offers at the two banks are similar and how they are different.
Answer:

Focus on math practices
Construct Arguments Gerard’s first deposit is $500. Which bank should he choose? Explain.

Essential Question
How does simple interest show proportional reasoning and relate to the percent equation?

Try It!

Victoria has another account at the bank that pays 2\(\frac{1}{2}\)% simple interest. How much interest will she earn in 8 years on an initial deposit of $250 assuming she neither adds to nor withdraws from the account?
2\(\frac{1}{2}\)% interest expressed as a decimal is ________.
Interest after 1 year: s = _______ • $ _______
= $ _______
Interest after 8 years: $ ______ • _______
= $ ______
Victoria will earn $ _______ in interest over 8 years.

Answer:
The victoria will earn $2000 in 8 years.

Explanation:
In the above-given question,
given that,
Victoria has another account at the bank that pays 2\(\frac{1}{2}\)% simple interest.
Interest after 1 year = 1 x 250.
Interest after 8 years = 8 x 250.
2000.
so victoria will earn $2000 in 8 years.

Convince Me! Would the interest for the second year be the same if it were calculated on the total after the first year? Why or why not?

Try It!

Another company will lend Maya’s older sister $4,000. Every month, she will pay $11.88 in interest. What is the interest rate, rounded to the nearest tenth of a percent, for 1 year?

Answer:
The interest rate rounded to the nearest tenth of a percent = $3857.44.

Explanation:
In the above-given question,
given that,
Another company will lend Maya’s older sister $4,000.
Every month, she will pay $11.88 in interest.
1 year = 12 months.
$11.88 x 12 = $142.56.
4000 – $142.56 = 3857.44.
so the interest rate rounded to the nearest tenth of a percent = $3857.44.

Try It!

Katelyn’s older brother borrowed money for school. He took out a loan that charges 6% simple interest. He will end up paying $720 in interest after 6 years. How much did Katelyn’s brother borrow for school?

Answer:
Katelyn’s brother borrows for school = $120.

Explanation:
In the above-given question,
given that,
Katelyn’s older brother borrowed money for school.
He took out a loan that charges 6% simple interest.
He will end up paying $720 in interest after 6 years.
$720/6 = 120.
so Katelyn’s brother borrows for school = $120.

KEY CONCEPT
Simple interest represents a proportional relationship between the yearly interest and the principal, or initial amount. The ratio of yearly interest to principal is the interest rate.
Envision Math Common Core Grade 7 Answer Key Topic 3 Analyze And Solve Percent Problems 64

Do You Understand?
Question 1.
Essential Question How does simple interest show proportional reasoning and relate to the percent equation?

Answer:
Simple interest represents a proportional relationship between the yearly interest and the principal or initial amount.

Explanation:
In the above-given question,
given that,
Simple interest represents a proportional relationship between the yearly interest and the principal or initial amount.
the ratio of yearly interest to principal to the interest rate.

Question 2.
Reasoning If the interest earned on an account after 2 years is $15, how much would it be after 10 years? Why?

Answer:
The interest earned on an account after 10 years = $60.

Explanation:
In the above-given question,
given that,
If the interest earned on an account after 2 years is $15.
2 years = $15.
10 years = $15 x $5.
10 years = $60.
so the interest earned on an account after 10 years = $60.

Question 3.
Be Precise Angelina is deciding which bank would give her the best simple interest rate on a $300 deposit. One bank says that she will have $320 in her account if she leaves the principal for 2 years. Is this enough information for Angelina to find the interest rate? Explain.

Answer:
Yes, it is enough information for Angelina to find the interest rate.

Explanation:
In the above-given question,
given that,
Angelina is deciding which bank would give her the best simple interest rate on a $300 deposit.
One bank says that she will have $320 in her account if she leaves the principal for 2 years.
2 years = 24 months.
for 1 month = 1 rs.
do this is enough information.

Do You Know How?
Question 4.
Find the missing value in each row. Use the percent equation.
Envision Math Common Core Grade 7 Answers Topic 3 Analyze And Solve Percent Problems 65

Answer:
$15.
4 years.
$50.
1%.

Explanation:
In the above-given question,
given that,
100 x 5/100 x 3 = $15.
$500 x 4/100 x 4 = $20.
$50 x 10/100 x 7 = $35.
$200 x 1/100 x 2 = $6.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-3-Analyze And Solve Percent Problems-18

Question 5.
Annika’s older cousin borrowed $800 to repair her car. She will pay off the loan after 2 years by paying back the principal plus 4.5% simple interest for each year.
a. How much will she pay in interest? Show your work.

Answer:
She will pay = $72.

Explanation:
In the above-given question,
given that,
Annika’s older cousin borrowed $800 to repair her car.
She will pay off the loan after 2 years by paying back the principal plus 4.5% simple interest for each year.
800 x 2 x 4.5%.
8 x 2 x 4.5.
$72.
so she will pay = $72.

b. How much will she pay back altogether?

Answer:
She will pay back = $72.

Explanation:
In the above-given question,
given that,
Annika’s older cousin borrowed $800 to repair her car.
She will pay off the loan after 2 years by paying back the principal plus 4.5% simple interest for each year.
800 x 2 x 4.5%.
8 x 2 x 4.5.
$72.
so she will pay back = $72.

Question 6.
J.D. opened a savings account with $425. After 2 years, the total interest he earned was $10.20. What was the annual interest rate?

Answer:
The annual interest rate = $870.4.

Explanation:
In the above-given question,
given that,
J.D. opened a savings account with $425.
After 2 years, the total interest he earned was $10.20.
$10.20 + $425 = $435.20.
$435.20 x 2 = $870.4.

Practice & Problem Solving

Leveled Practice In 7-8, fill in the boxes to solve.
Question 7.
Edward deposited $6,000 into a savings account 4 years ago. The simple interest rate is 3%. How much money did Edward earn in interest?
Interest = $ ______ • ______ • ______ years
Edward earned $ _______ in interest

Answer:
Edward earned = $720.

Explanation:
In the above-given question,
given that,
Edward deposited $6,000 into a savings account 4 years ago.
The simple interest rate is 3%.
interest = $6000 x 4 x 3/100.
interest = $60 x 4 x 3.
interest = $720.
so Edward earned = $720.

Question 8.
The interest on $2,000 for 2 years is $320. What is the simple interest rate?
$ ______ = $2,000 • r • ______
$ _____ = $ _____ • r
______ = r
_______% = r

Answer:
The simple interest rate = 8%.

Explanation:
In the above-given question,
given that,
The interest on $2,000 for 2 years is $320.
$320 = $2000 x r x 2.
$320 x 2 = $640.
$640 x $2000 = 8.

Question 9.
Suppose you deposited $100 in a savings account 4 years ago with a simple interest rate of 2.2%. The interest that you earned in those 4 years is $8.80. Which of the following is true? Select all that apply.
☐ The interest rate is 0.022.
☐ The principal was $100.
☐ The interest earned is $4.
☐ The account was opened 8 years and 8 months ago

Answer:
Option B, C, and D are correct.

Explanation:
In the above-given question,
given that,
Suppose you deposited $100 in a savings account 4 years ago with a simple interest rate of 2.2%.
The interest that you earned in those 4 years is $8.80.
the principal was $100.
the interest earned is $4.
the account was opened 8 years and 8 months ago.

Question 10.
A new bank customer with $3,000 wants to open a money market account. The bank is offering a simple interest rate of 1.1%.
a. How much interest will the customer earn in 20 years?

Answer:
The interest will the customer earn in 20 years = $660.

Explanation:
In the above-given question,
given that,
A new bank customer with $3,000 wants to open a money market account.
The bank is offering a simple interest rate of 1.1%.
3000 x 11/1000 x 20.
60 x 11 = 660.
so the interest will the customer earn in 20 years = 660.

b. What will be the account balance after 20 years?

Answer:
The interest will the customer earn in 20 years = $660.

Explanation:
In the above-given question,
given that,
A new bank customer with $3,000 wants to open a money market account.
The bank is offering a simple interest rate of 1.1%.
3000 x 11/1000 x 20.
60 x 11 = 660.
so the interest will the customer earn in 20 years = 660.

Question 11.
Boden’s account has a principal of $500 and a simple interest rate of 3.3%. Complete the double number line. How much money will be in the account after 4 years, assuming Boden does not add or take out any money?
Envision Math Common Core Grade 7 Answers Topic 3 Analyze And Solve Percent Problems 66

Answer:
The money will be in the account after 4 years = $66.

Explanation:
In the above-given question,
given that,
Boden’s account has a principal of $500 and a simple interest rate of 3.3%.
$500 x 4 x 3.3.
500 x 4 x 33/1000.
20 x 3.3 = $66.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-3-Analyze And Solve Percent Problems-19

Question 12.
Critique Reasoning Monica deposits $100 into a savings account that pays a simple interest rate of 3.4%. Paul deposits $200 into a savings account that pays a simple interest rate of 2.2%. Monica says that she will earn more interest in one year because her interest rate is higher. Do you agree? Justify your response.

Answer:
Monica says that she will earn more interest in one year because her interest is higher.

Explanation:
In the above-given question,
given that,
Monica deposits $100 into a savings account that pays a simple interest rate of 3.4%.
Paul deposits $200 into a savings account that pays a simple interest rate of 2.2%.
100 x 3.4 =
34.
200 x 2.2 = 8.8.

Question 13.
Construct Argument Tommy earned $76.00 in interest after 5 years on a principal of $400. Jane earned $82.00 in interest after 2 years on a principal of $1,000. Which bank would you rather use, Tommy’s or Jane’s? Why?

Answer:
I will use Jane’s bank.

Explanation:
In the above-given question,
given that,
Tommy earned $76.00 in interest after 5 years on a principal of $400.
Jane earned $82.00 in interest after 2 years on a principal of $1,000.
400 x 5 x $76 = $152000.
1000 x 82 x 2 = $164000.
Jane’s bank will get the higher principal.

Question 14.
Reasoning A bank manager wants to encourage new customers to open accounts with initial deposits of at least $3,000. He has posters made for the promotion.
Envision Math Common Core Grade 7 Answers Topic 3 Analyze And Solve Percent Problems 67
a. Under the new promotion, what is the minimum amount of interest a new account would make in one year if there were no withdrawls from the account?

Answer:
There were no withdrawals from the account = $144.

Explanation:
In the above-given question,
given that,
A bank manager wants to encourage new customers to open accounts with initial deposits of at least $3,000.
simple interest of 4.8%.
4.8 x 1 x 3000.
3000 x 48/1000 x 1.
$144.

b. The manager wants to add the sentence, “Open an account with $3,000 and earn at least $120 interest each year!” to the poster. Do you agree? Explain.

Answer:
Yes, the sentence was correct.

Explanation:
In the above-given question,
given that,
The manager wants to add the sentence, “Open an account with $3,000 and earn at least $120 interest each year!.
4.8 x 1 x 3000.
3000 x 48/1000 x 1.
$144.
so 144 is greater than $120.
the sentence was correct.

Question 15.
Suppose you deposit $100 in Account A with a simple interest rate of 3.4%, and $300 in Account B with a simple interest rate of 1.8%. One year later, you get a bank statement that shows the interest for Account A is $3.40 and the interest for Account B is $540.00.
a. Which account statement is incorrect?

Answer:
Yes, the account B statement is incorrect.

Explanation:
In the above-given question,
given that,
Suppose you deposit $100 in Account A with a simple interest rate of 3.4%, and $300 in Account B with a simple interest rate of 1.8%.
One year later, you get a bank statement that shows the interest for Account A is $3.40 and the interest for Account B is $540.00.
3.4 x 100.
34/1000 x 100.
3.4 x 1.
3.4.
1.8% x 300.
18/1000 x 300.
1.8 x 3.
5.4.
so the bank statement is incorrect for Account B.

b. What may have been the bank’s error?

Answer:
The error is account B.

Explanation:
In the above-given question,
given that,
Suppose you deposit $100 in Account A with a simple interest rate of 3.4%, and $300 in Account B with a simple interest rate of 1.8%.
One year later, you get a bank statement that shows the interest for Account A is $3.40 and the interest for Account B is $540.00.
3.4 x 100.
34/1000 x 100.
3.4 x 1.
3.4.
1.8% x 300.
18/1000 x 300.
1.8 x 3.
5.4.

Question 16.
Higher Order Thinking You have two different savings accounts. For Account A, the interest earned after 18 months is $12.00. For Account B, the interest earned after 27 months is $27.00.
a. If the interest rate is 3.2% for Account A, how much is the principal?

Answer:
The principle is $6.912.

Explanation:
In the above-given question,
given that,
You have two different savings accounts.
For Account A, the interest earned after 18 months is $12.00.
For Account B, the interest earned after 27 months is $27.00.
$12 x 18 x 3.2%.
$12 x 18 x 0.032.
$12 x 0.576.
$6.912.

b. If the interest rate is 2.4% for Account B, how much is the principal?

Answer:
The principle is $17.496.

Explanation:
In the above-given question,
given that,
You have two different savings accounts.
For Account A, the interest earned after 18 months is $12.00.
For Account B, the interest earned after 27 months is $27.00.
27 x $27 x 0.024.
$27 x 0.648.
17.496.

c. Which account earned you the most interest in the first year? Explain.

Answer:
Account B has earned the most interest in the first year.

Explanation:
In the above-given question,
given that,
You have two different savings accounts.
For Account A, the interest earned after 18 months is $12.00.
For Account B, the interest earned after 27 months is $27.00.
27 x $27 x 0.024.
$27 x 0.648.
17.496.

Assessment Practice

Question 17.
A certificate of deposit with principal $150 earns 4% annual interest for 2 years. Select all the options that would earn the same amount of interest.
☐ $300 at 2% for 2 years
☐ $150 at 6% for 18 months
☐ $400 at 3% for 1 year
☐ $100 at 1% for 8 years
☐ $50 at 48% for 6 months

Answer:
Option A and option C are correct.

Explanation:
In the above-given given question,
given that,
A certificate of deposit with a principal of $150 earns 4% annual interest for 2 years.
$150 x 4% x 2.
$150 x 4/100 x 2.
$15 x 0.4 x 2.
$12.
so option A and option C are correct.

Question 18.
Dakota earned $15.75 in interest in Account A and $28.00 in interest in Account B after 21 months. If the simple interest rate is for 3.0% for Account A and 4.0% for Account B, which account had the greater principal? Explain.

Answer:
Account B had the greater principal.

Explanation:
In the above-given question,
given that,
Dakota earned $15.75 in interest in Account A and $28.00 in interest in Account B after 21 months.
If the simple interest rate is 3.0% for Account A and 4.0% for Account B, which account had the greater principal.
$15.75 x 3% x 21.
15.75 x 0.003 x 21.
0.99225.
$28 x 21 x 4%.
$28 x 21 x 0.04.
$23.52.
so account B had the greater principal.

Topic 3 Review

Topic Essential Question
How can percents show proportional relationships between quantities and be used to solve problems?

Vocabulary Review
Complete each definition and then provide an example of each word.
Envision Math Common Core Grade 7 Answers Topic 3 Analyze And Solve Percent Problems 68

Answer:
1: Markup simple interest.
2: Percent change.
3: Markup principal.
4: Vocabulary.

Explanation:
In the above-given question,
given that,
the markup simple interest is the decrease from the original price of an item to its sale price.
the accuracy of a measured or estimated value compared to its actual value is described as the percent change.
an initial amount of money that is deposited in an account is called the markup principal.
the vocabulary describes how much a quantity has changed relative to its original amount.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-3-Analyze And Solve Percent Problems-20

Use Vocabulary in Writing
Birute deposits $500 in a savings account with a simple interest rate of 1.3%. How could you use this information to find the interest she would earn in 4 years and determine the percent change in her savings account?

Answer:
The percent change in her savings account.

Explanation:
In the above-given question,
given that,
Birute deposits $500 in savings account with a simple interest rate of 1.3%
How could you use this information to find the interest she would earn in 4 years and determine the percent change.
$500 x 1.3% x 4.
$500 x 13/1000 x 4.
5 x 1.3 x 4.
26.
so the percent change in her savings account.

Concepts and Skills Review

Lesson 3.1 Analyze Percents of Numbers

Quick Review
You can use equivalent ratios to find the percent of a number. Remember that a percent is a ratio that relates a number to 100.

Example
Find 57% of 690.
\(\frac{57}{100}=\frac{x}{690}\)
\(\frac{57}{100} \cdot 690=\frac{x}{690} \cdot 690\)
393.3 = x

Practice
Question 1.
Find 0.8% of 1,046.

Answer:
8.368.

Explanation:
In the above-given question,
given that,
0.8% of 1046.
0.8/100 x 1046.
0.008 x 1046.
8.368.

Question 2.
Find 160% of 98.

Answer:
156.8.

Explanation:
In the above-given question,
given that,
160% of 98.
160/100 x 98.
1.6 x 98.
156.8.

Question 3.
A company charges a shipping fee that is 4.5% of the purchase price for all items it ships. What is the fee to ship an item that costs $56?

Answer:
The fee to ship an item that costs = $2.52.

Explanation:
In the above-given question,
given that,
company charges a shipping fee that is 4.5% of the purchase price for all items it ships.
$56 x 4.5%.
56 x 45/1000.
$56 x 0.045.
$2.52.

Lesson 3.2 Connect Percent and Proportion

Quick Review
You can use proportions to solve different types of percent problems. There are three values in a percent problem–the percent, the part, and the whole. If you know two of these values, you can set up a proportion to find the third value.

Example
What percent of 19 is 4.75?
The whole is 19 and the part is 4.75. Write a proportion, and solve for the percent, p.
\(\frac{\text { part }}{\text { whole }}=\frac{p}{100}\)
\(\frac{4.75}{19}=\frac{p}{100}\)
\(\frac{4.75}{19} \cdot 100=\frac{p}{100} \cdot 100\)
25 = p
So, 4.75 is 25% of 19.

Practice
Question 1.
24.94 is 29% of what number?

Answer:
The number is 7.3.

Explanation:
In the above-given question,
given that,
24.94 is 29% of what number.
24.94 / p = 29/100.
p = 0.29 x 24.94.
p = 7.3.

Question 2.
On Thursday, a restaurant serves iced tea to 35 of its 140 customers. What percent of the customers order iced tea?

Answer:
The percent of the customers who order iced tea = 25%.

Explanation:
In the above-given question,
given that,
On Thursday, a restaurant serves iced tea to 35 of its 140 customers.
35/140 = p/100.
0.25 x 100 = p.
p = 25%.

Question 3.
Liam puts $40 in savings in March and 175% of this amount in savings in April. How much does Liam put in savings in April?

Answer:

whole = 100
4,75 = 100 4,175. 100 = 70100

Lesson 3.3 Represent and Use the Percent Equation

Quick Review
You can use the percent equation to solve percent problems.
part = percent • whole
Substitute two of the three values to solve for the unknown value.

Example
Michael earns a 6% commission on each house he sells. If he sells a house for $180,000, how much does he earn in commission, c?
c = 0.06. 180,000
c = 10,800
Michael earns $10,800 in commission.

Practice
Question 1.
Sharon paid $78 sales tax on a new camera. If the sales tax rate is 6.5%, what was the cost of the camera?

Answer:

Question 2.
There are 45 students who play a woodwind instrument in the school band. Of these, 18 play the saxophone. What percent of these students play the saxophone?
Answer:

Lesson 3.4 Solve Percent Change and Percent Error Problems

Quick Review
A percent change can be an increase or a decrease. You can use an equation to find a percent change.
change = percent change • original amount
A percent error is always a nonnegative value. You can use an equation to find a percent error.
difference = percent error • actual

Example
Juan’s puppy weighed 16 pounds at the age of 2 months. The puppy weighed 60 pounds at the age of 8 months. What is the percent change in the puppy’s weight?
change in pounds = 60 – 16 = 44
44 = P • 16
\(\frac{44}{16}\) = P
2.75 = P
The puppy’s weight increased by 275%.

Practice
Question 1.
In 2014, the attendance at Jefferson School’s Fall Festival was 650. In 2015, the attendance was 575. What was the percent change in attendance from 2014 to 2015? Round to the nearest whole percent.
Answer:

Question 2.
Melissa estimated that she would read 250 pages last week. She read 290 pages. What is the percent error of Melissa’s estimate? Round to the nearest whole percent.
Answer:

Lesson 3.5 Solve Markup and Markdown Problems

Quick Review
You can use the percent equation to solve markup and markdown problems.
markup = percent markup • cost
markdown = percent markdown • selling price

Example
Bree buys a purse for $90. She sells it at her store for $135. What is the percent markup on the purse?
markup = $135 – $90 = $45
45 = P • 90
45 = P
0.5 =P The decimal 0.5 is equivalent to 50%.
The percent markup on the purse is 50%.

Practice
Question 1.
Hank buys a used car for $7,200 and plans to sell it on his used car lot. What is the price of the car after a markup of 15%?

Answer:
The price of the car after a markup of 15% =

Explanation:
In the above-given question,
given that,
Hank buys a used car for $7,200 and plans to sell it on his used car lot.
part = percent x whole.
part = 15 x 7200.
part = 15/100 x 7200.
part = 1080.
so the price of the car after a markup of 15% = 1080.

Question 2.
Nya wants to buy a sweater that had an original price of $55. The sweater is now discounted 20% and the sales tax rate is 5.5%. How much will Nya pay for the sweater?

Answer:
Nya pays for the sweater.

Explanation:
In the above-given question,
given that,
Nya wants to buy a sweater that had an original price of $55.
The sweater is now discounted 20% and the sales tax rate is 5.5%.
$55 x 20/100 x 5.5%.
55 x 0.2 x 55/1000.
55 x 0.2 x 0.055.
0.605.
so Nya pays for the sweater = 0.605.

Lesson 3.6 Solve Simple Interest Problems

Quick Review
You consider four quantities when solving problems involving simple interest.
initial amount, or principal, p
interest rate, r
time, t amount of simple interest, I

Example
Yuni loaned $400 to her brother. He will repay the loan by paying 2.5% simple interest for 3 years. How much will he pay in interest?
I = 400 • 0.025 • 3 = $30
He will pay $30 in interest.

Practice
Question 1.
Ethan put $700 into a Certificate of Deposit (CD) account that earns 1.8% interest each year. What will the interest of the CD account be after 6 years?

Answer:
The interest of the CD account is after 6 years = $75.6.

Explanation:
In the above-given question,
given that,
Ethan put $700 into a Certificate of Deposit (CD) account that earns 1.8% interest each year.
$700 x 1.8% x 6.
$700 x 1.8/100 x 6.
$700 x 0.018 x 6.
$75.6.

Question 2.
Kelly opened a bank account that earns 1.2% simple interest each year. After 7 years, Kelly will earn $126 in interest. How much did Kelly deposit when she opened the account?

Answer:
The kelly deposit when she opened the account = $10.584.

Explanation:
In the above-given question,
given that,
Kelly opened a bank account that earns 1.2% simple interest each year.
After 7 years, Kelly will earn $126 in interest.
$126 x 7 x 1.2%.
$126 x 7 x 1.2/100.
$126 x 7 x 0.012.
$126 x 0.084.
$10.584.
so the kelly deposit when she opened the account = $10.584.

Topic 3 Fluency Practice

Riddle Rearranging
Find the value of x in each unit rate. Then arrange the answers in order from least to greatest. The letters will spell out the answer to the riddle below.
I can… find unit rates with ratios of fractions.
Envision Math Common Core Grade 7 Answers Topic 3 Analyze And Solve Percent Problems 69

Answer:
0.75 xc = 0.3h, 0.4 min = 0.6 ml, and 0.25 mi = 0.625 h.

Explanation:
In the above-given question,
given that,
3/4c/1/3h = xc/1h.
0.75c/0.3h = xc/h.
0.75c x xc = 0.3h x h.
0.75xc = 0.3h.
3/5ml/2/5 min = x ml /1min.
0.6ml/0.4 min = x ml / 1 min.
0.4 x 1 min = x ml x 0.6 ml.
0.4 min = 0.6ml.
1/4mi/5/8 h = x mi / 1h.
0.25 mi/ 0.625 h = x mi / 1h.
x mi /1 h = 0.25 mi/0.625 h.

enVision Math Common Core Grade 4 Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers

Go through the enVision Math Common Core Grade 4 Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers regularly and improve your accuracy in solving questions.

enVision Math Common Core 4th Grade Answers Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers

Essential Questions:
How can you use a model to multiply? How can you use the Distributive Property to multiply? How can you use multiplication to solve problems?
Answer:
Multiplication using  the “Area model”:
We know that,
An “Area” is the amount of flat space occupied by a figure, You can find the area of a figure by filling it with unit squares and then counting how many squares entirely fill the shape
The total area of the figure is the multiplication of the number of rows and columns
Multiplication using the Distributive Property:
Let the three numbers be a, b, c
Now,
According to the Distributive Property of Multiplication,
a × (b + c) = (a × b) + (a × c)
(a + b) × c = (a × c ) + (b × c)
Ex:
We have to find the product of 12 × 4
Now,
The steps to find the product of the given expression using the Distributive Property of Multiplication are:
Step 1:
Simplify the terms
So,
12 × 4 = (10 + 2) × 4
Step 2:
Use the Distributive Property of Multiplication on the simplified expression that we obtained in Step 1
So,
(10 + 2) × 4 = (10 × 4) + (2 × 4)
= 40 + 8
Step 3:
Add the values that we obtained in the last step of Step 2
So,
40 + 8 = 48
Hence,
The value of the given expression is: 48

Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 1

enVision STEM Project: Renewable Energy and Multiplication
Do Research
Use the Internet or other sources to find information about different sources of renewable energy?
Answer:
The major types of renewable energy sources are:
A. Biomass:
Obtained from wood and wood waste, municipal solid waste etc
B. Hydropower energy:
Obtained from water
C. Geothermal energy:
Obtained from land
D. Wind energy:
Obtained from wind turbines
E. Solar energy:
Obtained from the sun

Use Strategies and Properties to Multiply by 2-Digit Numbers 1

Journal: Write a Report that Includes what you found. Also in your report:
A)
A wind farm is an area of land with a large number of turbines. Draw an array with 15 rows to show the turbines in a wind farm. How many turbines are in your wind farm?
Answer:
It is given that
A wind farm is an area of land with a large number of turbines
Now,
The representation of an array with 15 rows to show the turbines in a wind farm is:

Hence, from the above figure,
We can conclude that there are 15 turbines in your wind farm

B)
Some turbines produce 63 megawatt-hours of energy every week. Find how much energy one of these turbines would produce in one year. Remember, one year is 52 weeks?
Answer:
It is given that
Some turbines produce 63 megawatt-hours of energy every week.
So,
The amount of energy these turbines produce in 1 year = (The amount of energy produced by one of these turbines in a week) × (The number of weeks in a year)
= 63 × 52
= 63 × (50 + 2) [By using the Distributive Property of Multiplication]
= (63 × 50) + (63 × 2)
= 3,150 + 126
= 3,276 megawatt-hours
Hence, from the above,
We can conclude that the amount of energy produced by one of these turbines in one year is: 3,276 megawatt-hours

Review What You Know

Vocabulary
Choose the best term from the box. Write it on the blank.

  • algorithm
  • product
  • variable
  • array

Question 1.
You multiply numbers to find a(n) __________.
Answer:
We know that,
You multiply numbers to find a “Product”
Hence, from the above,
We can conclude that the best term that fits the given definition is a “Product”

Question 2.
A(n) _________ shows the number of objects in rows and columns.
Answer:
We know that,
An “Array” shows the number of objects in rows and columns.
Hence, from the above,
We can conclude that the best term that fits the given definition is an “Array”

Question 3.
A symbol or letter that stands for a number is called a(n) ______________.
Answer:
We know that,
A symbol or letter that stands for a number is called a “Variable”
Hence, from the above,
We can conclude that the best term that fits the given definition is a “Variable”

Multiplication

Find each product.
Question 4.
4 × 8
Answer:
The given expression is: 4 × 8
Now,
By using the Distributive Property of Multiplication,
4 × 8
= 4 × (4 + 4)
= (4 × 4) + (4 × 4)
= 16 + 16
= 32
Hence, from the above,
We can conclude that the value of the given expression is: 32

Question 5.
2 × 9
Answer:
The given expression is: 2 × 9
Now,
By using the Distributive Property of Multiplication,
2 × 9
= 2 × (6 + 3)
= (2 × 6) + (2 × 3)
= 12 + 6
= 18
Hence, from the above,
We can conclude that the value of the given expression is: 18

Question 6.
9 × 5
Answer:
The given expression is: 9 × 5
Now,
By using the Distributive Property of Multiplication,
9 × 5
= 9 × (4 + 1)
= (9 × 4) + (9 × 1)
= 36 + 9
= 45
Hence, from the above,
We can conclude that the value of the given expression is: 45

Question 7.
6 × 8
Answer:
The given expression is: 6 × 8
Now,
By using the Distributive Property of Multiplication,
6 × 8
= 6 × (4 + 4)
= (6 × 4) + (6 × 4)
= 24 + 24
= 48
Hence, from the above,
We can conclude that the value of the given expression is: 48

Question 8.
16 × 4
Answer:
The given expression is: 16 × 4
Now,
By using the Distributive Property of Multiplication,
16 × 4
= (8 + 8) × 4
= (4 × 8) + (4 × 8)
= 32 + 32
= 64
Hence, from the above,
We can conclude that the value of the given expression is: 64

Question 9.
6 × 68
Answer:
The given expression is: 6 × 68
Now,
By using the Distributive Property of Multiplication,
6 × 68
= (3 + 3) × 68
= (3 × 68) + (3 × 68)
= 204 + 204
= 408
Hence, from the above,
We can conclude that the value of the given expression is: 408

Question 10.
87 × 5
Answer:
The given expression is: 87 × 5
Now,
By using the Distributive Property of Multiplication,
87 × 5
= 87 × (3 + 2)
= (87 × 3) + (87 × 2)
= 261 + 174
= 435
Hence, from the above,
We can conclude that the value of the given expression is: 435

Question 11.
19 × 9
Answer:
The given expression is: 19 × 9
Now,
By using the Distributive Property of Multiplication,
19 × 9
= 19 × (3 + 6)
= (19 × 3) + (19 × 6)
= 57 + 114
= 171
Hence, from the above,
We can conclude that the value of the given expression is: 171

Question 12.
128 × 6
Answer:
The given expression is: 128 × 6
Now,
By using the Distributive Property of Multiplication,
128 × 6
= (100 + 28) × 6
= (100 × 6) + (28 × 6)
= 600 + 168
= 768
Hence, from the above,
We can conclude that the value of the given expression is: 768

Rounding

Round each number to the nearest hundred.
Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 2
Question 13.
164
Answer:
The given number is: 164
Hence, from the above,
We can conclude that the number that is to the nearest hundred for the given number is: 200

Question 14.
8,263
Answer:
The given number is: 8,263
Hence, from the above,
We can conclude that the number that is to the nearest hundred for the given number is: 8,300

Question 15.
527
Answer:
The given number is: 527
Hence, from the above,
We can conclude that the number that is to the nearest hundred for the given number is: 500

Question 16.
2,498
Answer:
The given number is: 2,498
Hence, from the above,
We can conclude that the number that is to the nearest hundred for the given number is: 2,500

Question 17.
7,892
Answer:
The given number is: 7,892
Hence, from the above,
We can conclude that the number that is to the nearest hundred for the given number is: 7,900

Use Strategies and Properties to Multiply by 2-Digit Numbers 2

Question 18.
472
Answer:
The given number is: 472
Hence, from the above,
We can conclude that the number that is to the nearest hundred for the given number is: 500

Round each number to the nearest thousand.
Question 19.
8,685
Answer:
The given number is: 8,685
Hence, from the above,
We can conclude that the number that is to the nearest thousand for the given number is: 9,000

Question 20.
4,991
Answer:
The given number is: 4,991
Hence, from the above,
We can conclude that the number that is to the nearest thousand for the given number is: 5,000

Question 21.
62,549
Answer:
The given number is: 62,549
Hence, from the above,
We can conclude that the number that is to the nearest thousand for the given number is: 62,000

Question 22.
167,241
Answer:
The given number is: 167,241
Hence, from the above,
We can conclude that the number that is to the nearest thousand for the given number is: 167,200

Question 23.
77,268
Answer:
The given number is: 77,268
Hence, from the above,
We can conclude that the number that is to the nearest thousand for the given number is: 77,000

Question 24.
34,162
Answer:
The given number is: 34,162
Hence, from the above,
We can conclude that the number that is to the nearest thousand for the given number is: 34,000

Question 25.
1,372
Answer:
The given number is: 1,372
Hence, from the above,
We can conclude that the number that is to the nearest thousand for the given number is: 1,000

Question 26.
9,009
Answer:
The given number is: 9,009
Hence, from the above,
We can conclude that the number that is to the nearest thousand for the given number is: 9,000

Question 27.
121,619
Answer:
The given number is: 121,619
Hence, from the above,
We can conclude that the number that is to the nearest thousand for the given number is: 122,000

Question 28.
Construct Arguments Explain how to round 608,149 to the nearest thousands place.
Answer:
The given number is: 608,149
Now,
From the given number,
We can observe that the total number is less than 608,500
Hence, from the above,
We can conclude that the number that is to the nearest thousand for the given number is: 608,000

Pick a Project

PROJECT 4A
How high would a stack of sabal palms be?
Project: Explain the Processes
Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 3

PROJECT 4B
Can you estimate the weight of dozens of birds?
“Project: Write a Report About the Northern Mockingbird
Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 4

PROJECT 4C
How many soccer players start in the Women’s World Cup ?
Project: Create an Array Poster
Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 5

PROJECT 4D
How much weight can shotputters throw?
Project: Compare Shot Masses
Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 6

Lesson 4.1 Multiply Multiples of 10

Solve & Share
The principal of a school needs to order supplies for 20 new classrooms. Each classroom needs the following items: 20 desks, 30 chairs, and 40 pencils. How many of each item does the principal need to order? Solve these problems using any strategy you choose.
I can … use place-value strategies or properties of operations to multiply by multiples of 10.
Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 7
Answer:
It is given that
The principal of a school needs to order supplies for 20 new classrooms. Each classroom needs the following items: 20 desks, 30 chairs, and 40 pencils
So,
The total number of desks the principal needs to order = (The total number of classrooms) × (The number of desks needed for each classroom)
= 20 × 20
= 2 tens × 2 tens [By using the Place-Value strategy]
= 4 hundred
= 400
So,
The total number of chairs the principal needs to order = (The total number of classrooms) × (The number of chairs needed for each classroom)
= 20 × 30
= 2 tens × 3 tens [By using the Place-Value strategy]
= 6 hundred
= 600
So,
The total number of pencils the principal needs to order = (The total number of classrooms) × (The number of pencils needed for each classroom)
= 20 × 40
= 2 tens × 4 tens [By using the Place-Value strategy]
= 8 hundred
= 800
Hence, from the above,
We can conclude that
The total number of desks the principal needs to order is: 400
The total number of chairs the principal needs to order is: 600
The total number of pencils the principal needs to order is: 800

Look Back! Look at the factors and products. What patterns do you notice?
Answer:
The patterns we noticed from the above factors and products are:
a. All multiples of 2 have a pattern of 2, 4, 6, 8, or 0 in the 1’s place.
b. When multiplying Any number by 2, the result is even
c. There are 5 even numbers and 5 odd numbers in each range of 10 numbers

Essential Question
How Can You Multiply on by Multiples of 10?
Answer:
When we multiply any number by 10, the last digit will always be 0
Ex:
2 × 20 = 2 × 2 × 10
= 4 × 10
= 40

Visual Learning Bridge
The number of visitors of each age group for the Sunny Day Amusement Park is shown below. How many children visit the park in 30 days?
Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 8

One Way
Find 30 × 80.
Use basic facts and place value.
30 × 80 = 3 tens × 8 tens
= 24 hundreds
= 2,400
2,400 children visit the park in 30 days.
10 × 10 = 100

Another Way
Find 30 × 80
Break apart numbers. Use the Commutative Property and the Associative Property of Multiplication.
30 × 80 = (3 × 10) (8 × 10)
= 3 × 8 × 10 × 10
= (3 × 8) (10 × 10)
= 24 × 100
= 2,400
2,400 children visit the park in 30 days.

Convince Me!
Look for Relationships Use place value or properties of operations to determine how many adults age 65 and older visit the park in 30 days.
Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 8
Answer:
From the given figure,
We can observe that
The number of visitors each day that are Adults 65 and over are: 40
So,
The number of visitors in 30 days that are Adults 65 and over = 30 × (The number of visitors each day that are Adults 65 and over)
= 30 × 40
= 3 tens × 4 tens [By using the Place-Value strategy]
= 12 hundred
= 1,200
Hence, from the above,
We can conclude that the number of visitors in 30 days that are Adults 65 and over is: 1,200

Use Strategies and Properties to Multiply by 2-Digit Numbers 3

Another Example!
Use the properties of operations to find 50 × 60.
50 × 60 = 5 × 10 × 6 × 10
= (5 × 6) × (10 × 10)
= 30 × 100
= 3,000
If the product of the basic fact ends in zero, the product has one more zero than you see in the factors.

Guided Practice

Do You Understand?
Question 1.
Find 50 × 20. How many zeros are in the product? Explain.
Answer:
The given expression is: 50 × 20
Now,
By using the Place-Value Property,
50 × 20
= 5 tens × 2 tens
= 10 hundred
= 10 × 100
= 1,000
Hence, from the above,
We can conclude that
The value of the product of the given expression is: 1,000
The number of zeroes in the obtained product is: 3

Question 2.
How many adults under 65 visit the park in 30 days?
Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 8
Answer:
From the given figure,
We can observe that
The number of visitors each day that are adults under 65 are: 60
So,
The number of visitors in 30 days that are Adults under 65 = 30 × (The number of visitors each day that are adults under 65)
= 30 × 60
= 3 × 10 × 6 × 10
= (3 × 6) × (10 × 10)
= 18 × 100
= 1,800
Hence, from the above,
We can conclude that
The number of visitors in 30 days that are Adults under 65 is: 1,800

Do You Know How?
For 3-8, use basic facts and place-value or properties of operations to find each product.
Question 3.
30 × 10
Answer:
The given expression is: 30 × 10
Now,
By using the Properties of Operations,
30 × 10
= 3 × 10 × 1 × 10
= (3 × 1) × (10 × 10)
= 3 × 100
= 300
Hence, from the above,
We can conclude that the value of the given expression is: 300

Question 4.
50 × 10
Answer:
The given expression is: 50 × 10
Now,
By using the Properties of Operations,
50 × 10
= 5 × 10 × 1 × 10
= (5 × 1) × (10 × 10)
= 5 × 100
= 500
Hence, from the above,
We can conclude that the value of the given expression is: 500

Question 5.
20 × 10
Answer:
The given expression is: 20 × 10
Now,
By using the Properties of Operations,
20 × 10
= 2 × 10 × 1 × 10
= (2 × 1) × (10 × 10)
= 2 × 100
= 200
Hence, from the above,
We can conclude that the value of the given expression is: 200

Question 6.
60 × 20
Answer:
The given expression is: 60 × 20
Now,
By using the Properties of Operations,
60 × 20
= 6 × 10 × 2 × 10
= (6 × 2) × (10 × 10)
= 12 × 100
= 1,200
Hence, from the above,
We can conclude that the value of the given expression is: 1,200

Question 7.
90 × 40
Answer:
The given expression is: 90 × 40
Now,
By using the Properties of Operations,
90 × 40
= 9 × 10 × 4 × 10
= (9 × 4) × (10 × 10)
= 36 × 100
= 3,600
Hence, from the above,
We can conclude that the value of the given expression is: 3,600

Use Strategies and Properties to Multiply by 2-Digit Numbers 4

Question 8.
80 × 50
Answer:
The given expression is: 80 × 50
Now,
By using the Properties of Operations,
80 × 50
= 8 × 10 × 5 × 10
= (8 × 5) × (10 × 10)
= 40 × 100
= 4,000
Hence, from the above,
We can conclude that the value of the given expression is: 4,000

Independent Practice

For 9-16, use basic facts and place-value or properties of operations to find each product.
Question 9.
20 × 70
Answer:
The given expression is: 20 × 70
Now,
By using the Properties of Operations,
20 × 70
= 2 × 10 × 7 × 10
= (2 × 7) × (10 × 10)
= 14 × 100
= 1,400
Hence, from the above,
We can conclude that the value of the given expression is: 1,400

Question 10.
70 × 90
Answer:
The given expression is: 70 × 90
Now,
By using the Properties of Operations,
70 × 90
= 7 × 10 × 9 × 10
= (7 × 9) × (10 × 10)
= 63 × 100
= 6,300
Hence, from the above,
We can conclude that the value of the given expression is: 6,300

Question 11.
40 × 20
Answer:
The given expression is: 40 × 20
Now,
By using the Properties of Operations,
40 × 20
= 4 × 10 × 2 × 10
= (4 × 2) × (10 × 10)
= 8 × 100
= 800
Hence, from the above,
We can conclude that the value of the given expression is: 800

Question 12.
40 × 30
Answer:
The given expression is: 40 × 30
Now,
By using the Properties of Operations,
40 × 30
= 4 × 10 × 3 × 10
= (4 × 3) × (10 × 10)
= 12 × 100
= 1,200
Hence, from the above,
We can conclude that the value of the given expression is: 1,200

Question 13.
70 × 40
Answer:
The given expression is: 70 × 40
Now,
By using the Properties of Operations,
70 × 40
= 7 × 10 × 4 × 10
= (7 × 4) × (10 × 10)
= 28 × 100
= 2,800
Hence, from the above,
We can conclude that the value of the given expression is: 2,800

Question 14.
20 × 30
Answer:
The given expression is: 20 × 30
Now,
By using the Properties of Operations,
20 × 30
= 2 × 10 × 3 × 10
= (2 × 3) × (10 × 10)
= 6 × 100
= 600
Hence, from the above,
We can conclude that the value of the given expression is: 600

Question 15.
60 × 40
Answer:
The given expression is: 60 × 40
Now,
By using the Properties of Operations,
60 × 40
= 6 × 10 × 4 × 10
= (6 × 4) × (10 × 10)
= 24 × 100
= 2,400
Hence, from the above,
We can conclude that the value of the given expression is: 2,400

Question 16.
60 × 90
Answer:
The given expression is: 60 × 90
Now,
By using the Properties of Operations,
60 × 90
= 6 × 10 × 9 × 10
= (6 × 9) × (10 × 10)
= 54 × 100
= 5,400
Hence, from the above,
We can conclude that the value of the given expression is: 5,400

For 17-22, find the missing factor.
Question 17.
10 × ______ = 100
Answer:
The given expression is: 10 × _______ = 100
Now,
Let the missing factor be x
So,
10 × x = 100
x = \(\frac{100}{10}\)
x = 10
Hence, from the above,
We can conclude that the missing factor is: 10

Question 18.
_______ × 20 = 1,600
Answer:
The given expression is: ______ × 20 = 1,600
Now,
Let the missing factor be x
So,
x × 20 = 1,600
x = \(\frac{1,600}{20}\)
x = 80
Hence, from the above,
We can conclude that the missing factor is: 80

Question 19.
_______ × 30 = 1,500
Answer:
The given expression is: ______ × 30 = 1,500
Now,
Let the missing factor be x
So,
x × 30 = 1,500
x = \(\frac{1,500}{30}\)
x = 50
Hence, from the above,
We can conclude that the missing factor is: 50

Question 20.
20 × ______ = 1,000
Answer:
The given expression is: 20 × _______ = 1,000
Now,
Let the missing factor be x
So,
20 × x = 1,000
x = \(\frac{1,000}{20}\)
x = 50
Hence, from the above,
We can conclude that the missing factor is: 50

Question 21.
_______ × 90 = 8,100
Answer:
The given expression is: _______ × 90 = 8,100
Now,
Let the missing factor be x
So,
x × 90 = 8,100
x = \(\frac{8,100}{90}\)
x = 90
Hence, from the above,
We can conclude that the missing factor is: 90

Question 22.
60 × _______ = 4,200
Answer:
The given expression is: 60 × _______ = 4,200
Now,
Let the missing factor be x
So,
60 × x = 4,200
x = \(\frac{4,200}{60}\)
x = 70
Hence, from the above,
We can conclude that the missing factor is: 70

Problem Solving

Question 23.
Reasoning
The product of two factors is 4,200. If one of the factors is 60, what is the other factor? Explain.
Answer:
It is given that
The product of two factors is 4,200 and one of the factors is 60
Now,
Let the other factor be z
So,
60 × z = 4,200
z = \(\frac{4,200}{60}\)
z = 70
Hence, from the above,
We can conclude that the other factor for the given product is: 70

Question 24.
Algebra There is 30 players on each high school football team. Explain how you can find the total number of players if there are 40 teams. Write and solve an equation.
Answer:
It is given that
There is 30 players on each high school football team and there are 40 teams
So,
The total number of players = (The number of players present in each team) × (The total number f teams)
= 30 × 40
= 3 × 10 × 4 × 10
= (3 × 4) × (10 × 10)
= 12 × 100
= 1,200
Hence, from the above,
We can conclude that the total number of players are: 1,200

Question 25.
Bob uses 2 gallons of water while brushing his teeth. He uses 10 gallons of water to wash clothes. How many more cups of water did Bob use while washing his clothes than brushing his teeth?
Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 9
Answer:
It is given that
Bob uses 2 gallons of water while brushing his teeth. He uses 10 gallons of water to wash clothes.
Now,
The given figure is:
Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 9
So,
The number of cups Bob uses to wash his clothes = (The number of gallons Bob uses to wash his clothes) × (The number of cups in each gallon)
= 10 × 16
= 10 × (8 + 8) [By using the Distributive Property of Multiplication]
= (10 × 8) + (10 × 8)
= 80 + 80
= 160 cups
So,
The number of cups Bob uses while brushing his teeth = (The number of gallons Bob uses to brush his teeth) × (The number of cups in each gallon)
= 2 × 16
= 2 (8 + 8) [By using the Distributive Property of Multiplication]
= (2 × 8) + (2 × 8)
= 16 + 16
= 32 cups
So,
The number of more cups that Bob uses to wash his clothes than while brushing his teeth
= (The number of cups Bob uses to wash his clothes) – (the number of cups Bob uses while brushing his teeth)
= 160 – 32
= 128 cups
Hence, from the above,
We can conclude that
The number of more cups that Bob uses to wash his clothes than while brushing his teeth is: 128 cups

Question 26.
James walked 30 minutes each day for 90 days. Show how you can use place value or properties to find how many minutes James walked.
Answer:
It is given that James walked 30 minutes each day for 90 days
So,
The total number of minutes James walked = (The number of minutes James walks each day) × (The total number of days James walked)
= 30 × 90
= 3 × 10 × 9 × 10
= (3 × 9) × (10 × 10)
= 27 × 100
= 2,700
Hence, from the above,
We can conclude that
The total number of minutes James walked is: 2,700 minutes

Question 27.
Higher-Order Thinking What is one example of a product that will have the same number of zeros in the factors and the product? What is one example of a product that will NOT have the same number of zeros in the factors as the product?
Answer:
The example of a product that will have the same number of zeros in the factors and the product is:
10 × 15
= 10 × (10 + 5) [By usingthe Distributive Property of Multiplication]
= (10 × 10) + (10 × 5)
= 100 + 50
= 150
The example of a product that will not have the same number of zeros in the factors as the product is:
40 × 5
= 4 × 10 × 5
= (4 × 5) × 10
= 20 × 10
= 2 × 10 × 10
= 200

Assessment Practice

Question 28.
Select all of the expressions that have a product of 1,600.
☐ 20 × 80
☐ 20 × 60
☐ 40 × 40
☐ 60 × 30
☐ 90 × 20
Answer:
It is given that
To have a product of 1,600,
The expressions must be even numbers as factors and if we neglect zeroes, the product must be equal to 16
Hence, from the above,
We can conclude that
All the expressions that have a product of 1,600 are:

Question 29.
Which expression has 50 as the missing factor?
A. 20 × ? = 1,000
B. 50 × ? = 3,000
C. 30 × ? = 1,800
D. 10 × ? = 1,000
Answer:
Let the missing factor be x
Now,
A.
20 × x = 1,000
x = \(\frac{1,000}{20}\)
= 50
B.
50 × x =3,000
x = \(\frac{3,000}{50}\)
= 60
C.
30 × x =1,800
x = \(\frac{1,800}{30}\)
= 60
D.
10 × x =1,000
x = \(\frac{1,000}{10}\)
= 100
Hence, from the above,
We can conclude that the expression that has 50 as a missing factor is:

Use Strategies and Properties to Multiply by 2-Digit Numbers 5

Lesson 4.2 Use Models to Multiply 2-Digit Numbers by Multiples of 10

Solve & Share
There are 10 teams in a baseball league. Each team has 25 players. How many players are in the league? Solve this problem using any strategy you choose.
I can .. use models and properties of operations to help multiply.
Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 10
Answer:
It is given that
There are 10 teams in a baseball league. Each team has 25 players
Now,
The area-model that represents the number of teams and the number of players is:

So,
From the above Area-Model,
The total number of players in the league = (The number of players in each team) × (The total number of teams)
= 25 × 10
= 250 players
Hence, from the above,
We can conclude that the total number of players present in the league is: 250

Look Back!
How many players are in the league if there are 30 teams?
Answer:
We know that,
The total number of players in the league = (The number of players in each team) × (The total number of teams)
Now,
If there are 30 teams and each team consists of the same number of players as mentioned above, then
The total number of players present in the league
= 25 × 30
= 25 × 3 × 10
= (25 × 3) × 10
= 750 players
Hence, from the above,
We can conclude that the total number of players present in the league is: 750

Explain how you can use your answer above to help solve this problem.
Answer:
Multiplication using  the “Area model”:
We know that,
An “Area” is the amount of flat space occupied by a figure, You can find the area of a figure by filling it with unit squares and then counting how many squares entirely fill the shape
The total area of the figure is the multiplication of the number of rows and columns i.e., the value of the product
Hence, from the above process,
We can conclude that we used the above process to solve the above-given problems

Essentials Question
How Can You Use an Array or an Area Model to Multiply?
Answer:
The steps to find the product that uses an Array or an Area Model to multiply is:
a. First, write each factor in expanded form.
b. Then, draw your model.
c. Next, multiply to find the area of each smaller rectangle.
d. Finally, add those products to find the total area.

Visual Learning Bridge
Max’s Moving Company has boxes for packing books. If each box holds 24 books, how many books would fit into 20 boxes?
Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 11

Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 12

Convince Me!
Reasoning in the problem above, is the product, 480, reasonable? Explain.
Answer:
From the above,
It is given that there are 24 books in each box
So,
The total number of books that would fit in 20 boxes = (The total number of boxes) × (The number of books in each box)
= 20 × 24
= 2 × 10 × 24
= 48 × 10
= 480 books
Hence, from the above,
We can conclude that the product 480 is reasonable that we got in the above problem

Guided Practice

Do You Understand?
Question 1.
Draw an area model to show 20 × 26. Then find the product.
Answer:
The given expression is: 20 × 26
Now,
By using the Area-Model,
20 × 26
= (10 + 10) × (20 + 6)
= (10 × 20) + (10 × 6) + (10 × 20) + (10 × 6)
= 200 + 60 + 200 + 60
= 400 + 120
= 520

Hence, from the above,
We can conclude that the value of the given expression using the Area-Model is: 520

Do You Know How?
Question 2.
The place-value block array shows 10 × 16. Find the product.
Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 13
Answer:
The given expression is: 10 × 16
Now,
By using the Area-Model,
10 × 16
= 10 × (10 + 6)
= (10 × 10) + (10 × 6)
= 100 + 60
= 160

Hence, from the above,
We can conclude that the value of the given expression by using the Area-Model is: 160

Independent Practice

Leveled Practice For 3-8, use place-value blocks, area models, or arrays to find each product.
Question 3.
10 × 22
Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 14
Answer:
The given expression is: 10 × 22
Now,
By using the Area-Model,
10 × 22
= 10 × (20 + 2)
= (10 × 20) + (10 × 2)
= 200 + 20
= 220

Hence, from the above,
We can conclude that the value of the given expression by using the Area-Model is: 220

Question 4.
10 × 13
Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 15
Answer:
The given expression is: 10 × 13
Now,
By using the Area-Model,
10 × 13
= 10 × (10 + 3)
= (10 × 10) + (10 × 3)
= 100 + 30
= 130

Hence, from the above,
We can conclude that the value of the given expression by using the Area-Model is: 130

Question 5.
20 × 35
Answer:
The given expression is: 20 × 35
Now,
By using the Area-Model,
20 × 35
= (10 + 10) × (30 + 5)
= (10 × 30) + (10 × 5) + (10 × 30) + (10 × 5)
= 300 + 50 + 300 + 50
= 600 + 100
= 700

Hence, from the above,
We can conclude that the value of the given expression by using the Area-Model is: 700

Question 6.
20 × 41
Answer:
The given expression is: 20 × 41
Now,
By using the Area-Model,
20 × 41
= (10 + 10) × (40 + 1)
= (10 × 40) + (10 × 1) + (10 × 40) + (10 × 1)
= 400 + 10 + 400 + 10
= 800 + 20
= 820

Hence, from the above,
We can conclude that the value of the given expression by using the Area-Model is: 820

Question 7.
30 × 29
Answer:
The given expression is: 30 × 29
Now,
By using the Area-Model,
30 × 29
= (20 + 10) × (20 + 9)
= (20 × 20) + (20 × 9) + (10 × 20) + (10 × 9)
= 400 + 180 + 200 + 90
= 600 + 270
= 870

Hence, from the above,
We can conclude that the value of the given expression by using the Area-Model is: 870

Question 8.
40 × 37
Answer:
The given expression is: 40 × 37
Now,
By using the Area-Model,
40 × 37
= (20 + 20) × (30 + 7)
= (20 × 30) + (20 × 7) + (20 × 30) + (20 × 7)
= 600 + 140 + 600 + 140
= 1,200 + 280
= 1,480

Hence, from the above,
We can conclude that the value of the given expression by using the Area-Model is: 1,480

Problem Solving

Camera Sales
Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 16
Question 9.
Algebra In the first 3 months of the year, an electronics store sold 1,446 cameras. How many cameras did the store sell in March? Write and solve an equation.
Answer:
It is given that
In the first 3 months of the year, an electronics store sold 1,446 cameras
Now,
The given data is:
Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 16
So,
The number of cameras sold in the three months = (The number of cameras sold in January) + (The number of cameras sold in February) + (The number of cameras sold in March)
Now,
Let the number of cameras sold in March be x
So,
1,446 = 486 + 385 + x
x = 1,446 – (486 + 385)
x = 1,446 – 871
x = 575 cameras
Hence, from the above,
We can conclude that the number of cameras that the store sold in March  is: 575

Question 10.
For every camera sold in February, the store donated $2 to a charity. How much did the store donate?
Answer:
It is given that
For every camera sold in February, the store donated $2 to a charity
Now,
The given data is:
Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 16
So,
The total amount of money the store donated in February to a charity = (The number of cameras sold in February) × (The amount of money the store donated to a charity for every camera)
= 385 × $2
= (300 + 80 + 5) × $2
= (300 × $2) + (80 × $2) + (5 × $2)
= $600 + $160 + $10
= $770
Hence, from the above,
We can conclude that
The total amount of money the store donated in Februaryto a charity is: $770

Question 11.
Model with Math
During a basketball game, 75 cups of fruit punch was sold. Each cup holds 20 fluid ounces. How many total fluid ounces of fruit punch were sold?
Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 17
Answer:
It is given that
During a basketball game, 75 cups of fruit punch was sold. Each cup holds 20 fluid ounces.
So,
The amount of total fluid ounces of fruit punch that were sold = (The total number of cups) × (The amount of fluid ounces that each cup holds)
= 75 × 20
= (70 + 5) × (10 + 10)
= (70 × 10) + (70 × 10) + (5 × 10) + (5 × 10)
= 700 + 700 + 50 + 50
= 1,400 + 100
= 1,500 fluid ounces
Hence, from the above,
We can conclude that the amount of fluid ounces of fruit punch that were sold is: 1,500 fluid ounces

Question 12.
Higher-Order Thinking Describe how to correct the partial product that is not correct in the worked problem below. What is the correct product?
Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 18
Answer:
The given expression is: 27 × 73
Now,
By using the Area-Model,
27 × 73
= (20 + 7) × (70 + 3)
= (20 × 70) + (20 × 3) + (7 × 70) + (7 × 3)
= 1,400 + 60 + 490 + 21
= 1,971
Hence, from the above,
We can conclude that
The partial product that is not correct is: 490
The correct product for the given expression is: 1,971

Assessment Practice

Question 13.
Which is greater?
15 × 30 or 22 × 20 Show your thinking.
Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 19
Answer:
The given expressions are:
a. 15 × 30     b. 22 × 20
Now,
By using the Area-Model,
a.
15 × 30
= (10 + 5) × (20 + 10)
= (10 × 20) + (10 × 10) + (5 × 20) + (5 × 10)
= 200 + 100 + 100 + 50
= 400 + 50
= 450
b.
22 × 20
= (20 + 2) × (10 + 10)
= (20 × 10) + (20 × 10) + (2 × 10) + (2 × 10)
= 200 + 200 + 20 + 20
= 400 + 40
= 440
Hence, from the above,
We can conclude that the value of the expression 15 × 30 is greater

Lesson 4.3 Estimate: Use Rounding or Compatible Numbers

Solve & Share
Choose two factors from the numbers below to find a product that is as close to 1,600 as possible. Solve this problem using any strategy you choose.
I can .. use rounding or compatible numbers to estimate.
Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 20
Answer:
The given numbers are: 24, 32, 61, and 78
So,
The numbers that are closest to the given numbers are: 20, 30, 60, and 80
Now,
To find the two factors from the given numbers to find a product that is as close to 1,600 as possible,
We have to find the product of the factors that are close to the given factors
So,
a.
20 × 30
= 2 × 10 × 3 × 10
= (2 × 3) × 100
= 6 × 100
= 600
b.
20 × 60
= 2 × 10 × 6 × 10
= (2 × 6) × 100
= 12 × 100
= 1,200
c.
20 × 80
= 2 × 10 × 8 × 10
= (2 × 8) × 100
= 16 × 100
= 1,600
d.
30 × 60
= 3 × 10 × 6 × 10
= (3 × 6) × 100
= 18 × 100
= 1,800
e.
30 × 80
= 3 × 10 × 8 × 10
= (3 × 8) × 100
= 24 × 100
= 2,400
f.
60 × 80
= 6 × 10 × 8 × 10
= (6 × 8) × 100
= 48 × 100
= 4,800
Hence, from the above,
We can conclude that
The two factors that are closest to the product 1,600 are: 24, 78

Look Back!
Construct Arguments Why did you choose the two factors that you did? How do you know your factors will give the closest estimate of the product?
Answer:
We know that,
The difference between the product of the two exact numbers and the product of the numbers that are closest to the exact numbers must be small
So,
When we observe the above products which we obtained in the above problem,
We can observe that there are two products that are closest to the given product
But,
When we find the difference between the exact product and the estimated product,
The product of the factors 25 and 60 is close to the product of the exact factors 24, 78

Essential Question
What Strategies Can I Use When Estimating?
Answer:
The front-end strategy is a method of estimating computations by keeping the first digit in each of the numbers and changing all the other digits to zeros. This strategy can be used to estimate sums, differences, products and quotients.

Visual Learning Bridge
The workers picked 14 dozen apples at Ms. Piper’s apple grove and 12 dozen apples at Mr. Stuart’s apple grove. There are 12 apples in one dozen. About how many apples did the workers pick?
14 + 12 = 26 dozen apples picked in total
Envision Math Common Core 4th Grade Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 21

One Way
Use rounding to estimate 26 × 12.
Round 26 to the nearest ten.
Round 12 to the nearest ten.
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 22
The workers picked about 300 apples.
Some problems do not need an exact answer.

Another Way
Use compatible numbers to estimate 26 × 12.
Replace the factors with numbers that are close and easy to multiply.
26 is close to 25.
12 is close to 10.
25 × 10 = n
250 = n
The workers picked about 250 apples.

Convince Me!
Reasoning Sue said she could find an estimate for 26 × 12 by rounding only 1 factor and multiplying 26 × 10. Do you agree? Explain.
Answer:
Sue said she could find an estimate for 26 × 12 by rounding only 1 factor and multiplying 26 × 10
Now,
We know that,
When we want to find an estimate of the factors, we have to estimate all the factors and it is possible when you do partial estimation i.e., rounding off 1 number and not rounding off another number
Hence, from the above,
We can conclude that we can agree with Sue

Guided Practice

Do You Understand?
Question 1.
In the example on the previous page, how do you know you only need an estimate and not an exact answer?
Answer:
We know that,
There are two types of estimations. They are:
a. Round-off method      b. Closest estimated method
Now,
When we use the above two types of eatimations,
The answers we will get may differ
So,
In the example of previous page,
We got different answers when we use different types of estimation methods
Now,
From the example of previous page,
We can observe that the number of fruits had been asked in terms of estimations but not an exact number
Hence, from the above,
We can conclude that we only need an estimate and not an exact answer because of the mentioned quaestion

Question 2.
In the example on the previous page, what is another way you can multiply using compatible numbers by changing only one of the factors? Explain how you would find the estimated product.
Answer:
The given expression on the previous page is: 26 × 12
Now,
By using the closest number estimation,
The factors will be changed into: 25 and 12; or 26 and 10
So,
The estimation of the product using compatible numbers will be:
26 × 12
= 25 × 12                     (or)                     26 × 10
= 300                           (or)                   = 260
Hence, from the above,
We can coclude that the estimated product is approximatelu 300 (or) 260

Do You Know How?
For 3-6, estimate each product. Explain why you chose the strategy you used to estimate the product.
Question 3.
24 × 18
Answer:
The given expression is: 24 × 18
Now,
By using the Compatible method estimation,
The given expression will become: 25 × 20
So,
25 × 20
= 25 × 2 × 10
= (25 ×2) × 10
= 50 ×10
= 500
Hence, from the above,
We can conclude that the estimated product for the given expression is: 500

Question 4.
33 × 31
Answer:
The given expression is: 33 × 31
Now,
By using the Compatible method estimation,
The given expression will become: 35 × 30
So,
35 × 30
= 35 × 3 × 10
= (35 × 3) × 10
= 105 × 10
= 1,050
Hence, from the above,
We can conclude that the estimated product for the given expression is: 1,050

Question 5.
38 × 22
Answer:
The given expression is: 38 × 32
Now,
By using the Compatible method estimation,
The given expression will become: 40 × 30
So,
40 × 30
= 4 × 10 × 3 × 10
= (4 × 3) × 100
= 12 × 100
= 1,200
Hence, from the above,
We can conclude that the estimated product for the given expression is: 1,200

Question 6.
45 × 48
Answer:
The given expression is: 45 × 48
Now,
By using the Rounding-off method estimation,
The given expression will become: 50 × 50
So,
50 × 50
= 5 ×10 × 5 × 10
= (5 × 5) × 100
= 25 × 100
= 2,500
Hence, from the above,
We can conclude that the estimated product for the given expression is: 2,500

Independent Practice

For 7-12, estimate each product. Explain why you chose the strategy you used to estimate the product.
Question 7.
39 × 19
Answer:
The given expression is: 39 × 19
Now,
By using the Rounding-off estimation method,
The given expression will become: 40 × 20
So,
40 × 20
= 4 × 10 × 2 × 10
= (4 × 2) ×100
= 8 × 100
= 800
Hence, from the above,
We can conclude that the estimated product for the given expression is: 800

Question 8.
28 × 27
Answer:
The given expression is: 28 × 27
Now,
By using the Rounding-off estimation method,
The given expression will become: 30 × 30
So,
30 × 30
= 3 × 10 × 3 × 10
= (3 × 3) ×100
= 9 × 100
= 900
Hence, from the above,
We can conclude that the estimated product for the given expression is: 900

Question 9.
64 × 13
Answer:
The given expression is: 64 × 13
Now,
By using the Compatible estimation method,
The given expression will become: 65 × 15
So,
65 × 15
= (60 + 5) × (10 + 5)
= (60 × 10) + (60 × 5) + (5 × 10) + (5 × 5)
= 600 + 300 + 50 + 25
= 975
Hence, from the above,
We can conclude that the estimated product for the given expression is: 975

Question 10.
42 × 17
Answer:
The given expression is: 42 × 17
Now,
By using the Rounding-off estimation method,
The given expression will become: 40 × 20
So,
40 × 20
= 4 × 10 × 2 × 10
= (4 × 2) ×100
= 8 × 100
= 800
Hence, from the above,
We can conclude that the estimated product for the given expression is: 800

Question 11.
82 × 36
Answer:
The given expression is: 82 × 36
Now,
By using the Compatible way estimation method,
The given expression will become: 80 × 35
So,
80 × 35
= (40 + 40) × (30 + 5)
= (40 × 30) + (40 × 5) + (40 × 30) + (40 × 5)
= 1,200 + 200 + 1,200 + 200
= 2,400 + 400
= 2,800
Hence, from the above,
We can conclude that the estimated product for the given expression is: 2,800

Question 12.
54 × 18
Answer:
The given expression is: 54 × 18
Now,
By using the Compaible way estimation method,
The given expression will become: 55 × 20
So,
55 ×20
= 55 × 2 × 10
= (55 × 2) × 10
= 110 × 10
= 1,100
Hence, from the above,
We can conclude that the estimated product for the given expression is: 1,100

For 13-14, estimate to check if the given answer is reasonable.
Compare your estimate to the given answer to check if the answer is reasonable.
Question 13.
66 × 41 = 2,706
Rounds to ________ × ________ = ________
Reasonable
Not Reasonable
Answer:
The given expression is: 66 × 41
Now,
By using the Rounding-off method,
66 × 41
= 70 × 40
= 7 × 10 × 4 × 10
= (7 × 4) × 100
= 28 × 100
= 2,800
Hence, from the above,
We can conclude that the Rounding is “Reasonable” since the difference between the estimated answer and the exact answer is small

Question 14.
34 × 52 = 2,288
Rounds to ________ × ________ = ________
Reasonable
Not Reasonable
Answer:
The given expression is: 34 × 52
Now,
By using the Rounding-off method,
34 × 52
= 30 × 50
= 3 × 10 × 5 ×10
= (3 × 5) × 100
= 15 × 100
= 1,500
Hence, from the above,
We can conclude that the Rounding is not “Reasonable” since the difference between the estimated answer and the exact answer is large

Problem Solving

For 15-16, use the table at the right.
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 23
Question 15.
About how many more Valencia orange trees than Temple orange trees does Mr. Gonzalez have? Explain.
Answer:
The given data is:
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 23
Now,
From the given data,
We can find:
a.
The number of Orange trees in Valencia = (The number of trees in each row) × (The number of rows)
= 46 × 31
Now,
By using the rounding-off method,
46 × 31
= 50 × 30
= 5 × 10 × 3 × 10
= (5 × 3) × 100
= 1,500
b.
The number of Orange trees in Temple = (The number of trees in each row) × (The number of rows)
= 38 × 28
Now,
By using the rounding-off method,
38 × 28
= 40 × 30
= 4 × 10 × 3 × 10
= (4 × 3) × 100
= 12 × 100
= 1,200
So,
The number of Valencia orange trees more than Temple orange trees does Mr. Gonzalez have
= (The number of Valencia Orange trees) – (The number of Temple Orange trees)
= 1,500 – 1,200
= 300
Hence, from the above,
We can conclude that Mr Gonzalez have about 300 Valencia Orange trees more than temple Orange trees

Question 16.
About how many orange trees does Mr. Gonzalez have? Explain.
Answer:
The given data is:
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 23
Now,
From the given data,
We can find that the total number of Mr. Gonzalez’s trees can be obtained by adding the given three types of Orange trees
So,
The number of Orange trees does Mr Gonzalez have = (The total number of trees in each row) × (The total number of rows)
= (38 + 28 + 46) × (28 + 38 + 31)
= 112 × 97
Now,
By using the Rounding-off method,
112 × 97
= 110 × 100
= 11 × 10 × 100
= 11,000
Hence, from the above,
We can conclude that there are about 11,000 Orange trees does Mr. Gonzalez have

Question 17.
Higher-Order Thinking How is using compatible numbers to estimate similar to using rounding? How is it different?
Answer:
A)
Compatible numbers:
Round the whole number to the closest compatible number. “Compatible numbers” are numbers that are close in value to the real number that would make it easier to find an estimate calculation.
B)
Rounding-off numbers:
Rounding off means a number is made simpler by keeping its value intact but closer to the next number. It is done for whole numbers, and for decimals at various places of hundreds, tens, tenths, etc. Rounding off numbers is done to preserve the significant figures

Question 18.
Construct Arguments Explain how you would use estimation to decide which has the greater product, 39 × 21 or 32 × 32.
Answer:
The given expressions are: 39 × 21 and 32 × 32
Now,
A)
By using the rounding-off method:
a.
39 × 21
= 40 × 20
= 4 × 10 × 2 × 10
= (4 × 2) × 100
= 8 × 100
= 800
b.
32 × 32
= 30 × 30
= 3 × 10 × 3 × 10
= (3 × 3) × 100
= 9 × 100
= 900
B)
By using the compatible nembers method:
a.
39 × 21
= 40 × 20
= 4 × 10 × 2 × 10
= 8 × 100
= 800
b.
32 × 32
= 30 × 30
= 3 × 10 × 3 × 10
= (3 × 3) × 100
= 9 × 100
= 900
Hence, from the above,
We can conclude that the estimated product of 32 × 32 is greater

Assessment Practice

Question 19.
Select all of the expressions that could be used to estimate 32 × 14.
☐ 30 × 20
☐ 30 × 10
☐ 14 × 32
☐ 40 × 10
☐ 32 × 10
Answer:
The given expression is: 32 × 14
Now,
By using the Rounding-off method,
The given expression will become: 30 × 10 and 32 × 10 and 30 × 14
Now,
By using the Compatible numbers method,
The given expression will become: 30 × 15 and 30 × 14 and 32 × 15
Hence, from the above,
We can conclude that the expressions that can be used to estimate 32 × 14 are:

Question 20.
Select all of the equations that use compatible numbers to find an estimate for 28 × 24.
☐ 30 × 25 = 750
☐ 28 × 10 = 280
☐ 28 × 25 = 700
☐ 25 × 20 = 500
☐ 30 × 300 = 9,000
Answer:
The given expression is: 28 × 24
Now,
By usin the compatible numbers method,
The given expression will become: 30 × 25 and 28 × 25 and 30 × 24
Hence, from the above,
We can conclude that
The equations that use compatible numbers to find an estimate for 28 × 24 are:

Lesson 4.4 Arrays and Partial Products

Solve & Share
A theater contains 14 rows of seats with 23 seats in each row. How many seats are in the theater? Solve this problem using any strategy you choose.
I can … use place-value concepts and properties to multiply.
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 24
Answer:
It is given that
A theater contains 14 rows of seats with 23 seats in each row.
So,
The total number of seats that are present in a theater = (The number of rows of seats a theater contains) × (The number of seats in each row)
= 14 × 23
Now,
By using the Partial multiplication method,
14 × 23
= (10 + 4) × (20 + 3)
= (10 × 20) + (10 × 3) + (4 × 20) + (4 × 3)
= 200 + 30 + 80 + 12
= 322

Hence, from the above,
We can conclude that
The total number of seats that are present in a theater is: 322 seats

Look Back!
Use Structure Theater seating is an example of objects that are arranged in rows and columns, or arrays. How do the number of rows and the number of seats in each row relate to the total number of seats?
Answer:
It is given that
Theater seating is an example of objects that are arranged in rows and columns or arrays
Now,
We know that,
The total number of seats in a theater can be given by the multiplication of the number of rows present in a theater with the number of seats that are present in a row
Hence, from the above,
We can conclude that
The total number of seats in a theater = (The number of rows) × (The number of seats in each row)

Essential Question
How Can You Multiply Using an Array?
Answer:
The steps that are to be followed when you multiply using an Array are:
STEP 1:
Label the height and width of a rectangle with the key numbers in the question.
STEP 2:
a. Break down the numbers into tens and ones.
b Draw a separate column and row for each of these numbers.
STEP 3:
Multiply each column and row with each other, until you get four numbers.

Visual Learning Bridge
There are 13 toy dogs in each row of a carnival booth. Twenty rows contain toy bulldogs and 4 rows contain toy huskies. How many toy dogs are there?
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 25

Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 26

Convince Me!
Model with Math What 2-digit by 2-digit multiplication is shown by the model at the right? What is the product? Explain how you used the model to find the product.
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 27
Answer:
The given model is:
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 27
Now,
The representation of the given model in the simplest form is:

Now,
From the above,
We can observe that
The two-digit numbers that are used for multiplication is:
(10 + 8) × (10 + 5)
= 18 × 15
Now,
By using the Partial Multiplication method,
18 × 15
= (10 + 8) × (10 + 5)
= (10 × 10) + (10 × 5) + (8 × 10) + (8 × 5)
= 100 + 50 + 80 + 40
= 270
Hence, from the above,
We can conclude that
The 2-digit by 2-digit multiplication that is shown in the given model is: 18 × 15
The product of the obtained 2-digit by 2-digit multiplication using the partial multiplication method is: 270

Guided Practice

Do You Understand?
Question 1.
In the example on the previous page, what four simpler multiplication problems were used to find 24 × 13?
Answer:
The given expression is: 24 × 13
Now,
By using the Partial Multiplication method,
24 × 13
= (20 + 4) × (10 + 3)
= (20 × 10) + ( 20 × 3) + (4 × 10) + (4 × 3)
= 200 + 60 + 40 + 12
= 312
Hence, from the above,
We can conclude that
The four simpler multiplication problems that were used to find 24 × 13 that is the example on the previous page is:
a. 20 × 10 = 200
b. 20 × 3 = 60
c. 4 × 10 = 40
d. 4 × 3 = 12

Question 2.
How can you use properties to help find the product of 24 × 13?
Answer:
The given expression is: 24 × 13
Now,
We know that,
Distributive Property of Multiplication:
a (b + c) = a.b + a.c
Now,
Divide 24 × 13 into parts
So,
24 × 13
= (20 + 4) × (10 + 3)
Now,
By using the Distributive Property of Multiplication,
20 × (10 + 3) = (20 × 10) + (20 × 3)
= 200 + 60
= 260
4 × (10 + 3) = (4 × 10) + (4 × 3)
= 40 + 12
= 52
So,
24 × 13 = 260 + 52
= 312
Hence, from the above,
We can conclude that the product of the given expression by using the properties is: 312

Do You Know How?
For 3, use the array drawn on a grid to find the product. Check if your answer is reasonable.
Question 3.
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 28
Answer:
The given expression is: 24 × 16
Now,
By using the estimation of the Compatible numbers,
The given expression will become: 24 × 15
Now,
By using the Partial Multiplication method,
24 × 16                                                                24 ×15
= (20 + 4) × (10 + 6)                                           = (20 + 4) × (10 + 5)
= (20 × 10) + (20 × 6) + (4 × 10) + (4 × 6)         = (20 × 10) + (20 × 5) + (4 × 10) + (4 × 5)
= 200 + 120 + 40 + 24                                        = 200 + 100 + 40 + 20
= 384                                                                   = 360
Hence, from the above,
We can conclude that
The product of the given expression is: 384
The product for the given expression is “Reasonable” since there is a small difference between the exact product and the estimated product

Independent Practice
For 4-7, use the array drawn on a grid to find each product.
Using the Commutative Property of Addition, you can add the partial products in any order.

Question 4.
14 × 21
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 29
Answer:
The given expression is: 14 × 21
Now,
By using the Partial Multiplication method,
14 × 21
= (10 + 4) × (20 + 1)
= (10 × 20) + (10 × 1) + (4 × 20) + (4 × 1)
= 200 + 10 + 80 + 4
= 294
Hence, from the above,
We can conclude that the product for the given expression is: 294

Question 5.
14 × 12
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 30
Answer:
The given expression is: 14 × 12
Now,
By using the Partial Multiplication Method,
14 × 12
= (10 + 4) × (10 + 2)
= (10 × 10) + (10 × 2) + (4 × 10) + (4 × 2)
= 100 + 20 + 40 + 8
= 168
Hence, from the above,
We can conclude that the product for the given expression is: 168

Question 6.
18 × 18
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 31
Answer:
The given expression is: 18 × 18
Now,
By using the Partial Multiplication method,
18 × 18
= (10 + 8) × (10 + 8)
= (10 × 10) + (10 × 8) + (8 × 10) + (8 × 8)
= 100 + 80 + 80 + 64
= 324
Hence, from the above,
We can conclude that the product for the given expression is: 324

Question 7.
15 × 13
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 32
Answer:
The given expression is: 15 × 13
Now,
By using the Partial Multiplication ethod,
15 × 13
= (10 + 5) × (10 + 3)
= (10 × 10) + (5 × 10) + (10 × 3) + (5 × 3)
= 100 + 50 + 30 + 15
= 195
Hence, from the above,
We can conclude that the product for the given expression is: 195

Problem Solving

Question 8.
Find the missing factor. 20 × ? = 420 Explain how you solved.
Answer:
The given expression is: 20 ×? = 420
Now,
Let the missing factor be x
So,
20 × x = 420
x = \(\frac{420}{20}\)
x = 21
Hence, from the above,
We can conclude that the missing factor for the given expression is: 21

Question 9.
The flagpole in front of City Hall in Lou’s town is 35 feet tall. How many inches tall is the flagpole? Remember, there are 12 inches in 1 foot.
Answer:
It is given that
The flagpole in front of City Hall in Lou’s town is 35 feet tall
Now,
We know that,
1 foot = 12 inches
So,
The height of the flagpole in inches = (The height of the flagpole in feet) × (The number of inches for 1 foot)
= 35 × 12
Now,
By using the Partial Multiplication method,
35 × 12
= (30 + 5) × (10 + 2)
= (30 × 10) + (30 × 2) + (5 × 10) + (5 × 2)
= 300 + 60 + 50 + 10
= 420
Hence, from the above,
We can conclude that the height of the flagpole in inches is: 420 inches

For 10-11, use the array at the right.
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 33

Question 10.
Model with Math Maggie is making a balloon game for the school fair. Students throw darts to pop the balloons. Draw lines on the array to separate each factor into tens and ones. How many balloons are used to set up the game?
Answer:
It is given that
Maggie is making a balloon game for the school fair. Students throw darts to pop the balloons
Now,
The given array is:
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 33
So,
The total number of balloons used to set up the game = (The total number of rows of balloons) × (The total number of columns of balloons)
= 13 × 14
Now,
By using the Partial Multiplication method,
13 × 14
= (10 + 3) × (10 + 4)
= (10 × 10) + (10 × 4) + (3 × 10) + (3 × 4)
= 100 + 40 + 30 + 12
= 182

Hence, from the above,
We can conclude that the total number of balloons that are used to set up the game is: 182 balloons

Question 11.
Higher-Order Thinking Maggie knows she will have to completely refill the balloon board about 15 times. Write an equation to show the number of balloons Maggie will need.
Answer:
From Problem 10,
We know that,
The total number of balloons that are used to set up the game is: 182
Now,
It is given that
Maggie knows she will have to completely refill the balloon board about 15 times.
So,
The total number of balloons maggie will need to refill = (The total number of balloons that are used to set up the game) × (The number of times the balloon board will have to refill)
= 182 × 15
Now,
By using the Partial Multiplication method,
182 × 15
= (100 + 80 + 2) × 15
= (100 × 15) + (80 × 15) + (2 × 15)
= 1,500 + 1,200 + 30
= 2,730
Hence, from the above,
We can conclude that
The total number of balloons that Maggie will need to refill the balloon board 15 times is: 2,730 balloons

Assessment Practice

Question 12.
Insert the missing partial products. Then add to find the product.
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 34
Answer:
The given expressions are:
a. 13 × 23     b 33 × 42
Now,
By using the Partial Multiplication Method,
a
13 × 23
= (10 + 3) × (20 + 3)
= (10 × 20) + (10 × 3) + (3 × 20) + (3 × 3)
= 200 + 30 + 60 + 9
= 299
b
33 × 42
= (30 + 3) × (40 + 2)
= (30 × 40) + (30 × 2) + (3 × 40) + (3 × 2)
= 1,200 + 60 + 120 + 6
= 1,386
Hence, from the above,
We can conclude that
The missing partial products and the addition of partial products for the given expressions are:

Question 13.
Insert the missing factor in each equation.
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 35
Answer:
Let the missing factor be x
Now,
The given expressions are with the representation of missing factor are:
a. 19 × x = 1,007
b. 62 × x = 1,736
c. x × 37 = 1,813
d. 15 × x = 1,260
So,
The missing factors for the given expressions will be:
a.
x = \(\frac{1,007}{19}\)
x = 53
b.
x = \(\frac{1,736}{62}\)
x = 28
c.
x = \(\frac{1,813}{37}\)
x = 49
d.
x = \(\frac{1,260}{15}\)
x = 84
Hence, from the above,
We can conclude that
The missing factor in each equation is:

Lesson 4.5 Area Models and Partial Products

Solve & Share
A playground is divided into four sections as shown in the diagram below. Find the area of the playground. Explain how you found the answer. Solve this problem using any strategy you choose.
I can … use area models and properties of operations to multiply two 2-digit numbers.
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 36
Answer:
It is given that
A playground is divided into four sections as shown in the diagram below.
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 36
Now,
From the given shape of the playground,
We can observe that the shape is like an “Area-Model” used for multiplication
So,
By using the Area-Model for multiplication,
The area of the playground = The sum of all the areas of the parts of the playground
Now,
By using the partial Multiplication method,
The area of the playground = (10 × 20) + (10 × 4) + (8 × 20) + (8 × 4)
= 200 + 40 + 160 + 32
= 432 feet²
Hence, from the above,
We can conclude that the area of the given playground is: 432 feet²

Look Back! Explain how you found the area of the blue part of the playground?
Answer:
The given figure is:
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 36
Now,
From the above problem,
We can observe that
We had found out the area of the playground by using the Partial Multiplication method
Now,
By using the Partial Multiplication Method,
The area of the blue part of the playground = (20 × 10) + (8 × 20)
= 200 + 160
= 360 feet²
Hence, from the above,
We can conclude that the area of the blue part of the playground is: 360 feet²

Essential Question
How Can You Use the Distributive Property to Multiply?
Answer:
The steps to use the Distributive Property of Multiplication are:
a. Simplify the numbers.
b. Split the problem into two easier problems.
c Take the number outside the parentheses and multiply it by each number inside the parentheses, one at a time.
d. Add the products.

Visual Learning Bridge
There are 15 players on each baseball team of the Strike Out Club. How many players are on all of the teams in the Strike Out Club?
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 37

One Way
Use an area model and the Distributive Property to find 25 × 15.
Break apart 15 into 10+ 5.
25 × (10+5)
(25 × 10) + (25 × 5)
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 38

Another Way
Use an area model and the Distributive Property to find 25 × 15.
Break apart 25 into 20 + 5.
(20 + 5) 15)
(20 × 15) + (5 × 15)
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 39

Another Way
Use an area model and the Distributive Property to find 25 × 15.
Break apart both factors.
Break apart 25 into 20 + 5.
Break apart 15 into 10+ 5.
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 40

Convince Me!
Use Structure How does an area model illustrate the Distributive Property?
Answer:
The distributive property says that multiplication distributes over addition.
The representation of the area model will be in two ways:
A) a(b + c)
B) ab + ac
The equality of the expressions,
a(b + c) = ab + ac
is the distributive property.

Guided Practice

Do You Understand?
Question 1.
What four simpler multiplication problems are used to find 24 × 23?
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 41
Answer:
The given expression is: 24 × 23
Now,
By using the Partial multiplication method,
24 × 23
= (20 + 4) × (20 + 3)
= (20 × 20) + (20 × 3) + (4 × 20) + (4 × 3)
= 400 + 60 + 80 + 12
= 552
Hence, from the above,
We can conclude that
The four simpler multiplication problems used to find 24 × 23 are:
a. 20 × 20 = 400
b. 20 × 3 = 60
c. 4 × 20 = 80
d. 4 × 3 = 12

Question 2.
How can an area model and the Distributive Property help you multiply? Use 12 × 16 to explain.
Answer:
We know that,
The distributive property says that multiplication distributes over addition.
The representation of the area model will be in two ways:
A) a(b + c)
B) ab + ac
The equality of the expressions,
a(b + c) = ab + ac
is the distributive property.
Now,
By using the Distributive Property of Multiplication,
12 × 16
= 12 × (10 + 6)
= (12 × 10) + (12 × 6)
= 120 + 72
= 192
Hence, from the above,
We can conclude that the product for the expression 12 × 16 is: 192

Do You Know How?
Question 3.
Use the area model and the Distributive Property to find 35 × 12.
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 42
35 × 12 =
30 × 10 =
5 × 10 =
30 × 2 =
5 × 2 =
______ + _____ + ______ + _______ = _________
Answer:
The given expression is: 35 × 12
Now,
By using the area model and the Distributive Property of Multiplication,
35 × 12
= (30 + 5) × (10 + 2)
= (30 × 10) + (30 × 2) + (5 × 10) + (5 × 2)
= 300 + 60 + 50 + 10
= 420
Hence, from the above,
We can conclude that the value of the product for the given expression is: 420

Independent Practice

For 4-10, draw an area model to find each product.
Question 4.
18 × 25
Answer:
The given expression is: 18 × 25
Now,
By using the area model,
18 × 25
= (10 + 8) × (20 + 5)
= (10 × 20) + (10 × 5) + (8 × 20) + (8 × 5)
= 200 + 50 + 160 + 40
= 450

Hence, from the above,
We can conclude that the product value for the given expression is: 450

Question 5.
14 × 25
Answer:
The given expression is: 14 × 25
Now,
By using the area model,
14 × 25
= (10 + 4) × (20 + 5)
= (10 × 20) + (10 × 5) + (4 × 20) + (4 × 5)
= 200 + 50 + 80 + 20
= 350

Hence, from the above,
We can conclude that the product value for the given expression is: 450

Question 6.
22 × 88
Answer:
The given expression is: 22 × 88
Now,
By using the area model,
22 × 88
= (20 + 2) × (80 + 8)
= (20 × 80) + (20 × 8) + (2 × 80) + (2 × 8)
= 1,600 + 160 + 160 + 16
= 1,936

Hence, from the above,
We can conclude that the product value for the given expression is: 1,936

Question 7.
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 43
Answer:
The given expression is: 41 ×12
Now,
By using the area model,
41 × 12
= (40 + 1) × (10 + 2)
= (40 × 10) + (40 × 2) + (1 × 10) + (1 × 2)
= 400 + 80 + 10 + 2
= 492

Hence, from the above,
We can conclude that the product value for the given expression is: 492

Question 8.
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 44
Answer:
The given expression is: 30 × 27
Now,
By using the area model,
30 × 27
= 30 × (20 + 7)
= (30 × 20) + (30 × 7)
= 600 + 210
= 810

Hence, from the above,
We can conclude that the product value for the given expression is: 810

Question 9.
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 45
Answer:
The given expression is: 58 × 19
Now,
By using the area model,
58 × 19
= (50 + 8) × (10 + 9)
= (50 × 10) + (50 × 9) + (8 × 10) + (8 × 9)
= 500 + 450 + 80 + 72
= 1,102

Hence, from the above,
We can conclude that the product value for the given expression is: 1,102

Question 10.
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 46
Answer:
The given expression is: 29 × 50
Now,
By using the area model,
29 × 50
= (20 + 9) × 50
= (20 × 50) + (9 × 50)
= 1,000 + 450
= 1,450

Hence, from the above,
We can conclude that the product value for the given expression is: 1,450

Problem Solving

Question 11.
Write 652,079 using number names and in expanded form.
Answer:
The given number is: 652,079
Now,
We know that,
“Expanded form” or the “Expanded notation” is a way of writing numbers to see the math value of individual digits. When numbers are separated into individual place values and decimal places they can also form a mathematical expression
Hence,
The representation of the given number in the word form is:
Six hundred fifty-two thousand, and Seventy-nine
The representation of the given number in the expanded form is:
600,000 + 50,000 + 2,000 + 0 + 70 + 9

Question 12.
Number Sense Sara estimated 23 × 43 using 20 × 40. Sam estimated 23 × 43 using 25 × 40. Will Sara’s or Sam’s method give an estimate closer to the exact answer? Explain.
Answer:
It is given that
Sara estimated 23 × 43 using 20 × 40. Sam estimated 23 × 43 using 25 × 40
Now,
The given expression is: 23 × 43
Now,
By using the Rounding method,
The given expression will become: 20 × 40
By using the Compatible numbers method,
The given expression will become: 25 × 40 (or) 25 × 45
Now,
The exact answer is:
23 × 43
= (20 + 3) × (40 + 3)
= (20 × 40) + (20 × 3) + (3 × 40) + (3 × 3)
= 800 + 60 + 120 + 9
= 989
Now,
The estimation of Sara is:
20 × 40
= 2 × 10 × 4 × 10
= (2 × 4) × 100
= 8 × 100
= 800
Now,
The estimation of Sam is:
25 × 40
= (20 + 5) × 40
= (20 × 40) + (5 × 40)
= 800 + 200
= 1,000
Hence, from the above,
We can conclude that
Sam’s method gives an estimate closer to the exact answer

Question 13.
Use Structure Each family of Florida Scrub-Jays inhabits 25 acres of land. No other Scrub-Jay families live within this area. How many acres of land are needed for 24 families of Florida Scrub-Jays? Show how you can use the Distributive Property to solve this problem.
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 47
Answer:
It is given that
Each family of Florida Scrub-Jays inhabits 25 acres of land. No other Scrub-Jay families live within this area.
So,
The acres of land that is needed for 24 families of Florida Scrub-jays = (The amount of land needed by each family of Florida Scrub-Jays) × (The total number of families)
= 25 × 24
Now,
By using the Distributive Property of Multiplication,
25 × 24
= (20 + 5) × (20 + 4)
= (20 × 20) + (20 × 4) + (5 × 20) + (5 × 4)
= 400 + 80 +100 + 20
= 600 acres of land
Hence, from the above,
We can conclude that
The total amount of acres of land needed for 24 families of Florida Scrub-Jays is: 600 acres of land

Question 14.
Marla wants to buy a new tablet that costs $565, including tax. She saved $15 per week for 30 weeks. Does Marla have enough money saved to buy the tablet? Explain.
Answer:
It is given that
Marla wants to buy a new tablet that costs $565, including tax. She saved $15 per week for 30 weeks.
Now,
The amount of money that Marla saved = (The amount of money she saved per week) × (Total number of weeks)
= $15 × 30
= (10 + 5) × 30
= (10 ×30) + (5 × 30)
= 300 + 150
= $450
So,
The amount of money saved by Marla < The cost of a new tablet
$450 < $565
Hence, from the above,
We can conclude that Marla does not have enough money saved to buy the tablet

Question 15.
Higher-Order Thinking Which costs less: 13 oranges that cost 29 cents each or 17 apples that cost 25 cents each? How much less?
Answer:
It is given that
13 oranges cost 29 cents each and 17 apples cost 25 cents each
Now,
The total cost of Oranges = (The number of Oranges) × (The cost of each Orange)
= 13 × 29
= (10 + 3) × (20 + 9)
= (10 × 20) + (10 × 9) + (3 × 20) + (3 × 9)
= 200 + 90 + 60 + 27
= $377
Now,
The total cost of apples = (The number of apples) × (The cost of each apple)
= 17 × 25
= (10 + 7) × (20 + 5)
= (10 × 20) + (10 × 5) + (7 × 20) + (7 × 5)
= 200 + 50 + 140 + 35
= $425
So,
The difference between the cost of Oranges and the cost of apples = $425 – $377
= $48
Hence, from the above,
We can conclude that
The cost of Oranges is less
The cost of Oranges is less by $48 than the cost of apples

Assessment Practice

Question 16.
Select all of the partial products which would be used to find 19 × 26.
☐ 45; 60; 180; 200
☐ 200; 180; 60; 54
☐ 6; 45; 180; 200
☐ 200; 60; 180; 54
☐ 0; 54; 180; 200
Answer:
The given expression is: 19 × 26
Now,
By using the area model,
19 × 26
= (10 + 9) × (20 + 6)
= (10 × 20) + (10 × 6) + (9 × 20) + (9 × 6)
= 200 + 60 + 180 + 54
Hence, from the above,
We can conclude that all the partial products that are used to find 19 × 26 are:

Question 17.
Select all of the ways you can use breaking apart and the Distributive Property to find the product of 35 × 12.
☐ 35 × (10 + 2)
☐ (12 × 30) + (12 × 5)
☐ (12 × 5) + (35 × 5)
☐ (30 × 5) + (10 × 2)
☐ (30 × 10) + (5 × 10) + (30 × 2) + (5 × 2)
Answer:
The given expression is: 35 × 12
Now,
By using the Distributive Property of Multiplication,
35 × 12
= 35 × (10 + 2) —- (1)
= (30 + 5) × 12
= (30 × 12) + (5 × 12) —- (2)
= (30 + 5) × (10 + 2)
= (30 × 10) + (30 × 2) + (5 × 10) + (5 × 2) —– (3)
Hence, from the above,
We can conclude that
All of the ways you can use breaking apart and the Distributive Property to find the product of 35 × 12 are:

Lesson 4.6 Use Partial Products to Multiply by 2-Digit Numbers

Solve & Share
There are 11 regular players and 5 substitute players on a soccer team. How many players are on 15 soccer teams? Solve this problem using any strategy you choose.
I can … use place value and partial products to multiply.
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 48
Answer:
It is given that
There are 11 regular players and 5 substitute players on a soccer team.
So,
The total number of players in 15 teams = (The number of players) × (The number of teams)
= (Regular players + Substitute players) × (The number of teams)
= (11 + 5) × 15
= (11 × 15) + (5 × 15)
= 165 + 75
= 240 players
Hence, from the above,
We can conclude that there are a total of 240 players in 15 teams

Look Back! How could you use an array and rounding or an array and compatible numbers to estimate the product for the problem above?
Answer:
From the above problem,
We can observe that
We have to use the area model or the Distributive property to find the total number of players because there are different types of players i.e., Regular players, Substitute players, etc.

Essential Question
How Can You Record Multiplication?
Answer:
The ways to record Multiplication are:
a. Grid method.
b. Long multiplication.
c. Lattice multiplication.
d. Binary or Peasant multiplication.
e. Binary multiplication in computers.
f. Shift and add.
g Quarter square multiplication.
h. Fast multiplication algorithms for large inputs

Visual Learning Bridge
Marcia put 7 oranges and 8 apples into each of 12 bags. How many pieces of fruit did Marcia put into all of the bags?
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 49

Find 12 × 15.
Estimate: 12 × 15 is about 10 × 15 = 150.
First, multiply the ones.
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 50

Then, multiply the tens.
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 51

Convince Me!
Reasoning Which partial products are incorrect? What is the correct final product?
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 52
Answer:
The given expression is: 26 × 12
Now,
By using the Partial multiplication method,
26 × 12
= (20 + 6) × (10 + 2)
= (20 × 10) + (20 × 2) + (6 × 10) + (6 × 2)
= 200 + 40 + 60 + 12
= 312
Hence, from the above,
We can conclude that
The given incorrect partial products are: 200, 60
The correct final product is: 312

Guided Practice

Do You Understand?
Question 1.
In the example on the previous page, why do you find 2 × 1 ten, rather than 2 × 1?
Answer:
We know that,
1 ten = 10 —–> 10’s place-value
1 = 1 ——> Unit’s place-value
Now,
From the above,
We can observe that there is a lot of difference 10’s position and unit’s position
Hence, from the above,
We can conclude that
We would write 2 × 1 ten rather than 2 × 1 as we observed in the example on the previous page

Question 2.
In the example on the previous page, could you record the 4 partial products in a different order? Explain.
Answer:
The given expression in the example on the previous page is: 15 × 12
Now,
By using the partial Products method,
15 × 13
= (10 + 5) × (10 + 2)
= (10 × 10) + (10 × 2) + (5 × 10) + (5 × 2)
= 100 + 20 + 50 + 10
= 180
Hence, from the above,
We can conclude that
We can record the 4 partial products in a different order for the example that is present in the previous page

Do You Know How?
For 3-4, find all the partial products. Then add to find the final product. Draw area models as needed.
Question 3.
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 53
Answer:
The given expression is: 23 × 14
Now,
By using the Partial Products method,
23 × 14
= (20 + 3) × (10 + 4)
= (20 × 10) + (20 × 4) + (3 × 10) + (3 × 4)
= 200 + 80 + 30 + 12
= 322
Hence, from the above,
We can conclude that the final product for the given expression is: 322

Question 4.
Envision Math Common Core 4th Grade Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 54
Answer:
The given expression is: 41 × 25
Now,
By using the Partial Products method,
41 × 25
= (40 + 1) × (20 + 5)
= (40 × 20) + (40 × 5) + (1 × 20) + (1 × 5)
= 800 + 200 + 20 + 5
= 1,025
Hence, from the above,
We can conclude that the final product for the given expression is: 1,025

Independent Practice

For 5-12, estimate. Find all the partial products. Then add to find the final product. Draw area models as needed.
Remember to check if your final product is reasonable.
Question 5.
Envision Math Common Core Grade 4 Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 55
Answer:
The given expression is: 34 × 51
Now,
By using the Partial Products method,
34 × 51
= (30 + 4) × (50 + 1)
= (30 × 50) + (30 × 1) + (4 × 50) + (4 × 1)
= 1,500 + 30 + 200 + 4
= 1,734
Estimate:
35 × 50
= (30 + 5) × 50
= (30 × 50) + (5 × 50)
= 1,500 + 250
= 1,750
Hence, from the above,
We can conclude that
The final product for the given expression is: 1,750
The final product is reasonalble when we compare to its estimate

Question 6.
Envision Math Common Core Grade 4 Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 56
Answer:
The given expression is: 73 × 81
Now,
By using the Partial Products method,
73 × 81
= (70 + 3) × (80 + 1)
= (70 × 80) + (70 × 1) + (3 × 80) + (3 × 1)
= 5,600 + 70 + 240 + 3
= 5,913
Estimate:
75 × 80
= (70 + 5) × 80
= (70 × 80) + (5 × 80)
= 5,600 + 400
= 6,000
Hence, from the above,
We can conclude that
The final product for the given expression is: 5,913
The final product is reasonalble when we compare to its estimate

Question 7.
Envision Math Common Core Grade 4 Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 57
Answer:
The given expression is: 64 × 32
Now,
By using the Partial products method,
64 × 32
= (60 + 4) × (30 + 2)
= (60 × 30) + (60 × 2) + (4 × 30) + (4 × 2)
= 1,800 + 120 + 120 + 8
= 2,048
Estimate:
65 × 30
= (60 + 5) × 30
= (60 × 30) + (5 × 30)
= 1,800 + 150
= 1,950
Hence, from the above,
We can conclude that
The final product for the given expression is: 2,048
The final product is reasonalble when we compare to its estimate

Question 8.
Envision Math Common Core Grade 4 Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 58
Answer:
The given expression is: 26 × 53
Now,
By using the Partial Products method,
26 × 53
= (20 + 6) × (50 + 3)
= (20 × 50) + (20 × 3) + (6 × 50) + (6 × 3)
= 1,000 + 60 + 300 + 18
= 1,378
Estimate:
25 × 50
= (20 + 5) × 50
= (20 × 50) + (5 × 50)
= 1,000 + 250
= 1,250
Hence, from the above,
We can conclude that
The final product for the given expression is: 1,378
The final product is not reasonalble when we compare to its estimate

Question 9.
Envision Math Common Core Grade 4 Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 59
Answer:
The given expression is: 38 × 17
Now,
By using the partial products method,
38 × 17
= (30  +8) × (10 + 7)
= (30 × 10) + (30 × 7) + (8 × 10) + (8 × 7)
= 300 + 210 + 80 + 56
= 646
Estimate:
40 × 15
= 40 × (10 + 5)
= (40 × 10) + (40 × 5)
= 400 + 200
= 600
Hence, from the above,
We can conclude that
The final product for the given expression is: 646
The final product is reasonalble when we compare to its estimate

Question 10.
Envision Math Common Core Grade 4 Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 60
Answer:
The given expression is: 34 × 30
Now,
By using the Partial Products method,
34 × 30
= (30 + 4) × 30
= (30 × 30) + (4 × 30)
= 900 + 120
= 1,020
Estimate:
35 × 30
= (30 + 5) × 30
= (30 × 30) + (5 × 30)
= 900 + 150
= 1,050
Hence, from the above,
We can conclude that
The final product for the given expression is: 1,020
The final product is reasonalble when we compare to its estimate

Question 11.
Envision Math Common Core Grade 4 Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 61
Answer:
The given expression is: 19 × 43
Now,
By using the Partial products method,
19 × 43
= (10 + 9) × (40 + 3)
= (10 × 40) + (10 × 3) + (9 × 40) + (9 × 3)
= 400 + 30 + 360 + 27
= 817
Estimate:
20 × 45
= 20 × (40 + 5)
= (20 × 40) + (20 × 5)
= 800 + 100
= 900
Hence, from the above,
We can conclude that
The final product for the given expression is: 817
The final product is reasonalble when we compare to its estimate

Question 12.
Envision Math Common Core Grade 4 Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 62
Answer:
The given expression is: 19 × 52
Now,
By using the Partial Products method,
19 × 52
= (10 + 9) × (50 + 2)
= (10 × 50) + (10 × 2) + (9 × 50) + (9 × 2)
= 500 + 20 + 450 + 18
= 988
Estimate:
20 ×50
= 2 × 10 × 5 × 10
= (2 × 5) × 100
= 10 × 100
= 1,000
Hence, from the above,
We can conclude that
The final product for the given expression is: 988
The final product is reasonalble when we compare to its estimate

Problem Solving

Question 13.
The Castillo de San Marcos is a Spanish fortress that was built between 1672 and 1695.
Envision Math Common Core Grade 4 Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 63
a. Rounded to the nearest ten thousand, how many pesos did it cost to build the fortress?
Answer:
It is given that
The Castillo de San Marcos is a Spanish fortress that was built between 1672 and 1695.
Now,
The given figure is:
Envision Math Common Core Grade 4 Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 63
Now,
From the given figure,
We can observe that
It is given that it costs 138,375 pesos to build the given fortress
Now,
From the given number,
We can observe that
From the left of the digits’ positions, the 5th position value is “Ten-Thousand”
So,
The given number when it is rounded to the nearest ten thousand is: 140,000
Hence, from the above,
We can conclude that
The number we can obtain when we rounded the given number to the nearest ten thousand is: 140,000
It costs about 140,000 pesos to build the fortress

b. How many years did it take to build the fortress?
Answer:
It is given that
The Castillo de San Marcos is a Spanish fortress that was built between 1672 and 1695.
So,
The number of years did it take to build the fortress = 1695 – 1672
= 23 years
Hence, from the above,
We can conclude that it took 23 years to build a fortress

Question 14.
Construct Arguments A school has 2 large patios. One is rectangular and is 24 feet long by 18 feet wide. The other is square and each side is 21 feet long. Which patio has a greater area? Explain.
Answer:
It is given that
A school has 2 large patios. One is rectangular and is 24 feet long by 18 feet wide. The other is square and each side is 21 feet long
Now,
We know that,
The area of a rectangle = Length × Width
The area of a square = Side × Side
Now,
The area of a rectangular patio = 24 × 18
= (20 + 4) × (10 + 8)
= (20 × 10) + (20 × 8) + (4 × 10) + (4 × 8)
= 200 + 160 + 40 + 32
= 432 feet²
Now,
The area of a square patio = 21 × 21
= (20 + 1) × (20 + 1)
= (20 ×20) + (20 × 1) + (1 × 20) + (1 × 1)
= 400 + 20 + 20 + 1
= 441 feet²
So,
The area of a square patio > The area of a rectangular patio
Hence, from the above,
We can conclude that square patio has a greater area

Question 15.
Higher Order Thinking Show two ways! to break apart the factors, and then find the product of 14 × 22.
Answer:
The given expression is: 14 × 22
Now,
By using the Distributive Property of Multiplication,
Way 1:
14 × 22
= (10 + 4) × (20 + 2)
= (10 × 20) × (10 × 2) + (4 × 20) + (4 × 2)
= 200 + 20 + 80 + 8
= 308
Way 2:
14 × 22
= 14 × (20 + 2)
= (14 × 20) + (14 × 2)
= 280 + 28
= 308
Hence, from the above,
We can conclude that the product value for the given expression is: 308

Assessment Practice

Question 16.
Which set of numbers has the missing partial product and the final product?
Envision Math Common Core Grade 4 Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 64
A. 210; 836
B. 18; 806
C. 180; 806
D. 18; 644
Answer:
The given expression is: 13 × 62
Now,
By using the Partial Products method,
13 × 62
= (10 + 3) × (60 + 2)
= (10 × 60) + (10 × 2) + (3 × 60) + (3 × 2)
= 600 + 20 + 180 + 6
= 806
Hence, from the above,
We can conclude that
The missing partial product and the final product are:

Question 17.
Select all the equations in which 12 is the missing factor
☐ b × 14 = 336
☐ b × 36 = 432
☐ 18 × b = 216
☐ 39 × b = 468
☐ 21 × b = 231
Answer:
The given equations are:
a. b × 14 = 336           b. b ×36 = 432               c. 18 × b = 216
d. 39 × b = 468           e. 21 × b = 231
Now,
a.
b × 14 = 336
b = \(\frac{336}{14}\)
b = 24
b.
b × 36 = 432
b = \(\frac{432}{36}\)
b = 12
c.
18 × b = 216
b = \(\frac{216}{18}\)
b = 12
d.
39 × b = 468
b = \(\frac{468}{39}\)
b = 12
e.
21 × b = 231
b = \(\frac{231}{21}\)
b = 11
Hence, from the above,
We can conclude that the expressions that have the missing factor 12 are:

Lesson 4.7 Problem Solving

Make Sense and Persevere
Solve & Share
Five students set a goal to raise $500 from their charity walk. Sponsors donated $25 for each mile walked. By how much did these students exceed or miss their goals? Solve this problem using any strategy you choose.
I can … make sense of problems and keep working if I get stuck.
Envision Math Common Core Grade 4 Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 65

Thinking Habits
Be a good thinker! These questions can help you.
1. What do I need to find?
2. What do I know?
3. What’s my plan for solving the problem?
4. What else can I try if I get stuck?
5. How can I check that my solution makes sense?
Answer:
It is given that
Five students set a goal to raise $500 from their charity walk. Sponsors donated $25 for each mile walked
Now,
The given table is:
Envision Math Common Core Grade 4 Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 65
Now,
From the given table,
The total number of miles walked by all the students = 5 + 5 + 3 + 4 + 4
= 21 miles
So,
The amount of money raised by the given students = (The total number of miles walked by all the students) × (The amount of money donated by the sponsor for each mile)
= 21 × 25
= (20 + 1) × (20 + 5)
= (20 × 20) + (20 × 5) + (1 × 20) + (1 × 5)
= 400 + 100 + 20 + 5
= $525
So,
$500 < $525
Now,
The amount of money exceeded = (The amount of money raised by all the students) – (The amount of money that have to be raised by the students)
= $525 – $500
= $25
Hence, from the above,
We can conclude that the amount of money exceeded is: $25

Look Back!
Make Sense and Persevere Is there more than one way to solve the problem? Explain.
Answer:
No, there is no other way to solve the above problem because we must find the total number of miles traveled by all the students and the amount of money raised by them by walking the number of miles

Essential Question
How Can You Make Sense of and Persevere in Solving Problems with More Than One Step?
Answer:
Mathematically, proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals

Visual Learning Bridge
The park has a large garden with a walkway around it. The park’s ground crew are going to paint the walkway. What is the area of the walkway?
Envision Math Common Core Grade 4 Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 66
What is a good plan for solving this problem?
I need to find the area of the garden and walkway together. Then I need to subtract to find only the area of the walkway.

How can I make sense of and solve this problem?
I can

  • identify the quantities given.
  • understand how the quantities are related.
  • choose and implement an appropriate strategy.
  • check to be sure my work and answer make sense.

First find the area of the whole park.
Envision Math Common Core Grade 4 Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 67
52 × 85 = p
4,420 = p
Next, find the area of the of the garden.
32 × 65 = g
2,080 = g
Subtract to find the area of the walkway. 4,420 – 2,080 = w
The area of the walkway is 2,340 square feet.

Convince Me!
Make Sense and Persevere Can you solve the problem a different way and still calculate the same answer? Explain.
Answer:
There is no other way to solve the above problem because we must calculate the area of the park and the area of the garden to find the area of the walkway since the walkway is enclosed between the park and the garden

Guided Practice

Make Sense and Persevere On her vacation, Julia filled 3 memory cards like the one shown. She printed 2 copies of each picture. How many pictures did Julia print?
Envision Math Common Core Grade 4 Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 68
Question 1.
What do you know and what do you need to find?
Answer:
It is given that
On her vacation, Julia filled 3 memory cards like the one shown. She printed 2 copies of each picture.
Now,
We know that,
a. The number of filled memory cards
b. The number of copies of each picture
c. The number of pictures in each memory card
We need to find:
a. The number of pictures in all the filled memory cards
b. The total number of copies of pictures

Question 2.
What steps might you take to solve the problem?
Answer:
The steps you have to take to solve the given problem are:
Step 1:
Find the number of pictures in all the filled cards by multiplying the number of filled memory cards and the number of pictures in each memory card
Step 2:
After finding the number of pictures in all the filled memory cards, find the number of copies of pictures by multiplying the number of pictures in all the filled memory cards with 2

Question 3.
How many pictures did Julia print? Explain.
Answer:
It is given that
On her vacation, Julia filled 3 memory cards like the one shown. She printed 2 copies of each picture.
Now,
The given figure is:
Envision Math Common Core Grade 4 Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 68
Now,
From the given figure,
We can observe that
A memory card holds 28 pictures
So,
The number of pictures present in 3 memory cards = (The number of filled memory cards) × (The number of pictures in each memory card)
= 28 × 3
= (20 + 8) × 3
= (20 × 3) + (8 × 3)
= 60 + 24
= 84 pictures
Now,
The  total number of copies of pictures = (The number of pictures present in 3 memory cards) × 2
= 84 × 2
= (80 + 4) × 2
= (80 × 2) + (4 × 2)
= 160 + 8
= 168 pictures
Hence, from the above,
We can conclude that
The total number of pictures did Julia print are: 168

Independent Practice

Make Sense and Persevere Jarrod delivers 63 newspapers each Monday through Saturday and 78 newspapers each Sunday. Last month consisted of 4 Sundays and 26 other days. How many newspapers did Jarrod deliver last month? Use Exercises 4-6 to solve this problem.
Question 4.
What strategies can you use to find how many newspapers Jarrod delivered last month?
Answer:
It is given that
Jarrod delivers 63 newspapers each Monday through Saturday and 78 newspapers each Sunday. Last month consisted of 4 Sundays and 26 other days
Now,
The strategy can you use to find the number of newspapers Jarrod delivered last month is:
Step 1:
Find the total number of newspapers Jarrod delivered on other days
Step 2:
Find the total number of newspapers Jarrod delivered only on Sunday
Step 3:
Add the results we obtained in Step 1 and Step 2

Question 5.
How are the quantities related?
Answer:
The given quantities are related in a way such as:
The total number of newspapers Jarrod delivered last month = (The total amount of newspapers Jarrod delivered on other days) + (The total number of newspapers Jarrod delivered on Sunday)

Question 6.
Explain how to solve the problem.
Answer:
The steps to solve the given problem are:
Step 1:
Find the total number of newspapers Jarrod delivered on other days
Step 2:
Find the total number of newspapers Jarrod delivered only on Sunday
Step 3:
Add the results we obtained in Step 1 and Step 2

Problem Solving

Performance Task
Uniforms
The Stillwater Storm soccer team consists of 16 players. Each player needs a uniform set. The uniform set includes two jerseys, a pair of shorts, and a pair of socks. The price for each separate item is shown. Nine of the players need a medium size and the others need a small size. If the team buys more than 10 sets, the team price for each set is $56. How much money is saved if the team buys all the uniform sets together rather than separately?
Envision Math Common Core Grade 4 Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 69
Question 7.
Reasoning What are the quantities in the problem and how are they related?
Answer:
It is given that
The Stillwater Storm soccer team consists of 16 players. Each player needs a uniform set. The uniform set includes two jerseys, a pair of shorts, and a pair of socks. The price for each separate item is shown. Nine of the players need a medium size and the others need a small size. If the team buys more than 10 sets, the team price for each set is $56
So,
The quantities related to the given problem are;
a. The price for 2 pairs of jerseys
b. The price for a pair of shorts
c. The price for a pair of socks
Hence,
The relation between the quantities are:
The total cost of a set = (The cost of 2 pairs of jerseys) + (The cost of a pair of shorts) + (The cost of a pair of socks)

Question 8.
Model with Math
Use the bar diagram to write an addition equation and find the total cost of the uniform sets when purchased separately. Then find the total cost if 10 sets or more are purchased.
When you make sense and persevere, you think about the amounts given.
Envision Math Common Core Grade 4 Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 70
Answer:

Question 9.
Be Precise What is the difference in cost if the team bought all the uniform sets together or if they bought them separately?
Answer:

Topic 4 Fluency Practice Activity

Find a Match
Work with a partner. Point to a clue. Read the clue. Look below the clues to find a match. Write the clue letter in the box next to the match. Find a match for every clue.
I can … add and subtract multi-digit whole numbers.

Clues
A. The difference is between 950 and 1,000.
B. The difference is exactly 913.
C. The sum is between 600 and 700.
D. The sum is exactly 500.
E. The difference is between 700 and 800.
F. The sum is greater than 300 but less than 400.
G. The sum is exactly 753.
H. The difference is exactly 413.

Topic 4 Vocabulary Review

Understand Vocabulary
Word List

  • array
  • compatible numbers
  • estimate
  • factors
  • partial product
  • product
  • rounding
  • variable

Question 1.
Cross out the numbers that are NOT factors of 12.
1
3
5
6
8
Answer:
We know that,
A “factor” is defined as a number or algebraic expression that divides another number or expression evenly i.e., with no remainder
Hence, from the above,
We can conclude that
The numbers that are not factors of 12 are:

Question 2.
Cross out the numbers that are NOT good estimates for the product of 17 × 23.
600
400
300
200
100
Answer:
The given expression is: 17 × 23
Now,
The estimate of the given expression is: 20 × 25 (or) 15 × 25 (or) 20 × 20
Hence, from the above,
We can conclude that
The numbers that are not good estimates for the product 17 × 23 are:

Question 3.
Cross out the numbers that are NOT partial products for 12 × 41.
2
10
18
80
400
Answer:
The given expression is: 12 × 41
Now,
By using the Partial products method,
12 × 41
= (10 + 2) × (40 +1)
= (10 × 40) + (10 × 1) + (2 × 40) + (2 × 1)
= 400 + 10 + 80 + 2
= 492
Hence, from the above,
We can conclude that
The numbers that are not the partial products for the given expression are:

Label each example with a term from the Word List.

Question 4.
Envision Math Common Core Grade 4 Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 72 _________
Answer:
The given figure is:
Envision Math Common Core Grade 4 Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 72
Now,
We know that,
An “Array” is a collection of rows and columns
Hence, from the above,
We can conclude that the term that is sutable for the given figure is: Array

Question 5.
n. __________
Answer:
We know that,
A “Variable”, in algebra, is a symbol (usually a letter) standing in for an unknown numerical value in an equation
Hence, from the above,
We can conclude that the term that is best suitable for the given problem is: Variable

Question 6.
2,318 to the nearest thousand is 2,000 _________
Answer:
We know that,
The “Rounding-off”  is a mathematical technique of adjusting the number’s digits to make the number easier to use during calculations
Hence, from the above,
We can conclude that the term that is suitable for the given sentence is: Rounding

Question 7.
3 × 4 = 12 _________
Answer:
We know that,
A “Product” is a final value when we multiply the number of factors
Hence, from the above,
We can conclude that the term that is suitable for the given problem is: product

Use Vocabulary in Writing
Question 8.
Alicia needs to find 23 × 47. Use at least 3 terms from the Word List to explain how Alicia might find 23 × 47.
Answer:
It is given that
Alicia needs to find 23 × 47
Hence, from the above,
We can conclude that
The three words from the given word list to explain how Alicia might find the product for the given expression is:
a. Compatible numbers
b. Estimate
c. partial Product

Topic 4 Reteaching

Set A pages 129-132

Use place-value strategies to find 20 × 80.
Think about the pattern.
20 × 80 = 2 tens × 8 tens
= 16 hundreds
= 1,600

50 × 40 = (5 × 10) × (4 × 10)
= (5 × 4) × (10 × 10)
= 20 × 100
= 2,000

Remember when the product of a basic fact has a zero, there is one more zero in the answer.

Use a pattern to find each product.
Question 1.
40 × 10
Answer:
The given expression is: 40 × 10
Now,
By using the Place-value method,
40 × 10
= 4 × 10 × 1 × 10
= (4 × 1) × 100
= 400
Hence, from the above,
We can conclude that the product value for the given expression is: 400

Question 2.
60 × 20
Answer:
The given expression is: 60 × 20
Now,
By using the Place-value method,
60 × 20
= 6 × 10 × 2 × 10
= (6 × 2) × 100
= 1,200
Hence, from the above,
We can conclude that the product value for the given expression is: 1,200

Question 3.
80 × 50
Answer:
The given expression is: 80 × 50
Now,
By using the Place-value method,
80 × 50
= 8 × 10 × 5 × 10
= (8 × 5) × 100
= 4,000
Hence, from the above,
We can conclude that the product value for the given expression is: 4,000

Question 4.
30 × 90
Answer:
The given expression is: 30 × 90
Now,
By using the Place-value method,
30 × 90
= 3 × 10 × 9 × 10
= (3 × 9) × 100
= 2,700
Hence, from the above,
We can conclude that the product value for the given expression is: 2,700

Question 5.
80 × 70
Answer:
The given expression is: 80 × 70
Now,
By using the Place-value method,
80 × 70
= 8 × 10 × 7 × 10
= (8 × 7) × 100
= 5,600
Hence, from the above,
We can conclude that the product value for the given expression is: 5,600

Question 6.
60 × 60
Answer:
The given expression is: 60 × 60
Now,
By using the Place-value method,
60 × 60
= 6 × 10 × 6 × 10
= (6 × 6) × 100
= 3,600
Hence, from the above,
We can conclude that the product value for the given expression is: 3,600

Question 7.
80 × 30
Answer:
The given expression is: 80 × 30
Now,
By using the Place-value method,
80 × 30
= 8 × 10 × 3 × 10
= (8 × 3) × 100
= 2,400
Hence, from the above,
We can conclude that the product value for the given expression is: 2,400

Question 8.
20 × 50
Answer:
The given expression is: 20 × 50
Now,
By using the Place-value method,
20 × 50
= 2 × 10 × 5 × 10
= (2 × 5) × 100
= 1,000
Hence, from the above,
We can conclude that the product value for the given expression is: 1,000

Set B pages 133-136

Use an array or an area model to multiply 20 × 14.
Envision Math Common Core Grade 4 Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 73
200 + 80 = 280
So, 20 × 14 = 280.

Remember you can draw arrays or area models to represent multiplication problems.

Draw a model to find each product.
Question 1.
10 × 23
Answer:
The given expression is: 10 × 23
Now,
By using the Area-model,
10 × 23
= 10 × (20 + 3)
= (10 × 20) + (10 × 3)
= 200 + 30
= 230

Hence, from the above,
We can conclude that the product value for the given expression is: 230

Question 2.
16 × 20
Answer:
The given expression is: 16 × 20
Now,
By using the Area-model,
16 × 20
= (10 + 6) × 20
= (10 × 20) + (6 × 20)
= 200 + 120
= 320

Hence, from the above,
We can conclude that the product value for the given expression is: 320

Set C pages 137-140

Use compatible numbers to estimate 28 × 19.
28 is close to 25.
19 is close to 20.
If 25 × 2 = 50, then
25 × 20 = 500.
So, 28 × 19 is about 500.

Remember that compatible numbers are numbers that are easy to compute mentally.

Estimate each product.
Question 1.
29 × 31
Answer:
The given expression is: 29 × 31
Now,
By using the estimation of the Compatible number,
The given expression will become: 30 × 30
So,
30 × 30
= 3 × 10 × 3 × 10
= (3 × 3) × 100
= 9 × 100
= 900
Hence, from the above,
We can conclude that the product value for the given expression is about 900

Question 2.
42 × 49
Answer:
The given expression is: 42 × 49
Now,
By using the estimation of the Compatible number,
The given expression will become: 40 × 50
So,
40 × 50
= 4 × 10 × 5 × 10
= (4 × 5) × 100
= 20 × 100
= 2,000
Hence, from the above,
We can conclude that the product value for the given expression is about 2,000

Question 3.
73 × 18
Answer:
The given expression is: 73 × 18
Now,
By using the estimation of the Compatible number,
The given expression will become: 70 × 20
So,
70 × 20
= 7 × 10 × 2 × 10
= (7 × 2) × 100
= 14 × 100
= 1,400
Hence, from the above,
We can conclude that the product value for the given expression is about 1,400

Question 4.
24 × 38
Answer:
The given expression is: 24 × 38
Now,
By using the estimation of the Compatible number,
The given expression will become: 25 × 40
So,
25 × 40
= (20  +5) × 40
= (20 × 40) + (5 × 40)
= 800 + 200
= 1,000
Hence, from the above,
We can conclude that the product value for the given expression is about 1,000

Question 5.
19 × 31
Answer:
The given expression is: 19 × 31
Now,
By using the estimation of the Compatible number,
The given expression will become: 20 × 30
So,
20 × 30
= 2 × 10 × 3 × 10
= (2 × 3) × 100
= 6 × 100
= 600
Hence, from the above,
We can conclude that the product value for the given expression is about 600

Question 6.
63 × 87
Answer:
The given expression is: 63 × 87
Now,
By using the estimation of the Compatible number,
The given expression will become: 60 × 90
So,
60 × 90
= 6 ×10 ×9 × 10
= (6 × 9) × 100
= 54 × 100
= 5,400
Hence, from the above,
We can conclude that the product value for the given expression is about 5,400

Set D pages 141-144

Find 14 × 12. Draw a 14 × 12 array.
Separate each factor into tens and ones. Color each section a different color. Add each part to find the product.
Envision Math Common Core Grade 4 Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 74

Remember when you break apart a multiplication problem, you can solve the simpler problems in any order and the answer will remain the same.

Question 1.
14 × 32
Answer:
The given expression is: 14 × 32
Now,
By using the Partial Products method,
14 × 32
= (10 + 4) × (30 + 2)
= (10 × 30) + (10 × 2) + (4 × 30) + (4 × 2)
= 300 + 20 + 120 + 8
= 448
Hence, from the above,
We can conclude that the product value for the given expression is: 448

Question 2.
64 × 12
Answer:
The given expression is: 64 × 12
Now,
By using the Partial Products method,
64 × 12
= (60 + 4) × (10 + 2)
= (60 × 10) + (60 × 2) + (4 × 10) + (4 × 2)
= 600 + 120 + 40 + 8
= 768
Hence, from the above,
We can conclude that the product value for the given expression is: 768

Question 3.
56 × 17
Answer:
The given expression is: 56 × 17
Now,
By using the Partial Products method,
56 × 17
= (50 + 6) × (10 + 7)
= (50 × 10) + (50 × 7) + (6 × 10) + (6 × 7)
= 500 + 350 + 60 + 42
= 952
Hence, from the above,
We can conclude that the product value for the given expression is: 952

Question 4.
72 × 15
Answer:
The given expression is: 72 × 15
Now,
By using the Partial Products method,
72 × 15
= (70 + 2) × (10 + 5)
= (70 × 10) + (70 × 5) + (2 × 10) + (2 × 5)
= 700 + 350 + 20 + 10
= 1,080
Hence, from the above,
We can conclude that the product value for the given expression is: 1,080

Question 5.
26 × 63
Answer:
The given expression is: 26 × 63
Now,
By using the Partial Products method,
26 × 63
= (20 + 6) × (60 + 3)
= (20 × 60) + (20 × 3) + (6 × 60) + (6 × 3)
= 1,200 + 60 + 360 + 18
= 1,638
Hence, from the above,
We can conclude that the product value for the given expression is: 1,638

Question 6.
47 × 27
Answer:
The given expression is: 47 × 37
Now,
By using the Partial Products method,
47 × 37
= (40 + 7) × (30 + 7)
= (40 × 30) + (40 × 7) + (7 × 30) + (7 × 7)
= 1,200 + 280 + 210 + 49
= 1,739
Hence, from the above,
We can conclude that the product value for the given expression is: 1,739

Question 7.
19 × 51
Answer:
The given expression is: 19 × 51
Now,
By using the Partial Products method,
19 × 51
= (10 + 9) × (50 + 1)
= (10 × 50) + (10 × 1) + (9 × 50) + (9 × 1)
= 500 + 10 + 450 + 9
= 969
Hence, from the above,
We can conclude that the product value for the given expression is: 969

Question 8.
12 × 56
Answer:
The given expression is: 12 × 56
Now,
By using the Partial Products method,
12 × 56
= (10 + 2) × (50 + 6)
= (10 × 50) + (10 × 6) + (2 × 50) + (2 × 6)
= 500 + 60 + 100 + 12
= 662
Hence, from the above,
We can conclude that the product value for the given expression is: 662

Question 9.
76 × 23
Answer:
The given expression is: 76 × 23
Now,
By using the Partial Products method,
76 × 23
= (70 + 6) × (20 + 3)
= (70 × 20) + (70 × 3) + (6 × 20) + (6 × 3)
= 1,400 + 210 + 120 + 18
= 1,748
Hence, from the above,
We can conclude that the product value for the given expression is: 1,748

Question 10.
84 × 37
Answer:
The given expression is: 84 × 37
Now,
By using the Partial Products method,
84 × 37
= (80 + 4) × (30 + 7)
= (80 × 30) + (80 × 7) + (4 × 30) + (4 × 7)
= 2,400 + 560 + 120 + 28
= 3,108
Hence, from the above,
We can conclude that the product value for the given expression is: 3,108

Question 11.
14 × 72
Answer:
The given expression is: 14 × 72
Now,
By using the Partial Products method,
14 × 72
= (10 + 4) × (70 + 2)
= (10 × 70) + (10 × 2) + (4 × 70) + (4 × 2)
= 700 + 20 + 280 + 8
= 1,008
Hence, from the above,
We can conclude that the product value for the given expression is: 1,008

Question 12.
21 × 51
Answer:
The given expression is: 21 × 51
Now,
By using the Partial Products method,
21 × 51
= (20 + 1) × (50 + 1)
= (20 × 50) + (20 × 1) + (1 × 50) + (1 × 1)
= 1,000 + 20 + 50 + 1
= 1,071
Hence, from the above,
We can conclude that the product value for the given expression is: 1,071

Set E pages 145-152

Use the Distributive Property to find 13 × 55.
13 × 55
= (10 + 3) (50 + 5)
= (10 + 3) 50 + (10 + 3) × 5
= (10 × 50) + (3 × 50) + (10 × 5) + (3 × 5)
= 500 + 150 + 50 + 15
= 715

Remember you can break numbers apart in more than one way when using the Distributive Property to solve multiplication problems.

Question 1.
12 × 19
Answer:
The given expression is: 12 × 19
Now,
By using the Distributive Property of Multiplication,
12 × 19
= (10 + 2) × (10 + 9)
= (10 × 10) + (10 × 9) + (2 × 10) + (2 × 9)
= 100 + 90 + 20 + 18
= 228
Hence, from the above,
We can conclude that the product value for the given expression is: 228

Question 2.
38 × 12
Answer:
The given expression is: 38 × 12
Now,
By using the Distributive Property of Multiplication,
38 × 12
= (30 + 8) × (10 + 2)
= (30 × 10) + (30 × 2) + (8 × 10) + (8 × 2)
= 300 + 60 + 80 + 16
= 456
Hence, from the above,
We can conclude that the product value for the given expression is: 456

Question 3.
19 × 25
Answer:
The given expression is: 19 × 25
Now,
By using the Distributive Property of Multiplication,
19 × 25
= (10 + 9) × (20 + 5)
= (10 × 20) + (10 × 5) + (9 × 20) + (9 × 5)
= 200 + 50 + 180 + 45
= 475
Hence, from the above,
We can conclude that the product value for the given expression is: 475

Question 4.
45 × 23
Answer:
The given expression is: 45 × 23
Now,
By using the Distributive Property of Multiplication,
45 × 23
= (40 + 5) × (20 + 3)
= (40 × 20) + (40 × 3) + (5 × 20) + (5 × 3)
= 800 + 120 + 100 + 15
= 1,035
Hence, from the above,
We can conclude that the product value for the given expression is: 1,035

Question 5.
62 × 11
Answer:
The given expression is: 62 × 11
Now,
By using the Distributive Property of Multiplication,
62 × 11
= (60 + 2) × (10 + 1)
= (60 × 10) + (60 × 1) + (2 × 10) + (2 × 1)
= 600 + 60 + 20 + 2
= 682
Hence, from the above,
We can conclude that the product value for the given expression is: 682

Question 6.
46 × 26
Answer:
The given expression is: 46 × 26
Now,
By using the Distributive Property of Multiplication,
46 × 26
= (40 + 6) × (20 + 6)
= (40 × 20) + (40 × 6) + (6 × 20) + (6 × 6)
= 800 + 240 + 120 + 36
= 1,196
Hence, from the above,
We can conclude that the product value for the given expression is: 1,196

Set F pages 153-156

Think about these questions to help you make sense and persevere in solving problems.
Thinking Habits

  • What do I need to find?
  • What do I know?
  • What’s my plan for solving the problem?
  • What else can I try if I get stuck?
  • How can I check that my solution makes sense?
    Envision Math Common Core Grade 4 Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 76

Remember to use the information given to solve the problem.

Rose visited 14 cities on her vacation. She bought 8 souvenirs in each city to send to her friends. Rose paid $2 in postage for each souvenir to be sent.
Question 1.
What hidden question do you need to answer to find how much it cost Rose to send all of the souvenirs?
Answer:
It is given that
Rose visited 14 cities on her vacation. She bought 8 souvenirs in each city to send to her friends. Rose paid $2 in postage for each souvenir to be sent.
So,
The hidden question you need to answer to find how much it cost Rose to send all of the souvenirs is:
a. What is the total number of souvenirs Rose bought from all the cities?

Question 2.
How much did Rose spend on the souvenirs and postage?
Answer:
It is given that
Rose visited 14 cities on her vacation. She bought 8 souvenirs in each city to send to her friends. Rose paid $2 in postage for each souvenir to be sent.
So,
The total number of souvenirs Rose bought from all the cities = (The total number of cities) × (The number of souvenirs bought by Rose in each city)
= 14 × 8
= (10  +4) × 8
= (10 × 8) + (4 × 8)
= 80 + 32
= 112 souvenirs
So,
The total cost to send all the souvenirs in postage by Rose = (The total number of souvenirs Rose bought) × (The cost of each souvenir to be sent by Rose in postage)
= 112 × $2
= (100 + 2) × 2
= (100 × 2) + (2 × 2)
= 200 + 4
= $204
Hence, from the above,
We can conclude that the total money spent by Rose on souvenirs and postage is: $204

Topic 4 Assessment Practice

Question 1.
Don works 18 hours a week. Which expression shows a good way to use rounding to estimate how many hours Don will work in 52 weeks?
A. 10 × 50
B. 10 × 600
C. 20 × 50
D. 18 × 60
Answer:
It is given that Don works 18 hours a week
So,
The number of hours Don will work in 52 weeks = (The number of hours Don works in a week) × (The number of weeks)
= 18 × 52
Now,
By using the rounding-off estimation,
The number of hours Don will work in 52 weeks is about 20 × 50
Hence,
The number of hours Don will work in 52 weeks is about:

Question 2.
Find the product 21 × 40. Show your work.
Answer:
The given expression is: 21 × 40
Now,
By using the Distributive Property of Multiplication,
21 × 40
= (20 + 1) × 40
= (20 × 40) + (1 × 40)
= 800 + 40
= 840
Hence, from the above,
We can conclude that the product value for the given expression is: 840

Question 3.
The product of two factors is 2,000. One of the factors is 50. What is the other factor?
A. 20
B. 40
C. 200
D. 400
Answer:
It is given that
The product of two factors is 2,000. One of the factors is 50
Now,
Let the other factor be x
So,
50 × x = 2,000
x = \(\frac{2,000}{50}\)
x = 40
Hence, from the above,
We can conclude that the other factor is:

Question 4.
Select all the expressions that are equal to 260.
☐ 23 × 10
☐ 20 × 10 + 3 × 10
☐ 13 × 20
☐ 1 × 20 + 3 × 20
☐ 10 × 20 + 3 × 20
Answer:
The given expressions are:
a. 23 × 10
b. (20 × 10) + (3 × 10)
c. 13 × 20
d. (1 × 20) + (3 × 20)
e. (10 × 20) + (3 × 20)
So,
The product values for the given expressions are:
a.
23 × 10
= (20 + 3) × 10
= (20 × 10) + (3 × 10)
= 200 + 30
= 230
b.
(20 × 10) + (3 × 10)
= 230
c.
13 × 20
= (10 + 3) × 20)
= (10 × 20) + (3 × 20)
= 200 + 60
= 260
d.
(1 × 20) + (3 × 20)
= 70
e.
(10 × 20) + (3 × 20)
= 200 + 60
= 260
Hence, from the above,
We can conclude that all of the expressions that are equal to 260 are:

Question 5.
A florist makes centerpieces. He puts 18 roses in each centerpiece. Which is the best way to use compatible numbers to estimate the number of roses the florist needs for 24 centerpieces? What is the exact number of roses?
A. 10 × 25 = 250; 360
B. 20 × 25 = 500; 432
C. 25 × 30 = 750; 672
D. 30 × 30 = 900; 784
Answer:
It is given that
A florist makes centerpieces. He puts 18 roses in each centerpiece
So,
The number of roses the florist needs for 24 centerpieces = (The number of centerpieces) × (The number of roses in each centerpiece)
= 18 × 24
Now,
By using the partial Products method,
18 × 24
= (10 + 8) × (20 + 4)
= (10 × 20) + (10 × 4) + (8 × 20) + (8 × 4)
= 200 + 40 + 160 + 32
= 432
Now,
Estimate:
18 × 24
= 20 × 25 [Estimate of the total number of roses]
Now,
By using the Partial Products method,
20 × 25
= 20 × (20 + 5)
= (20 × 20) + (20 × 5)
= 400 + 100
= 500
Hence, from the above,
We can conclude that
The estimated and the exact number of the total number of roses in 24 centerpieces are:

Question 6.
Margo hiked 12 miles 13 times last month. She hiked 14 miles 12 times this month.
A. Draw arrays or area models to find the number of miles Margo hiked during the past two months.
Answer:
It is given that
Margo hiked 12 miles 13 times last month. She hiked 14 miles 12 times this month.
So,
The number of miles Margo hiked during the past month = (The number of miles margo hiked past month) × (The number of times Margo hiked during the past month)
= 12 × 13
= (10 + 2) × (10 + 3)
The number of miles Margo hiked during this month = (The number of miles margo hiked this month) × (The number of times Margo hiked during this month)
= 14 × 12
= (10 + 4) × (10 + 2)
Hence, from the above,
The representations of the area models for the number of miles Margo hiked this month and the past month is:

B. Write and solve equations to represent your arrays or area models.
Answer:
From part (A),
We know that,
The number of miles Margo hiked during the past month = (The number of miles margo hiked past month) × (The number of times Margo hiked during the past month)
= 12 × 13
The number of miles Margo hiked during this month = (The number of miles margo hiked this month) × (The number of times Margo hiked during this month)
= 14 × 12
So,
By using the Partial Products method,
12 × 13
= (10 + 2) × (10 + 3)
= (10 × 10) + (10 × 3) + (2 × 10) + (2 × 3)
= 100 + 30 + 20 + 6
= 156 miles
So,
By using the Partial products method,
14 × 12
= (10 + 4) × (10 + 2)
= (10 × 10) + (10 × 2) + (4 × 10) + (4 × 2)
= 100 + 20 + 40 + 8
= 168 miles
Hence, from the above,
We can conclude that
The number of miles Margo hiked the past month is: 156 miles
The number of miles Margo hiked this month is: 168 miles

Question 7.
Liz makes necklaces. Each has 16 beads. Use the numbers in the box to complete the table.
Envision Math Common Core Grade 4 Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 77
Answer:
It is given that
Liz makes necklaces. Each has 16 beads
So,
The total number of beads = (The number of necklaces) × (The number of beads in each necklace)
The number of beads in each necklace = \(\frac{The total number of beads}{The number of encklaces}\)
Hence from the above,
We can conclude that
The completed table is:

Question 8.
Justine’s plant shop has 7 shelves. Each shelf holds 18 plants. Suppose she adds 2 more shelves to the shop. Write and solve equations to find the number of plants all of the shelves can hold. Check if your answer is reasonable.
Answer:
It is given that
Justine’s plant shop has 7 shelves. Each shelf holds 18 plants. Suppose she adds 2 more shelves to the shop.
So,
The total number of shelves in the shop = 7 + 2
= 9 shelves
Now,
The total number of plants all of the shelves can hold = (The total number of shelves) × (The number of plants each shelf can hold)
= 9 × 18
= 9 × (10 + 8)
= (9 × 10) + (9 × 8)
= 90 + 72
= 162 plants
Estimate:
10 × 20
= 1× 10 × 2 × 10
= (1 × 2) × 100
= 2 × 100
= 200 plants
Hence, from the above,
We can conclude that the total number of plants all of the shelves can hold is exactly: 162 plants
As the estimated and the exact answer does not have much difference, the answer is reasonable

Question 9.
Jack’s landscape service charges $78 to plant a tree. What is the total cost to plant 18 trees on Tuesday and 23 trees on Wednesday? Write and solve equations.
Answer:
It is given that
Jack’s landscape service charges $78 to plant a tree
So,
The total cost to plant the trees on Tuesday = (The total number of trees planted on Tuesday) × (The cost to plant each tree)
= 18 × 78
= (10 + 8) × (70 + 8)
= (10 × 70) + (10 × 8) + (8 × 70) + (8 × 8)
= 700 + 80 +560 + 64
= $1,404
So,
The total cost to plant the trees on Wednesday = (The total number of trees planted on Wednesday) × (The cost to plant each tree)
= 23 × 78
= (20 + 3) × (70 + 8)
= (20 × 70) + (20 × 8) + (3 × 70) + (3 × 8)
= 1,400 + 160 +210 + 24
= $1,794
Hence, from the above,
We can conclude that
The total cost to plant trees on Tuesday is: $1,404
The total cost to plant trees on Wednesday is: $1,794

Question 10.
Which of the following uses properties of operations to help find 27 × 14?
A. 20 + 7 + 10 + 4
B. 20 × 7 × 10 × 4
C. 27 + 14
D. (20 + 7) (10 + 4)
Answer:
The given expression is: 27 × 14
Now,
By using the Distributive Property of Multiplication,
27 × 14
= (20 + 7) × (10 + 4)
Hence, from the above,
We can conclude that the properties of operations used to find 27 × 14 are:

Question 11.
Lorin drew an area model to find 19 × 15. Write the partial product for each rectangle in the area model.
Envision Math Common Core Grade 4 Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 78
Answer:
The given expression is: 19 × 15
Now,
By using the Partial products method,
19 × 15
= (10 + 9) × (10 + 5)
= (10 × 10) + (10 × 5) + (9 × 10) + (9 × 5)
= 100 + 50 + 90 + 45
= 285
Hence, from the above,
We can conclude that the partial products for the given expression are:

Question 12.
The biggest painting in an art show is 31 inches long and 28 inches wide. The smallest painting is 14 inches long and 11 inches wide. What is the difference between the areas of the two paintings? Use equations to show your work.
________ square inches
Answer:
It is given that
The biggest painting in an art show is 31 inches long and 28 inches wide. The smallest painting is 14 inches long and 11 inches wide.
Now,
The area of the biggest painting = Length × Width
= 31 × 28
= (30 + 1) × (20 + 8)
= (30 × 20) + (30 × 8) + (1 × 20) + (1 × 8)
= 600 +240 + 20 + 8
= 868 inches²
Now,
The area of the samllest painting = Length × Width
= 14 × 11
= (10 + 4) × (10 + 1)
= (10 × 10) + (10 × 1) + (4 × 10) + (4 × 1)
= 100 + 10 + 40 + 4
= 154 inches²
So,
The difference between the areas of the biggest painting and the smallest painting = 868 – 154
= 714 square inches
Hence, from the above,
We can conclude that the difference between the areas between the two paintings is: 714 square inches

Topic 4 Performance Task

Raising Money
The fourth-grade students at Skyline School sold candles to raise money. The money raised will be used to buy toys for children who live in a group home. The Candle Fundraiser table shows how many candles each class sold.

Envision Math Common Core Grade 4 Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 79 Beehive candles sell for $20 each. The class raised $12 for each candle sold.
Envision Math Common Core Grade 4 Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 79 Soybean candles sell for $22 each. The class raised $14 for each candle sold.
Envision Math Common Core Grade 4 Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 80

Question 1.
About how much money did the fourth-grade students raise? Explain.
Answer:
It is given that
Beehive candles sell for $20 each. The class raised $12 for each candle sold.
Soybean candles sell for $22 each. The class raised $14 for each candle sold.
So,
The total cost of each Beehive candle = $20 + $12
= $32
The total cost of each Soybean candle = $22 + $14
= $36
Now,
The given table is:
Envision Math Common Core Grade 4 Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 80
So,
The money raised by the fourth-grade students = (The total number of candles sold) + (The amount raised for each candle)
= (32 × 32) + (25 × 36) + (18 × 36) + (47 × 32)
= 1,024 + 900 + 648 + 1,504
= $4,076
Hence, from the above,
We can conclude that
The total money raised by the fourth-grade students is: $4,076

Question 2.
The school secretary needs to know exactly how much money was raised.
Part A
How much money did Ms. Schmidt’s class raise? Use an area model and partial products to find the product.
Answer:
The given data is:
Envision Math Common Core Grade 4 Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 80
Now,
From the given table,
We can observe that
The amount of money raised by Mr. Schmidt’s class = (The number of candles sold by Mr.Schmidt’s class) × (The total amount raised for each candle sold)
= 25 × 36
Now,
By using the area-Model,
25 × 36
= (20 + 5) × (30 + 6)
= (20 × 30) + (20 × 6) + (5 × 30) + (5 × 6)
= 600 + 120 + 150 + 30
= 900

Hence, from the above,
We can conclude that the money raised by Mr. Schmidt’s class is: $900

Part B
How much money did Ms. Picard’s class raise? Use the Distributive Property to find the product.
Answer:
The given data is:
Envision Math Common Core Grade 4 Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 80
Now,
From the given table,
We can observe that
The amount of money raised by Ms. Picard’s class = (The number of candles sold by Ms.Picord’s class) × (The total amount raised for each candle sold)
= 18 × 36
Now,
By using the Distributive Property of Multiplication,
18 × 36
= (10 + 8) × (30 + 6)
= (10 × 30) + (10 × 6) + (8 × 30) + (8 × 6)
= 300 + 60 + 240 + 48
= 648
Hence, from the above,
We can conclude that the amount of money raised by Ms. Picard’s class is: $648

Part C
How much money did the fourth-grade students raise? Explain.
Answer:
It is given that
Beehive candles sell for $20 each. The class raised $12 for each candle sold.
Soybean candles sell for $22 each. The class raised $14 for each candle sold.
So,
The total cost of each Beehive candle = $20 + $12
= $32
The total cost of each Soybean candle = $22 + $14
= $36
Now,
The given table is:
Envision Math Common Core Grade 4 Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 80
So,
The money raised by the fourth-grade students = (The total number of candles sold) + (The amount raised for each candle)
= (32 × 32) + (25 × 36) + (18 × 36) + (47 × 32)
= 1,024 + 900 + 648 + 1,504
= $4,076
Hence, from the above,
We can conclude that
The total money raised by the fourth-grade students is: $4,076

Part D
Is the total amount of money raised from Part C reasonable based on the estimate you found in Exercise 1? Explain.
Answer:
From part (C) and Exercise 1,
We can observe that
In Exercise 1,
We found out the total amount of money raised by the fourth-grade students
In part (C),
We found out the total amount of money raised by the fourth-grade students
So,
From the above,
We can observe that
We found the same value in both
Hence, from the above,
We can conclude that
The total amount of money raised from Part C is reasonable based on the estimate you found in Exercise 1

The fourth-grade students at Skyline School decide to buy the toys shown.
Envision Math Common Core Grade 4 Answers Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers 81
Buying Toys

  • They buy 68 toys in all.
  • They buy at least 25 of each type.
  • They cannot spend more money than was raised.

Question 3.
How many of each toy can be bought following the directions from the Buying Toys list? Find the total cost for the numbers you choose.
Answer:
It is given that
The cost of each teddy bear is: $17
The cost of each scooter is: $28
The total number of toys are: 68
Now,
Let there be an equal number of teddy bears and scooters
So,
The number of each toy = \(\frac{68}{2}\)
= 34
So,
The total cost of the toys = (34 × 17) + (34 × 28)
= $1,530
Hence, from the above,
We can conclude that
The total cost for the number of toys you choose is: $1,530

enVision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100

enVision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100

Go through the enVision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 regularly and improve your accuracy in solving questions.

enVision Math Common Core 2nd Grade Answers Key Topic 4 Fluently Add Within 100

Essential Question:
What are strategies for adding numbers to 100?
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 1

enVision STEM Project: Making and Using Models
Find Out Find and share books about Hawaii and volcanoes. Make a model of a volcano that becomes an island. Tell about how the island can change over a long time.
Journal: Make a Book Show what you learn in a book. In your book, also:

  • Draw pictures to show how volcanoes can become islands.
  • Show how you can use models to help you add numbers to 100.

Review What You Know

Vocabulary
Question 1.
Circle the tens digit in each number.
73
53
82
Answer:
The tens digit in each number is 7,5 and 8.

Explanation:
Given that the digits are 73, 53, 82. So the tens digit in each number is 7,5 and 8.

Fluently Add Within 100 1

Question 2.
Circle the ones digit in each number.
34
43
97
Answer:
The ones digit in each number is 4,3 and 7.

Explanation:
Given that the digits are 73, 53, 82. So the one digit in each number is 4,3 and 7.

Question 3.
Break apart 23 into tens and ones.
23 = _________ tens and ______ ones
Answer:
23 = 2 tens and 3 ones.

Explanation:
Given that the number is 23, so after a break apart of 23 it will be 2 tens and 3 ones.

Mental Math
Question 4.
Use mental math to find each sum.
34 + 10 = _________
50 + 5 = _________
20 + 40 = _________
Answer:
44, 55, 60.

Explanation:
The sum of
34 + 10 = 44
50 + 5 = 55
20 + 40 = 60.

Open Number Line
Question 5. Use the open number line to find 39 + 15.
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 2
39 + 15 = _________
Answer:

Math Story
Question 6.
Stacy has 17 marbles. Diana gives her 22 marbles. How many marbles does Stacy have now?
______ marbles
Answer:
The number of marbles does Stacy has now is 39 marbles.

Explanation:
Given that Stacy has 17 marbles and Diana gives her 22 marbles. So the number of marbles does Stacy has now is 17 + 22 which is 39 marbles.

Pick a Project

PROJECT 4A
Where can you bike near home?
Project: Make a Bike Trail Brochure
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 3

PROJECT 4B
What kinds of coral grow in Florida?
Project: Build a Coral Model
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 4

PROJECT 4C
What do you collect?
Project: Display a Rock and Leaf Collection
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 5

PROJECT 4D
How much does it cost to visit the Kennedy Space Center?
Project: Make a Space Center Poster
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 6

Lesson 4.1 Add 2-Digit Numbers Using Models

Solve & Share
Leslie collects 36 rocks. Her brother collects 27 rocks. How many rocks do they collect in all? Use place-value blocks to help you solve. Show your place-value blocks.
I can … use models to add 2-digit numbers and then explain my work.
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 7

Visual Learning Bridge
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 8

Convince Me! When do you need to regroup when adding?

Guided Practice

Add. Use place-value blocks to find each sum. Regroup if needed.
Question 1.
32 + 29 = ________
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 9
_______ tens ______ ones
Answer:
6 tens and 1 ones.

Explanation:
Given that the sum of 32 + 29 is 61. So there are 6 tens and 1 one.

Fluently Add Within 100 2

Question 2.
24 + 52 = _______
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 10
_______ tens ______ ones
Answer:
7 tens and 6 ones.

Explanation:
Given that the sum of 24 + 52 is 76. So there are 7 tens and 6 ones.

Question 3.
15 + 38 = _______
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 11
_______ tens ______ ones
Answer:
5 tens and 3 ones.

Explanation:
Given that the sum of 15 + 38 is 53. So there are 5 tens and 3 ones.

Independent Practice

Add. Use place-value blocks to find each sum. Regroup if needed.
Question 4.
36 + 29 = _______
Answer:
6 tens and 5 ones.

Explanation:
Given that the sum of 36 + 29 is 65. So there are 6 tens and 5 ones.

Question 5.
27 + 23 = _______
Answer:
5 tens and 0 ones.

Explanation:
Given that the sum of 27 + 23 is 50. So there are 5 tens and 0 ones.

Question 6.
59 + 13 = _______
Answer:
7 tens and 2 ones.

Explanation:
Given that the sum of 59 + 13 is 72. So there are 7 tens and 2 ones.

Question 7.
24 + 35 = _______
Answer:
5 tens and 9 ones.

Explanation:
Given that the sum of 24 + 35 is 59. So there are 5 tens and 9 ones.

Question 8.
58 + 25 = _______
Answer:
8 tens and 3 ones.

Explanation:
Given that the sum of 58 + 25 is 83. So there are 8 tens and 3 ones.

Question 9.
15+ 46 = _______
Answer:
6 tens and 1 one.

Explanation:
Given that the sum of 15 + 46 is 61. So there are 6 tens and 1 one.

Question 10.
75 + 19 = ________
Answer:
9 tens and 4 ones.

Explanation:
Given that the sum of 75 + 19 is 94. So there are 9 tens and 4 ones.

Question 11.
44 + 47 = _______
Answer:
9 tens and 1 one.

Explanation:
Given that the sum of 44 + 47 is 91. So there are 9 tens and 1 one.

Higher Order Thinking Read the sum. Circle all of the number pairs in the box that match that sum.
Question 12.
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 12
Answer:
12 + 10 = 22
18 + 4 = 22.

Explanation:
The number pairs to get the sum as 22 are 12 + 10 and 18 + 4.
Envision-Math-Common-Core-2nd-Grade-Answer-Key-Topic-4-Fluently-Add-Within-100-12-1

Question 13
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 13
Answer:
25 + 30
45 + 10.

Explanation:
The number pairs to get the sum as 55 are 25 + 30 and 45 + 10.
Envision-Math-Common-Core-2nd-Grade-Answer-Key-Topic-4-Fluently-Add-Within-100-13-1

Question 14.
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 14
Answer:
56 + 27
19 + 64.

Explanation:
The number pairs to get the sum as 83 are 56 + 27 and 19 + 64.
Envision-Math-Common-Core-2nd-Grade-Answer-Key-Topic-4-Fluently-Add-Within-100-14-1

Problem Solving

Solve the problems below. Show your work.
Question 15.
Model Chen counts 47 buttons. Then he counts 20 more buttons. How many buttons does Chen count in all?
________ buttons
Answer:
The total number of buttons does Chen count in all is 67 buttons.

Explanation:
Given that Chen counts 47 buttons and then he counts 20 more buttons. So the total number of buttons does Chen count in all is 47 + 20 which is 67 buttons.

Question 16.
en Vision® STEM There were 24 earthquakes in the U.S. one year. There were 28 earthquakes the next year. How many earthquakes were there in those two years?
________ earthquakes
Answer:
The number of earthquakes in those two years is 52 earthquakes.

Explanation:
Given that there were 24 earthquakes in the U.S in one year and in other year there were 28 earthquakes the next year. So the number of earthquakes is 24 + 28 which is 52 earthquakes.

Question 17.
Higher Order Thinking Write an addition story about the desks and chairs in your classroom. Use pictures, numbers, or words.
Answer:

Question 18.
Assessment Practice Do you have to regroup to find each sum? Choose Yes or No.
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 15
Answer:

Lesson 4.2 Continue to Add 2-Digit Numbers Using Models

Solve & Share
Wendy picked 35 pears. Toni picked 49 pears. How many pears did they pick in all? Use place-value blocks to help explain your work.
I can … add numbers using place value and properties of operations.
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 16

Answer:
The total number of pears that they picked up is 84 pears.

Explanation:
Given that Wendy picked 35 pears andToni picked 49 pears. So the total number of pears that they picked up is 35 + 49 which is 84 pears.
Envision-Math-Common-Core-2nd-Grade-Answer-Key-Topic-4-Fluently-Add-Within-100-16-1

Visual Learning Bridge
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 17

Convince Me! Ken adds 43 + 27. His sum is 60. Is he correct? Explain.

Answer:
No, Ken is not correct.

Explanation:
No, Ken is not correct, because 43 + 27 is 70 but he got the sum as 60.

Guided Practice

Add. Use place value. Draw blocks or use another way.
Question 1.
12 + 23 = ________
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 18

Answer:
The sum of 12 + 23 is 35.

Explanation:
The sum of
12
+23
——
30
5
——
35

Question 2.
18 + 42 = _______
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 19
Answer:
The sum of 18 + 42 is 60.

Question 3.
33 + 48 = ________
Answer:
The sum of 33 +48 is 81

Explanation:
The sum of
33
+48
——
70
11
——
81

Independent Practice

Add. Use place value. Draw blocks or use another way.
Question 4.
18 + 24 = ________
Answer:
The sum of 18 + 24 is 42

Explanation:
The sum of
18
+24
——
30
12
——
42

Question 5.
47 + 38 = ________
Answer:
The sum of 47 + 38 is 85

Explanation:
The sum of
47
+38
——
70
15
——
85

Question 6.
26 + 47 = ________
Answer:
The sum of 26 + 47 is 73

Explanation:
The sum of
26
+47
——
60
13
——
73

Question 7.
34 + 58 = ________
Answer:
The sum of 34 + 58 is 92

Explanation:
The sum of
34
+58
——
80
12
——
92

Question 8.
Higher Order Thinking Draw the second addend.
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 20
Answer:

Problem Solving

Solve each problem. Show your work.
Question 9.
Toby planted 28 trees. Juan planted 36 trees. How many trees did they plant in all?
________ trees
Answer:
The number of trees did they plant in all is 64 trees.

Explanation:
Given that Toby planted 28 trees and Juan planted 36 trees. So the number of trees did they plant in all is 28 + 36 which is 64 trees.

Question 10.
Make Sense Jenny has 13 tennis balls. Sal had 16 tennis balls, but he gave 7 of them to Joe. How many tennis balls do Jenny and Sal have now?
________ tennis balls
Answer:
The number of tennis balls does Jenny and Sal have now is 22 tennis balls.

Explanation:
Given that Jenny has 13 tennis balls and Sal had 16 tennis balls. So the total number of tennis balls is 13 + 16 = 29. He gave 7 of them to Joe, so the number of tennis balls do Jenny and Sal have now is 29 – 7 which is 22 tennis balls.

Question 11.
Higher Order Thinking Write an addition story using two 2-digit numbers. Then solve the problem for your story.
Answer:

Question 12.
Assessment Practice which is the same amount as 28 + 16? Choose all that apply.
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 21
Answer:

Lesson 4.3 Add with Partial Sums

Solve & Share
Ms. Kim’s class has 25 students. Mr. Will’s class has 36 students. Both classes go on a field trip. How many students are on the trip?
Draw place-value blocks to help you solve the problem.
I can … add using place value and partial sums.
___________ students

Visual Learning Bridge
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 22

Convince Me! Can you add the ones first and then the tens | when finding 57 + 28 using partial sums? Explain.

Guided Practice

Add. Use partial sums. Draw blocks if you need to.
Question 1.
24 + 13
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 23
Answer:
The sum of the given numbers is 37.

Explanation:
Given that the numbers to add is 24 and 13. Now we will perform a partial sum of those numbers,
Envision-Math-Common-Core-2nd-Grade-Answer-Key-Topic-4-Fluently-Add-Within-100-23-1

Question 2.
68 + 7
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 24
Answer:
The sum of the given numbers is 75.

Explanation:
Given that the numbers to add is 68 and 7. Now we will perform a partial sum of those numbers,
Envision-Math-Common-Core-2nd-Grade-Answer-Key-Topic-4-Fluently-Add-Within-100-24-1

Independent Practice

Add. Use partial sums. Draw blocks if you need to.
Question 3.
34 + 25
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 25
Answer:
The sum of the given numbers is 59.

Explanation:
Given that the numbers to add is 34 and 25. Now we will perform a partial sum of those numbers,
Envision-Math-Common-Core-2nd-Grade-Answer-Key-Topic-4-Fluently-Add-Within-100-25Envision-Math-Common-Core-2nd-Grade-Answer-Key-Topic-4-Fluently-Add-Within-100-25Envision-Math-Common-Core-2nd-Grade-Answer-Key-Topic-4-Fluently-Add-Within-100-25Envision-Math-Common-Core-2nd-Grade-Answer-Key-Topic-4-Fluently-Add-Within-100-25Envision-Math-Common-Core-2nd-Grade-Answer-Key-Topic-4-Fluently-Add-Within-100-25-1

Question 4.
68 + 18
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 26
Answer:
The sum of the given numbers is 86.

Explanation:
Given that the numbers to add is 68 and 18. Now we will perform a partial sum of those numbers,

Question 5.
37 + 8
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 27
Answer:
The sum of the given numbers is 45.

Explanation:
Given that the numbers to add is 37 and 8. Now we will perform a partial sum of those numbers,
Envision-Math-Common-Core-2nd-Grade-Answer-Key-Topic-4-Fluently-Add-Within-100-27-1

Question 6.
52 + 38
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 28
Answer:
The sum of the given numbers is 90.

Explanation:
Given that the numbers to add is 52 and 38. Now we will perform a partial sum of those numbers,
Envision-Math-Common-Core-2nd-Grade-Answer-Key-Topic-4-Fluently-Add-Within-100-28-1Envision-Math-Common-Core-2nd-Grade-Answer-Key-Topic-4-Fluently-Add-Within-100-28-1Envision-Math-Common-Core-2nd-Grade-Answer-Key-Topic-4-Fluently-Add-Within-100-28-1Envision-Math-Common-Core-2nd-Grade-Answer-Key-Topic-4-Fluently-Add-Within-100-28-1Envision-Math-Common-Core-2nd-Grade-Answer-Key-Topic-4-Fluently-Add-Within-100-28-1

Question 7.
45 + 29
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 29
Answer:
The sum of the given numbers is 74.

Explanation:
Given that the numbers to add is 45 and 29. Now we will perform a partial sum of those numbers,
Envision-Math-Common-Core-2nd-Grade-Answer-Key-Topic-4-Fluently-Add-Within-100-29-1

Question 8.
28 + 39
Envision Math Common Core 2nd Grade Answer Key Topic 4 Fluently Add Within 100 30
Answer:
The sum of the given numbers is 67.

Explanation:
Given that the numbers to add is 28 and 39. Now we will perform a partial sum of those numbers,
Envision-Math-Common-Core-2nd-Grade-Answer-Key-Topic-4-Fluently-Add-Within-100-30-1

Number Sense Write each missing tens or ones digit.
Question 9.
23 + 1☐ = 37
Answer:
23 + 14 = 37.

Explanation:
To find the missing digit, we will do subtraction
37 – 23  = 14.
Here, the missing digit is 4.

Question 10.
59 = ☐8 + 31
Answer:
59 = 28 + 31.

Explanation:
To find the missing digit, we will do subtraction
59 – 31 = 28
Here, the missing digit is 2.

Problem Solving

Solve each problem. Show your work.
Question 11.
Model 34 students are on the playground. 17 students are in the gym. How many students are there in all?
_________ students
Answer:
The number of students are there in all is 51 students.

Explanation:
Given that 34 students are on the playground and 17 students are in the gym. So the number of students are there in all is 34 + 17 which is 51 students.

Question 12.
Sarah put 8 white roses in a vase. Then she added 7 red roses and 12 yellow roses to the vase. How many roses are in the vase now?
_______ roses
Answer:
The number of roses is in the vase now is 27 roses.

Explanation:
Given that Sarah put 8 white roses in a vase and then she added 7 red roses and 12 yellow roses to the vase. So the number of roses are in the vase now is 8 + 7 + 12 which is 27 roses.

Envision Math Common Core 2nd Grade Answers Topic 4 Fluently Add Within 100 31

Question 13.
Higher Order Thinking Write each missing tens or ones digit. What strategy did you use?
12 + 3☐ = 50
24 + 3☐ = 60
35 + 3☐ = 70
Answer:
The missing digits are 8, 6, 5.

Explanation:
To find the missing digit, we will do subtraction
50 – 12 = 38.
Here, the missing digit is 8.
To find the missing digit, we will do subtraction
60 – 24 = 36.
Here, the missing digit is 6.
To find the missing digit, we will do subtraction
70 – 35 = 35
Here, the missing digit is 5.

Question 14.
Assessment Practice which is the sum of 28 + 36? Use partial sums to solve.
A. 8
B. 12
C. 54
D. 64
Answer:
The sum of 28 + 36 is 64.

Explanation:
The partial sum of 28 + 36 is
28
+ 36
——————–
20 + 30 = 50
8 + 6     = 14
——————–
64

Lesson 4.4 Add Using Mental Math and Partial Sums

Solve & Share
Add 46 + 26. Explain how you solved the problem.

I can … add numbers using place value and partial sums.

Visual Learning Bridge
Envision Math Common Core 2nd Grade Answers Topic 4 Fluently Add Within 100 32

Convince Me! Roger found 54 + 27. His sum was 71. Is he correct? Why or why not? Use partial sums to check.

Answer:
Roger is not correct.

Explanation:
Given that Roger found 54 + 27. His sum was 71, let’s check using the partial sum
54
+ 27
——————–
50 + 20 = 70
4 + 7     = 11
——————–
81
Roger is not correct, because the sum of 54 + 27 is 81.

Guided Practice

Write the addition problem. Use partial sums. Add any way you choose. Show your work.
Question 1.
34 + 17
Envision Math Common Core 2nd Grade Answers Topic 4 Fluently Add Within 100 33
Answer:
The sum of 34 + 17 is 51.

Explanation:
The partial sum of 34 + 17 is 51.
Envision-Math-Common-Core-2nd-Grade-Answers-Topic-4-Fluently-Add-Within-100-33-1

Question 2.
52 + 31
Envision Math Common Core 2nd Grade Answers Topic 4 Fluently Add Within 100 33
Answer:
The sum of 52 + 31 is 83.

Explanation:
The partial sum of 52 + 31 is 83.
Envision-Math-Common-Core-2nd-Grade-Answers-Topic-4-Fluently-Add-Within-100-33-2

Question 3.
35 + 26
Envision Math Common Core 2nd Grade Answers Topic 4 Fluently Add Within 100 33
Answer:
The sum of 35 + 26 is 61.

Explanation:
The partial sum of 35 + 26 is 61.
Envision-Math-Common-Core-2nd-Grade-Answers-Topic-4-Fluently-Add-Within-100-33-3

Independent Practice

Write the addition problem. Use partial sums. Add any way you choose. Show your work.
Question 4.
15 + 28
Envision Math Common Core 2nd Grade Answers Topic 4 Fluently Add Within 100 33
Answer:
The sum of 15 + 28 is 43.

Explanation:
The partial sum of 15 + 28 is 43.
Envision-Math-Common-4Core-2nd-Grade-Answers-Topic-4-Fluently-Add-Within-100-33-4

Question 5.
29 + 20
Envision Math Common Core 2nd Grade Answers Topic 4 Fluently Add Within 100 33
Answer:
The sum of 29 + 20 is 49.

Explanation:
The partial sum of 29 + 20 is 49.
Envision-Math-Common-Core-2nd-Grade-Answers-Topic-4-Fluently-Add-Within-100-33-5

Question 6.
63 + 29
Envision Math Common Core 2nd Grade Answers Topic 4 Fluently Add Within 100 33
Answer:
The sum of 63 + 29 is 92.

Explanation:
The partial sum of 63 + 29 is 92.
Envision-Math-Common-Core-2nd-Grade-Answers-Topic-4-Fluently-Add-Within-100-33-6

Question 7.
37 + 48
Envision Math Common Core 2nd Grade Answers Topic 4 Fluently Add Within 100 33
Answer:
The sum of 37 + 48 is 85.

Explanation:
The partial sum of 37 + 48 is 85.
Envision-Math-Common-Core-2nd-Grade-Answers-Topic-4-Fluently-Add-Within-100-33-7

Add any way you choose. Show your work.
Question 8.
67 + 17 = ________
Answer:
The sum of 67 + 17 is 84.

Explanation:
The partial sum of 67 + 17 is 84.
Envision-Math-Common-Core-2nd-Grade-Answers-Topic-4-Fluently-Add-Within-100-33-8

Question 9.
15 + 18 = ________
Answer:
The sum of 15 + 18 is 33.

Explanation:
The partial sum of 15 + 18 is 33.
Envision-Math-Common-Core-2nd-Grade-Answers-Topic-4-Fluently-Add-Within-100-33-9

Question 10.
43 + 49 = ________
Answer:
The sum of 43 + 49 is 92.

Explanation:
The partial sum of 43 + 49 is 92.
Envision-Math-Common-Core-2nd-Grade-Answers-Topic-4-Fluently-Add-Within-100-33-10

Question 11.
62 + 28 = ________
Answer:
The sum of 62 + 28 is 90.

Explanation:
The partial sum of 62 + 28 is 90.
Envision-Math-Common-Core-2nd-Grade-Answers-Topic-4-Fluently-Add-Within-100-33-11

Number Sense Write the missing ones or tens digits.
Question 12.
27 + 2☐ = 50
Answer:
The missing digit is 3.

Explanation:
To find the missing digit, we will do subtraction
50 – 27 = 23.
Here, the missing digit is 3.

Question 13.
3☐ + 16 = 48
Answer:
The missing digit is 2.

Explanation:
To find the missing digit, we will do subtraction
48 – 16 = 32.
Here, the missing digit is 2.

Question 14.
☐4 + 49 = 93
Answer:
The missing digit is 4.

Explanation:
To find the missing digit, we will do subtraction
93 – 49 = 44.
Here, the missing digit is 4.

Problem Solving

Solve each problem. Show your work. Use drawings, if needed.
Question 15.
Amir plants 25 trees. Juan plants 27 trees. How many trees do they plant in all?
________ trees
Answer:
The total number of trees do they plant in all is 52 trees.

Explanation:
Given that Amir plants 25 trees and Juan plant 27 trees, so the total number of trees do they plant in all is 25 + 27 which is 52 trees.

Question 16.
On Monday, Sasha puts 32 pennies in her bank. On Tuesday, she puts 57 more pennies in her bank. How many pennies does she put in her bank on both days?
________ pennies
Answer:
The number of pennies does she puts in her bank on both days is 89 pennies.

Explanation:
Given that, on Monday, Sasha puts 32 pennies in her bank, and on Tuesday, she puts 57 more pennies in her bank. So the number of pennies does she puts in her bank on both days is 32 + 57 which is 89 pennies.

Question 17.
Higher Order Thinking Write an addition story using 2 two-digit numbers. Then solve the problem for your story.
Answer:
The number of toffies is there in all is 86 toffies.

Explanation:
There are 60 toffies in a jar. Harry adds 26 toffies more. How many toffies are there in all?

Answer:
Given that there are 60 toffies in a jar and Harry adds 26 toffies more. So the number of toffies are there in all is 60 +26 which is 86 toffies.

Question 18.
Assessment Practice Find 52 + 39 using partial sums. Show how you solved the problem.
52 + 39 = ________
Answer:
The sum of 52 + 39 is 91.

Explanation:
The partial sum of 52 + 39 is 91
52
+ 39
——————–
50 + 30 = 80
2 + 9     = 11
——————–
91

Lesson 4.5 Break Apart Numbers and Add Using Mental Math

Solve & Share
Monica has 24 crayons. Paul has 64 crayons. How many crayons do they have in all? Use place-value blocks or draw a picture to explain your work.

I can … add within 100 using place value strategies.
Envision Math Common Core 2nd Grade Answers Topic 4 Fluently Add Within 100 34

Answer:
The number of crayons do they have is 88 crayons.

Explanation:
Given that Monica has 24 crayons and Paul has 64 crayons. So the number of crayons do they have is 64 + 24 which is 88 crayons.

Visual Learning Bridge
Envision Math Common Core 2nd Grade Answers Topic 4 Fluently Add Within 100 35

Convince Me! Explain one way you can break apart numbers to find 14 + 32.

Guided Practice

Find each sum using mental math. Draw pictures of blocks if needed.
Question 1.
Envision Math Common Core 2nd Grade Answers Topic 4 Fluently Add Within 100 36
Answer:
The sum of 17 + 52 is 69.

Explanation:
The sum of 17 + 52 is
17 + (50 + 2)
= (17 + 50) +2
= 67 + 2
= 69.

Question 2.
Envision Math Common Core 2nd Grade Answers Topic 4 Fluently Add Within 100 37
Answer:
The sum of 69 + 23 is 92.

Explanation:
The sum of 69 + 23 is
69 + (20 + 3)
= (69 + 20) +3
= 89 + 3
= 92.
Envision-Math-Common-Core-2nd-Grade-Answers-Topic-4-Fluently-Add-Within-100-37-1

Independent Practice

Find each sum using mental math. Draw pictures of blocks if needed.
Question 3.
Envision Math Common Core 2nd Grade Answers Topic 4 Fluently Add Within 100 38
Answer:
The sum of 23 + 26 is 49.

Explanation:
The sum of 23 + 26 is
23 + (20 + 6)
= (23 + 20) +6
= 43 + 6
= 49.
Envision-Math-Common-Core-2nd-Grade-Answers-Topic-4-Fluently-Add-Within-100-38-1

Question 4.
Envision Math Common Core 2nd Grade Answers Topic 4 Fluently Add Within 100 39
Answer:
The sum of 9 + 42 is 51.

Explanation:
The sum of 9 + 42 is
9 + (40 + 2)
= (9 + 40) + 2
= 49 + 2
= 51.
Envision-Math-Common-Core-2nd-Grade-Answers-Topic-4-Fluently-Add-Within-100-39-1

Question 5.
Envision Math Common Core 2nd Grade Answers Topic 4 Fluently Add Within 100 40
Answer:
The sum of 51 + 16 is 67.

Explanation:
The sum of 51 + 16 is
51 + (10 + 6)
= (51 + 10) + 6
= 61 + 6
= 67.
Envision-Math-Common-Core-2nd-Grade-Answers-Topic-4-Fluently-Add-Within-100-40-1

Question 6.
56 + 15 = _______
Answer:
The sum of 56 + 15 is 71.

Explanation:
The sum of 56 + 15 is
56 + (10 + 5)
= (56 + 10) + 5
= 66 + 5
= 71.

Question 7.
_______ = 76 + 18
Answer:
The sum of 76 + 18 is 94.

Explanation:
The sum of 76 + 18 is
76 + (10 + 8)
= (76 + 10) + 8
= 86 + 8
= 94.

Question 8.
33 + 49 = ________
Answer:
The sum of 33 + 49 is 82.

Explanation:
The sum of 33 + 49 is
33 + (40 + 9)
= (33 + 40) + 9
= 73 + 9
= 82.

Add tens and ones to solve.
Question 9.
Algebra One number makes both equations true. Find the missing number.
17 + ☐ = 28
☐ + 28 = 39
Answer:
The missing digit is 11.

Explanation:
To find the missing digit, we will do subtraction
28 – 17 = 11.
Here, the missing digit is 11.
To find the missing digit, we will do subtraction
39 – 28 = 11.
Here, the missing digit is 11.

Problem Solving

Generalize Solve each problem mentally. Show your work. Think about the steps you do every time you break apart numbers to add.
Question 10.
Billy puts 34 skateboard wheels in a pile. He puts 34 more wheels in another pile. How many wheels does Billy have in all?
_________ wheels
Answer:
The number of wheels does Billy has in all is 68 wheels.

Explanation:
Given that Billy puts 34 skateboard wheels in a pile and he puts 34 more wheels in another pile. So the number of wheels does Billy has in all is 34 + 34 which is 68 wheels.

Question 11.
enVision® STEM A new office and house were built in a town. 24 windows are needed for the office. 18 windows are needed for the house. How many windows are needed in all?
_________ windows
Answer:
The number of windows is 42 windows.

Explanation:
Given that a new office and house were built in a town with 24 windows are needed for the office and 18 windows are needed for the house. So the number of windows 24 + 18 which 42 windows.

Question 12.
Higher Order Thinking Write a story problem about 14 + 41. Then solve the problem.
14 + 41 = ________
Answer:

Question 13.
Assessment Practice Use mental math to find 15 + 12. Explain why the strategy works.
Answer:
The sum of 15 + 12 is 27.

Explanation:
The sum of 15 + 12 is
15 + (10 + 2)
= (15 + 10) + 2
= 25 + 2
= 27.

Lesson 4.6 Add More than Two 2-Digit Numbers

Solve & Share
12 + 34 + 28 = ?
Tom says he can find the sum by adding 28 and 12 first. He says he can add 34 to that sum to find the total.
Do you agree? Use pictures, words, and numbers to make a math argument. Then solve the problem. Show your work.

I can … add three or four 2-digit numbers.
Envision Math Common Core 2nd Grade Answers Topic 4 Fluently Add Within 100 41

Visual Learning Bridge
Envision Math Common Core 2nd Grade Answers Topic 4 Fluently Add Within 100 42

Convince Me! Look above. Why do you think the order of the addends changed? Why do you think 24 and 16 were added together?

Guided Practice

Add. Use any strategy. Show your work.
Question 1.
18+ 15 + 12 = _______
Envision Math Common Core 2nd Grade Answers Topic 4 Fluently Add Within 100 43

Question 2.
11 + 14 + 39 = ________
Answer:
11 + 14 + 39 = 64.

Explanation:
Given that 11 + 14 + 39
= 25 + 39
= 64.

Question 3.
21 + 14 + 41 + 22 = ________
Answer:
21 + 14 + 41 + 22 = 98.

Explanation:
Given that 21 + 14 + 41 + 22
= 35 + 63
= 98.

Question 4.
21 + 15 + 32 + 25 = ________
Answer:
21 + 15 + 32 + 25 = 93.

Explanation:
Given that 21 + 15 + 32 + 25
= 36 + 57
= 93.

Independent Practice

Add. Use any strategy. Show your work.
Question 5.
27 + 21 + 13 = ________
Answer:
27 + 21 + 13 = 61.

Explanation:
Given that 27 + 21 + 13
= 48 +13
= 61.

Question 6.
16 + 32 + 28 = ________
Answer:
16 + 32 + 28 = 76.

Explanation:
Given that 16 + 32 + 28
= 48 + 28
= 76.

Question 7.
32 + 14 + 42 = ________
Answer:
32 + 14 + 42 = 88.

Explanation:
Given that 32 + 14 + 42
= 46 + 42
= 88.

Question 8.
25 + 17 + 24 + 15 = ________
Answer:
25 + 17 + 24 + 15 = 81.

Explanation:
Given that 25 + 17 + 24 + 15
= 42 + 39
= 81.

Question 9.
32 + 16 + 18 + 31 = ________
Answer:
32 + 16 + 18 + 31 = 97.

Explanation:
Given that 32 + 16 + 18 + 31
= 48 + 49
= 97.

Question 10.
37 + 11 + 13 + 38 = ________
Answer:
37 + 11 + 13 + 38 = 99.

Explanation:
Given that 37 + 11 + 13 + 38
= 48 + 51
= 99.

Algebra Find the missing numbers.
Question 11.
8 + 3 + Envision Math Common Core 2nd Grade Answers Topic 4 Fluently Add Within 100 44 + 2 = 18
Answer:
The missing digit is 5.

Explanation:
Let the missing number be X, 8 + 3 + X + 2 = 18 which is 13 + X = 18.
To find the missing digit, we will do subtraction
X = 18 – 13
= 5.
Here, the missing digit is 5.

Question 12.
5 + Envision Math Common Core 2nd Grade Answers Topic 4 Fluently Add Within 100 44 + 6 + 5 = 19
Answer:
The missing digit is 3.

Explanation:
Let the missing number be X, 5 + X + 6 + 5 = 19 which is 16 + X = 19.
To find the missing digit, we will do subtraction
X = 19 – 16
= 3.
Here, the missing digit is 3.

Question 13.
7 + 27 + 23 + Envision Math Common Core 2nd Grade Answers Topic 4 Fluently Add Within 100 44 = 61
Answer:
The missing digit is 4.

Explanation:
Let the missing number be X, 7 + 27 + 23 + X = 61 which is 57 + X = 61.
To find the missing digit, we will do subtraction
X = 61 – 57
= 4.
Here, the missing digit is 4.

Question 14.
Envision Math Common Core 2nd Grade Answers Topic 4 Fluently Add Within 100 44 + 24 + 18+ 4 = 52
Answer:
The missing digit is 6.

Explanation:
Let the missing number be X, X + 24 + 18 + 4 = 52 which is 46 + X = 52.
To find the missing digit, we will do subtraction
X = 52 – 46
= 6.
Here, the missing digit is 6.

Problem Solving

Reasoning Solve each problem. Show your work.
Question 15.
28 trucks are blue. 32 trucks are yellow. 17 trucks are green. Il trucks are pink. How many trucks are there in all?
Envision Math Common Core 2nd Grade Answers Topic 4 Fluently Add Within 100 45
_______ trucks
Answer:
The total number of trucks is 88 trucks.

Explanation:
Given that there are 28 blue trucks, 32 yellow trucks and 17 green trucks, and 11 pink trucks. So the total number of trucks are 28 + 32 + 17 + 11 which is 88 trucks.

Question 16.
Joel won 19 tickets in March, 24 tickets in April, 23 tickets in May, and 16 tickets in June. How many tickets does Joel have in all?
_________ tickets
Answer:
The number of tickets Joel has in all is 82 tickets.

Explanation:
Given that Joel won 19 tickets in March, 24 tickets in April, 23 tickets in May, and 16 tickets in June. So the number of tickets Joel have in all is 19 + 24 + 23 + 16 =  82 tickets.

Question 17.
Higher Order Thinking Henry is adding the numbers 24, 18, and 36. He makes a ten to add. Which ones digits does Henry add first? Explain.
Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 46
Answer:

Question 18.
Assessment Practice Find the sum. Use any strategy. Show your work.
25 + 16 + 15 + 38 = _________
Answer:
The sum of 25 + 16 + 15 + 38 = 94.

Explanation:
Given that 25 + 16 + 15 + 38
= 41 + 53
= 94.

Lesson 4.7 Practice Adding Using Strategies

Solve & Share
Maria has 39 stickers. Sally has 28 stickers. They found 14 more stickers. How many stickers do they have in all? Solve. Use any strategy. Explain how you found the answer.
I can … use mental math, strategies, and models to add two, three, or four numbers.
Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 47

Visual Learning Bridge
Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 48

Convince Me! Find the sum of 14 + 28 +33 + 22. Explain.

Guided Practice

Add. Use any strategy. Show your work.
Question 1.
18 + 43 + 12 = _______
Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 49
Answer:

Question 2.
29 + 47 = _______
Answer:
The sum of 29 + 47 = 76.

Explanation:
Given that 29 + 47 which is
20 + 40 = 60
9 + 7 = 16
60 + 16 = 76.

Question 3.
9 + 34 + 21 = _______
Answer:
The sum of 9 + 34 + 21 is 64.

Explanation:
Given that 9 + 34 + 21 which is
30 + 20 = 50
9 + 4 + 1 = 14
50 + 14 = 64.

Question 4.
33 + 27 + 18 + 13 = _______
Answer:
The sum of 33 + 27 + 18 + 13 = 91.

Explanation:
Given that 33 + 27 + 18 + 13 which is
30 + 20 + 10 + 10 = 70
3 + 7 + 8 + 3 = 21
70 + 21 = 91.

Independent Practice

Add. Use any strategy. Show your work.
Question 5.
31 + 5 + 29 = _______
Answer:
The sum of 31 + 5 + 29 = 65.

Explanation:
Given that 31 + 5 + 29 which is
30 + 20 = 50
1 + 5 + 9 =15
50 + 15 = 65.

Question 6.
20 + 18 + 16 = _______
Answer:
The sum of 20 + 18 + 16 = 54.

Explanation:
Given that 20 + 18 + 16 which is
20 + 10 + 10 = 40
8 + 6 = 14
40 + 14 = 54.

Question 7.
8 + 17 = _______
Answer:
The sum of 8 + 17 is 25.

Explanation:
Given that 8 + 17
8 + 7 = 15
10 + 15 = 25.

Add. Use partial sums. Think about compatible numbers.
Question 8.
Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 50
Answer:
The sum of 25 + 13 + 12 + 5 = 55.

Explanation:
Given that 25 + 13 + 12 + 5 which is
20 + 10 + 10 = 40
5 + 3 + 2 + 5 = 15
40 + 15 = 55.

Question 9.
Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 51
Answer:
The sum of 9 + 29 + 5 + 35 = 70.

Explanation:
The sum of 9 + 29 + 5 + 35 which is
20 + 30 = 50
9 + 9 + 5 + 5 = 28
50 + 28 = 78.

Question 10.
Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 52
Answer:
The sum of 18 + 24 + 42 + 6 is 90.

Explanation:
The sum of 18 + 24 + 42 + 6 which is
10 + 20 + 40 = 70
8 + 4 + 2 + 6 = 20
70 + 20 = 90.

Question 11.
Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 53
Answer:
The sum of 27 + 15 + 33 + 24 = 99.

Explanation:
The sum of 27 + 15 + 33 + 24 is
20 + 10 + 30 + 20 = 80
7 + 5 + 3 + 4 = 19
80 + 19 = 99.

Question 12.
Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 54
Answer:
The sum of 13 + 7 + 20 + 55 = 95.

Explanation:
The sum of 13 + 7 + 20 + 55 is
10 + 20 + 50 = 80
3 + 7 + 5 = 15
80 + 15 = 95.

Number Sense Solve each problem. In which order did you add the numbers? Explain.
Question 13.
22 + 17 + 8 + 3 = ________
Answer:
The sum of 22 + 17 + 8 + 3 is 50.

Explanation:
The sum of 22 + 17 + 8 + 3 is
20 + 10 = 30
2 + 7 + 8 + 3 = 20
30 + 20 = 50.

Question 14.
5 + 12 + 15 + 3 = ________
Answer:
The sum of 5 + 12 + 15 + 3 = 35.

Explanation:
The sum of 5 + 12 + 15 + 3 is
10 + 10 = 20
5 + 2 + 5 + 3 = 15
20 + 15 = 35.

Problem Solving

Solve each problem. Show your work.
Question 15.
Model Kim has 38 seashells. Mike has 27 seashells. Use partial sums to find how many shells they have in all. Then check your answer by adding another way.
__________ seashells
Answer:
The number of shells they have in all is 65 seashells.

Explanation:
Given that Kim has 38 seashells and Mike has 27 seashells. So the number of shells they have in all is 38 + 27 which is
30 + 20 = 50
8 + 7 = 15.
50 + 15 = 65.

Question 16.
en Vision® STEM Fossils form slowly over millions of years. Many fossils come from the sea. Kyle has 9 fossils, Jorie has 12 fossils, Leah has 6 fossils, and Joshua has 8 fossils. How many fossils do they have in all?
Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 55
__________ fossils
Answer:
The total number of fossils do they have in all is 35 fossils.

Explanation:
Given that Kyle has 9 fossils, Jorie has 12 fossils, Leah has 6 fossils, and Joshua has 8 fossils. So the total number of fossils do they have in all is 9 + 12 + 6 + 8 which is
9 + 6 + 8 + 2 = 25
10 + 25 = 35.

Question 17.
Higher Order Thinking Find the sum of 42 + 13 + 18 + 7 = _________ . Explain why your strategy works.
Answer:
The sum of 42 + 13 + 18 + 7 is 80.

Explanation:
The sum of 42 + 13 + 18 + 7 is
40 + 10 + 10 = 60
2 + 3 + 8 + 7 = 20
60 + 20 = 80.

Question 18.
Assessment Practice Find the sum of 17 + 29 + 46. Use any strategy. Show your work.
17 + 29 + 46 = __________
Answer:
The sum of 17 + 29 + 46 = 92.

Explanation:
The sum of 17 + 29 + 46 is
10 + 20 + 40 = 70
7 + 9 + 6 = 22
22 + 70 = 92.

Lesson 4.8 Solve One-Step and Two-Step Problems

Solve & Share
The second graders take a trip to a nature center. The Green Class sees 23 animals. The Blue Class sees 14 animals. The Yellow Class sees 32 animals. How many animals do they see in all?
Solve using drawings, models, or an equation. Be prepared to explain your work.

I can … use drawings, models, and equations to solve one- and two-step problems.
Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 56
_________ animals

Answer:
The total number of animals do they see all is 69 animals.

Explanation:
Given that the second graders take a trip to a nature center. The Green Class sees 23 animals and the Blue Class sees 14 animals and the Yellow Class sees 32 animals. So the total number of animals do they see all is 23 + 14 + 32 which is
= 37 + 32
= 69 animals.

Visual Learning Bridge
Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 57

Convince Me! What is a different way you can use mental math to find 36 + ? = 53?

Answer:
36 + 17 = 53.

Explanation:
The different way we can use mental math is to find 36 + ? = 53 is first we will add 10 to the number 36 then we will get 36 + 10 which is 46 and here the result is 53, so we need to add 7 more. So it will be 46 + 7 which is 53.

Guided Practice

Use the bar diagram and mental math to solve each problem. Then check your work.
Question 1.
Bruce buys 18 eggs. He uses 10 of them for baking. Then he buys 36 more eggs. How many eggs does Bruce have now?
Check:
Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 58
________ eggs
Answer:
The total number of eggs is 44 eggs.

Explanation:
Given that Bruce buys 18 eggs and he uses 10 of them for baking, so the number of eggs will be 18 – 10 which is 8 eggs. And then he buys 36 more eggs. So the number of eggs will be 8 + 36 which is 44 eggs.

Independent Practice

Use the bar diagram(s) and mental math to solve each problem. Then check your work.
Question 2.
Ella has 34 more buttons than Julio. Julio has 49 buttons. How many buttons does Ella have?
Check
Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 59
________ buttons
Answer:
The number of buttons does Ella has now is 83 buttons.

Explanation:
Given that Ella has 34 more buttons than Julio and Julio has 49 buttons. So the number of buttons does Ella has now is 49 + 34 which is 83 buttons.
Envision-Math-Common-Core-Grade-2-Answer-Key-Topic-4-Fluently-Add-Within-100-59-1

Question 3.
20 students are on the bus. Then 10 students get off the bus. Next, 15 students get on the bus. How many students are on the bus now?
Check:
Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 60
________ students
Answer:
The total number of students will be 25 students.

Explanation:
Given that there are 20 students are on the bus and then 10 students get off the bus. So the number of students will be 20 – 10 which is 10 students. And then 15 students get on the bus, so the number of students now is 10 + 15  which is 25 students. So the total number of students will be 25 students.
Envision-Math-Common-Core-Grade-2-Answer-Key-Topic-4-Fluently-Add-Within-100-60-1

Question 4.
Wendy has 14 more crayons than Oscar. Oscar has 54 crayons. How many crayons does Wendy have?
Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 61
_________ crayons
Answer:
The number of crayons Wendy has is 68 crayons.

Explanation:
Given that Wendy has 14 more crayons than Oscar and Oscar has 54 crayons. So the number of crayons Wendy has is 54 + 14 which is 68 crayons.
Envision-Math-Common-Core-Grade-2-Answer-Key-Topic-4-Fluently-Add-Within-100-61-1

Problem Solving

Solve each problem. Show your work.
Question 5.
Make Sense Mariah has 17 figs. Kendra has 20 more figs than Mariah. Toby has 33 more figs than Kendra. How many figs do Kendra and Toby each have?
Step 1:
How many figs does Kendra have?
________ Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 62 _______ = ________ figs
Step 2: How many figs does Toby have?
________ Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 62 _______ = ________ figs
Solve one step at a time.
Answer:
Kendra has 17 + 20 = 37 figs.
Toby has 37 + 33 = 70 figs.

Explanation:
Given that Mariah has 17 figs and Kendra has 20 more figs than Mariah, so the number of figs does Kendra has now is 17 + 20 which is 37 figs.
Toby has 33 more figs than Kendra, so the number of figs does Toby has now is 37 + 33 which is 70 figs.

Question 6.
Higher Order Thinking 8 girls and some boys are in the pool. In all, 17 children are in the pool. Then some more boys jump in the pool. Now there are 13 boys in the pool. How many more boys jumped in the pool?
Step 1:
________ Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 62 _______ = ________
Step 2:
________ Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 62 _______ = ________
_________ more boys jumped in the pool
Answer:
4 more boys jumped in the pool.

Explanation:
Given that 8 girls and some boys are in the pool. In all, 17 children are in the pool, so the number of boys in the pool is 17 – 8 which is 9 boys. As some boys jump into the pool and the boys in the pool are 13. So the number of boys who jumped in the pool is 13 – 9 which is 4 boys.

Question 7.
Assessment Practice The soccer coach has 18 shirts. She gives 9 of the shirts to some of her players. Then she gets 11 more shirts. How many shirts does the coach have left?
Show how you can solve the problem in two steps.
Step 1:
Step 2:
The coach has ________ shirts left.
Answer:
The coach has 20 shirts left.

Explanation:
Given that the soccer coach has 18 shirts and she gives 9 of the shirts to some of her players, so the number of shirts does coach had is 18 – 9 which is 9 shirts. And then gets 11 more shirts, so the total number of shirts does the coach has left is 9 + 11 which is 20 shirts.

Lesson 4.9 Problem Solving

Model with Math
Solve & Share
Kim puts 25 toys into an empty toy box. Then she puts 17 more toys in the toy box. How many toys are in the box in all?
Use a model to show the problem. Be ready to explain how your model helps you solve the problem.

I can … make models to help solve math problems.
Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 63

Answer:
The total number of toys in the box is 42 toys.

Explanation:
Given that Kim puts 25 toys into an empty toy box and then she puts 17 more toys in the toy box. So the total number of toys in the box is 25 + 17 which is 42 toys.
Envision-Math-Common-Core-Grade-2-Answer-Key-Topic-4-Fluently-Add-Within-100-63-1

Thinking Habits
Can I write an equation or use a drawing, diagram, table, graph, or objects to show the problem?

Visual Learning Bridge
Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 64

Convince Me! How does drawing a bar diagram or writing an equation model a problem? Is there another way to model the problem?

Answer:
Here, both the bar diagram and writing an equation model for a problem works.

Guided Practice

Complete the bar diagram. Write an equation to model the problem. Then solve.
Question 1.
Flora has 24 books about birds. She has 18 books about bugs. How many books is that?
Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 65

Answer:
The number of books is 42 books.

Explanation:
Given that Flora has 24 books about birds and she has 18 books about bugs. So the number of books is 24 + 18 which is 42 books.

Question 2.
Barb saw 14 cars on Smith Street. She saw 8 cars on Alfred Street. How many more cars did Barb see on Smith Street?
Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 66
_______ Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 62 _______ = ________ cars
Answer:
The number of cars did Barb see on Smith Street is 22 cars.

Explanation:
Given that Barb saw 14 cars on Smith Street and she saw 8 cars on Alfred Street. So the number of cars did Barb see on Smith Street is 14 + 8 which is 22 cars.
Envision-Math-Common-Core-Grade-2-Answer-Key-Topic-4-Fluently-Add-Within-100-66-1

Independent Practice

Make a model to show each problem. Then use the model to solve. Show your work.
Question 3.
Avi takes 16 pictures. Then he takes 17 more pictures. How many pictures does Avi take?
Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 67
Answer:
The number of pictures does Avi take is 33 pictures.

Explanation:
Given that Avi takes 16 pictures and then he takes 17 more pictures. So the number of pictures does Avi take is 16 + 17 which is 33 pictures.

Question 4.
Tina picks 55 blueberries. Next, Tina eats 6 of these blueberries. Then she picks 27 more. How many blueberries does she have now?
Answer:
The number of blueberries does she have now is 76 blueberries.

Explanation:
Given that Tina picks 55 blueberries and Tina eats 6 of these blueberries, so the number of blueberries is 55 – 6 which is 49 blueberries. Now she picks 27 more, so the number of blueberries does she have now is 49 + 27 which is 76 blueberries.

Question 5.
Raj finds 47 acorns in his front yard. He finds 29 acorns in his back yard. How many acorns does Raj find in all?
Answer:
The number of acorns does Raj finds in all is 76 acorns.

Explanation:
Given that Raj finds 47 acorns in his front yard and he finds 29 acorns in his back yard. So the number of acorns does Raj finds in all is 47 + 29 which is 76 acorns.

Problem Solving

Performance Task
African Safari The Santos family is on an African safari. The chart at the right shows the number of animals they see. How many animals do they see?
Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 68

Question 6.
Make Sense What do you know? What are you asked to find?
Answer:
Here, we know that how many animals they saw of each type of animal and we need to find the total.

Question 7.
Model Make a model to help you find the number of animals they see. Be ready to explain why you chose the model you did.
Answer:
The number of animals they see is 51 animals.

Explanation:
Given that there are 15 giraffes, 9 elephants, 16 lions and 11 zebras. So the number of animals they see are 15 + 9 + 16 + 11
= 24 + 27
= 51.

Question 8.
Model What other model could you use to show the problem? Make another model. Explain which model you think is better.
Answer:

Topic 4 Fluency Practice Activity

Point & Tally
Find a partner. Get paper and a pencil. Each partner chooses a different color: light blue or dark blue.
Partner I and Partner 2 each point to a black number at the same time. Both partners add those numbers.
If the answer is on your color, you get a tally mark. Work until one partner gets twelve tally marks.

I can … add within 20.
Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 69

Topic 4 Vocabulary Review

Understand Vocabulary
Word List

  • addends
  • compatible numbers
  • ones
  • partial sum
  • regroup
  • sum
  • tens

Use the problem at the right. Write partial sum or sum for each.
Envision Math Common Core Grade 2 Answer Key Topic 4 Fluently Add Within 100 70
Question 1.
70 is a _________ .
Answer:
70 is a partial sum.

Explanation:
Here, the number 70 is a partial sum, because partial sum is just a sum of part of sequence.

Question 2.
17 is a ___________.
Answer:
17 is a partial sum.

Explanation:
Here, the number 17 is a partial sum, because partial sum is just a sum of part of sequence.

Question 3.
87 is the __________.
Answer:
87 is a sum.

Explanation:
Here, the number 87 is a sum, because it is the result of the addition.

Question 4.
Use the ones column in the problem at the right. Which compatible numbers can you add to make a ten two different ways?
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 71
Write two different equations.
Answer:
8 + 2 = 10,
6 + 4 = 10.

Explanation:
The two different equations which are compatible numbers can we add to make a ten is
8 + 2 = 10,
6 + 4 = 10.

Use Vocabulary in Writing
Question 5.
Solve 27 + 35 by breaking apart both addends. Tell how you solved the problem. Use terms from the Word List.
Answer:
27 + 35 = 65.

Explanation:
Here, we have added the ones and then we will regroup and added to find the sum.
27 + 35 = 65.

Topic 4 Reteaching

Set A

Find 55 + 17. Use place-value blocks. Show the addends. Join the tens and ones. Regroup if needed.
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 72

Add. Use place-value blocks to show your work. Regroup if needed.
Question 1.
36 + 23 = ________
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 73

Answer:
The sum of 36 + 23 is 59 which is 5 tens and 9 ones.

Explanation:
The sum of 36 + 23 is
30 + 20 = 50
6 + 3 = 9
50 + 9 = 59.

Question 2.
19 + 44 = _______
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 74
_______ tens _______ ones
Answer:
The sum of 19 + 44 is  63 which is 6 tens and 3 ones.

Explanation:
The sum of 19 + 44 is
10 + 40 = 50
9 + 4 = 13
50 + 13 = 63.

Set B

Find 24 + 49. Draw place-value blocks to show the addends. Join the tens and ones. Regroup if needed.
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 75

Add. Use place value and draw blocks.
Question 3.
64 + 13 = _______
Answer:
The sum of 64 + 13 is 77 which is 7 tens and 7 ones.

Explanation:
The sum of 64 + 13 is
60 + 10 = 70
4 + 3 = 7
70 + 7 = 77.

Question 4.
27 + 56 = ________
Answer:
The sum of 23 + 56 is 79 which is 7 tens and 9 ones.

Explanation:
The sum of 23 + 56 is
20 + 50 = 70
3 + 6 = 9
70 + 9 = 79.

Set C

You can draw blocks and use partial sums to find 46 + 37.
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 76

Add. Draw place-value blocks and use partial sums.
Question 5.
33 + 57
Drawings:
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 77
Answer:
The partial sum of 33 + 57 is 90.

Explanation:
Given that 33 + 57, the partial sum of 33 + 57 is
Envision-Math-Common-Core-Grade-2-Answers-Topic-4-Fluently-Add-Within-100-77-1

Set D

Find 29 + 63. You can break apart the addends and use partial sums.
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 78

Add. Break apart the addends and use partial sums.
Question 6.
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 79
Answer:
The partial sum of 17 + 77 is 94.

Explanation:
Given that 17 + 77, the partial sum of 17 + 77 is
Envision-Math-Common-Core-Grade-2-Answers-Topic-4-Fluently-Add-Within-100-79-1

Question 7.
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 80
Answer:
The partial sum of 24 + 38 is 62.

Explanation:
Given that 24 + 38, the partial sum of 24 + 38 is
Envision-Math-Common-Core-Grade-2-Answers-Topic-4-Fluently-Add-Within-100-80-1

Set E

Find 32 + 19. Break apart the second addend and add mentally.
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 81

Break apart the second addend to add. Show your work.
Question 8.
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 82
Answer:
The sum of 24 + 55 is 79.

Explanation:
The sum of 24 + 55 is 79.
Envision-Math-Common-Core-Grade-2-Answers-Topic-4-Fluently-Add-Within-100-82-1

Question 9.
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 83
Answer:
The sum of 64 + 27 is 91.

Explanation:
The sum of 64 + 27 is 91.
Envision-Math-Common-Core-Grade-2-Answers-Topic-4-Fluently-Add-Within-100-83-1

Set F

Use partial sums to find 65 + 7 + 13.
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 84

Add using partial sums.
Question 10.
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 85
Answer:
The partial sum is 70.

Explanation:
Given that 22 + 21 + 9 + 18, the partial sum of 22 + 21 + 9 + 18 is
Envision-Math-Common-Core-Grade-2-Answers-Topic-4-Fluently-Add-Within-100-85-1

Question 11.
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 86
Answer:
The partial sum of 41 + 15 + 29 + 10 is 95.

Explanation:
Given that 41 + 15 + 29 + 10, the partial sum of 41 + 15 + 29 + 10 is
Envision-Math-Common-Core-Grade-2-Answers-Topic-4-Fluently-Add-Within-100-86-1

Set G

27 students are eating lunch. More students join them. Now 63 students are eating lunch. How many students joined them?
Write an equation: 27 + ? = 63 Count on to find the missing addend:
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 87
So, 36 students joined them.

Complete the bar diagram, write an equation, and solve.
Question 12.
Lana has 24 crayons. Then she gets some more crayons. Now she has 42 crayons. How many crayons does Lana get?
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 88
Answer:
The number of crayons does Lana gets is 66 crayons.

Explanation:
Given that Lana has 24 crayons and then she gets some more crayons and now she has 42 crayons. So the number of crayons does Lana gets is 24 + 42 which is 66 crayons.
Envision-Math-Common-Core-Grade-2-Answers-Topic-4-Fluently-Add-Within-100-88-1

Set H

Thinking Habits
Model with Math
Can I use a drawing, diagram, table, graph, or objects to show the problem?
Can I write an equation to show the problem?
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 89

Make a model and solve the problem.
Question 13.
Students ride buses to the museum. 28 students ride in Bus A. 27 students ride in Bus B. How many students ride in both buses?
Answer:
The total number of students ride in both buses is 55 students.

Explanation:
Given that 28 students ride in Bus A and 27 students ride in Bus B. So the total number of students ride in both buses is 28 + 27 which is 55 students.

Topic 4 Assessment Practice

Question 1.
Kelly has 46 beads. Her sister gives her 28 more beads. How many beads does Kelly have now?
A. 18
B. 64
C. 22
D. 74
Answer:
The total number of beads does Kelly have now is 74 beads.

Explanation:
Given that Kelly has 46 beads and her sister gives her 28 more beads. So the total number of beads does Kelly have now is 46 + 28 which is 74 beads.

Question 2.
Which are equal to 45 + 38? Choose all that apply.
☐ 40 + 30 + 5+ 8
☐ 93
☐ 45 + 30 + 5 + 3
☐ 40 + 30 + 10
☐ 83
Answer:
Option A and Option E.

Explanation:
Given that 45 + 38, so 45 + 38 is equal to 40 + 30 + 5 + 8 which is 83.

Question 3.
Is the sum equal to 81? Choose Yes or No.
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 90
Answer:
Option B and Option D.

Explanation:
Given that the sum equal to 81, so the addends will be 52  + 29 and 63 + 18 which is 81.
Envision-Math-Common-Core-Grade-2-Answers-Topic-4-Fluently-Add-Within-100-90-1

Question 4.
Find 27 + 44 using partial sums. Show how you solved the problem.
Answer:
The sum of 27 + 44 is 71.

Explanation:
The partial sum of 27 + 44 is
= 27 + 40 + 4
= 67 + 4
= 71.

Question 5.
Which is the sum of 57 + 15?
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 91
A. 82
B. 72
C. 62
D. 42
Answer:
Option B.

Explanation:
The sum of 57 + 15 is 72.

Question 6.
Which are the same amount as 13 + 8 + 7? Choose all that apply.
☐ 20 + 8
☐ 10 + 10 + 8
☐ 18
☐ 28
☐ 20 + 7
Answer:
Option A, Option B and Option D.

Explanation:
The same amount as 13 + 8 + 7 is 20 + 8, 10 + 10 + 8 and 28.

Question 7.
What is the sum of 26 + 53 + 7? Show your work.
26 + 53 + 7 = ________
Answer:
The sum is 86.

Explanation:
The sum of 26 + 53 + 7 is
= 79 + 7
=  86.

Question 8.
Find the sum of 23 and 19. Then subtract 10 from that sum. Show and explain your work.
The answer is __________
Answer:
The answer is 32.

Explanation:
Given that the sum of 23 and 19 is 42 and should subtract 10 from the sum which is 42 – 10 = 32.

Question 9.
A. Find the sum. Use a strategy. Show your work.
25 + 37 + 15 + 8 = __________
Answer:
25 + 37 + 15 + 8 = 85.

Explanation:
The sum of 25 + 37 + 15 + 8 is
= 25 + 60
= 85.

B. Explain how you found the sum and why your strategy works.
Answer:
Here, we will first add ones and got 25 and then we will regroup the 2 tens and then we will add to get the sum of 85.

Question 10.
Ms. Wise has 12 tablets. Then she gets 9 more tablets. Ms. Wise has 20 students in her class. She gives each student one tablet. How many tablets does she have left? Use equations to solve the problem.
________ tablet(s)
Answer:
The number of tablets left with Ms. Wise is 1 tablet.

Explanation:
Given that Ms. Wise has 12 tablets and she gets 9 more tablets, so the total tablets will be 12 + 9 which is 21 tablets. Ms. Wise has 20 students in her class and she gives each student one tablet. So the number of tablets left with Ms. Wise is 21 – 20 which is 1 tablet.

Question 11.
What is the sum of 59 + 33?
Draw place-value blocks and show your work in the blue box at the right.
59 + 33 = _________
Answer:

Question 12.
Each equation has a missing addend. Find it in the number cards. Draw lines to match. Then write the missing addend in the gray box.
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 92
Answer:
The missing addends are 7, 8, 5, 6.

Explanation:
Given that 6 + 14 +– + 3 = 30, so the missing addend is 23 + — = 30 and the missing addend is 30 -23 = 7.
Given that 3 + 32 + 7 +– = 50, so the missing addend is 42+ — = 50 and the missing addend is 50 – 42= 8.
Given that 25 + 25  +– + 5 = 60, so the missing addend is 55 + — = 60 and the missing addend is 60 – 55 = 5.
Given that 5 +– + 15 + 14 = 40, so the missing addend is 34 + — = 40 and the missing addend is 40 – 34 = 6.

Envision-Math-Common-Core-Grade-2-Answers-Topic-4-Fluently-Add-Within-100-92-1

Question 13.
Corey has 17 more marbles than Tony. Tony has 64 marbles. How many marbles does Corey have?
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 93
Use the bar diagram to model and solve the problem. Show your work.
_________ marbles
Answer:
The total number of marbles is 81 marbles.

Explanation:
Given that Corey has 17 more marbles than Tony and Tony has 64 marbles. So the total number of marbles is 17 + 64 which is 81 marbles.

Topic 4 Performance Task

Our Pets
Students draw pictures of their pets. The chart shows the number of pets the students have.
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 94
Question 1.
How many dogs and cats do the students have? Show your work.
_________ dogs and cats
Answer:
&0 students have dogs and cats.

Explanation:
The total number of dogs and cats is 41 + 29 which is 70 students.

Question 2.
How many cats and fish do the students have? Show your work.
__________ cats and fish
Answer:
53 students have cats and fish

Explanation:
The total number of cats and fish is 29 + 24 which is 53 students.

Question 3.
Use partial sums to find how many cats, rabbits, and fish the students have. Be sure to check your work.
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 95
How many cats, rabbits, and fish in all?
__________ cats, rabbits, and fish
Answer:
59 students have cats, rabbits, and fish.

Explanation:
The total number of cats, rabbits, and fish is 29 + 6 + 24 which is 59 students.
Envision-Math-Common-Core-Grade-2-Answers-Topic-4-Fluently-Add-Within-100-95-1

Question 4.
Explain how to use partial sums to add.
Answer:

Question 5.
The students also draw pictures of 10 hamsters, 19 birds, and 5 mice.
Part A
Complete the model to show how to find the total number of hamsters, birds, and mice.
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 96
Answer:

Part B
Complete the equation to show how many hamsters, birds, and mice they draw. Then write the total.
Envision Math Common Core Grade 2 Answers Topic 4 Fluently Add Within 100 97
__________ hamsters, birds, and mice
Answer:

enVision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8

enVision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties: Multiplication Facts for 3, 4, 6, 7, 8

Go through the enVision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties: Multiplication Facts for 3, 4, 6, 7, 8 regularly and improve your accuracy in solving questions.

enVision Math Common Core 3rd Grade Answers Key Topic 3 Apply Properties: Multiplication Facts for 3, 4, 6, 7, 8

Essential Question:
How can you use known multiplication facts to solve unknown facts?
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 1

enVision STEM Project: Inherited Traits
Do Research Some characteristics of organisms are inherited. The traits are passed from generation to generation. In flowers, one of the inherited traits is color. Use the Internet or other sources to make a list of other traits that flowers inherit from their parent plants.

Journal: Write a Report Include what you found. Also in your report:

  • Compare your list of traits with lists of traits other students have made. If there is a trait you do not have, add it to your list.
  • Draw flowers or animals with similar traits in an array. Show how to break apart the array and use multiplication facts to find the total number.

Review What You Know
Vocabulary
Choose the best term from the box. Write it on the blank.

  • skip counting
  • The Commutative (Order) Property of Multiplication
  • The Identity (One) Property of Multiplication
  • The Zero Property of Multiplication

 

Question 1.
________ says that the product of any number and zero is zero.
Answer:
The zero property of multiplication  says that the product of any number and zero is zero.
Explanation:
The multiplication property states that the product of any number and zero is zero. It doesn’t matter what the number is, when you multiply it to zero, you get zero as the answer.

Apply Properties Multiplication Facts 1

Question 2.
________ says that 1 times any number is that number.
Answer:
The Identity (One) Property of Multiplication says that 1 times any number is that number.
Explanation:
The identity property of 1 says that any number multiplied by 1 keeps its identity. The reason the number stays the same is because multiplying by 1 means we have 1 copy of the number.

 

Question 3.
_________ says that you can multiply factors in any order, and the product stays the same.
Answer:
The Commutative (Order) Property of Multiplication says that you can multiply factors in any order, and the product stays the same.
Explanation:
The math rule that says the order in which we multiply the factors does not change the product is the commutative property.

 

 

Multiplying Use multiplication to solve.

Question 4.
10 × 1 = ________
Answer:
10 × 1 = 10.

Question 5.
2 × 10 = _______
Answer:
2 × 10 = 20.

Question 6.
0 × 5 = _______
Answer:
0 × 5 = 0.

Question 7.
9 × 5 = _______
Answer:
9 × 5 = 45.

Question 8.
2 × 7 = _________
Answer:
2 × 7 = 14.

Question 9.
1 × 8 = _______
Answer:
1 × 8 = 8.

Question 10.
5 × 7 = ?
A. 7 + 5
B. 5 + 7
C. 7 × 5
D. 7 ÷ 5
Answer:
5 × 7 = 35.

Explanation:
5 × 7 = ?
=> 5 × 7 = 35.

 

 

Adding 2-Digit Numbers
Find the sum.
Question 11.
16 + 12 = _______
Answer:
16 + 12 = 28.
Explanation:
16 + 12 = ??
Ten’s place: 2+1=3.
Ones place: 6+2=8.

 

Question 12.
21 + 14 = _______
Answer:
21 + 14 = 35.
Explanation:
21 + 14 = ??
Ten’s place: 2+1=3.
Ones place: 1+4=5.

 

Question 13.
24 + 12 = _______
Answer:
24 + 12 = 36.
Explanation:
24 + 12 = ??
Ten’s place: 2+1=3.
Ones place: 4+2=6.

 

 

Arrays
Question 14.
How can you represent 3 × 6 using an array? Draw an array, and explain how to use it to find the product.
Answer:
3 × 6 = 18 or Eighteen.

Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Arrays-14
Explanation:
Number of rows = 3
Number of columns = 6
Multiplication:
Number of rows × Number of columns
= 3 × 6
= 18 or Eighteen.

 

 

Pick a Project

PROJECT ЗА
How many points can you score?
Project: Make a Basketball Data Display
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 2

Answer:
It is calculated by taking the difference in the score when the player enters the game and subtracting it from the score when the player exits the game. These differences are added up over the entire game to give the score.

Explanation:
It is calculated by taking the difference in the score when the player enters the game and subtracting it from the score when the player exits the game.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Pick a Project-PROJECT ЗА

 

 

PROJECT 3B
Do you like collecting seashells?
Project: Draw a Shell Array
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 3

Answer:
Yes, I like collecting seashells.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Pick a Project-PROJECT 3B
Explanation:
Number of rows of shells = 3
Number of columns of the shells = 2
Multiplication:
Number of rows of shells × Number of columns of the shells
= 3 × 2
= 6.

Apply Properties Multiplication Facts 1

PROJECT 3C
Would you like to run for president?
Project: Write a Presidential Report
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 4

Answer:
Yes, I would like to run for president.

Explanation:
Writing a Report on a US President.
Getting Started:
First, read about the President you will be writing the report on. Read as much information about the President
as you can find. Try the Internet and the library; try Whitehouse.gov, presidentsusa.net, .ipl.org/div/potus/, an
encyclopedia, books, or even videos.
As you’re finding out about your President, take notes on key information, such as important dates, critical
events in the President’s life, people who influenced his life, major world events during his life, etc.

 


3-ACT MATH PREVIEW
Math Modeling
Thirsty Students
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 5
I can .. model with math to solve a problem that involves using multiplication facts and computing

Lesson 3.1 The Distributive Property

Solve & Share
Find two ways to break the array below into two smaller arrays. What multiplication equation can you write for each array? What is the total? Tell how you decided.
I can … break apart unknown facts into known facts and solve multiplication problems.
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 6

Look Back! Find the total number of items in both of the smaller arrays. Compare their combined total to the total number of items in the one large array. Why are the totals the same even though the arrays are different?
Answer:
The totals are the same even though the arrays are different because Distributive law, in mathematics,  the law relating the operations of multiplication and addition, stated symbolically, a(b + c) = ab + ac; that is, the monomial factor a is distributed, or separately applied, to each term of the binomial factor b + c, resulting in the product ab + ac.

Explanation:
Array:1:
Number of Rows = 4 or Four.
Number of columns = 5 or Five.
Number of items = Number of Rows × Number of columns
= 4 × 5
= 20 or Twenty.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.1 The Distributive Property-Solve & Share

Array:2:
Number of Rows = 5 or Five.
Number of columns = 4 or Four.
Number of items = Number of Rows × Number of columns
= 5 × 4
= 20 or Twenty.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.1 The Distributive Property-Solve & Share..

 

 

Essential Question
How Can You Break Up a Multiplication Fact?
Answer:
You can Break Up a Multiplication Fact using distributive property for multiplication.

Explanation:
To “distribute” means to divide something or give a share or part of something. According to the distributive property, multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together.


Visual Learning Bridge

Maria wants to set up 7 rows of 4 chairs for a meeting. She wants to know how many chairs she needs but does not know the product of 7 4.
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 7

What You Think
Maria thinks of 7 rows of 4 chairs as 5 rows of 4 chairs and another 2 rows of 4 chairs.
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 8

What You Write
The Distributive Property says that a multiplication fact can be broken apart into the sum of two other multiplication facts.
Maria knows the two new facts.
7 × 4 = (5 × 4) + (2 × 4)
7 × 4 = 20 + 8
7 × 4 = 28
So, 7 × 4 = 28.
Maria needs 28 chairs.

Convince Me! Use Structure What are two ways that Maria could break up the array for 7 × 4? Draw a picture of the two new arrays and write the new facts.
Answer:
The new facts are :
Way:1:
7 × 4 = (4 × 4) + (3 × 4)
7 × 4 = 16 + 12
7 × 4 = 28
So, 7 × 4 = 28.
Way:2:
7 × 4 = (6 × 4) + (1 × 4)
7 × 4 = 24 + 4
7 × 4 = 28
So, 7 × 4 = 28.

Explanation:
Way:1:
7 × 4 = (4 × 4) + (3 × 4)
7 × 4 = 16 + 12
7 × 4 = 28
So, 7 × 4 = 28.
Maria needs 28 chairs.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.1 The Distributive Property-Visual Learning Bridge

Way:2:
7 × 4 = (6 × 4) + (1 × 4)
7 × 4 = 24 + 4
7 × 4 = 28
So, 7 × 4 = 28.
Maria needs 28 chairs.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.1 The Distributive Property-Visual Learning Bridge..

 

 

Guided Practice
Do You Understand?
Question 1.
Rafael broke up an array for 6×3 into two new arrays. Both of his new arrays are the same. What were the two arrays?
Answer:
The two arrays are :
Way:1:
6 × 3 = (5 × 3) + (1 × 3)
6 × 3 = 15 + 3
6 × 3 = 18
So, 6 × 3 = 18.

Way:2:
6 × 3 = (4 × 3) + (2 × 3)
6 × 3 = 12 + 6
6 × 3 = 18
So, 6 × 3 = 18.

Explanation:
The two arrays are :
Way:1:
6 × 3 = (5 × 3) + (1 × 3)
6 × 3 = 15 + 3
6 × 3 = 18
So, 6 × 3 = 18.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.1 The Distributive Property-Guided Practice-Do You Understand-1..

Way:2:
6 × 3 = (4 × 3) + (2 × 3)
6 × 3 = 12 + 6
6 × 3 = 18
So, 6 × 3 = 18.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.1 The Distributive Property-Guided Practice-Do You Understand-1

 

Question 2.
Ann broke up a large array into two smaller arrays. The two smaller arrays show 1 × 8 and 4 × 8. What was the large array that Ann started with?
Answer:
The large array that Ann started with :
5 × 8 = 40
So, 5 × 8 = 40..

Explanation:
The large array that Ann started with :
5 × 8 = (5 × 8)
5 × 8 = 40.
So, 5 × 8 = 40.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.1 The Distributive Property-Guided Practice-2

 

 

Do You Know How?
In 3 and 4, use the smaller arrays and the Distributive Property to find each missing factor. You may use counters to help
Question 3.
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 9
Answer:
__4__ × 8 = (__2__ × 8) + (2 × 8)
4 × 8 = 16 + 16
4 × 8 = 32.

Explanation:
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 9
4 × 8 = (2 × 8) + (2 × 8)
4 × 8 = 16 + 16
4 × 8 = 32.
So, 4 × 8 = 32.

 

 

Question 4.
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 10
Answer:
__3__ × __5__ = (__2__ × 5) + (1 × __5__)
3 × 5 = 10 + 5
3 × 5 = 15.

Explanation:
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 10
3 × 5 = (2 × 5) + (1 × 5)
3 × 5 = 10 + 5
3 × 5 = 15.
So, 3 × 5 = 15.

 

 

Independent Practice
In 5 and 6, separate the rows in the large array into two smaller arrays. Write the new facts.
Question 5.
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 11
Answer:
4 × 5 = ( __2__ × __5__) + ( __2__ × __5__)
4 × 5 = 10 + 10
4 × 5 = 20.

Explanation:
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 11
4 × 5 = ( 2 × 5) + ( 2 × 5)
4 × 5 = 10 + 10
4 × 5 = 20.
So, 4 × 5 = 20.

 

 

 

Question 6.
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 12
Answer:
5 × 6 = (__3__ × __6__) + ( __2__ × __6__)
5 × 6 = 18 + 12
5 × 6 = 30.

Explanation:
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 12
5 × 6 = (3 × 6) + ( 2 × 6)
5 × 6 = 18 + 12
5 × 6 = 30.
So, 5 × 6 = 30.

In 7-10, use the Distributive Property to find each missing factor. Use counters and arrays to help.

Apply Properties Multiplication Facts 2
Question 7.
6 × 8 = (4 × _____) +(2 × 8)
Answer:
6 × 8 = (4 × ___8____) +(2 × 8)
6 × 8 = 32 + 16
6 × 8 = 48.

Explanation:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.1 The Distributive Property-Independent Practice-7
6 × 8 = (4 × 8) +  (2 × 8)
6 × 8 = 32 + 16
6 × 8 = 48.
So, 6 × 8 = 48.

 

Question 8.
10 × 3= (____ × 3) + (2 × 3)
Answer:
10 × 3= (__8__ × 3) + (2 × 3)
10 × 3 = 24 + 6
10 × 3 = 30.

Explanation:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.1 The Distributive Property-Independent Practice-8
10 × 3 = (__8__ × 3) + (2 × 3)
10 × 3 = 24 + 6
10 × 3 = 30.
So, 10 × 3 = 30.

 

Question 9.
(_____ × 7) = (3 × 7) + (2 × _____)
Answer:
(___5__ × 7) = (3 × 7) + (2 × __7___)
5 × 7 = 21 + 14
5 × 7 = 35.

Explanation:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.1 The Distributive Property-Independent Practice-9
5 × 7 = (3 × 7) + (2 × 7)
5 × 7 = 21 + 14
5 × 7 = 35.
So, 5 × 7 = 35.

 

 

Question 10.
(8 × _____) = (_____ × 8) + (4 × 8)
Answer:
(8 × __8___) = (___4__ × 8) + (4 × 8)
(8 × 8 ) = 32 + 32
(8 × 8 ) = 64.

Explanation:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.1 The Distributive Property-Independent Practice-8
(8 × 8 ) = (4 × 8) + (4 × 8)
(8 × 8 ) = 32 + 32
(8 × 8 ) = 64.
So, (8 × 8 ) = 64.

 

 

Problem Solving
Question 11.
Paige bakes 5 cupcakes. She puts 7 jelly beans on each cupcake. How many jelly beans does Paige need? Use the bar diagram to help write an equation.
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 13
Answer:
Total number of jelly beans she needs = 35 or Thirty Five.
Explanation:
Number of cupcakes Paige bakes = 5 or Five.
Number of jelly beans on each cupcake she puts = 7 or Seven.
Total number of jelly beans she needs = Number of cupcakes Paige bakes × Number of jelly beans on each cupcake she puts
= 5 × 7
= 35 or Thirty Five.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.1 The Distributive Property-Problem Solving-11

 

Question 12.
Critique Reasoning
Fred wants to separate the rows of the array below into a 2 × 4 array and a 3 × 4 array. Can Fred do this? Explain.
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 14
Answer:
No, Fred cannot separate the rows of the array below into a 2 × 4 array and a 3 × 4 array because number of rows are four yet number of columns are 4 not five.

Explanation:
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 14
No, Fred cannot separate the rows of the array below into a 2 × 4 array and a 3 × 4 array because number of rows are four yet number of columns are 4 not five. He can separate into 2 × 4 array  and 2 × 4 array  yet not 2 × 4 array  and 3 × 4 array .

 

Question 13.
Lane uses counters to make a 4 × 7 array and a 1 × 7 array. What size array can he make using all of these counters?
Answer:
The two arrays given = 4 × 7 and 1 × 7.
=> (4+1)  × 7
=> 5  × 7.
The size array he make using all of these counters = 5  × 7.

Explanation:
The two arrays given = 4 × 7 and 1 × 7.
=> (4+1)  × 7
=> 5  × 7.
The size array he make using all of these counters = 5  × 7.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.1 The Distributive Property-Critique Reasoning-13

 

Apply Properties Multiplication Facts 3

Question 14.
Gavin had $75 on Monday. On Tuesday, he spent $23. Then he spent $14 on Wednesday. How much money does Gavin have left?
Answer:
Amount Gavin has left = $38.

Explanation:
Total amount Gavin has on Monday = $75.
Amount Gavin spent on Tuesday = $23.
Amount Gavin spent on Wednesday = $14
Amount Gavin has left = Total amount Gavin has on Monday – ( Amount Gavin spent on Tuesday + Amount Gavin spent on Wednesday )
= $75 – ($23 + $14)
= $75 – $37
= $38.

 

 

Question 15.
Vocabulary
Explain how you can use the Distributive Property to solve 9 × 6.
Answer:
9 × 6 = (5 × 6) + (4 × 6).
To “distribute” means to divide something or give a share or part of something. According to the distributive property, multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together.

Explanation:
To “distribute” means to divide something or give a share or part of something. According to the distributive property, multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together.
9 × 6 = (5 × 6) + (4 × 6).

 

 

Question 16.
Higher Order Thinking
How can you use 3 × 5 = 15 to help find 6 × 5?
Answer:
6 × 5 = (3 × 5) + (3 × 5)
6 × 5 = 15 + 15
6 × 5 = 30.
So, 6 × 5 = 30.

Explanation:
We can use 3 × 5 = 15 to help find 6 × 5 by distributive property.
6 × 5 = (3 × 5) + (3 × 5)
6 × 5 = 15 + 15
6 × 5 = 30.
So, 6 × 5 = 30.

 

Assessment Practice
Question 17.
Using the Distributive Property, which of the following expressions are equivalent to 7 × 7? Select all that apply.
☐ (7 × 7) + (7 × 7)
☐ (5 × 7) + (2 × 7)
☐ (2 × 7) + (5 × 7)
☐ (7 × 7) + (1 × 7)
☐ (7 × 7) + (2 × 7)
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.1 The Distributive Property-Assessment Practice-17

Explanation:
Expressions equivalent to 7 × 7 are as below:
7 × 7 = (4 × 7)+ (3 × 7)
7 × 7 = (2 × 7)+ (5 × 7)
7 × 7 = (6 × 7)+ (1 × 7)
7 × 7 = (5 × 7)+ (2 × 7)
7 × 7 = (3 × 7)+ (4 × 7)
7 × 7 = (1 × 7)+ (6 × 7)

 

 

Question 18.
An equation is shown. Select all the ways you can use the Distributive Property to find the missing factors. 7 × 3 = ( × 3) + ( × 3)
☐ 7 × 3 = (5 × 3) + (2 × 3)
☐ 7 × 3 = (2 × 3) + (5 × 3)
☐ × 3 = (7 × 3) + (1 × 3)
☐ × 3 = (1 × 3) + (6 × 3)
☐ × 3 = (6 × 3) + (1 × 3)
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.1 The Distributive Property-Assessment Practice-18

Explanation:
All the ways you can use the Distributive Property to find the missing factors:
7 × 3 = ( 5 × 3) + ( 2 × 3)
7 × 3 = ( 2 × 3) + ( 5 × 3)
7 × 3 = ( 1 × 3) + ( 6 × 3)
7 × 3 = ( 6 × 3) + ( 1 × 3)
7 × 3 = ( 4 × 3) + ( 3 × 3)
7 × 3 = ( 3 × 3) + ( 4 × 3)

 

 

Lesson 3.2 Apply Properties: 3 and 4 as Factors

Solve & Share
There are 3 rows of pictures on a wall. Each row has 7 pictures. How many pictures are on the wall?
I can … use tools and properties strategically to solve problems when I multiply by 3 or 4.
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 15

Look Back! How can you use what you know about multiplication facts for 1s facts and 2s facts to solve multiplication facts for 3s facts?
Answer:
Number of rows of Pictures = 3 or Three.
Number of Pictures in each row = 7 or Seven.
Number of pictures are on the wall = Number of rows of Pictures  × Number of Pictures in each row
= 3 × 7
= 21.

Explanation:
Number of rows of Pictures = 3 or Three.
Number of Pictures in each row = 7 or Seven.
Number of pictures are on the wall = Number of rows of Pictures  × Number of Pictures in each row
= 3 × 7
= 21.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.2 Apply Properties-3 and 4 as Factors

 


Essential Question

How Can You Break Apart Arrays to Multiply with 3?
Answer:
You can break Arrays to Multiply with 3 are:
3 = 1 + 2
3 = 2 + 1

Explanation: You can break Arrays to Multiply with 3 are:
3 = 1 + 2
3 = 2 + 1

 

Visual Learning Bridge
The Park District has canoes stored in 3 rows. There are 6 canoes in each row. What is the total number of canoes stored?
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 16

What You Show
Find 3 × 6.
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 17

What You Think
3 × 6 is 3 rows of 6. That is 2 sixes plus 1 more six.
2 sixes are 12. 1 six is 6.
12 + 6 = 18
3 × 6 = 18
There are 18 canoes.

Convince Me! Use Structure Suppose there were 7 canoes in each of 3 rows. How can 2 × 7 = 14 help you find the total number of canoes?

Answer:
The total number of canoes = Number of rows of canons + Number of canons in each row
= 3 × 7
= (2 × 7) + (1 × 7)
= 14 + 7
= 21.

Explanation:
Number of rows of canons = 3 or Three.
Number of canons in each row = 7 or Seven.
The total number of canoes = Number of rows of canons + Number of canons in each row
= 3 × 7
= (2 × 7) + (1 × 7)
= 14 + 7
= 21.

Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.2 Apply Properties-3 and 4 as Factors-Visual Learning Bridge

 

 

Another Example !
Find 4 × 7.
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 18

What You Think
4 × 7 is 4 rows of 7.
That is 2 sevens plus 2 sevens. 2 sevens are 14.
14 + 14 = 28
So, 4 × 7 = 28

Apply Properties Multiplication Facts 4

Guided Practice
Do You Understand?
Question 1.
Besides using a 2s fact and doubling it, what is another way to break apart 4 × 7 using facts you already know?
Answer:
Another ways to break 4 × 7 using facts known by me:
4 × 7 = (1 × 7) + (3 × 7)
4 × 7 = (3 × 7) + (1 × 7 )

Explanation:
Another ways to break 4 × 7 using facts known by me:
4 × 7 = (1 × 7) + (3 × 7)
4 × 7 = (3 × 7) + (1 × 7 )

 

 

Question 2.
Selena arranged 3 rows of plants in her garden. She put 9 plants in each row. How many plants did Selena arrange?
Answer:
Number of plants Selena arrange = 27 or Twenty Seven.

Explanation:
Number of Rows of plants = 3 or Three.
Number of plants in each row = 9 or Nine.
Number of plants Selena arrange = Number of Rows of plants × Number of plants in each row
= 3 × 9
= 27 or Twenty Seven.

 

 

 

Do You Know How?
In 3-8, multiply. You may use counters or pictures to help.
Question 3.
3 × 10 = ______
Answer:
3 × 10 = __30__.

Explanation:
3 × 10 = 30.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.2 Apply Properties-3 and 4 as Factors-Do You Know How-3

 

 

 

Question 4.
______ = 5 × 4
Answer:
___20___ = 5 × 4.

Explanation:
5 × 4 = 20.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.2 Apply Properties-3 and 4 as Factors-Do You Know How-4

 

 

Question 5.
______ = 3 × 8
Answer:
__24____ = 3 × 8.

Explanation:
3 × 8 = 24.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.2 Apply Properties-3 and 4 as Factors-Do You Know How-5

 

 

 

 

Question 6.
1 × 4 = ______
Answer:
1 × 4 = __4___.
Explanation:
1 × 4 = ______.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.2 Apply Properties-3 and 4 as Factors-Do You Know How-6

 

 

Question 7.
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 19
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.2 Apply Properties-3 and 4 as Factors-Do You Know How-7..
Explanation:
3 × 7 = 21.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.2 Apply Properties-3 and 4 as Factors-Do You Know How-7

 

Question 8.
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 20
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.2 Apply Properties-3 and 4 as Factors-Do You Know How-8..
Explanation:
10 × 4 = 40.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.2 Apply Properties-3 and 4 as Factors-Do You Know How-8

 

 

 

 

Independent Practice

Leveled Practice In 9-13, multiply. You may use counters or pictures to help.
Question 9.
Find 3 × 4.
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 21
So, 3 × 4 = _______
Answer:
So, 3 × 4 = ___12____.

Explanation:
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 21
3 × 4 = (2 × 4) + (1 × 4)
3 × 4 = 8 + 4
3 × 4 = 12.
So, 3 × 4 = 12.

 

 

 

Question 10.
Find 4 × 9.
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 22
So, 4 × 9 = ________
Answer:
So, 4 × 9 = ____36____.

Explanation:
Envision Math Common Core 3rd Grade Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 22

 

4 × 9 = (2 × 9) + (2 × 9)
4 × 9 = 18 + 18
4 × 9 = 36.
So, 4 × 9 = 36.

Apply Properties Multiplication Facts 6

Question 11.
7 × 3 = _______
Answer:
7 × 3 = ____21___.
Explanation:
7 × 3 = 21.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.2 Apply Properties-3 and 4 as Factors-Independent Practice-11

 

Question 12.
_____ = 4 × 3
Answer:
__12___ = 4 × 3.
Explanation:
__12___ = 4 × 3.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.2 Apply Properties-3 and 4 as Factors-Independent Practice-12

 

 

 

Question 13.
10 × 3 = ______
Answer:
10 × 3 = __30___.
Explanation:
10 × 3 = __30____.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.2 Apply Properties-3 and 4 as Factors-Independent Practice-13

 

 

 

Problem Solving
Question 14.
Make Sense and Persevere James needs to buy supplies for his trail walk. What is the total number of cereal bars James needs to buy? Explain how you used the table to find the answer.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 23
Answer:
Number of Cereal bars James needed = 4 or Four.
The tables is used in knowing the number of packages needed James needed for trial walk supplies.

Explanation:
Number of Cereal bars James needed = 4 or Four.
The tables is used in collecting data of cereal bars. It helps in knowing the number of packages needed James needed for trial walk supplies.

 

Question 15.
How many more apples than juice drinks does James need? Show how you found the answer.
Answer:
Number of more apples than juice drinks does James need = 5 or Five.

Explanation:
Number of Apples James needed = 8 or Eight.
Number of juice drinks James needed =3 or Three.
Number of more apples than juice drinks does James need = Number of juice drinks James needed – Number of Apples James needed
= 8 – 3
= 5 or Five.

 

 

Question 16.
Allison bought 10 packages of energy bars. Each package contains 6 bars. Allison says she has a total of 65 energy bars. Is her answer reasonable? Why or why not?
Answer:
Her answer is not reasonable because the total number of energy bars Allison bought = 60 or Sixty not 65.

Explanation:
Number of packages of energy bars Allison bought = 10 or Ten.
Number of bars each package contains = 6 or Six.
Total number of energy bars = Number of packages of energy bars Allison bought  × Number of bars each package contains
= 10 × 6
= 60 or Sixty.

 

 

Question 17.
Higher Order Thinking What two multiplication facts can help you find 3 × 9? How could you use 3 × 9 to find 9 × 3?
Answer:
First multiplication fact:
3 × 9 = (2 × 9) + (1 × 9)
3 × 9 = 18 + 9
3 × 9 = 27.
Second multiplication fact:
3 × 9 = (1 × 9) + (2 × 9)
3 × 9 = 9 + 18
3 × 9 = 27.
You cannot use 3 × 9  for finding 9 × 3 because number of rows and number of items it , does not match.

Explanation:
First multiplication fact:
3 × 9 = (2 × 9) + (1 × 9)
3 × 9 = 18 + 9
3 × 9 = 27.
So, 3 × 9 = 27.
Second multiplication fact:
3 × 9 = (1 × 9) + (2 × 9)
3 × 9 = 9 + 18
3 × 9 = 27.
So, 3 × 9 = 27.

 

 

Assessment Practice

Question 18.
Bess has 6 boxes of candles. There are 4 candles in each box. Which equation can be used to find the number of candles Bess has?
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 24
A. (3 × 4) + (3 × 4) = ?
B. (6 × 2) + (6 × 6) = ?
C. (6 × 2) + (3 × 2) = ?
D. (2 × 2) + (6 × 6) = ?
Answer:
Total number of candles Bess has = 24 or Twenty Four.
A. (3 × 4) + (3 × 4) = 12 + 12 = 24 can be used to find the number of candles Bess has.

Explanation:
Number of boxes of candles Bess has = 6 or Six.
Number of  candles each box has = 4 or Four.
Total number of candles Bess has = Number of boxes of candles Bess has × Number of  candles each box has
= 6 × 4
= (3 × 4) + (3 × 4)
= 12 + 12
= 24 or Twenty Four.

 

 

Question 19.
Which of the following shows one way to use the Distributive Property to find 9 × 3?
A. (3 × 2) + (3 × 1)
B. (9 × 2) + (9 × 1)
C. (9 × 2) + (3 × 2)
D. (9 + 2) × (9 + 1)
Answer:
B. (9 × 2) + (9 × 1) shows one way to use the Distributive Property to find 9 × 3.

Explanation:
9 × 3 = ??
9 × 3 = (9 × 2) + (9 × 1)
9 × 3 = 18 + 9
9 × 3 = 27.
So, 9 × 3 = 27.

 

 

 

Lesson 3.3 Apply Properties: 6 and 7 as Factors

Solve & Share
Students set up 6 rows of seats for a music concert. They put 6 seats in each row. What is the total number of seats?
I can … make and use models to solve multiplication problems that have 6 and 7 as factors.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 25
Answer:
Total number of seats = 36 or Thirty Six.

Explanation:
Number of rows of seats Students set up for a music concert = 6 or Six.
Number of seats in each row Students set up for a music concert = 6 or Six.
Total number of seats = Number of rows of seats Students set up for a music concert  × Number of seats in each row Students set up for a music concert
= 6 × 6
= 36 or Thirty Six.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties

 

 

Essential Question
How Can You Break Apart Arrays to Multiply?
Answer:
You can break apart Arrays to multiply by using the Distributive Property.

Explanation:
Yes, you can Break Apart Arrays to Multiply. It says that you can break apart a multiplication fact into the sum of two other multiplication facts.

 

 

Visual Learning Bridge
The members of the band march in 6 equal rows. There are 8 band members in each row. How many are in the band?
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 26

What You Show
Find 6 × 8.
Use 5s facts and 1s facts.
Make an array for each multiplication sentence.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 27

What You Think
6 × 8 is 6 rows of 8. That is 5 eights plus 1 more eight.
5 eights are 40.
8 more is 48.
40 + 8 = 48
So, 6 × 8 = 48.
The band has 48 members.

Convince Me! Use Structure Use a 5s fact and a 1s fact to find 6 x 9. Draw two arrays. Explain your drawings.

Another‌ ‌Example‌ ‌!‌ ‌
Find‌ ‌7‌ ‌×‌ ‌8.‌ ‌
Use‌ ‌5s‌ ‌facts‌ ‌and‌ ‌2s‌ ‌facts‌ ‌to‌ ‌help‌ ‌multiply‌ ‌by‌ ‌7.‌ ‌
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 28
7‌ ‌×‌ ‌8‌ ‌equals‌ ‌7‌ ‌rows‌ ‌of‌ ‌8.‌ ‌
That‌ ‌is‌ ‌5‌ ‌eights‌ ‌plus‌ ‌2‌ ‌eights.‌
‌5‌ ‌eights‌ ‌are‌ ‌40.‌ ‌
2‌ ‌eights‌ ‌are‌ ‌16.‌ ‌
40+‌ ‌16‌ ‌ =‌ ‌56‌ ‌
So,‌ ‌7‌ ‌×‌ ‌8‌ ‌=‌ ‌56.‌ ‌

 

Guided‌ ‌Practice‌ ‌

Do‌ ‌You‌ ‌Understand?‌ ‌
Question 1.‌
‌The‌ ‌students‌ ‌who‌ ‌are‌ ‌graduating‌ ‌are‌ ‌ standing‌ ‌in‌ ‌7‌ ‌equal‌ ‌rows.‌ ‌There‌ ‌are‌ ‌9‌ ‌students‌ ‌in‌ ‌each‌ ‌row.‌ ‌How‌ ‌many‌ ‌students‌ ‌are‌ ‌graduating?‌ ‌Use‌ ‌a‌ ‌5s‌ ‌fact‌ ‌and‌ ‌a‌ ‌2s‌ ‌fact.‌ ‌
Answer:
Total number of students‌ ‌are‌ ‌graduating = 63 or Sixty Three.

Explanation:
Number of rows students‌ ‌who‌ ‌are‌ ‌graduating‌ ‌are‌ ‌standing‌ ‌in‌ ‌= 7‌ ‌or Seven.
Number of students in each row = 9 or nine.
Total number of students‌ ‌are‌ ‌graduating = Number of rows students‌ ‌who‌ ‌are‌ ‌graduating‌ ‌are‌ ‌standing‌ ‌in‌ × Number of students in each row
= 7 × 9
= (5 × 9) + ( 2 × 9)
= 45 + 18
= 63 or Sixty Three.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Guided‌ ‌Practice‌ ‌-1

 

 

 

Question 2.‌
Chrissy‌ ‌bakes‌ ‌3‌ ‌cherry‌ ‌pies.‌ ‌She‌ ‌cuts‌  each‌ ‌pie‌ ‌into‌ ‌6‌ ‌slices.‌ ‌How‌ ‌many‌ ‌slices‌ ‌does‌ ‌Chrissy‌ ‌have?‌ ‌
Answer:
Number of slices Chrissy has = 18 or Eighteen.

Explanation:
Number of cherry pies Chrissy‌ ‌bakes‌ ‌= 3‌ ‌or Three.
Number of slices she cuts each pie = 6 or Six.
Number of slices Chrissy has = Number of cherry pies Chrissy‌ ‌bakes‌ × Number of slices she cuts each pie
= 3 × 6
= (2 × 6) + (1 × 6)
= 12 + 6
= 18 or Eighteen.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Guided‌ ‌Practice‌ ‌-2

 

 

 

Do‌ ‌You‌ ‌Know‌ ‌How?‌ ‌
In‌ ‌3-8,‌ ‌multiply.‌ ‌You‌ ‌may‌ ‌draw‌ ‌pictures‌ ‌or‌ ‌use‌ ‌counters‌ ‌to‌ ‌help.‌ ‌
Question 3.‌
‌6‌ ‌×‌ ‌10‌ ‌=‌ ‌________
Answer:
‌6‌ ‌×‌ ‌10‌ ‌=‌ ‌___60_____.

Explanation:
‌6‌ ‌×‌ ‌10‌ ‌=‌ ( 3 ×10) + ( 3 ×10)
‌6‌ ‌×‌ ‌10‌ ‌=‌  30 + 30
‌6‌ ‌×‌ ‌10‌ ‌=‌  60.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Guided‌ ‌Practice‌ ‌-3

 

 

 

Question 4.‌
‌7‌ ×‌ ‌6 =‌ ________
Answer:
‌7‌ ×‌ ‌6 =‌ ___42_____.
Explanation:
‌7‌ ×‌ ‌6 = (6 × 6) + (1 × 6)
‌7‌ ×‌ ‌6 = 36 + 6
‌7‌ ×‌ ‌6 =‌ 42.
So, ‌7‌ ×‌ ‌6 =‌ 42.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Guided‌ ‌Practice‌‌-4

 

 


Question 5.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 29
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Guided‌ ‌Practice‌‌-5..

Explanation:
7 × 7 = (3 ×7) + (4 ×7)
7 × 7 = 21 + 28
7 × 7 = 49.
So, 7 × 7 = 49.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Guided‌ ‌Practice‌‌-5

 

 

Question 6.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 30
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Guided‌ ‌Practice‌‌-6
Explanation:
9 × 7 = (5 × 7) + (4 × 7)
9 × 7 = 35 + 28
9 × 7 = 63.
So, 9 × 7 = 63.

Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Guided‌ ‌Practice‌‌-6..

 

Question 7.‌
‌Find‌ ‌4‌ ‌times‌ ‌7.‌ ‌ ______
Answer:
4 × 7 = 28.

Explanation:
4 × 7 = 28.
4 × 7 = ( 2 × 7) + ( 2 × 7)
4 × 7 =14 + 14
4 × 7 = 28.
So, 4 × 7 = 28.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Guided‌ ‌Practice‌‌-7

 

 

Question 8.‌
‌Multiply‌ ‌6‌ ‌times‌ ‌5.‌ ‌ ________
Answer:
‌Multiply‌ ‌6‌ ‌times‌ ‌5.‌ ‌
=> 6 × 5 = 30.

Explanation:
6 × 5 = (3 × 5) + (3 × 5)
6 × 5 = 15 + 15
6 × 5 = 30.
So, 6 × 5 = 30.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Guided‌ ‌Practice‌‌-8

 

 

Independent‌ ‌Practice‌ ‌
‌In‌ ‌9-16,‌ ‌find‌ ‌the‌ ‌product.‌ ‌You‌ ‌may‌ ‌draw‌ ‌pictures‌ ‌to‌ ‌help.‌
Question 9.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 31
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Independent‌ ‌Practice‌-9

Explanation:
5 × 7 = (3 × 7) + (2 × 7)
5 × 7 = 21 + 14
5 × 7 = 35.
So, 5 × 7 = 35.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Independent‌ ‌Practice‌-9..

 

 

 

Question 10.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 32
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Independent‌ ‌Practice‌-10..

Explanation:
3 × 6 = (2 × 6) + (1 × 6)
3 × 6 = 12 + 6
3 × 6 = 18.
So, 3 × 6 = 18.

Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Independent‌ ‌Practice‌-10

 

 

Question 11.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 33
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Independent‌ ‌Practice‌-11..

Explanation:
7 × 8 = (4 × 8) + (3 × 8)
7 × 8 = 32+ 24
7 × 8 = 56.
So, 7 × 8= 56.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Independent‌ ‌Practice‌-11

 

 

 

Question 12.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 34
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Independent‌ ‌Practice‌-12..

Explanation:
1 × 7 = 7.
So, 1 × 7 = 7.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Independent‌ ‌Practice‌-12

 

 

Question 13.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 35
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Independent‌ ‌Practice‌-13..

Explanation:
10 × 6 = (5 × 6) + (5 × 6)
10 × 6 = 30+ 30
10 × 6 = 60.
So, 10 × 6 = 60.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Independent‌ ‌Practice‌-13

 

 

 

Question 14.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 36
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Independent‌ ‌Practice‌-14..

Explanation:4
4 × 7 = (2 × 7) + (2 × 7)
4 × 7 = 14+ 14
4 × 7 = 28.
So, 4 × 7 = 28.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Independent‌ ‌Practice‌-14

 

 

Question 15.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 37
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Independent‌ ‌Practice‌-15..

Explanation:
7 × 3 = (4 × 3) + (3 × 3)
7 × 3 = 12 + 9
7 × 3 = 21.
So, 7 × 3 = 21.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Independent‌ ‌Practice‌-15

 

 

Question 16.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 38
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Independent‌ ‌Practice‌-16..

Explanation:
8 × 6 = (4 × 6) + (4 × 6)
8 × 6 = 24+ 24
8 × 6 = 48.
So, 8 × 6 = 48.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.3 Apply Properties-Independent‌ ‌Practice‌-16

 

 

 

Problem Solving
Question 17.
The National Toy Train Museum has 5 exhibits for trains. In one of the exhibits, the trains are on 5 tracks. How many trains are on display at that exhibit? Write an equation to solve the problem.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 39
Answer:
Number of trains are on display at that exhibit = 25 or Twenty five.

Explanation:
Number of trains exhibits the National Toy Train Museum has = 5 or Five.
Number of trains each exhibits track the National Toy Train Museum has = 5 or Five.
Number of trains are on display at that exhibit = Number of trains exhibits the National Toy Train Museum has × Number of trains each exhibits track the National Toy Train Museum has
= 5 × 5
= (3 × 5) + (2 × 5)
= 15 + 10
= 25 or Twenty five.

 

Question 18.
Tracy used the flat surface of a cube to draw a plane shape. What plane shape did Tracy draw? How do you know?
Answer:
A cube is comprised of 6 congruent squares. Therefore, the only shape Tracy could have drawn is a square

Explanation:
In geometry a cube is a three- dimensional  solid object bounded by six square faces. So, Tracy could have drawn is a square.

 

 

Question 19.
The dance team lines up in 4 rows of 6 dancers each. How many dancers are on the dance team?
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 40
Answer:
Number of dancers are on the dance team = 24 or Twenty Four.

Explanation:
Number of rows the dance team lines up = 4 or Four.
Number of dancers in each row = 6 or Six.
Number of dancers are on the dance team = Number of rows the dance team lines up × Number of dancers in each row
= 4 × 6
= (2 × 6) + (2 × 6)
= 12 + 12
= 24 or Twenty Four.

 

 

 

Question 20.
Higher Order Thinking
Marge says 7 × 0 is equal to 7 + 0. Is Marge correct? Why or why not?
Answer:
No, Marge is not correct because 7 × 0 = 0 where as  7 + 0 = 7.

Explanation:
Marge says 7 × 0 is equal to 7 + 0.
7 × 0 = 0.
7 + 0 = 7.

 

 

 

Question 21.
Use Structure : There are 7 rows of 10 seats. How can you use the Distributive Property to find the total number of seats?
Answer:
Total number of  seats = Number of rows × Number of seats in each row
= 7 × 10
= (3 × 10) + (4 × 10)
= 30 + 40
= 70 or Seventy.

Explanation:
The distributive property tells us how to solve expressions in the form of a(b + c).  The distributive property is sometimes called the distributive law of multiplication and division.
Number of rows = 7 or Seven.
Number of seats in each row = 10 or Ten.
Total number of  seats = Number of rows × Number of seats in each row
= 7 × 10
= (3 × 10) + (4 × 10)
= 30 + 40
= 70 or Seventy.

 

 

 

Assessment Practice
Question 22.
Select numbers to create a different expression that is equal to 7 × 8.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 41
Answer:
Different expression that is equal to 7 × 8.
8 × 7 = (5 × 7) + (3 × 7)
8 × 7 = 35 + 21
8 × 7 = 56 or Fifty Six.
So, 8 × 7 = 56 or Fifty Six.

Explanation:
7 × 8 = (4 × 8) + (3 × 8)
7 × 8 = 32 + 24
7 × 8 = 56.
So, 7 × 8 = 56 or Fifty Six.

Different expression that is equal to 7 × 8.
8 × 7 = (5 × 7) + (3 × 7)
8 × 7 = 35 + 21
8 × 7 = 56 or Fifty Six.
So, 8 × 7 = 56 or Fifty Six.

 

 

 

Question 23.
Select numbers to create a different expression that is equal to 8 × 6.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 42
Answer:
Different expression that is equal to 8 × 6.
6 × 8 = (5 × 8) + (1 × 8)
6 × 8 = 40 + 8
6 × 8 = 48.
So, 6 × 8 = 48 or Forty Eight.

Explanation:
8 × 6 = (4 × 6) + (4 × 6)
8 × 6 = 24 + 24
8 × 6 = 48 or Forty Eight.
So, 8 × 6 = 48 or Forty Eight.

Different expression that is equal to 8 × 6.
6 × 8 = (5 × 8) + (1 × 8)
6 × 8 = 40 + 8
6 × 8 = 48.
So, 6 × 8 = 48 or Forty Eight.

 

 

 

Lesson 3.4 Apply Properties: 8 as a Factor

Solve & Share
There are 8 rows of prizes. There are 6 prizes in each row. How many prizes are there?
I can … use known facts and properties to multiply by 8.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 43

Look Back! Tell how you can use 2s, 3s, or 4s facts to solve the problem.

Essential Question
How Can You Use Doubles on to Multiply with 8?
Answer:
Doubles on to Multiply with 8:
8 = ( 2 × 2 × 2)
Explanation:
Doubles multiplication on 8:
8 = 4 × 2 = 2 × 2 × 2.

Visual Learning Bridge
At a school fun fair, students try to toss a table tennis ball into a bowl. There are 8 rows of bowls. There are 8 bowls in each row. How many bowls are there?
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 44

One Way
Use 2s facts to find 8 × 8.
8 × 8 equals 4 groups of 2 eights.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 45
16 + 16 + 16 + 16 = 64
So, 8 × 8 = 64.

Another Way
Double a 4s fact to find 8 × 8.
8 × 8 equals 4 eights plus 4 eights.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 46
32 + 32 = 64
So, 8 × 8 = 64.

Convince Me! Use Structure How does knowing 5 × 8 = 40 help you find 8 × 8?

Guided Practice

Do You Understand?
Question 1.
Multiply 8 times 3. Write and solve a multiplication equation.
Answer:
Multiply 8 times 3.
=> 8 × 3 = (5 × 3) + (3 × 3)
=> 8 × 3 = 15 + 9
=> 8 × 3 = 24 or Twenty Four.

Explanation:
Multiply 8 times 3.
=> 8 × 3 = (5 × 3) + (3 × 3)
=> 8 × 3 = 15 + 9
=> 8 × 3 = 24 or Twenty Four.
So,  8 × 3 = 24 or Twenty Four.

 

 

 

Question 2.
Multiply 5 times 8. Write and solve a multiplication equation.
Answer:
Multiply 5 times 8.
=> 5 × 8 = (3 × 8) + (2 × 8)
=> 5 × 8 = 24 + 16
=> 5 × 8 = 40 or Forty.

Explanation:
Multiply 5 times 8.
=> 5 × 8 = (3 × 8) + (2 × 8)
=> 5 × 8 = 24 + 16
=> 5 × 8 = 40 or Forty.
So, 5 × 8 = 40 or Forty.

 

 

Question 3.
Multiply 8 times 1. Write and solve a multiplication equation.
Answer:
Multiply 8 times 1.
=> 8 × 1 = (4 × 1) + (4 × 1)
=> 8 × 1= 4 + 4
=> 8 × 1 = 8 or Eight.

Explanation:
Multiply 8 times 1.
=> 8 × 1 = (4 × 1) + (4 × 1)
=> 8 × 1= 4 + 4
=> 8 × 1 = 8 or Eight.
So, 8 × 1 = 8 or Eight.

 


Do You Know How?

In 4-9, multiply. You may draw pictures or use counters to help.
Question 4.
8 × 7 = ______
Answer:
8 × 7 = __56____.

Explanation:
8 × 7 = ( 4 × 7) + ( 4 × 7)
8 × 7 = 28 + 28
8 × 7 = 56 or Fifty Six.
So, 8 × 7 = 56 or Fifty Six.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-4

 

 

 

Question 5.
______ = 8 × 4
Answer:
__32____ = 8 × 4.

Explanation:
8 × 4 = (4 × 4) + (4 × 4)
8 × 4 = 16 + 16
8 × 4 = 32 or Thirty Two.
So, 8 × 4 = 32 or Thirty Two.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-5

 

 

Question 6.
6 × 8 = ________
Answer:
6 × 8 = ___48_____.

Explanation:
6 × 8 = (3 × 8) + (3 × 8)
6 × 8 = 24 + 24
6 × 8 = 48 or Forty Eight.
So, 6 × 8 = 48 or Forty Eight.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-6

 

 

 

Question 7.
______ = 10 × 8
Answer:
__80____ = 10 × 8.

Explanation:
10 × 8 = (5× 8) + (5 × 8)
10 × 8 = 40 + 40
10 × 8 = 80.
So, 10 × 8 = 80 or Eighty.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-7

 

 

 

Question 8.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 47
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-8

Explanation:
9 × 8 =  (5 × 8) + (4 × 8)
9 × 8 =  40 + 32
9 × 8 = 72.
So, 9 × 8 = 72 or Seventy Two.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-8...

 

 

 

Question 9.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 48
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-9

Explanation:
8 × 3 =  (5 × 3) + (3 × 3)
8 × 3 = 15+ 9
8 × 3 = 24.
So, 8 × 3 = 24 or Twenty Four.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-9...

 

 

Independent Practice

In 10-23, find the product. You may draw pictures to help.
Question 10.
8 × 4 = ________
Answer:
8 × 4 = ___32_____.

Explanation:
8 × 4 = (4 × 4) +  (4 × 4)
8 × 4 = 16+ 16
8 × 4 = 32.
So, 8 × 4 = 32 or Thirty Two.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-10

 

 

Question 11.
1 × 8 = _______
Answer:
1 × 8 = ___8____.

Explanation:
1 × 8 = 8.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-11

 

 

Question 12.
_____ = 2 × 8
Answer:
__16___ = 2 × 8.

Explanation:
2 × 8 = (1 × 8) + (1 × 8)
2 × 8 = 8 + 8
2 × 8 = 16.
So, 2 × 8 = 16 or Sixteen.

Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-12

 

 

Question 13.
___ = 5 × 8
Answer:
__40__ = 5 × 8.

Explanation:
5 × 8 = (3 × 8) + (2 × 8)
5 × 8 = 24 + 16
5 × 8 = 40.
So, 5 × 8 = 40 or Forty.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-13

 

 

 

Question 14.
8 × 2 = ________
Answer:
8 × 2 = ___16_____.

Explanation:
8 × 2 = (4 × 2) + (4 × 2)
8 × 2 = 8 + 8
8 × 2 = 16.
So, 8 × 2 = 16 or Sixteen.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-14

 

 

Question 15.
8 × 6 = _______
Answer:
8 × 6 = __48_____.

Explanation:
8 × 6 = (4 × 6) + (4 × 6 )
8 × 6 = 24 + 24
8 × 6 = 48.
So, 8 × 6 = 48 or Forty Eight.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-15

 

 

Question 16.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 49
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-16..

Explanation:
8 ×8 = (4 × 8) + (4 × 8)
8 ×8 = 32 + 32
8 ×8 = 64.
So, 8 ×8 = 64 or Sixty Four.

Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-16

 

 

Question 17.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 50
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-17..

Explanation:
8 × 5 = (5 × 5) + (3 × 5)
8 × 5 = 25 + 15
8 × 5 = 40.
So, 8 × 5 = 40 or Forty.

Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-17

 

 

 

Question 18.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 51
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-18

Explanation:
0 × 8 = 0 or Zero.

 

 

Question 19.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 52
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-19

Explanation:
4 × 8 = (2 × 8) + (2 × 8)
4 × 8 = 16 + 16
4 × 8 = 32.
So, 8 × 5 = 32 or Thirty Two..

Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-19..

 

Question 20.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 53
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-20

Explanation:
10 × 8 = (5 × 8) + (5 × 8)
10 × 8= 40+ 40
10 × 8 = 80.
So, 10 × 8 = 80 or Eighty.

Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-20..

 

 

Question 21.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 54
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-21..

Explanation:
8 × 1 = (5 × 1) + (3 × 1)
8 × 1 = 5 + 3
8 × 1 = 8.
So, 8 × 1 = 8 or Eight.

Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-21

 

Question 22.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 55
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-22

Explanation:
3 × 8 = (2 × 8) + (1 × 8)
3 × 8 = 16+ 8
3 × 8 = 24.
So, 3 × 8 = 24 or Twenty Four.

Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-22..

 

 

Question 23.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 56
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-23

Explanation:
7 × 8 = (2 × 8) + (5 × 8)
7 × 8 = 16+ 40
7 × 8 = 56.
So, 7 × 8 = 56 or Fifty Six.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-23..

 

 

 

Problem Solving

Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 57
Question 24.
Use Structure Ming bought 8 belts for gifts. How much money did Ming spend? Show how you can use a 4s fact to find the answer.
Answer:
Amount spent by Ming in buying belts = $72.

Explanation:
Number of belts bought by Ming for gifts = 8 or Eight.
Cost of each belt bought by Ming for gifts = $9
Amount spent by Ming in buying belts =  Number of belts bought by Ming for gifts × Cost of each belt bought by Ming for gifts
= 8 × $9
= (4 × $9) + (4 × $9) { use a 4s fact }
= $36 + $36
= $72.

 

 

Question 25.
Willa bought a shirt and a sweater. She had $14 left. How much money did Willa start with? How do you know?
Answer:
Amount of money Willa stated with = $75.

Explanation:
Cost of the shirt = $23
Cost of a sweater = $38
Cost of the money left with her = $14
Amount of money Willa stated with = Cost of the shirt + Cost of a sweater + Cost of the money left with her
= $23 + $38 + $14
= $61 + $14
= $75.

 

 

 

Question 26.
Mr. Garner spends $52 on groceries and $24 on gas. How much does Mr. Garner spend? Write an equation and solve.
Envision Math Common Core 3rd Grade Answers Topic 3 Apply Propertie Multiplication Facts for 3, 4, 6, 7, 8 58
Answer:
Total amount spent by Mr. Garner = $76.

Explanation:
Amount spent on groceries by Mr. Garner = $52
Amount spent on gas by Mr. Garner = $24
Total amount spent by Mr. Garner = Amount spent on groceries by Mr. Garner + Amount spent on gas by Mr. Garner
= $52 + $24
= $76.

Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.4 Apply Properties-8 as a Factor-Essential Question-Guided Practice-26

 

 

 

Question 27.
Algebra Mischa bought 7 boxes of orange tiles. There are 8 tiles in each box. How many tiles did Mischa buy? Write an equation and solve. Use ? to represent the unknown quantity of tiles.
Answer:
Number of tiles Mischa buys = 56 or Fifty Six.

Explanation:
Number of tiles Mischa buys = ??
Amount of orange tiles boxes Algebra Mischa bought = 7 or Seven.
Number of orange tiles in each box = 8 or Eight.
Number of tiles Mischa buys = Amount of orange tiles boxes Algebra Mischa bought × Number of orange tiles in each box
= 7 × 8
= 56 or Fifty Six.

 

 

Question 28.
Aaron bought 6 packs of sports cards. There are 7 cards in each pack. How many sports cards did Aaron buy in all? Use properties to solve the problem.
Answer:
Number of sports cards did Aaron buy in all = 42 or Forty Two.

Explanation:
Number of packs of sports cards Aaron bought = 6 or Six.
Number of cards in each pack = 7 or Seven.
Number of sports cards did Aaron buy in all = Number of packs of sports cards Aaron bought × Number of cards in each pack
= 6 × 7
= (3 × 7) + (3 × 7)
= 21 + 21
= 42 or Forty Two.

 

 

 

Question 29.
Higher Order Thinking
Sophie says, “To find 8 × 8, I can find 8 × (4 + 4).” Do you agree? Explain.
Answer:
Yes, agree with what Sophie says “To find 8 × 8, I can find 8 × (4 + 4).” Because both are the same.

Explanation:
Sophie says, “To find 8 × 8, I can find 8 × (4 + 4).”
8 × 8 = 64
8 × (4 + 4)
=> 8 × 8 = 64.

 

Assessment Practice
Question 30.
Ms. Vero has 8 boxes of crayons. Each box has 8 crayons. Select all the correct equations that could show how many crayons Ms. Vero has.
☐ 8 × 8 = 64
☐ 4 × 8 = 32
☐ 2 × 8 = 15
☐ 3 × 8 = 24
☐ 6 × 8 = 84
Answer:
Total number of crayons Ms. Vero has =  8 × 8 = 64 or Sixty Four.
☐ 8 × 8 = 64 is the expression that can be used to find 8 × 7.

Explanation:
Number of crayons boxes Ms. Vero has = 8 or Eight.
Number of crayons in each box = 8 or Eight.
Total number of crayons Ms. Vero has = Number of crayons boxes Ms. Vero has  × Number of crayons in each box
= 8 × 8
= 64 or Sixty Four.

 

 

 

Question 31.
Select all the expressions that can be used to find 8 × 7.
☐ (4 × 7) + (4 × 7)
☐ (4 × 6) + (4 × 1)
☐ (4 × 5) + (4 × 2)
☐ (8 × 5) + (8 × 2)
☐ (2 × 7) + (2 × 7) + (2 × 7) + (2 × 7)
Answer:
8 × 7 = (4 × 7) + (4 × 7)
8 × 7 = (2 × 7) + (2 × 7) + (2 × 7) + (2 × 7) are the expressions that can be used to find 8 × 7.
So, 8 × 7 = 56 or Fifty Six.

Explanation:
8 × 7 = (4 × 7) + (4 × 7)
8 × 7 = 28 + 28
8 × 7 = 56 or Fifty Six.
8 × 7 = (2 × 7) + (2 × 7) + (2 × 7) + (2 × 7)
8 × 7 = 14 + 14 + 14 + 14
8 × 7 = 28 + 14 + 14
8 × 7 = 42 + 14
8 × 7 = 56 or Fifty Six.

 

Lesson 3.5 Practice Multiplication Facts
Solve & Share
Jermaine has 7 coolers. Each cooler contains 8 bottles of sports drink. How many bottles of sports drink does Jermaine have in all?
I can … use strategies and tools to represent and solve multiplication facts.
Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 59

Look Back! Jermaine now has 8 coolers with 7 bottles of sports drink in each cooler. Does that change the total number of bottles of sports drink that Jermaine has? Explain why or why not.

Answer:
Equation:1:
Number of coolers Jermaine has = 7 or Seven.
Number of bottles of sports drink in each cooler = 8 or Eight.
Total number of bottles she has = Number of coolers Jermaine has  × Number of bottles of sports drink in each cooler
= 7 × 8
= 56 or Fifty Six.

Equation:2:
Number of coolers Jermaine has = 8 or Eight.
Number of bottles of sports drink in each cooler = 7 or Seven.
Total number of bottles she has = Number of coolers Jermaine has  × Number of bottles of sports drink in each cooler
= 8 × 7
= 56 or Fifty Six.

 

 

Essentials Question
How Do You Use Strategies on to Multiply?
Answer:
We use Strategies on to Multiply accordingly to the process which it is required to get the correct solution.

Explanation:
We use Strategies on to Multiply accordingly to the process which it is required to get the correct solution. It is an important foundational strategy that leads to more efficient strategies (and helps with conceptual understanding).

 

 

Visual Learning Bridge
Justin and Dolores made a dragon float for a parade. They connected 9 equal sections to make the dragon’s body. What is the total length of the dragon’s body in feet?
Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 60

One Way
Draw a picture to find 9 × 3.
9 × 3 means 9 groups of 3. Combine the groups to find the product.
Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 61
9 × 3 = 27
The dragon’s body is 27 feet long.

Another Way
Use known facts to find 9 × 3.
Use 4s facts and 5s facts to help.
Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 62
12 + 15 = 27
The dragon’s body is 27 feet long.

Convince Me! Make Sense and Persevere What two other facts can you use to find 9 × 3? Explain.

Answer:
The two other facts can you use to find 9 × 3 are:
First way:
9 × 3 = (6 × 3) + (3 × 3)
9 × 3 = 18 + 9
9 × 3 = 27 or Twenty Seven.
Second way:
9 × 3 = (8 × 3) + (1 × 3)
9 × 3 = 24 + 3
9 × 3 = 27 or Twenty Seven.

Explanation:
Two other facts can you use to find 9 × 3 are:
First way:
9 × 3 = (6 × 3) + (3 × 3)
9 × 3 = 18 + 9
9 × 3 = 27 or Twenty Seven.
Second way:
9 × 3 = (8 × 3) + (1 × 3)
9 × 3 = 24 + 3
9 × 3 = 27 or Twenty Seven.

 

Guided Practice
Do You Understand?
Question 1.
What known facts can you use to find 7 × 5?
Answer:
Known facts which can be used to find 7 × 5 are 3s, 5s, 2s, 4s.

Explanation:
7 × 5 = (5 × 5) + ( 2 × 5)
7 × 5 = 25 + 10
7 × 5 = 35 or Thirty Five.

7 × 5 = (4 × 5) + ( 3 × 5)
7 × 5 = 20 + 15
7 × 5 = 35 or Thirty Five.

 

Question 2.
To find 8 × 6, how does knowing 6 × 6 = 36 help you?
Answer:
8 × 6 = ( 6 × 6) + ( 2 × 6) = 36 + 12 = 48 or Forty Eight.

Explanation:
8 × 6 = ( 6 × 6) + ( 2 × 6)
8 × 6 = 36 + 12
8 × 6 = 48 or Forty Eight.

 

 

Do You Know How?
In 3-8, multiply.
Question 3.
3 × 7 = _______
Answer:
3 × 7 = __21___.

Explanation:
3 × 7 = (1 × 7) + (2 × 7)
3 × 7 = 7 + 14
3 × 7 = 21 or Twenty One.

 

Question 4.
______ = 6 × 5
Answer:
__30____ = 6 × 5

Explanation:
6 × 5 = (3 × 5)  + (3 × 5)
6 × 5 = 15 + 15
6 × 5 = 30 or Thirty.

 

Question 5.
9 × 4 = _______
Answer:
9 × 4 = ___36____.

Explanation:
9 × 4 = (5 × 4) + (4 × 4)
9 × 4 = 20 + 16
9 × 4 = 36 or Thirty Six.

 

Question 6.
____= 3 × 0
Answer:
__0__= 3 × 0.

Explanation:
3 × 0 = 0 or Zero.

 

Question 7.
Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 63
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.5 Practice Multiplication Facts-Guided Practice-7

Explanation:
1 × 7 = 7 or Seven.

 

Question 8.
Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 64
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.5 Practice Multiplication Facts-Guided Practice-8

Explanation:
10 × 8 = (5× 8) + (5× 8)
10 × 8 = 40 + 40
10 × 8 = 80 or Eighty.

 

Independent Practice
In 9-25, use known facts and strategies to find the product.
Question 9.
7 × 7 = _______
Answer:
7 × 7 = ___49____.

Explanation:
7 × 7 = (4 × 7) + (3 × 7)
7 × 7 = 28 + 21
7 × 7 = 49 or Forty Nine.

 

Question 10.
8 × 2 = _______
Answer:
8 × 2 = __16_____.

Explanation:
8 × 2 = (6 × 2) + (2 × 2)
8 × 2 = 12 + 4
8 × 2 = 16 or Sixteen.

 

Question 11.
3 × 10 = ______
Answer:
3 × 10 = __30____.

Explanation:
3 × 10 = (2 × 10 ) + (1 × 10 )
3 × 10 = 20 + 10
3 × 10 = 30 or Thirty.

 

Question 12.
_____ = 8 × 9
Answer:
__72___ = 8 × 9.

Explanation:
8 × 9 = (6 × 9) + (2 × 9)
8 × 9 = 54 + 18
8 × 9 = 72 or Seventy Two.

 

Question 13.
_____ = 4 × 6
Answer:
__24___ = 4 × 6.

Explanation:
4 × 6 = (2 × 6) + (2 × 6)
4 × 6 = 12 + 12
4 × 6 = 24 or Twenty Four.

 

Question 14.
_____ = 4 × 4
Answer:
__16___ = 4 × 4.

Explanation:
4 × 4 = (2 × 4) + (2 × 4)
4 × 4 = 8 + 8
4 × 4 = 16 or sixteen.

 

Question 15.
Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 65
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.5 Practice Multiplication Facts-Guided Practice-15

Explanation:
10 × 7 = (6× 7) + (4× 7)
10 × 7 = 42 + 28
10 × 7 = 70 or Seventy.

 

Question 16.
Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 66
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.5 Practice Multiplication Facts-Guided Practice-16

Explanation:
2 × 6 = (1 × 6) + (1 × 6)
2 × 6 = 6 + 6
2 × 6 = 12 or Twelve.

 

Question 17.
Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 67
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.5 Practice Multiplication Facts-Guided Practice-17

Explanation:
1 × 3 = 3 or Three.

 

Question 18.
Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 68
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.5 Practice Multiplication Facts-Guided Practice-18

Explanation:
2 × 7 = (1 × 7) + (1 × 7)
2 × 7 = 7 + 7
2 × 7 = 14 or Fourteen.

 

Question 19.
Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 69
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.5 Practice Multiplication Facts-Guided Practice-19
Explanation:
8 × 0 = 0 or zero.

 

Question 20.
Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 70
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.5 Practice Multiplication Facts-Guided Practice-20

Explanation:
10 × 6 = (5 × 6) + (5 × 6)
10 × 6 = 30 + 30
10 × 6 = 60 or Sixty.

 

Question 21.
Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 71
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.5 Practice Multiplication Facts-Guided Practice-21

Explanation:
4 × 7 = (2 × 7) + (2 × 7)
4 × 7 =  14 + 14
4 × 7 = 28 or Twenty Eight.

 

Question 22.
Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 72
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.5 Practice Multiplication Facts-Guided Practice-22

Explanation:
8 × 9 = (3 × 9) + (5 × 9)
8 × 9 = 27 + 45
8 × 9 = 72 or Seventy Two.

 

Question 23.
What is 6 × 9?
Answer:
6 × 9 = 54 or Fifty Four.

Explanation:
6 × 9 = (3 × 9) + (3 × 9)
6 × 9 = 27 + 27
6 × 9 = 54 or Fifty Four.

 

Question 24.
What is 7 × 2?
Answer:
7 × 2 = 14 or Fourteen.

Explanation:
7 × 2 = (4 × 2) + (3 × 2)
7 × 2 = 8 + 6
7 × 2 = 14 or Fourteen.

 

Question 25.
What is 8 × 1?
Answer:
8 × 1 = 8 or Eight.

Explanation:
8 × 1 = (4 × 1) + (4 × 1)
8 × 1 = 4 + 4
8 × 1 = 8 or Eight.

 

Problem Solving
Question 26.
Reasoning Mr. Ling walks 5 miles each day. How many total miles does he walk in one week? Explain.
Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 73
Answer:
Total number of miles he walks in one week = 35 or Thirty Five.

Explanation:
Number of miles each day Mr. Ling walks = 5 or Five.
Number of days in a week = 7 or Seven.
Total number of miles he walks in one week = Number of miles each day Mr. Ling walks × Number of days in a week
= 5 × 7
= (3 × 7) + (2 × 7)
= 21 + 14
= 35 or Thirty Five.

 

Question 27.
David wants to buy new shoes and a jersey. The shoes cost $56. The jersey costs $42. How much money does David need to buy both items?
Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 74
Answer:
Amount of money David needs to buy both items = $98.

Explanation:
Cost of shoes David buys = $56
Cost of jersey David buys = $42
Amount of money David needs to buy both items = Cost of shoes David buys + Cost of jersey David buys
= $56 + $42
= $98.
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.5 Practice Multiplication Facts-Guided Practice-27

Question 28.
Ms. Wilson drank three 8-ounce glasses of tea before lunch. Then she drank three 8-ounce glasses of water before dinner. How many ounces of liquid did she drink in all? Write an equation to help solve.
Answer:
Number of 8-ounces of liquid did she drink in all = Number of 8-ounce glasses of tea before lunch Ms. Wilson drank  + Number of 8-ounce glasses of water before dinner Ms. Wilson drank
= 3 + 3 = 6 or Six.

Explanation:
Number of 8-ounce glasses of tea before lunch Ms. Wilson drank = 3 or Three.
Number of 8-ounce glasses of water before dinner Ms. Wilson drank = 3 or Three.
Number of 8-ounces of liquid did she drink in all = Number of 8-ounce glasses of tea before lunch Ms. Wilson drank  + Number of 8-ounce glasses of water before dinner Ms. Wilson drank
= 3 + 3
= 6 or Six.

 

 

 

Question 29.
Higher Order Thinking
Show how you can use known facts to find 4 × 11. Explain how you chose the known facts.
Answer:
In this problem, 4 × 11 represents multiplication, or finding the product of the two numbers. In this case, this problem represents four multiplied eleven times, or adding the number four eleven times. So the number four added to the number four a total of eleven times is Forty-four.

Explanation:
4 × 11 = (2 × 11) + (2 × 11)
4 × 11 = 22 + 22
4 × 11 = 44 or Forty Four.
So, 4 × 11 = 44.

 

 

Question 30.
Mr. Evans needs to assign 32 students into 8 equal groups. He says, “I can use repeated subtraction. Because I subtract 3 times, each group has 3 students.” Do you agree with Mr. Evans? Explain why or why not.
32 – 16 = 16
16 – 8 = 8
8 – 8 = 0
Answer:
No, I does not agree with Mr. Evans because dividing 32 students equally into 8 groups means in each group four students will be.
Number of members in each group =  32 ÷ 8 = 4 or Four.

Explanation:
Number of students needs to assign = 32.
Number of equal groups to be assigned = 8.
Number of members in each group = Number of students needs to assign ÷ Number of equal groups to be assigned
= 32 ÷ 8
= 4 or Four.

 

 

Assessment Practice
Question 31.
Select the possible ways to display 20 counters in equal groups.
☐ 2 groups of 10
☐ 4 groups of 5
☐ 5 groups of 4
☐ 4 groups of 6
☐ 10 groups of 2
Answer:
The possible ways to display 20 counters in equal groups.
☐ 2 groups of 10.
☐ 4 groups of 5.
☐ 5 groups of 4.
☐ 10 groups of 2.

Explanation:
The possible ways to display 20 counters in equal groups.
☐ 2 groups of 10 = 2 × 10 = 20.
☐ 4 groups of 5 = 4 × 5 = 20.
☐ 5 groups of 4 = 5 × 4 = 20.
☐ 10 groups of 2 = 10 × 2 = 20.

 

Question 32.
Select the possible ways to display 24 counters in an array.
☐ 8 rows of 4
☐ 3 rows of 6
☐ 6 rows of 4
☐ 3 rows of 8
☐ 2 rows of 9
Answer:
The possible ways to display 24 counters in an array:
☐ 3 rows of 6
☐ 6 rows of 4
☐ 3 rows of 8

Explanation:
The possible ways to display 24 counters in an array:
☐ 3 rows of 6 = 3 × 6 = 24.
☐ 6 rows of 4 = 6 × 4 = 24.
☐ 3 rows of 8 = 3 × 8 = 24.

 

 

Lesson 3.6 The Associative Property: Multiply with 3 Factors
Solve & Share

Gina has 2 quilts. Each quilt has 5 rows with 3 squares in each row. How many squares are in both quilts? Solve this problem any way you choose. Then find another way to solve the problem.
I can … multiply 3 factors in any order to find a product.
Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 75
You can make sense and persevere in solving problems. You can solve this problem in more than one way.

Look Back! Did you get a different answer when you solved the problem a different way? Explain why or why not.

Answer:
Number of squares in both quilts = 30 or Thirty.

Explanation:
Number of quilts Gina has = 2 or Two.
Number of rows each quilt has = 5 or Five.
Number of  squares in each row = 3 or Three.
Number of squares in both quilts = Number of quilts Gina has × ( Number of rows each quilt has × Number of  squares in each row )
= 2 × ( 5 × 3)
= 2 × 15
= 30 or Thirty.

 

Essential Question
How Can You Multiply 3 Numbers?
Answer:
Multiply the first number by the second number.
Multiply the product of the first multiplication by the third number.

Explanation:
How to multiply three numbers:
Multiply the first number by the second number.
Multiply the product of the first multiplication by the third number.

 

Visual Learning Bridge
Drew is joining 3 sections of a quilt. Each section has 2 rows with 4 squares in each row. How many squares are in these 3 sections? Find 3 × 2 × 4.
You can multiply to find the total for an array
Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 76

One Way
Find 3 × 2 first.
Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 77
6 rows, 4 squares in each row
There are 24 squares in all.

Another Way
Find 2 × 4 first.
Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 78
3 sections, 8 squares in each section
There are 24 squares in Drew’s quilt.

Convince Me! Generalize Use the Associative Property of Multiplication to show two different ways to find c Did you get the same answer both ways? What can you generalize?

Answer:
First Way:
5 × 2 × 3 = 5 × (2 × 3) =  5 × 6 =  30 or Thirty..
Second Way:
5 × 2 × 3 = (5 × 2) × 3 = 10 × 3 = 30 or Thirty.
It is generalize that the result for both ways, the answers are same.

Explanation:
First Way:
5 × 2 × 3 = 5 × (2 × 3)
5 × 2 × 3 = 5 × 6
5 × 2 × 3 = 30 or Thirty..
Second Way:
5 × 2 × 3 = (5 × 2) × 3
5 × 2 × 3 = 10 × 3
5 × 2 × 3 = 30 or Thirty.

 

Guided Practice
Do You Understand?
Question 1.
Sarah has 4 pages of stickers in an album. Each page has 3 rows with 2 stickers in each row. How many stickers are in Sarah’s album? You may use objects to help.
Answer:
Number of stickers are in Sarah’s album = 24 or Twenty Four.

Explanation:
Number of pages of stickers in an album Sarah has = 4 or Four.
Number of rows each page has = 3 or Three.
Number of stickers in each row = 2 or Two.
Number of stickers are in Sarah’s album = Number of pages of stickers in an album Sarah has × Number of rows each page has × Number of stickers in each row
= (4 × 3)× 2
= 12 ×  2
= 24 or Twenty Four.

 

Question 2.
Billy concludes the product of (2 × 3) × 5 is not equal to the product of 2 × (3 × 5). Is Billy correct? Explain.
Answer:
Well, Billy concludes the product of (2 × 3) × 5 is not equal to the product of 2 × (3 × 5) is not correct because both the answers are same.

Explanation:
(2 × 3) × 5 = 6 × 5 = 30 or Thirty.
2 × (3 × 5) = 2 × 15 = 30 or Thirty.

 

 

Do You Know How?
In 3-6, use the Associative Property of Multiplication to find the missing number. You may use objects or draw a picture to help.
Question 3.
2 × (4 × 2) = (2 × 4) × ______
Answer:
2 × (4 × 2) = (2 × 4) × ______
2 × (4 × 2) = (2 × 4) × __2____
2 × 8 = 8 × 2
16.

Explanation:
2 × (4 × 2) = (2 × 4) × ?
2 × (4 × 2) = (2 × 4) × 2
2 × 8 = 8 × 2
16 = 16 or Sixteen.

 

 

Question 4.
(3 × 4) × 3 = 3 × (_____ × 3)
Answer:
(3 × 4) × 3 = 3 × (_____ × 3)
(3 × 4) × 3 = 3 × (__4___ × 3)
12× 3 = 3 × 12
36 or Thirty Six.

Explanation:
(3 × 4) × 3 = 3 × (?× 3)
(3 × 4) × 3 = 3 × (4× 3)
12× 3 = 3 × 12
36= 36 or Thirty Six.

 

Question 5.
2 × (2 × 3) = (2 × 2) × _______
Answer:
2 × (2 × 3) = (2 × 2) × ?
2 × (2 × 3) = (2 × 2) × ___3____
2 × 6 = 4 × 3
12 or Twelve.

Explanation:
2 × (2 × 3) = (2 × 2) × _______.
2 × (2 × 3) = (2 × 2) × ?.
2 × (2 × 3) = (2 × 2) × 3.
2 × 6 = 4 × 3
12 = 12 or Twelve.

 

 

Question 6.
(3 × 2) × 4 = ______ × (2 × 4)
Answer:
(3 × 2) × 4 = ______ × (2 × 4)
(3 × 2) × 4 = __3____ × (2 × 4)
6 × 4 = 3 × 8
24 or Twenty Four.

Explanation:
(3 × 2) × 4 = ______ × (2 × 4)
(3 × 2) × 4 = ? × (2 × 4)
(3 × 2) × 4 = 3 × (2 × 4)
6 × 4 = 3 × 8
24 = 24 or Twenty Four.

 

 

 

Independent Practice
In 7-12, use the Associative Property of Multiplication to find the missing number. You may use objects or draw a picture to help.
Question 7.
8 × (3 × 6) = (8 × 3) × ______
Answer:
8 × (3 × 6) = (8 × 3) × ______
8 × (3 × 6) = (8 × 3) × __6____.
8 × 18 = 24 × 6
144 or One hundred and Forty Four.

Explanation:
8 × (3 × 6) = (8 × 3) × ______.
8 × (3 × 6) = (8 × 3) × ?
8 × (3 × 6) = (8 × 3) × 6
8 × 18 = 24 × 6
144 = 144 or One hundred and Forty Four.

 

Question 8.
5 × (6 × 9) = (5 × 6) × _______
Answer:
5 × (6 × 9) = (5 × 6) × _______
5 × (6 × 9) = (5 × 6) × ?
5 × 54 = 30 × 9
270 = 270 or Two hundred and seventy.

Explanation:
5 × (6 × 9) = (5 × 6) × _______.
5 × (6 × 9) = (5 × 6) × ?
5 × (6 × 9) = (5 × 6) × 9
5 × 54 = 30 × 9
270 = 270 or Two hundred and seventy.

 

Question 9.
5 × (7 × 2) = (5 × 7) × _______
Answer:
5 × (7 × 2) = (5 × 7) × _______
5 × (7 × 2) = (5 × 7) × ?
5 × 14 = 35 × 2
70 or 70 or Seventy.

Explanation:
5 × (7 × 2) = (5 × 7) × _______.
5 × (7 × 2) = (5 × 7) × ?
5 × (7 × 2) = (5 × 7) × 2
5 × 14 = 35 × 2
70 or 70 or Seventy.

 

Question 10.
5 × (2 × 9) = (5 × _____) × 9
Answer:
5 × (2 × 9) = (5 × _____) × 9
5 × (2 × 9) = (5 × ?) × 9
5 × (2 × 9) = (5 × 2) × 9
5 × 18 =  10 × 9
90 = 90 or Ninety.

Explanation:
5 × (2 × 9) = (5 × _____) × 9
5 × (2 × 9) = (5 × ?) × 9
5 × (2 × 9) = (5 × 2) × 9
5 × 18 =  10 × 9
90 = 90 or Ninety.

 

 

 

Question 11.
3 × (2 × 5) = (3 × 2) × ______
Answer:
3 × (2 × 5) = (3 × 2) × ?
3 × (2 × 5) = (3 × 2) × 5
3 × 10 = 6 × 5
30 = 30 or Thirty.

Explanation:
3 × (2 × 5) = (3 × 2) × ______
3 × (2 × 5) = (3 × 2) × ?
3 × (2 × 5) = (3 × 2) × 5
3 × 10 = 6 × 5
30 = 30 or Thirty.

 

 

 

Question 12.
4 × (2 × 2) = (4 × _____) × 2
Answer:
4 × (2 × 2) = (4 × _____) × 2
4 × (2 × 2) = (4 × ?) × 2
4 × (2 × 2) = (4 × 2) × 2
4 × 4 = 8 × 2
16 = 16 or Sixteen.

Explanation:
4 × (2 × 2) = (4 × _____) × 2
4 × (2 × 2) = (4 × ?) × 2
4 × (2 × 2) = (4 × 2) × 2
4 × 4 = 8 × 2
16 = 16 or Sixteen.

 

 

 

In 13-18, use the Associative Property of Multiplication to find the product. You may use objects or draw a picture to help.
Question 13.
2 × 3 × 2 = ______
Answer:
2 × 3 × 2 = __12 or Twelve____.

Explanation:
2 × 3 × 2 = ??
= (2 × 3) × 2
= 6 × 2
= 12 or Twelve.

 

 

 

Question 14.
3 × 6 × 2 = ______
Answer:
3 × 6 × 2 = ______
Explanation:
3 × 6 × 2 = ??
3 × 6 × 2
= (3 × 6) × 2
= 18 × 2
= 36 or Thirty Six.

 

 

 

Question 15.
2 × 6 × 2 = ______
Answer:
2 × 6 × 2 = __24 or Twenty Four____.

Explanation:
2 × 6 × 2 = ??
(2 × 6) × 2
= 12 × 2
= 24 or Twenty Four.

 

 

 

Question 16.
5 × 2 × 4 = _______
Answer:
5 × 2 × 4 = __ 40 or Forty_____.

Explanation:
5 × 2 × 4 = ??
(5 × 2) × 4
= 10 × 4
= 40 or Forty.

 

 

 

Question 17.
5 × 2 × 2 = ________
Answer:
5 × 2 × 2 = __20 or Twenty.______.

Explanation:
5 × 2 × 2 = ??
(5 × 2) × 2
= 10 × 2
= 20 or Twenty.

 

 

Question 18.
3 × 3 × 2 = _______
Answer:
3 × 3 × 2 = __18 or Eighteen.____.

Explanation:
3 × 3 × 2 = ??
(3 × 3) × 2
= 9 × 2
= 18 or Eighteen.

 

 

 

Problem Solving

Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 79
Question 19.
Reasoning There are 7 mockingbird nests at a park with eggs in them. What is the greatest number of eggs there could be at this park? What is the least number of eggs there could be?
Answer:
The greatest number of eggs there could be at this park = 35 or Thirty Five.
The least number of eggs there could be at this park = 21 or Twenty One.

Explanation:
Case:1: If mockingbird  lays 3 eggs in the nests at a park.
Number of mockingbird nests at a park  = 7 or Seven.
Number of mockingbird nests at a park with eggs = 3 or Three.
Number of eggs there could be at this park = Number of mockingbird nests at a park  × Number of mockingbird nests at a park with eggs
= 7 × 3
= 21 or Twenty One.

Case:2: If mockingbird  lays 3 eggs in the nests at a park.
Number of mockingbird nests at a park  = 7 or Seven.
Number of mockingbird nests at a park with eggs = 4 or Four.
Number of eggs there could be at this park = Number of mockingbird nests at a park  × Number of mockingbird nests at a park with eggs
= 7 × 4
= 28 or Twenty Eight.

Case:3: If mockingbird  lays 5 eggs in the nests at a park.
Number of mockingbird nests at a park  = 7 or Seven.
Number of mockingbird nests at a park with eggs = 5 or Five.
Number of eggs there could be at this park = Number of mockingbird nests at a park  × Number of mockingbird nests at a park with eggs
= 7 × 5
= 35 or Thirty Five.

 

Question 20.
At another park, there are 3 mockingbird nests with 4 eggs in each nest and 1 more nest with 3 eggs. How many eggs are there at this park?
Answer:
Total number of eggs  at this park = 15 or Fifteen.

Explanation:
Number of mockingbird nests at a park  = 3 or Three.
Number of mockingbird nests at a park with eggs = 4 or Four.
Number of mockingbird nests at a park  extra = 1 or One.
Number of mockingbird nests at a park with eggs extra = 3 or Three.
Total number of eggs  at this park = (Number of mockingbird nests at a park  × Number of mockingbird nests at a park with eggs ) + (Number of mockingbird nests at a park  extra × Number of mockingbird nests at a park with eggs extra)
= (3 × 4) + (1 × 3)
= 12 + 3
= 15 or Fifteen.

 

Question 21.
Maria says she can find the product for 2 × 3 × 4 by solving 3 × 2 × 4. Is Maria correct? Explain.
Answer:
Yes, Maria is correct because both the equations are same.

Explanation:
2 × 3 × 4
= (2 × 3) × 4
= 6 × 4
= 24 or Twenty Four.

3 × 2 × 4
= (3 × 2) × 4
= 6 × 4
= 24 or Twenty Four.

 

 

Question 22.
Anita has 2 arrays. Each array has 3 rows of 3 counters. Explain why Anita can use the Associative Property to find the total number of counters in two different ways.
Answer:
Way:1:
Total number of counters = Number of Arrays Anita has × Number of rows in each Array × Number of counters in each Array
= 2 × 3 × 3
= 6 × 3
= 18 or Eighteen.

Way:2:
Total number of counters = Number of counters in each Array × Number of rows in each Array × Number of Arrays Anita has
= 3 × 3 × 2
= 9 × 2
= 18 or Eighteen.

Explanation:
Way:1:
Number of Arrays Anita has = 2 or Two.
Number of rows in each Array = 3 or Three.
Number of counters in each Array = 3 or Three.
Total number of counters = Number of Arrays Anita has × Number of rows in each Array × Number of counters in each Array
= 2 × 3 × 3
= (2 × 3) × 3
= 6 × 3
= 18 or Eighteen.

Way:2:
Number of counters in each Array = 3 or Three.
Number of rows in each Array = 3 or Three.
Number of Arrays Anita has = 2 or Two.
Total number of counters = Number of counters in each Array × Number of rows in each Array × Number of Arrays Anita has
= 3 × 3 × 2
= (3 × 3) × 2
= 9 × 2
= 18 or Eighteen.

 

 

 

Question 23.
Algebra Which number makes both equations true?
4 × (3 × 2) = (4 × ?) × 2
3 × (5 × 2) = (? × 5) × 2
Answer:
3 or Three number makes both equations true.
Explanation:
4 × (3 × 2) = (4 × ?) × 2
= 4 × (3 × 2) = (4 × 3) × 2
= 4 × 6 = 12 ×2
= 24 = 24.

3 × (5 × 2) = (? × 5) × 2
= 3 × (5 × 2) = (3 × 5) × 2
= 3 × 10 = 15 × 2
= 30 = 30.

 

Question 24.
Higher Order Thinking
How do you know that 4 × 2 × 2 is the same as 4 × 4? Explain.
Answer:
4 × 2 × 2 is the same as 4 × 4 because their product value is same.

Explanation:
4 × 2 × 2
= (4 × 2) × 2
= 8 × 2
= 16 or Sixteen.

4 × 4
= 16 or Sixteen.

 

Assessment Practice
Question 25.
Use properties of operations to select all the expressions that could be used to find 7 × 1 × 3.
☐ (7 × 1) × 3
☐ 7 × (1 × 3)
☐ (1 × 7) × 3
☐ 7 × 1 × 1
☐ 7 × (3 × 1)
Answer:
Expressions that could be used to find 7 × 1 × 3:
☐ (7 × 1) × 3
☐ 7 × (1 × 3)
☐ (1 × 7) × 3
☐ 7 × (3 × 1)

Explanation:
Expressions that could be used to find 7 × 1 × 3:
(7 × 1) × 3
7 × (1 × 3)
(7 × 3) × 1
7 × (3 × 1)
(1 × 7) × 3
1 × (7 × 3)

 

Question 26.
An expression is shown. Select all the equivalent expressions.
4 × 2 × 3
You can use properties to solve problems in different ways.
☐ 4 × (2 × 3)
☐ (4 × 2) × 3
☐ (2 × 4) × 3
☐ (4 × 2) × 4
☐ (3 × 2) × 2.
Answer:
Expressions that could be used to find 4 × 2 × 3:
☐ 4 × (2 × 3)
☐ (4 × 2) × 3
☐ (2 × 4) × 3

Explanation:
Expressions that could be used to find 4 × 2 × 3:
(4 × 2) × 3
4 × (2 × 3)
(4 × 3) × 2
4 × (3 × 2)
(2 × 4) × 3:
2 × (4 × 3)

 

 

Lesson 3.7 Problem Solving
Repeated Reasoning
Solve
You have learned that you can use known facts to find unknown facts. For each of the 4 multiplication facts below, list two multiplication facts from the box that can be added to find the given product. The first solution is completed for you.
What do you notice about the facts you used to find the products when 6 or 7 is a factor?

I can.. use reasoning to look for and describe general strategies for finding products.
Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 80

Thinking Habits
Be a good thinker! These questions can help you.

  • Are any calculations repeated?
  • Can I generalize from examples?
  • What shortcuts do I notice?
    Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 81

Look Back! Generalize Use your observations from your work above to complete these facts.
(______ × 6) + (_____ × 6) = 36
(______ × 7) + (_____ × 7) = 49
Answer:
(______ × 6) + (_____ × 6)
= (___5___ × 6) + (__1___ × 6)
= 30 + 6
= 36.

(______ × 7) + (_____ × 7)
= (___6___ × 7) + (__1___ × 7)
= 42 + 7
= 49.

Explanation:
(______ × 6) + (_____ × 6) = 36.
(___5___ × 6) + (__1___ × 6) = 36.
30 + 6= 36.

(______ × 7) + (_____ × 7) = 49.
(___6___ × 7) + (__1___ × 7) = 49.
42 + 7= 49.

 


Essential Question

How Can You Use Repeated Reasoning When Multiplying?
Answer:
“Repeated reasoning” means, why looking for and expressing regularity in it is such a valuable mathematical habit of mind, and how that differs from analyzing structure (MP 7) and from finding patterns in numerical results.

Explanation:
To make sense of the Common Core State Standards mathematical practice (MP) 8, this article illustrates what “repeated reasoning” means, why looking for and expressing regularity in it is such a valuable mathematical habit of mind, and how that differs from analyzing structure (MP 7) and from finding patterns in numerical results.

 

Visual Learning Bridge
Ellie wrote the equations below to find the total number of squares in each of these rectangles. Look at the equations. Known facts with which factors are used repeatedly to find the products?
Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 82
A. 3 × 6 = (2 × 6) + (1 × 6) = 12 + 6 = 18
B. 4 × 9 = (2 × 9) + (2 × 9) = 18 + 18 = 36
C. 6 × 8 = (5 × 8) + (1 × 8) = 40 + 8 = 48
D. 7 × 7 = (5 × 7) + (2 × 7) = 35 + 14 = 49
What do I need to do to complete the task?
I need to see if there are known facts that can be used repeatedly to find other facts.

How can I make a generalization from repeated reasoning?
I can

  • look for repeated calculations.
  • make generalizations about the repeated calculations.
  • test whether my generalizations work for other numbers.

Here’s my thinking…
I see that the factors
1, 2, and 5 are used repeatedly.
I see two generalizations.
I can break facts with 3 or 4 into 2s and 1s facts.
3 × 6 = (2 × 6) + (1× 6)
4 × 9 = (2 × 9) + (2 × 9)

I can break facts with 6 or 7 into 5s, 25, and 1s facts.
6 × 8 = (5 × 8) + (1 × 8)
7 × 7 = (5 × 7) + (2 × 7)

I can test this with other facts.
3 × 5 = (2 × 5) + (1 × 5)
6 × 7 = (5 × 7) + (1 × 7)

Convince Me! Generalize Use the generalizations above to complete each of the following. Tell how you decided.
7 × 5 = (____) + (______)
7 × 6 = (____) + (______)

Answer:
7 × 5 = (____) + (______)
7 × 5 = (4 × 5) + (3 × 5) = 35 or Thirty Five.

7 × 6 = (____) + (______)
7 × 6 = (2 × 6) + (5 × 6) = 42 or Forty Two.

Explanation:
7 × 5 = (4 × 5) + (3 × 5)
7 × 5 = 20 + 15
7 × 5 = 35 or Thirty Five.

7 × 6 = (____) + (______)
7 × 6 = (2 × 6) + (5 × 6)
7 × 6 = 12 + 30
7 × 6 = 42 or Forty Two.

 

 

Guided Practice
Generalize Ricardo wrote the equations below.
Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 83
Question 1.
Which factors did Ricardo use repeatedly to find the products? Make a generalization.
3 × 8 = (2 × 8) + (1 × 8) = 24
3 × 7 = (2 × 7) + (1 × 7) = 21
6 × 3 = (6 × 1) + (6 × 2) = 18
Answer:
3 × 8 = (2 × 8) + (1 × 8) = 24 True.
3 × 7 = (2 × 7) + (1 × 7) = 21 True.
6 × 3 = (6 × 1) + (6 × 2) = 18 False.

Explanation:
Factors which Ricardo used repeatedly to find the products:
3 × 8 = (2 × 8) + (1 × 8) = 24 True.
3 × 7 = (2 × 7) + (1 × 7) = 21 True.
6 × 3 = (6 × 1) + (6 × 2) = 18False.

 

 

 

Question 2.
Complete this equation to test whether your generalization is true for other facts. Explain.
3 × 9 = (__ × ____) + (____ × ____) = _____
Answer:
3 × 9 = (_2__ × __9__) + (__1__ × __9__) = __ 27 or Twenty Seven___.

Explanation:
3 × 9 = (2 × 9 ) + (1 × 9)
3 × 9 = 18 + 9
3 × 9 = 27 or Twenty Seven.

 

 

Independent Practice
Generalize
Mary wrote the equations at the right.
Question 3.
Which factors did Mary use repeatedly to find the products? Make a generalization.
8 × 7 = (5 × 7) + (3 × 7) = 56
6 × 8 = (6 × 5) + (6 × 3) = 48
8 × 9 = (3 × 9) + (5 × 9) = 72
Answer:
8 × 7 = (5 × 7) + (3 × 7) = 56. True.
6 × 8 = (6 × 5) + (6 × 3) = 48. False.
8 × 9 = (3 × 9) + (5 × 9) = 72. True.

Explanation:
8 × 7 = (5 × 7) + (3 × 7) = 35 + 21 = 56. True.
6 × 8 = (6 × 5) + (6 × 3) =  30 + 18 = 48. False.
8 × 9 = (3 × 9) + (5 × 9) = 27 + 45 = 72. True.

 

 

 

Question 4.
Complete this equation to test whether your generalization is true for other facts. Explain.
8 × 3 = (____ × _____) + (____ × _____) = ______
Answer:
8 × 3 = (__5__ × __3___) + (__3__ × _3____) = __24 or Twenty Four____.

Explanation:
8 × 3 = (5 × 3) + (3 × 3)
8 × 3 = 15 + 9
8 × 3 = 24 or Twenty Four.

 

 

Question 5.
What is another way you can use known facts to solve 8 × 3? What generalization can you make from this way?
Answer:
Another way:
8 × 3 = (2 ×3) + (2 ×3) + (2 ×3) + (2 ×3)
= 6 + 6 + 6 + 6
= 12 + 6 + 6
= 18 + 6 = 24 or Twenty Four.

Explanation:
Another way:
8 × 3 = (2 ×3) + (2 ×3) + (2 ×3) + (2 ×3)
8 × 3 = 6 + 6 + 6 + 6
8 × 3 = 12 + 6 + 6
8 × 3 = 18 + 6
8 × 3 = 24 or Twenty Four.

 

Problem Solving
Performance Task
Baking Pizzas
Adam is baking 4 pizzas. Each pizza is a rectangle. It takes Adam 35 minutes to bake the pizzas. He divides each pizza into the equal-size square slices shown.
Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 84

Question 6.
Make Sense and Persevere Adam multiplies to find the total number of square slices for each pizza. For each pizza, tell the factors Adam multiplies.
Answer:
Number of square slices in pizza 1 = 6 × 5  = 30 or Thirty.
Number of square slices in pizza 2 = 6 × 7  = 42 or Forty Two.
Number of square slices in pizza 3 = 4 × 7  = 28 or Twenty Eight.
Number of square slices in pizza 4 = 4 × 8  = 32 or Thirty Two.

Explanation:
Number of pizzas Adam bakes = 4 or Four.
Number of square slices in pizza 1 = 6 × 5 = (3 × 5) + (3 × 5) = 15 + 15 = 30 or Thirty.
Number of square slices in pizza 2 = 6 × 7 = (3 × 7) + (3 × 7) = 21+ 21= 42 or Forty Two.
Number of square slices in pizza 3 = 4 × 7 = (2 × 7) + (2 × 7) = 14 + 14 = 28 or Twenty Eight.
Number of square slices in pizza 4 = 4 × 8 = (2 × 8) + (2 × 8) = 16 + 16 = 32 or Thirty Two.

 

 

Question 7.
Use Structure Look at the facts you wrote in Exercise 6. Break apart these facts into 1s, 2s, or 5s facts to find the total number of slices for each pizza.
Pizza 1
(____ × _____) = (____ × _____) + (_____ × ______) = ______
Pizza 2
(____ × _____) = (____ × _____) + (_____ × ______) = ______
Pizza 3
(____ × _____) = (____ × _____) + (_____ × ______) = ______
Pizza 4
(____ × _____) = (____ × _____) + (_____ × ______) = ______
Answer:
Pizza 1
(__6__ × __5___) = (__3__ × __5___) + (___3__ × ___5___) = __30____.
Pizza 2
(__6__ × _7____) = (__3__ × __7___) + (__3___ × ___7___) = __42____.
Pizza 3
(__4__ × __7___) = (__2__ × ___7__) + (__2___ × __7____) = __28____.
Pizza 4
(__4__ × __8___) = (__2__ × __8___) + (__2___ × ___8___) = ___32___.

Explanation:
Pizza 1
6 × 5 = (3 × 5) + (3 × 5) = 15 + 15 = 30 or Thirty.
Pizza 2
(3 × 7) + (3 × 7) = 21+ 21 = 42 or Forty Two.
Pizza 3
4 × 7 = (2 × 7) + (2 × 7) = 14 + 14 = 28 or Twenty Eight.
Pizza 4
4 × 8 = (2 × 8) + (2 × 8) = 16 + 16 = 32 or Thirty Two.

 

 

 

Question 8.
Generalize Look at how you used the 1s, 2s, and 5s facts above. What generalizations can you make? Test your generalizations with another fact.
Envision Math Common Core Grade 3 Answer Key Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 85
Answer:
The generalizations we can make is that using1s,2s,3s,4s and 5s , we can multiply the numbers and get the correct answer asked in the problem.

Explanation:
By using 1s,2s,3s,4s,5s,6s we can get the correct solution for the problem using multiplication.

 

Question 9.
Critique Reasoning
Look at the model for Pizza 3. Adam says he can use 2s facts to solve 4 × 7 or 7 × 4. Is he correct? Explain.
Answer:
He is correct, we can use  2s facts to solve 4 × 7 or 7 × 4.

Explanation:
4 × 7 = (2 × 7) + (2 × 7)
4 × 7 = 14 + 14
4 × 7 = 28 or Twenty Eight.

7 × 4 = (2 × 4) + (2 × 4) + (3 × 4)
7 × 4 = 8+ 8+ 12
7 × 4 = 16+ 12
7 × 4 = 28 or Twenty Eight.

 

 

Topic 3 Fluency Review Activity

Follow the path
Shade a path from START to FINISH. Follow the differences that are correct. You can only move up, down, right, or left.
I can … subtract within 100.
Envision Math Common Core Grade 3 Answers Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 86
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.7 Problem Solving-Topic 3 Fluency Review Activity

Explanation:
Correct Subtraction equations which are shaded:
Start: 75 – 13 = 62.
99 – 36 = 63.
59 – 37 = 22.
46 – 27 = 19.
65 – 59 = 6.
81 – 29 = 52.
67 – 19 = 48.
78 – 35 = 43. Finish
Other equations are wrongly subtracted.

Topic 3 Vocabulary Review

Understand Vocabulary
Word List

  • Associative (Grouping) Property of Multiplication
  • Commutative (Order) Property of Multiplication
  • Distributive Property
  • factor
  • Identity (One) Property of Multiplication
  • multiple
  • product
  • Zero Property of Multiplication

Match the example to the term.
Envision Math Common Core Grade 3 Answers Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 87
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.7 Problem Solving-Topic 3 Vocabulary Review
Explanation:
The  Associative property is a math rule that says that the way in which factors are grouped in a multiplication problem does not change the product.
Commutative property states that the change in the order of numbers in an addition or multiplication operation does not change the sum or the product. The Commutative property of addition is written as: A + B = B + A.
The distributive property tells us how to solve expressions in the form of a(b + c). The distributive property is sometimes called the distributive law of multiplication.
The  identity property of 1 says that any number multiplied by 1 keeps its identity . In other words, any number multiplied by 1 stays the same. The reason the number stays the same is because multiplying by 1 means we have 1 copy of the number

 

 

Write T for true or F for false.
Question 5.
______ 3 and 8 are multiples of 24.
Answer:
___T___ 3 and 8 are multiples of 24.

Explanation:
3 × 8 = 24.

Question 6.
________ You can multiply factors in any order.
Answer:
___T_____ You can multiply factors in any order.

Explanation:
True you can multiply factors in any order.
Example:
3 × 8 = 24.
8 × 3 = 24.

 

Question 7.
________ The product of zero and any number is that number.
Answer:
___F_____ The product of zero and any number is that number.

Explanation:
0 × 2 = infinity.
2 × 0 = 0 or Zero.

 

 

Question 8.
________ There are 3 factors in the equation 5 × 3 × 2 = 30.
Answer:
____T____ There are 3 factors in the equation 5 × 3 × 2 = 30.

Explanation:
2 × 15 = 30.
3 × 10 = 30.
5 × 6 = 30.

 

 

Use Vocabulary in Writing
Question 9.
Explain how to use 8 × 5 = 40 to find 8 × 6. Use at least 2 terms from the Word List in your explanation.
Answer:
8 × 5 = 40.
8 × 6 = (8 × 5 ) + (8 × 1 ) = 48 or Forty Eight.

Explanation:
8 × 5 = 40.
8 × 6 = (8 × 5 ) + (8 × 1 )
8 × 6 = 40 + 8
8 × 6 = 48 or Forty Eight.

 

 

 

Topic 3 Reteaching
Set A pages 77-80

You can break an array into 2 smaller arrays.
Envision Math Common Core Grade 3 Answers Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 88
You can write an unknown fact as the sum of 2 known facts.
4 × 8 = (3 × 8) + (1 × 8)

Remember that the Distributive Property says that a multiplication fact can be broken apart into the sum of two other multiplication facts.

In 1 and 2, find the missing value.
Question 1.
_____ × 4 = (2 × 4) + (2 × 4)
Answer:
__4___ × 4 = (2 × 4) + (2 × 4).

Explanation:
_____ × 4 = (2 × 4) + (2 × 4).
? × 4 = (2 × 4) + (2 × 4)
4 × 4 = (2 × 4) + (2 × 4)
4 × 4 = 8 + 8
4 × 4 = 16 or Sixteen.

 

 

 

Question 2.
6 × 5 = (4 × 5) + (______ × 5)
Answer:
6 × 5 = (4 × 5) + (__2____ × 5)

Explanation:
6 × 5 = (4 × 5) + (? × 5)
6 × 5 = (4 × 5) + (2 × 5)
6 × 5 = 20 + 10
6 × 5 = 30 or Thirty.

 

 

Set B pages 81-84
Find 3 × 4.
You can use a 2s fact to help multiply by 3.
Envision Math Common Core Grade 3 Answers Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 89
You can also find 4s facts.
3 × 4 = (3 × 2) + (3 × 2) = 6 + 6 = 12

Remember that to find a 3s fact, add a 2s fact and a 1s fact. To find a 4s fact, double the product of a 2s fact.

Question 1.
3 × 7 = _______
Answer:
3 × 7 = __(3 × 2) + (3 × 2) + (3 × 3)_= 6 + 6 + 9_= 21 or Twenty One__.

Explanation:
3 × 7 = (3 × 2) + (3 × 2) + (3 × 3)
3 × 7 = 6 + 6 + 9
3 × 7 = 12 + 9
3 × 7 = 21 or Twenty One.

 

 

 

Question 2.
4 × 9 = _______
Answer:
4 × 9 = _ (4 × 5) + (4 × 2) + (4 × 2)__= 20 + 8 + 8__= 36 or Thirty Six__.

Explanation:
4 × 9 = (4 × 5) + (4 × 2) + (4 × 2)
4 × 9 = 20 + 8 + 8
4 × 9 = 28 + 8
4 × 9 = 36 or Thirty Six.

 

Question 3.
4 × 10 = _______
Answer:
4 × 10 = _(4 × 2) + (4 × 2) + (4 × 2) + (4 × 2) + (4 × 2)_= 8 + 8 + 8 + 8 + 8_= 40 or Forty____.

Explanation:
4 × 10 = (4 × 2) + (4 × 2) + (4 × 2) + (4 × 2) + (4 × 2)
4 × 10 = 8 + 8 + 8 + 8 + 8
4 × 10 = 16 + 8 + 8 + 8
4 × 10 = 24 + 8 + 8
4 × 10 = 32 + 8
4 × 10 = 40 or Forty.

 

 

Question 4.
3 × 10 = ______
Answer:
3 × 10 =  _(3 × 2) + (3 × 2) + (3 × 2) + (3 × 2) + (3 × 2)_= 6 + 6 + 6 + 6 + 6_= 30 or Thirty____.

Explanation:
3 × 10 = (3 × 2) + (3 × 2) + (3 × 2) + (3 × 2) + (3 × 2)
3 × 10 = 6 + 6 + 6 + 6 + 6
3 × 10 = 12 + 6 + 6 + 6
3 × 10 = 18 + 6 + 6
3 × 10 = 24 + 6
3 × 10 = 30 or Thirty.

 

 

 

Question 5.
3 × 8 = ______
Answer:
3 × 8 = __ (3 × 2)+ (3 × 2) + (3 × 2) + (3 × 2)_= 6 + 6 + 6 + 6 =_ 24 or Twenty Eight___

Explanation:
3 × 8 = (3 × 2)+ (3 × 2) + (3 × 2) + (3 × 2)
3 × 8 = 6 + 6 + 6 + 6
3 × 8 = 12 + 6 + 6
3 × 8 = 18 + 6
3 × 8 = 24 or Twenty Eight.

 

 

Question 6.
8 × 4 = ______
Answer:
8 × 4 = _ (8 × 2) + (8 × 2) _= 16 + 16_= 32 or Thirty Two___.

Explanation:
8 × 4 = (8 × 2) + (8 × 2)
8 × 4 = 16 + 16
8 × 4 = 32 or Thirty Two.

 

Question 7.
9 × 3 = _______
Answer:
9 × 3 = __(9 × 2) + (9 × 1)_= 18 + 9_= 27 or Twenty Seven___.

Explanation:
9 × 3 = (9 × 2) + (9 × 1)
9 × 3 = 18 + 9
9 × 3 = 27 or Twenty Seven.

 

 

Question 8.
10 × 4 = ______
Answer:
10 × 4 = _ (10 × 2) + (10 × 2)_= 20 + 20_= 40 or Forty.___.

Explanation:
10 × 4 = (10 × 2) + (10 × 2)
10 × 4 = 20 + 20
10 × 4 = 40 or Forty.

 

 

 

Set C pages 85-88
You can use known facts to help multiply. Find 6 × 9.
6 × 9 = (5 × 9) + (1 × 9)
6 × 9 = 45 + 9
6 × 9 = 54

Find 7 × 4.
7 × 4 = (5 × 4) + (2 × 4)
7 × 4 = 20 + 8
7 × 4 = 28

 

Remember that you can break a multiplication problem into two smaller problems.

Question 1.
6 × 6 = _____
Answer:
6 × 6 = _(3 ×6) + (3 ×6)_= 18 + 18_= 36 or Thirty Six.__.

Explanation:
6 × 6 = (3 ×6) + (3 ×6)
6 × 6 = 18 + 18
6 × 6 = 36 or Thirty Six.

 

Question 2.
7 × 9 = _____
Answer:
7 × 9 = _(2×9) + (2×9) + (3×9)  = 18 + 18 + 27 = 63 or Sixty Three___.

Explanation:
7 × 9 = (2×9) + (2×9) + (3×9)
7 × 9 = 18 + 18 + 27
7 × 9 = 36 + 27
7 × 9 = 63 or Sixty Three.

 

Question 3.
7 × 7 = _____
Answer:
7 × 7 = (2×7) + (2×7) + (3×7)_= 14 + 14 + 21_= 49 or Forty Nine___.

Explanation:
7 × 7 = (2×7) + (2×7) + (3×7)
7 × 7 = 14 + 14 + 21
7 × 7 = 28 + 21
7 × 7 = 49 or Forty Nine.

 

 

Question 4.
6 × 8 = _______
Answer:
6 × 8 = _(3×8) + (2×8) + (1×8)_= 24 + 16 + 8__= 48 or Forty Eight___.

Explanation:
6 × 8 = (3×8) + (2×8) + (1×8)
6 × 8 = 24 + 16 + 8
6 × 8 = 40 + 8
6 × 8 = 48 or Forty Eight.

 

 

 

Question 5.
Envision Math Common Core Grade 3 Answers Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 90
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.7 Problem Solving-Set C pages 85-88-5

Explanation:
6 × 5 = (3×5) + (3×5)
6 × 5 = 15 + 15
6 × 5 = 30 or Thirty.

 

 

Question 6.
Envision Math Common Core Grade 3 Answers Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 91
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.7 Problem Solving-Set C pages 85-88-6

Explanation:
6 × 3 = (2×3) + (2×3) + (2×3)
6 × 3 = 6 + 6 + 6
6 × 3 = 12 + 6
6 × 3 = 18 or Eighteen.

 

 

Question 7.
Envision Math Common Core Grade 3 Answers Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 92
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.7 Problem Solving-Set C pages 85-88-7

Explanation:
10 × 7 = (5×7) + (3×7) + (2×7)
10 × 7 = 35 + 21 + 14
10 × 7 = 56 + 14
10 × 7 = 70 or Seventy.

 

 

Set D pages 89-96
Find 8 × 9.
You can use 2s facts.
8 × 9 = (2 × 9) + (2 × 9) + (2 × 9) + (2 × 9)
8 × 9 = 18 + 18 + 18 + 18
8 × 9 = 72
You can use skip counting.
8, 16, 24, 32, 40, 48, 56, 64, 72

Remember to use patterns, known facts, or skip counting to find products.

Question 1.
8 × 6 = _______
Answer:
8 × 6 = _ (2 × 6) + (2 × 6) + (2 × 6) + (2 × 6)_= 12 + 12 + 12 + 12_= 48 or Forty Eight____.

Explanation:
8 × 6 = (2 × 6) + (2 × 6) + (2 × 6) + (2 × 6)
8 × 6 = 12 + 12 + 12 + 12
8 × 6 = 24 + 12 + 12
8 × 6 = 36 + 12
8 × 6 = 48 or Forty Eight.
Skip counting:
8, 16, 24, 32, 40, 48.

 

Question 2.
8 × 8 = ______
Answer:
8 × 8 = _(2 × 8) + (2 × 8) + (2 × 8) + (2 × 8)_= 16 + 16 + 16 + 16_= 64 or Sixty Four___.

Explanation:
8 × 8 = (2 × 8) + (2 × 8) + (2 × 8) + (2 × 8)
8 × 8 = 16 + 16 + 16 + 16
8 × 8 = 32 + 16 + 16
8 × 8 = 48 + 16
8 × 8 = 64 or Sixty Four.
Skip counting:
8, 16, 24, 32, 40, 48, 56, 64.

 

Question 3.
8 × 7 = _____
Answer:
8 × 7 = (2 × 7) + (2 × 7) + (2 × 7) + (2 × 7)_= 14 + 14 + 14 + 14_= 56 or Fifty Six___.

Explanation:
8 × 7 = (2 × 7) + (2 × 7) + (2 × 7) + (2 × 7)
8 × 7 = 14 + 14 + 14 + 14
8 × 7 = 28 + 14 + 14
8 × 7 = 42 + 14
8 × 7 = 56 or Fifty Six.
Skip counting:
8, 16, 24, 32, 40, 48, 56.

 

 

Question 4.
8 × 10 = ______
Answer:
8 × 10 = _(2 × 10) + (2 × 10) + (2 × 10) + (2 × 10)_= 20 + 20 + 20 + 20 _= 80 or Eighty___.

Explanation:
8 × 10 = (2 × 10) + (2 × 10) + (2 × 10) + (2 × 10)
8 × 10 = 20 + 20 + 20 + 20
8 × 10 = 40 + 20 + 20
8 × 10 = 60 + 20
8 × 10 = 80 or Eighty.
Skip counting:
8, 16, 24, 32, 40, 48, 56 , 64 , 72 , 80.

 

Question 5.
1 × 8 = _______
Answer:
1 × 8 = __8 or Eight_____.

Explanation:
1 × 8 = 8 or Eight.
Skip counting:
8.

 

 

Question 6.
0 × 8 = ______
Answer:
0 × 8 = __infinity____.

Explanation:
0 × 8 = infinity.

 

Question 7.
Envision Math Common Core Grade 3 Answers Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 93
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.7 Problem Solving-Set D pages 89-96-7

Explanation:
8 × 5 = (2×5) + (2×5) + (2×5) + (2×5)
8 × 5 = 10 + 10 + 10 + 10
8 × 5 = 20 + 10 + 10
8 × 5 = 30+ 10
8 × 5 = 40 or Forty.
Skip counting:
8, 16, 24, 32, 40.

 

 

Question 8.
Envision Math Common Core Grade 3 Answers Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 94
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.7 Problem Solving-Set D pages 89-96-8

Explanation:
8 × 3 = (2×3) + (2×3) + (2×3) + (2×3)
8 × 3 = 6 + 6 + 6 + 6
8 × 3 = 12 + 6 + 6
8 × 3 = 18 + 6
8 × 3 = 24 or Twenty Four.
Skip counting:
8, 16, 24.

 

Question 9.
Envision Math Common Core Grade 3 Answers Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 95
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.7 Problem Solving-Set D pages 89-96-9

Explanation:
8 × 2 = (2×2) + (2×2) + (2×2) + (2×2)
8 × 2 = 4 + 4 + 4 + 4
8 × 2 = 8 + 4 + 4
8 × 2 = 12 + 4
8 × 2 = 16 or Sixteen.
Skip counting:
8, 16.

 

 

Set E pages 97-100
You can use the Associative Property to group the factors. The product does not change.
Find 4 × 2 × 2.
Envision Math Common Core Grade 3 Answers Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 96

Remember that you can use properties to write unknown facts as known facts.

In 1-3, find the product. Show how you grouped the factors.

Question 1.
4 × 2 × 5 = _____ × ______ = _______
Answer:
4 × 2 × 5 = (4 × 2) × 5 = __8___ × ___5___ = __40 or Forty_____.

Explanation:
4 × 2 × 5 = (4 × 2) × 5 = 8 × 5 = 40 or Forty.

 

 

Question 2.
3 × 3 × 7 = _____ × ______ = _______
Answer:
3 × 3 × 7 = __3___ × ___21___ = __63 or Sixty Three_____.

Explanation:
3 × 3 × 7 = 3 × (3 × 7) =__3___ × __21____ = ___63 or Sixty Three____.

 

 

Question 3.
5 × 5 × 2 = _____ × ______ = _______
Answer:
5 × 5 × 2 = (5 × 5) × 2 = __25___ × __2____ = __50 or Fifty_____.

Explanation:
5 × 5 × 2 = (5 × 5) × 2 =__25___ × ___2___ = ___50 or Fifty____.

 

 

Set F pages 101-104
Think about these questions to help you use repeated reasoning.
Thinking Habits

  • Are any calculations repeated?
  • Can I generalize from examples?
  • What shortcuts do I notice?
    Envision Math Common Core Grade 3 Answers Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 97

Remember that patterns can help you make a generalization.

Question 1.
What is repeated in these equations? Use what you see to make a generalization.
6 × 6 = (6 × 3) + (6 × 3) = 18 + 18 = 36
7 × 6 = (7 × 3) + (7 × 3) = 21 + 21 = 42
8 × 6 = (8 × 3) + (8 × 3) = 24 + 24 = 48
Answer:
In these equations, 6 number is repeated.

Explanation:
6 × 6 = (6 × 3) + (6 × 3) = 18 + 18 = 36.
7 × 6 = (7 × 3) + (7 × 3) = 21 + 21 = 42.
8 × 6 = (8 × 3) + (8 × 3) = 24 + 24 = 48.
6 number is repeated in all the equations.

 

 

 

Question 2.
Solve this equation to test whether your generalization is true.
10 × 6 = ?
______ × _______ = (_____ × ______) + (______ × ______)
= ______ + ______ = ______
Answer:
10 × 6 = ?
__10____ × ___6____ = (___10__ × ___3___) + (___10___ × ___3___)
= __30____ + __30____ = __Sixty____.

Explanation:
10 × 6 = ?
10 × 6 = (10 × 3) + (10 × 3)
10 × 6 = 30 + 30
10 × 6 = 60 or Sixty.

 

Topic 3 Assessment Practice

Question 1.
Krista arranged her buttons in an array. Which two expressions can be used to find the total number of buttons?
Envision Math Common Core Grade 3 Answers Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 98
A. 6 × 7 and 6 × 1
B. 4 × 7 and 2 × 7
C. 4 × 4 and 2 × 3
D. 3 × 7 and 4 × 7
Answer:
B. 4 × 7 and 2 × 7 is the expression used to find the total buttons.

Explanation:
Number of rows of buttons Krista arranged = 7 or Seven.
Number of columns of buttons Krista arranged = 6 or Six.
Total number of buttons =  Number of columns of buttons Krista arranged × Number of rows of buttons Krista arranged
= 6 × 7
= (4 × 7) + (2 × 7)
= 28 + 14
= 42 or Forty Two.

 

 

 

Question 2.
Choose Yes or No to tell if the Distributive Property is being used.
Envision Math Common Core Grade 3 Answers Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 99
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.7 Problem Solving-Topic 3 Assessment Practice-2

Explanation:
2a. 4 × (2 + 3) = (4 × 2) + (4 ×3) = 8 + 12 = 20. True.
2b. (4 + 5) × 2 = (4 × 2) + (5 ×2) = 8 + 10 = 18. True.
2c. 2 × 4 = 2 × (3 + 1) False.
2d. 7 × (5 – 2) = (7 × 5) – (7 × 2) = 35 – 14 = 21. True.

 

 

Question 3.
Jeff makes the generalization that a 10s fact can be broken into two 5s facts. Write an equation to test his generalization.
Answer:
10 × 2 = (5 × 2) + (5 × 2) = 20 or Twenty.

Explanation:
10 × 2 = (5 × 2) + (5 × 2)
10 × 2 = 10 + 10
10 × 2 = 20 or Twenty.

 

 

 

Question 4.
June broke up a large array into a 3 × 4 array and a 5 × 4 array. What was the large array? Show your work.
Answer:
Larger array = (First smaller array) + (Second smaller array)
Larger array = (3 × 4) + (5 × 4 )
Larger array = (8 × 4)

Explanation:
First smaller array = 3 × 4
Second smaller array = 5 × 4
Larger array = (First smaller array) + (Second smaller array)
Larger array = (3 × 4) + (5 × 4 )
Larger array = (8 × 4)

 

 

 

Question 5.
Which facts can you use to find 4 x 8? Select all that apply.
☐ 2 × 8 and 2 × 9
☐ 2 × 8 and 2 × 8
☐ 2 × 4 and 1 × 8
☐ 4 × 5 and 4 × 3
☐ 3 × 8 and 1 × 8
Answer:
The facts used to apply to find 4 x 8 are:
☐ 2 × 8 and 2 × 8
☐ 4 × 5 and 4 × 3
☐ 3 × 8 and 1 × 8

Explanation:
4 × 8 = (2 × 8) + (2 × 8) = 16 + 16 = 32 or Thirty Two.
4 × 8 = (4 × 5) + (4 × 3) = 20 + 12 = 32 or Thirty Two.
4 × 8 = (3 ×8) + (1 × 8) = 24 + 8 = 32 or Thirty Two.

 

 

Question 6.
A bakery uses 3 cups of flour for each loaf of bread. There are 3 loaves of bread on a tray. There are 6 trays on a cart. How many cups of flour are used to make the bread on the cart? Show your work.
Answer:
Number of cups of flour are used to make the bread on the cart = 54 or Fifty Four.

Explanation:
Number of cups of flour for each loaf of bread a bakery uses = 3 or Three.
Number of loaves of bread on a tray = 3 or Three.
Number of trays on a cart = 6 or Six.
Number of cups of flour are used to make the bread on the cart = (Number of cups of flour for each loaf of bread a bakery uses) × Number of loaves of bread on a tray × Number of trays on a cart
= (3 × 3) × 6
= 9 × 6
= (5 × 6) + (4 × 6)
= 30 + 24
= 54 or Fifty Four.

 

 

Question 7.
Find the number that makes the equation correct. Explain your reasoning.
(3 × 4) + (3 × 4) = ______
Answer:
(3 × 4) + (3 × 4) = ____(6 × 4)___= 24 or Twenty Four.___.

Explanation:
(3 × 4) + (3 × 4) = (6 × 4) = 24 or Twenty Four.
Two small arrays combined together to make a large array.

 

Question 8.
Casey has 3 bags of baseballs. Each bag has 6 baseballs. How many baseballs does he have? Show your work.
☐ baseballs
Answer:
Total number of baseballs Casey has = 18 or Eighteen.

Explanation:
Number of bags of baseballs Casey has = 3 or Three.
Number of each bag has baseballs = 6 or Six.
Total number of baseballs Casey has = Number of bags of baseballs Casey has  ×  Number of each bag has baseballs
= 3 × 6
= (1 × 6) + (2 × 6)
= 6 + 12
= 18 or Eighteen.

 

 

Question 9.
Jonathan organizes his pictures into a 6 × 4 array. Kim organizes her pictures into a 7 × 5 array. How can Kim and Jonathan break apart their arrays? Write each pair of facts in the correct space.
Envision Math Common Core Grade 3 Answers Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 100
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.7 Problem Solving-Topic 3 Assessment Practice-9

Explanation:
Jonathan pictures array = 6 × 4.
6 × 4 = (5 × 4) + (1 × 4) = 20 + 4 = 24 or Twenty Four.
6 × 4 = (3 × 4) + (3 × 4)  = 12 + 12 = 24 or Twenty Four.
6 × 4 = (1 × 4) + (5 × 4)  = 4 + 20 = 24 or Twenty Four.

Kim pictures array =  7 × 5.
7 × 5 = (4 × 5) + (3 × 5) = 20 + 15 = 35 or Thirty Five.
7 × 5 = (5 × 5) + (2 × 5) = 25 + 10 = 35 or Thirty Five.

 

Question 10.
A farm stand has cherries on 2 shelves. Each shelf has 4 boxes. Each box has 8 ounces of cherries. How many ounces of cherries are displayed in all? Write an expression that represents the amount.
☐ ounces
Answer:
Total number of ounces of cherries are displayed in all = 64 or Sixty Four.

Explanation:
Number of cherries on shelves on a farm stand has = 2 or Two.
Number of boxes on each shelves = 4 or Four.
Number of ounces of cherries on each box = 8 or Eight.
Total number of ounces of cherries are displayed in all = Number of cherries on shelves on a farm stand has × (Number of boxes on each shelves  × Number of ounces of cherries on each box)
= 2 × (4 × 8)
= 2 × 32
= 64 or Sixty Four.

 

 

 

Question 11.
Amy arranged her counters into this array.
Envision Math Common Core Grade 3 Answers Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 101
A. What two facts could Amy use to write an equation for the array?
Answer:
3 × 9 = (1 × 9) + (2 × 9) = 9 + 18 = 27 or Twenty Seven.

Explanation:
Two facts could Amy use to write an equation for the array:
3 × 9 = (1 × 9) + (2 × 9)
3 × 9 = 9 + 18
3 × 9 = 27 or Twenty Seven.

 

 

B. If Amy adds one more row of 9 counters to her array, can she still use the facts you wrote in Part A to find the total number? Explain why or why not.
Answer:
She can use the two facts which are wrote in Part A to find the total number.

Explanation:
Number of rows added = 1 or One.
Total number of array = (4 × 9)
(4 × 9) = (1 × 9)+ (3 × 9) = 9 + 27 = 36 or Thirty six.
(4 × 9) = (2 × 9) + (2 × 9) = 18 + 18 = 36 or Thirty six.

 

 

 

Question 12.
Tim’s family rented a canoe for 6 hours on Monday and 2 hours on Tuesday. How much did they spend? Show any equations you used.
Envision Math Common Core Grade 3 Answers Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 102
Answer:
Total cost they spent = $56 or $Fifty Six.

Explanation:
Number of hours Tim’s family rented a canoe on Monday = 6 or Six.
Number of hours Tim’s family rented a canoe on Tuesday = 2 or Two.
Cost of the canoe Tim’s family rented on Monday for each hour = $7
Total cost they spent = (Number of hours Tim’s family rented a canoe on Monday × Cost of the canoe Tim’s family rented on Monday for each hour ) + (Number of hours Tim’s family rented a canoe on Tuesday + Cost of the canoe Tim’s family rented on Monday for each hour )
= (6 ×$7) + (2 ×$7)
= (2 ×$7) + (2 ×$7)+ (2 ×$7)+ (2 ×$7)
= $14 + $14 + $14 + $14
= $28 + $14 + $14
= $42 + $14
= $56 or $Fifty Six.

 

 

 

Topic 3 Performance Task

School Fair
Kay and Ben are helping to organize the School Fair. Kay is organizing the school band. Ben is organizing the bake sale.

The 3 × 7 array at the right shows how chairs have been set up for the school band. Use the array to answer Exercises 1 and 2.

Question 1.
Kay wants to have chairs in a 6 × 7 array. Add to the array to show how the new array will look.
Answer:
The new array will look  6 × 7  = (3 × 7) + (3 × 7).

Explanation:
Kay wants to have chairs in a 6 × 7 array.
The new array:
=> 6 × 7  = (3 × 7) + (3 × 7).

 

Question 2.
Kay needs a path between two of the rows, so she separates the chairs into two smaller arrays.
Part A
Draw a line to show one way Kay can separate the chairs into two smaller arrays.
Envision Math Common Core Grade 3 Answers Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 103
Answer:
3 × 7 = (2 × 7) + (1 ×7)
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.7 Problem Solving-Topic 3 Performance Task-2

Explanation:
Chairs array:
3 × 7 = (2 × 7) + (1 ×7)

 

 

Part B
Kay wants to know the number of chairs in each new array. Write a multiplication fact for each of the new arrays to show how she can find this.
Answer:
Number of chairs in each array = 21 or Twenty One.

Explanation:
The new array:
=> 6 × 7  = (3 × 7) + (3 × 7)
=> 6 × 7  = 21 + 21

 

 

Part C
Kay wants to find the total number of chairs that are being used. Show how to use the facts in Part B to find the total number of chairs.
Answer:
The total number of chairs = 6 × 7  = (3 × 7) + (3 × 7) = 42 or Forty Two.

Explanation:
The total number of chairs = 6 × 7 .
=> 6 × 7  = (3 × 7) + (3 × 7)
=> 6 × 7  = 21 + 21
=> 6 × 7  = 42 or Forty Two.

 

 

The Bake Sale table shows the baked goods Ben has for sale at the School Fair. Use the Bake Sale table to answer Exercises 3-5.
Envision Math Common Core Grade 3 Answers Topic 3 Apply Properties Multiplication Facts for 3, 4, 6, 7, 8 104

Question 3.
Ben sells 4 trays of granola bars in the morning and 4 trays of granola bars in the afternoon. How much money does this raise? Show your work.
Answer:
Amount of money this raise = $24 or $Twenty Four.

Explanation:
Number of trays of granola bars in the morning Ben sells = 4 or Four.
Number of trays of granola bars in the afternoon Ben sells = 4 or Four.
Cost the each trays of granola bars  Ben sells = $3.
Amount of money this raise = (Number of trays of granola bars in the morning Ben sells × Cost the each trays of granola bars  Ben sells ) + (Number of trays of granola bars in the afternoon Ben sells  × Cost the each trays of granola bars  Ben sells )
= (4 × $3) + (4 × $3)
= $12 + $12
= $24 or $Twenty Four.

 

 

Question 4.
Ben organizes the blueberry muffins into a 4 × 7 array
Part A
Ben breaks up the array of blueberry muffins into 2 arrays that look the same. At the right, draw the 2 arrays of blueberry muffins.
Answer:
Envision-Math-Common-Core-3rd-Grade-Answers-Key-Topic-3-Apply-Properties-Multiplication-Facts-for 3, 4, 6, 7, 8-Lesson 3.7 Problem Solving-Topic 3 Performance Task-4

Explanation:
Ben organizes the blueberry muffins into a 4 × 7 array.
=> 4 × 7 = ( 2 × 7) + ( 2 × 7).

 

 

Part B
Ben wants to check the total number of blueberry muffins. He knows 2 × 7 = 14. How can he use this to find the total number of blueberry muffins?
Answer:
Total number of blueberry muffins = 4 × 7 = (2 × 7)  + (2 × 7) = 14 + 14 = 28  or Twenty Eight.

Explanation:
Ben organizes the blueberry muffins into a 4 × 7 array.
Total number of blueberry muffins = 4 × 7
4 × 7 = (2 × 7)  + (2 × 7)
4 × 7 = 14 + 14
4 × 7 = 28 or Twenty Eight.

 

 

Question 5.
Two friends each bought 3 trays of strawberry tarts. Ben says they spent more than $20 in total. Do you agree? Explain.
Answer:
Yes, I agree with what Ben says “Ben says they spent more than $20 in total” because they spent $24 in total.

Explanation:
Cost of each trays of strawberry tarts = $4
Number of trays of strawberry tarts two friends each bought = 3 or Three.
Cost of trays of strawberry tarts Two friends each bought = (Cost of each trays of strawberry tarts × ( 2 × Number of trays of strawberry tarts two friends each bought)
= $4 × (2 × 3)
= $4 × 6
= $24 or $Twenty Four.

 

Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships

enVision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships

Go through the enVision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships regularly and improve your accuracy in solving questions.

enVision Math Common Core 8th Grade Answers Key Topic 3 Use Functions To Model Relationships

Topic 3 GET READY!

Review What You Know!

Vocabulary
Choose the best term from the box to complete each definition.

Question 1.
The ____ is the ratio of the vertical change to the horizontal change of a line.
Answer:
We know that,
The “Slope” is the ratio of the vertical change to the horizontal change of a line
Hence, from the above,
We can conclude that the best term from the box to complete the given definition is “Slope”

Question 2.
A relationship that can be modeled by the equation y = mx is a ___
Answer:
We know that,
A relationship that can be modeled by the equation y = mx is a “Proportional relationship”
Hence, from the above,
We can conclude that the best term from the box to complete the given definition is “Proportional relationship”

Question 3.
y-value at which a line of a graph crosses the y-axis is called the ___
Answer:
We know that,
y-value at which a line of a graph crosses the y-axis is called the “y-intercept”
Hence, from the above,
We can conclude that the best term from the box to complete the given definition is “y-intercept”

Question 4.
An equation written in the form y = mx + b is called the ___
Answer:
We know that,
An equation written in the form y = mx + b is called the “Linear equation” or the “Slope-intercept form”
Hence, from the above,
We can conclude that the best term from the box to complete the given definition is “Linear equation”(or) the “Slope-intercept form”

Use Functions To Model Relationships 1

Slope and y-Intercept

Find the slope and y-intercept of a line that passes through these points.

Question 5.
(2, 2) and (3, 0)
Answer:
The given points are:
(2, 2), and (3, 0)
Compare the given points with (x1, y1), (x2,y2)
We know that,
Slope(m) = y2 – y1 / x2 – x1
So,
m = \(\frac{0 – 2}{3 – 2}\)
= \(\frac{-2}{1}\)
= -2
We know that,
The linear equation in the slope-intercept form is:
y = mx + c
Where,
m is the slope
c is the y-intercept
We know that,
We can obtain the y-intercept by putting the value of x equal to 0
So,
y = -2x + c
Substitute (3, 0) or (2, 2) in the above equation
So,
0 = -6 + c
So,
c = 6
Hence, from the above,
We can conclude that
The slope of a line that passes through the given points is: -2
The y-intercept of a line is: 6

Question 6.
(1, 5) and (4, 10)
Answer:
The given points are:
(1, 5), and (4, 10)
Compare the given points with (x1, y1), (x2,y2)
We know that,
Slope(m) = y2 – y1 / x2 – x1
So,
m = \(\frac{10 – 5}{4 – 1}\)
= \(\frac{5}{3}\)
We know that,
The linear equation in the slope-intercept form is:
y = mx + c
Where,
m is the slope
c is the y-intercept
We know that,
We can obtain the y-intercept by putting the value of x equal to 0
So,
y = \(\frac{5}{3}\)x + c
Substitute (4, 10) or (1, 5) in the above equation
So,
5 = \(\frac{5}{3}\) + c
So,
c = \(\frac{10}{3}\)
Hence, from the above,
We can conclude that
The slope of a line that passes through the given points is: \(\frac{5}{3}\)
The y-intercept of a line is: \(\frac{10}{3}\)

Question 7.
(8, 2) and (-8,6)
Answer:
The given points are:
(8, 2), and (-8, 6)
Compare the given points with (x1, y1), (x2,y2)
We know that,
Slope(m) = y2 – y1 / x2 – x1
So,
m = \(\frac{6 – 2}{-8 – 8}\)
= \(\frac{4}{-16}\)
= –\(\frac{1}{4}\)
We know that,
The linear equation in the slope-intercept form is:
y = mx + c
Where,
m is the slope
c is the y-intercept
We know that,
We can obtain the y-intercept by putting the value of x equal to 0
So,
y = –\(\frac{1}{4}\)x + c
Substitute (-8, 6) or (8, 2) in the above equation
So,
2 = –\(\frac{1}{4}\) (8) + c
So,
c = 4
Hence, from the above,
We can conclude that
The slope of a line that passes through the given points is: –\(\frac{1}{4}\)
The y-intercept of a line is: 4

Use Functions To Model Relationships 2

Compare Proportional Relationships

Jenna’s mother is shopping for energy drinks in 12-ounce bottles for Jenna’s soccer team. Store A sells a case of 18 bottles for $10. Store B sells a case of 12 bottles for $6. Which store sells the drinks for less? Use the graph to compare the unit costs of the drinks.

Question 8.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 1
Answer:
Jenna’s mother is shopping for energy drinks in 12-ounce bottles for Jenna’s soccer team. Store A sells a case of 18 bottles for $10. Store B sells a case of 12 bottles for $6
Now,
The unit cost rate of a bottle in store A = \(\frac{The cost of 18 bottles}{18}\)
= \(\frac{$10}{18}\)
= $0.55
The unit cost rate of a bottle in store B = \(\frac{The cost of 12 bottles}{12}\)
= \(\frac{$6}{12}\)
= $0.5
So,
The representation of the unit cost rate of a bottle in both stores is:

So,
From the above graph,
We can observe that
The unit cost rate of a bottle in store B < The unit cost rate of a bottle in store A
Hence, from the above,
We can conclude that store B sells the drinks for less cost

Linear Equations

Question 9.
Write the equation for the graph of the line shown.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 3.2
Answer:
The given graph is:
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 3.2
We know that,
The y-intercept is the value of the point that passes through the y-axis
So,
From the graph,
The point that passes through the y-axis is: (0, -6)
So,
The y-intercept is: -6
Now,
To find the slope,
The points are: (2, 2), and (0, -6)
Now,
Compare the given points with (x1, y1), (x2,y2)
We know that,
Slope(m) = y2 – y1 / x2 – x1
So,
m = \(\frac{-6 – 2}{0 – 2}\)
= \(\frac{-8}{-2}\)
= 4
We know that,
The linear equation in the slope-intercept form is:
y = mx + c
So,
y = 4x – 6
Hence, from the above,
We can conclude that the equation of the line for the given graph is:
y = 4x – 6

Language Development

Write key words or phrases associated with each representation. Then write function or not a function on the given lines.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 3.3
Answer:

Topic 3 PICK A PROJECT

PROJECT 3A
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 3.10
What machine could be invented to make your life better?
PROJECT: BUILD A RUBE GOLDBERG MACHINE

PROJECT 3B
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 3.11
What games can you play indoors?
PROJECT: MAKE A MATH
CARD GAME

Use Functions To Model Relationships 3

PROJECT 3C
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 101
What are the steps for fixing a leaky pipe?
PROJECT: PLAN A MAINTENANCE ROUTE

PROJECT 3D
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 102
If you were to make a video game, what kind of game would it be?
PROJECT: DESIGN A VIDEO GAME ELEMENT

Lesson 3.1 Understand Relations and Functions

Solve & Discuss It!

The 10 members of Photography Club want to raise $500, so they will hold a raffle with donated prizes. Jesse proposes that to reach their goal, each member should sell 50 raffle tickets. Alexis proposes that each member should raise $50.
Whose plan would you recommend? Explain.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 3.15
RAFFLE TICKETS
$1 1 ticket
$5 6 tickets
$20 25 tickets
Answer:
It is given that
The 10 members of the Photography Club want to raise $500, so they will hold a raffle with donated prizes. Jesse proposes that to reach their goal, each member should sell 50 raffle tickets. Alexis proposes that each member should raise $50.
It is also given that
$1              –       1 ticket
$5              –        6 tickets
$20            –        25 tickets
Now,
According to Jesse’s goal,
Each member should sell 50 raffle tickets to make the total amount of $500
The possible combinations may be:
50 $1 tickets will be sold by each member
Any other combination is not possible
According to Alexis’s plan,
Each member should raise $50 to make the total amount of $500
It is possible and very easy because $50 by each member can be raised in many ways
Hence, from the above,
We can conclude that Alexis’s plan would be recommended

Focus on math practices
Reasoning How are the two plans different? How are they similar?
Answer:
According to Jesse’s goal,
Each member should sell 50 raffle tickets to make the total amount of $500
The possible combinations may be:
50 $1 tickets will be sold by each member
Any other combination is not possible
Now,
According to Alexis’s plan,
Each member should raise $50 to make the total amount of $500
It is possible and very easy because $50 by each member can be raised in many ways

Use Functions To Model Relationships 4

? Essential Question
when is a relation a function?
Answer:
A “Relation” from a set X to a set Y is called a “Function” if and only if each element of X is related to exactly one element in Y.

Try It!
Joe needs to advertise his company. He considers several different brochures of different side lengths and areas. He presents the data as ordered pairs (side length, area).
(4, 24), (5, 35), (8, 24), (2, 20), (9, 27)
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 4.1
Complete the arrow diagram. Is the area of a brochure a function of the side length? Explain.
Answer:
Joe needs to advertise his company. He considers several different brochures of different side lengths and areas. He presents the data as ordered pairs (side length, area).
(4, 24), (5, 35), (8, 24), (2, 20), (9, 27)
We know that,
The ordered pairs can be represented in the form of (x, y)
Where,
x is the input
y is the output
Now,
From the given ordered pairs,
We can observe that for different values of the input, there are different values of output
Note:
If there are the same outputs for the different inputs, then also a relationship is considered a function
So,
The complete arrow diagram for the given ordered pairs are:

Hence, from the above,
We can conclude that the area of the brochure is a function of the side length

Convince Me!
There are two outputs of 24. Does this help you determine whether the relation is a function? Explain.
Answer:
We know that,
If there are the same outputs for the different inputs, then also a relationship is considered a function
Hence, from the above
We can conclude that even for the two outputs of 24, the given relationship is considered a function

Try It!

Frank reverses the ordered pairs to show the heights and ages of the same six students. Is age a function of height? Explain.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 4.2
Answer:
It is given that
Frank reverses the ordered pairs to show the heights and ages of the same six students.
Now,
The given table is:
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 4.2
Now,
From the given table,
We can observe that
For different values of age, there are different values of height
Where,
Age —-> Input
Height —> Output
Hence, from the above,
We can conclude that age is a function of height

Try It!
Heather claims that she can tell exactly how long a family was at the museum by how much the family pays for parking. Is Heather correct? Explain.
Answer:
It is given that
Heather claims that she can tell exactly how long a family was at the museum by how much the family pays for parking.
Now,
The table for the given situation is: (Example 3)

Now,
From the table,
We can observe that
There are different costs for the different times in hours
So,
We can say that cost is a function of time
Hence, from the above,
We can conclude that Heather’s claim is correct

KEY CONCEPT
A relation is a function if each input corresponds to exactly one output. You can use an arrow diagram or a table to determine whether a relation is a function.
This relation is a function.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 4.3
This relation is not a function.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 4.4

Do You Understand?

Question 1.
? Essential Question
when is a relation a function?
Answer:
A relation from a set X to a set Y is called a function if and only if each element of X is related to exactly one element in Y

Question 2.
Model with Math
How can you use different representations of a relation to determine whether the relation is a function?
Answer:
Relations can be displayed as a table, a mapping, or a graph. In a table, the x-values and y-values are listed in separate columns. Each row represents an ordered pair: Displaying a relation as a table

Question 3.
Generalize
Is a relation always a function? Is a function always a relation? Explain.
Answer:
All functions are relations, but not all relations are functions. A function is a relation that for each input, there is only one output. Here are mappings of functions. The domain is the input or the x-value, and the range is the output or the y-value.

Question 4.
Is the relation shown below a function? Explain.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 4.5
Answer:
The given relation is:
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 4.5
From the given relation,
We can observe that there is the same input for the different outputs,
We know that,
A relation can be considered as a function when the different inputs have different outputs
Hence, from the above,
We can conclude that the given relation is not a function

Question 5.
Is the relation shown below a function? Explain.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 4.6
Answer:
The given relation is:
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 4.6
From the given relation,
We can observe that there are different outputs for different inputs
We know that,
A relation can be considered as a function when the different inputs have different outputs
Hence, from the above,
We can conclude that the given relation is a function

Use Functions To Model Relationships 5

Question 6.
Is the relation shown below a function? Explain.
(4,16), (5, 25), (3,9), (6, 36), (2, 4), (1, 1)
Answer:
The given relation is:
(4,16), (5, 25), (3,9), (6, 36), (2, 4), (1, 1)
From the given relation,
We can observe that there are different outputs for different inputs
We know that,
A relation can be considered as a function when the different inputs have different outputs
Hence, from the above,
We can conclude that the given relation is a function

Practice & Problem Solving

Question 7.
The set of ordered pairs (1, 19), (2, 23), (3, 23), (4, 29), (5, 31) represents the number of tickets sold for a fundraiser. The input values represent the day and the output values represent the number of tickets sold on that day.
a. Make an arrow diagram that represents
Answer:
The arrow diagram for the given relation is:

b. is the relation a function? Explain.
Answer:
It is given that
The set of ordered pairs (1, 19), (2, 23), (3, 23), (4, 29), (5, 31) represents the number of tickets sold for a fundraiser. The input values represent the day and the output values represent the number of tickets sold on that day.
Now,
The given relation is:
(1, 19), (2, 23), (3, 23), (4, 29), (5, 31)
From the given relation,
We can observe that there are different outputs for different inputs
We know that,
A relation can be considered as a function when the different inputs have different outputs
Hence, from the above,
We can conclude that the given relation is a function

Question 8.
Does the relation shown below represent a function? Explain.
(-2, 2), (-7, 1), (-3, 9), (3, 4), (-9,5), (-6, 8)
Answer:
The given relation is:
(-2, 2), (-7, 1), (-3, 9), (3, 4), (-9,5), (-6, 8)
From the given relation,
We can observe that there are different outputs for different inputs
We know that,
A relation can be considered as a function when the different inputs have different outputs
Hence, from the above,
We can conclude that the given relation is a function

Question 9.
Is the relation shown in the table a function? Explain.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 4.10
Answer:
The given relation is:
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 4.10

From the given relation,
We can observe that there are different outputs for the same inputs
We know that,
A relation can be considered as a function when the different inputs have different outputs
Hence, from the above,
We can conclude that the given relation is not a function

Question 10.
Construct Arguments
During a chemistry experiment, Sam records how the temperature changes over time using ordered pairs (time in minutes, temperature in °C).
(0, 15), (5, 20), (10,50) (15, 80). (20, 100), (25, 100) Is the relation a function? Explain.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 5.1
Answer:
It is given that
During a chemistry experiment, Sam records how the temperature changes over time using ordered pairs (time in minutes, temperature in °C).
(0, 15), (5, 20), (10,50) (15, 80). (20, 100), (25, 100)
We know that,
An ordered pair can be represented as (x, y)
Where,
x is the time
y is the temperature in °C
Now,
The given relation is:
(0, 15), (5, 20), (10,50) (15, 80). (20, 100), (25, 100)
From the given relation,
We can observe that there are different outputs for the different inputs
We know that,
A relation can be considered as a function when the different inputs have different outputs
Hence, from the above,
We can conclude that the given relation is a function

Question 11.
Reasoning
Taylor has tracked the number of students in his grade since third grade. He records his data in the table below. Is the relation a function? Explain.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 5.12
Answer:
It is given that
Taylor has tracked the number of students in his grade since third grade. He records his data in the table
Now,
The given table is:
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 5.12
From the given table,
We can observe that there are different outputs (People) for the different inputs (Grade)
We know that,
A relation can be considered as a function when the different inputs have different outputs
Hence, from the above,
We can conclude that the given table is a function

Question 12.
James raises chickens. He tracks the number of eggs his chickens lay at the end of each week. Is this relation a function? Explain.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 5.3
Answer:
It is given that
James raises chickens. He tracks the number of eggs his chickens lay at the end of each week
Now,
The given relation is:
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 5.3
From the given relation,
We can observe that there are different outputs (Eggs) for the different inputs (Weeks)
We know that,
A relation can be considered as a function when the different inputs have different outputs
Hence, from the above,
We can conclude that the given relation is a function

Question 13.
Relations P and Q are shown below.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 5.4
a. Make an arrow diagram to represent Relation P.
Answer:
The given relation is:
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 5.4
Hence,
The arrow diagram to represent the relation P is:

Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 5.5
b. Make an arrow diagram to represent Relation Q.
Answer:
The given relation is:
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 5.5
Hence,
The arrow diagram to represent the relation Q is:

c. Which relation is a function? Explain.
Answer:
From relation P,
We can observe that there are different outputs for the different inputs
From relation Q,
We can observe that there are different outputs for the same inputs
Hence, from the above,
We can conclude that relation P is a function

Question 14.
Higher-Order Thinking
On a recent test, students had to determine whether the relation represented by the ordered pairs (1, 2), (6, 12), (12, 24), (18, 36) is a function. Bobby drew the arrow diagram on the right and said the relationship was not a function. What error did Bobby most likely make?
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 5.6
Answer:
It is given that
On a recent test, students had to determine whether the relation represented by the ordered pairs (1, 2), (6, 12), (12, 24), (18, 36) is a function. Bobby drew the arrow diagram on the right and said the relationship was not a function.
Now,
From the given arrow diagram and ordered pairs,
We can observe that
In the arrow diagram, inputs and outputs are reversely represented
Hence, from the above,
We can conclude that the error Bobby most likely made is the reversal of inputs and outputs

Assessment Practice

Question 15.
Write the set of ordered pairs that is represented by the arrow diagram at the right. Is the relation a function? Explain.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 5.7
Answer:
The given arrow diagram is:
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 5.7
In the arrow diagram,
The left side represented the inputs and the right side represented the outputs
So,
The representation of the arrow diagram in the form of the ordered pairs (Input, Output) are:
(49, 13), (61, 36), (10, 27), (76, 52), (23, 52)
From the above relation,
We can observe that there are different outputs for the different inputs
We know that,
A relation can be considered as a function when the different inputs have different outputs
Hence, from the above,
We can conclude that the given relation is a function

Question 16.
Which of these relations are functions? Select all that apply.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 5.8
Answer:
We know that,
A relation can be considered as a function when the different inputs have different outputs
So,
From the given relations,
Relation 2, Relation 3 are the functions
Hence, from the above,
We can conclude that Relation 2 and Relation 3 are the functions

Lesson 3.2 Connect Representations of Functions

Solve & Discuss It!

Eliza volunteers at a nearby aquarium, where she tracks the migratory patterns of humpback whales from their feeding grounds to their breeding grounds. She recorded the distance, in miles, traveled by the whales each day for the first 7-day period of their migration. Based on Eliza’s data, how long will it take the humpback whales to travel the 3,100 miles to their breeding grounds?
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 5.9

Focus on math practices
Construct Arguments How does finding an average distance the whales travel in miles help with finding a solution to this problem?

? Essential Question
What are different representations of a function?
Answer:
Relationships and functions can be represented as graphs, tables, equations, or verbal descriptions. Each representation gives us certain information. A table of values, mapping diagram, or set of ordered pairs gives us a list of input values and their corresponding output values.

Try It!
As the pump is pumping water, the amount of water in the pool decreases at a constant rate. Complete the statements below. Then graph the function.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 5.10
The amount of water remaining in the pool is Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 5.11 gallons.
The amount of water pumped each hour is Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 5.11 gallons.
The equation is Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 5.11
Answer:
It is given that
As the pump is pumping water, the amount of water in the pool decreases at a constant rate
Now,
Let the initial amount of water present in the pool is: 9,000 gallons
So,
The rate of the amount of water that pumped each hour = \(\frac{The initial amount of water present in the pool}{The time that is present where the initial amount of water present}\)
= \(\frac{9,000}{12}\)
= 750 gallons per hour
So,
The amount of water remaining in the pool = The initial amount of water present in the pool – The amount of water that pumped each hour
= 9,000 – 1,500
= 7,500 gallons
Let the number of hours be x
We know that,
The linear equation is in the form of
y = mx + c
So,
The total amount of water present in the pool = The rate at which the water pumps out + The amount of water that pumped each hour
9,000 = 750x + 7,500
Hence, from the above,
We can conclude that
The amount of water remaining in the pool is 7,500 gallons.
The amount of water pumped each hour is 1,500 gallons.
The equation is:
9,000 = 750x + 7,500

Convince Me!
How is the rate of change of this function different from that in Example 1? Explain.
Answer:
The rate of change of the function present in Example 1 is increasing at a constant rate whereas the rate of change of the function in this situation is decreasing at a constant rate

Try It!
Draw a graph that represents a linear function?
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 5.111
Answer:
We know that,
The representation of the linear equation is:
y = mx —–> Slope form
y = mx + c —-> Slope-intercept form
Now,
Let the linear equation in the slope-intercept form be:
y = x + 3
Hence,
The graph of the above linear equation in the coordinate plane is:

KEY CONCEPT

You can represent a function in different ways: in a table, in a graph, or as an equation.
A day at the amusement park costs $10 for an entrance fee and $2.50 for each ride ticket.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 5.122

Do You Understand?

Question 1.
?Essential Question What are different representations of a function?
Answer:
Relationships and functions can be represented as graphs, tables, equations, or verbal descriptions. Each representation gives us certain information. A table of values, mapping diagram, or set of ordered pairs gives us a list of input values and their corresponding output values.

Question 2.
Use Appropriate Tools How can you use a graph to determine that a relationship is NOT a function?
Answer:
Use the vertical line test to determine whether or not a graph represents a function. If a vertical line is moved across the graph and, at any time, touches the graph at only one point, then the graph is a function. If the vertical line touches the graph at more than one point, then the graph is not a function.

Question 3.
Construct Arguments Must the ordered pairs of a function be connected by a straight line or a curve on a graph? Explain.
Answer:
The points can be connected by a straight line. Thus, the ordered pairs represent a linear function.

Do You Know How?

Question 4.
Each week, Darlene tracks the number of party hats her company has in stock. The table shows the weekly stock. Is the relationship a linear function? Use the graph below to support your answer.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 4.12
Answer:
It is given that
Each week, Darlene tracks the number of party hats her company has in stock. The table shows the weekly stock.
We know that,
A relation is defined as a function only when there are different outputs for different inputs
Now,
From the given table,
We can observe that the outputs (party hats) are different for different inputs (Weeks)
Now,
The given function can be called “Linear function” if the rate of change is constant
The given function can be called a “Non-linear function” if the rate of change is not constant
Now,
The rate of change = Initial value – Next value
Hence, from the above,
We can conclude that the given relationship is a linear function

Question 5.
How can Darlene use the graph above to know when to order more party hats?
Answer:
From the graph,
Darlene know when to order more hats when there is no stack when observing the track sheet

Practice & Problem Solving

Leveled Practice In 6-7, explain whether each graph represents a function.

Question 6.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 6.2
Answer:
The given graph is:
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 6.2
From the above graph,
We can observe that each input has a different output and the rate of change is constant
Hence, from the above,
We can conclude that the given graph represents a function

Question 7.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 6.3
Answer:
The given graph is:
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 6.3
From the given graph,
We can observe that each input has a different output but the rate of change is not constant
Hence, from the above,
We can conclude that the given graph represents a function but a non-linear function

Question 8.
Hannah approximates the areas of circles using the equation A = 3r2 and records areas of circles with different radius lengths in a table.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 6.4
a. Graph the ordered pairs from the table.
Answer:
The given table is:
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 6.4
So,
From the table,
The representation of the ordered pairs (in, in²) are:
(1, 3), (2, 12), (3, 27), (4, 48), (5, 75)
Hence,
The representation of the ordered pairs in the coordinate plane is:

b. Is the relation a function? Explain.
Answer:
From part (a),
We can observe that there are different outputs for different inputs
Hence, from the above,
We can conclude that the given relation is a function

Question 9.
Model with Math
The relationship between the number of hexagons, x, and the perimeter of the figure they form, y, is shown in the graph. is the perimeter of the figure a function of the number of hexagons? Explain.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 6.5
Answer:
It is given that
The relationship between the number of hexagons, x, and the perimeter of the figure they form, y, is shown in the graph
Now,
From the graph,
We can observe that for different values of perimeters, the number of hexagons is also different
Hence, from the above,
We can conclude that the perimeter of a figure is a function of the number of hexagons

Question 10.
Construct Arguments Do the ordered pairs plotted in the graph below represent a function? Explain.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 6.6
Answer:
The given graph is:
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 6.6
From the given graph,
We can observe that for the different values of x, there are different values of y
Hence, from the above,
We can conclude that the given ordered pairs in the graph represents a function

Question 11.
A train leaves the station at time t = 0. Traveling at a constant speed, the train travels 360 kilometers in 3 hours.
a. Write a function that relates the distance traveled, d, to the time, t.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 6.7
Answer:
It is given that
A train leaves the station at time t = 0. Traveling at a constant speed, the train travels 360 kilometers in 3 hours.
We know that,
Speed = \(\frac{Distance}{Time}\)
Here,
Speed is constant
So,
Distance = m (Time)
d = mt
Where,
m is the rate of change or proportionality constant
Now,
m = \(\frac{Distance}{Time}\)
= \(\frac{360}{3}\)
= 120 km / hour
Hence, from the above,
We can conclude that the function that relates to the distance d, and time t is:
d = 120t

b. Graph the function and tell whether it is a linear function or a nonlinear function.
The function is a Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 6.8 function.
Answer:
From part (a),
The function that relates the distance d, and time t is:
d = 120t
Now,
Compare the above equation with y = mx
Hence,
The graph of the above function is:

Hence, from the above,
We can conclude that the given function is a linear function from the above graph

Question 12.
Higher-Order Thinking Tell whether each graph is a function and justify your answer. Which graph is not a good representation of a real-world situation? Explain.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 6.88
Answer:
The given graphs are:
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 6.88
From graph A,
We can observe that there are different outputs for the same input
So,
Graph A does not represent the function
From graph B,
We can observe that there are different outputs for the different inputs
So,
Graph B does represent a function
Hence, from the above,
We can say that graph A does not represent the real-world situation

Assessment Practice

Question 13.
You have an ant farm with 22 ants. The population of ants on your farm doubles every 3 months.
PART A
Complete the table.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 7.1
Answer:
It is given that
You have an ant farm with 22 ants. The population of ants on your farm doubles every 3 months.
Hence,
The completed table for the given situation is:

PART B
Is the relation a function? If so, is it a linear function or a nonlinear function? Explain.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 7.2
Answer:
From part (a),
We can observe that the ant population is different for the different number of months
Now,
Rate of change = \(\frac{44}{22}\)
= 2
So,
The rate of change is also constant for all the table
Hence, from the above,
We can conclude that the given relation is a linear function

Question 14.
Use the function y = \(\frac{3}{2}\)x + 3 to complete the table of values.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 7.3
Answer:
The given function is:
y = \(\frac{3}{2}\)x + 3
Hence,
The completed table for the given values of y is:

Lesson 3.3 Compare Linear and Nonlinear Functions

Solve & Discuss It!

Two streaming video subscription services offer family plans with different monthly costs, as shown in the ads below. What do the two plans have in common? How are they different? When is Movies4You a better deal than Family Stream?
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 7.4
Answer:
It is given that
Two streaming video subscription services offer family plans with different monthly costs, as shown in the ads
Now,
Let the number of devices be x
Let the total subscription cost be y
So,
For Movies 4 U,
The total subscription cost = The subscription cost of the first device + The subscription fee for additional devices
y = $10 + $2x
For Family Stream,
The total subscription cost = The subscription cost of the first device + The subscription fee for additional devices
y = \(\frac{$12}{4}\) + $1x
y = $3 + $1x
Now,
The above equations are in the form of slope-intercept form
We know that,
The slope-intercept form of the equation is:
y = mx + c
Now,
The common points in the two plans are:
A) The subscription cost of the first device plan
B) The additional fee plan
The different points in the two plans are:
A) The first plan consists of the additional fee of each device
B) The second plan consists of the additional fee for greater than 5 devices
Now,
Movies 4 U is better than Family Stream when the subscription cost of the first device will be less

Model with Math
How can you represent the relationship between cost and number of devices?
Answer:
The representation of the relationship between the cost and the number of devices is:
The total subscription cost = The subscription cost of the first device + The subscription fee for additional devices

Focus on math practices
Look for Relationships Describe the relationship between the cost and the number of devices for each service. What do you notice about each relationship?
Answer:
The relationship between the cost and the number of devices for each service is:
The total subscription cost = The subscription cost of the first device + The subscription fee for additional devices
In the service of Family Stream,
The subscription fee is given for up to 4 devices and the additional subscription fee is for greater than 5 devices

? Essential Question
How can you compare two functions?
Answer:
The two functions can be compared by:
A) Identify the rate of change for the first function
B) Identify the rate of change for the second function
C) Identify the y-intercept of the first function
D) Identify the y-intercept of the second function
E) Compare the properties of each function

Try It!
The welding rate of a third robot is represented by the equation t = 10.8w, where t represents the time in minutes and w represents the number of welding tasks. How does it compare to the other two?
Answer:
It is given that
The welding rate of a third robot is represented by the equation t = 10.8w, where t represents the time in minutes and w represents the number of welding tasks (Refer to Example 1)
Now,
For a third robot,
The wielding rate = \(\frac{The number of wielding tasks (w)}{Time in minutes (t)}\)
= 10.8 (From the equation t = 10.8w)
Now,
When we compare the wielding rates of the three robots,
The wielding rate of the first robot (10.4) < The wielding rate of the third robot (10.8) < The wielding rate of the second robot (11.2)
Hence, from the above,
We can conclude that the comparison of the wielding rates of the three robots is:
The first robot’s wielding rate < The third robot’s wielding rate < The second robot’s wielding rate

Convince Me!
How can linear equations help you compare linear functions?
Answer:
While all linear equations produce straight lines when graphed, not all linear equations produce linear functions. In order to be a linear function, a graph must be both linear (a straight line) and a function (matching each x-value to only one y-value).

Try It!
Compare the properties of these two linear functions.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 8.1
Answer:
The given functions are:
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 8.1
Now,
For function 1,
Find out the rate of change and the y-intercept i.e., the initial value
We know that,
Rate of change = \(\frac{y}{x}\)
Now,
For y = 1 and x = 2.
Rate of change = 0.5
For y = 5.5 and x = 5,
Rate of change = 1.1
SO,
From the above values,
We can say that the rate of change is not constant
So,
The given function is a non-linear function and it does not have any initial value i.e., the y-intercept is 0
Now,
For function 2,
Compare the given equation with
y = mx + c
Where,
m is the slope or the rate of change
c is the initial value or the y-intercept
So,
From the given equation,
Rate of change (m): 2
The y-intercept is: -4
Hence, from the above 2 functions,
We can conclude that
The y-intercept of function1 > The y-intercept of function 2

KEY CONCEPT
You can compare functions in different representations by using the properties of functions.
Compare the constant rate of change and the initial value.
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 8.2

Do You Understand?

Question 1.
? Essential Question
How can you compare two functions?
Answer:
The two functions can be compared by:
A) Identify the rate of change for the first function
B) Identify the rate of change for the second function
C) Identify the y-intercept of the first function
D) Identify the y-intercept of the second function
E) Compare the properties of each function

Question 2.
Reasoning Anne is running on a trail at an average speed of 6 miles per hour beginning at mile marker 4. John is running on the same trail at a constant speed, shown in the table. How can you determine who is running faster?
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 9.1
Answer:
It is given that
Anne is running on a trail at an average speed of 6 miles per hour beginning at mile marker 4. John is running on the same trail at a constant speed, shown in the table.
So,
For Anne,
The rate of change is defined as the average speed
The y-intercept is defined as the beginning point
Hence,
For Anne,
The rate of change is: 6 miles per hour
The y-intercept is: 4
Now,
The given table is:
Envision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships 9.1
From the given table,
For John,
Rate of change = \(\frac{y}{x}\) = \(\frac{Mile marker}{Time (hours)}\)
Now,
The total distance traveled by John (y) = Final value – Initial value
= 11.5 – 1
= 10.5 miles
The total time took by John (x) = 1.5 hours
So,
Rate of change = \(\frac{y}{x}\)
= \(\frac{10.5}{1.5}\)
= 7 miles per hour
Now,
We know that,
The y-intercept is the value of y when x = 0
So,
The y-intercept is: 1
Hence, from the above,
By comparing the values of the rate of change,
We can conclude that John is running faster

Question 3.
Reasoning In Item 2, how do Anne and John’s starting positions compare? Explain.
Answer:
We know that,
The starting positions are nothing but the initial positions of both Anne and John i.e., the y-intercepts of both Anne and John
So,
The y-intercept of Anne is: 4
The y-intercept of John is: 1
Hence, from the above,
By comparing the y-intercepts,
We can conclude that
The starting position of Anne > The starting position of John

Do You Know How?

Felipe and Samantha use a payment plan to buy musical instruments. Felipe writes the equation y = -30x + 290 to represent the amount owed, y, after x payments. The graph shows how much Samantha owes after each payment.
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.2
Answer:
It is given that
Felipe and Samantha use a payment plan to buy musical instruments. Felipe writes the equation y = -30x + 290 to represent the amount owed, y, after x payments. The graph shows how much Samantha owes after each payment.
Now,
Compare the given equation with
y = mx + c
Where,
m is the rate of change
c is the initial value or the y-intercept
So,
From the given equation
For Felipe,
The rate of change is: -30
The initial value is: 290
Now,
The given graph is:
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.2
For Samantha,
From the given graph,
The initial value is: 240
The rate of change = \(\frac{Any value of y}{The value of x that corresponds to the value of y}\)
= \(\frac{120}{6}\)
= 20

Question 4.
Whose musical instrument costs more, Felipe’s or Samantha’s? Explain.
Answer:
We know that,
If the amount owed (y) is less i.e, the rate of change is negative, then the cost of the instrument will also be less
So,
When we compare the rate of change of Felip and Samantha,
The rate of change of Felip < The rate of change of Samantha
Hence, from the above comparison,
We can conclude that the instrument of Samantha costs more

Question 5.
Who will pay more each month? Explain.
Answer:
When we compare the rate of change of Felip and Samantha,
The rate of change of Felip < The rate of change of Samantha
Hence, from the above comparison,
We can conclude that Samantha will pay more each month

Practice & Problem Solving

Question 6.
Two linear functions are shown below. Which function has the greater rate of change?
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.3
Answer:
The given linear functions are:
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.3
We know that,
The rate of change = \(\frac{y}{x}\)
So,
For Function A,
The rate of change = \(\frac{Any value of y}{The value of x that corresponds to y}\)
= \(\frac{4}{2}\)
= 2
For Function B,
The rate of change = \(\frac{y}{x}\)
=  \(\frac{3}{2}\)
= 1.5
Hence, from the above,
We can conclude that Function A has a greater rate of change

Question 7.
Two linear functions are shown below. Which function has the greater initial value?
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.4
Answer:
The given functions are:
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.4
We know that,
The initial value is also known as the y-intercept
The y-intercept is the value of y when x = 0
So,
For Function A,
The initial value (y-intercept) is: 4
For function B,
Compare the given equation with
y = mx + c
Where,
m is the rate of change
c is the initial value or the y-intercept
So.
The initial value (y-intercept) is: 3
Hence, from the above,
We can conclude that Function A has the greater initial value

Question 8.
Tell whether each function is linear or nonlinear.
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.5
Answer:
The given functions are:
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.5
We know that,
To find whether the given function is linear or non-linear, we have to verify whether the rate of change is constant or not
If the rate of change is constant, then the function is linear
If the rate of change is not constant, then the function is non-linear
Now,
For Function A,
The rate of change = \(\frac{y}{x}\)
For x = 1 and y = 2,
The rate of change = 2
For x = 2 and y = 5,
The rate of change = 2.5
Hence,
Function A is a non-linear function
Now,
For Function B,
The rate of change = \(\frac{y}{x}\)
For x = 1 and y = 4,
The rate of change = 4
For x = 1.5 and y = 3,
The rate of change = 2
Hence,
Function B is a non-linear function

Question 9.
Tell whether each function is linear or nonlinear.
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.6
Answer:
The given functions are:
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.6
We know that,
For a relation to be a graph, each input has a different output but the same input will not have different outputs
Now,
From Function A,
We can observe that there are different inputs for different outputs i.e., the values of x and y are different
So,
The rate of change is not constant since the graph is non-linear
Hence,
Function A is a non-linear function
Now,
From Function B,
Compare the given equation with
y = mx + c
Where,
m = 1
c = 0
So,
The rate of change is constant for Function B
Hence,
Function B is a linear function

Question 10.

Determine whether each function is linear or nonlinear from its graph.
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.7
Answer:
The given graphs are:
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.7
We know that,
If the graph is a straight line, then the function is a linear function
If the graph is not a straight line, then the function is not a linear function
Hence, from the above,
We can conclude that
The function 1 is a linear function
The function 2 is a non-linear function

Question 11.
Look for Relationships Justin opens a savings account with $4. He saves $2 each week. Does a linear function or a nonlinear function represent this situation? Explain.
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.8
Answer:
It is given that
Justin opens a savings account with $4. He saves $2 each week.
Now,
The given table is:
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.8
From the given table,
We can observe that there is a constant rate of change
Now,
For weeks,
We can obtain the number of weeks by adding 1 i.e., 0 + 1, 1 + 1, etc
So,
The rate of change is constant i.e., 1
For money in account,
We can obtain the money by adding 2 to the initial amount of money i.e., 4 + 2, 6 + 2, etc
So,
The rate of change is constant i.e., 2
Hence, from the above,
We can conclude that since the rate of change is constant, the given situation represents a linear function

Question 12.
Reasoning The function y = 4x + 3 describes Player A’s scores in a game of trivia, where x is the number of questions answered correctly and y is the score. The function represented in the table shows Player B’s scores. What do the rates of change tell you about how each player earns points?
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.9
Answer:
It is given that
The function y = 4x + 3 describes Player A’s scores in a game of trivia, where x is the number of questions answered correctly and y is the score. The function represented in the table shows Player B’s scores.
Now,
For Player A,
The given equation is:
y = 4x + 3
Compare the givene quation with
y = mx + c
Where,
m is the rate of change
c is the y-intercept
So,
The rate of change of Player A is: 4
Now,
The given table is:
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.9
For Player B,
From the given table,
We can observe that the rate of change is constant for both the values of x and y
The rate of change for both the values of x and y is: 1
So,
The rate of change of Player B is: 1
So,
The rate of change of Player A > The rate of change of Player B
Hence, from the above,
We can conclude that Player A earns more points

Question 13.
Two athletes are training over a two-week period to increase the number of push-ups each can do consecutively. Athlete A can do 16 push-ups to start, and increases his total by 2 each day. Athlete B’s progress is charted in the table. Compare the initial values for each. What does the initial value mean in this situation?
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.10
Answer:
It is given that
Two athletes are training over a two-week period to increase the number of push-ups each can do consecutively. Athlete A can do 16 push-ups to start, and increases his total by 2 each day. Athlete B’s progress is charted in the table
Now,
For Athlete A,
The starting point is: 16
So,
The initial point for Athlete A is: 16
Now,
For Athlete B,
The given table is:
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.10
We know that,
The initial point or the y-intercept is the value of y when x = 0
So,
The initial point for Athlete B is: 12
Hence, from the above,
We can conclude that
The initial points in the given situation describe the number of pushups one can do at a time without stop
The initial point for Athlete A > The initial point for Athlete B

Question 14.
Higher-Order Thinking The equation y = 4x – 2 and the table and graph shown at the right describe three different linear functions. Which function has the greatest rate of change? Which has the least? Explain.
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.11
Answer:
It is given that
The equation y = 4x – 2 and the table and graph shown at the right describe three different linear functions
Now,
a)
The given equation is:
y = 4x – 2
Compare the given equation with
y = mx + c
Where,
m is the constant rate of change
So,
For the given equation,
The rate of change is: 4
b)
The given table and graph are:
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.11
Now,
From the given table,
The rate of change = \(\frac{y}{x}\)
For x = 1 and y = 5,
The rate of change is: 5
For x = 2 and y = 10
The rate of change is: 5
Now,
Since the rate of change is constant for all the cases,
The rate of change for the given table is: 5
Now,
From the given graph,
The given points to find the slope are: (0, 4), and (2, 0)
So,
Slope (or) The rate of change = \(\frac{0 – 4}{2 – 0}\)
= \(\frac{-4}{2}\)
= -2
Now,
When we compare the rate of change for all the three linear functions,
The rate of change of the table > The rate of change of the equation < The rate of change of the graph
Hence, from the above,
We can conclude that
The function that has the greatest rate of change is: Table
The function that has the least rate of change is: Graph

Assessment Practice

Question 15.
The students in the After-School Club ate 12 grapes per minute. After 9 minutes, there were 32 grapes remaining. The table shows the number of carrots remaining after different amounts of time. Which snack did the students eat at a faster rate? Explain.
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.12
Answer:
It is given that
The students in the After-School Club ate 12 grapes per minute. After 9 minutes, there were 32 grapes remaining. The table shows the number of carrots remaining after different amounts of time.
Now,
The rate of change of grapes consumption is: 12 grapes per minute
Now,
The given table is:
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.12
Now,
The rate of change of carrots consumption = \(\frac{The difference between any 2 values of carrots remaining}{The values of the tie elapsed corresponding to the carrots remaining}\)
= \(\frac{118 – 136}{8 – 6}\)
= –\(\frac{18}{2}\)
= -9 carrots per minute
So,
The consumption rate of grapes > The consumption rate of carrots
Hence, from the baove,
We can conclude that grapes can be eaten at a faster rate

Question 16.
The height of a burning candle can be modeled by a linear function. Candle A has an initial height of 201 millimeters, and its height decreases to 177 millimeters after 4 hours of burning. The height, h, in millimeters, of Candle B, can be modeled by the function h = 290 – 5t, where t is the time in hours. Which of the following statements are true?
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.13 The initial height of Candle A is greater than the initial height of Candle B.
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.13 The height of Candle A decreases at a faster rate than the height of Candle B.
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.13 Candle B will burn out in 58 hours.
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.13 After 10 hours, the height of Candle A is 110 millimeters.
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 9.13 Candle A will burn out before Candle B.
Answer:
Let the given options be named as A, B, C, D, and E
It is given that
The height of a burning candle can be modeled by a linear function. Candle A has an initial height of 201 millimeters, and its height decreases to 177 millimeters after 4 hours of burning. The height, h, in millimeters, of Candle B, can be modeled by the function h = 290 – 5t, where t is the time in hours
Now,
The rate of change of Candle A = \(\frac{201 – 177}{4}\)
= \(\frac{24}{4}\)
= 6 millimeters per hour
Now,
For Candle B,
The time to burn the Candle B = \(\frac{290}{5}\)
= 58 hours
Hence, from the above,
We can conclude that B, C, and E match with the given situation

Topic 3 MID-TOPIC CHECKPOINT

Question 1.
Vocabulary How can you determine whether a relation is a function? Lesson 3.1
Answer:
Identify the output values. If each input value leads to only one output value, classify the relationship as a function. If any input value leads to two or more outputs, do not classify the relationship as a function.

Question 2.
Can an arrow or arrows be drawn from 10.3 so the relation in the diagram is a function? Explain your answer. Lesson 3.1
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 102
Answer:
The given diagram is:
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 102
For the relation in the diagram to be a function,
The arrows from the right side to the left side can be many but the arrows from the left side to the right side can only be one

Question 3.
Two linear functions are shown below. Which function has the greater rate of change? Justify your response. Lesson 3.3
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 10.1
Answer:
The given linear functions are:
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 10.1
We know that,
For a linear function,
The rate of change is constant
So,
For Function A,
The rate of change = \(\frac{y}{x}\)
= \(\frac{3}{6}\)
= \(\frac{1}{2}\)
= 0.5
For Function B,
The given equation is:
y = \(\frac{1}{2}\)x – 1
y = 0.5x – 1
Compare the above equation with
y = mx + c
Where,
m is the rate of change
So,
The rate of change is: 0.5
Hence, from the above,
We can conclude that the two linear functions have the same rate of change

Question 4.
Neil took 3 math tests this year. The number of hours he spent studying for each test and the corresponding grades he earned is shown in the table. Is the relation of hours of study time to the grade earned on a test a function? Explain why. Use the graph to justify your answer. Lesson 3.2
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 10.2
Answer:
It is given that
Neil took 3 math tests this year. The number of hours he spent studying for each test and the corresponding grades he earned is shown in the table.
Now,
For the number of hours as input and the Grades as the output,
We can observe that
For each value of the hours, there are the same values of the Grade
So,
The relation of hours of study time to the grade earned on a test is not considered a function
Now,
The representation of the relation in the form of the ordered pairs is:
(4, 75), (6, 75), and (6, 82)
Hence,
The representation of the ordered pairs in the coordinate plane is:

Question 5.
Is the function shown linear or nonlinear? Explain your answer. Lesson 3.3
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 10.3
Answer:
The given graph is:
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 10.3
From the given graph,
We can observe that
For each value of x, there is only 1 value of y
So,
The given graph is a function
We know that,
A function is called a linear if the graph is a straight line
A function is called non-linear if the graph is in any shape other than the straight line
Hence, from the above,
We can conclude that the given graph is a non-linear function

How well did you do on the mid-topic checkpoint? Fill in the stars.Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 10.5

Topic 3 MID-TOPIC PERFORMANCE TASK

Sarah, Gene, and Paul are proposing plans for a class fundraiser. Each presents his or her proposal for the amount of money raised, y, for x number of hours worked, in different ways.
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 10.4
Answer:
The given graphs are:
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 10.4
Now,
a)
From the given graph,
We can observe that
For each value of x, there is only 1 value of y
The given graph is a straight line
Now,
The rate of change of the given graph = \(\frac{y}{x}\)
= \(\frac{0 – 10}{1 – 0}\)
= -10
Hence,
The given graph is considered a linear function
b)
From the given table,
To consider a function linear, verify whether the rate of change is constant or not
Now,
For all the values of x,
The rate of change is: 5
For all the values of y,
The rate of change is: 35
So,
The rate of change of the given table = \(\frac{y}{x}\)
= \(\frac{5}{5}\)
= 7
Hence,
The given table is considered a linear function
c)
The given equation is:
y = 10x + 7
Compare the given equation with
y = mx + c
Where,
m is the rate of change
So,
The rate of change for the given equation is: 10
Hence,
The given equation is considered a linear function

PART A
Is each of the proposals represented by linear functions? Explain.
Answer:
Yes, all the proposals are represented by linear functions

PART B
Does the class have any money in the account now? How can you tell?
Answer:
From the given graph,
We can observe that the straight line does not start from 0 but from 10
So,
The initial value of the graph will be: 10
We know that,
The initial value is considered the y-intercept
Hence, from the above,
We can conclude that the class have the money in the account now i.e., $10

PART C
Which fundraising proposal raises money at the fastest rate? Explain.
Answer:
Since the rate of change is the highest for Paul’s proposal,
Paul’s Proposal raises money at the fastest rate

PART D
If Sarah and her classmates are hoping to raise $200, which proposal do you recommend that Sarah and her classmates choose? Explain why you recommend that proposal.
Answer:
It is given that Sarah and her classmates are hoping to raise $200
So,
To raise the money,
We have to choose the plan which has the highest rate of change
Hence, from the above,
We can conclude that Sarah and her classmates choose Paul’s proposal

3-Act Mathematical Modeling: Every Drop Counts

3-ACT MATH

Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 10.55

АСТ 1

Question 1.
After watching the video, what is the first question that comes to mind?
Answer:
After watching the video,
The first question that comes to mind is:
How much amount of water people waste brushing their teeth?

Question 2.
Write the Main Question you will answer.
Answer:
The main question you will answer is:
How much amount of water people waste brushing their teeth?

Question 3.
Construct Arguments Predict an answer to this Main Question. Explain your prediction.
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 10.6
Answer:
The answer to the main question is: 4 gallons
The prediction of the answer for the main question is according to the surveys done by International Organisations

Question 4.
On the number line below, write a number that is too small to be the answer. Write a number that is too large.
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 10.7
Answer:
From the above,
We can observe that
The maximum amount (Too large) of water used to brush teeth is: 4 gallons
The minimum amount (Too small) of water used to brush the teeth is: 2 gallons
Hence,
The representation of the amounts of water used to brush teeth in this situation is:

Question 5.
Plot your prediction on the same number line.
Answer:
From the above,
We can observe that there are minimum and maximum amounts of water used to brush the teeth

Now,
Let x be the amount of water used to brush the teeth
So,
The prediction will be: 2 < x < 4
Hence,
The representation of the prediction on the number line is:

ACT 2

Question 6.
What information in this situation would be helpful to know? How would you use that information?
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 10.8
Answer:
The information in this situation that would be helpful to know is:
How much time did it take to completely brush your teeth?
From the above information,
We can estimate the amount of water used to brush your teeth

Question 7.
Use Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.
Answer:

Question 8.
Model with Math
Represent the situation using mathematics.
Use your representation to answer the Main Question.
Answer:

Question 9.
What is your answer to the Main Question? Is it higher or lower than your prediction? Explain why.
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 11.1
Answer:

ACT 3

Question 10.
Write the answer you saw in the video.
Answer:

Question 11.
Reasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 11.3
Answer:

Question 12.
Make Sense and Persevere Would you change your model now that you know the answer? Explain.
Answer:

ACT 3

Reflect

Question 13.
Model with Math
Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 11.4
Answer:

Question 14.
Be Precise How do the units you chose and the method you used help you communicate your answer?
Answer;

SEQUEL

Question 15.
Use Structure How much water will he save in a year?
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 11.5
Answer:

Lesson 3.4 Construct Functions to Model Linear Relationships

ACTIVITY

Explore It!
Erick wants to buy a new mountain bike that costs $250. He has already saved $120 and plans to save $20 each week from the money he earns for mowing lawns. He thinks he will have saved enough money after seven weeks.
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 11.6
Answer:
It is given that
Erick wants to buy a new mountain bike that costs $250. He has already saved $120 and plans to save $20 each week from the money he earns for mowing lawns. He thinks he will have saved enough money after seven weeks.
So,
The total amount he saved = The amount he saved already + The amount he planned to save each week
Let,
The number of weeks —-> x
The total amount he saved —-> y
So,
y = $20x + $120
Compare the above equation with y = mx + c
Where,
m s the rate of change (or) slope
c is the y-intercept
So,
For the given equation,
The rate of change (m) is: 20
The y-intercept is: 120

A. Complete the table. Then graph the data.
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 11.7
Answer:
The equation is:
y = $20x + $120
So,
The completed table for the above equation is:

So,
The representation of the given equation in the coordinate plane is:

B. How can you tell that the relationship is a linear function from the table? How can you tell from the graph?
Answer:
We know that,
A relation is said to be a function when an input value matches only with an output value
A function is said to be a linear function when the graph of that function is a straight line
So,
From part (a),
From the table,
We can observe that for each value of the week, there is a different amount saved
So,
From the table,
The given relation is said to be a function
From the graph of the equation,
We can observe that the graph is a straight line
So,
We can say that the function is a linear function

Focus on math practices
Generalize How can the different representations help you determine the properties of functions?
Answer:
The different representations of the functions are:
A) Symbolic or Algebraic representation – The properties can be found out by comparing with the standard form
B) Numerical (Tables) representation – The properties can be found out by the rate of change and the initial values
C) Graphical representation – The properties can be found out by the values of x and y
D) Verbal representation – The properties can be found out by the keywords

? Essential Question
How can you use a function to represent a linear relationship?
Answer:
Another approach to representing linear functions is by using function notation. One example of function notation is an equation written in the form known as the slope-intercept form of a line, where x is the input value, m is the rate of change, and c is the initial value of the dependent variable.

Try It!
How will the height of the ramp change if the plan shows that for every 3 inches of height, the triangle should have a base that is 15 inches long?
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 11.8
Graph the function. The slope of the function shown in the graph is Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 11.9. The equation of the function is y = Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 11.9
x. If the base length is 110 inches, then the height of the ramp will be Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 11.9 inches.
Answer:
It is given that
The plan shows that for every 3 inches of height, the triangle should have a base that is 15 inches long
So,
The rate of change (m) = \(\frac{Rise}{Run}\)
m = \(\frac{3}{15}\)
m = \(\frac{1}{5}\)
So,
The slope of the function shown in the graph is: \(\frac{1}{5}\)
Now,
We know that,
The representation of the linear equation is:
y = mx
So,
y = \(\frac{1}{5}\)x
Where,
y is the height of the ramp
x is the base length of the triangle
Now,
For the base length of 110 inches,
y = \(\frac{1}{5}\) (110)
y = 22 inches
Hence, from the above,
We can conclude that the height of the ramp will be 22 inches for the base length of 110 inches

Convince Me!
Explain why the initial value and the y-intercept are equivalent.
Answer:
An equation in slope-intercept form of a line includes the slope and the initial value of the function. The initial value, or y-intercept, is the output value when the input of a linear function is zero. It is the y-value of the point where the line crosses the y-axis.

Try It!
Jin is tracking how much food he feeds his dogs each week. After 2 weeks, he has used 8\(\frac{1}{2}\) cups of dog food. After 5 weeks, he has used 21\(\frac{1}{4}\) cups. Construct a function in the form y = mx + b to represent the amount of dog food used, y, after x weeks.
Answer:
It is given that
Jin is tracking how much food he feeds his dogs each week. After 2 weeks, he has used 8\(\frac{1}{2}\) cups of dog food. After 5 weeks, he has used 21\(\frac{1}{4}\) cups.
Now,
Let x be the number of weeks
Let y be the number of cups of dog food used
So,
For x = 2, y = 8.5 (The value of 8\(\frac{1}{2}\))
For x = 5, y = 21.25 (The value of 21\(\frac{1}{4}\))
We know that,
The equation in the slope-intercept form is:
y = mx + c
So,
8.5 = 2m + c —–> (1)
21.25 = 5m + c ——> (2)
Solve eq (1) and eq (2)
So,
8.5 = 2m + 21.25 – 5m
8.5 – 21.25 = 2m – 5m
-12.75 = -3m
3m = 12.75
m = \(\frac{12.75}{3}\)
m = 4.25
Now,
For the value of c,
Substitute the value of m either in eq (1) or in eq (2)
So,
8.5 = 2m + c
8.5 = 2 (4.25) + c
8.5 – 8.5 = c
c = 0
Hence, from the above,
We can conclude that the representation of the amount of dog used y, after x weeks in the equation form is:
y = 4.25x

Try It!
The graph shows the relationship between the number of pages printed by a printer and the warm-up time before each printing. What function in the form y = mx + b represents this relationship?
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 13.1
Answer:
It is given that
The graph shows the relationship between the number of pages printed by a printer and the warm-up time before each printing
Now,
The given graph is:
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 13.1
From the given graph,
We can observe that there is an initial value or y-intercept
So,
The y-intercept from the given graph is: 1
Now,
To find the slope from the given graph,
The given points are: (30, 4), and (10, 2)
So,
Slope (m) = \(\frac{2 – 4}{10 – 30}\)
m = \(\frac{-2}{-20}\)
m = \(\frac{1}{10}\)
We know that,
The representation of the equation in the slope-intercept form is:
y = mx + c
So,
y = \(\frac{1}{10}\)x + 1
Hence, from the above,
We can conclude that the equation that represents the given situation is:
y = \(\frac{1}{10}\)x + 1

KEY CONCEPT

A function in the form y= mx + b represents a linear relationship between two quantities, x, and y.
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 13.2

Do You Understand?

Question 1.
?Essential Question How can you use a function to represent a linear relationship?
Answer:
Another approach to representing linear functions is by using function notation. One example of function notation is an equation written in the form known as the slope-intercept form of a line, where x is the input value, m is the rate of change, and c is the initial value of the dependent variable.

Question 2.
Make Sense and Persevere Tonya is looking at a graph that shows a line drawn between two points with a slope of -5. One of the points is smudged and she cannot read it. The points as far as she can tell are (3, 5) and (x, 10). What must the value of x be? Explain.
Answer:
It is given that
Tonya is looking at a graph that shows a line drawn between two points with a slope of -5. One of the points is smudged and she cannot read it. The points as far as she can tell are (3, 5) and (x, 10)
Now,
We know that,
Slope (m) = y2 – y1 / x2 – x1
Compare the given points with (x1, y1), (x2, y2)
So,
–5 = \(\frac{10 – 5}{x – 3}\)
-5 = \(\frac{5}{x – 3}\)
-5 (x – 3) = 5
-5 (x) + 5 (3) = 5
-5x + 15 = 5
-5x = 5 – 15
-5x = -10
5x = 10
x = \(\frac{10}{5}\)
x = 2
Hence, from the above,
We can conclude that the value of x is: 2

Question 3.
Reasoning What is the initial value of all linear functions that show a proportional relationship?
Answer:
We know that,
The representation of the proportional relationship is:
y = mx + 0
Where,
m is the slope (or) rate of change
So,
From the above equation,
We can say that the initial value (or) the y-intercept is: 0
Hence, from the above,
We can conclude that the initial value of all linear functions that show a proportional relationship is: 0

Do You Know How?

Question 4.
Write a function in the form y = mx + b for the line that contains the points (-8.3, -5.2) and (6.4, 9.5).
Answer:
The given points are:
(-8.3, -5.2) and (6.4, 9.5)
Now,
We know that,
Slope (m) = y2 – y1 / x2 – x1
So,
Compare the given points with (x1, y1), (x2, y2)
So,
m = \(\frac{9.5 + 5.2}{6.4 + 8.3}\)
m = \(\frac{14.7}{14.7}\)
m = 1
We know that,
The representation of the equation in the slope-intercept form is:
y = mx + c
So,
y = x + c
Now,
To find the value of c,
Substitute any one of the points in the above equation
So,
-5.2 = -8.3 + c
c = 8.3 – 5.2
c = 3.1
Hence, from the above,
We can conclude that the representation of the linear equation for the given points is:
y = x + 3.1

Question 5.
The data in the table below represent a linear relationship. Fill in the missing data.
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 14.1
Answer:
It is given that the data in the table represent a linear relationship
So,
For a linear relationship, for each value of x, there is only 1 value of y
Now,
To find the missing data,
Find the rate of change for x and y
For all the values of x,
The rate of change = 20 – 10
= 10
For all the values of y,
The rate of change = 15 – 10
= 5
Hence,
The completed table with the missing data is:

Question 6.
What is an equation that represents the linear function described by the data in Item 5?
Answer:
From the data in Item 5,
Slope (m) = \(\frac{The rate of change of y}{The rate of change of x}\)
m = \(\frac{5}{10}\)
m = \(\frac{1}{2}\)
We know that,
The equation that represents a linear relationship is:
y = mx
So,
y = \(\frac{1}{2}\)x
Hence, from the above,
The equation that represents the linear function described by the data in Item 5 is:
y = \(\frac{1}{2}\)x

Practice & Problem Solving

Question 7.
A line passes through the points (4, 19) and (9, 24). Write a linear function in the form y = mx + b for this line.
Answer:
The given points are:
(4, 19) and (9, 24)
Now,
We know that,
Slope (m) = y2 – y1 / x2 – x1
So,
Compare the given points with (x1, y1), (x2, y2)
So,
m = \(\frac{24 – 19}{9 – 4}\)
m = \(\frac{5}{5}\)
m = 1
We know that,
The representation of the equation in the slope-intercept form is:
y = mx + c
So,
y = x + c
Now,
To find the value of c,
Substitute any one of the points in the above equation
So,
19 = 4 + c
c = 19 – 4
c = 15
Hence, from the above,
We can conclude that the representation of the linear equation for the given points is:
y = x + 15

Question 8.
What is a linear function in the form y = mx + b for the line passing through (4.5, -4.25) with y-intercept 2.5?
Answer:
It is given that
A line passing through (4.5, -4.25) with y-intercept 2.5
We know that,
The y-intercept is the value of y when x = 0
Now,
The given points are:
(4.5, -4.25) and (0, 2.5)
Now,
We know that,
Slope (m) = y2 – y1 / x2 – x1
So,
Compare the given points with (x1, y1), (x2, y2)
So,
m = \(\frac{2.5 + 4.25}{0 – 4.5}\)
m = \(\frac{6.75}{-4.5}\)
m = -1.5
We know that,
The representation of the equation in the slope-intercept form is:
y = mx + c
So,
y = -1.5x + 2.5
Hence, from the above,
We can conclude that the representation of the linear equation for the given points is:
y = -1.5x + 2.5

Question 9.
A car moving at a constant speed passes a timing device at t = 0. After 8 seconds, the car has traveled 840 feet. What linear function in the form y = mx + b represents the distance in feet, d, the car has traveled any number of seconds, t, after passing the timing device?
Answer:
It is given that
A car moving at a constant speed passes a timing device at t = 0. After 8 seconds, the car has traveled 840 feet
We know that,
Speed = \(\frac{Distance}{Time}\)
So,
For a constant speed,
Time is considered as input and distance is considered as the output
It is also given that time will start from (0, 0) i..e, at t = 0
So,
The representation of the equation that passes through the origin is:
y = mx
Where,
m is the rate of change (or) slope
In this situation,
The rate of change = \(\frac{Distance}{Time}\)
So,
y = \(\frac{840}{8}\)x
y = 105x
Hence, from the above,
We can conclude that the linear equation that represents the distance traveled by a car after passing the timing device is:
y = 105x

Question 10.
At time t = 0, water begins to drip out of a pipe into an empty bucket. After 56 minutes, 8 inches of water are in the bucket. What linear function in the form y = mx + b represents the amount of water in inches, w, in the bucket after t minutes?
Answer:
It is given that
At time t = 0, water begins to drip out of a pipe into an empty bucket
Now,
In this situation,
Time is considered as output and the amount of water in the bucket is considered as the input
It is also given that time will start from (0, 0) i..e, at t = 0
So,
The representation of the equation that passes through the origin is:
y = mx
Where,
m is the rate of change (or) slope
In this situation,
The rate of change = \(\frac{Time}{The amount of water}\)
So,
y = \(\frac{56}{8}\)x
y = 7x
Hence, from the above,
We can conclude that the linear equation that represents the amount of water in inches, w, in the bucket after t minutes
y = 7x

Question 11.
The graph of the line represents the cost of renting a kayak. Write a linear function in the form y = mx + b to represent the relationship of the total cost, c, of renting a kayak for t hours.
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 16.1
Answer:
It is given that
The graph of the line represents the cost of renting a kayak
Now,
The given graph is:
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 16.1
From the given graph,
We can observe that the line does not pass through the origin and it has the initial value i..e, the y-intercept
So,
From the graph,
The y-intercept is: 8
Now,
To find the slope,
The points from the graph are: (2, 12), and (4, 16)
We know that,
Slope (m) = y2 – y1 / x2 – x1
So,
Compare the given points with (x1, y1), (x2, y2)
So,
m = \(\frac{16 – 12}{4 – 2}\)
m = \(\frac{4}{2}\)
m = 2
We know that,
The representation of the equation in the slope-intercept form is:
y = mx + c
So,
y = 2x + 8
Hence, from the above,
We can conclude that the relationship to represent the total cost, c, of renting a kayak for t hours is:
y = 2x + 8

Question 12.
An online clothing company sells custom sweatshirts. The company charges $6.50 for each sweatshirt and a flat fee of $3.99 for shipping.
a. Write a linear function in the form y = mx + b that represents the total cost, y, in dollars, for a single order of x sweatshirts.
Answer:
It is given that
An online clothing company sells custom sweatshirts. The company charges $6.50 for each sweatshirt and a flat fee of $3.99 for shipping.
So,
The total cost of the sweatshirts = The cost of each sweatshirt + The flat fee of the sweatshirt for shipping
Let the number of sweatshirts be x
let the total cost of the sweatshirts be y
So,
y = $3.99 + $6.50x
Now,
We know that,
The linear equation representation in the slope-intercept form is:
y = mx + c
Hence, from the above,
We can conclude that the equation that represents the total cost, y, in dollars, for a single order of x sweatshirts is:
y = $3.99 + 6.50x

b. Describe how the linear function would change if the shipping charge applied to each sweatshirt.
Answer:
From part (a),
We know that,
The equation that represents the total cost, y, in dollars, for a single order of x sweatshirts, is:
y = $3.99 + $6.50x
Where,
$3.99 is the shipping charge for x sweatshirts,
Now,
If the shipping charge applied to each sweatshirt, then
The representation of the linear equation which we obtained in part (a) is:
y = $6.50x + \(\frac{$3.99}{x}\)
Hence, from the above,
We can conclude that the linear function that represents if the shipping charge applied to each sweatshirt is:
y = $6.50x + \(\frac{$3.99}{x}\)

Question 13.
A store sells packages of comic books with a poster.
a. Model with Math Write a linear function in the form y = mx + b that represents the cost, y, of a package containing any number of comic books, x.
b. Construct Arguments Suppose another store sells a similar package, modeled by a linear function with initial value $7.99. Which store has the better deal? Explain.
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 16.3
Answer:
a)
It is given that a store seller sells packages of comic books with a poster
Now,
Let x be the number of comics
Let y be the amount obtained by selling comics & poster
So,
For x = 6, y = $12.75
For x = 13, y = $19.75
We know that,
The representation of the linear equation in the slope-intercept form is:
y = mx + c
Where,
m is the slope
c is the y-intercept
Now,
To find the slope,
The points are: (6, 12.75), and (13, 19.75)
Now,
We know that,
Slope (m) = y2 – y1 / x2 – x1
So,
Compare the given points with (x1, y1), (x2, y2)
So,
m = \(\frac{19.75 – 12.75}{13 – 6}\)
m = \(\frac{7}{7}\)
m = 1
So,
y = x + c
Now,
To find the value of c,
Substitute any one of the points in the above equation
So,
12.75 = 6 + c
c = 12.75 – 6
c = 6.75
Hence, from the above,
We can conclude that the representation of the linear equation that represents the cost, y, of a package containing any number of comic books, x is:
y = x + 6.75
b)
It is given that another store sells a similar package, modeled by a linear function with an initial value of $7.99
Now,
From part (a),
The initial value is: $6.75
From the above initial values,
We can observe that
$6.75 < $7.99
Hence, from the above,
We can conclude that another store has the better deal

Question 14.
Higher-Order Thinking Recommendations for safely thawing a frozen turkey are provided on the packaging.
a. What is the thaw rate of the turkey in hours per pound for refrigerator thawing? For cold water thawing?
b. Write a linear function in the form y = mx + b to represent the time, t, in hours it takes to thaw a turkey in the refrigerator as a function of the weight, w, in pounds of the turkey.
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 16.4
Answer:
a)
We know that,
The thaw rate is nothing but the rate of change
So,
For refrigerator thawing,
Rate of change = \(\frac{The number of pounds}{Time}\)
= 4 pounds per day
We know that
1 day = 24 hours
So,
Rate of change = \(\frac{4}{24}\)
= \(\frac{1}{6}\) pound per hour
For Cold water thawing,
Rate of change = \(\frac{The number of pounds}{Time}\)
= 1 pound per 30 minutes
We know that
1 hour = 60minutes
So,
Rate of change = \(\frac{1 (2)}{1}\)
= 2 pounds per hour
Hence, from the above,
We can conclude that
The rate of change for refrigerator thawing is: \(\frac{1}{6}\) pounds per hour
The rate of change for cold water thawing is: 2 pounds per hour
b)
We know that,
The representation of the linear equation in the slope-intercept form is:
y = mx + c
So,
For refrigerator thawing,
The representation of the linear equation as a function of weight w in pounds is:
y = \(\frac{1}{6}\)x + c
For the value of c,
Substitute (24, 4) in the above equation [ The time as x in hours and the weight as y]
So,
4 = \(\frac{24}{6}\) + c
4 = 4 + c
c = 0
Hence, from the above,
We can conclude that the representation of the linear equation as a function of weight w in pounds is:
y = \(\frac{1}{6}\)x

Question 15.
Reasoning The graph shows the relationship between the number of cubic yards of mulch ordered and the total cost of the mulch delivered.
a. What is the constant rate of change? What does it represent?
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 16.5
Answer:
It is given that
The graph shows the relationship between the number of cubic yards of mulch ordered and the total cost of the mulch delivered.
Now,
The given graph is:
Envision Math Common Core 8th Grade Answers Topic 3 Use Functions To Model Relationships 16.5
From the given graph,
To find the rate of change,
The points are: (20, 450), and (10, 300)
Now,
We know that,
Slope (m) = y2 – y1 / x2 – x1
So,
Compare the given points with (x1, y1), (x2, y2)
So,
m = \(\frac{300 – 450}{10 – 20}\)
m = \(\frac{150}{10}\)
m = 15
Hence, from the above,
We can conclude that the constant rate of change for the given graph is: 15

b. What is the initial value? What might that represent?
Answer:
We know that,
The initial value is nothing but the y-intercept
Hence, from the above,
We can conclude that the initial value is 50 and this value represents the initial cost of the mulch

Assessment Practice

Question 16.
An international food festival charges for admission and for each sample of food. Admission and 3 samples cost $5.75. Admission and 6 samples cost $8.75. Which linear function represents the cost, y, for any number of samples, x?
A. y = x + 2.75
B. y = 3x + 2.75
C. y = x + 3
D. y = 3x + 3
Answer:
It is given that
An international food festival charges for admission and for each sample of food. Admission and 3 samples cost $5.75. Admission and 6 samples cost $8.75
Now,
Let x be the number of samples
Let y be the cost of samples &Admission
So,
For x = 3, y = $5.75
For x = 6, y = $8.75
We know that,
The representation of the linear equation in the slope-intercept form is:
y = mx + c
Where,
m is the slope
c is the y-intercept
Now,
To find the slope,
The points are: (3, 5.75), and (6, 8.75)
Now,
We know that,
Slope (m) = y2 – y1 / x2 – x1
So,
Compare the given points with (x1, y1), (x2, y2)
So,
m = \(\frac{8.75 – 5.75}{6 – 3}\)
m = \(\frac{3}{3}\)
m = 1
So,
y = x + c
Now,
To find the value of c,
Substitute any one of the points in the above equation
So,
5.75 = 3 + c
c = 5.75 – 3
c = 2.75
So,
y = x + 2.75
Hence, from the above,
We can conclude that option A matches the given situation

Question 17.
Some eighth-graders are making muffins for a fundraiser. They have already made 200 muffins and figure they can make 40 muffins in an hour.

PART A
Write a linear function in the form y = mx + b that represents the total number of muffins the students will make, y, and the number of additional hours spent making the muffins, x.
Answer:
It is given that
Some eighth-graders are making muffins for a fundraiser. They have already made 200 muffins and figure they can make 40 muffins in an hour.
Now,
Let x be the number of hours
So,
The total number of muffins students will make = The number of muffins that the students had already made + The number of muffins the students will make in x hours if they made 40 muffins in an hour
y = 40x + 200
We know that,
The representation of the linear equation in the slope-intercept form is:
y = mx + c
Hence, from the above,
We can conclude that
A linear function that represents the total number of muffins the students will make, y, and the number of additional hours spent making the muffins, x is:
y = 40x + 200

PART B
How many additional hours would the students spend to make 640 muffins?
Answer:
From part (a),
A linear function that represents the total number of muffins the students will make, y, and the number of additional hours spent making the muffins, x is:
y = 40x + 200
It is given that the number of muffins students made is: 640
So,
y = 640
So,
640 = 40x + 200
40x = 640 – 200
40x = 440
4x = 44
x = \(\frac{44}{4}\)
x = 11 hours
Hence, from the above,
We can conclude that the additional hours the students would spend to make 640 muffins is: 11 hours

Lesson 3.5 Intervals of Increase and Decrease

Solve & Discuss It!

Martin will ride his bike from his house to his aunt’s house. He has two different routes he can take. One route goes up and down a hill. The other route avoids the hill by going around the edge of the hill. How do you think the routes will differ? What do you think about the relationship between speed and time?
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 20.1
Answer:
It is given that
Martin will ride his bike from his house to his aunt’s house. He has two different routes he can take. One route goes up and down a hill. The other route avoids the hill by going around the edge of the hill.
We know that,
Speed = \(\frac{Distance}{Time}\)
In this situation,
The distance is constant for the 2 routes
So,
As speed increases, the time decreases
Now,
For route 1:
Route 1 is divided into 2 parts
For the first part (Going up),
As there is friction,
The speed decreases as time increases
For the second part (Going down),
The speed increases as time decreases
For route 2:
Route 2 is divided into 2 parts
For the first part (Going down),
The speed increases as time decreases
For the second part,
The speed is constant as time increases

Focus on math practices
Reasoning How do the characteristics of each route affect Martin’s travel time and speed?
Answer:
We know that,
For the constant distance,
Speed ∝ \(\frac{1}{Time}\)
Now,
Route 1 is in the shape of a parabola
In route 1,
For the first part,
The speed increases as time decreases
For the second part,
The speed decreases as time increases
Now,
Route 2 is increasing for some time and later becomes constant
In route 2,
For the first part,
The speed increases as time decreases
For the second part,
The speed is constant as time increases

? Essential Question
How does a qualitative graph describe the relationship between quantities?
Answer:
The formal term to describe a straight-line graph is linear, whether or not it goes through the origin, and the relationship between the two variables is called a linear relationship. Similarly, the relationship shown by a curved graph is called non-linear.

Try It!
The graph at the right shows another interval in the train’s travel. Which best describes the behavior of the train in the interval shown?
As time Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 20.2, the speed of the train
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 20.3
The function is Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 20.2
Answer:
It is given that the graph shows another interval in the train’s travel
Now,
The given graph is:
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 20.3
From the given graph,
As speed decreases, the time increases
Hence,
The function of the given graph is decreasing in nature

Convince Me!
How would the graph of the function change if the speed of the train was increasing?
Answer:
We know that,
If speed increases, then the time decreases for a constant distance
So,
For a constant distance,
Speed ∝ \(\frac{1}{Time}\)
Hence,
The graph of the function is decreasing in nature as the speed of the train is increased

Try It!
Write a scenario that the graph above could represent. (Example 3)
Answer:
The scenario that the given graph could represent is:
The traveling of a vehicle on a hill

KEY CONCEPT
You can describe the relationship between two quantities by analyzing the behavior of the function relating the quantities in different intervals on a graph.
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 20.4

Do You Understand?

Question 1.
Essential Question How does a qualitative graph describe the relationship between quantities?
Answer:
The formal term to describe a straight-line graph is linear, whether or not it goes through the origin, and the relationship between the two variables is called a linear relationship. Similarly, the relationship shown by a curved graph is called non-linear.

Question 2.
Look for Relationships How would knowing the slope of a linear function help determine whether a function is increasing or decreasing?
Answer:
The graph of an increasing function has a positive slope. A line with a positive slope slants upward from left to right. For a decreasing function, the slope is negative. The output values decrease as the input values increase.

Question 3.
Use Structure What kind of graph of a function shows the same output values, or y-values, for each input value, or x-value?
Answer:
The vertical line test can be used to determine whether a graph represents a function. A vertical line includes all points with a particular x value. The y value of a point where a vertical line intersects a graph represents an output for that input x value.

Do You Know How?

Question 4.
What does the graph of the function at each interval represent?
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 30.1
Answer:
The given graph is:
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 30.1
Now,
The given graph is divided into 2 intervals
In the first interval,
The height increases with the increase of time
In the second interval,
The height decreases with the increase of time

Question 5.
In which intervals is the function increasing, decreasing, or constant?
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 30.2
Answer:
The given graph is:
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 30.2
We know that,
The graph is said to be increasing in nature when the line moves from left to right or the slope is positive
The graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
The graph is said to be constant in nature when the line is parallel to any one of the coordinate axes
Hence, from the above,
We can conclude that
The intervals that the function is constant – 1, 5
The intervals that the function is increasing – 3, 4
The intervals that the function is decreasing – 2, 6

Practice & Problem Solving

Question 6.
Use the graph to complete the statements.
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 30.3
The function is Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 30.4 in intervals 1, 3, and 6.
The function is Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 30.4 in intervals 2 and 5.
The function is constant in interval Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 30.5
Answer:
The given graph is:
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 30.3
We know that,
The graph is said to be increasing in nature when the line moves from left to right or the slope is positive
The graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
The graph is said to be constant in nature when the line is parallel to any one of the coordinate axes
Hence, from the above,
We can conclude that
The intervals that the function is constant – 4
The intervals that the function is increasing – 1, 3, 6
The intervals that the function is decreasing – 2, 5

Question 7.
The graph below shows the temperature in Paula’s house over time after her mother turned on the air conditioner. Describe the relationship between the two quantities.
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 30.44
Answer:
It is given that
The graph below shows the temperature in Paula’s house over time after her mother turned on the air conditioner
Now,
The given graph is:
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 30.44
From the given graph,
We can observe that
As time increases, the temperature decreases
Hence, from the above,
We can conclude that
Temperature (°F) ∝ \(\frac{1}{Time}\)

Question 8.
You have a device that monitors the voltage across a lamp over time. The results are shown in the graph. Describe the behavior of the function in each interval.
In interval (a), the function is Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 30.66
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 30.55
In the interval (b), the function is Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 30.66
In the interval (c), the function is Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 30.66
In interval (d), the function is Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 30.66
Answer:
It is given that
You have a device that monitors the voltage across a lamp over time. The results are shown in the graph
Now,
The given graph is:
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 30.55
We know that,
The graph is said to be increasing in nature when the line moves from left to right or the slope is positive
The graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
The graph is said to be constant in nature when the line is parallel to any one of the coordinate axes
Hence, from the above,
We can conclude that
In interval (a), the function is increasing
In the Interval (b), the function is constant
In the interval (c), the function is decreasing
In the interval (d), the function is constant

Question 9.
The graph below shows the height of a roller coaster over time during a single ride. Circle the intervals in which the function is increasing. In which interval is the increase the greatest?
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 30.6
Answer:
It is given that
The graph below shows the height of a roller coaster over time during a single ride.
Now,
The given graph is:
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 30.6
We know that,
The graph is said to be increasing in nature when the line moves from left to right or the slope is positive
The graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
The graph is said to be constant in nature when the line is parallel to any one of the coordinate axes
Now,
The given graph with the intervals is:

So,
From the given graph,
The intervals which are increasing in nature are: 1, 3, 5
Hence, from the above,
We can conclude that the increase is the greatest in the 1st interval

Question 10.
Reasoning The graph shows the speed of a car over time. What might the constant intervals in the function represent?
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 30.7
Answer:
It is given that
The graph shows the speed of a car over time
Now,
The given graph is:
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 30.7
We know that,
The graph is said to be increasing in nature when the line moves from left to right or the slope is positive
The graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
The graph is said to be constant in nature when the line is parallel to any one of the coordinate axes
We know that,
Speed = \(\frac{Distance}{Time}\)
Now,
From the given graph,
We can observe that
The constant lines represent that the speed becomes constant even though the time is increasing
Hence, from the above,
We can conclude that the constant lines in the given graph represent the constant speed

Question 11.
Higher-Order Thinking A signal generator is used to generate signals for a lab experiment over time. The graph shows the frequency of the signal generated.
a. In how many intervals is the function decreasing?
b. How are the decreasing intervals alike?
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 30.8
Answer:
It is given that
A signal generator is used to generate signals for a lab experiment over time. The graph shows the frequency of the signal generated.
Now,
The given graph is:
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 30.8
We know that,
The graph is said to be increasing in nature when the line moves from left to right or the slope is positive
The graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
The graph is said to be constant in nature when the line is parallel to any one of the coordinate axes
Now,
The given graph with the marked intervals are:

a)
The intervals that the function is decreasing are: 3, 7, 11
b)
The decreasing intervals are all decreasing in nature and have a negative slope

c. How are the decreasing intervals different?
Answer:
The sizes of the decreasing intervals are different
So,
The values of the negative slopes for decreasing intervals will also be different

Question 12.
Critique Reasoning The graph shows the speed of a person riding his stationary exercise bicycle over time. a. A student claims that the function is constant in two intervals. Do you agree? Explain.
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 30.9
Answer:
It is given that
The graph shows the speed of a person riding his stationary exercise bicycle over time. a. A student claims that the function is constant in two intervals.
We know that,
The graph is said to be increasing in nature when the line moves from left to right or the slope is positive
The graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
The graph is said to be constant in nature when the line is parallel to any one of the coordinate axes
So,
From the given graph,
We can observe that
The number of intervals that are constant in nature is: 4
Hence, from the above,
We can conclude that the claim of the student is not correct

b. What error might the student have made?
Answer:
The student considered only the highest constant intervals but not the lowest intervals
The student also did not consider the last constant interval due to the misconception that it is decreasing in nature but it is constant after that decreasing in nature
So,
The above are the errors the student has made

Question 13.
Look for Relationships The graph shows the speed of a roller coaster over time. Describe the relationship of speed as a function of time.
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 40.1
Answer:
It is given that
The graph shows the speed of a roller coaster over time. Describe the relationship of speed as a function of time.
Now,
The given graph is:
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 40.1
From the given graph,
We can observe that
First, the speed of a roller coaster increases with time, and then it fluctuates between increasing and decreasing as time increases further, and at last, the speed of the roller coaster decreases

Assessment Practice

Question 14.
Which statements about the graph are true? Select all that apply.
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 40.2 The graph is decreasing in intervals (1) and (4).
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 40.2 The graph shows a constant function in interval (2).
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 40.2 The graph is increasing in intervals (2) and (4).
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 40.2 The graph has a constant rate of change.
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 40.2 The graph shows a constant function in interval (3).
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 40.3
Answer:
Let the given options be named as A, B, C, D, and E respectively
Now,
The given graph is:
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 40.3
We know that,
The graph is said to be increasing in nature when the line moves from left to right or the slope is positive
The graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
The graph is said to be constant in nature when the line is parallel to any one of the coordinate axes
Hence, from the above,
We can conclude that the options that match with the given situation are: A and E

Lesson 3.6 Sketch Functions from Verbal Descriptions

ACTIVITY

Explain It!

The Environmental Club is learning about oil consumption and energy conservation around the world. Jack says oil consumption in the United States has dropped a lot. Ashley says oil consumption in China is the biggest problem facing the world environment.

Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 40.4

A. Do you agree or disagree with Jack’s statement? Construct an argument based on the graph to support your position.
Answer:
The given graph is:
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 40.4
Now,
The statement of Jack is:
Oil consumption in the United States has dropped a lot
Now,
From the given graph,
We can observe that the oil consumption of the United States (The first line in the graph) is constant from 2000 to 2003 and the consumption is constant from 2004 to 2008 and the consumption decreased abruptly from 2008 to 2011
Hence, from the above,
We can conclude that we can agree with Jack’s statement

B. Do you agree or disagree with Ashley’s statement? Construct an argument based on the graph to support your position.
Answer:
The given graph is:
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 40.4
Now,
The statement of Ashley is:
Oil consumption in China is the biggest problem facing the world environment.
Now,
From the given graph,
We can observe that the oil consumption of China (The third line in the graph) is constant from 2000 to 2004 and the consumption is increased abruptly from 2004 to 2008 and the consumption increased minimally from 2008 to 2011
Hence, from the above,
We can conclude that we can agree with Ashley’s statement since China’s consumption of oil will make the other countries suffer

Focus on math practices
Look for Relationships What trend do you see in oil consumption in the United States and in Europe?
Answer:
The given graph is:
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 40.4
From the given graph,
We can observe that
The trend in the oil consumption of the United States and Europe (The first line and the second line in the graph) is:
a) The consumption increases gradually for both the United States and Europe from 2000 to 2003
b) From 2004 to 2008, the oil consumption of the United States becomes constant but the consumption of Europe increases gradually
c) From 2009 to 2011, the oil consumption of the United States and Europe decreases gradually

? Essential Question
How does the sketch of a graph of a function help describe its behavior?
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 40.5

Answer:
From the graph,
We can observe that,
At t = 0,
The level of oxygen is full
At t = 22 minutes,,
The level of oxygen decreases
Hence, from the above,
We can conclude that
The behavior of the time and the oxygen level is:
Time ∝ \(\frac{1}{Oxygen level}\)

Try It!

The weight of the water exerts pressure on a diver. At a depth of 10 feet, the water pressure is 19.1 pounds per square inch (psi) and at a depth of 14 feet, the water pressure is 20.9 psi. Complete the statements, and then sketch the qualitative graph of this function.
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 40.6
The input, or x-variable, is Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 40.7
The output, or y-variable, is Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 40.7
Answer:
It is given that
The weight of the water exerts pressure on a diver. At a depth of 10 feet, the water pressure is 19.1 pounds per square inch (psi) and at a depth of 14 feet, the water pressure is 20.9 psi.
So,
From the above statements,
We can observe that
As the depth of the water increases, the water pressure also increases
So,
The input or x-variable for the given situation is: Depth
The output or y-variable for the given situation is: Water pressure
Now,
To draw the graph,
The required points are: (10, 19.1), and (14, 20.9)
We know that,
the equation of the straight line in the slope-intercept form is:
y = mx + c
Now,
We know that,
Slope (m) = y2 – y1 / x2 – x1
So,
Compare the given points with (x1, y1), (x2, y2)
So,
m = \(\frac{20.9 – 19.1}{14 – 10}\)
m = \(\frac{1.8}{4}\)
m = \(\frac{9}{20}\)
So,
y = \(\frac{9}{20}\)x + c
Now,
To find the value of c,
Substitute any one of the points in the above equation
So,
19.1 = \(\frac{9}{20}\) (10) + c
c = 19.1 – 4.5
c = 14.6
So,
y = \(\frac{9}{20}\)x + 14.6
Hence,
The representation of the above equation in the coordinate plane is:

Convince Me!
Generalize How are the sketches of the two functions similar? How are they different?
Answer:
From Example 1 and the above item’s graph,
We can observe that both graphs have the equation in the slope-intercept form
But,
The graph in Example 1 is decreasing in nature
The graph in the above item is increasing in nature

Try It!
Haru rides his bike from his home for 30 minutes at a fast pace. He stops to rest for 20 minutes and then continues in the same direction at a slower pace for 30 more minutes. Sketch a graph of the relationship of Haru’s distance from home over time.
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 40.77
Answer:
It is given that
Haru rides his bike from his home for 30 minutes at a fast pace. He stops to rest for 20 minutes and then continues in the same direction at a slower pace for 30 more minutes.
Hence,
The graph of the relationship of Haru’s distance from the house over time is:

KEY CONCEPT

You can sketch a graph of a function to describe its behavior. When sketching a function, follow these steps:
1. Identify the two variables (input, output) that have a relationship.
2. Analyze the situation. Look for keywords that indicate that the function is increasing, decreasing, or constant.
3. Sketch the graph.
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 40.8

Do You Understand?

Question 1.
? Essential Question How does the sketch of a graph of a function help describe its behavior?

Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 40.5

Answer:
From the graph,
We can observe that,
At t = 0,
The level of oxygen is full
At t = 22 minutes,,
The level of oxygen decreases
Hence, from the above,
We can conclude that
The behavior of the time and the oxygen level is:
Time ∝ \(\frac{1}{Oxygen level}\)

Question 2.
Make Sense and Persevere How do you know which variable goes with which axis when you graph?
Answer:
The independent variable belongs on the x-axis (horizontal line) of the graph and the dependent variable belongs on the y-axis (vertical line).

Question 3.
Reasoning How can you determine the shape of a graph?
Answer:
The four ways to describe shape are whether it is symmetric, how many peaks it has if it is skewed to the left or right, and whether it is uniform. A graph with a single peak is called unimodal. A single peak over the center is called bell-shaped. And, a graph with two peaks is called bimodal.

Do You Know How?

Question 4.
A class plants a tree. Sketch the graph of the height of the tree over time.
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 70.1
a. Identify the two variables.
Answer:
It is given that a class plants a tree
Now,
From the given figure,
We can observe that
As time increases, the height of the tree increases
Hence, from the above,
We can conclude that
Input variable (or) x-coordinate: Time
Output variable (or) y-coordinate: Height of the tree

b. How can you describe the relationship between the two variables?
Answer:
From part (a),
We can observe that
As time increases, the height of the tree increases
Hence, from the above,
We can conclude that the relationship between the variables is:
Time ∝ Height of the tree

c. Sketch the graph.
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 70.2
Answer:
To draw the graph,
The required points are: (0, 3), and (3, 7)
We know that,
the equation of the straight line in the slope-intercept form is:
y = mx + c
Now,
We know that,
Slope (m) = y2 – y1 / x2 – x1
So,
Compare the given points with (x1, y1), (x2, y2)
So,
m = \(\frac{7 – 3}{3 – 0}\)
m = \(\frac{4}{3}\)
So,
y = \(\frac{4}{3}\)x + c
Now,
To find the value of c,
Substitute any one of the points in the above equation
So,
3 = \(\frac{4}{3}\) (0) + c
c = 3 – 0
c = 3
So,
y = \(\frac{4}{3}\)x + 3
Hence,
The representation of the above equation in the coordinate plane is:

Question 5.
An airplane takes 15 minutes to reach its cruising altitude. The plane cruises at that altitude for 90 minutes, and then descend for 20 minutes before it lands. Sketch the graph of the height of the plane over time.
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 70.3
Answer:
It is given that
An airplane takes 15 minutes to reach its cruising altitude. The plane cruises at that altitude for 90 minutes, and then descend for 20 minutes before it lands.
Hence,
The graph of the height of the plane over time is (Height of the plane is any value) is:

Practice & Problem Solving

Question 6.
What relationship between money (in dollars) and time (in months) does this graph show? Write a description of the given graph.
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 70.4
Answer:
The given graph is:
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 70.4
From the given graph,
We can observe that
The amount of money is in an increasing trend for the first half of the month
The amount of money is in a decreasing trend for the second half of the month

Question 7.
When a new laptop became available in a store, the number sold in the first week was high. Sales decreased over the next two weeks and then they remained steady over the next two weeks. The following week, the total number sold by the store increased slightly. Sketch the graph that represents this function over the six weeks.
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 70.5
Answer:
It is given that
When a new laptop became available in a store, the number sold in the first week was high. Sales decreased over the next two weeks and then they remained steady over the next two weeks. The following week, the total number sold by the store increased slightly
Hence,
The graph which represents the given function over the six weeks is:

Question 8.
Aaron’s mother drives to the gas station and fills up her tank. Then she drives to the market. Sketch the graph that shows the relationship between the amount of fuel in the gas tank of her car and time.
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 70.6
Answer:
It is given that
Aaron’s mother drives to the gas station and fills up her tank. Then she drives to the market.
So,
At first, the amount of fuel in the gas tank will be of some initial value and it is in an increasing trend after filling up the tank
After that, when she drives to the market, the amount of fuel will be in a decreasing trend over time
Hence,
The graph that shows the relationship between the amount of fuel in the gas tank of her car and time is:

Question 9.
Melody starts at her house and rides her bike for 10 minutes to a friend’s house. She stays at her friend’s house for 60 minutes. Sketch a graph that represents this description.
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 70.7
Answer:
It is given that
Melody starts at her house and rides her bike for 10 minutes to a friend’s house. She stays at her friend’s house for 60 minutes
Hence,
The graph that represents the above description is:

Question 10.
Which description best represents the graph shown?
A. People are waiting for a train. A train comes and some people get on. The other people wait for the next train. As time goes by, people gradually leave the station.
B. One train arrives and some people get off the train and wait in the station.
C. People are waiting for a train. Everyone gets on the first train that comes.
D. People are waiting for a train. A train comes and some people get on the train. The other people wait for the next train. Another train arrives and all of the remaining people get on.
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 70.8
Answer:
The given graph is:
Envision Math Common Core Grade 8 Answers Topic 3 Use Functions To Model Relationships 70.8
From the given graph,
We can observe that
At first, there are some people and after some time, the number of people decreased
After some time, there are some people other than the people that decreased and after some further time, the total number of people becomes zero
Hence, from the above,
We can conclude that option D matches the above description

Question 11.
A baker has already made 10 cakes. She can make the same number of cakes each hour, which she does for 5 hours. Sketch the graph of the relationship between the number of cakes made and time.
Envision Math Common Core 8th Grade Answer Key Topic 3 Use Functions To Model Relationships 70.9
Answer:
It is given that
A baker has already made 10 cakes. She can make the same number of cakes each hour, which she does for 5 hours.
So,
The initial value for the given graph is: 10
Hence,
The graph of the relationship between the number of cakes made and time is:

Question 12.
Model with Math An air cannon launches a T-shirt upward toward basketball fans. It reaches a maximum height and then descends for a couple seconds until a fan grabs it. Sketch the graph that represents this situation.
Envision Math Common Core 8th Grade Answer Key Topic 3 Use Functions To Model Relationships 70.10
Answer:
It is given that
An air cannon launches a T-shirt upward toward basketball fans. It reaches a maximum height and then descends for a couple of seconds until a fan grabs it
We know that,
The projectile (Graph) of an air cannon is always like a “Parabola”
Hence,
The graph that represents the given situation is:

Question 13.
Higher-Order Thinking Write a verbal description of how these two variables are related. The description must suggest at least two intervals. Sketch the graph that represents the verbal description.
Envision Math Common Core 8th Grade Answer Key Topic 3 Use Functions To Model Relationships 70.11
Answer:
The verbal description of the relationship between the total people and time in hours is:
People are waiting for a train. A train comes and some people get on the train. The other people wait for the next train. Another train arrives and all of the remaining people get on.
Hence,
The graph that describes the above verbal description is:

Assessment Practice

Question 14.
A baseball team scores the same number of runs in each of the first 4 innings. After that, the team did not score a run for the rest of the game, which lasts 9 innings. Let x represent the innings of the game, and y represent the total number of runs.
PART A
Sketch the graph of this situation below.
Envision Math Common Core 8th Grade Answer Key Topic 3 Use Functions To Model Relationships 70.12
Answer:
It is given that
A baseball team scores the same number of runs in each of the first 4 innings. After that, the team did not score a run for the rest of the game, which lasts 9 innings.
Hence,
The graph for the above situation is:

PART B
How would the graph change if the innings in which the team scores runs changes?
Answer:
If the innings in which the team scores runs change, then
The graph will be either in an increasing trend or in a decreasing trend

Topic 3 REVIEW

? Topic Essential Question

How can you use functions to model linear relationships?
Answer:
Another approach to representing linear functions is by using function notation. One example of function notation is an equation written in the form known as the slope-intercept form of a line, where x is the input value, m is the rate of change, and b is the initial value of the dependent variable.

Vocabulary Review

Match each vocabulary term with its definition.
Envision Math Common Core 8th Grade Answer Key Topic 3 Use Functions To Model Relationships 82.1
Answer:
Each vocabulary term with its definition is:

Use Vocabulary in Writing

Explain how to write a linear function in the form y = mx + b by using the two points given below. Use vocabulary words in your explanation. (0, -2), (2, 6)
Answer:
The given points are:
(0, -2) and (2, 6)
Now,
We know that,
Slope (m) = y2 – y1 / x2 – x1
So,
Compare the given points with (x1, y1), (x2, y2)
So,
m = \(\frac{6 + 2}{2 – 0}\)
m = \(\frac{8}{2}\)
m = 4
We know that,
The representation of the equation in the slope-intercept form is:
y = mx + c
So,
y = 4x + c
Now,
To find the value of c,
Substitute any one of the points in the above equation
So,
-2 = 0 + c
c = -2 – 0
c = -2
Hence, from the above,
We can conclude that the representation of the linear equation for the given points is:
y = 4x – 2

Concepts and Skills Review

LESSON 3.1 Understand Relations and Functions

Quick Review
A relation is a set of ordered pairs. A relation is a function if each input, or x-value, has exactly one unique output, or y-value.

Practice

Question 1.
Is the relation shown in the table a function? Explain.
Envision Math Common Core 8th Grade Answer Key Topic 3 Use Functions To Model Relationships 85.1
Answer:
The given relation is:
Envision Math Common Core 8th Grade Answer Key Topic 3 Use Functions To Model Relationships 85.1
We know that,
A relation is said to be a function if each input has only 1 output
So,
From the given relation,
We can observe that each input has only 1 output
Hence, from the above,
We can conclude that the given relation is a function

Question 2.
Does the relation {(-5, -3), (7, 2), (3,8), (3, -8), (5, 10)} represent a function? Use the arrow diagram. Then explain your answer.
Envision Math Common Core 8th Grade Answer Key Topic 3 Use Functions To Model Relationships 85.2
Answer:
The given ordered pairs are:
(-5, -3), (7, 2), (3,8), (3, -8), (5, 10)
We know that,
The ordered pairs are in the form of (input (x), output (y))
In the arrow diagram,
The left side represents the values of the input
The right side represents the values of the output
So,
The representation of the ordered pairs in the arrow diagram is:

We know that,
A relation is said to be a function only when each input corresponds with only 1 output
Now,
From the arrow diagram,
We can observe that
The same input corresponds with 2 outputs
Hence, from the above,
We can conclude that the given relation is not a function

LESSON 3.2 Connect Representations of Functions

Quick Review
You can represent a function in a table, in a graph, or as an equation. The graph of a linear function is a straight line.

Practice
Mark has a $100 gift card to buy apps for his smartphone. Each week, he buys one new app for $4.99.
1. Write an equation that relates the amount left on the card, y, over time, x.
Answer:
It is given that
Mark has a $100 gift card to buy apps for his smartphone. Each week, he buys one new app for $4.99.
Now,
Let y be the amount left on the card
Let x be the number of weeks
So,
The amount left on the card = The total amount of the card – The amount of money he used to buy a new app for x weeks
y = $100 – $4.99x
Hence, from the above,
We can conclude that
The equation that relates the amount left on the card, y, over time, x is:
y = $100 – $4.99x

2. Make a graph of the function.
Envision Math Common Core 8th Grade Answer Key Topic 3 Use Functions To Model Relationships 85.3
Answer:
From part (a),
The equation that relates the amount left on the card, y, over time, x is:
y = $100 – $4.99x
Hence,
The representation of the above equation in the coordinate plane is:

LESSON 3.3 Compare Linear and Nonlinear Functions

Quick Review
You can compare functions in different representations by looking at the properties of functions: the constant rate of change and the initial value.

Practice
Two linear functions are shown.
Envision Math Common Core 8th Grade Answer Key Topic 3 Use Functions To Model Relationships 90.1
Answer:
The given linear functions are:
Envision Math Common Core 8th Grade Answer Key Topic 3 Use Functions To Model Relationships 90.1
Function A:
Compare the given equation with
y = mx + c
Where,
m is the rate of change (or) slope
c is the initial value (or) y-intercept
So,
From the above equation,
The rate of change is: -3
The initial value is: 2
Function B:
From the given table,
We can observe that
The rate of change for all the values of x is: 1
The rate of change for all the values of y is: 2
So,
The rate of change for the table = \(\frac{y}{x}\)
= \(\frac{2}{1}\)
= 2
We know that,
The initial value is the value of y when x = 0
So,
For the given table,
The initial value is: 0
The rate of change is: 2

Question 1.
Which function has the greater initial value? Explain.
Answer:
Function A has the greater initial value

Question 2.
Which function has the greater rate of change?
Answer:
Function B has the greater rate of change

LESSON 3.4 Construct Functions to Model Linear Relationships

Quick Review
A function in the form y = mx + b represents a linear relationship between two quantities, x and y, where m represents the constant rate of change and b represents the initial value.

Practice

Question 1.
What is the equation of a line that passes through (0.5, 4.25) and (2, 18.5) and has a y-intercept of -0.5?
Answer:
It is given that
A line passing through (0.5, 4.25), and (2, 18.5) and has a y-intercept of  -0.5
We know that,
The y-intercept is the value of y when x = 0
Now,
The given points are:
(0.5, 4.25) and (2, 18.5)
Now,
We know that,
Slope (m) = y2 – y1 / x2 – x1
So,
Compare the given points with (x1, y1), (x2, y2)
So,
m = \(\frac{18.5 – 4.25}{2 – 0.5}\)
m = \(\frac{14.25}{1.5}\)
m = 9.5
We know that,
The representation of the equation in the slope-intercept form is:
y = mx + c
So,
y = 9.5x – 0.5
Hence, from the above,
We can conclude that the representation of the linear equation for the given points is:
y = 9.5x – 0.5

Question 2.
The graph shows the relationship of the number of gallons being drained from an aquarium over time. What function models the relationship?
Envision Math Common Core 8th Grade Answer Key Topic 3 Use Functions To Model Relationships 95.1
Answer:
It is given that
The graph shows the relationship of the number of gallons being drained from an aquarium over time.
Now,
The given graph is:
Envision Math Common Core 8th Grade Answer Key Topic 3 Use Functions To Model Relationships 95.1
From the given graph,
We can observe that there is an initial value or y-intercept
So,
The y-intercept from the given graph is: 90
Now,
To find the slope from the given graph,
The given points are: (0, 90), and (9, 0)
So,
Slope (m) = \(\frac{0 – 90}{9 – 0}\)
m = \(\frac{-90}{9}\)
m = -10
We know that,
The representation of the equation in the slope-intercept form is:
y = mx + c
So,
y = -10x + 90
Hence, from the above,
We can conclude that the equation that represents the given relationship is:
y = -10x + 90

LESSON 3-5 Intervals of Increase and Decrease

Quick Review
You can describe the relationship between two quantities by looking at the behavior of the line at different intervals on a qualitative graph. The function is increasing if both x- and y-values increase. The function is decreasing if the y-values decrease as the x-values increase.

Practice

The graph shows the altitude of an airplane over time.
Envision Math Common Core 8th Grade Answer Key Topic 3 Use Functions To Model Relationships 95.3
Answer:
We know that,
The graph is said to be increasing in nature when the line moves from left to right or the slope is positive
The graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
The graph is said to be constant in nature when the line is parallel to any one of the coordinate axes

Question 1.
In which intervals is the graph of the function constant? Explain.
Answer:
The intervals in which the graph of the function is constant are: 2, 4, 6

Question 2.
In which intervals is the graph of the function decreasing? Explain.
Answer:
The intervals in which the graph of the function is decreasing in nature are: 5, 7

LESSON 3.6 Sketch Functions from Verbal Descriptions

Quick Review
You can sketch a graph of a function to describe its behavior. When sketching a function, identify the variables (input, output) that have a relationship, analyze the situation, and then sketch the graph.

Practice

Question 1.
Jack’s mother brings him a bowl of carrots as a snack. At first he does not eat any; then he eats one at a time until half of the carrots are gone. Then he does not eat any more. Sketch a graph that shows the number of carrots in the bowl over time.
Envision Math Common Core 8th Grade Answer Key Topic 3 Use Functions To Model Relationships 96.1
Answer:
It is given that
Jack’s mother brings him a bowl of carrots as a snack. At first, he does not eat any; then he eats one at a time until half of the carrots are gone. Then he does not eat anymore
So,
At first,
The number of carrots present in a bowl is: An arbitrary number
The number of carrots Jack consumed is: 0
Hence,
The graph that shows the number of carrots in the bowl over time is:

Topic 3 Fluency Practice

What’s the Message?

In each row, determine which equation has the greater solution. Circle the letter next to the equation with the greater solution in each row.

Envision Math Common Core 8th Grade Answer Key Topic 3 Use Functions To Model Relationships 96.3
Answer:
The row of equations that has great solutions are:

What does the zero say to the eight?
Envision Math Common Core 8th Grade Answer Key Topic 3 Use Functions To Model Relationships 96.4
Answer:
The statement said by zero to eight is:
You are greater than me

Envision Math Common Core Grade 7 Answer Key Topic 1 Rational Number Operations

enVision Math Common Core Grade 7 Answer Key Topic 1 Rational Number Operations

Go through the enVision Math Common Core Grade 7 Answer Key Topic 1 Rational Number Operations regularly and improve your accuracy in solving questions.

enVision Math Common Core 7th Grade Answers Key Topic 1 Rational Number Operations

Topic 1 Essential Question

How can the properties of operations be used to solve problems involving integers and rational numbers?

3-ACT MATH

Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 1
Win Some, Lose Some Are you the kind of person who has a lot of knowledge about history, literature, or science? What about pop culture, music, sports, and current events? Some schools have an academic bowl team that competes in tournaments against other schools. The teams are made up of members with strengths in different subject areas.
In any quiz competition, it’s important to understand the rules and scoring. Think about this during the 3-Act Mathematical Modeling lesson.
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 2

Topic 1 enVision STEM Project

Did You Know?
The lowest recorded temperature in the world, -136°F (-93.2°C), occurred in Antarctica.
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 3
The highest recorded temperature in the world, 134°F (56.7°C), occurred in Death Valley, California.
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 4

Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 5 The Celsius scale (°C) is commonly used for temperature measurement in most of the world.
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 6 Only a small number of nations, including the United States, regularly use the Fahrenheit scale (°F).

Windchill, based on the rate of heat loss from exposed skin, can make it feel colder outside than the actual air temperature indicates. Wind chills in some places of the world can dip into the – 100°F range.
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 7

Your Task: How Cold is Too Cold?
There are many regions of the world with cold temperatures and extreme conditions. How do the inhabitants of these regions adapt and thrive? Do conditions exist that make regions too cold for human living? You and your classmates will explore and describe the habitability of regions with low temperatures.
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 8

Topic 1 Get Ready!

Review What You Know!

Vocabulary
Choose the best term from the box. Write it on the blank.

  • absolute value
  • Associative Property
  • Commutative Property
  • Distributive Property
  • integers
  • rational number

Rational Number Operations 1

Question 1.
The __________ explains why a × b = b × a and a + b = b + a.

Answer:
The commutative property explains why a x b = b x a and a + b = b + a.

Explanation:
In the above-given question,
given that,
The commutative property explains why a x b = b x a and a + b = b + a.
the commutative property states that the numbers on which we operate can be moved or swapped from their position without making any difference to the answer.
the property holds for addition and multiplication, but not for subtraction and division.
for example:
4 + 3 = 7.
3 + 4 = 7.
2 x 6 = 12.
6 x 2 = 12.

Question 2.
The __________ of -6 is 6, because it is 6 units from zero on the number line.

Answer:
The absolute value of -6 is 6, because it is 6 units from zero on the number line.

Explanation:
In the above-given question,
given that,
The absolute value of -6 is 6, because it is 6 units from zero on the number line.
for example:
6 and -6 are at the same distance from zero on the number line.

Question 3.
The number \(\frac{5}{3}\) is a _________ because 5 and 3 are integers and 3 ≠ 0.

Answer:
The number 5/3 is a rational number because 5 and 3 are integers and 3 is not equal to 0.

Explanation:
In the above-given question,
given that,
The number 5/3 is a rational number because 5 and 3 are integers and 3 is not equal to 0.
for example:
3, 5, 6, and 7 are rational numbers.
rational numbers are also integers.

Question 4.
The set of _________ consists of the counting numbers, their opposites, and zero.

Answer:
The set of integers consists of the counting numbers, their opposites, and zero.

Explanation:
In the above-given question,
given that,
The set of integers consists of the counting numbers, their opposites, and zero.
Integers are counting numbers, their opposites, and zero.
for example:
integers can either be negative, positive, or zero.
1, -1, 2, -2, and 3, -3.

Question 5.
The sum of (a + b) + c is equal to the sum of a + (b + c) as explained by the _________

Answer:
The sum of (a + b) + c is equal to the sum of a + (b + c) as explained by the distributive property.

Explanation:
In the above-given question,
given that,
The sum of (a + b) + c is equal to the sum of a + (b + c) as explained by the distributive property.
the distributive property is sometimes called the distributive law of multiplication and division.
for example:
4(x – 3) = 20.
4x – 12 = 20.
4x – 12 + 12 = 20 + 12.
4x = 32.
x = 32/4.
x = 8.

Rational Number Operations 2

Question 6.
If you evaluate n × (v + z) by writing it as (n × y) + (n × z), you have used the __________.

Answer:
If you evaluate n x (v + z) by writing it as (n x y) + (n x z), you have used the distributive property.

Explanation:
In the above-given question,
given that,
If you evaluate n x (v + z) by writing it as (n x y) + (n x z), you have used the distributive property.
for example:
4(x – 3) = 20.
4x – 12 = 20.
4x – 12 + 12 = 20 + 12.
4x = 32.
x = 32/4.
x = 8.

Add and Subtract Fractions and Decimals

Add or subtract.
Question 7.
\(2 \frac{1}{3}+6 \frac{3}{5}\)

Answer:
2(1/3) + 6(3/5) = 8.9.

Explanation:
In the above-given question,
given that,
2(1/3) + 6(3/5).
7/3 + 33/5.
2.3 + 6.6.
8.9.
2(1/3) + 6(3/5) = 8.9.

Question 8.
\(9 \frac{1}{10}-4 \frac{3}{4}\)

Answer:
9(1/10) – 4(3/4) = 4.35.

Explanation:
In the above-given question,
given that,
9(1/10) – 4(3/4).
91/10 – 19/4.
9.1 – 4.75.
4.35.
9(1/10) – 4(3/4) = 4.35.

Question 9.
19.86 + 7.091

Answer:
19.86 + 7.091 = 26.951.

Explanation:
In the above-given question,
given that,
add the numbers.
19.86 + 7.091.
19.86 + 7.091 = 26.951.

Question 10.
57 – 10.62

Answer:
57 – 10.62 = 46.38.

Explanation:
In the above-given question,
given that,
subtract the numbers.
57 – 10.62.
46.38.
57 – 10.62 = 46.38.

Multiply and Divide Fractions and Decimals

Multiply or divide.
Question 11.
4.08 × 29.7

Answer:
4.08 x 29.7 = 121.176.

Explanation:
In the above-given question,
given that,
multiply the numbers.
4.08 x 29.7 = 121.176.

Question 12.
15,183.3 ÷ 473

Answer:
15,183.3 / 473 = 32.1.

Explanation:
In the above-given question,
given that,
divide the numbers.
15,183.3 / 473 = 32.1.

Question 13.
\(\frac{15}{16} \times 9 \frac{1}{5}\)

Answer:
15/16 x 41/5 = 7.6875.

Explanation:
In the above-given question,
given that,
multiply the numbers.
15/16 x 41/5.
0.9375 x 8.2.
7.6875.
15/16 x 41/5 = 7.6875.

Rational Number Operations 3

Question 14.
\(4 \frac{7}{9} \div 1 \frac{7}{12}\)

Answer:
4(7/9) / 1(7/12) = 2.974.

Explanation:
In the above-given question,
given that,
divide the numbers.
4(7/9) / 1(7/12).
43/9 / 19/12.
4.7 / 1.58.
2.974.
4(7/9) / 1(7/12) = 2.974.

Question 15.
Byron has \(1 \frac{7}{10}\) kilograms of black pepper. He uses of the pepper and splits it between 7 pepper shakers. How much pepper will be in each shaker?
A. \(\frac{119}{80} \mathrm{~kg}\)
B. \(\frac{1}{8} \mathrm{~kg}\)
C. 1.4125 kg
D. \(\frac{17}{80} \mathrm{~kg}\)

Answer:
The quantity of pepper will be in each shaker = 1/8.

Explanation:
In the above-given question,
given that,
Byron has \(1 \frac{7}{10}\) kilograms of black pepper.
He uses the pepper and splits it between 7 pepper shakers.
1(7/10).
17/10 = 1.7.

Language Development
Fill in the word map with new terms, definitions, and supporting examples or illustrations.
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 9

Answer:
Distributive property.
commutative property.
Integer.
Rational number.

Explanation:
In the above-given question,
given that,
The commutative property explains why a x b = b x a and a + b = b + a.
the commutative property states that the numbers on which we operate can be moved or swapped from their position without making any difference to the answer.
the property holds for addition and multiplication, but not for subtraction and division.
for example:
4 + 3 = 7.
3 + 4 = 7.
2 x 6 = 12.
6 x 2 = 12.
The number 5/3 is a rational number because 5 and 3 are integers and 3 is not equal to 0.
for example:
3, 5, 6, and 7 are rational numbers.
rational numbers are also integers.
The set of integers consists of the counting numbers, their opposites, and zero.
Integers are counting numbers, their opposites, and zero.
for example:
integers can either be negative, positive, or zero.
1, -1, 2, -2, and 3, -3.

Pick A Project

PROJECT 1A
What is something you can make?
PROJECT: DESIGNATIANDMADE IM TO SIIT
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 10

PROJECT 1B
How old were you when petroglyphs were being painted?
PROJECT: MAKE A TIMELINE
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 11

PROJECT 1C
What makes an obstacle course fun?
PROJECT: BUILD A MODEL OF AN OBSTACLE COURSE
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 12

PROJECT 1D
What are your favorite ways to exercise ?
PROJECT: FILM AN EXERCISE VIDEO
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 13

Lesson 1.1 Relate Integers and Their Opposites

Solve & Discuss It!
When preparing for a rocket launch, the mission control center uses the phrase “T minus” before liftoff.
…T minus 3, T minus 2, T minus 1, …
After the rocket has launched, “T plus” is used while the rocket is in flight.
…T plus 1, T plus 2, T plus 3, …
When does the rocket launch? What could “T” represent?

I can… relate integers, their opposites, and their absolute values.
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 14

Reasoning
What integers can you use to represent this situation?

Rational Number Operations 4

Focus on math practices
Reasoning How are “T minus 4” and “T plus 4” related?

Answer:
The situation is T plus.

Explanation:
In the above-given question,
given that,
the rocket is launching.
so we have to use the “T plus”.
“T plus 4”.

Essential Question
How are integers and their opposites related?

Try It!

Xavier climbs 9 feet up into an apple tree. What integer represents the direction and how far he will climb to get back down to the ground? What does the integer 0 represent in this situation?
The integer ________ represents Xavier’s climb down.
The integer 0 represents __________.

Answer:
The integer -9 represents Xavier’s climb down.
the integer 0 represents that he did not move.

Explanation:
In the above-given question,
given that,
Xavier climbs 9 feet up into an apple tree.
Xavier climbs feet down into an apple tree is -9.
the integer 0 represents that he did not move.
so the integer 0 is related to +0ft or -0ft.

Convince Me! How are the absolute values of opposite integers related?

Try It!

The temperature was 75°. At noon, the temperature increased 7°. By evening, the temperature decreased by 7°. How did the temperature change?

Answer:
The temperature change by 75° + 7 and 75° – 7.

Explanation:
In the above-given question,
given that,
The temperature was 75°. At noon, the temperature increased 7°
75° + 7.
increased means plus.
decreased means minus.
the temperature change by 75° + 7 and 75° – 7.

Try It!

Shaniqua has $45 in her wallet. She spends $4 on snacks and $8 on a movie ticket. What integer represents the change in the amount of money in Shaniqua’s wallet? How much money does she have left?

Answer:
The money does she have left = $33.

Explanation:
In the above-given question,
given that,
Shaniqua has $45 in her wallet.
She spends $4 on snacks and $8 on a movie ticket.
$45 – $4 = $41.
$41 – $8 = $33.
we are using the minus operator.
so the money does she have left = $33.

KEY CONCEPT
An integer, n, and its opposite, -n, combine to make 0.
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 15

Do You Understand?
Question 1.
Essential Question How are integers and their opposites related?

Answer:
The integers and their opposites are related because an integer’s opposite has to be the same number away from zero as the integer in question.

Explanation:
In the above-given question,
given that,
The integers and their opposites are related because an integer’s opposite has to be the same number away from zero as the integer in question.
for example:
67 is the integer and its opposite side is -67.
its opposite has to be on the other side of zero and the same amount of numbers away from zero.

Rational Number Operations 5

Question 2.
Reasoning In order for an atom to have a zero charge, every proton, which has a charge of +1, must be matched with an electron, which has a charge of -1. A helium atom has 2 protons and 2 electrons. Explain why a helium atom has a zero charge.
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 16

Answer:
Helium atom has integers are +2 and -2.

Explanation:
In the above-given question,
atom has a zero charge, every proton, which has a charge of +1, must be matched with an electron, which has a charge of -1.
A helium atom has 2 protons and 2 electrons.
+2, and -2.
so helium atom has integers that are +2 and -2.

Question 3.
Model with Math Explain how to use a number line to show that opposite quantities combine to make 0.

Answer:
The number line shows the + 3 and – 3.

Explanation:
In the above-given question,
given that,
there are some numbers on the number line.
-3, 0, and 3.
so the number line shows the integers +3 and -3.

Do You Know How?
Question 4.
Marcus dives from the surface of the ocean to a reef 18 meters below sea level. What integer represents Marcus’s location relative to the surface? How far does Marcus have to go to return to the surface?

Answer:
The Marcus have to go to return to the surface = -18 meters.

Explanation:
In the above-given question,
given that,
Marcus dives from the surface of the ocean to a reef 18 meters below sea level.
-18 represents Marcus’s location relative to the surface.
-18, 0, and 18.
so the Marcus has to go to return to the surface = -18 meters.

Question 5.
The temperature of the water in Emily’s fish tank was 78°F on Sunday. The water temperature changed by -3° on Monday, and then by 3o on Tuesday. What integer represents the temperature change of the water from Sunday to Tuesday? What was the water temperature on Tuesday?

Answer:
The temperature change of the water from Sunday to Tuesday = plus.
The water temperature on Tuesday = 105°F.

Explanation:
In the above-given question,
given that,
The temperature of the water in Emily’s fish tank was 78°F on Sunday.
The water temperature changed by -3° on Monday.
75°F.
and then by 3o on Tuesday.
105°F.
the temperature change of the water from Sunday to Tuesday = plus.
the water temperature on Tuesday = 105°F.

Question 6.
The scores of players on a golf team are shown in the table. The team’s combined score was 0. What was Travis’s score?
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 17

Answer:
Travis’s score = -4.

Explanation:
In the above-given question,
given that,
The scores of players on a golf team are shown in the table.
The team’s combined score was 0.
Travis’s score is -4.

Practice & Problem Solving

Leveled Practice In 7-9, write the integer that represents the situation.
Question 7.
Max spent $53 and now has no money left. He had $ _______ before his purchase.

Answer:
He had $53 before his purchase.

Explanation:
In the above-given question,
given that,
Max spent $53 and now has no money left.
$53 – $53 = 0.
so he had $53 before his purchase.

Question 8.
The temperature was 8°F. It dropped so that the temperature was 0°F. ________ °F represents the change in temperature.

Answer:
The change in temperature represents = -8°F.

Explanation:
In the above-given question,
given that,
The temperature was 8°F.
It dropped so that the temperature was 0°F.
8 – 0 = 8.
so the change in the temperature represents = -8°F.

Question 9.
An airplane descended 4,000 feet before landing. The integer that represents how many feet the airplane was above the ground before its descent is _________.

Answer:
The number of feet the airplane was above the ground before its descent = -4000.

Explanation:
In the above-given question,
given that,
An airplane descended 4,000 feet before landing.
before landing = 4000 meters.
after landing = -4000 m.
so the number of feet the airplane was above the ground before its descent = -4000 m.

Question 10.
Carolyn says that point A and point B represent opposite integers.
a. What is the opposite of the integer represented by point A? By point B?
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 18

Answer:
The opposite of the integer represented by point A is -7 and by point, B is 8.

Explanation:
In the above-given question,
given that,
Carolyn says that point A and point B represent opposite integers.
in the number line, the numbers are -10, -5, 0, 5, and 10.
so the missing numbers are points A and B.
The opposite of the integer represented by point A is -7 and by point, B is 8.

b. Construct Arguments Do you agree with Carolyn? Explain.

Answer:
Yes, Arguments agree with Carolyn.

Explanation:
In the above-given question,
given that,
Carolyn says that point A and point B represent opposite integers.
in the number line, the numbers are -10, -5, 0, 5, and 10.
so the missing numbers are points A and B.
The opposite of the integer represented by point A is -7 and by point, B is 8.

Question 11.
A football team lost 9 yards during a play. The team had a combined gain or loss of 0 yards after the next play. What integer represents the yards gained or lost on the next play? Show this on the number line.
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 19

Answer:
The integer represents the yards gained = 9 yards.

Explanation:
In the above-given question,
given that,
A football team lost 9 yards during a play.
The team had a combined gain or loss of 0 yards after the next play.
the integer represents the yards gained  = 9 yards.
-9, 0, and 9.
so the integer represents the yards gained = 9 yards.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational Number Operations-2

Question 12.
A roller coaster car goes above and below ground. Use the number line to show its changes in height. What is the height of the car at the end of the ride?
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 20

Answer:
The height of the car at the end of the ride = 5 m.

Explanation:
In the above-given question,
given that,
A roller coaster car goes above and below ground.
The roller coaster starts at 1 meter.
drops 4 meters down.
rises 13 meters up.
drops 6 meters.
13 – 6 = 7.
so the height of the car at the end of the ride = 5 m.

Question 13.
Dimitri is buying a car. He chooses Option 1 to add a new sound system to his car. What integer represents the change from the base price of the car to its final price?
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 21

Answer:
The integer represents the change from the base price of the car to its final price = $1400.

Explanation:
In the above-given question,
given that,
Dimitri is buying a car.
He chooses Option 1 to add a new sound system to his car.
the base price of the car = -$700.
option 1 = $1400.
so the integer represents the change from the base price of the car to its final price = $1400.

Question 14.
Make Sense and Persevere What values do x and y have if |x| = 16, |y| = 16, and when x and y are combined they equal 0? Explain your reasoning.

Answer:
x and y are equal to 0.

Explanation:
In the above-given question,
given that,
if |x| = 16.
|y| = -16.
16 – 16 = 0.
so x and y are equal to 0.

Question 15.
Write a situation that can be represented by the opposite of -42.

Answer:
The situation that can be represented by the opposite of -42 is 42.

Explanation:
In the above-given question,
given that,
the situation that can be represented by the opposite of -42 is 42.
42 is the positive integer.
-42 is the negative integer.
so the situation that can be represented by the opposite 0f -42 is 42.

Question 16.
Higher Order Thinking Three friends all live on the same street that runs west to east. Beth lives 5 blocks from Ann. Carl lives 2 blocks from Beth. If the street is represented by a number line and Ann’s house is located at 0, what are the possible locations for Carl’s house? Assume that each unit on the number line represents 1 block.

Answer:
The possible locations for Carl’s house = 3 and 5.

Explanation:
In the above-given question,
given that,
Three friends all live on the same street that runs west to east.
Beth lives 5 blocks from Ann.
Carl lives 2 blocks from Beth.
5 – 2 = 3.
so the location of Ann is 0.
the location of Carl is 3.
the location of Beth is 5.
so the possible locations for Carl’s house = 3 and 5.

Assessment Practice

Question 17.
Which of these situations can be represented with an integer that when combined with -9 makes 0? Select all that apply.
☐ You walk down 9 flights of stairs.
☐ You climb up 9 flights of stairs.
☐ The temperature drops 9°F.
☐ You spend $9 on a book.
☐ You earn $9 from your job.

Answer:
You climb up 9 flights of stairs, you spend $9 on a book, and you earn $9 from your job.

Explanation:
In the above-given question,
given that,
an integer that when combined with -9 makes 0.
You climb up 9 flights of stairs.
you spend $9 on a book.
you earn $9 from your job.

Question 18.
Which of these situations can be represented by the opposite of 80? Select all that apply.
☐ An airplane descends 80 m.
☐ An elevator ascends 80 m.
☐ The cost of a train ticket drops by $80.
☐ You remove 80 songs from an MP3 player.
☐ Suzy’s grandmother is 80 years old.

Answer:
An airplane descends 80m, the cost of a train ticket drops by $80, and you remove 80 songs from an MP3 player.

Explanation:
In the above-given question,
given that,
the situations can be represented by the opposite of 80.
An airplane descends 80m = -80m.
the cost of a train ticket drops by $80 = -$80.
You remove 80 songs from an MP3 player = -80.

Lesson 1.2 Understand Rational Numbers

Solve & Discuss It!
Calvin wants to customize his surfboard so that it is wider than the 82 model but narrower than the 92 model. What measurement could be the width of his surfboard? Explain.

I can… recognize rational numbers and write them in decimal form.
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 22

Answer:
The width of his surfboard = 1 wide.

Explanation:
In the above-given question,
given that,
Calvin wants to customize his surfboard so that it is wider than the 82 models but narrower than the 92 models.
82 model has a wide of 221/2 and 3(1/4) of thick.
92 model has a wide of 23(1/4) and 3(1/2) thick.
23 – 22 = 1.
so the width of his surfboard = 1 wide.

Focus on math practices
Use Structure Lindy’s surfboard is 23 inches wide. Between which two surfboard models is her custom surfboard’s width? How do you know?

Answer:
The two surfboard models are her custom surfboard’s width = surfboards 82 and 92.

Explanation:
In the above-given question,
given that,
Lindy’s surfboard is 23 inches wide.
Between which, two surfboard models are her custom surfboard’s width.
so the two surfboard models is her custom surfboard’s width = surfboards 82 and 92.

Essential Question
How are rational numbers written as decimals?

Try It!

In the next several games, the pitcher threw a total of 384 pitches and used a fastball 240 times. What decimal should Juanita use to update her report?
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 23
Juanita should use the decimal ________ to update her report.

Answer:
Juanita should use the decimal 0 to update her report.

Explanation:
In the above-given question,
given that,
the pitcher threw a total of 384 pitches and used a fastball 240 times.
380 / 240.
380 x 6 = 2304.
380 x 2 = 768.
380 x 5 = 1920.
1920 – 1920 = 0.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational Number Operations-3

Convince Me! How do you know that the answer is a terminating decimal?

Try It!

What is the decimal form of \(\frac{100}{3}, \frac{100}{5}\) and \(\frac{100}{6}\)? Determine whether each decimal repeats or terminates.

Answer:
The decimal form of 100/3 = 33.3.
the decimal form of 100/5 = 20.
the decimal form of 100/6 = 16.6.

Explanation:
In the above-given question,
given that,
the decimal form of 100/3, 100/5, and 100/6.
100/3 = 33.3.
100/5 = 20.
100/6 = 16.6.
the decimal 100/3 and 100/6 repeats.
the decimal 100/5 terminates.

Try It!

Is -0.3 a rational number? Is 3.14144144414444… a rational number? Explain your reasoning.

Answer:
Yes, both are rational numbers.

Explanation:
In the above-given question,
given that,
the numbers are -0.3 and 3.14144144414444…
-0.3 is a rational number.
3.14144144414444 is a rational number.

KEY CONCEPT
To convert from the fraction form of a rational number to its decimal form, divide the numerator by the denominator. The decimal form of a rational number either terminates in 0s or eventually repeats.
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 24

Do You Understand?
Question 1.
Essential Question How are rational numbers written as decimals?

Answer:
Rational numbers can be written as decimals in two ways.
they are terminating decimal and repeating decimal.

Explanation:
In the above-given question,
given that,
the rational numbers can be written as a decimal in two ways.
for example:
100/5 = 20.
20 is a terminating decimal.
100/3 = 33.3.
33.3 is a repeating decimal.

Question 2.
Reasoning How can you use division to find the decimal equivalent of a rational number?

Answer:
100/3 = 33.3.

Explanation:
In the above-given question,
given that,
100/3 = 33.3.
33.3 is a terminating decimal.
100/5 = 20.
20 is a repeating decimal.

Question 3.
Be Precise What is the difference between a terminating decimal and a repeating decimal?

Answer:
The difference is one decimal will stop and one decimal will continue.

Explanation:
In the above-given question,
given that,
the two terms are terminating and repeating decimals.
100/3 = 33.3.
it is a repeating decimal.
100/5 = 20.
20 is a terminating decimal.

Do You Know How?
Question 4.
What is the decimal equivalent of each rational number?
a. \(\frac{7}{20}\)

Answer:
7/20 = 0.35.

Explanation:
In the above-given question,
given that,
the number is 7/20.
7/20 = 0.35.
0.35 is a terminating decimal.

b. –\(\frac{23}{20}\)

Answer:
-23/20 = -1.15.

Explanation:
In the above-given question,
given that,
the number is -23/20.
-23/20 = -1.15.
it is a terminating decimal.

c. \(\frac{1}{18}\)

Answer:
1/18 = 0.055.

Explanation:
In the above-given question,
given that,
the number is 1/18.
1/18 = 0.055.
0.055 is a repeating decimal.

d. –\(\frac{60}{22}\)

Answer:
-60/22 = 2.727 is a repeating decimal.

Explanation:
In the above-given question,
given that,
the number is -60/22.
-60/22 = 2.727.
2.727 is a repeating decimal.

Question 5.
There are 5,280 feet in a mile. What part of a mile, in decimal form, will you drive until you reach the exit?
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 25

Answer:
5280/1000 = 5.28 miles.

Explanation:
In the above-given question,
given that,
There are 5,280 feet in a mile.
5280/1000 = 5.28.
so the part of the mile in decimal form = 5.28 miles.

Practice & Problem Solving

Leveled Practice In 6-8, write the decimal equivalent for each rational number. Use a bar over any repeating digits.
Question 6.
\(\frac{2}{3}\)

Answer:
2/3 = 0.666.

Explanation:
In the above-given question,
given that,
the number is 2/3.
2/3 = 0.666.
0.666 is a repeating decimal.

Question 7.
\(\frac{3}{11}\)

Answer:
3/11 = 0.272727.

Explanation:
In the above-given question,
given that,
the number is 3/11.
3/11 = 0.272727.
0.272727 is a repeating decimal.

Question 8.
\(8\frac{4}{9}\)

Answer:
8(4/9) = repeating decimal.

Explanation:
In the above-given question,
given that,
the number is 8(4/9).
9 x 8 = 72.
72 + 4 = 76.
76/9 = 8.4444.
8.444 is a repeating decimal.

Question 9.
Is 1.0227 a rational number? Explain.

Answer:
Yes, 1.0227 is a rational number.

Explanation:
In the above-given question,
given that,
the number is 1.0227.
1.0227 is a rational number.

Question 10.
Which should Aaron use to convert a fraction to a decimal?
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 26

Answer:
Option A is correct.

Explanation:
In the above-given question,
given that,
numerator/denominator is used to convert a fraction to a decimal.
so option A is correct.

Question 11.
Is the fraction \(\frac{1}{3}\) equivalent to a terminating decimal or a decimal that does not terminate?

Answer:
1/3 is a repeating decimal.

Explanation:
In the above-given question,
given that,
the number is 1/3.
1/3 = 0.333.
so 1/3 is a repeating decimal.

Question 12.
Determine whether the given number belongs to each set.
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 27

Answer:
-34 is an Integer.

Explanation:
In the above-given question,
given that,
they have given the whole numbers, integers, and rational numbers.
so -34 is an integer.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational Number Operations-4

Question 13.
Ariel incorrectly says that 2\(\frac{5}{8}\) is the same as 2.58.
a. Convert 2\(\frac{5}{8}\) to a decimal.

Answer:
2(5/8) = 2.625.

Explanation:
In the above-given question,
given that,
Ariel incorrectly says that 2\(\frac{5}{8}\) is the same as 2.58.
the number is 2(5/8).
8 x 2 = 16.
16 + 5 = 21.
21/8 = 2.625.

b. What was Ariel’s likely error?

Answer:
Ariel’s error was the decimal.

Explanation:
In the above-given question,
given that,
Ariel incorrectly says that 2\(\frac{5}{8}\) is the same as 2.58.
the number is 2(5/8).
8 x 2 = 16.
16 + 5 = 21.
21/8 = 2.625.

Question 14.
Use Structure Consider the rational number \(\frac{3}{11}\)
a. What are the values of a and b in \(a\sqrt {b}\) when you use division to find the decimal form?

Answer:
3/11 = 0.272727.

Explanation:
In the above-given question,
given that,
a = 3, and b = 11.
3/11 = 0.272727.
so 3/11 is a repeating decimal.
3/11 = 0.272727.

b. What is the decimal form for \(\frac{3}{11}\)?

Answer:
The decimal form is 0.272727.

Explanation:
In the above-given question,
given that,
a = 3, and b = 11.
3/11 = 0.272727.
so 3/11 is a repeating decimal.
3/11 = 0.272727.

Question 15.
At a grocery store, Daniel wants to buy 3\(\frac{1}{5}\) lb of ham. What decimal should the digital scale show?
Write 3\(\frac{1}{3}\) as a fraction and then divide.
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 28
The scale should read ________ lb.

Answer:
The scale should read 3.3 lb.

Explanation:
In the above-given question,
given that,
At a grocery store, Daniel wants to buy 3\(\frac{1}{5}\) lb of ham.
3(1/3).
3 x 3 = 9.
9 + 1 = 10.
10/3 = 3.333.
3.333 is a repeating decimal.
so the scale should read 3.3 lb.

Question 16.
Reasoning At a butcher shop, Hilda bought beef and pork. She left with 18\(\frac{8}{25}\) pounds of meat. Express the number of pounds of pork she bought using a decimal.
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 29

Answer:
The number of pounds of pork she bought using a decimal = 9.47 pounds.

Explanation:
In the above-given question,
given that,
At a butcher shop, Hilda bought beef and pork.
She left with 18\(\frac{8}{25}\) pounds of meat.
18(8/25).
18 x 25 = 450.
458/25 = 18.32.
the cost of the beef is 8(17/20).
20 x 8 = 160.
177/20 = 8.85.
18.32 – 8.85 = 9.47.

Question 17.
Be Precise Is 9.373 a repeating decimal? Is it rational? Explain your reasoning,

Answer:
Yes, it is a repeating decimal.

Explanation:
In the above-given question,
given that,
the number is 9.373 is a rational number.
it is also a repeating decimal.

Question 18.
Reasoning Aiden has one box that is 3\(\frac{3}{11}\), feet tall and a second box that is 3.27 feet tall. If he stacks the boxes, about how tall will the stack be?

Answer:
The tall will the stack be = 6.54 feet.

Explanation:
In the above-given question,
given that,
Aiden has one box that is 3\(\frac{3}{11}\), feet tall and a second box that is 3.27 feet tall.
3(3/11).
33 + 3 = 36.
36/11 = 3.272727.
3.27 + 3.27 = 6.54,
so the heght of the stack = 6.54 feets.

Question 19.
You are adding air to a tire. The air pressure in the tire should be \(32\frac{27}{200}\) pounds per square inch. What decimal should you watch for on the digital pressure gauge?

Answer:
The decimal should I watch for on the digital pressure gauge = terminating decimal.

Explanation:
In the above-given question,
given that,
You are adding air to a tire.
The air pressure in the tire should be \(32\frac{27}{200}\) pounds per square inch.
32(27/200).
32 x 200 = 6400.
6400 + 27 = 6427.
6427/200 = 32.135.
so the decimal should I watch for on the digital pressure gauge = terminating decimal.

Question 20.
Higher Order Thinking Dion has a pizza with a diameter of 10\(\frac{1}{3}\) in. Is the square box shown big enough to fit the pizza inside? Justify your answer.
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 30

Answer:
The square box shown is big enough to fit the pizza inside.

Explanation:
In the above-given question,
given that,
Dion has a pizza with a diameter of 10\(\frac{1}{3}\) in.
10(1/3).
30 + 1/3.
31/3 = 10.33.
so the square box shown is big enough to fit the pizza inside.

Assessment Practice

Question 21.
Which of the following mixed numbers has the same decimal value as \(110 \frac{147}{168}\)?
A. \(110 \frac{49}{56}\)
B. \(110 \frac{170}{180}\)
C. \(110 \frac{56}{72}\)
D. \(110 \frac{247}{268}\)

Answer:
Option A is the correct.

Explanation:
In the above-given question,
given that,
110/ 147/168.
110/ 0.875.
so option A is correct.

Question 22.
Select all the true statements about the negative fractions –\(\frac{4}{5}\) and –\(\frac{5}{6}\)
–\(\frac{4}{5}\) can be expressed as a repeating decimal.
o –\(\frac{5}{6}\) can be expressed as a repeating decimal.
Both fractions can be expressed as repeating decimals.
The digit that repeats is 3.
The digit that repeats is 8.

Answer:
-5/6 can be expressed as a repeating decimal.
the digit that repeats is 3.

Explanation:
In the above-given question,
given that,
the two numbers are -4/5 and -5/6.
-4/5 and -5/6.
-4/5 = -0.8.
-5/6 = -0.8.
so -5/6 can be expressed as a repeating decimal.

Lesson 1.3 Add Integers

Explore It!
Rain increases the height of water in a kiddie pool, while evaporation decreases the height. The pool water level is currently 2 inches above the fill line.

I can… add integers.
Envision Math Common Core 7th Grade Answer Key Topic 1 Rational Number Operations 31

A. Look for patterns in the equations in the table so you can fill in the missing numbers. Describe any relationships you notice.

Answer:
The missing numbers are 2, -1, -2.

Explanation:
In the above-given question,
given that,
the top of the pool is 4.
the bottom of the pool is -4.
we can add integers.
so the missing numbers are 2, -1, and -2.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational Number Operations-5

B. Will the sum of 2 and (-6) be a positive or negative number? Explain.

Answer:
The sum of 2 and -6 be a negative number.

Explanation:
In the above-given question,
given that,
2 + (-6).
2 – 6 = -4.
the sum of two +ve numbers will always be a positive number.
the sum of two -ve numbers will also be positive number.
the sum of one +ve and one -ve is a negative number.
so the sum of 2 and -6 be a negative number.

Focus on math practices
Look for Relationships Suppose the water level of the pool started at 2 inches below the fill line. Make a table to show the starting height of the water, the change in inches, and the new final height of the water.

Essential Question
How do you use what you know about the absolute value to add integers?

Try It!

Dana recorded a temperature drop of 2o and a second temperature drop of 3°. What is the total change in temperature?
_______ + ________ = _________
The sign of the sum is _______. The total change in temperature is ________°.

Answer:
The sign of the sum is negative.
the total change in temperature is 5°.

Explanation:
In the above-given question,
given that,
Dana recorded a temperature drop of 2o and a second temperature drop of 3°.
-2 – 3 = -5.
so the sign of the sum is negative.
the total change in temperature is 5°.

Convince Me! Would the sum of two positive integers be positive or negative? Explain.

Answer:
The sum of two positive integers is positive.

Explanation:
In the above-given question,
given that,
the sum of two positive integers is positive.
2 + 3 = 5.
so the sum is also a positive number.

Try It!

Find the sum for each expression.
a. -66 + 42

Answer:
-66 + 42 = -24.

Explanation:
In the above-given question,
given that,
the numbers are -66 and 42.
add the numbers.
the sum of one positive and one negative is also a negative integer.
-66 + 42 = -24.

b. -57 + 57

Answer:
-57 + 57 = 0.

Explanation:
In the above-given question,
given that,
the numbers are -57 and 57.
add the numbers.
the sum of one positive and one negative is also a negative integer.
-57 + 57 = 0.

c. 29 + (-28)

Answer:
29 + (-28) = 1.
Explanation:
In the above-given question,
given that,
the numbers are -28 and 29.
add the numbers.
the sum of one positive and one negative is also a negative integer.
29 + (-28) = 1.

KEY CONCEPT
When adding integers with the same sign, find the sum of the absolute values.
(-36) + (-12)
|-36| = 36 and |-12| = 12
36 + 12 = 48
So, (-36) + (-12) = -48
Use the same sign as the addends.

When adding integers with different signs, find the difference of the absolute values.
18+ (-14)
|18| = 18 and |-14| = 14
18 – 14 = 4
So, 18 + (-14) = 4
Use the sign of the greater absolute value.

Do You Understand?
Question 1.
Essential Question How do you use what you know about absolute value to add integers?

Answer:
When adding the integers with the same sign, we will find the sum of absolute values.

Explanation:
In the above-given question,
given that,
When adding the integers with same sign, we will find the sum of absolute values.
for example:
(-36) + (-12)
|-36| = 36 and |-12| = 12
36 + 12 = 48
So, (-36) + (-12) = -48.

Question 2.
Reasoning How can you tell the sign of the sum of a positive and negative integer without doing any calculations?

Answer:
The sum of positive and negative integers is always negative.

Explanation:
In the above-given question,
given that,
the sum of positive and negative integers is always negative.
for example:
-33 + 22.
-11.
so the sum of positive and negative integers is always negative.

Question 3.
Model with Math How would you use a number line to determine the sum of two negative integers?

Answer:
The sum of two negative integers is also a positive integer.

Explanation:
In the above-given question,
given that,
the sum of two negative integers is also a positive integer.
for example:
-33 + -22.
-33 – 22.
55.

Do You Know How?
Question 4.
Sarah bought a bike that cost $260. She had a coupon that was worth $55 off the cost of any bike. Use the expression 260 + (-55) to find how much Sarah paid for her bike.

Answer:
The amount Sarah paid for her bike = $205.

Explanation:
In the above-given question,
given that,
Sarah bought a bike that cost $260.
She had a coupon that was worth $55 off the cost of any bike.
260 + (-55).
260 – 55.
205.
so the amount Sarah paid for her bike = $205.

Question 5.
A shark is swimming 60 feet below the surface of the ocean. There is a fish that is 25 feet deeper in the water. Use the expression (-60) + (-25) to describe the fish’s location relative to the surface of the ocean.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 32

Answer:
The fish’s location is relative to the surface of the ocean = -85 feet.

Explanation:
In the above-given question,
given that,
A shark is swimming 60 feet below the surface of the ocean.
There is a fish that is 25 feet deeper in the water.
(-60) + (-25).
-60 – 25 = -85.
so the fish’s location relative to the surface of the ocean = -85 feet.

Question 6.
The high temperature one day was 30°F. Then the temperature dropped 23 degrees during the night. Does the expression 30 + (-23) represent the temperature at night? Explain.

Answer:
The expression 30 + (-23) represent the temperature at night = 7 degrees.

Explanation:
In the above-given question,
given that,
The high temperature one day was 30°F.
Then the temperature dropped 23 degrees during the night.
30 + (-23).
30 – 23 = 7.
so the expression represents the temperature at night = 7 degrees.

Practice & Problem Solving

Leveled Practice For 7-9, use the number lines to help find each sum.

Question 7.
5 + (-3) is ________ units from 5, in the ________ direction.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 33

Answer:
5 + (-3) is ____2____ units from 5, in the _positive____ direction.

Explanation:
In the above-given question,
given that,
5 + (-3).
5 – 3 = 2.
2 units from 5, in the positive direction.

Question 8.
-1 + (-3) is _______ units from -1, in the ________ direction.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 33

Answer:
-1 + (-3) is __-4_____ units from -1, in the ___negative_____ direction

Explanation:
In the above-given question,
given that,
-1 + (-3).
-1 – 3 = -4.
-4 units from -1.
in the negative direction.

Question 9.
In City A, the temperature rises 9° from 8 A.M. to 9 A.M. Then the temperature drops 8° from 9 A.m. to 10 A.M. In City B, the temperature drops 5° from 8 A.M. to 9 A.M. Then the temperature drops 4° from 9 A.M. to 10 A.M.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 34
a. What expression represents the change in temperature for City A?

Answer:
The expression represents the change in temperature for City A is 9° – 8° = 1°.

Explanation:
In the above-given question,
given that,
In City A, the temperature rises 9° from 8 A.M. to 9 A.M.
Then the temperature drops 8° from 9 A.m. to 10 A.M.
9° – 8° = 1°.
so the expression represents the change in temperature for City A is 9° – 8° = 1°.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational Number Operations-6

b. What integer represents the change in temperature for City A?

Answer:
The positive integer represents the change in temperature for City A.

Explanation:
In the above-given question,
given that,
In City A, the temperature rises 9° from 8 A.M. to 9 A.M.
Then the temperature drops 8° from 9 A.m. to 10 A.M.
9° – 8° = 1°.
so the positive integer represents the change in temperature for City A.

c. What expression represents the change in temperature for City B?

Answer:
The expression represents the change in temperature for City B is 5° – 4° = 1°.

Explanation:
In the above-given question,
given that,
In City B, the temperature rises 5° from 8 A.M. to 9 A.M.
Then the temperature drops 4° from 9 A.m. to 10 A.M.
5° – 4° = 1°.
so the expression represents the change in temperature for City B is 5° – 4° = 1°.

d. What integer represents the change in temperature for City B?

Answer:
A positive integer represents the change in temperature for City B.

Explanation:
In the above-given question,
given that,
In City B, the temperature rises 5° from 8 A.M. to 9 A.M.
Then the temperature drops 4° from 9 A.m. to 10 A.M.
5° – 4° = 1°.
so the positive integer represents the change in temperature for City B.

e. Which city has the greater change in temperature from 8 A.M. to 10 A.M.?

Answer:
Both the cities have an equal change in temperature.

Explanation:
In the above-given question,
given that,
In City A, the temperature rises 9° from 8 A.M. to 9 A.M.
Then the temperature drops 8° from 9 A.m. to 10 A.M.
9° – 8° = 1°.
5° – 4° = 1°.
so both the cities have an equal change in temperature.

Question 10.
An airplane flying at an altitude of 30,000 feet flies up to avoid a storm. Immediately after passing the storm, the airplane returns to its original altitude.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 35
a. What integer represents the airplane’s change in altitude to avoid the storm?

Answer:
The positive integer represents the airplane’s change in altitude to avoid the storm.

Explanation:
In the above-given question,
given that,
An airplane flying at an altitude of 30,000 feet flies up to avoid a storm.
the airplane flies up to 38,000 feet to avoid a storm.
38000 – 30000.
8000.
so the positive integer represents the airplane’s change in altitude to avoid the storm.

b. What integer represents the airplane’s change in altitude immediately after passing the storm?

Answer:
The positive integer represents the airplane’s change in altitude immediately after passing the storm.

Explanation:
In the above-given question,
given that,
An airplane flying at an altitude of 30,000 feet flies up to avoid a storm.
the airplane flies up to 38,000 feet to avoid a storm.
38000 – 30000.
8000.
so the positive integer represents the airplane’s change in altitude immediately after passing the storm.

c. Use Appropriate Tools Draw a number line to represent the airplane’s change in altitude.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 36

Answer:
The change in altitude is 38000.

Explanation:
In the above-given question,
given that,
An airplane flying at an altitude of 30,000 feet flies up to avoid a storm.
the airplane flies up to 38,000 feet to avoid a storm.
30000 + 8000 = 38000.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational Number Operations-7

Question 11.
A deep-sea diver dives 81 feet from the surface. He then dives 14 more feet. The diver’s depth can be represented by -81 + (-14). What is the diver’s present location?

Answer:
The diver’s present location is -95  feet below the sea.

Explanation:
In the above-given question,
given that,
A deep-sea diver dives 81 feet from the surface.
He then dives 14 more feet.
The diver’s depth can be represented by -81 + (-14).
-81 + (-14).
-81 – 14 = -95.
so the diver’s present location is -95 feet below the sea.

Question 12.
Rena’s rowboat drifts 23 feet from shore, followed by 9 more feet. The rowboat’s current position can be represented by –23 + (-9). What integer represents the rowboat’s position?

Answer:
The integer represents the rowboat’s position as negative.

Explanation:
In the above-given question,
given that,
Rena’s rowboat drifts 23 feet from shore, followed by 9 more feet.
The rowboat’s current position can be represented by –23 + (-9).
-23 – 9 = -32.
so the integer represents the rowboat’s position is negative.

Question 13.
Critique Reasoning A submarine traveling 200 meters below the surface of the ocean increases its depth by 45 meters. Adam says that the new location of the submarine is -155 meters. Describe an error Adam could have made that would result in the answer he gave.

Answer:
Yes, Adam was incorrect.

Explanation:
In the above-given question,
given that,
A submarine traveling 200 meters below the surface of the ocean increases its depth by 45 meters.
200 + 45 = 245.
but Adam says that the new location of the submarine is -155 meters.
so Adam was incorrect.

Question 14.
Kim has $45 to spend for a day at the zoo. She pays $17 for admission, $8 for lunch, and $4 for a snack.
a. Model with Math Use integers to write an addition expression that represents the amount of money Kim has left.

Answer:
The amount of money Kim has left = $16.

Explanation:
In the above-given question,
given that,
Kim has $45 to spend for a day at the zoo.
She pays $17 for admission, $8 for lunch, and $4 for a snack.
$45 + (-$17) + (-$8) + (-$4).
$45 – $17 – $8 – $4.
$45 – $29.
$16.
so the amount of money Kim has left = $16.

b. Kim goes to the gift shop and finds a T-shirt she likes for $19. Does she have enough money to buy the T-shirt? Explain.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 37

Answer:
No, she does not have enough money to buy the T-shirt.

Explanation:
In the above-given question,
given that,
Kim has $45 to spend for a day at the zoo.
She pays $17 for admission, $8 for lunch, and $4 for a snack.
she has left $16.
so she does not have enough money to buy the T-shirt.

Question 15.
Higher Order Thinking Samantha has $300 for guitar lessons to learn her favorite song. Mrs. Jones charges $80 per lesson and requires three lessons to teach Samantha the song. Mr. Beliz charges $62 per lesson and will require four lessons to teach Samantha the song. Use integers to represent what each teacher charges. Which is the better deal for Samantha?

Answer:
The better deal for Samantha is Mr. Beliz charges $62 per lesson.
$62 + $62 + $62 + $62 = $248.

Explanation:
In the above-given question,
given that,
Samantha has $300 for guitar lessons to learn her favorite song.
Mrs. Jones charges $80 per lesson and requires three lessons to teach Samantha the song.
Mr. Beliz charges $62 per lesson and will require four lessons to teach Samantha the song.
$80 + $80 + $80 = $240.
$62 + $62 + $62 + $62 = $248.
so the better deal for Samantha is Mr. Beliz charges $62 per lesson.

Assessment Practice

Question 16.
A fish swims at 10 ft below sea level, and then swims another 10 ft deeper to avoid a shark. Write an addition expression that represents this situation.

Answer:
The addition expression that represents this situation is (-10) + (-10) = -20.

Explanation:
In the above-given question,
given that,
A fish swims at 10 ft below sea level and then swims another 10 ft deeper to avoid a shark.
(-10) + (-10) = -20.
below sea level = -10.
so the addition expression that represents this situation is (-10) + (-10) = -20.

Question 17.
The temperature drops 10 degrees and then rises 10 degrees. Write an addition expression that represents this situation.

Answer:
The addition expression that represents this situation is 10 + (-10) = 0.

Explanation:
In the above-given question,
given that,
The temperature drops 10 degrees and then rises 10 degrees.
10 + (-10).
10 – 10 = 0.
so the addition expression that represents this situation is 10 + (-10) = 0.

Lesson 1.4 Subtract Integers

Solve & Discuss It!
A library database shows the total number of books checked out at any given time as a negative number. What are the possible numbers of books that were checked out and checked in on Monday? Explain.
I can.. subtract integers.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 38

Make Sense and Persevere
How can you use the data to understand what happened during the day?

Focus on math practices
Reasoning Suppose the library database showed 0 for Monday evening. What do you know about the number of books checked out and checked in that day?

Answer:
The number of books checked out and checked in that day = 45 books.

Explanation:
In the above-given question,
given that,
0 + (-45).
0 – 45 = -45.
so the number of books checked out and checked in that day = -45.

Essential Question
How is subtracting integers related to adding integers?

Try It!

On the next play, the team gained 5 yards and then lost 6 yards. What is the total change in yards?
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 39
5 – ______
= 5 + ______
= _______
The total change in yards is ________, so they had a total loss of ________ yard.

Answer:
The total change in yards is 1.
so they had a total loss of 1 yard.

Explanation:
In the above-given question,
given that,
On the next play, the team gained 5 yards and then lost 6 yards.
5 – 6 = -1.
5 + (-6) = -1.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational Number Operations-8

Convince Me! Is the additive inverse of an integer always negative? Explain.

Try It!

Subtract. Use a number line to help you find the answer.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 40
a. -4 – 6

Answer:
-4 -6 = 10.

Explanation:
In the above-given question,
given that,
the numbers are -4 and -6.
-4 + (-6).
-4 – 6 = -10.

b. -6 – (-4).

Answer:
-6 – (-4) = -2.

Explanation:
In the above-given question,
given that,
the numbers are -4 and -6.
-6 – (-4).
-6 + 4.
-2.

c. 4 – (-6)

Answer:
4 – (-6) = 10.

Explanation:
In the above-given question,
given that,
the numbers are 4 and -6.
4 – (-6).
4 + 6.
10.

d. 6 – 4

Answer:
6 – 4 = 2.

Explanation:
In the above-given question,
given that,
the numbers are 6 and -4.
6 – 4 = 2.

e. 4 – 6

Answer:
4 – 6 = -2.

Explanation:
In the above-given question,
given that,
the numbers are 4 and -6.
4 – 6 = -2.

f. -4 – (-6)

Answer:
-4 + 6 = 2.

Explanation:
In the above-given question,
given that,
the numbers are -4 and -6.
-4 – (-6).
-4 + 6 = 2.

KEY CONCEPT
When subtracting integers, such as a – b, you can use the additive inverse to write subtraction as an equivalent addition expression.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 41

Do You Understand?
Question 1.
Essential Question How is subtracting integers related to adding integers?

Answer:
Subtracting integers related to additive inverse as an equivalent addition.

Explanation:
In the above-given question,
given that,
Subtracting integers related to additive inverse as an equivalent addition.
for example:
a + (-b) = a -b.

Question 2.
Reasoning Explain how to simplify the expression -98 – 31 using the additive inverse.

Answer:
-98 + (-31) = – 129.

Explanation:
In the above-given question,
given that,
the numbers are -98 and -31.
-98 + (-31).
-129.

Question 3.
Model with Math How can you use a number line to represent the subtraction between two integers?

Answer:
-5 – 2 = -7.

Explanation:
In the above-given question,
given that,
the two numbers on the number line = -5 and -2.
-5 + (-2).
-5 – 2 = -7.

Do You Know How?
Question 4.
It was 12°C when Preston got home from school. The weather report shows a storm front moving in that will drop the temperature by 17°C. What is the expected temperature?

Answer:
The expected temperature is -5 degrees c.

Explanation:
In the above-given question,
given that,
It was 12°C when Preston got home from school.
The weather report shows a storm front moving in that will drop the temperature by 17°C.
12 – 17 = -5.
so the expected temperature is -5°C.

Question 5.
Complete the equation.
-67 – _____ = 0

Answer:
-67 – (-67) = 0.

Explanation:
In the above-given question,
given that,
-67 – (-67).
-67 + 67 = 0.

Question 6.
Find the difference.
a. 41 – 275

Answer:
41 – 275 = -234.

Explanation:
In the above-given question,
given that,
the two numbers are 41 and 275.
subtract the numbers.
41 – 275.
– 234.

b. -15 – 47

Answer:
-15 – 47 = -32.

Explanation:
In the above-given question,
given that,
the two numbers are -15 and -47.
subtract the numbers
-15 – 47.
47 – 15 = -32.

c. -72 – (-151)

Answer:
-72 – (-151) = 79.

Explanation:
In the above-given question,
given that,
the two numbers are -72 and -151.
subtract the numbers.
-72 – (-151).
-72 + 151.
79.

d. 612 – (-144)

Answer:
612 – (-144) = 756.

Explanation:
In the above-given question,
given that,
the two numbers are 612 and -144.
subtract the numbers.
612 – (-144).
612 + 144 = 756.

Practice & Problem Solving

Leveled Practice In 7-8, fill in the boxes to solve.
Question 7.
What subtraction expression does the number line model show?
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 42

Answer:
-2 + (-9) = 7.

Explanation:
In the above-given question,
given that,
the lines are moving from -2 and -9.
-2 + (-9).
-2 – 9 = -11.

Question 8.
What is the value of the expression -9 – (-5)?
-9 – (-5)
= -9 ______ 5
= _______

Answer:
The value of the expression -9 – (-5) = -4.

Explanation:
In the above-given question,
given that,
the two numbers are -9 and -5.
subtract the numbers.
-9 – (-5) = -4.

Question 9.+e
The temperature at the beginning of the day was 6°F. The temperature dropped 9°F by the end of the day. Use the number line to find the temperature at the end of the day.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 43

Answer:
The temperature at the end of the day = -3.

Explanation:
In the above-given question,
given that,
The temperature at the beginning of the day was 6°F.
The temperature dropped 9°F by the end of the day.
6 + (- 9) = -3.
so the temperature at the end of the day = -3.

Question 10.
Murphy and Naryam do their math homework together. When they find 9- (-8), they get different answers. Murphy claims the difference is 17. Naryam claims the difference is – 1.
a. Who is correct?

Answer:
Murphy was correct.

Explanation:
In the above-given question,
given that,
Murphy and Naryam do their math homework together.
When they find 9- (-8), they get different answers.
9 – (-8).
9 + 8 = 17.
so Murphy was correct.

b. What error likely led to the incorrect answer?

Answer:
The sign likely led to the incorrect answer.

Explanation:
In the above-given question,
given that,
Murphy and Naryam do their math homework together.
When they find 9- (-8), they get different answers.
9 – (-8).
9 + 8 = 17.

Question 11.
The news reports that today’s high temperature is 16°F colder than yesterday’s high temperature. Yesterday’s high temperature was -2°F.
a. Write an expression to represent today’s high temperature.

Answer:
The expression is -2°F + 16°F = 14°F.

Explanation:
In the above-given question,
given that,
The news reports that today’s high temperature is 16°F colder than yesterday’s high temperature.
Yesterday’s high temperature was -2°F.
so the expression is -2°F + 16°F = 14°F.

b. Reasoning is today’s high temperature positive or negative? Why?

Answer:
Today’s high temperature is positive.

Explanation:
In the above-given question,
given that,
The news reports that today’s high temperature is 16°F colder than yesterday’s high temperature.
Yesterday’s high temperature was -2°F.
so the expression is -2°F + 16°F = 14°F.

Question 12.
Max sprints forward 10 feet and then stops and sprints back 15 feet. Use subtraction to explain where Max is relative to where he started.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 44

Answer:
He started at -5.

Explanation:
In the above-given question,
given that,
Max sprints forward 10 feet and then stops and sprints back 15 feet.
10 + (-15).
10 – 15.
-5.
so he started at -5.

Question 13.
Higher Order Thinking Use the number line at the right.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 45
a. What subtraction equation does the number line represent?

Answer:
The subtraction equation is -2 + (-6) = -8.

Explanation:
In the above-given question,
given that,
the starting point is -2.
he then increases by -6.
-2 + (-6).
-2 -6 = -8.

b. Use the number line to represent a different subtraction equation that has the same difference shown in the number line. Write the subtraction equation.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 46

Answer:
0 – 2 = -2.
-4 – (- 2) = -2.

Explanation:
In the above-given question,
given that,
the subtraction equations are:
0 – 2 = -2.
-4 – (-2).
-4 + 2 = -2.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational Number Operations-9

Question 14.
A crane lifts a pallet of concrete blocks 8 feet from the back of a truck. The truck drives away and the crane lowers the pallet 13 feet: What is the final position of the pallet relative to where it started in the back of the truck?
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 47

Answer:
The final position of the pallet relative to where it started in the back of the truck = 5 feet.

Explanation:
In the above-given question,
given that,
A crane lifts a pallet of concrete blocks 8 feet from the back of a truck.
The truck drives away and the crane lowers the pallet 13 feet:
13 – 8 = 5.
so the final position of the pallet relative to where it started in the back of the truck = 5 feet.

Question 15.
Make Sense and Persevere At its highest point, the elevation of a county is 5,762 feet above sea level. At its lowest point, the elevation of the county is 9 feet below sea level.
a. Write an expression using integers to represent the difference between the elevations.

Answer:
The expression using integers to represent the difference between the elevations is 5753.

Explanation:
In the above-given question,
given that,
At its highest point, the elevation of a county is 5,762 feet above sea level.
At its lowest point, the elevation of the county is 9 feet below sea level.
5762 + (-9) = 5753.
so the expression using integers to represent the difference between the elevations is 5753.

b. Will the answer be written as a positive or negative integer?

Answer:
The answer will be written as a positive number.

Explanation:
In the above-given question,
given that,
At its highest point, the elevation of a county is 5,762 feet above sea level.
At its lowest point, the elevation of the county is 9 feet below sea level.
5762 + (-9) = 5753.
so the answer will be written as a positive number.

c. What is the difference between the highest and lowest points of the county?

Answer:
The difference between the highest and lowest points of the county is 5753.

Explanation:
In the above-given question,
given that,
At its highest point, the elevation of a county is 5,762 feet above sea level.
At its lowest point, the elevation of the county is 9 feet below sea level.
5762 + (-9) = 5753.
so the difference between the highest and lowest points of the county is 5753.

Assessment Practice

Question 16.
Which number line model shows the subtraction 2 – 4?
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 48

Answer:
Option D is the correct answer.

Explanation:
In the above-given question,
given that,
the number line from 0 to -2 and -2 to -6.
-2 + (-6).
-2 -6 = -8.
so option D is the correct answer.

Lesson 1.5 Add and Subtract Rational Numbers

Solve & Discuss It!
Malik hikes Castle Trail from point A to point B. The elevation at point A is below sea level. What are possible beginning and ending elevations of Malik’s hike?

I can… add and subtract rational numbers.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 49

Look for Relationships
How are elevation values of point A and point B related?

Focus on math practices
Reasoning What would be different about the hike from point B to point A?

Answer:
The difference between the hike from point B to point A is 120.5 meters.

Explanation:
In the above-given question,
given that,
Malik hikes Castle Trail from point A to point B.
The elevation at point A is below sea level.
120(1/2).
120 x 2 = 240.
240 + 1 = 241.
241/2 = 120.5.
so the difference between the hike from point B to point A is 120.5 meters.

Essential Question
How are adding and subtracting integers related to adding and subtracting other rational numbers?

Try It!

A dolphin is at the surface of the water and then descends to a depth of 4\(\frac{1}{2}\) feet. Then the dolphin swims down another 2\(\frac{3}{4}\) feet. What is the location of the dolphin relative to the surface of the water?
-4\(\frac{1}{2}\) – ________
-4\(\frac{1}{2}\) + _______ = __________
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 50
The location of the dolphin relative to the surface of the water is ________ feet.

Answer:
The location of the dolphin relative to the surface of the water is -7.25 feet.

Explanation:
In the above-given question,
given that,
A dolphin is at the surface of the water and then descends to a depth of 4\(\frac{1}{2}\) feet.
Then the dolphin swims down another 2\(\frac{3}{4}\) feet.
-4(1/2) – 2(3/4).
-9/2 – 11/4.
-4.5 – 2.75.
-7.25.
so the location of the dolphin relative to the surface of the water is -7.25 feet.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational Number Operations-10

Convince Me! How are adding and subtracting two rational numbers with different signs related to adding and subtracting two integers with different signs?

Try It!

Find the sum or difference of the rational numbers.
a. \(-2.5+\left(-5 \frac{6}{10}\right)\)

Answer:
-2.5 + (-5 x 6/10) = -8.1.

Explanation:
In the above-given question,
given that,
the sum of two numbers.
-2.5 – 5 x 6/10.
-2.5 – 56/10.
-2.5 – 5.6.
-8.1.

b. \(-4.4-\left(-1 \frac{1}{2}\right)\)

Answer:
-5.9.

Explanation:
In the above-given question,
given that,
the difference between two numbers.
-4.4 – 3/2.
-4 . 4 – 1.5.
-5.9.

c. \(-135.4+78 \frac{1}{2}\)

Answer:
-56.9.

Explanation:
In the above-given question,
given that,
-135.4 + 78(1/2).
-135.4 + 157/2.
-135.4 + 78.5.
-56.9.

Try It!

Two divers are swimming at different depths below sea level. One diver is at -25.5 feet. The other diver is at –40.75 feet. How much farther below sea level is the diver who is farthest below sea level?

Answer:
The much farther below sea level is the diver who is farthest below sea level = -15.25 feet.

Explanation:
In the above-given question,
given that,
Two divers are swimming at different depths below sea level.
One diver is at -25.5 feet.
The other diver is at –40.75 feet.
-25.5 – 40.75.
– 15.25.
so the much farther below sea level is the diver who is farthest below sea level = -15.25 feet.

KEY CONCEPT
The rules for adding and subtracting all rational numbers are the same as those for adding and subtracting integers.
The distance between any two rational numbers p and q on a number line is the absolute value of their difference.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 51

Do You Understand?
Question 1.
Essential Question How are adding and subtracting integers related to adding and subtracting other rational numbers?

Answer:
The integers are related to adding and subtracting other rational numbers is the absolute value of their difference.

Explanation:
In the above-given question,
given that,
The integers are related to adding and subtracting other rational numbers is the absolute value of their difference.
for example:
-5 – (-2).
-2 – (-5).
-2 + 5 = 3.

Question 2.
Reasoning When finding the distance between two rational numbers on a number line, does the order of the numbers you subtract matter? Explain.

Answer:
The distance between two rational numbers on a number line does the order of the numbers you subtract matter.

Explanation:
In the above-given question,
given that,
The integers are related to adding and subtracting other rational numbers is the absolute value of their difference.
for example:
-5 – (-2).
-2 – (-5).
-2 + 5 = 3.

Question 3.
Critique Reasoning Gwen says that the sum of – 1\(\frac{3}{4}\) and 2\(\frac{1}{2}\) is the same as the difference between 2\(\frac{1}{2}\) and 1\(\frac{3}{4}\). Is Gwen correct? Explain why or why not.

Answer:
Gwen was correct.

Explanation:
In the above-given question,
given that,
Gwen says that the sum of – 1\(\frac{3}{4}\) and 2\(\frac{1}{2}\).
-1(3/4) and 2(1/2).
-4 + 3 = -7/4.
4 + 1 = 5/2.
-7/4 = -1.75.
5/2 = 2.5.
1(3/4) = 7/4.
7/4 = 1.75.

Do You Know How?
Question 4.
What is the distance between the top of the fishing pole and the fish?
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 52

Answer:
The distance between the top of the fishing pole and the fish = 4.

Explanation:
In the above-given question,
given that,
the top of the pole = 8(1/2).
the distance of the fish = 4(1/2).
8(1/2) – 4(1/2).
8 x 2 = 16.
16 + 1 = 17/2.
4 x 2 = 8.
8 + 1 = 9.
9/2 = 4.5.
17/2 = 8.5.
8.5 – 4.5 = 4.

Question 5.
A shark began at 172.5 meters below sea level and then swam up 137.1 meters. Where is the shark’s location now in relation to sea level?

Answer:
The shark’s location now in relation to sea level = 35.4 meters.

Explanation:
In the above-given question,
given that,
A shark began at 172.5 meters below sea level and then swam up 137.1 meters.
172.5 – 137.1.
35.4.
so the shark’s location now in relation to sea level = 35.4 meters.

Question 6.
Find the sum or difference.
a. -12\(\frac{1}{2}\) + 4\(\frac{1}{2}\)

Answer:
-8.

Explanation:
In the above-given question,
given that,
the two numbers.
add the numbers.
-12(1/2) + 4(1/2).
-25/2 + 9/2.
-12.5 + 4.5.
-8.

b. -0.35 – (-0.25)

Answer:
-0.10.

Explanation:
In the above-given question,
given that,
the two numbers.
subtract the numbers.
-0.35 – (-0.25).
-0.35 + 0.25.
-0.10.

Practice & Problem Solving

Leveled Practice In 7-8, complete the expressions to find the sum or difference.
Question 7.
3.2 – (-5.7)
= 3.2 + _____
= ______

Answer:
3.2 – (-5.7) = 8.9.

Explanation:
In the above-given question,
given that,
the two numbers.
subtract the numbers.
3.2 – (-5.7).
3.2 + 5.7.
8.9.

Question 8.
\(\frac{12}{13}+\left(\frac{-1}{13}\right)\)
= _____ – ______
= ______

Answer:
12/13 + (-1/13) = 0.85.

Explanation:
In the above-given question,
given that,
the two numbers.
add the numbers.
12/13 = 0.92.
1/13 = 0.07.
0.92 – 0.07 = 0.85.

Question 9.
Reasoning When Tom simplified the expression -2.6 + (-5.4), he got 2.8. What mistake did Tom likely make?

Answer:
The mistake did Tom likely make = -8.

Explanation:
In the above-given question,
given that,
-2.6 + (-5.4).
-2.6 – 5.4.
– 8.

Question 10.
The temperature in a town is 36.6°F during the day and -12.6°F at night. What is the temperature change from day to night?

Answer:
The temperature change from day to night = 24°F.

Explanation:
In the above-given question,
given that,
The temperature in a town is 36.6°F during the day and -12.6°F at night.
36.6 + (-12.6).
36.6 – 12.6.
24.
so the temperature change from day to night = 24°F.

Question 11.
Simplify each expression.
a. 50\(\frac{1}{2}\) + (-12.3)

Answer:
50(1/2) + (-12.3) = 38.2.

Explanation:
In the above-given question,
given that,
the two numbers are 50(1/2) and -12.3.
add the numbers.
101/2 and -12.3.
50.5 + (-12.3).
50.5 – 12.3 = 38.2.

b. -50\(\frac{1}{2}\) + (-12.3)

Answer:
-50(1/2) + (-12.3) = -62.8.

Explanation:
In the above-given question,
given that,
the two numbers are -50(1/2) and -12.3.
add the numbers.
-101/2 and -12.3.
-50.5 – 12.3.
– 62.8.

C. –50\(\frac{1}{2}\) + 12.3

Answer:
-50(1/2) + 12.3 = -32.8.

Explanation:
In the above-given question,
given that,
the two numbers are -50(1/2) and 12.3.
add the numbers.
-101/2 + 12.3.
-50.5 + 12.3.
-38.2.

Question 12.
At the beginning of the day, the stock market goes up 30\(\frac{1}{2}\) points. At the end of the day, the stock market goes down 120\(\frac{1}{4}\) points. What is the total change in the stock market from the beginning of the day to the end of the day?

Answer:
The total change in the stock market from the beginning of the day to the end of the day = 89.75 points.

Explanation:
In the above-given question,
given that,
At the beginning of the day, the stock market goes up 30\(\frac{1}{2}\) points.
At the end of the day, the stock market goes down 120\(\frac{1}{4}\) points.
30(1/2) = 61/2.
61/2 = 30.5.
120(1/4) = 481/4.
481/4 = 120.25.
120.25 – 30.5 = 89.75.
so the total change in the stock market from the beginning of the day to the end of the day = 89.75 points.

Question 13.
A dolphin is swimming 18 feet below the surface of the ocean. There is a coast guard helicopter 75.5 feet above the surface of the water that is directly above the dolphin. What is the distance between the dolphin and the helicopter?

Answer:
The distance between the dolphin and the helicopter = 57.5 feet.

Explanation:
In the above-given question,
given that,
A dolphin is swimming 18 feet below the surface of the ocean.
There is a coast guard helicopter 75.5 feet above the surface of the water that is directly above the dolphin.
75.5 – 18 = 57.5 feet.
so the distance between the dolphin and the helicopter = 57.5 feet.

Question 14.
A bird flies from its nest to the bottom of the canyon. How far did the bird fly?
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 53

Answer:
The far did the birds fly = 438.6.

Explanation:
In the above-given question,
given that,
A bird flies from its nest to the bottom of the canyon.
the top of the nest is 528(1/5).
528 x 1/5.
528 x 5 = 2640.
2641/5 = 528.2.
the height of canyon floor = -89(3/5).
-89 x 5 = 445.
-448 / 5 = -89.6.
528.2 – 89.6 = 438.6.

Question 15.
A scuba diving instructor takes a group of students to a depth of 54.96 feet. Then they ascend 22.38 feet to see some fish. Where are the fish in relation to the surface?

Answer:
The fish in relation to the surface = 32.58 feet.

Explanation:
In the above-given question,
given that,
A scuba diving instructor takes a group of students to a depth of 54.96 feet.
Then they ascend 22.38 feet to see some fish.
54.96 – 22.38 = 32.58.
so the fish in relation to the surface = 32.58 feet.

Question 16.
Model with Math Write an addition expression that is represented by the number line.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 54

Answer:
The addition expression that is represented by the number line = 0.5.

Explanation:
In the above-given question,
given that,
0 + 1 = 1.
0 + 0.5 = 0.5.
1 – 0.5 = 0.5.
so the addition expression that is represented by the number line is 1 + (-0.5).

Question 17.
The roots of a plant reach down 3\(\frac{3}{4}\) inches below ground. How many inches is the plant above the ground?
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 55

Answer:
The number of inches is the plant above the ground = 8.75 inches.

Explanation:
In the above-given question,
given that,
The roots of a plant reach down 3\(\frac{3}{4}\) inches below ground.
3(3/4) = 15/4.
15/4 = 3.75.
12(1/2) = 25/2.
25/2 = 12.5
12.5 – 3.75 = 8.75.
so the number of inches is the plant above the ground = 8.75 inches.

Question 18.
Higher Order Thinking
a. Simplify the expression (-13.2) + 8.1.

Answer:
-13.2 + 8.1 = -5.1.

Explanation:
In the above-given question,
given that,
the two numbers are -13.2 and 8.1.
add the numbers.
-13.2 + 8.1.
– 5.1.
-13.2 + 8.1 = -5.1.

b. How are (-13.2) + 8.1 and 13.2 + (-8.1) related? Explain without computing.

Answer:
-13.2 + 8.1 = -5.1.
13.2 – 8.1 = 5.1.

Explanation:
In the above-given question,
given that,
the two numbers are 13.2 and 8.1.
add the numbers.
they have an equal value but the sign is different.

c. Using a property of operations, what can you say about the sum of the two expressions?

Answer:
The sum of the two expressions is also positive.

Assessment Practice

Question 19.
The temperatures at sunrise and sunset are shown in the table.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 56
PART A
Write an expression that represents the change in temperature for Day 1. Show how you can use properties of operations to find the value of the expression.

Answer:
The properties of operations to find the value of the expression is -11.31 + 13.49.

Explanation:
In the above-given question,
given that,
the temperatures at sunrise and sunset are shown.
sunrise on day 1 is -11.31 and day 2 is -7.69.
sunset on day 1 is 13.49 and day 2 is 25.25.
-11.31 + (13.49).
-11.31 + 13.49.
2.18.
so the properties of operations to find the value of the expression is -11.31 + 13.49 is 2.18.

PART B
On which day did the temperature change more? Explain your reasoning.

Answer:
On day 2 the temperature change more.

Explanation:
In the above-given question,
given that,
the temperatures at sunrise and sunset are shown.
sunrise on day 1 is -11.31 and day 2 is -7.69.
sunset on day 1 is 13.49 and day 2 is 25.25.
-11.31 + (13.49).
-11.31 + 13.49.
2.18.
-7.69 + 25.25.
17.56.
so on day 2 the temperature change more.

Question 20.
Mischa dives from a platform that is 5 meters above water. Her dive takes her 2.1 meters below the surface of the water. Which expression could represent the distance, in meters, that Mischa dives? Select all that apply.
☐ |5 – (-2.1)|
☐ |-(2.1) – (-5)|
☐ |2.1 – 5|
☐ |-(2.1) – 5|
☐ |5 + (-2.1)|

Answer:
The expression could represent the distance in meters is 5 – (-2.1).

Explanation:
In the above-given question,
given that,
Mischa dives from a platform that is 5 meters above the water.
Her dive takes her 2.1 meters below the surface of the water.
5 – (-2.1).
5 + 2.1 = 7.1.
so the expression could represent the distance in meters is 7.1.

Topic 1 Mid-Topic Checkpoint

Question 1.
Vocabulary How do you find the additive inverse of a number? Give an example of a number and its additive inverse. Lesson 1-3

Answer:
The additive inverse of a number is always negative.

Explanation:
In the above-given question,
given that,
the numbers are -4 and -6.
-4 + (-6).
-4 – 6 = -10.

Question 2.
A plastic toy submarine is held 15 centimeters below the water surface in a bath tub. The submarine is let go and rises 15 centimeters. What integer represents the toy submarine’s position with respect to the surface of the water? Lesson 1-1

Answer:
The positive integer represents the toy submarine’s position with respect to the surface of the water.

Explanation:
In the above-given question,
given that,
A plastic toy submarine is held 15 centimeters below the water surface in a bathtub.
The submarine is let go and rises 15 centimeters.
-15 + (15).
-15 + 15 = 0.
so the integer is zero.

Question 3.
The temperature in the late afternoon was -7.5°C. It dropped 5 degrees by early evening and then dropped another 8.5 degrees by midnight. What was the temperature at midnight? Lessons 1-3, 1-4, and 1-5

Answer:
The temperature at the midnight = -21°C.

Explanation:
In the above-given question,
given that,
The temperature in the late afternoon was -7.5°C.
It dropped 5 degrees by early evening and then dropped another 8.5 degrees by midnight.
-7.5 – 5 – 8.5.
-7.5 – 13.5.
-21.
so the temperature at the midnight is -21°C.

Question 4.
The floor of an elevator in a building is 30 feet above ground level. It travels down to the lower level of the building, where the floor is 10 feet below ground level. What distance has the elevator’s floor traveled? Lessons 1-4 and 1-5

Answer:
The distance has the elevator’s floor traveled = 20 feet.

Explanation:
In the above-given question,
given that,
The floor of an elevator in a building is 30 feet above ground level.
It travels down to the lower level of the building, where the floor is 10 feet below ground level.
30 + (-10).
30 – 10.
20.
so the distance has the elevator’s floor traveled = 20 feet.

Question 5.
Greg says that 3.3. is a rational number. Kari says 3.3 is not a terminating decimal. Who is correct and why? Lesson 1-2

Answer:
Yes. both of them are correct.

Explanation:
In the above-given question,
given that,
Greg says that 3.3. is a rational number.
3.3 is a rational number.
Kari says 3.3 is not a terminating decimal.
3.3 is also a terminating decimal.
so both of them are correct.

Question 6.
Cece is hiking on a mountain and stops at 15\(\frac{5}{8}\) feet above sea level. The base of the mountain is 10.2 feet below sea level. What is the vertical distance between Cece and the base of the mountain? Lesson 1-5
A. 5.425 feet
B. 25.825 feet
C. 25\(\frac{3}{8}\) feet
D. 5\(\frac{1}{4}\) feet

Answer:
Option A is correct.

Explanation:
In the above-given question,
given that,
Cece is hiking on a mountain and stops at 15\(\frac{5}{8}\) feet above sea level.
The base of the mountain is 10.2 feet below sea level.
15(5/8) – 10.2.
125/8 – 10.2.
15.625 – 10.2.
5.425.
so option A is correct.

Topic 1 Mid-Topic Performance Task

An oceanographer, Dr. Price, is studying the types of sea life at various depths.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 57

PART A

Dr. Price uses a table to organize the types of sea life and the positions relative to sea level of each location.

Complete each sentence.
The difference between Location A and Location B is ________ meters.
The difference between Location B and Location C is ________ meters.
The difference between Location C and Location D is ________ meters.

Answer:
The difference between Location A and Location B is 202.25 meters.
The difference between Location B and Location C is 1686.6 meters.
The difference between Location C and Location D is474.875 meters.

Explanation:
In the above-given question,
given that,
location A = -895.9.
location B = – 1098(3/20).
location C = -2784.75.
location D = – 3259(5/8).
-895.9 – (-1098(3/20).
-895.9 + 21963/20.
-895.9 + 1098.15.
202.25.
-1098.15 – (-2784.75).
-1098.15 + 2784.75.
1686.6.
-2784.75 + 3259.625.
474.875.

PART B
After observing Location B, Dr. Price returns to Location A before descending to Location C. What is the total distance she travels?

Answer:
The total distance she travels = -2987 meters.

Explanation:
In the above-given question,
given that,
location B = -1098.15.
location A = – 895.9.
location C = -2784.75.
-1098.15 + (-895.9) + (- 2784.75).
-1098.15 – 895.9 – 2784.75.
-1098.15 – 1888.85.
-2987.
so the total distance she travels = -2987 meters.

PART C
Dr. Price descends to Location:D to observe shrimp. She then ascends and stops to observe sea life that is halfway between Location B and Location C. What is the total distance between Location D and where Dr. Price stopped to observe?

Answer:
The distance between Location D and where Dr. Price stopped to observe = -272.625 meters.

Explanation:
In the above-given question,
given that,
Dr. Price descends to Location:D to observe shrimp.
She then ascends and stops to observe sea life that is halfway between Location B and Location C.
-3259.625 – (-2987).
-3259.625 + 2987.
-272.625.
so the distance between Location D and where Dr. Price stopped to observe = -272.625 meters.

Lesson 1.6 Multiply Integers

Explore It!
A popular beach erodes 4 inches per year on average.
I can… multiply integers.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 58

A. How many years will it take for the coastline to erode one foot?

Answer:
The number of years will it take for the coastline to erode one foot = 12 years.

Explanation:
In the above-given question,
given that,
A popular beach erodes 4 inches per year on average.
1 year = 4 inches.
one feet = 12 inches.
so the number of years will it take for the coastline to erode one foot = 12 years.

B. The number line below shows the expected change in the coastline as years pass. How could you use the number line to show the erosion after 10 years?
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 59

Answer:
The erosion after 10 years = 10.

Explanation:
In the above-given question,
given that,
The number line below shows the expected change in the coastline as the years pass.
coastline this year = 0.
0 + 10 = 10.
so the erosion after 10 years = 10.

Focus on math practices
Be Precise What expression could you use to represent the change in the coastline in 5 years?

Essential Question
How do the signs of factors affect their product?

Try It!

A race car game takes 6 points from a player each time the player hits a cone. What integer represents the change in total points if the player hits 10 cones?
10 • ______ = ______
The change in total points is ______.

Answer:
If the player hits 10 cones = 60 points.

Explanation:
In the above-given question,
given that,
A race car game takes 6 points from a player each time the player hits a cone.
10 x 6 = 60.
so the change in total points is 60 points.

Convince Me! Could the product of a positive integer and a negative integer be positive? Explain.

Try It!

Find each product.
a. -7 • (-2)

Answer:
The product is 14.

Explanation:
In the above-given question,
given that,
the two numbers are -7 and -2.
multiply the numbers.
-7 x -2.
14.
so the product is 14.

b. 7 • (-13)

Answer:
The product is -91.

Explanation:
In the above-given question,
given that,
the two numbers are 7 and -13.
multiply the numbers.
7 x -13.
-91.
so the product is -91.

c.-6 • 8

Answer:
The product is -48.

Explanation:
In the above-given question,
given that,
the two numbers are -6 and 8.
multiply the numbers.
-6 x 8.
-48.
so the product is -48.

d.(-1) • (-1)

Answer:
The product is 1.

Explanation:
In the above-given question,
given that,
the two numbers are -1 and -1.
multiply the numbers.
-1 x -1.
1.
so the product is 1.

KEY CONCEPT
When multiplying two integers, the sign of the product depends on the sign of the factors.
If the signs of the factors are the same, the product is positive.
7 • 3 = 21
– 7 • (-3) = 21

If the signs of the factors are different, the product is negative.
-4 • 5 = -20
4 • (-5) = -20

Do You Understand?
Question 1.
Essential Question How do the signs of factors affect their product?

Answer:
The signs of factors affect their product according to the sign.

Explanation:
In the above-given question,
given that.
If the signs of the factors are the same, the product is positive.
7 x 3 = 21
– 7 x (-3) = 21
If the signs of the factors are different, the product is negative.
-4 x 5 = -20
4 x (-5) = -20

Question 2.
Construct Arguments What is the sign of the product if you multiplied three negative integers? Explain your answer.

Answer:
The product is also a negative integer.

Explanation:
In the above-given question,
given that.
the numbers are -7, -8, and -3.
-7 x -8 x -3.
-7 -24.
-168.
so the product is also a negative integer.

Question 3.
Reasoning Explain why the product of two negative integers is not negative. Use (-1)(-1) as an example.

Answer:
yes, the product of two negative integers is positive.

Explanation:
In the above-given question,
given that,
the product of two negative integers is positive.
for example:
-1 x -1 = 1.

Question 4.
Use Structure is the product the same when multiplying 22 × (-5) and multiplying (-5) × 22? Explain.

Answer:
Yes, the product is the same.

Explanation:
In the above-given question,
given that,
the numbers are 22 and -5.
22 x -5 = 110.
the numbers are -5 and 22.
-5 x 22 = 110.
so both the product is the same.

Do You Know How?
Question 5.
Represent 2 • (-3) on the number line.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 60

Answer:
The number is -6.

Explanation:
In the above-given question,
given that,
the numbers are 2 and -3.
2 x -3 = -6.
so the number is -6.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational Number Operations-11

Question 6.
Which of these products is negative? Select all that apply.
☐ -8 • (-3)
☐ -2 • 8
☐ 0 • (-2)
☐ 15 • (-5)
☐ -8 • (-9)

Answer:
-2 x 8 = -16.
15 x -5 = -75.

Explanation:
In the above-given question,
given that,
the product is negative.
the numbers are – 2 and 8.
-2 x 8 = -16.
15 x -5 = -75.

Question 7.
Find each product.
a. -9 • (-4)
b.-7 • 12
c. 8 • (-8)
d. 9 • 15

Answer:
The products are 36, -84, -64, and 135.

Explanation:
In the above-given question,
given that,
-9 x -4 = 36.
12 x -7 = -84.
8 x -8 = 64.
15 x 9 = 135.
so the products are 36, -84, -64, and 135.

Question 8.
A game show contestant starts a game by answering two questions incorrectly. Each incorrect answer costs the contestant $600. Use a product of two integers to show the point total that would appear for the contestant.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 61

Answer:
The numbers are $20 x $30.

Explanation:
In the above-given question,
given that,
A game show contestant starts a game by answering two questions incorrectly.
Each incorrect answer costs the contestant $600.
the product of two integers is $600.
$20 x $30.
so the numbers are $20 and $30.

Practice & Problem Solving

In 9-14, multiply.
Question 9.
(-6) • (-2)

Answer:
The product is 12.

Explanation:
In the above-given question,
given that,
the numbers are -6 and -2.
-6 x -2.
12.
so the product is 12.

Question 10.
4 • (-8)

Answer:
The product is -32.

Explanation:
In the above-given question,
given that,
the numbers are -8 and 4.
-8 x 4.
-32.
so the product is -32.

Question 11.
7 • (-5)

Answer:
The product is -35.

Explanation:
In the above-given question,
given that,
the numbers are -5 and -7.
-5 x 7.
-35.
so the product is -35.

Question 12.
-5 • 2

Answer:
The product is -10.

Explanation:
In the above-given question,
given that,
the numbers are -5 and 2.
-5 x 2.
-10.
so the product is -10.

Question 13.
-1 • (-24)

Answer:
The product is 24.

Explanation:
In the above-given question,
given that,
the numbers are -1 and -24.
-1 x -24.
24.
so the product is 24.

Question 14.
(5) • (-9) • (-2)

Answer:
The product is 90.

Explanation:
In the above-given question,
given that,
the numbers are 5, -9, and -2.
-9 x -2.
18.
18 x 5 = 90.
so the product is 90.

Question 15.
A football team lost the same number of yards on each of 3 consecutive plays. What is the total change in yards from where the team started?
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 62

Answer:
The total change in yards from where the team started = 18 yards.

Explanation:
In the above-given question,
given that,
A football team lost the same number of yards on each of 3 consecutive plays.
6 x 3 = 18.
so the total change in yards from where the team started = 18 yards.

Question 16.
a. Find the product.
-41 • (-1)

Answer:
The product is 41.

Explanation:
In the above-given question,
given that,
the numbers are -41 and -1.
multiply the numbers.
-41 x -1 = 41.
so the product is 41.

b. Construct Arguments Describe how you use the properties of multiplication to find the product.

Answer:
The properties of multiplication depend on the sign.

Explanation:
In the above-given question,
given that,
If the signs of the factors are the same, the product is positive.
7 x 3 = 21
– 7 x (-3) = 21
If the signs of the factors are different, the product is negative.
-4 x 5 = -20
4 x (-5) = -20

Question 17.
Alex is working to simplify 5 • (-8) • 2.
a. What is the product?

Answer:
The product is -80.

Explanation:
In the above-given question,
given that.
the numbers are 5, 2, and -8.
multiply the numbers.
5 x -8 x 2 = -80.
so the product is -80.

b. Suppose Alex found the opposite of the correct product. Describe an error he could have made that resulted in that product.

Answer:
The opposite of the correct product is 80.

Explanation:
In the above-given question,
given that,
the numbers are 5, 2, and -8.
multiply the numbers.
5 x -8 x 2 = -80.
so the product is -80.

Question 18.
Which product is greater, (-4) • (-6) or (-7) • (-8)? Explain.

Answer:
-7 x -8 = 56.

Explanation:
In the above-given question,
given that,
the product of two numbers.
-4 x -6 = 24.
-7 x -8 = 56.
so 56 is the greatest product.

Question 19.
Make Sense and Persevere While playing a board game, Cecilia had to move back 6 spaces 9 times. What integer represents Cecilia’s movement on the board for those 9 turns?

Answer:
The integer represents Cecilia’s movement on the board for those 9 turns = 54 times.

Explanation:
In the above-given question,
given that,
While playing a board game, Cecilia had to move back 6 spaces 9 times.
6 x 9 = 54.
so the integer represents Cecilia’s movement on the board for those 9 turns = 54 times.

Question 20.
Anya makes withdrawals from and deposits into her bank account.
a. What integer represents the change in the amount in her account if Anya withdraws $12 once each day for four days?

Answer:
The integer represents the change in the amount in her account = $48.

Explanation:
In the above-given question,
given that,
Anya makes withdrawals and deposits into her bank account.
$12 x 4 = $48.
so the integer represents the change in the amount in her account = $48.

b. What integer represents the change in the amount in her account if Anya deposits $12 once each day for four days?

Answer:
The integer represents the change in the amount in her account = $48.

Explanation:
In the above-given question,
given that,
Anya makes withdrawals and deposits into her bank account.
$12 x 4 = $48.
so the integer represents the change in the amount in her account = $48.

c. Look for Relationships Explain the difference between the integer for the withdrawals and the integer for the deposits.

Answer:
The difference is zero.

Explanation:
In the above-given question,
given that,
Anya makes withdrawals and deposits into her bank account.
$12 x 4 = $48.
so the integer represents the change in the amount in her account = $48.
$48 – $48 = 0.

Question 21.
Higher Order Thinking A gold mine has two elevators, one for equipment and one for miners One day, the equipment elevator begins to descend. After 28 seconds, the elevator for the miners begins to descend. What is the position of each elevator relative to the surface after another 14 seconds? At that time, how much deeper is the elevator for the miners?
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 63

Answer:
At 14 seconds the elevator of the miners is 210 feet.

Explanation:
In the above-given question,
given that,
A gold mine has two elevators, one for equipment and one for miners One day, the equipment elevator begins to descend.
After 28 seconds, the elevator for the miners begins to descend.
14 x 4 = 56.
14 x 15 = 210.
so the elevator of the miners is 210 feet.

Assessment Practice

Question 22.
A number line is shown.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 64
Write a multiplication equation that is represented by the number line.

Answer:
The equation is -35 x 2 = -70.

Explanation:
In the above-given question,
given that,
the number line is moving from 0 to -35.
-35 to -70.
-35 x 2 = -70.
so the equation is -35 x 2 = -70.

Question 23.
Which of these expressions have the same product as (-6) • 7? Select all that apply.
☐ (-3) • 14
☐ 16 • (-3)
☐ -6 • (-7)
☐ 7 • (-6)
☐ 14 • (-3)

Answer:
-3 x 14, 7 x -6 and 14 x -3.

Explanation:
In the above-given question,
given that,
the product is -6 x 7 = -42.
-3 x 14 = -42.
7 x -6 = -42.
14 x -3 = -42.

Lesson 1.7 Multiply Rational Numbers

Solve & Discuss It!
Stella is making the United States flag. She has blue fabric, red fabric, and white fabric. Choose a length for the flag. What length of blue fabric would Stella need to make this flag? Explain your thinking.

I can… multiply rational numbers.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 65

Focus on math practices
Be Precise The blue region of the flag is \(\frac{7}{13}\) the width and \(\frac{2}{5}\) the length of the flag. What part of the total area is the blue region of the flag?

Answer:
The total area is the blue region of the flag = 0.2152.

Explanation:
In the above-given question,
given that,
The blue region of the flag is \(\frac{7}{13}\) the width.
\(\frac{2}{5}\) the length of the flag.
7/13 x 2/5.
0.538 x 0.4.
0.2152.
so the total area is the blue region of the flag = 0.2152.

Essential Question
How is multiplying rational numbers like multiplying integers?

Try It!

Meghan’s bank account is charged $9.95 per month for an online newspaper subscription. How could you represent the change in her account balance after three months of charges?
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 66
_____ • -9.95 = ______
After three months, the change in her account balance is $ ________.

Answer:
The change in her account balance after three months of charges = -20.05.

Explanation:
In the above-given question,
given that,
Meghan’s bank account is charged $9.95 per month for an online newspaper subscription.
$9.95 x -30 = -20.05.
so the change in her account balance after three months of charges = -20.05.

Convince Me! Meghan’s bank account is charged 3 times. Without calculating, how can you determine whether this is a negative or positive change to her account? Explain.

Try It!

Find each product.
a. -5.3 • (-2.6)

Answer:
The product is 13.78.

Explanation:
In the above-given question,
given that,
the numbers are -5.3 and -2.6.
multiply the numbers.
-5.3 x – 2.6.
13.78.

b. \(-\frac{3}{5} \cdot 4 \frac{1}{6}\)

Answer:
-3/5 x 4(1/6) = -2.46.

Explanation:
In the above-given question,
given that,
the numbers are -3/5 and 25/6.
-0.6 x 4.1.
-2.46.
the product is -2.46.

c. 0.2 • (-1.78)

Answer:
0.2 x – 1.78 = -0.356.

Explanation:
In the above-given question,
given that,
the numbers are 0.2 x -1.78.
multiply the numbers.
0.2 x -1.78 = -0.356.

d. -2.5 • (70)

Answer:
The product is -175.

Explanation:
In the above-given question,
given that,
the numbers are 70 x -2.5.
multiply the numbers.
-2.5 x 70 = -175.

KEY CONCEPT
The same rules for multiplying integers apply to multiplying all rational numbers. When multiplying two rational numbers:

  • If the signs of the factors are the same, the product is positive.
  • If the signs of the factors are different, the product is negative.

Do You Understand?
Question 1.
Essential Question How is multiplying rational numbers like multiplying integers?

Answer:
The same rules for multiplying integers apply to multiplying all rational numbers.

Explanation:
In the above-given question,
given that,
If the signs of the factors are the same, the product is positive.
If the signs of the factors are different, the product is negative.

Question 2.
How do you multiply a decimal greater than 0 and a fraction less than 0?

Answer:
The product is negative.

Explanation:
In the above-given question,
given that,
the decimal greater than 0 is 1.5.
a fraction less than 0 is – 2.2.
1.5 x – 2.2 = -3.3.
so the product is negative.

Question 3.
Model with Math How does this number line represent multiplication of a negative number by a positive number? Explain.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 67

Answer:
-2/3 x -1/3 = 0.108.

Explanation:
In the above-given question,
given that,
the two numbers are -2/3 and -1/3.
multiply the numbers.
-2/3 x -1/3.
-0.6 x – 0.33.
0.108.
-2/3 x -1/3 = 0.108.

Do You Know How?
Question 4.
Use the number line to find the product
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 68

Answer:
The product is -4.5.

Explanation:
In the above-given question,
given that,
the numbers are 3 and -1(1/2).
multiply the numbers.
3 x -3/2.
3 x – 1.5.
-4.5.
so the product is -4.5.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational Number Operations-12

Question 5.
Which of these products is positive? Select all that apply.
☐ -0.2 • (12.5)
☐ \(\frac{1}{12}\) • \(\left(-6 \frac{1}{2}\right)\)
☐ 3.2 • \(\left(-\frac{1}{900}\right)\)
☐ -3\(\frac{1}{2}\) • 0
☐ -4.7 • (-1)

Answer:
-4.7 x -1 = 4.7.

Explanation:
In the above-given question,
given that,
the numbers are -4.7 and -1.
multiply the product.
-4.7 x -1 = 4.7.
so the product is 4.7.

Question 6.
Find the product.
a. -3.1 • (-2.9)

Answer:
The product is 8.99.

Explanation:
In the above-given question,
given that,
the numbers are -3.1 and -2.9.
multiply the product.
-3.1 x -2.9 = 8.99.
so the product is 8.99.

b. 1\(\frac{1}{2}\) • \(\left(-\frac{5}{3}\right)\)

Answer:
1(1/2) x -5/3 = -2.4.

Explanation:
In the above-given question,
given that,
the numbers are 3/2 and -5/3.
multiply the product.
3/2 x – 5/3.
1.5 x – 1.6.
-2.4.
so the product is -2.4.

c. -3\(\frac{1}{2}\) • 0.5

Answer:
-3(1/2) x 0.5 = – 1.75.

Explanation:
In the above-given question,
given that,
the numbers are -7/2 and 0.5.
multiply the product.
-7/2 x 0.5.
-3.5 x 0.5 = -1.75.

d. –\(\frac{4}{5}\) • –\(\frac{1}{8}\)

Answer:
-4/5 x – 1/8 = 0.1.

Explanation:
In the above-given question,
given that,
the numbers are -4/5 and -1/8.
multiply the product.
-4/5 x -1/8.
– 0.8 x – 0.125 = 0.1.

Practice & Problem Solving

In 7-14, multiply.
Question 7.
(-2.655) • (18.44)

Answer:
-2.655 x 18.44 = -48.9582.

Explanation:
In the above-given question,
given that,
the numbers are -2.655 and 18.44.
multiply the product.
-2.655 x 18.44 = -48.9582.

Question 8.
-1\(\frac{5}{6}\) • 6\(\frac{1}{2}\)

Answer:
-1(5/6) x 6(1/2) = -11.895.

Explanation:
In the above-given question,
given that,
the numbers are -1(5/6) and 6(1/2).
multiply the product.
-11/6 x 13/2.
-1.83 x 6.5.
-11.895.

Question 9.
-2\(\frac{1}{2}\) • (-1\(\frac{2}{3}\))

Answer:
-2(1/2) x -1(2/3) = 4.1.

Explanation:
In the above-given question,
given that,
the numbers are -(5/2) and -(5/3).
multiply the product.
-5/2 = -2.5.
-5/3 = – 1.6.
so the product is 4.1.

Question 10.
-3\(\frac{7}{8}\) • (-5\(\frac{3}{4}\))

Answer:
-3(7/8) x -5(3/4) = 22.28125.

Explanation:
In the above-given question,
given that,
the numbers are -31/8 and -23/4.
multiply the product.
-31/8 x -23/4.
3.875 x 5.75.
22.28125.

Question 11.
-7.5 • -2\(\frac{3}{4}\)

Answer:
-7.5 x -2(3/4) = 20.625.

Explanation:
In the above-given question,
given that,
the numbers are -7.5 and -11/4.
multiply the product.
-7.5 x – 2.75.
20.625.

Question 12.
-0.6 • (-0.62)

Answer:
-0.6 x -0.62 = 0.372.

Explanation:
In the above-given question,
given that,
the numbers are -0.6 and -0.62.
multiply the product.
-0.6 x -0.62 = 0.372.

Question 13.
-0.2 • –\(\frac{5}{6}\)

Answer:
-0.2 x – 5/6 = 0.166.

Explanation:
In the above-given question,
given that,
the numbers are -0.2 and -5/6.
multiply the product.
-0.2 x – 0.83.
0.166.

Question 14.
–\(\frac{5}{6}\) • \(\frac{1}{8}\)

Answer:
-5/6 x 1/8 = 0.0375.

Explanation:
In the above-given question,
given that,
the numbers are -5/6 and -1/8.
multiply the product.
-5/6 x -1/8.
-0.83 x – 0.125 = 0.0375.

Question 15.
At the beginning of the season, Jamie pays full price for a ticket to see the Panthers, her favorite baseball team.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 69
a. Represent the total change in the cost of a ticket given their losses.

Answer:
The total change in the cost of a ticket given their losses = $20.3524.

Explanation:
In the above-given question,
given that,
At the beginning of the season, Jamie pays full price for a ticket to see the Panthers, her favorite baseball team.
ticket prices decrease by $0.41 for every game the panthers lose this season.
$0.41 x $49.64.
$20.3524.
so the total change in the cost of a ticket given their losses = $20.3524.

b. What is the cost of a ticket for the next game they play?

Answer:
The cost of a ticket for the next game they play = $20.3524.

Explanation:
In the above-given question,
given that,
At the beginning of the season, Jamie pays full price for a ticket to see the Panthers, her favorite baseball team.
ticket prices decrease by $0.41 for every game the panthers lose this season.
$0.41 x $49.64.
$20.3524.
so the total change in the cost of a ticket given their losses = $20.3524.

Question 16.
The price per share of ENVX stock is dropping at a rate of $1.45 each hour.
Envision Math Common Core 7th Grade Answers Topic 1 Rational Number Operations 70
a. Write the rate as a negative number.

Answer:
The rate as a negative number = $55.274.

Explanation:
In the above-given question,
given that,
The price per share of ENVX stock is dropping at a rate of $1.45 each hour.
$1.45 x 38.12 = $55.274.
so the rate as a negative number = $55.274.

b. What rational number represents the change in the price per share after 5 hours?

Answer:
The rational number represents the change in the price per share after 5 hours = 190.6.

Explanation:
In the above-given question,
given that,
the rational number is 38.12.
38.12 x 5 = 190.6.
so the rational number represents the change in the price per share after 5 hours = 190.6.

c. What is the price per share after 5 hours?

Answer:
The price per share after 5 hours = 953.

Explanation:
In the above-given question,
given that,
the price per share after 5 hours is:
190.6 x 5 = 953.
so the price per share after 5 hours = 953.

Question 17.
Ming incorrectly says that this product is \(\frac{4}{63}\).
\(-\left(-\frac{4}{9}\right)\) • \(\left(-\frac{1}{7}\right)\)
a. What is the correct product?

Answer:
The correct product is 0.0616.

Explanation:
In the above-given question,
given that,
Ming incorrectly says that this product is \(\frac{4}{63}\).
4/63 = 0.063.
-4/9 x -1/7.
– 0.44 x – 0.14.
0.0616.
so the correct product is 0.0616.

b. What error could Ming have made?

Answer:
The error could ming have made = 0.0616.

Explanation:
In the above-given question,
given that,
Ming incorrectly says that this product is \(\frac{4}{63}\).
4/63 = 0.063.
-4/9 x -1/7.
– 0.44 x – 0.14.
0.0616.
so the correct product is 0.0616.

Question 18.
Higher Order Thinking place the products in order from least to greatest.
\(4 \frac{4}{7}\) • \(4 \frac{4}{7}\)
\(5 \frac{6}{7}\) • \(\left(-6 \frac{6}{7}\right)\)
\(-5 \frac{1}{8}\) • \(\left(-2 \frac{1}{4}\right)\)

Answer:
-39.44, 11.53, and 20.894.

Explanation:
In the above-given question,
given that,
4(4/7) x 4(4/7).
32/7 x 32/7.
4.571 x 4.571.
20.894.
41/7 x -48/7.
5.8 x – 6.8.
-39.44.
-41/8 x -9/4.
-5.125 x -2.25.
11.53.
so the numbers from least to greatest are -39.44, 11.53, and 20.894.

Assessment Practice

Question 19.
Suppose there is a 1.3°F drop in temperature for every thousand feet that an airplane climbs into the sky. The temperature on the ground is -2.8°F.
PART A
Write a multiplication equation to represent the change in temperature after the plane ascends 10,000 feet.

Answer:
The equation represents the change in temperature after the plane ascends 10,000 feet = 4.1°F.

Explanation:
In the above-given question,
given that,
Suppose there is a 1.3°F drop in temperature for every thousand feet that an airplane climbs into the sky.
The temperature on the ground is -2.8°F.
1.3°F – (-2.8°F).
1.3°F + 2.8°F.
4.1°F.

PART B
What will the temperature be when the plane reaches an altitude of 10,000 feet?
A. -15.8
B. -10.2
C. 10.2
D. 15.8

Answer:
The temperature is when the plane reaches an altitude of 10,000 feet = -15.8.

Explanation:
In the above-given question,
given that,
The temperature on the ground is -2.8°F.
1.3°F – (-2.8°F).
1.3°F + 2.8°F.
4.1°F.
so option B is correct.

Lesson 1.8 Divide Integers

Explain It!
The shapes below are used to show the relationship between each of the four equations in the same fact family.
I can… divide integers.
Envision Math Common Core Grade 7 Answer Key Topic 1 Rational Number Operations 71

A. Suppose the star represents -24. What values could the other shapes represent?

Answer:
Square represents 8.
the circle represents 3.

Explanation:
In the above-given question,
given that,
8 x 3 = 24.
3 x 8 = 24.
24 / 3 = 8.
24 / 8 = 3.
so the square represents 8.
the circle represents 3.

B. What do you know about the square and circle if the star represents a negative number?

Answer:
The square and circle if the star represents a negative number.

Explanation:
In the above-given question,
given that,
8 x 3 = 24.
3 x 8 = 24.
24 / 3 = 8.
24 / 8 = 3.
so the square represents 8.
the circle represents 3.

C. What do you know about the star if the square and circle both represent a negative number?

Answer:
The square and circle both represent a positive numbers.

Explanation:
In the above-given question,
given that,
if the square and circle both represent a negative number.
so the square and circle both represent positive numbers.
for example:
– 3 x -4 = 12.

Focus on math practices
Use Structure Suppose the square represents -8 and the circle represents 3. Use what you know about integer multiplication and the relationship between multiplication and division to write the complete fact family.

Essential Question
How does dividing integers relate to multiplying integers?

Try It!

Suppose the machine drilled the same distance into the ground for 3 days and reached water at 84 feet below ground level. What was the change in the location of the bottom of the hole each day?
Envision Math Common Core Grade 7 Answer Key Topic 1 Rational Number Operations 72
Each day, the location of the bottom of the hole changed by _______ feet, or decreased by ________ feet.

Answer:
-84/3 = -28.
3 x 9 = 28.
3 x -28 = -84.
-84/3 = -28.

Explanation:
In the above-given question,
given that,
the machine drilled the same distance into the ground for 3 days and reached water at 84 feet below ground level.
84/3 = 28.
so the location of the bottom of the hole changed by 28 feet or decreased by 28 feet.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational Number Operations-13

Convince Me! Explain why the quotient of two integers with different signs is negative.

Try It!

Simplify.
a. -40 ÷ (-5)

Answer:
-40 / -5 = 8.

Explanation:
In the above-given question,
given that,
the numbers are -40 and -5.
divide the numbers.
-40 / -5 = 8.

b. 40 ÷ (-5)

Answer:
40 / -5 = -8.

Explanation:
In the above-given question,
given that,
the numbers are 40 and -5.
divide the numbers.
40 / -5 = -8.

c. 0 ÷ -40

Answer:
0 / 40 = 0.

Explanation:
In the above-given question,
given that,
the numbers are 0 and 40.
divide the numbers.
0 / 40 = 0.

Try It!

Which of the following are equivalent to -5?
Envision Math Common Core Grade 7 Answer Key Topic 1 Rational Number Operations 73

Answer:
The following are equivalent to -5 is -(55/11), -55/11, 55/-11, and -(-55/-11).

Explanation:
In the above-given question,
given that,
the numbers are 55 and -11.
divide the numbers.
-55/11 = -5.
-(55/11) = -5.
55/-11 = -5.
-(-55/-11) = -5.

KEY CONCEPT
The rules for dividing integers are related to the rules for multiplying integers.
If the signs of the dividend and the divisor are the same, the quotient is positive.
24 ÷ 4 = 6
-24 ÷ (-4) = 6

If the signs of the dividend and the divisor are different, the quotient is negative.
-15 ÷ 3 = -5
15 ÷ (-3) = -5

Do You Understand?
Question 1.
Essential Question How does dividing integers relate to multiplying integers?

Answer:
The integers are related to multiplying integers depending on their signs.

Explanation:
In the above-given question,
given that,
If the signs of the dividend and the divisor are the same, the quotient is positive.
24 ÷ 4 = 6
-24 ÷ (-4) = 6
If the signs of the dividend and the divisor are different, the quotient is negative.
-15 ÷ 3 = -5
15 ÷ (-3) = -5

Question 2.
Reasoning Why is the quotient of two negative integers positive?

Answer:
Yes, the quotient of two negative integers is positive.

Explanation:
In the above-given question,
given that,
the numbers are -30 and -3.
divide the numbers.
-30 / -3 = 10.
on both sides, the signs get canceled.

Question 3.
Helen wrote the following facts to try to show that division by 0 results in 0. Explain her error.
Envision Math Common Core Grade 7 Answer Key Topic 1 Rational Number Operations 74

Answer:
Yes, the division by 0 results in 0.

Explanation:
In the above-given question,
given that,
0 x -7 = 0.
-7 / 0 = 0.
so the division by 0 results in 0.

Do You Know How?
Question 4.
Find each quotient.
a. –\(\frac{18}{3}\)

Answer:
-18/3 = -6.

Explanation:
In the above-given question,
given that,
the number is -18/3.
divide the number.
-18/3 = -6.

b. \(\frac{-5}{-1}\)

Answer:
-5/-1 = 5.

Explanation:
In the above-given question,
given that,
the number is -5/-1.
divide the number.
-5/-1 = 5.

c. \(\frac{24}{-6}\)

Answer:
24/-6 = -4.

Explanation:
In the above-given question,
given that,
the number is 24/-6.
divide the number.
24/-6 = -4.

d. \(\frac{-10}{-1}\)

Answer:
-10/-1 = 10.

Explanation:
In the above-given question,
given that,
the number is -10/-1.
divide the number.
-10/-1 = 10.

e. \(\frac{-25}{5}\)

Answer:
-25/5 = -5.

Explanation:
In the above-given question,
given that,
the number is -25/5.
divide the number.
-25/5 = -5.

f. –\(\frac{8}{2}\)

Answer:
-8/2 = -4.

Explanation:
In the above-given question,
given that,
the number is -8/2.
divide the number.
-8/2 = -4.

Question 5.
A scuba diver descends 63 feet in 18 seconds. What integer represents the change in the diver’s position in feet per second?

Answer:
The integer represents the change in the diver’s position in feet per second = 3.5 feet.

Explanation:
In the above-given question,
given that,
A scuba diver descends 63 feet in 18 seconds.
63/18 = 3.5.
so the integer represents the change in the diver’s position in feet per second = 3.5 feet.

Question 6.
Which of the following are equivalent to -7?
Envision Math Common Core Grade 7 Answer Key Topic 1 Rational Number Operations 75

Answer:
49/-7 = -7 and -21/3 = -7.

Explanation:
In the above-given question,
given that,
the quotient is -7.
49/-7 = -7.
-21/3 = -7.

Practice & Problem Solving

Leveled Practice In 7-8, fill in the boxes to find each quotient.
Question 7.
-16 ÷ 4 = ?
4 • ? = ______
4 • _____ = _______
So, -16 ÷ 4 = ________.

Answer:
-16/4 = -4.

Explanation:
In the above-given question,
given that,
the numbers are -16 and 4.
divide the numbers.
-16 / 4 = -4.
the quotient is -4.

Question 8.
-56 ÷ – 7 = ?
______ • ? = _______
______ • ______ = ________
So, -56 ÷ -7 = ________

Answer:
-56 / -7 = 8.

Explanation:
In the above-given question,
given that,
the numbers are -56 and -7.
divide the numbers.
-56/ -7 = 8.

Question 9.
Classify the quotient -50 ÷ 5 as positive, negative, zero, or undefined.

Answer:
The quotient is -10.

Explanation:
In the above-given question,
given that,
the numbers are -50 and -5.
divide the numbers.
-50/5 = -10.

Question 10.
Is the expression \(\frac{42}{-7}\) undefined? If not, find the quotient.

Answer:
The quotient is -6.

Explanation:
In the above-given question,
given that,
the numbers are 42 and -7.
divide the numbers.
42/-7 = -6.
so the quotient is -6.

Question 11.
A company loses $780 as a result of a shipping delay. The 6 owners of the company must share the loss equally.
a. Write an expression to show the change in profit for each owner.

Answer:
The change in profit for each owner = $130.

Explanation:
In the above-given question,
given that,
A company loses $780 as a result of a shipping delay.
The 6 owners of the company must share the loss equally.
$780 / 6 = $130.
so the change in profit for each owner = $130.

b. Evaluate the expression.

Answer:
The change in profit for each owner = $130.

Explanation:
In the above-given question,
given that,
A company loses $780 as a result of a shipping delay.
The 6 owners of the company must share the loss equally.
$780 / 6 = $130.
so the change in profit for each owner = $130.

Question 12.
Which of the quotients are equivalent to 2.5? Select all that apply.
Envision Math Common Core Grade 7 Answer Key Topic 1 Rational Number Operations 76

Answer:
-5/12, 10/4, -10/ -4, and 5/2 are equivalent to 2.5.

Explanation:
In the above-given question,
given that,
the quotient equal to 2.5 is
10/-4 = 5/-2 = -2.5.
-5/-2 = 2.5.
10/4 = 2.5.
-10/-4 = 2.5.
5/2 = 2.5.

Question 13.
Use Structure The price of a stock steadily decreased by a total of $127 over 15 months. Which expression shows the change in the stock’s value?
Envision Math Common Core Grade 7 Answer Key Topic 1 Rational Number Operations 77

Answer:
Option A is correct.

Explanation:
In the above-given question,
given that,
The price of a stock steadily decreased by a total of $127 over 15 months.
-$127/ -15 months.
so option A is correct.

Question 14.
Zak goes parachuting and descends at the rate shown. If he maintains a steady descent, what integer represents Zak’s change in elevation in feet per second?
Envision Math Common Core Grade 7 Answer Key Topic 1 Rational Number Operations 78

Answer:
The integer represents Zak’s change in elevation in feet per second = 12 feet.

Explanation:
In the above-given question,
given that,
Zak goes parachuting and descends at the rate shown.
24 feet in 2 seconds.
24/2 = 12.
so the integer represents Zak’s change in elevation in feet per second = 12 feet.

Question 15.
Model with Math Find each quotient and plot it on the number line. Which of the expressions are undefined?
Envision Math Common Core Grade 7 Answer Key Topic 1 Rational Number Operations 79

Answer:
The expressions are undefined = 0/-8.

Explanation:
In the above-given question,
given that,
-8/4 = 2.
-21/ -7 = 3.
-4/0 = 0.
-25/-5 = 5.
36/-9 = -4.
9/0 = 9.
0/-8 = 0.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational Number Operations-14

Question 16.
Use Structure The temperature in a town increased 16°F in 5 hours. The temperature decreased 31°F in the next 8 hours. Which of the expressions shows the rate of the total change in temperature?
A. \(\frac{-15^{\circ} \mathrm{F}}{13 \text { hours }}\)
B. \(\frac{47^{\circ} \mathrm{F}}{13 \text { hours }}\)
C. \(\frac{15^{\circ} \mathrm{F}}{13 \text { hours }}\)
D. \(\frac{47{\circ} \mathrm{F}}{-13 \text { hours }}\)

Answer:
Option C is correct.

Explanation:
In the above-given question,
given that,
The temperature in a town increased by 16°F in 5 hours.
The temperature decreased 31°F in the next 8 hours.
31 – 16 = 15.
5 + 8 = 13.
so option C is correct.

Question 17.
Camille takes a rock-climbing class. On her first outing, she rappels down the side of a boulder in three equal descents. What integer represents Camille’s change in altitude in feet each time she descends?
Envision Math Common Core Grade 7 Answer Key Topic 1 Rational Number Operations 80

Answer:
The integer represents Camille’s change in altitude in feet each time she descends = 165 feet.

Explanation:
In the above-given question,
given that,
Camille takes a rock-climbing class.
On her first outing, she rappels down the side of a boulder in three equal descents.
165 feet.
so the integer represents Camille’s change in altitude in feet each time she descends = 165 feet.

Question 18.
Higher Order Thinking If the fraction \(\frac{396}{x-10}\) is equivalent to -22, find the value of x. Show your work.

Answer:
x = 9.94.

Explanation:
In the above-given question,
given that,
396 x x – 10 = -22.
396x – 3960 = -22.
396x = -22 + 3960.
396x = 3938.
x = 3938/396.
x = 9.94.

Assessment Practice

Question 19.
Which of the quotients is equivalent to \(-\frac{5}{8}\)? Select all that apply.
☐ \(\frac{-5}{8}\)
☐ \(\frac{5}{8}\)
☐ \(\frac{5}{-8}\)
☐ \(-\left(\frac{5}{-8}\right)\)
☐ \(\frac{-5}{-8}\)

Answer:
-5/8, and 5/-8.

Explanation:
In the above-given question,
given that,
-5/8 = -0.625.
5/-8 = -0.625.

Question 20.
Which of the following pairs of quotients are equivalent?
A. \(\frac{-4}{5}\) and \(-\left(\frac{20}{25}\right)\)
B. \(-\left(\frac{2}{-3}\right)\) and \(\frac{-4}{6}\)
C. \(\frac{-5}{7}\) and \(\frac{35}{-40}\)
D. \(\frac{1}{5}\) and \(-\left(\frac{-2}{-10}\right)\)

Answer:
Option A is correct.

Explanation:
In the above-given question,
given that,
the pairs of quotients are equivalent.
-4/5 = -20/25.
5 x 4 = 20.
5 x 5 = 25.
so option A is correct.

Question 21.
An elevator descends 36 feet in 3 seconds. What integer represents the elevator’s change in elevation in feet per a second?

Answer:
The integer represents the elevator’s change in elevation in feet per second = 12 seconds.

Explanation:
In the above-given question,
given that,
An elevator descends 36 feet in 3 seconds.
36/3 = 13.
so the integer represents the elevator’s change in elevation in feet per second = 12 seconds.

Lesson 1.9 Divide Rational Numbers

Explore It!
The number line shows the movement of a glacier that retreats 8 meters every year.
I can… divide rational numbers.
Envision Math Common Core Grade 7 Answer Key Topic 1 Rational Number Operations 81

A. How could you use division to represent the yearly change in the glacier’s position over the next 4 years?

Answer:
The change in the glacier’s position over the next 4 years = -32.

Explanation:
In the above-given question,
given that,
The number line shows the movement of a glacier that retreats 8 meters every year.
4 x -8 = -32.
-32/-8 = 4.
so the change in the glacier’s position over the next 4 years = -32.

B. How could you use division to represent the yearly change in the glacier’s position over the past 4 years?

Answer:
The change in the glacier’s position over the past 4 years = -32.

Explanation:
In the above-given question,
given that,
The number line shows the movement of a glacier that retreats 8 meters every year.
4 x -8 = -32.
-32/-8 = 4.
so the change in the glacier’s position over the past 4 years = -32.

C. Suppose the glacier retreated 8.25 meters every year. Draw a number line to represent this movement.

Answer:

The change in the glacier’s position over the past 4 years = 33.

Explanation:
In the above-given question,
given that,
The number line shows the movement of a glacier that retreats 8 meters every year.
4 x 8.25 = 33.
33/8.25 = 4.
so the change in the glacier’s position over the past 4 years

Focus on math practices
Reasoning If the number of meters the glacier retreats each year changes, does it affect the signs of each part of the division statement in Part A? Explain.

Essential Question
How is dividing rational numbers like dividing integers?

Try It!

Suppose that the volume of water in the rain barrel decreased by 4\(\frac{5}{8}\) gallons in 4 minutes. What will be the change in the volume of water after 1 minute?
Envision Math Common Core Grade 7 Answer Key Topic 1 Rational Number Operations 82
The rain barrel will lose _________ gallons in 1 minute.

Answer:
The rain barrel will lose -37/32 gallons in 1 minute.

Explanation:
In the above-given question,
given that,
suppose that the volume of water in the rain barrel decreased by 4\(\frac{5}{8}\) gallons in 4 minutes.
-4(5/8) /4.
-37/8 x 4.
-37/32.
so the rain barrel will lose -37/32 gallons in 1 minute.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational Number Operations-15

Convince Me! How are multiplicative inverses used in division with rational numbers?

Try It!

Find each quotient.
a. \(\frac{1 \frac{2}{5}}{-\frac{1}{5}}\)

Answer:
1(2/5) – 1/5 = 1.2.

Explanation:
In the above-given question,
given that,
7/5 – 1/5.
1.4 – 0.2.
1.2.

b. -0.4 – 0.25

Answer:
-0.4 – 0.25 = 0.65.

Explanation:
In the above-given question,
given that,
the two numbers are -0.4 and -0.25.
subtract the numbers.
-0.4 – 0.25.
0.65.

c. \(\frac{7}{8} \div-\frac{3}{4}\)

Answer:
7/8 – 3/4 = 0.125.

Explanation:
In the above-given question,
given that,
the two numbers are 7/8 and -3/4.
subtract the numbers.
7/8 = 0.875.
-3/4 = 0.75.
0.875 – 0.75.
0.125.

d. \(0.7 \div-1 \frac{1}{6}\)

Answer:
0.7 / 1(1/6) = 0.6.

Explanation:
In the above-given question,
given that,
the two numbers are 0.7 and -7/6.
divide the numbers.
0.7/ 1.16.
0.6.

Try It!

Find each quotient.
a. \(-1 \frac{1}{3} \div(-1.6)\)

Answer:
The quotient is 0.83.

Explanation:
In the above-given question,
given that,
the two numbers are -1(1/3) and -1.6.
divide the numbers.
-4/3 / -1.6.
-1.33/ -1.6.
0.83.

b. \(\frac{-\frac{2}{3}}{-\frac{1}{4}}\)

Answer:
The quotient is 2.4.

Explanation:
In the above-given question,
given that,
the two numbers are -(2/3) and -1/4.
divide the numbers.
-2/3 / -1/4.
-0.6/-0.25.
2.4.

c. \(-\frac{9}{10} \div\left(-\frac{3}{10}\right)\)

Answer:
The quotient is 3.

Explanation:
In the above-given question,
given that,
the two numbers are -9/10 and -3/10.
divide the numbers.
-9/10/-3/10.
-0.9/-0.3.
3.

d. \(-0.5 \div\left(-\frac{3}{13}\right)\)

Answer:
The quotient is 2.17.

Explanation:
In the above-given question,
given that,
the two numbers are -0.5 and -3/13.
divide the numbers.
-0.5/ -0.23.
2.17.

KEY CONCEPT
The same rules for dividing integers apply to dividing rational numbers. When dividing two rational numbers:

  • If the signs of the dividend and divisor are the same, the quotient is positive.
  • If the signs of the dividend and divisor are different, the quotient is negative.

Do You Understand?
Question 1.
Essential Question How is dividing rational numbers like dividing integers?

Answer:
The division of rational numbers like dividing integers depends upon size.

Explanation:
In the above-given question,
given that,
If the signs of the dividend and divisor are the same, the quotient is positive.
If the signs of the dividend and divisor are different, the quotient is negative.
for example:
12/3 = 4.
-12/3 = -4.

Question 2.
Use Structure How do you know the sign of the quotient \(-\frac{4}{5} \div \frac{1}{6}\)?

Answer:
-4/5 / 1/6 = -8.

Explanation:
In the above-given question,
given that,
the two numbers are -4/5 and 1/6.
divide the numbers.
-4/5 / 1/6.
-0.8 / 0.1.
-8.

Question 3.
Reasoning When -4 is divided by a rational number between 0 and 1, where would the quotient be located on the number line? Why?

Answer:
The quotient will be located on the number line is -4.

Explanation:
In the above-given question,
given that,
When -4 is divided by a rational number between 0 and 1.
-4/0 = -4.
-4/1 = -4.

Do You Know How?
Question 4.
Find each quotient.
a. \(-\frac{7}{12} \div \frac{1}{7}\)

Answer:
-7/12 / 1/7 = -4.14.

Explanation:
In the above-given question,
given that,
the two numbers are -7/12 and 1/7.
divide the numbers.
-7/12 / 1/7.
-0.58 / 0.14.
-4.14.

b. \(-0.05 \div\left(-\frac{5}{8}\right)\)

Answer:
-0.05 / -5/8 =0.08.

Explanation:
In the above-given question,
given that,
the two numbers are -0.05 and -5/8.
divide the numbers.
-0.05 / -5/8.
-0.05 / -0.625.
0.08.

c. \(6 \frac{1}{4} \div\left(-\frac{5}{16}\right)\)

Answer:
6(1/4) / -5/16 = -20.03.

Explanation:
In the above-given question,
given that,
the two numbers are 6(1/4) and -5/16.
divide the numbers.
25/4 / -5/16.
6.25 / -0.3125.
-20.03.

d. \(-1 \div\left(-\frac{10}{13}\right)\)

Answer:
-1 / -10/3 = 0.300.

Explanation:
In the above-given question,
given that,
the two numbers are -1 and -10/3.
divide the numbers.
-1/ -10/3.
-1 / -3.33.
0.300.

Question 5.
Simplify the complex fraction.
a. \(\frac{-\frac{2}{7}}{1 \frac{1}{3}}\)

Answer:
-2/7 / 4/3 = -0.21.

Explanation:
In the above-given question,
given that,
the two numbers are -2/7 and 1(1/3).
divide the numbers.
-2/7 / 4/3.
-0.28 / 1.3.
-0.21.

b. \(\frac{-\frac{3}{5}}{2 \frac{1}{4}}\)

Answer:
-3/5 / 9/4 = -0.26.

Explanation:
In the above-given question,
given that,
the two numbers are -3/5 and 2(1/4).
divide the numbers.
-3/5 / 9/4.
– 0.6 / 2.25.
-0.26.

c. \(\frac{-\frac{9}{10}}{1 \frac{3}{5}}\)

Answer:
-9/10 / 8/5 = – 0.5625.

Explanation:
In the above-given question,
given that,
the two numbers are -9/10 and 8/5.
divide the numbers.
-9/10 / 8/5.
-0.9 / 1.6.
-0.5625.

Practice & Problem Solving

Leveled Practice In 6-7, fill in the boxes to find the quotient.
Question 6.
Find the quotient \(\frac{5}{7} \div\left(-\frac{11}{5}\right)\).
Envision Math Common Core Grade 7 Answer Key Topic 1 Rational Number Operations 83
Answer:
5/7 / -11/5 = -0.32.

Explanation:
In the above-given question,
given that,
the two numbers are 5/7 and -11/5.
divide the numbers.
5/7/ – 11/5.
0.71 / -2.2.
– 0.32.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational Number Operations-16

Question 7.
Simplify the complex fraction \(\frac{-\frac{4}{5}}{\frac{3}{10}}\)
Rewrite the complex fraction ______ ÷ _______
Write the division as multiplication: ______ • _______
The product is _______ .

Answer:
The product is -0.24.

Explanation:
In the above-given question,
given that,
the two numbers are -4/5 and 3/10.
divide the numbers.
-4/5 x 3/10.
-0.8 x 0.3.
-0.24.
so the product is -0.24.

Question 8.
Which multiplication expression is equivalent to the division expression \(-\frac{7}{17} \div \frac{13}{34}\)?
Envision Math Common Core Grade 7 Answer Key Topic 1 Rational Number Operations 84
Answer:
Option A is correct.

Explanation:
In the above-given question,
given that,
the two numbers are -7/17 and 13/34.
divide the numbers.
-7/17 x 13/34.
-0.411 x 0.38.
-0.156.
so option A is correct.

Question 9.
Derek says that the quotient
\(-\frac{2}{7} \div\left(-\frac{2}{21}\right) \text { is }-\frac{1}{3} \text { . }\)
a. What is the correct quotient?

Answer:
The correct quotient is 3.11.

Explanation:
In the above-given question,
given that,
Derek says that the quotient
\(-\frac{2}{7} \div\left(-\frac{2}{21}\right) \text { is }-\frac{1}{3} \text { . }\).
-2/7 / -2/21.
-0.28/ -0.09.
3.11.

b. What mistake did Derek likely make?

Answer:
The two expressions are not equal.

Explanation:
In the above-given question,
given that,
Derek says that the quotient
\(-\frac{2}{7} \div\left(-\frac{2}{21}\right) \text { is }-\frac{1}{3} \text { . }\).
-2/7 / -2/21.
-0.28/ -0.09.
3.11.

Question 10.
The water level of a lake fell by \(1 \frac{1}{2}\) inches during a \(1 \frac{2}{3}\)-week-long dry spell. Simplify the complex fraction below to find the average rate at which the water level changed every week.
\(\frac{-1 \frac{1}{2}}{1 \frac{2}{3}}\) inches/week
Envision Math Common Core Grade 7 Answer Key Topic 1 Rational Number Operations 85

Answer:
The average rate at which the water level changed every week = 0.9375 meters.

Explanation:
In the above-given question,
given that,
The water level of a lake fell by \(1 \frac{1}{2}\) inches during a \(1 \frac{2}{3}\)-week-long dry spell.
1(1/2) / 1(2/3).
3/2 / 5/3.
1.5 / 1.6.
0.9375.
so the average rate at which the water level changed every week = 0.9375 meters.

Question 11.
Complete the table. Simplify expressions.
Envision Math Common Core Grade 7 Answer Key Topic 1 Rational Number Operations 86

Answer:
The quotients are the same.

Explanation:
In the above-given question,
given that,
-3/4 / 2/5.
-0.75 / 0.4 = -1.875.
-0.75/ 0.4.
-1.875.
3/4 / -2/5 = -1.875.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational Number Operations-17

Question 12.
a. Find the reciprocal of \(-1 \frac{1}{17}\).

Answer:
The reciprocal of -1(1/17) = 1.05.

Explanation:
In the above-given question,
given that,
-1(1/17).
-18/17.
-1.05.

b. Find the reciprocal of \(-\frac{17}{18}\).

Answer:
The reciprocal of -17/18 = -1.05.

Explanation:
In the above-given question,
given that,
the reciprocal of -17/18 is -18/17.
-18/17 = -1.05.

c. Reasoning Explain why the answer for part a is the multiplicative inverse of the answer for part b.

Answer:
The answer for part a is the multiplicative inverse of the answer for part b.

Explanation:
In the above-given question,
given that,
the reciprocal of -17/18 is -18/17.
-18/17 = -1.05.

Question 13.
Use numbers \(-\frac{7}{13}, 1 \frac{6}{7},-1 \frac{6}{7}, \frac{7}{13}\)
a. Which is the reciprocal of \(1 \frac{6}{7}\)?

Answer:
The reciprocal of 1(6/7) = 7/13.

Explanation:
In the above-given question,
given that,
1(6/7).
13/7.
13/7 = 1.85.

b. Which is the reciprocal of \(\frac{7}{13}\)?

Answer:
The reciprocal of 7/13 = 13/7.

Explanation:
In the above-given question,
given that,
the numbers are 7 and 13.
divide the numbers.
7/13 = 0.538.
13/7 = 1.85.

c. Reasoning What do you notice about the reciprocals of \(1 \frac{6}{7}\) and \(\frac{7}{13}\)?

Answer:
The reciprocals of 1(6/7) and 7/13 is

Explanation:
In the above-given question,
given that,
the number is 1(6/7).
13/7 = 7/13.
the reciprocals are the same.

Question 14.
A water tank in Stewart’s home had a small, steady leak.
Envision Math Common Core Grade 7 Answer Key Topic 1 Rational Number Operations 87
a. Use a complex fraction to represent the change in the volume of water in 1 minute.
Envision Math Common Core Grade 7 Answer Key Topic 1 Rational Number Operations 88

Answer:
The change in the volume of water in 1 minute = 1.6 ml.

Explanation:
In the above-given question,
given that,
A water tank in Stewart’s home had a small, steady leak.
1(3/5) = 8/5.
8/5 = 1.6.
so the change in the volume of water in 1 minute = 1.6 ml.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational Number Operations-18

b. Simplify the complex fraction to find the change in the volume of water in the tank in 1 minute.

Answer:
The change in the volume of water in 1 minute = 1.6 ml.

Explanation:
In the above-given question,
given that,
A water tank in Stewart’s home had a small, steady leak.
1(3/5) = 8/5.
8/5 = 1.6.
so the change in the volume of water in 1 minute = 1.6 ml.

Question 15.
Find the quotient. Express your answer as a simplified fraction.
\(\frac{3}{10} \div 3.8\)

Answer:
The quotient is 1.14.

Explanation:
In the above-given question,
given that,
the two numbers are 3/10 and 3.8.
divide the numbers.
3/10 = 0.3.
0.3 x 3.8 = 1.14.

Question 16.
Higher Order Thinking Between 10 P.M. and 7:45 A.M., the water level in a swimming pool decreased by \(\frac{13}{16}\) inch.
Assuming that the water level decreased at a constant rate, how much did it drop each hour?
The water level decreased by ______ inch each hour.

Answer:
The water level decreased by 0.8125 inches each hour.

Explanation:
In the above-given question,
given that,
Between 10 P.M. and 7:45 A.M., the water level in a swimming pool decreased by \(\frac{13}{16}\) inch.
13/16.
0.8125.
so the water level decreased by 0.8125 inches each hour.

Question 17.
Critique Reasoning Kayla wants to find \(2 \frac{2}{3} \div\left(-1 \frac{3}{7}\right)\). She first rewrites the division as \(\left(2 \frac{2}{3}\right)\left(-1 \frac{7}{3}\right)\) What is wrong with Kayla’s reasoning?

Answer:
The wrong with Kayla’s reasoning = -1.8309.

Explanation:
In the above-given question,
given that,
Kayla wants to find \(2 \frac{2}{3} \div\left(-1 \frac{3}{7}\right)\).
2(2/3) / -1(3/7).
8/3 / -10/7.
2.6 / – 1.42.
-1.8309.

Assessment Practice

Question 18.
Which is an equivalent multiplication expression for \(\frac{-\frac{3}{8}}{\left(-\frac{7}{54}\right)}\)?
Envision Math Common Core Grade 7 Answer Key Topic 1 Rational Number Operations 89

Answer:
Option A is correct.

Explanation:
In the above-given question,
given that,
\(\frac{-\frac{3}{8}}{\left(-\frac{7}{54}\right)}\).
-3/8 / -7/54.
-3/8 x 7/54.
so option A is correct.

Question 19.
Which is NOT a step you perform to divide \(-2 \frac{1}{8} \div 6 \frac{4}{5}\). Select all that apply.
☐ Rewrite the mixed numbers as fractions.
☐ Divide 8 by 4.
☐ Multiply by the multiplicative inverse of a \(\frac{34}{5}\)
☐ Multiply by the multiplicative inverse of \(\frac{17}{8}\)
☐ Multiply 8 and 34.

Answer:
Rewrite the mixed numbers as fractions.

Explanation:
In the above-given question,
given that,
\(-2 \frac{1}{8} \div 6 \frac{4}{5}\).
-2(1/8) / 6(4/5).
-17/8 / 34/5.
-2.125 / 6.8.
-0.3125.

Lesson 1-10 Solve Problems with Rational Numbers

Solve & Discuss It!
Stefan estimates the income and expenses for renting a phone accessory store in the mall. He enters the amounts in the table below. Should Stefan rent a phone accessory store? Explain.

I can… solve problems with rational numbers.
Envision Math Common Core Grade 7 Answer Key Topic 1 Rational Number Operations 90

Focus on math practices
Reasoning How can you assess the reasonableness of your solution using mental math or estimation strategies?

Essential Question
How do you decide which rational number operations to use to solve problems?

Try It!

A weather balloon ascended from an elevation of 18 feet below sea level to an elevation of 19\(\frac{1}{2}\) feet above sea level. What distance did the weather balloon rise?
The distance between two points is the absolute value of their __________.
Envision Math Common Core Grade 7 Answer Key Topic 1 Rational Number Operations 91
The weather balloon rose a distance of ________ feet.

Answer:
The weather balloon rose a distance of 1.5 feet.

Explanation:
In the above-given question,
given that,
A weather balloon ascended from an elevation of 18 feet below sea level to an elevation of 19\(\frac{1}{2}\) feet above sea level.
-18 + 19(1/2).
-18 + 39/2.
-18 + 19.5.
1.5.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational Number Operations-19

Convince Me! How can you decide which operation to use to solve a problem?

Try It!

Rashida had 18 correct answers and 12 incorrect answers. What was Rashida’s score?

Answer:
Rashida’s score is 6.

Explanation:
In the above-given question,
given that,
Rashida had 18 correct answers and 12 incorrect answers.
18 – 12 = 6.
so Rashida’s score is 6.

Try It!

The temperature at 10:00 A.M. was -3°F and increased 2.25°F each hour for the next 5 hours. What was the temperature at 3:00 P.M.?

Answer:
The temperature at 3:00 P.M = -1.25°F.

Explanation:
In the above-given question,
given that,
The temperature at 10:00 A.M. was -3°F and increased 2.25°F each hour for the next 5 hours.
-3 + 2.25.
-1.25.
so the temperature at 3:00 P.M = – 1.25°F.

KEY CONCEPT
You can solve a problem with rational numbers by making sense of the problem and deciding which operations to use.

Do You Understand?
Question 1.
Essential Question How do you decide which rational number operations to use to solve problems?

Question 2.
Reasoning A truck’s position relative to a car’s position is -60 feet. The car and the truck move in the same direction, but the car moves 5 feet per second faster for 8 seconds. What operations could be used to find the truck’s relative position after 8 seconds? Explain.

Answer:
The operations could be used to find the truck’s relative position after 8 seconds = -47 feet.

Explanation:
In the above-given question,
given that,
A truck’s position relative to a car’s position is -60 feet.
The car and the truck move in the same direction, but the car moves 5 feet per second faster for 8 seconds.
-60 + 5 + 8.
-47.
so the operations could be used to find the truck’s relative position after 8 seconds = -47 feet.

Question 3.
Construct Arguments Emilio used addition of two rational numbers to solve a problem. Jim used subtraction to solve the same problem. Is it possible that they both solved the problem correctly? Use a specific example to explain.

Answer:
Yes, they both have solved the problem correctly.

Explanation:
In the above-given question,
given that,
Emilio used the addition of two rational numbers to solve a problem.
for example:
4 + 3 = 7.
Jim used subtraction to solve the same problem.
4 – 3 = 1.

Do You Know How?
Question 4.
Kara had a savings account balance of $153 on Monday. On Tuesday, she had six withdrawals of $15.72 and a deposit of $235.15. What was her account balance after these transactions?

Answer:
The account balance after these transactions = $66.43.

Explanation:
In the above-given question,
given that,
Kara had a savings account balance of $153 on Monday.
On Tuesday, she had six withdrawals of $15.72 and a deposit of $235.15.
$153 + $15.72 = $168.72.
$235.15 – $168.72 = $66.43.
so the account balance after these transactins = $66.43.

Question 5.
A scuba diver is swimming at the depth shown, and then swims 0.5 foot toward the surface every 3 seconds. What is the location of the scuba diver, relative to the surface, after 15 seconds?
Envision Math Common Core Grade 7 Answers Topic 1 Rational Number Operations 92

Answer:
The location of the scuba diver, relative to the surface, after 15 seconds = 22.5 feet.

Explanation:
In the above-given question,
given that,
A scuba diver is swimming at the depth shown and then swims 0.5 feet toward the surface every 3 seconds.
0.5 x 3 = 1.5.
1.5 x 15 = 22.5.
so the location of the scuba diver, relative to the surface, after 15 seconds = 22.5 feet.

Question 6.
The temperature of a cup of coffee changed by -54°F over 22\(\frac{1}{2}\) minutes. What was the change in temperature each minute?

Answer:
The change in temperature each minute = 22.5 minutes.

Explanation:
In the above-given question,
given that,
The temperature of a cup of coffee changed by -54°F over 22\(\frac{1}{2}\) minutes.
22(1/2).
44 + 1 = 45.
45/2 = 22.5.
so the change in temperature each minute = 22.5 minutes.

Practice & Problem Solving

Question 7.
Suppose there is a 1.1°F drop in temperature for every thousand feet that an airplane climbs into the sky. If the temperature on the ground is 59.7°F, what will be the temperature at an altitude of 11,000 ft?
Envision Math Common Core Grade 7 Answers Topic 1 Rational Number Operations 93

Answer:
The temperature at an altitude of 11,000 ft = 656700°F.

Explanation:
In the above-given question,
given that,
Suppose there is a 1.1°F drop in temperature for every thousand feet that an airplane climbs into the sky.
If the temperature on the ground is 59.7°F.
11000 x 59.7°F.
656700°F
so the temperature at an altitude of 11,000 ft = 656700°F.

Question 8.
A farmer sells an average of 15\(\frac{3}{5}\) bushels of corn each day. What integer represents the change in bushels of corn in his inventory after 6 days?

Answer:
The integer represents the change in bushels of corn in his inventory after 6 days = 93.6 bushels.

Explanation:
In the above-given question,
given that,
A farmer sells an average of 15\(\frac{3}{5}\) bushels of corn each day.
15(3/5).
75 + 3 = 78.
78/5 = 15.6.
15.6 x 6 = 93.6.
so the integer represents the change in bushels of corn in his inventory after 6 days = 93.6 bushels.

Question 9.
A certain plant grows 1\(\frac{1}{6}\) inches every week. How long will it take the plant to grow 6\(\frac{1}{6}\) inches?

Answer:
The long will it take the plant to grow = 7.1456 inches.

Explanation:
In the above-given question,
given that,
A certain plant grows 1\(\frac{1}{6}\) inches every week.
1(1/6) = 7/6.
7/6 = 1.16.
6(1/6) = 37/6.
37/6 = 6.16.
6.16 x 1.16 = 7.1456.
so the longer will it take the plant to grow = 7.1456 inches.

Question 10.
An object is traveling at a steady speed of 8\(\frac{2}{3}\) miles per hour. How long will it take the object to travel 5\(\frac{1}{5}\) miles?

Answer:
The long will it take for the object o travel is 5(1/5) miles = 45.032.

Explanation:
In the above-given question,
given that,
An object is traveling at a steady speed of 8\(\frac{2}{3}\) miles per hour.
8(2/3) = 26/3.
26/3 = 8.66.
8.66 x 26/5.
8.66 x 5.2.
45.032.

Question 11.
Brianna works as a customer service representative. She knows that the amount of her yearly bonus is $155, but $2.50 is taken away for each customer complaint about her during the year. What is her bonus if there are 12 complaints about her in the year?

Answer:
The bonus if there are 12 complaints about her in the year = $1890.

Explanation:
In the above-given question,
given that,
Brianna works as a customer service representative.
She knows that the amount of her yearly bonus is $155, but $2.50 is taken away for each customer complaint about her during the year.
$155 + $2.50 = 157.5.
$157.5 x 12 = $1890.
so the bonus if there are 12 complaints about her in the year = $1890.

Question 12.
Make Sense and Persevere There are ten birdbaths in a park. On the first day of spring, the birdbaths are filled. Several weeks later, the overall change in the water level is found. The results are shown in the table. What is the range of the data?
Envision Math Common Core Grade 7 Answers Topic 1 Rational Number Operations 94

Answer:
The range of the data =

Explanation:
In the above-given question,
given that,
There are ten birdbaths in a park.
On the first day of spring, the birdbaths are filled.
2.4 – 0.9 = 1.5.
so the range of the data = 1.5.

Question 13.
Model with Math Marcelo played a carnival game at the Interstate Fair 6 times. He spent 3 tokens to play each game, and he won 7 tokens each game. Write two different expressions that can be used to find the total profit in tokens that Marcelo made.

Answer:
The different expressions that can be used to find the total profits in tokens that Marcelo made = 25.

Explanation:
In the above-given question,
given that,
Marcelo played a carnival game at the Interstate Fair 6 times.
He spent 3 tokens to play each game.
and he won 7 tokens each game.
6 x 3 = 18
18 + 7 = 25.
so the different expressions that can be used to find the total profits in tokens that Marcelo made = 25.

Question 14.
The temperature of a pot of water is shown. The temperature of the water changed -2.5°F per minute.
Envision Math Common Core Grade 7 Answers Topic 1 Rational Number Operations 95
a. What was the temperature after 20 minutes?

Answer:
The temperature of the water after 20 minutes = -3.606°F.

Explanation:
In the above-given question,
given that,
The temperature of a pot of water is 180.3°F.
-2.5°F x 20 = -50°F .
180.3°F /-50°F.
-3.606°F.
so the temperature of the water after 20 minutes = -3.606°F.

b. Make Sense and Persevere How many minutes did it take to cool to 100.3°F?

Answer:
The number of minutes did it take to cool to 100.3°F = 40 minutes.

Explanation:
In the above-given question,
given that,
-2.5 x 40 = 100.3°F.
so the number of minutes did it take to cool to 100.3°F = 40 minutes.

Question 15.
Higher Order Thinking the table shows the relationship between a hedgehog’s change in weight and the number of days of hibernation.
Envision Math Common Core Grade 7 Answers Topic 1 Rational Number Operations 96
a. What number represents the change in weight for each day of hibernation?

Answer:
The number represents the change in weight for each day of hibernation = -0.06.

Explanation:
In the above-given question,
given that,
the change in weight is -0.24 – 0.84 – 2.25 -2.79.
-0.6 – 2.25 – 2.79.
-2.85 – 2.79.
-0.06.
so the number represents the change in weight for each day of hibernation = -0.06.

b. What number represents the change in weight in ounces for the hedgehog in 115 days of hibernation?

Answer:

Assessment Practice

Question 16.
A basketball team played six games. In those games, the team won by 7 points, lost by 20, won by 8, won by 11, lost by 3, and won by 9. Which was the mean amount by which the team won or lost over the six games?
A. -3 points
B. 2 points
C. 3 points
D. 6 points

Answer:
Option D is correct.

Explanation:
In the above-given question,
given that,
A basketball team played six games.
In those games, the team won by 7 points, lost by 20, won by 8, won by 11, lost by 3, and won by 9.
won points = 7 + 8 + 11 + 9.
35.
lost points = -20 -3.
35/6 = 5.8.
so option D is correct.

Question 17.
In digging a hole, the construction crew records the location of the bottom of the hole relative to ground level. After 3 hours the hole is 8.25 feet deep.
PART A
What number represents the change in location in feet after 1 hour?

Answer:
The change in location in feet after 1 hour = 2.75 feet.

Explanation:
In the above-given question,
given that,
In digging a hole, the construction crew records the location of the bottom of the hole relative to ground level.
After 3 hours the hole is 8.25 feet deep.
8.25/3 = 2.75.
2.75 + 2.75 + 2.75 = 8.25.
so the change in locate in feet after 1 hour = 2.75 feet.

PART B
If the crew were to continue digging at the same rate, what number would they record for the location in feet after 8 hours?

Answer:
The number would they record for the location in feet after 8 hours = 1.03 feet.

Explanation:
In the above-given question,
given that,
In digging a hole, the construction crew records the location of the bottom of the hole relative to ground level.
After 8 hours the hole is 8.25 feet deep.
8.25/8 = 1.03.
so the number would they record for the location in feet after 8 hours = 1.03 feet.

3-ACT MATH

3-Act Mathematical Modeling: Win Some, Lose Some
Envision Math Common Core Grade 7 Answers Topic 1 Rational Number Operations 97
ACT 1
Question 1.
After watching the video, what is the first question that comes to mind?

Answer:
If the winning price is more than the loose price.

Explanation:
In the above-given question,
given that,
win some and lose some.
for example:
if we win rs100  and if we lose rs50.
so the profit is rs50.

Question 2.
Write the Main Question you will answer.
Answer:

Question 3.
Make a prediction to answer this Main Question.
The person who will win is _________.
Answer:

Question 4.
Construct Arguments Explain how you arrived at your prediction.
Envision Math Common Core Grade 7 Answers Topic 1 Rational Number Operations 98
Answer:

ACT 2
Question 5.
What information in this situation would be helpful to know? How would you use that information?
Envision Math Common Core Grade 7 Answers Topic 1 Rational Number Operations 99
Answer:

Question 6.
Use Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.
Answer:

Question 7.
Model with Math Represent the situation using mathematics. Use your representation to answer the Main Question.
Answer:

Question 8.
What is your answer to the Main Question? Does it differ from your prediction? Explain.
Envision Math Common Core Grade 7 Answers Topic 1 Rational Number Operations 100
Answer:

ACT 3
Question 9.
Write the answer you saw in the video.
Answer:

Question 10.
Reasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
Envision Math Common Core Grade 7 Answers Topic 1 Rational Number Operations 101
Answer:

Question 11.
Make Sense and Persevere Would you change your model now that you know the answer? Explain.
Answer:

Reflect
Question 12.
Model with Math Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?
Envision Math Common Core Grade 7 Answers Topic 1 Rational Number Operations 102
Answer:

Question 13.
Reasoning How is each person’s starting score related to their final score?
Answer:

SEQUEL
Question 14.
Construct Arguments If there were one final round where each contestant chooses how much to wager, how much should each person wager? Explain your reasoning.
Envision Math Common Core Grade 7 Answers Topic 1 Rational Number Operations 103

Answer:
I think that each person would wager at least $200.

Explanation:
In the above-given question,
given that,
If there were one final round where each contestant chooses how much to wager.
so I am thinking that each person would wager at least $200.

Topic 2 Review

Topic Essential Question
How can the properties of operations be used to solve problems involving integers and rational numbers?

Vocabulary Review
Complete each definition and then provide an example of each vocabulary word.

Envision Math Common Core Grade 7 Answers Topic 1 Rational Number Operations 104

Answer:
A Rational number is a fraction a/b where a and/or b are fractions and b is not equal to 0.
A terminating decimal that ends a(n).
Two numbers that have a sum of 0 are additive inverse.

Explanation:
In the above-given question,
given that,
A Rational number is a fraction a/b where a and/or b are fractions and b is not equal to 0.
for example:
all fractions and decimals are rational numbers.
A terminating decimal that ends a(n).
for example:
2.30 is a terminating decimal.
Two numbers that have a sum of 0 are additive inverse.
for example.
3 and -3.
Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational Number Operations-20

Use Vocabulary in Writing
Explain how you could determine whether \(\frac{\frac{21}{3}}{\frac{120}{12}}\) and \(\frac{7}{9}\) have the same decimal equivalent. Use vocabulary words in your explanation.

Answer:
No, both of them are not equal.

Explanation:
In the above-given question,
given that,
\(\frac{\frac{21}{3}}{\frac{120}{12}}\).
21/3 / 120/12.
7/ 10.
7/10 and 7/9 are not equal.
so both of them are not equal.

Concepts and Skills Review

Lesson 1.1 Relate Integers and Their Opposites

Quick Review
Integers are the counting numbers, their opposites, and 0. Opposite integers are the same distance from 0 in opposite directions. Opposite quantities combine to make 0.

Example
A climber descends 3 miles into a canyon. What integer represents the descent of her climb? How far does she have to climb to return to her starting point?
The descent of her climb is represented by -3. She has to climb 3 miles to return to her starting point.

Practice
Question 1.
On a cold winter morning, the temperature was -4°F By noon, the temperature increased 4o. What was the temperature at noon?

Answer:
The temperature at noon = 4°F.

Explanation:
In the above-given question,
given that,
On a cold winter morning, the temperature was -4°F By noon, the temperature increased by 40.
the temperature at noon is 4°F.

Question 2.
Audrey deposits $27 in her account. Then she makes two withdrawals, one for $15 and one for $12. What is the total change to the balance of Audrey’s account? Explain.

Answer:
The total change chance to the balance of Audrey’s account = $0.

Explanation:
In the above-given question,
given that,
Audrey deposits $27 in her account.
Then she makes two withdrawals, one for $15 and one for $12.
$15 + $12 = $27.
$27 – $27 = 0.
so the total change chance to the balance of Audrey’s account = $0.

Lesson 1.2 Understand Rational Numbers

Quick Review
All rational numbers have an equivalent decimal form. The decimal equivalent will be either a terminating decimal or a repeating decimal. A terminating decimal ends in repeating zeros. A repeating decimal has a never-ending pattern of the same digits.

Example
Write the decimal equivalents for and i Are the decimals terminating or repeating?
The decimal equivalent for \(\frac{5}{8}\) = 0.625, which is a terminating decimal.
The decimal equivalent for \(\frac{8}{11}\) = 0.7272… = 0.72,
which is a repeating decimal.

Practice
Question 1.
Which fractions have a decimal equivalent that is a repeating decimal? Select all that apply.
☐ \(\frac{13}{65}\)
☐ \(\frac{141}{47}\)
☐ \(\frac{11}{12}\)
☐ \(\frac{19}{3}\)

Answer:
Options B and C are repeating decimals.

Explanation:
In the above-given question,
given that,
13/65 = 0.2.
141/47 = 3.
11/12 = 0.916.
19/3 = 6.33.
so options B and C are repeating decimals.

Question 2.
Greg bought 19\(\frac{11}{16}\) gallons of gas. What decimal should the meter on the gas pump read?

Answer:
The decimal should the meter on the gas pump read = 19.6875.

Explanation:
In the above-given question,
given that,
Greg bought 19\(\frac{11}{16}\) gallons of gas.
19(11/16) = 315/16.
315/16 = 19.6875.
so the decimal should the meter on the gas pump read = 19.6875.

Question 3.
What is the decimal equivalent of each rational number?
a. \(\frac{9}{11}\)
b. \(\frac{4}{5}\)
c. –\(\frac{17}{5}\)
d. \(\frac{5}{9}\)

Answer:
0.81, 0.8, 3.4, and 0.55.

Explanation:
In the above-given question,
given that,
9/11 = 0.81.
4/5 = 0.8.
17/5 = 3.4.
5/9 = 0.55.

Lesson 1.3 Add Integers

Quick Review
To add integers with the same sign, add the absolute value of each integer. The sign of the sum will be the same as the sign of the addends. To add integers with different signs, find the difference of the absolute value of each integer. The sign of the sum will be the same as the sign of the greater addend.

Example
Find the sum of (-28) + (-19)
|-28| + |-19| = 28 + 19 = 47
The sum of (-28) + (-19) = -47.
Find the sum of (-28) + 19.
|-28| – |19| = 28 – 19 = 9
The sum of (-28) + 19 = -9, because
|-28| > |19|

Practice
Question 1.
Jonah’s cell phone came with 64 GB of memory. He has used 15 GB. He then uses 5 MB of memory to record photos and videos from a trip. Use the addition expression 64 + (-15) + (-5) to find how much memory is left on his phone.

Answer:
The memory is left on his phone = 44 MB.

Explanation:
In the above-given question,
given that,
Jonah’s cell phone came with 64 GB of memory.
He has used 15 GB.
He then uses 5 MB of memory to record photos and videos from a trip.
64 + (-15) + (-5).
64 -15 -5.
49 – 5.
44.
so the memory is left on his phone = 44 MB.

Question 2.
Stella walks down a flight of stairs to the basement. Then she walks back up the stairs and up another flight of stairs to the second floor of her house. Each flight of stairs represents a change of 12 feet in height. How far is Stella above the ground?

Answer:
The far is Stella above the ground = 24 feet.

Explanation:
In the above-given question,
given that,
Stella walks down a flight of stairs to the basement.
Then she walks back up the stairs and up another flight of stairs to the second floor of her house.
Each flight of stairs represents a change of 12 feet in height.
12 +12 = 24.
so the far is stella above the ground = 24 feet.

Question 3.
Find the sum.
a. 64 + (-15)

Answer:
The sum is 49.

Explanation:
In the above-given question,
given that,
the two numbers are 64 and -15.
add the numbers.
64 + (-15).
64 – 15.
49.

b. -121 + (-34)

Answer:
The sum is -155.

Explanation:
In the above-given question,
given that,
the two numbers are -121 and -34.
add the numbers.
-121 – 34.
-155.

c. -86 + 92

Answer:
The sum is 6.

Explanation:
In the above-given question,
given that,
the two numbers are -86 and 92.
add the numbers.
-86 + 92 = 6.
so the sum is 6.

d. 109 + (-162)

Answer:
The sum is -53.

Explanation:
In the above-given question,
given that,
the two numbers are 109 and -53.
add the numbers.
109 + (-162).
-53.
so the sum is -53.

Lesson 1.4 Subtract Integers

Quick Review
To subtract integers, use the additive inverse to write an equivalent addition expression. Then follow the rules for addition. When the signs are the same, find the sum of the absolute values. When the signs are different, find the difference. Use the sign of the number with the greater absolute value.

Example
Find -7 – (8).
-7 + (-8) = -15
The signs are the same, so the sum has the same sign as the addends.

Find -7 – (-8).
-7 + 8 = 1
The signs are different, so the sign of the difference is the same sign as the integer (8) with the greater absolute value, which is positive.

Practice
Question 1.
The temperature is 1°F at dusk. It is 8 degrees colder at dawn. What is the temperature at dawn?

Answer:
The temperature at the dawn = -7°F.

Explanation:
In the above-given question,
given that,
The temperature is 1°F at dusk.
It is 8 degrees colder at dawn.
1- 8 = -7.
so the temperature at the dawn = -7°F.

Question 2.
Kyle and Nadim are on the same space on a board game they are playing. Kyle moves back 2 spaces in one turn and moves back 3 more spaces in his second turn. Nadim has remained in the same place. What integer represents Kyle’s location relative to Nadim’s location on the game board?

Answer:
The integer represents Kyle’s location relative to Nadim’s location on the game board = -1.

Explanation:
In the above-given question,
given that,
Kyle and Nadim are in the same space on a board game they are playing.
Kyle moves back 2 spaces in one turn and moves back 3 more spaces in his second turn.
Nadim has remained in the same place.
2 – 3 = -1.
so the integer represents Kyle’s location relative to Nadim’s location on the game board = -1.

Question 3.
Find the difference.
a. 82 – (-14)

Answer:
The difference is 96.

Explanation:
In the above-given question,
given that,
the two numbers are 82 – (-14).
subtract the numbers.
82 – (-14).
82 + 14 = 96.

b. -18 – (-55)

Answer:
The difference is 37.

Explanation:
In the above-given question,
given that,
the two numbers are -18 – (-55).
subtract the numbers.
-18 – (-55).
-18 + 55.
37.

c. – 17 – 44

Answer:
The difference is -61.

Explanation:
In the above-given question,
given that,
the two numbers are -17 – 44.
subtract the numbers.
-17 – 44.
-61.

d. 70 – (-101)

Answer:
The difference is 171.

Explanation:
In the above-given question,
given that,
the two numbers are 70 – (-101).
subtract the numbers.
70 – (-101).
70 + 101.
171.

Lesson 1.5 Add and Subtract Rational Numbers

Quick Review
Positive and negative rational numbers and decimals can be added and subtracted following the same rules as adding and subtracting integers.

Example
Find -5\(\frac{1}{2}\) – 1.75.
Convert 1.75 to an equivalent fraction, 1 \(\frac{3}{4}\)
\(-5 \frac{1}{2}-1 \frac{3}{4}\)
= \(-5 \frac{2}{4}+\left(-1 \frac{3}{4}\right)\)
= \(-6 \frac{5}{4}\)
= \(-7 \frac{1}{4}\)

Practice
Question 1.
Doug digs a hole that is 1.7 feet below ground level. He plants a bush that is 37 feet tall from the bottom of the root to the top branch. How much of the bush is above the ground?

Answer:
Much of the bush is above the ground = 38.7 feet.

Explanation:
In the above-given question,
given that,
Doug digs a hole that is 1.7 feet below ground level.
He plants a bush that is 37 feet tall from the bottom of the root to the top branch.
1.7 + 37 = 38.7.
so much of the bush is above the ground = 38.7feet.

Question 2.
Penelope has a birdhouse that is 4 feet above the roof of her garage. She has a second birdhouse that is 5.36 feet below the roof of her garage. What is the distance between the birdhouses?

Answer:
The distance between the birdhouses = -1.36.

Explanation:
In the above-given question,
given that,
Penelope has a birdhouse that is 4 feet above the roof of her garage.
She has a second birdhouse that is 5.36 feet below the roof of her garage.
4 + (-5.36).
4 – 5.36.
-1.36.
so the distance between the birdhouses = -1.36.

Question 3.
Find the sum or difference.
a. -2.63 + 3\(\frac{1}{4}\)

Answer:
The difference is 0.62.

Explanation:
In the above-given question,
given that,
-2.63 + 3\(\frac{1}{4}\)
-2.63 + 3 (1/4).
-2.63 + 13/4.
-2.63 + 3.25.
0.62.

b. -4\(\frac{1}{2}\) – (-1.07)

Answer:
The difference is -3.43.

Explanation:
In the above-given question,
given that,
-4\(\frac{1}{2}\) – (-1.07).
-4(1/2) – (-1.07).
-9/2 + 1.07.
-4.5 + 1.07.
-3.43.

c. 0.74 + (-\(\frac{3}{5}\))

Answer:
The difference is 0.14.

Explanation:
In the above-given question,
given that,
0.74 + (-\(\frac{3}{5}\)).
0.74 – 3/5.
0.74 – 0.6.
0.14.

d. –\(\frac{1}{8}\) – 0.356

Answer:
The difference is -0.481.

Explanation:
In the above-given question,
given that,
–\(\frac{1}{8}\) – 0.356.
-1/8 – 0.356.
-0.125 – 0.356.
-0.481.
so the difference is -0.481.

Lesson 1.6 Multiply Integers

Quick Review
Multiply integers the same way you multiply whole numbers. If the signs of the factors are the same, the product is positive. If the signs of the factors are different, the product is negative.

Example
-9 • -8 = 72
-9 • 8 = -72

Practice
Question 1.
Marisa buys 4 books at $13 per book. What integer represents the total change in the amount of money Marisa has?

Answer:
The integer represents the total change in the amount of money Marisa has = $52.

Explanation:
In the above-given question,
given that,
Marisa buys 4 books at $13 per book.
4 x $13 = $52.
so the integer represents the total change in the amount of money Marisa has = $52.

Question 2.
Which expressions have a product of -18? Select all that apply.
☐ -2 • -9
☐ -6 • 3
☐ -3 • 6
☐ -9 • 2

Answer:
Options B, C, and D are correct.

Explanation:
In the above-given question,
given that,
expressions have a product of -18.
-2 x -9 = 18.
-6 x 3 = -18.
-3 x 6 = -18.
-9 x 2 = -18.
so options B, C, and D are correct.

Question 3.
Find the product.
a. -7 • -14
b. –15 • 12
c. 9 • -20
d. -11 • -16

Answer:
The products are 98, -180, -180, and 176.

Explanation:
In the above-given question,
given that,
-7 x -14 = 98.
-15 x 12 = -180.
9 x -20 = -180.
-11 x -16 = 176.

Lesson 1.7 Multiply Rational Numbers

Quick Review
The same rules for multiplying integers apply to multiplying rational numbers. If the signs of the factors are the same, the product will be positive. If the signs of the factors are different, the product will be negative.

Example
-9.6 • 1.8 = -17.28
-9.6 • -1.8 = 17.28

Practice
Question 1.
Jason spends $2.35 to buy lunch at school. If he buys a lunch on 9 days, what number represents the total change in the amount of money Jason has?

Answer:
The total change in the amount of money Jason has = $21.15.

Explanation:
In the above-given question,
given that,
Jason spends $2.35 to buy lunch at school.
If he buys lunch on 9 days.
$2.35 x 9 = $21.15.
so the total change in the amount of money Jason has = $21.15.

Multiply.
Question 2.
-2\(\frac{2}{3}\) • -4\(\frac{3}{7}\)

Answer:
-2(2/3) x -4(3/7) = 11.492.

Explanation:
In the above-given question,
given that,
-2\(\frac{2}{3}\) • -4\(\frac{3}{7}\)
-2(2/3) x -4(3/7).
-8/3 x -31/7.
-2.6 x -4.42.
11.492.

Question 3.
-3\(\frac{4}{9}\) • 5\(\frac{2}{5}\)

Answer:
-3(4/9) x 5(2/5) = -18.36.

Explanation:
In the above-given question,
given that,
-3\(\frac{4}{9}\) • 5\(\frac{2}{5}\)
-3(4/9) x 5(2/5).
-31/9 x 27/5.
-3.4 x 5.4.
-18.36.

Question 4.
6\(\frac{2}{3}\) • -4\(\frac{1}{5}\)

Answer:
6(2/3) x -4(1/5) = 2.4.

Explanation:
In the above-given question,
given that,
6\(\frac{2}{3}\) • -4\(\frac{1}{5}\).
6(2/3) x -4(1/5).
20/3 – 21/5.
6.6 – 4.2.
2.4.

Lesson 1.8 Divide Integers

Quick Review
Divide integers the same way you divide whole numbers. The quotient is positive if the signs of the dividend and divisor are the same. The quotient is negative if the signs of the dividend and divisor are different.

Example
-39 ÷ 3 = -13
-39 ÷ 3 = 13

Practice
Question 1.
Which expressions have a quotient of -4? Select all that apply.
☐ \(\frac{-24}{6}\)
☐ -36 ÷ -9
☐ -72 ÷ 18
☐ \(\frac{84}{-21}\)

Answer:
Options A,C, and D are correct.

Explanation:
In the above-given question,
given that,
the expressions have a quotient of -4 are:
-24/6 = -4.
-36/-9 = 4.
-72/ 18 = -4.
84/-21 = -4.
so options A, C, and D are correct.

Question 2.
Whitney rolls a ball down a ramp that is 18 feet long. If the ball rolls down 2 feet each second, what integer represents the amount of time, in seconds, the ball takes to reach the end of the ramp?

Answer:
The integer represents the amount of time in seconds, the ball takes to reach the end of the ramp = 9 seconds.

Explanation:
In the above-given question,
given that,
Whitney rolls a ball down a ramp that is 18 feet long.
If the ball rolls down 2 feet each second.
18/2 = 9.
so the integer represents the amount of time in seconds, the ball takes to reach the end of the ramp = 9 seconds.

Question 3.
Find the quotient.
a. \(\frac{81}{-9}\)
b. -123 ÷ -4
c. –\(\frac{94}{4}\)
d. 65 ÷ (-5)

Answer:
The quotients are -9, 30.75, -23.5, and -13.

Explanation:
In the above-given question,
given that,
a: 81/-9 = -9.
-123/-4 = 30.75.
-94/4 = -23.5.
65 / -5 = -13.
so the quotients are -9, 30.75, -23.5, and -13.

Lesson 1.9 Divide Rational Numbers

Quick Review
The same rules for dividing integers apply to dividing all rational numbers. The quotient is positive when the numbers being divided have the same signs. The quotient is negative when the numbers being divided have different signs. Complex fractions have a fraction in the numerator, the denominator, or both. To divide by a fraction, rewrite as multiplication by its multiplicative inverse, or reciprocal.

Example
Simplify \(\frac{-\frac{3}{4}}{\frac{15}{24}}\)
\(-\frac{3}{4} \div \frac{15}{24}=-\frac{3}{4} \cdot \frac{24}{15}=-\frac{72}{60}=-\frac{6}{5}=-1 \frac{1}{5}\)

Practice
Find the quotient.
Question 1.
\(\frac{8}{9}\) ÷ -1\(\frac{4}{15}\)

Answer:
8/9 / -1(4/15) = -0.38.

Explanation:
In the above-given question,
given that,
divide the numbers.
\(\frac{8}{9}\) ÷ -1\(\frac{4}{15}\).
8/9 – 1(4/15).
0.88 – 19/15.
0.88 – 1.26.
-0.38.

Question 2.
–3.6 ÷ 2\(\frac{1}{7}\)

Answer:
–3.6 ÷ 2\(\frac{1}{7}\) = -1.68.

Explanation:
In the above-given question,
given that,
divide the numbers.
–3.6 ÷ 2\(\frac{1}{7}\).
-3.6 / 2(1/7).
-3.6 / 2.14.
-1.68.

Question 3.
A boat drops an anchor 17.5 feet to the bottom of a lake. If the anchor falls at a rate of 0.07 feet each second, how long will it take the anchor to reach the bottom of the lake?

Answer:
The long will it take the anchor to reach the bottom of the lake = 25 feet.

Explanation:
In the above-given question,
given that,
A boat drops an anchor 17.5 feet to the bottom of a lake.
If the anchor falls at a rate of 0.07 feet each second.
1.75 / 0.07 = 25.
so the longer will it take the anchor to reach the bottom of the lake = 25 feet.

Lesson 1.10 Solve Problems with Rational Numbers

Quick Review
You can use rational numbers to solve problems in the same way that you use whole numbers. Be sure to make sense of the problem you are solving to help you choose the correct operations and determine which values will be positive and which will be negative.

Example
During a 15-day dry spell, the water level in a lake changed by -2\(\frac{3}{8}\) inches. What rational number represents the average change in the water level per day?
-2\(\frac{3}{8}\) ÷ 15
= –\(\frac{19}{8}\) • \(\frac{1}{15}\)
= –\(\frac{19}{120}\) inch

Practice
Question 1.
In 5 rounds of a game, Jill scored -3, 8, 9, -7, and 13. What integer represents her average score for the 5 rounds?

Answer:
The integer represents her average score for the 5 rounds = 2.

Explanation:
In the above-given question,
given that,
In 5 rounds of a game, Jill scored -3, 8, 9, -7, and 13.
-3, 8, 9, -7, and 13.
10/5 = 2.
so the integer represents her average score for the 5 rounds = 2.

Question 2.
Peter signed up for a program that costs $10.50 per month to stream movies to his computer. He decided to cancel his service after \(\frac{5}{6}\) month. He only has to pay for the amount of time he used the service. What number represents the total change in the amount of money Peter has after paying for the service?

Answer:
The number represents the total change in the amount of money Peter has after paying for the service =$8.715.

Explanation:
In the above-given question,
given that,
Peter signed up for a program that costs $10.50 per month to stream movies to his computer.
He decided to cancel his service after \(\frac{5}{6}\) month.
He only has to pay for the amount of time he used the service.
$10.50 x 5/6.
$10.50 x 0.83.
$8.715.
so the number represents the total change in the amount of money Peter has after paying for the service = $8.715.

Question 3.
Maggie spent $4.05 on cheese and fruit at the farmer’s market. She bought \(\frac{1}{8}\) pound of apples, \(\frac{1}{4}\) pound of pears, and 1.25 pounds of bananas. If fruit cost $0.80 per pound, how much did Maggie spend on cheese?

Answer:
Maggie spends on cheese = 5.675 pounds.

Explanation:
In the above-given question,
given that,
Maggie spent $4.05 on cheese and fruit at the farmer’s market.
She bought \(\frac{1}{8}\) pound of apples, \(\frac{1}{4}\) pound of pears, and 1.25 pounds of bananas.
If fruit cost $0.80 per pound.
$4.05 + 1/8 + 1/4 + 1.25 = 5.3 + 0.125 + 0.25.
5.3 + 0.375.
5.675.
so the Maggie spend on cheese = 5.675 pounds.

Topic 1 Fluency Practice

Crisscrossed
Find each sum, difference, product, or quotient. Write your answers in the cross-number puzzle below. Each digit and negative sign in your answers goes in its own box.
I can… add, subtract, multiply, and divide integers.
Envision Math Common Core Grade 7 Answers Topic 1 Rational Number Operations 105

Across
2. 248 + (-1,027)
5. 818 – (-1,021)
6. -516 + 774
8. 242 + (-656)
9. 2,087 + (-1,359)
10. 631 – 897
11. –342 + 199
12. -49 • -27
13. –321 – 987
14. 2,988 ÷ -3
15. 2,580 ÷ 6
16. 4,592 ÷ -82
17. 48 • -27
18. -24 • 83
21. -118 + 1,201
22. -45 • -59

Answer:

Explanation:
In the above-given question,
given that,
248 – 1027 =

Down
1. 246 + 173
2. 22 • -22
3. 726 – (-219)
4. 501 – 699
7. -10,740 ÷ 15
8. 6,327 ÷ -9
10. 144 • -16
11. 15 • -67
12. 7,164 ÷ 4
13. –33 • 63
14. -2,695 ÷ 55
17. -1,032 – (-285)
18. 512 – 720
19. –729 + 951
20. –17 • -25

enVision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Value

enVision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Value

Go through the enVision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Value regularly and improve your accuracy in solving questions.

enVision Math Common Core 5th Grade Answers Key Topic 1 Understand Place Value

Envision Math Common Core 5th Grade Answers Key Topic 1 Understand Place Value has all the topics covered , which are related to the basics of Math by giving live examples of our daily life. This chapter is loaded with Decimals, Fractions, and how to do rounding to the nearest numbers, which are quite helpful for students to deal with Math basics. Experience the most satisfying and understandable answers and easy methods with Envision Math Common Core 5th Grade Answers Key Topic 1 Understand Place Value.

Envision Math Common Core 5th Grade Answers Key Topic 1 Understand Place Value

Are you facing difficulties to understand the simple Math problems or numbers , that we have to use in day to day life. Then you have come to the right place to grasp the simple tricks of small calculations. Envision Math Common Core 5th Grade Answers Key Topic 1 Understand Place Value. will help you with rounding the numbers , understand the place values , converting decimal into fractions .These are the major topics covered, get on the track with your kids to know them a better way of learning.

Envision STEM Project: Pollinating Insects

Lesson 1 Patterns with exponents and Powers of 10

Lesson 2 Understand Whole Number Place Value

Lesson 3 Decimals to Thousandths

Lesson 4 Understand Decimal Place Value

Lesson 5 Compare Decimals

Lesson 6 Round Decimals

Lesson 7 Look For and Use Structure

 Performance Task

Essential Question: How are whole numbers and decimals written, compared, and ordered?

Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 1
Envision STEM Project Pollinating Insects
Do Research Use the Internet or other sources to find out more about pollinating insects in the United States. What types of insects are they? How many are there of each type? How many crops and flowering plants depend on pollinating insects in order to produce the foods we eat?
Journal: Write a Report Include what you found. Also in your report:

  • Choose two of the pollinating insects. Estimate how many crop plants each type of insect pollinates.
  • Estimate how many of your favorite foods and beverages come from pollinated plants.
  • Make up and solve ways to compare and order your data.

Answer: Pollinated insects are nothing but, the insects which are helpful to carry the pollinated grains along with them with the help of their legs or wings from the flower to promote the vast growth of new plants and are commonly known as insect pollinators.

Report for the project :

  •  Insect pollinators include bees, flies, butterflies, beetles, wasps, moths, midges and ants, among others. Of these, bees are the most important group, with both wild and managed species acting as pollinators.
  • Pollinators are essential for continued plant growth in the wild. There are seven insect pollinators other than bees and butterflies that also help spread plant seeds and enable plant growth.
  • Three-fourths of the world’s flowering plants and about 35 percent of the world’s food crops depend on animal pollinators to reproduce. More than 3,500 species of native bees help increase crop yields.
  • More than 75 percent of the world’s food crops depend on pollination . When a seed forms in flowering plants, a fruit is able to grow to protect the seed.

Among these two major pollinators are honey bees and butterflies,
Honey bees are the commonly known pollinators which are helpful for the crops like Okra, kiwifruit, Onion, cashew, strawberry, Broccoli, Cauliflower , Cabbage etc.
Butterflies are mostly the pollinators of vegetables and herbs and are helpful for the crops like especially those in the carrot family (dill, fennel, celery, cilantro, parsnip), sunflower family (artichokes, lettuce, chicory, chamomile), legume family (peas, beans), mint family (lavender, basil).

Review What You Know

Vocabulary

Choose the best term from the box. Write it on the blank.

• digits
• place value
• period
• whole numbers

Question 1.
____ are the symbols used to show numbers.
Answer: Digits are the symbols used to show numbers.

Explanation:
Because, A digit is a single symbol used to make numerals or numbers ,
That is 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are the ten digits we use in everyday numerals.
For example, the number 78 is a two digit number which is made up of two digits that is ‘7’ and ‘8’.

Question 2.
A group of 3 digits in a number is a ____
Answer: A group of 3 digits in a number is a Period.

Explanation:
Every number consists of digits. The place value is the position of each digit in a number.
Digits are separated into groups of three by commas. So, a period is a group of three digits,
For example, The number 13,456 is a 5 digit number and the ending 3 digits are called a period, because it is separated by the commas in the number which is referred to a period.

Question 3.
____ is the position of a digit in a number that is used to determine the value of the digit.
Answer: Place value is the position of a digit in a number that is used to determine the value of the digit.

Explanation:
The value of a digit depends on its place, or position, in the number.
Place value is the value of a digit according to its position in the number such as ones, tens, hundreds, and so on.
For example, In the number 3,548
3 is in thousands place and its place value is 3,000,
5 is in hundreds place and its place value is 500,
4 is in tens place and its place value is 40,
8 is in ones place and its place value is 8.

Comparing

Compare. Use <,>, or = for each Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 2.

Question 4.
869 Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 2 912
Answer: 869 < 912.

Explanation:
Because 869 is less than 912.
For the number 869
8 is in hundreds place and its place value is 800,
6 is in tens place and its place value is 60,
9 is in ones place and its place value is 9.

And For the number 912
9 is in hundreds place and its place value is 900,
1 is in tens place and its place value is 10,
2 is in ones place and its place value is 2.

So, 869 is less than 912.

Question 5.
9,033 Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 2 9,133
Answer: 9,033 < 9,133

Explanation:
Because 9,033 is less than 9,133
For the number 9,033
9 is in thousands place and its place value is 9,000,
0 is in hundreds place and its place value is 000,
3 is in tens place and its place value is 30,
3 is in ones place and its place value is 3.

And For the number 9,133
9 is in thousands place and its place value is 9,000,
1 is in hundreds place and its place value is 100,
3 is in tens place and its place value is 30,
3 is in ones place and its place value is 3.

So, 9,033 is less than 9,133

Question 6.
1,338 Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 2 1,388
Answer:  1,338 < 1,388

Explanation:
Because 1,338 is less than 1,388
For the number 1,338
1 is in thousands place and its place value is 1,000,
3 is in hundreds place and its place value is 300,
3 is in tens place and its place value is 30,
8 is in ones place and its place value is 8.

And For the number 1,388
1 is in thousands place and its place value is 1,000,
3 is in hundreds place and its place value is 300,
8 is in tens place and its place value is 80,
8 is in ones place and its place value is 8.

So, 1,338 is less than 1,388

Question 7.
417,986 Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 2 417,986
Answer: 417,986 = 417,986

Explanation:
Because, Numbers given are the same digits having the same place value .

So, 417,986 Equal to 417,986

Question 8.
0.25 Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 2 0.3
Answer:  0.25 < 0.3

Explanation:
Because, regarding with its place value the number  0.25 is less than 0.3

Question 9.
0.5 Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 2 0.50
Answer:  0.5 = 0.50

Explanation:
Because the 0 after the number next to decimal is exactly the same number with or without 0
Both the given numbers are in their same places with respect to each other
So, 0.5 is equal to 0.50.

Question 10.
Kamal has 7,325 songs on his computer. Benito has 7,321 songs on his computer. Who has more songs?
Answer:  Kamal has more songs than Benito.

Explanation:
Given , Kamal has 7,325 songs on his computer.
Benito has 7,321 songs on his computer.
Then 7,325 > 7,321 or 7,325 is greater than 7,321

So, Kamal has more songs than Benito.

Adding Whole Numbers

Find each sum.

Question 11.
10,000 + 2,000 + 60 + 1
Answer: 12,061

Explanation:

Adding all the given numbers by placing them in order of their place value of the digit
So, the total sum is 12,061.

Question 12.
20,000 + 5,000 + 400 + 3
Answer: 25,403

Explanation:

Adding all the given numbers by placing them in order of their place value of the digit
So, the total sum is 25,403.

Question 13.
900,000 + 8,000 + 200 + 70 + 6
Answer: 9,08,276

Explanation:

Adding all the given numbers by placing them in order of their place value of the digit
So, the total sum is 9,08,276.

Question 14.
7,000,000 + 50,000 + 900 + 4
Answer: 70,50,904

Explanation:

Adding all the given numbers by placing them in order of their place value of the digit
So, the total sum is 70,50,904.

Place Value

Question 15.
The largest playing card structure was made of 218,792 cards. What is the value of the digit 8 in 218,792?
A. 80
B. 800
C. 8,000
D. 80,000
Answer: C

Explanation:
Given, The largest playing card structure was made of 218,792 cards.
According to the place value method, the value of the digit 8 in 218,792 is 8,000

Question 16.
Construct Arguments In the number 767, does the first 7 have the same value as the final 7? Why or why not?
Answer: In the number  767, The first 7 does not have the same value as the final 7.

Explanation:
Given , number is 767 , The first 7 does not have the same value as the final 7.
Because, the place value is counted from right to left by ones, tens, hundreds, thousand and so on.
So, in this number 767, the first  number 7 holds the hundreds place and the final number 7 holds the ones place, giving it a different value for both the numbers in their place respectively.

Pick a Project

PROJECT 1A
Manatees or sea cows?
Project: Create a Manatee Poster
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 4.1

Answer:

PROJECT 1B
What makes a game fun?
Project: Design a Game with Place-Value Blocks
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 4.2

Answer:

PROJECT 1С
How far are we from the sun?
Project: Research Measurements in Our Solar System
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 4.3

Answer: The Sun is at an average distance of about 93,000,000 miles (150 million kilometers) away from Earth. It is so far away that light from the Sun, traveling at a speed of 186,000 miles (300,000 kilometers) per second, takes about 8 minutes to reach us.

Lesson 1.1 Patterns with exponents and Powers of 10

Solve & share
A store sells AA batteries in packages of 10 batteries. They also sell boxes of 10 packages, cases of 10 boxes, and cartons of 10 cases. How many AA batteries are in one case? One carton? 10 cartons? Solve these problems any way you choose.

You can use appropriate tools, such as place-value blocks, to help solve the problems. However you choose to solve it, show your work!
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 4.4

Look Back! How many 10s are in 100? How many 10s are in 1,000? Write equations to show your work.

Answer: There are 1000 AA batteries in one case , 10,000 batteries in one carton and 1,00,000 batteries in 10 cartons.

Explanation:
Given, A store sells AA batteries in packages of 10 batteries, boxes of 10 packages, cases of 10 boxes, and cartons of 10 cases.
Then we have 10 batteries for one package and given that 1 box contains 10 packages ,
10 × 10 = 100 , that is 100 batteries for each box.
Next for 10 boxes we have a case which means 100 × 10 = 1000 batteries ,
That is each case contains 1000 batteries.
Now for 10 cases we have a carton which means 1000 × 10 = 10,000 batteries ,
That is each carton contains 10,000 batteries.
And now they asked for 10 cartons ,
So, for 10 cartons we have 10,000 × 10 = 1,00,000 batteries .

Visual Learning Bridge

Essential Question
How Can You Explain Patterns in the Number of Zeros in a Product?
Answer: An exponent identifies a quantity representing the power to which a given number or expression is to be raised.

Explanation:
For example , 4 × 60
= 4 x (6 x 10)
= (4 x 6) x 10
= 24 x 10
=240
or
50 x 700
= (5 x 10) + (7 x 100)
= (5 x 7) x (10 x 100)
= 35 x 1,000
= 35,000.

Math Grade 5th Understand Place Value 1

A.
Tamara’s new horse weighs about 1,000 pounds. How can you show 1,000 as a power of 10 using an exponent?

The exponent is the number that tells how many times a base number is used as a factor.
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 4.5

Answer: 10³

Explanation:
Given, horse weighs about 1,000 pounds. then show 1,000 as a power of 10 using an exponent
So we can write 1000 as 10 × 10 × 10 = 10³.
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 4.6

B.
Write 1,000 as a product using 10 as a factor.
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 4.7
The exponent, 3, shows that the base number, 10, is multiplied 3 times.
So, 1,000 is written as 103 using exponents.

C.
Tamara estimates that her horse will eat about 5,000 pounds of hay each year. How can you write 5,000 using exponents? 5 × 101 = 5 × 10 = 50
5 × 102 = 5 × 10 × 10 = 500
5 × 103 = 5 × 10 × 10 × 10 = 5,000
The number of zeros in the product is the same as the exponent.
So, 5,000 is written as 5 × 103 using exponents.

Convince Me! Look for Relationships What pattern do you notice in the number of zeros in the products in box C above?

Guided Practice

Do You Understand?

Question 1.
Why are there three zeros in the product of 6 × 103?
Answer: The product of 6 × 10³ is 6,000 so it contains three zeros.

Explanation:
The product of 6 × 10³ is 6,000,
Because, 6 × 10 × 10 × 10 = 6,000.
So it contains three zeros.

Question 2.
Susan said that 105 is 50. What mistake did Susan make? What is the correct answer?
Answer: The correct answer is 1,00,000.

Explanation:
10 × 10 × 10 × 10 × 10 = 1,00,000.
The mistake Susan made is he multiplied the exponent with the base.

Do You Know How?

In 3 and 4, complete the pattern.

Question 3.
101 =
102 =
103 =
104 =
Answer:
101 = 10
102 = 100
103 = 1000
104 =  10,000

Explanation:
10  = 10
10 × 10  =  100
10 × 10 × 10  =  1000
10 × 10 × 10 × 10  = 10,000.

Question 4.
= 7 × 101
= 7 × 102
= 7 × 103
= 7 × 104
Answer:
70 = 7 × 101
700 = 7 × 102
7,000 = 7 × 103
70,000  = 7 × 104

Explanation:
Given, 7 ×10  = 70
7 × 10 × 10  =  700
7 ×10 × 10 × 10  =  7,000
7 × 10 × 10 × 10 × 10  = 70,000.

Independent Practice

In 5-15, find each product. Use patterns to help.

Question 5.
3 × 101 =
3 × 102 =
3 × 103 =
3 × 104 =
Answer:
3 × 101 = 30
3 × 102 = 300
3 × 103 = 3,000
3 × 104 = 30,000

Explanation:
3 ×10  = 30
3 × 10 × 10  =  300
3 ×10 × 10 × 10  =  3,000
3 × 10 × 10 × 10 × 10  = 30,000.

Question 6.
2 × 10 =
2 × 100 =
2 × 1,000 =
2 × 10,000 =
Answer:
2 × 10 =  20
2 × 100 = 200
2 × 1,000 = 2,000
2 × 10,000 = 20,000

Explanation:
2 × 10 =  2 × 101 = 20
2 × 100 = 2 ×  102 =  2 × 10 × 10 = 200
2 × 1,000 = 2 × 103  =  2 × 10 × 10 × 10 = 2,000
2 × 10,000 = 2 × 104  = 2 × 10 × 10 × 10  × 10 = 20,000

Math Grade 5th Understand Place Value 2

Question 7.
= 9 × 101
= 9 × 102
= 9 × 103
= 9 × 104
Answer:

90 = 9 × 101
900 = 9 × 102
9,000 = 9 × 103
90,000  = 9 × 104

Explanation:
Given, 7 ×10  = 70
7 × 10 × 10  =  700
7 ×10 × 10 × 10  =  7,000
7 × 10 × 10 × 10 × 10  = 70,000.

Question 8.
8 × 104
Answer:  8 × 104 = 80,000

Explanation:
8 × 104 = 8 × 10 × 10 × 10 × 10  = 80,000

Question 9.
4 × 1,000
Answer:  4 × 1,000 = 4,000

Explanation:
4 × 103  = 4 ×10 × 10 × 10  =  4,000

Question 10.
5 × 102
Answer:  5 × 102 = 500

Explanation:
5 × 102 =  5 × 10 × 10  = 500

Question 11.
6 × 10,000
Answer:  6 × 10,000 = 60,000

Explanation:
6 × 10,000 =  6 × 10 × 10 × 10 × 10  = 6 × 104 = 60,000

Question 12.
4 × 101
Answer:  4 × 101  =  40

Explanation:
4 × 101  =  40

Question 13.
100 × 9
Answer: 100 × 9 = 900

Explanation:
100 × 9  = 10 × 10  × 9 = 900

Question 14.
103 × 6
Answer:  103 × 6 = 6,000

Explanation:
103 × 6 = 10 × 10 × 10 × 6  = 6,000

Question 15.
8 × 105
Answer: 8 × 105  = 8,00,000

Explanation:
8 × 105  = 8 × 10 × 10 × 10 × 10 × 10 = 8,00,000

Question 16.
Write 10 × 10 × 10 × 10 × 10 × 10 with an exponent. Explain how you decided what exponent to write.
Answer: 106

Explanation:
As Given there are 6 tens together
So, 10 × 10 × 10 × 10 × 10 × 10  = 106

Problem Solving

Question 17.
One box of printer paper has 3 × 102 sheets of paper. Another box has 103 sheets of paper. What is the total number of sheets in both boxes?
Answer: The total number of sheets in both boxes is 403

Explanation:
Given, One box of printer paper has 3 × 102 sheets of paper. then , 3 × 102 = 300
Another box has 103 sheets of paper.
So, 300 + 103  = 403
The total number of sheets in both boxes is 403

Question 18.
A post is put every 6 feet along a fence around a rectangular field that is 42 ft long and 36 ft wide. How many posts are needed?
Answer: 26 posts are needed.

Explanation:
42 feet for one side, 42 feet for the parallel side
36 feet for one side, 36 feet for the parallel side
42 ÷ 6 = 7 ,One of these posts is a corner, so make it 6 because we do not want to count a corner post more than once.
36 ÷ 6 = 6 ,One of these posts is a corner, so make it 5 because we do not want to count a corner post more than once
We now add these: 2 x 6 + 2 x 5 = 12 + 10 = 22
Add in the four corner posts: 22 + 4 = 26 posts.
So , 26 posts are needed.

Question 19.
Number Sense A company had 9 × 106 dollars in sales last year. Explain how to find the product 9 × 106.
Answer:  9 × 106   = 9,000,000.

Explanation:
9 × 106  =  9 ×10 × 10 × 10 × 10 × 10 × 10  = 9,000,000.

Question 20.
An aquarium has the same shape as the solid figure shown below. What is the name of this solid figure?
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 15
Answer: A Rectangular prism.

Explanation:
a solid figure that has six sides, called faces, that are rectangles.
So, they are known as Rectangular prism.

Question 21.
Model with Math Isaac takes 5 minutes to ride his bike down the hill to school and 10 minutes to ride up the hill from school. He attends school Monday through Friday. How many minutes does he spend biking to and from school in two weeks? Write an equation to model your work.
Answer:  It takes 150 minutes

Explanation:
Given, Issac take 5 minutes to ride his bike down the hill to school
And 10 minutes to ride up the hill from school .
So, it takes 15 min each day to and from school He attends school Monday through Friday,
So, 5 days in a week for 5 days it takes  15 × 5 = 75 minutes
For 2 weeks , 75 × 2 = 150 minutes.

Question 22.
Higher Order Thinking Santiago hopes to buy a 4-horse trailer for about $12,000. Describe all the numbers that when rounded to the nearest hundred are 12,000.
Answer: Any number from 11,950 to 12,049, will result to 12,000 when rounded to the nearest hundred. 

Explanation:
Since we have given that,
Number of horse trailer = 4 Cost of 4 horse trailer = $12000 ,
As we know about the “Estimation”, if there is a number greater or equal to 5 in tens place then it will be rounded off to the nearest next greatest integer.
So, if this number is $11950 to 12049.
So, the possible numbers that when rounded to the nearest hundred are 12000 are from 11500 to 12049.

Assessment Practice

Question 23.
Choose all the equations that are true.
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 16 10 × 10 × 10 × 10 = 40
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 16 10 × 10 × 10 × 10 = 104
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 16 10 × 10 × 10 × 10 = 1,000,
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 16 10 × 10 × 10 × 10 = 10,000
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 16 10 × 10 × 10 × 10 = 4 × 104
Answer:  10 × 10 × 10 × 10 = 104  and  10 × 10 × 10 × 10 = 10,000

Explanation:
There are 4 tens in 10 × 10 × 10 × 10 = 104
So, 10 × 10 × 10 × 10 = 10,000.

Question 24.
Choose all the equations that are true.
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 16 6 × 105 = 6 × 100,000
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 16 6 × 105 = 6 × 10,000
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 16 6 × 105 = 600,000
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 16 6 × 105 = 60,000
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 16 6 × 105 = 650,000
Answer: 6 × 105 = 6 × 100,000 and  6 × 105 = 600,000

Explanation:
6 × 105 = 6 × 10 × 10 × 10 × 10  × 10 = 6,00,000. or
6 × 105 = 6 × 10 × 10 × 10 × 10  × 10 = 6 × 100,000.

Math Grade 5th Understand Place Value 3

Lesson 1.2 Understand Whole Number Place Value

Activity

Solve & Share

The population of a city is 1,880,000. What is the value of each of the two 8s in this number? How are the two values related? Use tools like this place-value chart to help solve the problem.

Answer: These are the values of each number

Explanation:
Add up all these values and we will have 1,880,000.

Use Structure You can use place value to analyze the relationship between the digits of a number. Show your work!
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 17

Look Back! Is the relationship between the value of the two 85 in 1,088,000 the same as the relationship between the value of the two 8s in the problem above? Explain.

Visual Learning Bridge

Essential Question
How Are Place-Value Positions Related?

A.
According to the 2010 U.S. Census, the population of Phoeni×, Arizona is about 1,440,000. What is the relationship between the value of the two 4s in this number?
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 18
Writing the number in expanded form can help.
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 19

C.
Look at the expanded form of 1,440,000. The value of the 4 in the hundred thousands place is 400,000. The value of the 4 in the ten thousands place is 40,000.
400,000 is 10 times as great as 40,000.
40,000 is \(\frac{1}{10}\) of 400,000.

Sometimes word form is used instead of number name.
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 20
Standard form
1,440,000
Expanded form:
1 × 1,000,000 + 4 × 100,000 + 4 × 10,000
Using exponents, this can be written as:
(1 × 106) + (4 × 105) + (4 × 104)
Number name:
one million, four hundred forty thousand

Convince Me! Construct Arguments is the value of the 1 in 1,440,000 10 times as great as the value of the 4 in the hundred thousands place? Explain.

Another example
When two digits next to each other in a number are the same, the digit on the left has 10 times the value of the digit to its right.
When two digits next to each other are the same, the digit on the right has to the value of the digit to its left.
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 20.1

Guided Practice

Do You Understand?

Question 1.
In 9,290, is the value of the first 9 ten times as great as the value of the second 9? Explain.
Answer:  No

Explanation:
The first 9 is on the place of the thousands, while the second 9 is on the place of the tens. so the first 9 is a hundred times as great as the second 9

Do You Know How?

Question 2.
Write 4,050 in expanded form.
Answer:  4,000 + 50

Explanation:
4,000 + 50  = 4,050.

In 3 and 4, write the values of the given digits.

Question 3.
the 7s in 7,700
Answer:  7,000 + 700

Explanation:
The first 7 is on the place of the thousands, while the second 7 is on the place of the hundreds.

Question 4.
the 2s in 522
Answer:  500 + 22

Explanation:
The first 2 is on the place of the tens, while the second 2 is on the place of the ones.

Independent Practice

In 5-7, write each number in standard form.

Question 5.
8,000,000 + 300 + 9
Answer: 8,000,309.

Explanation:
8,000,000 + 300 + 9 = 8,000,309.

Question 6.
(4 × 104) + (6 × 102)
Answer: 40,600

Explanation:
4 × 104 =4 × 10 × 10 × 10 × 10  =  40,000
6 × 102  = 6 × 10 × 10 =  600
So, 40,000 + 600 = 40,600

Question 7.
10,000 + 20 + 3
Answer:   10023

Explanation:
10,000 + 20 + 3 = 10023.

In 8-10, write each number in expanded form.

Question 8.
5,360
Answer: (5 × 10³) + (3 × 10²) + (6 × 10)

Explanation:
5,360 =  5000 + 300 + 60  = (5 × 10³) + (3 × 10²) + (6 × 10)

Question 9.
102,200
Answer:  105 + (2 × 10³) + ( 2 × 10²)

Explanation:
102,200 = 1,00,000 + 2000 + 200 =  105 + (2 × 10³) + ( 2 × 10²)

Question 10.
85,000,011
Answer:  (85 ×  106) + 10 + 1

Explanation:
85,000,011 = 85,000,000 + 10 + 1 = (85 ×  106) + 10 + 1

In 11-13, write the values of the given digits.

Question 11.
the 7s in 6,778
Answer: 6,000 + 700 + 70 + 8

Explanation:
The first 7 is on the place of the hundreds, while the second 7 is on the place of the tens.

Question 12.
the 9s in 990,250
Answer: 9,00,000 + 90,000 + 200 + 50

Explanation:
The first 9 is on the place of the hundred thousands, while the second 9 is on the place of the thousands.

Question 13.
the 1s in 2,011,168
Answer: 2,000,000 + 11,000 + 100 + 60 + 8

Explanation:
The first 1 is on the place of the ten thousands, while the second 1 is on the place of the thousands and the third 1 is on the place of hundreds.

Problem Solving

Question 14.
Write the number name and expanded form for the number of driver ants that could be in two colonies.

Up to 22,000,000 driver ants can live in a single colony.
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 20.5

Answer:  For 2 colonies  44,000,000
The number name is Forty four million
The expanded form is (4 ×106) + (4 ×106)

Explanation:
Given , Up to 22,000,000 driver ants can live in a single colony.
For 2 colonies 22,000,000 + 22,000,000 = 44,000,000
The expanded form is 40,000,000 + 4,000,000 = (4 ×106) + (4 ×106)

Question 15.
enVision® STEM A queen ant can produce about nine million ants in her lifetime. Write this number in standard form.
Answer: The number in standard form is 9,000,000

Explanation:
Given, A queen ant can produce about nine million ants in her lifetime.
The number in standard form is 9,000,000.

Math Grade 5th Understand Place Value 4

Question 16.
Critique Reasoning Paul says that in the number 6,367, one 6 is 10 times as great as the other 6. Is he correct? Explain why or why not.
Answer: No

Explanation:
Given in the number 6,367, one 6 is 10 times as great as the other 6.
The first 6 is on the place of the thousands, while the second 6 is on the place of the tens. so the first 6 is a hundred times as great as the second 6.

Question 17.
Jorge drew a square that had a side length of 8 inches. What is the perimeter of Jorge’s square?
Remember, the perimeter of a shape is the distance around it.
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 20.6
Answer: The perimeter of Jorge’s square is 32 inches.

Explanation:
Given, Jorge drew a square that had a side length of 8 inches.
Square has all equal sides, so perimeter of the square P = 4a = 4 × 8 = 32 inches

Question 18.
Higher Order Thinking
Dan wrote (2 × 106) + (3 × 104) + (5 × 103) + 4 for the expanded form of two million, three hundred fifty thousand, four. What error did he make in the expanded form? What is the standard form of the number?

Answer: The standard form of the number is 20,35,004.
The number name is two million, thirty five thousand and four.

Explanation:
(2 × 106) + (3 × 104) + (5 × 103) + 4
= 2,000,000 + 30,000 + 5,000 + 4
= 20,35,004.
The standard form of the number is 20,35,004.
The number name is two million, thirty five thousand and four.
He made mistake with the three hundred fifty thousand, instead of  thirty five thousand

Assessment Practice

Question 19.
Colleen says she is thinking of a 4-digit number in which all the digits are the same. The value of the digit in the hundreds place is 200.
Part A
What is the number? Explain.

Part B
Describe the relationship between the values of the digits in the number.
Answer:  The number is 2,222.
The first 2 is on the place of the thousands, while the second 2 is on the place of the hundreds, The third 2 is on the place of the tens, and the last 2 is on the place of the ones.

Explanation:
Given,  Colleen says she is thinking of a 4-digit number in which all the digits are the same. The value of the digit in the hundreds place is 200.
The number is 2,222.
The first 2 is on the place of the thousands, while the second 2 is on the place of the hundreds, The third 2 is on the place of the tens, and the last 2 is on the place of the ones.

Lesson 1.3 Decimals to Thousandths

Solve & Share
At Suzie’s Sticker City, customers can buy a book of stickers, a page, a strip, or a single sticker. Provide the missing fractions in the boxes below.

How can you use what you know about powers of 10 to help you fill in the boxes?
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 20.7

Look Back! Use Structure Describe any patterns you notice in the fractions.

Visual Learning Bridge

Essential Question
How Can You Read and Write Question Decimals to the Thousandths?

A.
A box is filled with 1,000 cubes. Amy picks out 4 cubes. How can you represent 4 out of 1,000 cubes as a decimal?
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 20.8

You can write 4 out of 1,00 as the fraction \(\frac{4}{1,000}\).
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 20.9

B.
The number name for \(\frac{4}{1,000}\) is four thousandths. A decimal place-value chart can help you determine the decimal. Notice that the thousandths place is three places to the right of the decimal point.
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 20.10
So, \(\frac{4}{1,000}\) can be represented by the decimal 0.004.

C.
How can \(\frac{444}{1,000}\) be represented by a decimal? \(\frac{444}{1,000}\) is read as four hundred forty-four thousandths and represented by the decimal 0.444.
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 20.11
The value of the digit 4 in the hundredths place has 10 times the value of the digit 4 in the thousandths place and \(\frac{1}{10}\) the value of the digit 4 in the tenths place.

Convince Me! Reasoning How is 0.004 the same as and different from 0.444?

Answer: 0.004 and 0.444 both have 4 in the thousandth place

Explanation:
0.004 and 0.444 both have 4 in the thousandth place,
both are in thousandth place
different: 0.444 is greater than 0.004 when divide by 1000
444/1000 is greater than 4/1000
if convert to percentage : 0.444 is 44.4% and 0.004 is 0.4%

Guided Practice

Do You Understand?

Question 1.
If four cubes are pulled from the box on the previous page, how would you write the fraction representing the cubes that are left? the decimal representing the cubes that are left?
Answer:  In decimal form we can write it as 0.996.

Explanation:
Given, A box is filled with 1,000 cubes, picking out 4 cubes,
1000 – 4 = 996,
Then \(\frac{996}{1000}\) are the remaining of the cubes which are left in the box
In decimal form we can write it as 0.996.

Do You Know How?

Question 2.
0.3 is 10 times as great as what decimal? 0.003 is \(\frac{1}{10}\) of what decimal?
Answer: 0.3 is 10 times as great as 0.03 and 0.003 is \(\frac{1}{10}\) of 0.03

Explanation:
0.3 × \(\frac{1}{10}\) = 0.03
0.03 = 0.003 × \(\frac{1}{10}\)
So, 0.3 is 10 times as great as 0.03 and 0.003 is \(\frac{1}{10}\) of 0.03

In 3-6, write each decimal as a fraction.

Question 3.
0.001 =
Answer: \(\frac{1}{1000}\)

Explanation:
Given 0.001 ,
In fraction form we can write it in to \(\frac{1}{1000}\)

Question 4.
0.05 =
Answer: \(\frac{5}{100}\)

Explanation:
Given 0.05 ,
In fraction form we can write it in to \(\frac{5}{100}\)

Question 5.
0.512 =
Answer: \(\frac{512}{1000}\)

Explanation:
Given 0.512 ,
In fraction form we can write it in to \(\frac{512}{1000}\)

Question 6.
0.309 =
Answer: \(\frac{309}{1000}\)

Explanation:
Given 0.309 ,
In fraction form we can write it in to \(\frac{309}{1000}\)

In 7-10, write each fraction as a decimal.

Question 7.
\(\frac{2}{1,000}\) =
Answer: 0.002

Explanation:
Given , \(\frac{2}{1,000}\)
In decimal form we can  write it into 0.002

Question 8.
\(\frac{34}{100}\) =
Answer: 0.34

Explanation:
Given , \(\frac{34}{100}\)
In decimal form we can  write it into 0.34

Question 9.
\(\frac{508}{1,000}\) =
Answer: 0.508

Explanation:
Given , \(\frac{508}{1,000}\)
In decimal form we can  write it into 0.508

Question 10.
\(\frac{99}{1,000}\) = =
Answer:

Explanation:
Given , \(\frac{99}{1,000}\)
In decimal form we can  write it into 0.099

Independent Practice

In 11-18, write each decimal as a fraction.

Question 11.
0.007
Answer: \(\frac{7}{1000}\)

Explanation:
Given 0.007 ,
In fraction form we can write it in to \(\frac{7}{1000}\)

Question 12.
0.08
Answer: \(\frac{8}{100}\)

Explanation:
Given 0.08 ,
In fraction form we can write it in to \(\frac{8}{100}\)

Question 13.
0.065
Answer: \(\frac{65}{1000}\)

Explanation:
Given 0.08 ,
In fraction form we can write it in to \(\frac{65}{1000}\)

Question 14.
0.9
Answer: \(\frac{9}{10}\)

Explanation:
Given 0.9 ,
In fraction form we can write it in to \(\frac{9}{10}\)

Question 15.
0.832
Answer: \(\frac{832}{1000}\)

Explanation:
Given 0.832 ,
In fraction form we can write it in to \(\frac{832}{1000}\)

Question 16.
0.203
Answer: \(\frac{203}{1000}\)

Explanation:
Given 0.203 ,
In fraction form we can write it in to \(\frac{203}{1000}\)

Question 17.
0.78
Answer:  \(\frac{78}{100}\)

Explanation:
Given 0.78 ,
In fraction form we can write it in to \(\frac{78}{100}\)

Question 18.
0.999
Answer: \(\frac{999}{1000}\)

Explanation:
Given 0.999 ,
In fraction form we can write it in to \(\frac{999}{1000}\)

In 19-26, write each fraction as a decimal.

Question 19.
\(\frac{434}{1,000}\) =
Answer:  0.434

Explanation:
Given, \(\frac{434}{1,000}\)
In decimal form we can  write it into 0.0.434

Question 20.
\(\frac{3}{10}\) =
Answer: 0.3

Explanation:
Given, \(\frac{3}{10}\)
In decimal form we can  write it into 0.3

Question 21.
\(\frac{873}{1,000}\) =
Answer: 0.873

Explanation:
Given, \(\frac{873}{1,000}\)
In decimal form we can  write it into 0.873

Question 22.
\(\frac{17}{1,000}\) =
Answer:0.017

Explanation:
Given, \(\frac{17}{1,000}\)
In decimal form we can  write it into 0.017

Question 23.
\(\frac{309}{1,000}\) =
Answer: 0.309

Explanation:
Given, \(\frac{309}{1,000}\)
In decimal form we can  write it into 0.309

Question 24.
\(\frac{5}{1,000}\) =
Answer: 0.005

Explanation:
Given, \(\frac{5}{1,000}\)
In decimal form we can  write it into 0.005

Question 25.
\(\frac{6}{100}\) =
Answer: 0.06

Explanation:
Given, \(\frac{6}{100}\)
In decimal form we can  write it into 0.06

Question 26.
\(\frac{999}{1,000}\) =
Answer: 0.999

Explanation:
Given, \(\frac{999}{1,000}\)
In decimal form we can  write it into 0.999

Question 27.
Look at the middle 9 in exercise 18. How is its value related to the value of the 9 to its left? to the value of the 9 to its right?
Answer: The value of the digit 9 in the hundredths place has 10 times the value of the digit 9 in the thousandths place and \(\frac{1}{10}\) the value of the digit 9 in the tenths place.

Explanation:
The value of the number given in exercise 18 is 0.999,
The first 9 is on the place of the tenths, while the second 9 is on the place of the hundredths and the second 9 is  on the place of thousandths.
So, The value of the digit 9 in the hundredths place has 10 times the value of the digit 9 in the thousandths place and \(\frac{1}{10}\) the value of the digit 9 in the tenths place.

Problem Solving

Question 28.
The Palmers’ property tax bill for the year is $3,513. In their first installment, they paid $1,757. How much do they still owe on their bill? Write an equation to model your work.
Answer:  They still owe $1,756 on their bill .

Explanation:
Given, The Palmers’ property tax bill for the year is $3,513.
In their first installment, they paid $1,757.
$3,513 – $1,757 = $1,756.
So, they still have to pay the half amount that is $1,756
Then $1,757 × 2 = $3,513.
They still owe $1,756 on their bill .

Question 29.
Write the fractions \(\frac{22}{100}\) and \(\frac{22}{1,000}\) as decimals. How are the values of the digit 2 related in each of the decimals?
Answer: 0.22 and 0.022

Explanation:
Given, \(\frac{22}{100}\)
In decimal form we can write it as 0.22,
The value of the digit 2 in the tenths place has 10 times the value of the digit 4 in the hundredths place .

And \(\frac{22}{1,000}\)
In decimal form we can write it as 0.022,
The value of the digit 2 in the hundredths place has 10 times the value of the digit 4 in the thousandths place.

Question 30.
Simon scored 4 × 102 points in a game. Joe scored 2 × 103 points in the same game. Whose score is higher? How much higher?
Answer: Joe scored 5 times as many as Simon.

Explanation:
Given, Simon scored 4 × 102 points in a game.
Joe scored 2 × 103 points in the same game.
Now we have  4 × 102 =  4 × 10 × 10 = 400
2 × 103 =  2 × 10 × 10 × 10 = 2000
Finally, Simon scored 400 point and Joe scored 2000 points
So, Joe scored 5 times as many as Simon.

Question 31.
Higher Order Thinking Kelly said that \(\frac{97}{1,000}\) can be written as 0.97. Is she correct? Explain.
Answer: No , \(\frac{97}{1,000}\)  should be 0.097 not 0.97

Explanation:
Given, Kelly said that \(\frac{97}{1,000}\) can be written as 0.97
The decimal form can written according to the number of zeros in the denominator
So , \(\frac{97}{1,000}\)  should be 0.097 not 0.97

Question 32.
Critique Reasoning Frank reasoned that in the number 0.555, the value of the 5 in the thousandths place is ten times as great as the 5 in the hundredths place. Is he correct? Explain.
Answer: NO

Explanation:
Given, the number 0.555
The value of the digit 5 in the hundredths place has 10 times the value of the digit 5 in the thousandths place

Question 33.
How many cubes are in the box? What fraction of the entire box do the 7 cubes represent? Explain your answer.
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 35.1
Answer: \(\frac{7}{1000}\)

Explanation:
Given, 10 × 10 × 10  = 1000
And we have the 7 cubes
So, the fraction will be  \(\frac{7}{1000}\)

Assessment Practice

Question 34.
0.04 is 10 times as great as which decimal?
A. 0.4
B. 0.1
C. 0.004
D. 0.001
Answer: C , 0.04 is 10 times as great as 0.004

Explanation:
0.04 × \(\frac{1}{10}\) = 0.004
So, 0.04 is 10 times as great as 0.004

Question 35.
0.009 is o of which decimal?
A. 0.01
B. 0.09
C. 0.1
D. 0.9
Answer:  B, 0.009 is \(\frac{1}{10}\) of 0.09

Explanation:
0.09 × \(\frac{1}{10}\) = 0.009
So, 0.009 is \(\frac{1}{10}\) of 0.09

Lesson 1.4 Understand Decimal Place Value

Solve & Share

A runner won a 100-meter race with a time of 9.85 seconds. How can you use place value to Explain this time? Complete a place-value chart to show this time.

Generalize You can use what you know about whole-number place value to help you think about place value of decimal numbers.
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 36.1

Look Back! In the decimal 9.85, what is the value of the 8? What is the value of the 5?

Visual Learning Bridge

Essential Question How Can You Represent Decimals?

A.
Jo picked a seed from her flower. The seed has a mass of 0.245 gram. What are some different ways you can represent 0.245?

You can write the standard form, expanded form, and number name for a decimal just like you can for a whole number.
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 36.2

B.
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 36.3
Number Name: two hundred forty-five

A place-value chart can help you identify the tenths, hundredths, and thousandths places in a decimal.
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 36.4

Convince Me! Use Structure How many hundredths are in one tenth? How many thousandths are in one hundredth? Tell how you know.

Another example
Equivalent decimals name the same amount.
What are two other decimals equivalent to 1.4?
One and four tenths is the same as one and forty hundredths.
1.4 = 1.40
One and four tenths is the same as one and four hundred thousandths.
1.4 = 1.400.
Envision Math Common Core Grade 5 Answer Key Topic 1 Understand Place Vale 37.1
So, 1.4 = 1.40 = 1.400.

Guided Practice

Do You Understand?

Question 1.
The number 3.453 has two 3s. Why does each 3 have a different value?
Answer: These are different due to the place values. 3 and 0.003

Explanation:
The given number is 3.453.
The first 3 that is before the decimal is at ones place, that is 3 × 1 = 3
But the second 3 which is after the decimal is at thousandth place, \(\frac{3}{1000}\) = 0.003
Hence, you can clearly see the different values of both three’s. These are different due to the place values.

Do You Know How?

In 2 and 3, write each number in standard form.

Question 2.
4 × 100 + 7 × 10 + 6 × 1 + 6 × \(\left(\frac{1}{10}\right)\) + 3 × \(\left(\frac{1}{100}\right)\) + 7 × \(\left(\frac{1}{1,000}\right)\)
Answer:  The standard form is 476.637.

Explanation:
Given,
(4 × 100)+ (7 × 10) + (6 × 1) + {6 × \(\left(\frac{1}{10}\right)\) + 3 × \(\left(\frac{1}{100}\right)\) + 7 × \(\left(\frac{1}{1,000}\right)\)}
Solve for the brackets,
we have 400 + 70 + 6 + [ 0.6 + 0.03 + 0.007]
= 476 + 0.637
= 476.637
The standard form is 476.637.

Question 3.
four and sixty-eight thousandths
Answer: The standard form is 0.468.

Explanation:
Given, four and sixty-eight thousandths
The expanded form is 4 × \(\left(\frac{1}{10}\right)\) + 6 × \(\left(\frac{1}{100}\right)\) + 8 × \(\left(\frac{1}{1,000}\right)\)
Then, 0.4 + 0.06 + 0.008 = 0.468
So, the standard form is 0.468

Independent Practice

In 4-6, write each number in standard form.

Question 4.
(2 × 1) + (6 × \(\frac{1}{1,000}\))
Answer: The standard form is 2.006

Explanation:
Given, (2 × 1) + (6 × \(\frac{1}{1,000}\))
= 2 + 0.006
= 2.006
The standard form is 2.006

Question 5.
(3 × 1) + (3 × \(\frac{1}{10}\)) + (9 × \(\frac{1}{1,000}\))
Answer: The standard form is 3.309

Explanation:
Given, (3 × 1) + (3 × \(\frac{1}{10}\)) + (9 × \(\frac{1}{1,000}\))
= 3 + 0.3 + 0.009
= 3.309
So, The standard form is 3.309

Question 6.
nine and twenty hundredths
Answer: The standard form is 0.920.

Explanation:
Given, nine and twenty hundredths
The expanded form is (9 × \(\frac{1}{10}\)) + (20 × \(\frac{1}{100}\))
= 0.9 + 0.020
= 0.920
The standard form is 0.920

In 7-10, write two decimals that are equivalent to the given decimal.

Question 7.
2.200
Answer: The two decimals that are equivalent to the given decimal are 2.2 and 2.20

Explanation:
Given, 2.200
Two and two tenths is the same as two and twenty hundredths.
2.2 = 2.20
Two and Two tenths is the same as two and twenty tenths.
2.2 = 2.2.
So, The two decimals that are equivalent to the given decimal are 2.2 and 2.20

Question 8.
8.1
Answer: The two decimals that are equivalent to the given decimal are 8.10 and 8.100

Explanation:
Given, 8.1
Eight and one tenths is the same as eight and one hundredths.
8.1 = 8.10
Eight and one tenths is the same as eight and one hundred thousandths.
8.1 = 8.100.
So, The two decimals that are equivalent to the given decimal are 8.10 and 8.100

Question 9.
9.50
Answer: The two decimals that are equivalent to the given decimal are 9.5 and 9.500

Explanation:
Given, 9.50
Nine and five tenths is the same as nine and  fifty tenths.
9.50 = 9.5.
Nine and five tenths is the same as nine and fifty hundred thousandths.
9.50 = 9.500.
So, The two decimals that are equivalent to the given decimal are 9.5 and 9.500

Question 10.
4.200
Answer: The two decimals that are equivalent to the given decimal are 4.2 and 4.20

Explanation:
Given, 4.200
Four and two tenths is the same as four and twenty hundredths.
4.200 = 4.20
Four and two tenths is the same as two and twenty tenths.
4.200 = 4.2.
So, The two decimals that are equivalent to the given decimal are 4.2 and 4.20

Problem Solving

Question 11.
The annual fundraising goal of a charity is $100,000. So far $63,482 has been raised. How much more money is needed to reach the goal?
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 40.1
Answer: To reach the goal they need $36,518.

Explanation:
Given, The annual fundraising goal of a charity is $100,000.
So far $63,482 has been raised.
Then, $100,000 – $63,482 = $36,518.
So, To reach the goal they need $36,518.

Question 12.
Santiago has a rope that measures 205.95 centimeters. Write this number in expanded form.
Answer:

Explanation:
Given, Santiago has a rope that measures 205.95 centimeters
The standard form is 205.95
Then, The expanded form is (2 × 100) + (5 × 1) + (9 × \(\frac{1}{10}\)) + (5 × \(\frac{1}{100}\))
= 200 + 5 +[ 0.9 + 0.05 ]
= 205 + 0.95
= 205.95
So, The expanded form is (2 × 100) + (5 × 1) + (9 × \(\frac{1}{10}\)) + (5 × \(\frac{1}{100}\))

Question 13.
How can you tell that 7.630 and 7.63 are equivalent decimals?
Answer: The two decimals that are equivalent to both the numbers

Explanation:
Given, 7.630 and 7.63
Seven and sixty three tenths is the same as Seven and sixty three hundredths.
7.63 = 7.630
So, The two decimals that are equivalent to both the numbers

Question 14.
In Justin’s school, 0.825 of the students participate in a sport. If there are one thousand students in Justin’s school, how many participate in a sport?
Answer: Totally 825 students are participating in the school.

Explanation:
Given, In Justin’s school, 0.825 of the students participate in a sport.
Since, there are 1 thousand students, and the decimal is 0.825,
So, there are 825 students participating.


Question 15.
Be Precise Maria incorrectly placed the decimal point when she wrote 0.65 inch for the width of her tablet computer. What is the correct decimal number for the width?
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 40.2
Answer: The correct decimal number for the width is 6.5.

Explanation:
Given, Maria incorrectly placed the decimal point when she wrote 0.65 inch for the width of her tablet computer.
As shown in the figure, the scale placed there to measure the width , as per the scale the width is 6.5.
So, The correct decimal number for the width is 6.5.

Question 16.
Higher Order Thinking Three boys cut out hundredths decimal models. Derrick does not shade any of his models. Ari shades half of one model. Wesley shades two models and one tenth of another model. What decimal represents the amount each boy shades?
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 42.2
Answer: Derrick’s value = 0,
Ari’s value = 0.5,
Wesley’s value = 2.1 ,

Explanation:
Given, Derrick does not shade any of his models. Ari shades half of one model. Wesley shades two models and one tenth of another model.
Derrick doesn’t shade anything, so his value is 0.
Ari shades half of one model, so 1/2 = 0.5 is his value Wesley shades 2 full models, plus 1/10 = 0.1 of another one, leading to 2+0.1 = 2.1 as his value. 
Derrick’s value = 0,
Ari’s value = 0.5,
Wesley’s value = 2.1 ,

Assessment Practice

Question 17.
Find two decimals that are equivalent to (4 × 10) + (7 × \(\frac{1}{100}\)). Write the decimals in the box.
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 42.3
Answer:  The two decimals that are equivalent to the 40.07 are 40.070 and 40.07

Explanation:
Given, (4 × 10) + (7 × \(\frac{1}{100}\)).
= 40 + 0.07
= 40.07
The standard form is 40.07
The two decimals that are equivalent to the 40.07 are 40.070 and 40.07 , from the given box of decimals.

Lesson 1.5 Compare Decimals

Activity

Solve & Share

The lengths of three ants were measured in a laboratory. The lengths were 0.521 centimeter, 0.498 centimeter, and 0.550 centimeter. Which ant was the longest? Which ant was the shortest?

How can you use the math you know to compare and order the decimals? Tell how you decided.
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 42.4

Look Back! Be Precise What are the lengths of the ants in order from least to greatest?

Visual Learning Bridge

Essential Question How Can You Compare Decimals?

A.
Scientists collected and measured the lengths of different cockroach species. Which cockroach had the greater length, the American or the Oriental cockroach?
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 42.5
Comparing decimals is like comparing whole numbers!
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 42.6

B.
Step 1
Line up the decimal points.
Start at the left.
Compare digits of the same place value.
3.576
3.432

C.
Step 2
Find the first place where the digits are different.
3.576
3.432

D.
Step 3
Compare.
5 > 4
0.5 > 0.4
So, 3.576 > 3.432.
The American cockroach is longer than the Oriental cockroach.

Convince Me! Critique Reasoning Valerie said, “12.68 is greater than 12.8 because 68 is greater than 8.” Is she correct? Explain.

Answer: No, 12.8 is greater than 12.68

Explanation:
Given  : 12.68 is greater than 12.8
We have given 12.68 and 12.8
In 12.68,
Tens place is 1,
Ones place is 2,
Tenth place is 6,
Hundredth place is 8.
In 12.8,
Tens place is 1 .
Ones place is 2,
Tenth place is 8,
So, We can see Tens place and ones place of both number are same but tenth place of 12.8 is greater than 12.68.
Therefore, 12.8 is greater than 12.68 . 

Another example
Order the cockroaches from least to greatest length.
Step 1
Write the numbers, lining up the decimal points. Start at the left. Compare digits of the same place value.
3.576
3.582
3.432 is the least.

Step 2
Write the remaining numbers, lining up the decimal points. Start at the left. Compare.
3.576
3.582
3.582 is greater than 3.576.

Step 3
Write the numbers from least to greatest.
3.432 3.576 3.582
From least to greatest lengths are the Oriental, the American, and the Australian.

Guided Practice

Do You Understand?

Question 1.
Scientists measured a Madeira cockroach and found it to be 3.44 centimeters long. Toby says that the Madeira is shorter than the Oriental because 3.44 has fewer digits than 3.432. Is he correct? Explain.
Answer: 3.44cm cockroach is the longest.

Explanation:
Given,  3.44  and  3.432
3.44
3.432
Comparing the numbers after decimals,
4 > 3
So, 3.44cm cockroach is the longest.

Do You Know How?

In 2 and 3, write >, <, or = for each Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 55.

Question 2.
3.692 Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 55 3.697
Answer: 3.692 < 3.697

Explanation:
Given,  3.692  and  3.697
3.692
3.697
Comparing the numbers after decimals, we have
2 < 7
So, 3.692 < 3.697.

Question 3.
7.216 Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 55 7.203
Answer:  7.216 > 7.203.

Explanation:
Given,  7.216  and  7.203
7.216
7.203
Comparing the numbers after decimals, we have
1 > 0
So, 7.216 > 7.203.

In 4 and 5, order the decimals from least to greatest.

Question 4.
5.540, 5.631, 5.625
Answer: The order from least to greatest is  5.540, 5.625, 5.631.

Explanation:
Given, 5.540, 5.631, 5.625
lining up the decimal points.
5.540, is the least
5.631,
5.625,
Comparing the numbers after decimals, we have
5.631,
5.625,
Compare digits of the same place value.
3 > 2
5.631 is the greatest
So, The order from least to greatest is  5.540, 5.625, 5.631.

Question 5.
0.675, 1.529, 1.35, 0.693
Answer: The order from least to greatest is 0.675, 0.693, 1.35, 1.529.

Explanation:
Given, 0.675, 1.529, 1.35, 0.693
lining up the decimal points.
0.675,
1.529,
1.35,
0.693
Comparing the numbers after decimals, we have
0.675,
0.693,
Compare digits of the same place value.
7 < 9
0.675 is the less than 0.693
Then 1.529,
1.35,
Compare digits of the same place value.
5 > 3
1.529 is greater than 1.35
So, The order from least to greatest is 0.675, 0.693, 1.35, 1.529.

Independent Practice

In 6-8, compare the two numbers. Write >, <, or = for each Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 55.

Question 6.
0.890 Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 55 0.890
Answer: 0.890 = 0.890

Explanation:
Given,  0.890  and 0.890
0.890
0.890
Comparing the numbers after decimals, we have
All the numbers are equal
So, 0.890 = 0.890.

Question 7.
5.733 Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 55 5.693
Answer: 5.733 > 5.693.

Explanation:
Given,  5.733  and  5.693
5.733
5.693
Comparing the numbers after decimals, we have
7 > 6
So, 5.733 > 5.693.

Question 8.
9.707 Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 55 9.717
Answer: 9.707 < 9.717

Explanation:
Given,  9.707  and  9.717
lining up the decimal points
9.707
9.717
Comparing the numbers after decimals, we have, Compare digits of the same place value.
0 < 1
So, 9.707 < 9.717.

In 9 and 10, order the decimals from greatest to least.

Question 9.
878.403, 887.304,887.043
Answer: The order from least to greatest is 878.403 ,887.043 , 887.304.

Explanation:
Given, 878.403, 887.304,887.043
lining up the decimal points.
878.403, is the least
887.304,
887.043
Comparing the numbers after decimals, we have
887.304,
887.043
Compare digits of the same place value.
3 > 0
887.304 is greater than 887.043
So, The order from least to greatest is 878.403 ,887.043 , 887.304.

Question 10.
435.566, 436.565, 435.665
Answer: The order from least to greatest is 435.566, 435.665 , 436.565.

Explanation:
Given, 435.566, 436.565, 435.665
lining up the decimal points.
435.566,
436.565, is the greatest
435.665
Comparing the numbers after decimals, we have
435.566,
435.665
Compare digits of the same place value.
5 < 6
435.566 is less than 435.665
So, The order from least to greatest is 435.566, 435.665 , 436.565.

Problem Solving

Question 11.
Critique Reasoning Explain why it is not reasonable to say that 4.23 is less than 4.135 because 4.23 has fewer digits after the decimal point than 4.135.
Answer: 4.23 is greater than 4.135.

Explanation:
Given, 4.23 and 4.135
lining up the decimal points.
4.23,
4.135
Comparing the numbers after decimals, we have ,Compare digits of the same place value.
2 < 1
So, 4.23 is greater than 4.135.

Question 12.
Number Sense Carlos wrote three numbers between 0.33 and 0.34. What numbers could Carlos have written?
Answer: there can be any three numbers mentioned Below between 0.33 and 0.34.

Explanation:
Given, three numbers between 0.33 and 0.34
The numbers between the 0.33 and 0.34 are 0.331, 0.332, 0.333, 0.334, 0.335, 0.336, 0.337, 0.338 and 0.339,
Then the next number will be 0.340 that is same as 0.34
So, there can be any three numbers mentioned above between 0.33 and 0.34.

Question 13.
Vocabulary Draw lines to match each decimal on the left to its equivalent decimal on the right.
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 58.2
Answer: The given decimals are compared to its nearest decimals by place value method.

Explanation:

Question 14.
Is 0.5 greater than or less than \(\frac{6}{10}\)? Draw a number line to show your answer.
Answer: 0.5 is < 0.6

Explanation:

Given,  0.5 and \(\frac{6}{10}\)
We can write \(\frac{6}{10}\) as 0.6 in decimal form,
Compare digits of the same place value.
5 < 6
So, 0.5 is < 0.6

Question 15.
Higher Order Thinking Ana’s gymnastics scores were posted on the scoreboard in order from highest to lowest score. One digit in her floor score is not visible. List all the possible digits for the missing number.
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 58.3
Answer: The possible digits are 1 , 2 , 3 , 4

Explanation:
Given,
The floor score must be higher than 15.133 and lower than 15.500
Then all the possible scores are: 15.166 , 15.266 , 15.366 and 15.466
Because 15.166 > 15.133 and 15.466 < 15.500
Then all the possible digits are 1 , 2 , 3 , 4 

Question 16.
Marcia’s vault score is 15.050. How does it compare to Ana’s vault score?
Answer: Marcia’s vault score is less than Ana’s vault score

Explanation:
Given, Marcia’s vault score is 15.050
Ana’s vault score is 15.500
Lining up the decimal points.
15.050
15.500
Comparing the numbers after decimals, we have ,Compare digits of the same place value.
0 < 5
So, 15.050 is less than 15.500.
Finally, Marcia’s vault score is less than Ana’s vault score

Assessment Practice

Question 17.
Which statements correctly compare two numbers?
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 59 0.1 <0.125
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 59 0.2 < 0.125
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 59 125 > 0.13
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 59 0.125 > 0.12
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 59 0.126 < 0.125
Answer: 0.1 < 0.125 , 125 > 0.13 , 0.125 > 0.12 .

Explanation:
Compare digits of the same place value.
0.1 < 0.125 , because 0 < 2
125 > 0.13 , because .0 > 0.
0.125 > 0.12 . because 5 > 0

Question 18.
Cara weighed 4.16 pounds of apples at the grocery store. Which numbers make the statement true? Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 60 > 4.16
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 59.2 04.15
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 59.2 04.19
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 59.2 4.2
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 59.2 4.09
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 59.2 4.1
Answer: 04.15

Explanation:
Given, Cara weighed 4.16 pounds of apples at the grocery store
4.16 and 04.15 are the closest decimals
Compare digits of the same place value.
5 < 6
So, the immediate number of 4.16 is 04.15.

Lesson 1.6 Round Decimals

Activity

Solve & Share

In science class, Marci recorded numbers from an experiment as 12.87, 12.13, 12.5, and 12.08. Which numbers are closer to 12? Which are closer to 13? How can you tell?

You can use structure to help determine what number is halfway between two whole numbers. Show your work!
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 60.1

Look Back! What is the halfway point between 12 and 13? Is that point closer to 12 or 13?

Visual Learning Bridge

Essential Question How Can You Round Decimals?

A.
Rounding replaces one number with another number that tells about how many or about how much. Round 2.36 to the nearest tenth. Is 2.36 closer to 2.3 or 2.4?

A number line can help you round a decimal.
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 60.2

B.
Step 1
Find the rounding place. Look at the digit to the right of the rounding place.
2.36

C.
Step 2
If the digit is 5 or greater, add 1 to the rounding digit. If the digit is less than 5, leave the rounding digit alone.
Since 6 >5, add 1 to the 3.

D.
Step 3
Drop the digits to the right of the rounding digit.
2.36 rounds to 2.4

Rounding can help (you find which tenth or hundredth a decimal is closest to.
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 60.3

Convince Me! Critique Reasoning Carrie said, “448 rounds to 500 because 448 rounds to 450 and 450 rounds to 500.” Is she correct? Explain. Use the number line in your explanation.
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 60.4

Answer: No

Explanation:
Given, 448 rounds to 500 because 448 rounds to 450 and 450 rounds to 500.”
The nearest number to 448 is 450 and that is correct but,
450 is not near the number 500 ,
The difference between these numbers is 50 ,
So, Carrie is wrong.

Another example
Round 3.2 to the nearest whole number. Is 3.2 closer to 3 or 4?
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 60.5

Step 1
Find the rounding place. Look at the digit to the right of the rounding place.
3.2
Step 2
If the digit is 5 or greater, add 1 to the rounding digit. If the digit is less than 5, leave the rounding digit alone. Since 2 < 5, leave 3 the same.
Step 3
Drop the digits to the right of the decimal point. Drop the decimal point.
3.2 rounds to 3.

Guided Practice

Do You Understand?

Question 1.
To round 74.58 to the nearest tenth, which digit do you look at? What is 74.58 rounded to the nearest tenth?
Answer: the nearest tenth to 74.58 is 74.6

Explanation:
Given, To round 74.58 to the nearest tenth,
Find the rounding place. Look at the digit to the right of the rounding place.
8 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point.
74.6
So, the nearest tenth to 74.58 is 74.6

Question 2.
A car-rental service charges customers for the number of miles they travel, rounded to the nearest whole mile. George travels 40.8 miles. For how many miles will he be charged? Explain.
Answer: He will be charged for 41 miles

Explanation:
Given, George travels 40.8 miles.
Find the rounding place. Look at the digit to the right of the rounding place.
8 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point.
The nearest to 40.8 is 41
So, He will be charged for 41 miles

Do You Know How?

In 3-10, round each number to the place of the underlined digit.

Question 3.
16.5
Answer: 17

Explanation:
Given, 16.5
6 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 17
The rounding number is 17.

Question 4.
56.1
Answer: 56

Explanation:
Given, 56.1
6 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 56
The rounding number is 56.

Question 5.
1.32
Answer: 1.3

Explanation:
Given, 1.32
3 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 1.3
The rounding number is 1.3.

Question 6.
42.78
Answer: 42.8

Explanation:
Given, 42.78
7 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 42.8
The rounding number is 42.8.

Question 7.
1.652
Answer: 1.65

Explanation:
Given, 1.652
5 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 1.65
The rounding number is 1.65.

Question 8.
582.04
Answer: 582.0

Explanation:
Given, 582.04
4 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 582.0
The rounding number is 582.0.

Question 9.
80,547.645
Answer: 80,547.65

Explanation:
Given, 80,547.645
4 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 80,547.65
The rounding number is 80,547.65.

Question 10.
135,701.949
Answer: 135,701.9

Explanation:
Given, 135,701.949
9 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 135,701.9
The rounding number is 135,701.9.

Independent Practice

In 11-14, round each decimal to the nearest whole number.

Question 11.
4.5
Answer: 5

Explanation:
Given, 4.5
4 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 5
The rounding number is 5.

Question 12.
57.3
Answer: 57

Explanation:
Given, 57.3
7 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 57
The rounding number is 57.

Question 13.
34.731
Answer: 34.73

Explanation:
Given, 34.731
3 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 34.73
The rounding number is 34.73.

Question 14.
215.39
Answer: 215.4

Explanation:
Given,215.39
3 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 215.4
The rounding number is 215.4.

In 15-18, round each number to the place of the underlined digit.

Question 15.
7.158
Answer: 7.2

Explanation:
Given, 7.158
1 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 7.2
The rounding number is 7.2.

Question 16.
0.758
Answer: 0.76

Explanation:
Given, 0.758
5 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 0.76
The rounding number is 0.76.

Question 17.
6.4382
Answer: 6.44

Explanation:
Given, 6.4382
3 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 6.44
The rounding number is 6.44.

Question 18.
84.732
Answer: 84.7

Explanation:
Given, 84.732
7 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 84.7
The rounding number is 84.7.

Problem Solving

Question 19.
The picture at the right shows the length of an average American alligator. What is the length of the alligator rounded to the nearest tenth?
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 62.1
Answer: 4.4 is the length of the alligator rounded to the nearest tenth.

Explanation:
Given, 4.39
3 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 4.4
The rounding number is 4.4.
So,4.4  is the length of the alligator rounded to the nearest tenth

Question 20.
Name two different numbers that round to 8.21 when rounded to the nearest hundredth.
Answer: 8.211 and 8.212

Explanation:
Given, 8.21 rounded to the nearest hundredth,
2 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 8.211 and 8.212
The rounding number is 8.211 and 8.212.

Question 21.
Number Sense To the nearest hundred, what is the greatest whole number that rounds to 2,500? the least whole number?
Answer: 2,499 and 2,491.

Explanation:
Given, 2,500
The greatest whole number to nearest hundred of 2,500 is 2,499
And, The less whole number to nearest hundred of 2,500 is 2,491

Question 22.
Draw all of the lines of symmetry in the figure shown below.
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 62.2
Answer: There are 2 line lines of symmetry.

Explanation:

The given figure is in the form of a Rectangle
So, it has 2 lines of symmetry.

Question 23.
Higher Order Thinking Emma needs 2 pounds of ground meat to make a meatloaf. She has one package with 2.36 pounds of ground meat and another package with 2.09 pounds of ground meat. She uses rounding and finds that both packages are close to 2 pounds. Explain how Emma can choose the package closer to 2 pounds.
Answer: She can use the package of 2.09 because it is close to 2 pounds.

Explanation:
Given, 2.09 and 2.36
For 2.09 , 0 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point. 2.1
For 2.36, 3 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point. 2.4
So, She can use the package of 2.09 because it is close to 2 pounds.

Question 24.
Make Sense and Persevere Robert slices a large loaf of bread to make 12 sandwiches. He makes 3 turkey sandwiches and 5 veggie sandwiches. The rest are ham sandwiches. What fraction of the sandwiches Robert makes are ham?
Answer: \(\frac{3}{12}\), \(\frac{5}{12}\), \(\frac{4}{12}\).

Explanation:
Given, Robert slices a large loaf of bread to make 12 sandwiches,
He makes 3 turkey sandwiches and 5 veggie sandwiches
The rest are ham sandwiches.
The number of turkey sandwiches are \(\frac{3}{12}\),
The number of veggie sandwiches are \(\frac{5}{12}\),
The number of ham sandwiches are \(\frac{4}{12}\),
Total 12 sandwiches are there made by Robert.

Question 25.
Algebra After buying school supplies, Ruby had $32 left over. She spent $4 on notebooks, $18 on a backpack, and $30 on a new calculator. How much money, m, did Ruby start with? Write an equation to show your work.
Answer: Ruby started with $84.

Explanation:
Given, Ruby had $32 left over.
She spent $4 on notebooks, $18 on a backpack, and $30 on a new calculator.
The total money spent =  $4 + $18  +  $30  = $52
Add the money she had left over and money she had spent
we have, $52 + $32 = $84
So, Ruby started with $84.

Assessment Practice

Question 26.
Find two numbers that round to 35.4 when rounded to the nearest tenth. Write the numbers in the box.
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 63.1
Answer: 35.45 and 35.391

Explanation:
Given, 35.40
4 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 35.45
if the digit is 5 or greater, add 1 to the rounding digit. 35.391
The rounding number is 35.45 and 35.391.

Lesson 1.7 Look For and Use Structure

Problem Solving

Activity

Solve & Share
Angie volunteers in the school library after school. The librarian gave her a stack of books and told her to use the number on each book to shelve it where it belongs.
How can Angie arrange the books in order from least to greatest to make shelving them easier?
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 63.2

Thinking Habits
Be a good thinker! These questions can help you.
• What patterns can I see and describe?
• How can I use the patterns to solve the problem?
• Can I see expressions and objects in different ways?
• What equivalent expressions can I use?
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 63.3

Look Back! Use Structure Explain why 323.202 is less than 323.21 even though 202 is greater than 21.

Answer: Because , when rounding the decimals we have to look for the place values to decide which is greater or least.

Explanation:
Given, 323.202
0 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 323.21
The rounding number is 323.21.

Visual Learning Bridge

Essential Question
How Can You Use Structure to Solve Problems?

A.
Analyze the chart. What do you notice that can help you complete the chart?
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 63.4
What do I need to do to solve this problem?

I can use the structure of the decimal place-value system to complete the chart.
Envision Math Common Core Grade 5 Answers Topic 1 Understand Place Vale 63.5
You can look for patterns to find the missing numbers.

B.
How can I make use of structure to solve this problem?
I can
• find and describe patterns.
• use the patterns to see how the numbers are organized.
• analyze patterns to see the structure in the table.
• break the problem into simpler parts.

C.
Solve
Here’s my thinking…
Envision Math Common Core Grade 5 Answers Topic 1 Understand Place Vale 63.6
As you move down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same, except for the last number, while the hundredths increase by 1.
Envision Math Common Core Grade 5 Answers Topic 1 Understand Place Vale 63.7

Convince Me! Use Structure Write the missing numbers. Explain how you can use structure to find the last number in the bottom row.
Envision Math Common Core 5th Grade Answers Topic 1 Understand Place Vale 63.8

Answer:   

Explanation:
Because, down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same,
except for the last number, while the hundredths increase by 1.

Guided Practice
Use Structure Each of these grids is a part of a decimal number chart similar to the one on page 30.

You can use what you know about place value when you look for patterns with decimals.
Envision Math Common Core Grade 5 Answers Topic 1 Understand Place Vale 63.9

Question 1.
Describe the pattern for moving from a pink square to a green square. Then write the missing numbers.
Envision Math Common Core Grade 5 Answers Topic 1 Understand Place Vale 63.10
Answer:

Explanation:
Because, down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same,
except for the last number, while the hundredths increase by 1.

Question 2.
How can you use patterns to find the number that would be in the box below 0.52?
Answer: the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same,
except for the last number, while the hundredths increase by 1. and completing the pattern.

Independent Practice

Use Structure Pamela is hiking. When she returns to camp, she passes the mile markers shown at the right.

Question 3.
Explain how you can use structure to find the decimal numbers that will be shown on the next four mile markers.
Envision Math Common Core Grade 5 Answers Topic 1 Understand Place Vale 63.11
Answer:  the next four mile markers will be 2.2 , 2.1 , 2.0 , 1.9

Explanation:

Moving from left to right in the rows, tenths stay the same,
except for the last number, while the hundredths increase by 1
So, the next four mile markers will be 2.2 , 2.1 , 2.0 , 1.9

Question 4.
Pamela stops at the 1.8 mile marker. Where will she be if she walks one tenth of a mile towards camp? one mile towards camp? Explain.
Answer: she will be on 0.18 if she walks one tenth of a mile towards camp and 1.7 if she walks one mile towards the camp.

Explanation:
Given, 1.8
8 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 1.7
The reducing mile as reaching out for camp is 1.7
If she will be on 0.18 if she walks one tenth of a mile towards camp, That is 1.8 × \(\frac{1}{10}\)
Finally, she will be on 0.18 if she walks one tenth of a mile towards camp and 1.7 if she walks one mile towards the camp.

Problem Solving

Performance Task

Thousandths Chart
The students in Ms. Lowell’s class wrote a thousandths decimal chart on the board. Some of the numbers got erased.
Envision Math Common Core Grade 5 Answers Topic 1 Understand Place Vale 63.12
Answer:

Explanation:
Because, down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same,
except for the last number, while the hundredths increase by 1.

Question 5.
Use Structure Describe the pattern for moving across a row from left to right.
Answer:

Explanation:
Moving from left to right in the rows, tenths stay the same, except for the last number, while the hundredths increase by 1.

You can use structure to decide if decimal numbers are following a pattern.
Envision Math Common Core Grade 5 Answers Topic 1 Understand Place Vale 69.9

Answer:

Explanation:
Above question has the figure, we have the decimal numbers following a pattern of  tenths stay the same, except for the last number, while the hundredths increase by 1.

Question 6.
Be Precise How does the pattern change in the last square of each row?
Answer: As you move down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.

Explanation:

As you move down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.

Question 7.
Use Structure Describe the pattern for moving down a column.
Answer: As you move down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.

Explanation:

As you move down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.

Question 8.
Use Repeated Reasoning Write the missing numbers in the decimal chart above.
Answer:

Explanation:
Because, down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same,
except for the last number, while the hundredths increase by 1.

Question 9.
Use Structure Suppose the students add to the chart. Write the missing numbers in the row and the column below.
Envision Math Common Core Grade 5 Answers Topic 1 Understand Place Vale 70.1
Answer:  The complete chart is as follows,

Explanation:
Because, down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same,
except for the last number, while the hundredths increase by 1.

Topic 1 Fluency Review

Activity

Find a Match

Work with a partner. Point to a clue. Read the clue. Look below the clues to find a match. Write the clue letter in the box above the match. Find a match for every clue.
Envision Math Common Core Grade 5 Answers Topic 1 Understand Place Vale 70.2

Clues
A The sum is between 15,000 and 20,000.
B The difference is less than 10,000.
C The difference is between 41,000 and 42,000.
D The sum is exactly 52,397.
E The difference is between 82,000 and 84,000.
F The sum is greater than 79,000.
G The sum is exactly 52,407,
H The difference is exactly 42,024.

Envision Math Common Core Grade 5 Answers Topic 1 Understand Place Vale 70.3

Answer:

Topic 1 Vocabulary Review

Understand Vocabulary

Glossary

Word List
• base
• equivalent decimals
• expanded form
• exponent
• power
• thousandths
• value

Choose the best term from the Word List. Write it on the blank.

Question 1.
Decimal numbers that name the same part of a whole or the same point on a number line are called ____
Answer: Decimal numbers that name the same part of a whole or the same point on a number line are called Equivalent decimals

Question 2.
The ____ of a digit in a number depends on its place in the number.
Answer: The value of a digit in a number depends on its place in the number.

Question 3.
The product that results from multiplying the same number over and over is a(n) ____ of that number.
Answer:  The product that results from multiplying the same number over and over is a(n) POWER of that number.

Question 4.
A digit in the hundredths place has ten times the value of the same digit in the ___ place.
Answer: A digit in the hundredths place has ten times the value of the same digit in the thousandths place.

Question 5.
In 105, the number 10 is the _____.
Answer: In 105, the number 10 is the BASE.

Draw a line from each number in Column A to the same number in Column B.

Envision Math Common Core Grade 5 Answers Topic 1 Understand Place Vale 72.3

Answer:

Use Vocabulary in Writing

Question 10.
Explain why each 8 in the number 8.888 has a different value. Use one or more terms from the Word List in your explanation.
Answer:  Because of its place value.

Explanation:
Given, 8.888
The first 8 has the place value of tenths, The second 8 has the place value of hundredths and the final 8 has the place value of thousandths.

Set A
pages 5-8
How can you write 7,000 using exponents?
7,000 = 7 × 10 × 10 × 10 = 7 × 103
So, using exponents, you can write 7,000 as 7 × 103

Remember the number of zeros in the product is the same as the exponent.

Find each product.

Question 1.
9 × 101
Answer: 90

Explanation:
9 × 101 = 9 × 10 = 90

Question 2.
8 × 1,000
Answer: 8,000

Explanation:
8 × 10 × 10 × 10 = 8,000

Question 3.
5 × 102
Answer: 5 × 102 = 500

Explanation:
5 × 102 =  5 × 10 × 10  = 500

Question 4.
2 × 105
Answer: 2 × 105  = 2,00,000

Explanation:
2 × 105  = 2 × 10 × 10 × 10 × 10 × 10 = 2,00,000

Set B
pages 9-12
Write the number name and tell the value of the underlined digit for 930,365.
Nine hundred thirty thousand, three hundred sixty-five
Since the 0 is in the thousands place, its value is 0 thousands, or 0.
Use digital tools to solve these and other problems.
Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 76.9

Remember you can find the value of a digit by its place in a number.

Write the number name and tell the value of the underlined digit.

Question 1.
9,000,000
Answer: Nine million

Explanation:
the value of the underlined digit is in millions place.

Question 2.
485,002,000
Answer: four hundred and eighty five million and two thousands

Explanation:
the value of the underlined digit is in millions place.

Question 3.
25,678
Answer: Twenty five thousand and six hundred and seventy eight

Explanation:
the value of the underlined digit is in thousands place.

Question 4.
17,874,000
Answer: seventeen million and eight hundred and seventy four thousand

Explanation:
the value of the underlined digit is in ten millions place.

Set C
pages 13-16, 17-20
A place-value chart can help you write the standard form, expanded form, and number name for a decimal.
Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 76.10
Standard form: 8.026
expanded form: 8 + 2 × \(\frac{1}{100}\) + 6 × \(\frac{1}{1,000}\)
Number name: Eight and twenty-six thousandths

Remember the word and is written for the decimal point.

Question 1.
How can you write 0.044 as a fraction?
How are the values of the two 4s related in 0.044?
Answer: \(\frac{44}{1,000}\) and The first 4 is in hundredths place and last 4 is in thousandths place

Explanation:
\(\frac{44}{1,000}\) = 0.044,
The first 4 is in hundredths place and last 4 is in thousandths place

Write each number in standard form.

Question 2.
eight and fifty-nine hundredths
Answer: 8.59

Question 3.
seven and three thousandths + 4. 3 + 2 × \(\frac{1}{10}\) + 4 × \(\frac{1}{1000}\)
Answer: 11.507

Explanation:
Given, seven and three thousandths + 4. 3 + 2 × \(\frac{1}{10}\) + 4 × \(\frac{1}{1000}\)
7.003 + 4.3 + [ 0.2 + 0.004]
= 11.303 + 0.204
= 11.507

Set D
pages 21-24 Compare. Write >, <, or =.

8. 45 Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 88 8.47
Line up the decimal points. Start at the left to compare. Find the first place where the digits are different.
8.45
8.47 0.05 < 0.07
So, 8.45 < 8.47.

Remember that equivalent decimals, such as 0.45 and 0.450, can help you compare numbers.
Compare. Write >, <, or =.

Question 1.
0.584 Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 88 0.58
Answer: 0.584 > 0.58

Explanation:
Given,  0.584 and 0.58
0.584
0.58
Comparing the numbers after decimals,
4 > 0
So, 0.584 > 0.58

Question 2.
9.327 Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 88 9.236
Answer: 9.327 > 9.236

Explanation:
Given, 9.327 and 9.236
9.327
9.236
Comparing the numbers after decimals,
3 > 2
So, 9.327 > 9.236

Question 3.
5.2 Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 88 5.20
Answer: 5.2 = 5.20

Explanation:
Given, 5.2 and 5.20
5.2
5.20
Comparing the numbers after decimals, we have
All the numbers are equal
So, 5.2 = 5.20

Question 4.
5.643 Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 88 5.675
Answer: 5.643 < 5.675

Explanation:
Given, 5.643 and 5.675
5.643
5.675
Comparing the numbers after decimals, we have
4 < 7
So, 5.643 < 5.675

Question 5.
0.07 Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 88 0.08
Answer: 0.07 < 0.08.

Explanation:
Given,  0.07 and 0.08
0.07
0.08
Comparing the numbers after decimals, we have
7 < 8
So, 0.07 < 0.08.

Set E
pages 25-28
Round 12.087 to the place of the underlined digit.
12.087 Look at the digit following the underlined digit. Look at 7.
Round to the next greater number of hundredths because 7 > 5.
12.087 rounded to the nearest hundredth is 12.09.

Remember that rounding a number means replacing it with a number that tells about how many or how much.
Round each number to the place of the underlined digit.

Question 1.
10.245
Answer: 10.2

Explanation:
Given, 10.245
2 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 10.2
The rounding number is 10.2.

Question 2.
73.4
Answer: 73

Explanation:
Given, 73.4
7 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 73
The rounding number is 73.

Question 3.
0.145
Answer: 0.1

Explanation:
Given, 0.145
1 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 0.1
The rounding number is 0.1.

Question 4.
3.999
Answer: 4

Explanation:
Given, 3.999
9 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 4
The rounding number is 4.

Question 5.
13.023
Answer: 13

Explanation:
2 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 13
The rounding number is 13.

Question 6.
45.398
Answer: 45

Explanation:
3 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 45
The rounding number is 45.

Set F
pages 29-32
Think about these questions to help you look for and use structure to understand and Explain patterns with decimal numbers.

Thinking Habits
• What patterns can I see and describe?
• How can I use the patterns to solve the problem?
• Can I see expressions and objects in different ways?
• What equivalent expressions can I use?
Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 89.1

Remember to check that all of your answers follow a pattern.
Each grid is part of a decimal number chart. Write the missing numbers to complete the grids.

Question 1.
Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 89.2
Answer:

Explanation:
As you move down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same, except for the last number, while the hundredths increase by 1.

Question 2.
Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 89.3
Answer:

Explanation:
As you move down the columns, tenths increase by 1 while the Column 1 hundredths stay the same.
Moving from left to right in the rows, tenths stay the same, except for the last number, while the hundredths increase by 1.

Topic 1 Assessment Practice

Question 1.
Complete the sentences to make true statements.
6 is 100 times ____.
0.06 is 10 times ______.
60 is \(\frac{1}{100}\) of ____.
Answer: 600, 0.6, 0.6

Explanation:
6 × 100 = 600
0.06 × 10 = 0.6
60 × \(\frac{1}{100}\) = 0.6

Question 2.
A national park has eighty thousand, nine-hundred twenty-three and eighty-six hundredths acres of land. Which shows this in standard form?
A. 80,923.086
B. 80,923.68
C. 80,923.806
D. 80,923.86
Answer: A , that is 80,923.086

Explanation:
Because, the number name of 80,923.086 is eighty thousand, nine-hundred twenty-three and eighty-six hundredths

Question 3.
Which numbers have a digit in the ones place that is \(\frac{1}{10}\) the value of the digit in the tens place? Select all that apply.
Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 90 9,077
Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 90 9,884
Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 90 1,303
Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 90 1,055
Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 90 3,222
Answer: 9,884 and  3,222

Explanation:
These numbers 9,884 and  3,222 have a digit in the ones place that is \(\frac{1}{10}\) the value of the digit in the tens place

Question 4.
Mrs. Martin has $7,000 in her savings account. Alonzo has \(\frac{1}{10}\) to as much money in his account as Mrs. Martin. How much money does Alonzo have in his account?
Answer: Amount Alonzo has is $ 700

Explanation:
he amount Martin has in her account is $ 7000
Alonzo has 1/10th the amount Martin has
Therefore Alonzo has \(\frac{1}{10}\) of 7000 ;
= 7000/10 = $ 700
Amount Alonzo has is $ 700 .

Question 5.
Select all the comparisons that are true.
Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 91 04.15 > 4.051
Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 91 1.054 > 1.45
Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 91 5.14 < 5.041
Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 91 5.104 < 5.41
Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 91 5.014 < 5.41
Answer:  04.15 > 4.051, 5.104 < 5.41 and 5.014 < 5.41

Explanation:
Because of their place value ,
04.15 > 4.051
1 > 0
So, 04.15 > 4.051

5.104 < 5.41
1 < 4
So, 5.104 < 5.41

5.014 < 5.41
0 < 4
So, 5.014 < 5.41

Question 6.
Luke shaded 20 squares on his hundredths grid. Bekka shaded 30 squares on her hundredths grid.
Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 91.1
A. Whose grid represents the larger decimal?
B. Write two decimals equivalent to Luke’s decimal.
Answer: 0.2 < 0.3, The two decimals equivalent to Luke’s decimal are 0.20 and 0.21

Explanation:
Given, Luke shaded 20 squares on his hundredths grid, \(\frac{20}{100}\) that is 0.2
Bekka shaded 30 squares on her hundredths grid. \(\frac{30}{100}\)that is 0.3
Bekka has largest decimal.

The two decimals equivalent to Luke’s decimal are 0.20 and 0.21

Question 7.
Which statements about the values of 2.044 and 20.44 are true? Select all that apply.
Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 93 2.044 is \(\frac{1}{10}\) of 20.44.
Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 93 2.044 is \(\frac{1}{100}\) of 20.44.
Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 93 20.44 is 10 times 2.044.
Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 93 20.44 is 100 times 2.044.
Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 93 2.044 is 10 times 20.44.
Answer:  20.44 is 10 times 2.044.

Explanation:
Given, 20.44 is 10 times 2.044.
20.44 × 10 = 2.044.

Question 8.
The weight of Darrin’s phone is 3.405 ounces. What is 3.405 written in expanded form?
A. 3 × 1 + 4 × \(\frac{1}{10}\) + 5 × \(\frac{1}{1,000}\)
B. 3 × 10 + 4 × \(\frac{1}{10}\) + 5 × \(\frac{1}{1,000}\)
C. 3 × 10 + 4 × \(\frac{1}{10}\) + 5 × \(\frac{1}{100}\)
D. 3 × 1 + 4 × \(\frac{1}{100}\) + 5 × \(\frac{1}{1,000}\)
Answer:  D

Explanation:
Given, 3.405
The expanded form, 3 × 1 + 4 × \(\frac{1}{100}\) + 5 × \(\frac{1}{1,000}\)
= 3 +0.04 + 0.005
= 3.405

Question 9.
Elaine has a piece of wire that is 2.16 meters long. Dikembe has a piece of wire that is 2.061 meters long. Whose piece of wire is longer? How can you tell?
Answer: Dikembe has the longest wire.

Explanation:
Given, 2.16 and 2.061
2.16
2.061
Comparing the place values of the digits after decimal
1 > 0
So, 2.16 > 2.061
Dikembe has the longest wire.

Question 10.
In a basketball tournament, Dimitri averaged 12.375 rebounds per game. What is 12.375 written in expanded form? How is it written with number names?
Answer: The expanded form is (12 × 1) + {3 × \(\frac{1}{10}\) + 7 × \(\frac{1}{100}\) + 5 × \(\frac{1}{1,000}\)}

Explanation:
Given, 12.375
The expanded form is (12 × 1) + {3 × \(\frac{1}{10}\) + 7 × \(\frac{1}{100}\) + 5 × \(\frac{1}{1,000}\)}
12 + 0.3 + 0.07 + 0.005
= 12.375.
The number name is twelve thousand and three hundred and seventy five.

Question 11.
The numbers below follow a pattern.
0.006 0.06 0.6 6 ____ _____
A. What are the next two numbers in the pattern?
B. What is the relationship between the terms in the pattern?
Answer:  60 and 600 , The terms relate to each other in that the next term is 10 times the previous term.

Explanation:
Relationship between term: The next term is ten times the previous term.
0.006 = 6/1000
0.06 = 6/100
0.6 = 6/10
6 = 6/1
This is a geometric sequence, where the common ratio is 10. 0.6/0.06 = 6/0.6 = 0.06/0.006 = 10
To get next number we simply multiply the previous number by 10.
The terms relate to each other in that the next term is 10 times the previous term. 

Question 12.
Kendra and her horse completed the barrel racing course in 15.839 seconds. What is this number rounded to the nearest tenth? Explain how you decided.
Answer: The rounding number is 16.

Explanation:
Given, 15.839
5 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 16
The rounding number is 16.

Topic 1 Performance Task

Fruits and Vegetables
Henry recorded how many pounds of fruits and vegetables his family bought during the past two months.

Question 1.
Pick four fruits and list them in the table below.
Part A
Round each fruit’s weight to the nearest 0.1 pound. Write the rounded weight in the next column.
Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 94.1

Answer:

Part B
Explain how you rounded the weights of the fruits.
Answer:  detailed explanation is given below.

Explanation:
For Apples, given 2.068
5 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 2
The rounding number is 2.

For Blueberries 1.07
0 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 1.1
The rounding number is 1.1.

For lemons 1.031
0 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 1
The rounding number is 1.

For oranges 3.502
0 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 3.5
The rounding number is 3.5.

For peaches 2.608
0 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 2.6
The rounding number is 2.6.

For pears 3.592
5 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 3.6
The rounding number is 3.6.

Question 2.
Pick four vegetables and list them in the table below.
Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 94.2

Part A
Round each vegetable’s weight to the nearest 0.01 pound. Write the rounded weight in the next column.
Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 95.2

Answer:

Part B
Explain how you rounded the weights of the vegetables.
Answer: Detailed explanation is given below

Explanation:

For Beets 1.862
6 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 1.86
The rounding number is 1.86.

For Celery 1.402
0 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 1.4
The rounding number is 1.4.

For Corn 2.556
5 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 2.56
The rounding number is 2.56.

For potatoes 3.441
4 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 3.44
The rounding number is 3.44.

Question 3.
Use <, >, or= to compare the weights of blueberries and lemons.
Answer:

Question 4.
When rounded to the nearest hundredth, two items will round to the same decimal. What two items are they?
Answer:

Question 5.
How does writing the weight for potatoes in expanded form show why the same digit can have different values?
Answer: Because of the place value of the numbers in the decimals

Explanation:
For potatoes 3.441
4 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 3.44
The rounding number is 3.44.

The first 4 has the place value of tenths and the last 4 has the place value of hundredths.

Envision Math Common Core 5th Grade Answer Key Topic 1 Understand Place Vale 100

Question 6.
What is the relationship between the values of the two 4s in the weight of the potatoes?
Answer: The first 4 has the place value of tenths and the last 4 has the place value of hundredths. and the first 4 is ten times the value of second 4.

Explanation:

For potatoes 3.441
4 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 3.44
The rounding number is 3.44.

The first 4 has the place value of tenths and the last 4 has the place value of hundredths.
and the first 4 is ten times the value of second 4.

Question 7.
Write the number of pounds of celery Henry’s family bought using number names and in expanded form.
Answer: The number name is 1.4 pounds
The expanded form is 14 × \(\frac{1}{10}\)

Explanation:
For Celery 1.402
0 is the rounding place, the digit is less than 5, leave the rounding digit alone.
Drop the digits to the right of the decimal point, that is 1.4
The rounding number is 1.4.

The number name is 1.4 pounds
The expanded form is 14 × \(\frac{1}{10}\)

Question 8.
The store where Henry’s family shops sold 103 times as many pounds of corn as Henry’s family bought.
Answer:263.68 pounds have sold.

Explanation:
For Corn 2.556
5 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 2.56
The rounding number is 2.56.

Given, Henry’s family shops sold 103 times as many pounds of corn as Henry’s family bought.
2.56 × 103 = 263.68

Part A
How many pounds of corn did the store sell? Write your answer in standard form and with number names.

Answer: The number form is twenty six thousand and three hundred and sixty eight.

Explanation:
Store has sold 263.68
The number form is twenty six thousand and three hundred and sixty eight.

Part B
Explain how you found your answer.
Answer:  detailed explanation is given below

Explanation:
For Corn 2.556
5 is the rounding place, the digit is 5 or greater, add 1 to the rounding digit.
Drop the digits to the right of the decimal point, that is 2.56
The rounding number is 2.56.

Given, Henry’s family shops sold 103 times as many pounds of corn as Henry’s family bought.
2.56 × 103 = 263.68