## Envision Math 6th Grade Textbook Answer Key Topic 2.7 Using Expressions to Describe Patterns

Using Expressions to Describe Patterns

How can you write expressions to describe patterns?
Delvin saves a part of everything he earns. The table at the right shows Delvin’s savings pattern.
The INPUT column shows the money he has earned.
The OUTPUT column shows the money he has saved.

Question.
Write an expression to describe the pattern.

Guided Practice

Do you know HOW?
Use the input/output table for 1 and 2.

Question 1.
If the input number is 8, what is the output number?
11
Explanation:

Question 2.
Write an algebraic expression that describes the output pattern.
x + 3
Explanation:

Do you UNDERSTAND?
Question 3.
Suppose that Delvin earned $36 mowing lawns. What input and output entries would you add to his table? Answer: IN:$36; OUT: $18 Explanation: Question 4. Reasonableness Is it reasonable for an output to be greater than the input in the table above? Explain. Answer: See margin. Explanation: Question 5. What is the algebraic expression that describes the output pattern for the table above if the input is x? Answer: $$\frac{1}{2}$$x. Explanation: Independent Practice Use this table for 6 and 7. Question 6. What is the cost of 4 lb, 5 lb, and 10 lb of apples? Answer: Question 7. Write an algebraic expression that describes the output pattern if the input is a variable a. Answer: 2a Explanation: Use this table for 8 and 9. Question 8. Copy and complete the table. Answer: Question 9. Write an algebraic expression that describes the relationship between the input and output values. Answer: x ÷ 3 Explanation: An input/output table is a table of related values. Identify the pattern. What is the relationship between the values? $$\frac{1}{2}$$ (84) = 42 → 42 is half of 84 $$\frac{1}{2}$$ (66) = 33 → 33 is half of 66 $$\frac{1}{2}$$ (50) = 25 → 25 is half of 50. The pattern is: $$\frac{1}{2}$$ (INPUT) = OUTPUT Let x= INPUT. So, the pattern is $$\frac{1}{2}$$ x. Use the pattern to find the missing values. $$\frac{1}{2}$$ (22) = 11 $$\frac{1}{2}$$ (30) = 15 Problem Solving Use the input/output table at right for 10 and 11. Question 10. Hazem keeps $$\frac{1}{3}$$ of the tips he earns. Also, he gets$1 each night to reimburse his parking fee. This information is shown in the input/output table. Write an algebraic expression that describes the output pattern if the input is the variable k.
(k ÷ 3) + 1 or $$\frac{1}{3}$$ k + 1
Explanation:

Question 11.
How much money would Hazem keep in a night if he takes in $36 in tips? Answer:$13
Explanation:

Use the input/output table at right for 12 and 13.

Question 12.
Ms. Windsor’s classroom has a tile floor. The students are making stars to put in the center of 4-tile groups. This input/ output chart shows the pattern. Write an algebraic expression that describes the output pattern if the input is the variable t.
t ÷ 4
Explanation:

Question 13.
Writing to Explain There are 30 rows with 24 tiles in each row on a floor. Explain how to find the number of stars needed to complete the pattern for the floor.
See margin.
Explanation:

Use the table at right for 14.

Question 14.
Think About the Process Which algebraic expression shows the cost of a chosen number of books b?
A. b + $2.50 B.$2.50b
C. $b –$2.50
D. b + $2.50 Answer: B.$2.50b

## Envision Math 6th Grade Textbook Answer Key Topic 2.5 Mental Math

Mental Math

How can you break apart numbers to compute mentally?
Jo has to read 45 history pages and 46 science pages by the end of next week.

Question.
How many total pages must Jo read?

Another Example
What other strategies can you use to compute mentally?
Look for compatible numbers and use properties of operations to compute mentally.

Use compensation to create compatible numbers that are easy to compute mentally.

Explain It
Question 1.
When you use compensation to subtract, why must you add to the difference you find?
You must add back to the difference the extra amount you subtracted.
Explanation:

Question 2.
When you use compensation to multiply, which property are you using?
Distributive Property
Explanation:

Use mental math.

Combine the sums. 80 + 11 = 91
So, 45 + 46 = 91.

Explain why breaking apart the numbers works.
40 + 5) + (40 + 6) ← Break apart numbers.
(40 + 40) + (5 + 6) ← Use Commutative and Associative Properties.
Jo has 91 pages to read.

Guided Practice

Do you know HOW?
Compute mentally.
Question 1.
89 + 32 + 8
(32 + 8) + 89 = 129
Explanation:

Question 2.
76 + 59 + 6
See margin.
Explanation:

Question 3.
2 × 9 × 20
(2 × 20)9 = 360
Explanation:

Question 4.
5 × 31 × 2
(5 × 2)31 = 310
Explanation:

Question 5.
8 × 39
See margin.
Explanation:

Question 6.
48 + 52
See margin.
Explanation:

Question 7.
453 – 397
See margin.
Explanation:

Question 8.
6(42)
6(40) + 6(2) = 252
Explanation:

Do you UNDERSTAND?
Question 9.
When you use the Compensation Strategy in a problem such as 5 x 698, you turn one multiplication problem into two and then combine them.
Why does this idea make sense?
See margin.
Explanation:

Question 10.
Jo has to read 2 science chapters for each of the next 5 weeks. Each chapter is 45 pages long. How many pages does she have to read?
2 × 5 × 45 = (2 × 5)45 = 450
Explanation:

Independent Practice

Compute mentally.
Question 11.
10 + 23 +130
10 + 130 + 23 = 163
Explanation:

Question 12.
721 – 395
(721 + 5) – (395 + 5) = 326
Explanation:

Question 13.
2 × 38 × 5
(2 × 5)38 = 380
Explanation:

Question 14.
28 + 26 + 32 + 14
See margin.
Explanation:

Question 15.
5 × 3 × 40
(5 × 40)3 = 600
Explanation:

Question 16.
856 – 403
See margin.
Explanation:

Question 17.
6(69)
6(70) – 6(1) = 414
Explanation:

Question 18.
80 × 10 × 5
(80 × 10)5 = 4,000
Explanation:

Question 19.
44 + 56
See margin
Explanation:

Question 20.
840 + 260 + 72
See margin.
Explanation:

Question 21.
495 + 75 + 14
See margin
Explanation:

Question 22.
397 + 255
See margin.
Explanation:

Question 23.
8(82)
8(80) + 8(2) = 656
Explanation:

Question 24.
4 × 5 × 25
(4 × 25) × 5 = 500
Explanation:

Problem Solving

Use the data table for 25 and 26. Find the answers mentally.

Question 25.
How much did Jacob make during the first three weeks?
$45 +$32 + $55 = ($45 + $55) +$32 = $100 +$32 = $132 Explanation: Question 26. Jamal earned$5 for every dollar Jacob earned during week 4 on his paper route. How much money did Jamal earn?
$5($64) = $5($60 + $4) =$300 + $20 =$320
Explanation:

Question 27.
Writing to Explain Avis swam her first lap in 32 seconds and her second lap in 45 seconds. Explain the steps you can use to mentally calculate her total time.
See margin.
Explanation:

Question 28.
Use the break-apart strategy to mentally solve 81 + 43 + 2.
(80 + 1) + (40 + 3) + 2 = 120 + 6 = 126
Explanation:

Question 29.
Draw It Copy and complete the bar diagram to show how to use the Distributive Property to mentally compute the problem below.

Four friends are having a snack. They each have 8 strawberries and 22 blueberries. How many berries do they have altogether?
The friends have 4 × 30 = 120 berries in all.
Explanation:

Question 30.
Number Sense What kind of numbers would make 3-factor multiplication problems easy to do mentally?
Numbers in which two of them combine to a multiple of 10
Explanation:

Question 31.
Writing to Explain Explain how you can use the Distributive Property to multiply 6 and 82.
See margin.
Explanation:

One of the world’s tallest fountains shoots water 171 m into the air once per hour. Use this information for 32 and 33.

Question 32.
Think About the Process Which expression shows a break-apart strategy to mentally calculate the total of the heights of the water this fountain shoots in the air in 5 hours?
A. 171 × 5
B. 5(100 × 71)
C. 170 × 5 + 1
D. 5(100 + 70 + 1)
D. 5(100 + 70 + 1)
Explanation:

Question 33.
The fountain sprays water for 15 minutes during every hour between 10:00 A.M. and 9:00 P.M. If the fountain sprayed water 10 times higher than it does, how high would the water shoot up? Solve mentally, and then check your answer.
1,710m
Explanation:

Mixed Problem Solving

Counting Calories
Calories come from the food you eat. Your body converts those calories into the energy you need every day.
After class, the sixth grade had the snack foods shown in the chart. For 1 through 5, write an expression to show how many calories the person ate. Then, evaluate each expression.

Tips: Make sure to watch the units on the chart compared to the units in the problems.

Question 1.
Wen-Wei ate a peach and an apple for lunch. After class, he ate 2 cups of celery and 8 tablespoons of peanuts. How many calories did he eat after class?
2(20) + 8(50) = 440 calories
Explanation:

Question 2.
Andrea ate 3 tablespoons of Brazil nuts. Then she ate 2 more tablespoons. Later, she ate another 4 tablespoons.
3(60) + 2(60) + 4(60) = 540 calories
Explanation:

Question 3.
Gloria ate 2 ounces of low fat cottage cheese, a cup of carrots, and 6 tablespoons of cashews. Then, she went back and got 4 more tablespoons of cashews and an orange. She ate everything but half of the orange.
See margin
Explanation:

Question 4.
Ayesha ate 4 cups of celery, 3 oranges, and 2 ounces of Swiss cheese.
4(20) + 3(60) + 2(105) = 470 calories
Explanation:

Question 5.
Daryl took 2 oz dried apricots, 4 tbsp almonds, and 2 cups of pretzels. He ate all of it, except for 1 oz of apricots.
2(40) + 4(55) + 2(60) – 40 = 380 calories
Explanation:

For 6 and 7, write an expression that describes the situation, and then evaluate the expression.
Question 6.
Paulo took 4 bananas, 6 ounces of dates, 4 ounces of cheddar cheese, and 2 ounces of Swiss cheese. He split it all evenly with Alex. How many calories did Paulo eat?
(4(105) + 6(70) + 4(105) + 2(105)) ÷ 2 = 735 calories
Explanation:

Question 7.
Kyle filled a basket with 10 cups of plain popcorn. He shared it equally with 4 other people. Then, he got an apple and shared it equally with 1 other person. On his last trip to the snack table, he took 3 ounces of raisins, but he tripped and dropped 1 ounce of them. How many calories did Kyle eat?
See margin.
Explanation:

## Envision Math 6th Grade Textbook Answer Key Topic 2.3 Order of Operations

Order of Operations

How do you know which operation to perform first?
Evaluate 14 + 8 × 6.

Other Examples
Evaluate 20 + (30 – 10) ÷ 5.
Using order of operations:
20 + (30 – 10) ÷ 5 ← Compute inside the parentheses first.
20 + 20 ÷ 5 ← Next, divide.
20 + 4 ← Finally, add.
24

Using a scientific calculator:
Press:

Use the order of operations to evaluate.

Evaluate 42 – (4 + 6) ÷ 2.
Using order of operations:
42 – (4 + 6) ÷ 2 ← Compute inside theparentheses first.
42 – 10 ÷ 2 ← Evaluate exponents.
16 – 10 ÷ 2 ← Then divide.
16 – 5 ← Finally, subtract.
11

Using a scientific calculator:
Press:

Explain It
Question 1.
To evaluate 3 × (7 + 5), what should you do first and why?
First, add 7 + 5 since that operation is inside the parentheses.
Explanation:

Question 2.
In the second example using the scientific calculator, what is the purpose of the key?
It is the key that is used to evaluate a base number raised to the power of an exponent.
Explanation:

Mathematicians use a set of rules known as order of operations , the order in which to perform operations in calculations.
1. Compute inside parentheses.
2. Evaluate terms with exponents.
3. Multiply and divide from left to right.
4. Add and subtract from left to right.

Using the correct order of operations, 14 + 8 × 6 = 62.
A scientific calculator uses order of operations.

Guided Practice

Do you know HOW?
Evaluate each expression.
Question 1.
36 ÷ 6 + 6
12
Explanation:

Question 2.
36 ÷ (6 + 6)
3
Explanation:

Question 3.
24 ÷ (4 + 8) + 2
4
Explanation:

Question 4.
48 ÷ (4 + 8) + 22
8
Explanation:

Question 5.
24 ÷ 4 + 8 + 2
16
Explanation:

Question 6.
48 ÷ 4 + 8 + 22
24
Explanation:

Do you UNDERSTAND?
Question 7.
Where could you insert parentheses to make this number sentence true? 80 ÷ 8 × 5 + 4= 90
See margin.
Explanation:

Question 8.
Donavan entered 12 + 4 × 36 into Lidia’s scientific calculator.The display showed 18. In what order did the calculator complete the operations?
See margin.
Explanation:

Independent Practice

Evaluate each expression.
Question 9.
33 – 8 × 3
3
Explanation:

Question 10.
(52 + 7) ÷ 4
8
Explanation:

Question 11.
6 × 4 – 4 + 2
8
Explanation:

Question 12.
18 – 3 × 5 + 2
5
Explanation:

Question 13.
49 – 4 × (49 ÷ 7)
21
Explanation:

Question 14.
(64 ÷ 8) × 3 + 6
30
Explanation:

Question 15.
72 ÷ (4 + 4) × 5
45
Explanation:

Question 16.
(3 × 3) × (2 × 2) ÷ 36
1
Explanation:

Use parentheses to make each number sentence true.
See margin.
Explanation:

Question 17.
5 + 4 × 3 × 3 = 41

Question 18.
9 × 0 + 4 = 36

Question 19.
52 – 6 × 0 = 25

Question 20.
8 × 9 – 2 – 3 = 32

Question 21 .
5 + 4 × 3 × 3 = 81

Question 22.
9 × 0 + 4 = 4

Question 23.
52 – 6 × 0 = 0

Question 24.
8 × 9 – 2 – 3 = 53

Question 25.
1 + 2 × 3 + 4 = 21

Question 26.
22 + 4 × 6 = 48

Question 27.
5 × 6 × 8 – 7 = 30

Question 28.
62 + 7 + 9 × 10 = 133

Question 29.
Number Sense Use the symbols +, -, ×, and ÷ to make the number sentence true.

Explanation:

Question 30.

Explanation:
Given Adult admission to the dog show is $16 and Children’s admission is$9 so for $16 for 3 Adults and 2 Children it will be 3 X$16 + 2 X $9 =$48 + $18 =$66, So the total cost will be $66 to enter the dog show. Question 13. Meg saved coins she found for a year. She found a total of 95 pennies, 13 nickels, 41 dimes, and 11 quarters. She would like to evenly divide the coins into 4 piggy banks. How many coins will go in each piggy bank? Answer: 40 coins will go in each piggy bank, Explanation: Given Meg saved coins she found for a year. She found a total of 95 pennies, 13 nickels, 41 dimes, and 11 quarters. She would like to evenly divide the coins into 4 piggy banks. So first total number of coins Meg have are 95 + 13 + 41 + 11 = 160 coins, As coins are evenly divide the coins into 4 piggy banks, each piggy bank will have 160/4 = 40 coins respectively. Pick a Project PROJECT 9A How do you follow a recipe? Project: Exploring Recipes Answer: Follow a recipe 1. Read the recipe. Take a good look at the recipe, 2. Know the assumptions, 3. Figure out the timing, 4. Plan ahead, 5. Bone up on new techniques, 6. Lay out our tools, too, 7. Make notes or highlight, Exploring Cookie Recipe: PROJECT 9B Would you like to be a code breaker? Project: Create a Fraction Code PROJECT 9C What is a farmers’ market? Project: Write and Perform a Skit A farmers’ market is a physical retail market place intended to sell foods directly by farmers to consumers. Farmers’ markets may be indoors or outdoors and typically consist of booths, tables or stands where farmers sell their produce, live animals and plants, and sometimes prepared foods and beverages. Farmers’ markets exist in many countries worldwide and reflect the local culture and economy. The size of the market may be just a few stalls or it may be as large as several city blocks. Due to their nature, they tend to be less rigidly regulated than retail produce shops. They are distinguished from public markets which are generally housed in permanent structures, open year-round, and offer a variety of non-farmer/non-producer vendors, packaged foods and non-food products. 3-ACT MATH PREVIEW Math Modeling Just Add Water I can … model with math to solve a problem that involves estimating, adding fractions and mixed numbers, and comparing quantities. ### Lesson 9.1 Model Addition of Fractions Solve & Share Kyle and Jillian are working on a sports banner. They painted $$\frac{3}{8}$$ of the banner green and $$\frac{4}{8}$$ purple. How much of the banner have they painted? Solve this problem any way you choose. I can … use tools such as fraction strips or area models to add fractions. Look Back! Use Appropriate Tools Kyle says $$\frac{1}{8}+\frac{1}{8}+\frac{1}{8}=\frac{3}{8}$$. Jillian says $$\frac{1}{8}+\frac{1}{8}+\frac{1}{8}=\frac{3}{24}$$. Use a tool to decide who is correct and explain. Answer: Kyle and Jillian painted $$\frac{7}{8}$$ banner, Kyle is correct, Explanation: Given Kyle and Jillian are working on a sports banner. They painted $$\frac{3}{8}$$ of the banner green and $$\frac{4}{8}$$ purple. Therefore the banner have they painted is $$\frac{3}{8}$$ + $$\frac{4}{8}$$ used area model to add fractions as shown above. Given Kyle says $$\frac{1}{8}+\frac{1}{8}+\frac{1}{8}=\frac{3}{8}$$. Jillian says $$\frac{1}{8}+\frac{1}{8}+\frac{1}{8}=\frac{3}{24}$$ but Kyle is correct as $$\frac{1}{8}+\frac{1}{8}+\frac{1}{8} is \frac{3}{8}$$ not Jillian says $$\frac{1}{8}+\frac{1}{8}+\frac{1}{8} ≠ \frac{3}{24}$$ as $$\frac{1}{8}+\frac{1}{8}+\frac{1}{8} is \frac{3}{8}$$ because as denominators are same we add numerators only so Jillian is incorrect. Essential Question How Can You Use Tools to Add Fractions? We can use tools such as fraction strips to add two or more fractions, Explanation: Visual Learning Bridge Ten canoeing teams are racing downriver. Five teams have silver canoes and two teams have brown canoes. What fraction of the canoes are either silver or brown? You can use tools such as fraction strips to add two or more fractions. Convince Me! Make Sense and Persevere What two fractions would you add to find the fraction of the canoes that are either green or brown? What is the sum? How do you know your sum is correct? Guided Practice Do You Understand? Question 1. In the problem on the previous page, would you get the same answer if you used an area model instead of fraction strips or a number line? Explain. Answer: Question 2. What two fractions are being added below? What is the sum? Answer: The two fractions are $$\frac{2}{8}$$ and $$\frac{3}{8}$$, The sum is $$\frac{5}{8}$$, Explanation: Given two fractions are $$\frac{2}{8}$$ and $$\frac{3}{8}$$, as denominators are same 8 we add numerators 2 and 3 we get 5, therefore sum is $$\frac{5}{8}$$. Do You Know How? For 3-4, find each sum. Question 3. Answer: $$\frac{3}{5}$$, Explanation : As $$\frac{2}{5}$$ and $$\frac{1}{5}$$ are like fraction as denominators are same , we can add numerators 2 and 1 = 3, therefore $$\frac{2}{5}$$+ $$\frac{1}{5}$$ =$$\frac{3}{5}$$. Question 4. Answer: $$\frac{2}{6}$$, Simplest form : $$\frac{1}{3}$$, Explanation : As $$\frac{1}{6}$$ and $$\frac{1}{6}$$ are like fraction, we can add numerators 1 and 1 so $$\frac{2}{6}$$ as it can be further simplified 2 and 6 goes in 3 table so answer in simplest form $$\frac{1}{3}$$. Independent Practice Leveled Practice For 5-16, find each sum. Use a tool as needed. Question 5. Answer: $$\frac{7}{12}$$, Explanation: As $$\frac{3}{12}$$ and $$\frac{4}{12}$$ are like fraction as denominators are same , we can add numerators 3 and 4 as 7, therefore $$\frac{3}{12}$$ + $$\frac{4}{12}$$ =$$\frac{7}{12}$$. Question 6. Answer: $$\frac{5}{10}$$, Simplest form = $$\frac{1}{2}$$, Explanation: As $$\frac{4}{10}$$ and $$\frac{1}{10}$$ are like fraction as denominators are same, we can add numerators 4 and 1 as 5, therefore $$\frac{4}{10}$$+ $$\frac{1}{10}$$ =$$\frac{5}{10}$$. As $$\frac{5}{10}$$ can be further simplified because 5 and 10 goes in 5 table so answer in the simplest form is $$\frac{1}{2}$$. Question 7. Answer: $$\frac{6}{12}$$, Simplest form = $$\frac{1}{2}$$, Explanation: As $$\frac{2}{12}$$ and $$\frac{4}{12}$$ are like fraction as denominators are same, we can add numerators 2 and 4 = 6, therefore $$\frac{2}{12}$$+ $$\frac{4}{12}$$ =$$\frac{6}{12}$$. As $$\frac{6}{12}$$ can be further simplified because 6 and 12 goes in 6 table so answer in the simplest form is $$\frac{1}{2}$$. Question 8. Answer: $$\frac{6}{6}$$, Simplest form = $$\frac{1}{1}$$ = 1, Explanation: As $$\frac{1}{2}$$, $$\frac{2}{6}$$ and $$\frac{3}{6}$$ are like fraction as denominators are same , we can add numerators 1, 2 and 3 = 6, therefore $$\frac{1}{2}$$+ $$\frac{2}{6}$$ + $$\frac{3}{6}$$ =$$\frac{6}{6}$$. As $$\frac{6}{6}$$ can be further simplified because 6 and 6 can be cancelled so answer in the simplest form is $$\frac{1}{1}$$ = 1. Question 9. Answer: $$\frac{3}{4}$$, Explanation: As $$\frac{1}{4}$$ and $$\frac{2}{4}$$ are like fraction as denominators are same , we can add numerators 1 and 2 =3, therefore $$\frac{1}{4}$$ + $$\frac{2}{4}$$ = $$\frac{3}{4}$$. Question 10. Answer: $$\frac{2}{3}$$, Explanation: Given $$\frac{1}{3}$$ and $$\frac{1}{3}$$ are like fraction as denominators are same , we can add numerators 1 and 1 = 2, therefore $$\frac{1}{3}$$ + $$\frac{1}{3}$$ = $$\frac{2}{3}$$. Question 11. $$\frac{5}{8}+\frac{1}{8}$$ Answer: $$\frac{6}{8}$$ or $$\frac{3}{4}$$, Explanation: As $$\frac{5}{8}$$ and $$\frac{1}{8}$$ are like fraction as denominators are same, we can add numerators 5 and 1 = 6, therefore $$\frac{5}{8}$$ + $$\frac{1}{8}$$ = $$\frac{6}{8}$$, As $$\frac{6}{8}$$ can be further simplified because 6 and 8 can be cancelled by 2, we get 3 X 2 =6 and 4 X 2 = 8, so answer in the simplest form is $$\frac{3}{4}$$. Question 12. $$\frac{1}{4}+\frac{3}{4}$$ Answer: $$\frac{4}{4}$$ or 1, Explanation: As $$\frac{1}{4}$$ and $$\frac{3}{4}$$ are like fraction as denominators are same, we can add numerators 1 and 3 = 4, therefore $$\frac{1}{4}$$ + $$\frac{3}{4}$$ = $$\frac{4}{4}$$, As $$\frac{4}{4}$$ can be further simplified because 4 and 4 can be cancelled by 4, so answer in the simplest form is 1. Question 13. $$\frac{7}{12}+\frac{2}{12}$$ Answer: $$\frac{9}{12}$$ or $$\frac{3}{4}$$, Explanation: As $$\frac{7}{12}$$ and $$\frac{2}{12}$$ are like fraction as denominators are same, we can add numerators 7 and 2 = 9, therefore $$\frac{7}{12}$$ + $$\frac{2}{12}$$ = $$\frac{9}{12}$$, As $$\frac{9}{12}$$ can be further simplified because 9 and 12 can be cancelled by 3, 3 X 3 = 9 and 4 X 3 = 12, we get (3,4) so answer in the simplest form is $$\frac{3}{4}$$. Question 14. $$\frac{1}{4}+\frac{1}{4}$$ Answer: $$\frac{2}{4}$$ or $$\frac{1}{2}$$, Explanation: As $$\frac{1}{4}$$ and $$\frac{1}{4}$$ are like fraction as denominators are same, we can add numerators 1 and 1 = 2, therefore $$\frac{1}{4}$$ + $$\frac{1}{4}$$ = $$\frac{2}{4}$$, As $$\frac{2}{4}$$ can be further simplified because 2 and 4 can be cancelled by 2, 1 X 2 = 2 and 2 X 2 = 4, we get (1,2) so answer in the simplest form is $$\frac{1}{2}$$. Question 15. $$\frac{2}{5}+\frac{2}{5}$$ Answer: $$\frac{4}{5}$$, Explanation: As $$\frac{2}{5}$$ and $$\frac{2}{5}$$ are like fraction as denominators are same, we can add numerators 2 and 2 = 4, therefore $$\frac{2}{5}$$ + $$\frac{2}{5}$$ = $$\frac{4}{5}$$. Question 16. $$\frac{1}{10}+\frac{2}{10}+\frac{1}{10}$$ Answer: $$\frac{4}{10}$$ or $$\frac{2}{5}$$, Explanation: As $$\frac{1}{10}$$, $$\frac{2}{10}$$ and $$\frac{1}{10}$$ are like fraction as denominators are same, we can add numerators 1,2 and 1 = 4, therefore $$\frac{1}{10}$$ + $$\frac{2}{10}$$ + $$\frac{1}{10}$$ = $$\frac{4}{10}$$, As $$\frac{4}{10}$$ can be further simplified because 4 and 10 can be cancelled by 2, 2 X 2 = 4 and 2 X 5 = 10, we get (2,5) so answer in the simplest form is $$\frac{2}{5}$$. Problem Solving Question 17. Number Sense Using three different numerators, write an equation in which three fractions, when added, have a sum of 1. Answer: $$\frac{1}{6}$$, $$\frac{2}{6}$$ and $$\frac{3}{6}$$, As $$\frac{1}{6}$$ + $$\frac{2}{6}$$ + $$\frac{3}{6}$$ = $$\frac{6}{6}$$, Simplest form = $$\frac{1}{1}$$ = 1, Explanation: As $$\frac{1}{6}$$, $$\frac{2}{6}$$ and $$\frac{1}{6}$$ are like fraction as denominators are same , we can add numerators 1, 2 and 3 = 6, therefore $$\frac{1}{6}$$+ $$\frac{2}{6}$$ + $$\frac{3}{6}$$ = $$\frac{6}{6}$$. As $$\frac{6}{6}$$ can be further simplified because 6 and 6 can be cancelled so answer in the simplest form is $$\frac{1}{1}$$ = 1. Question 18. Use Appropriate Tools Diane added $$\frac{3}{8}$$ to $$\frac{1}{8}$$. Draw a picture to show $$\frac{1}{8}$$ + $$\frac{3}{8}$$ = $$\frac{4}{8}$$. Answer: $$\frac{4}{8}$$= $$\frac{1}{2}$$, Explanation : $$\frac{3}{8}$$ + $$\frac{1}{8}$$ = $$\frac{4}{8}$$ further it can be simplified as $$\frac{1}{2}$$ as 4 and 8 goes in 4 table. Question 19. A bakery sells about 9 dozen bagels per day. About how many bagels does the bakery sell in a typical week? Explain. Answer: 756 bagels, Explanation: 1 dozen = 12, so 9 dozen = 9 times 12 = 108, so the bakery sells 108 bagels per day. now to calculate for a typical week one week = 7 days, so bagels sold in a week = 108 times 7 = 756 bagels. Therefore the bakery sells in a typical week about 756 bagels. Question 20. During a field trip to a baseball game, $$\frac{3}{8}$$ of the students are wearing red caps and $$\frac{3}{8}$$ are wearing blue caps. Write and solve an equation to find the number of students, s, who are wearing red or blue caps. Answer: Equation: $$\frac{3}{8}$$ + $$\frac{3}{8}$$ + (Number of red + Number of blues) = s, Number of red and Number of blues wearing caps are $$\frac{2}{8}$$ or $$\frac{1}{4}$$, Explanation: Given during a field trip to a baseball game, $$\frac{3}{8}$$ of the students are wearing red caps and $$\frac{3}{8}$$ are wearing blue caps. Equation to find the number of students, s, who are wearing red or blue caps is $$\frac{3}{8}$$ + $$\frac{3}{8}$$ + (Number of red + Number of blues) = s, as s = 1, Substituting above we get $$\frac{3}{8}$$ + $$\frac{3}{8}$$ + (Number of red + Number of blues) = 1, solving above (Number of red + Number of blues) = 1 – ($$\frac{3}{8}$$ + $$\frac{3}{8}$$), (Number of red + Number of blues) = 1 – $$\frac{6}{8}$$, (Number of red + Number of blues) = $$\frac{8-6}{8}$$, (Number of red + Number of blues) = $$\frac{2}{8}$$ or $$\frac{1}{4}$$. Question 21. Higher Order Thinking Terry ran $$\frac{1}{10}$$ of the distance from school to home. He walked more $$\frac{3}{10}$$ of the distance and then skipped $$\frac{2}{10}$$ more of the distance. What fraction of the distance home does Terry still have to go? Answer: The fraction of the distance home does Terry still have to go is $$\frac{8}{10}$$ or $$\frac{4}{5}$$, Explanation: Given Terry ran $$\frac{1}{10}$$ of the distance from school to home. He walked more $$\frac{3}{10}$$ of the distance and then skipped $$\frac{2}{10}$$ more of the distance. So let x be fraction of the distance home does Terry still have to go therefore $$\frac{1}{10}$$ + $$\frac{3}{10}$$ – $$\frac{2}{10}$$ + x = 1, $$\frac{4}{10}$$ – $$\frac{2}{10}$$ + x = 1, $$\frac{4-2}{10}$$ + x = 1, $$\frac{2}{10}$$ + x = 1, x = 1- $$\frac{2}{10}$$, x = $$\frac{10 – 2}{10}$$, x = $$\frac{8}{10}$$ or $$\frac{4}{5}$$. Assessment Practice Question 22. Which is the sum of $$\frac{3}{12}$$ + $$\frac{7}{12}$$? Answer: B, $$\frac{10}{12}$$, Explanation: As $$\frac{3}{12}$$ and $$\frac{7}{12}$$ are like fraction as denominators are same, we can add numerators 3 and 7 so answer is 3 + 7 = 10, therefore $$\frac{3}{12}$$+ $$\frac{7}{12}$$ =$$\frac{10}{12}$$. Question 23. Lindsay had $$\frac{5}{10}$$ cup of flour in the mixing bowl. She added $$\frac{2}{10}$$ cup of cocoa powder and $$\frac{3}{10}$$ cup of sugar. What is the total amount of dry ingredients in the mixing bowl? A. 1 cup B. $$\frac{7}{10}$$ C. $$\frac{5}{10}$$ D. $$\frac{1}{10}$$ Answer: A, 1 cup, Explanation: As $$\frac{5}{10}$$, $$\frac{2}{10}$$ and $$\frac{3}{10}$$ are like fraction as denominators are same , we can add numerators 5, 2 and 3 so answer is 5 + 2 + 3 = 10, therefore $$\frac{5}{10}$$+ $$\frac{2}{10}$$ + $$\frac{3}{10}$$ =$$\frac{10}{10}$$. $$\frac{10}{10}$$ as it can be further simplified 10 and 10 can be cancelled so answer in the simplest form is $$\frac{1}{1}$$ = 1. ### Lesson 9.2 Decompose Fractions Solve & Share Karyn has 1 pounds of chili to put into three bowls. The amount of chili in each bowl does not have to be the same. How much chili could Karyn put into each bowl? Solve this problem any way you choose. | can…use number lines, area models, or drawings to decompose fractions. Look Back! Use a drawing or fraction strips to help write equivalent fractions for the amount of chili in one of the bowls. Essential Question How Can You Represent a Fraction in a Variety of Ways? Answer: In 5 ways we can represent fractions, Explanation: 1. Sales: If you get reduction of 20%, 20% is a fraction: 20/100, 2. Comparisons: Ship A is half as big as Ship B. Half is a fraction: 1/2, 3. Taking a part of something: You eat two slices of pizza. Two out of eight is a fraction: 2/8, 4. Division of integers: If the quotient is not an integer, it will necessarily be a fraction, 5. Ratios: One of the definitions of π is that it is the ratio of the Circumference to the Diameter. It is a fraction, although it can’t be expressed with both terms of it being integers. Visual Learning Bridge Charlene wants to leave $$\frac{1}{6}$$ of her garden empty. What are some different ways Charlene can plant the rest of her garden? Decompose means to break into parts. Compose means to combine parts. The fraction of the garden that Charlene will plant can be decomposed in more than one way. One Way Charlene could plant four $$\frac{1}{6}$$ sections with blue flowers and one $$\frac{1}{6}$$ section with red peppers. Another Way Charlene could plant one $$\frac{1}{6}$$ section with green beans, one $$\frac{1}{6}$$ section with yellow squash, one $$\frac{1}{6}$$ section with red peppers, and two $$\frac{1}{6}$$ sections with blue flowers. Convince Me! Use Appropriate Tools Draw pictures or use fraction strips to show why these equations are true. $$\frac{5}{6}=\frac{3}{6}+\frac{2}{6} \quad \frac{5}{6}=\frac{1}{6}+\frac{2}{6}+\frac{2}{6}$$ Another Example! What is one way you can decompose 3$$\frac{1}{8}$$? A mixed number has a whole number part and a fraction part. 3 $$\frac{1}{8}$$ is 1 whole + 1 whole + 1 whole + $$\frac{1}{8}$$. Each whole can also be shown as eight equal parts. Guided Practice Do You Understand? Question 1. What is another way to decompose 3$$\frac{1}{8}$$? Answer: Question 2. Look at the area model above. What fraction with a greater numerator than denominator is equivalent to 3$$\frac{1}{8}$$? Explain. Answer: $$\frac{25}{8}$$, Explanation: Given the area model of 3$$\frac{1}{8}$$ which is equivalent to $$\frac{25}{8}$$ because we convert mixed fraction into fraction as (3 X 8 + 1) by 8 = $$\frac{25}{8}$$, where numerator is greater than denominator. Do You Know How? For 3-4, decompose each fraction or mixed number in two different ways. Use a tool if needed. Question 3. Answer: , Explanation: Given to find $$\frac{3}{5}$$ equivalent to $$\frac{1}{5}$$ + $$\frac{2}{5}$$ and $$\frac{3}{5}$$ equivalent to $$\frac{1}{5}$$ + $$\frac{1}{5}$$ + $$\frac{1}{5}$$. Question 4. Answer: , Explanation: Given to find 1$$\frac{3}{4}$$ equivalent to as 1$$\frac{3}{4}$$= (1 X 4 + 3) by 4 = $$\frac{7}{4}$$ which is equivalent to 1 + $$\frac{3}{4}$$ and $$\frac{2}{4}$$ + $$\frac{5}{4}$$. Independent Practice Leveled Practice For 5-10, decompose each fraction or mixed number in two different ways. Use a tool if needed. Question 5. $$\frac{4}{6}$$ = Answer: $$\frac{4}{6}$$ = $$\frac{2}{3}$$, Explanation: Given $$\frac{4}{6}$$ to decompose each fraction or mixed number so as 4 and 6 can be divided by 2, we get 2 and 3, so $$\frac{4}{6}$$ = $$\frac{2}{3}$$. Question 6. $$\frac{7}{8}$$ = Answer: $$\frac{7}{8}$$ =$$\frac{3}{8}$$ + $$\frac{4}{8}$$, Explanation: Given $$\frac{7}{8}$$ to decompose each fraction or mixed number, so $$\frac{7}{8}$$ = $$\frac{3}{8}$$ + $$\frac{4}{8}$$. Question 7. 1$$\frac{3}{5}$$ = Answer: 1$$\frac{3}{5}$$ = $$\frac{8}{5}$$ or $$\frac{5}{5}$$ + $$\frac{3}{5}$$, Explanation: Given 1$$\frac{3}{5}$$ to decompose each fraction or mixed number, so 1$$\frac{3}{5}$$ = $$\frac{1 X 5 + 3}{5}$$ = $$\frac{8}{5}$$ or 1$$\frac{3}{5}$$ = $$\frac{5}{5}$$ + $$\frac{3}{5}$$. Question 8. 2$$\frac{1}{2}$$ = Answer: 2$$\frac{1}{2}$$ = $$\frac{5}{2}$$ or $$\frac{4}{2}$$ + $$\frac{1}{2}$$, Explanation: Given 2$$\frac{1}{2}$$ to decompose each fraction or mixed number, so 2$$\frac{1}{2}$$ = $$\frac{2 X 2 + 1}{5}$$ = $$\frac{5}{2}$$ or 2$$\frac{1}{2}$$ = $$\frac{4}{2}$$ + $$\frac{1}{2}$$. Question 9. $$\frac{9}{12}$$ = Answer: $$\frac{9}{12}$$ = $$\frac{3}{4}$$, Explanation: Given $$\frac{9}{12}$$ to decompose each fraction or mixed number, as $$\frac{9}{12}$$ can be divided by 3, we get 3 and 4, so $$\frac{3}{4}$$ therefore $$\frac{9}{12}$$ = $$\frac{3}{4}$$. Question 10. 1$$\frac{1}{3}$$ = Answer: 1$$\frac{1}{3}$$ = $$\frac{4}{3}$$ or $$\frac{3}{3}$$ + $$\frac{1}{3}$$ Explanation: Given 1$$\frac{1}{3}$$ to decompose each fraction or mixed number, so 1$$\frac{1}{3}$$ = $$\frac{1 X 3 + 1}{3}$$ = $$\frac{4}{3}$$ or 1$$\frac{1}{3}$$ = $$\frac{3}{3}$$ + $$\frac{1}{3}$$. Problem Solving Question 11. Jackie ate $$\frac{1}{5}$$ of a bag of popcorn. She shared the rest with Enrique. List three ways they could have shared the remaining popcorn. Answer: The list of three ways they shared the remaining popcorn is 1. $$\frac{1}{5}$$ and $$\frac{3}{5}$$ = $$\frac{1+3}{5}$$= $$\frac{4}{5}$$, 2. $$\frac{2}{5}$$ and $$\frac{2}{5}$$ = $$\frac{2+2}{5}$$= $$\frac{4}{5}$$, 3. $$\frac{3}{5}$$ and $$\frac{1}{5}$$ = $$\frac{3+1}{5}$$= $$\frac{4}{5}$$, Explanation: Given Jackie ate $$\frac{1}{5}$$ of a bag of popcorn. She shared the rest with Enrique, So the rest bag of popcorn is 1 – $$\frac{1}{5}$$ = $$\frac{1 X 5 – 1}{5}$$ = $$\frac{4}{5}$$, The list of three ways they shared the remaining popcorn is 1. $$\frac{1}{5}$$ and $$\frac{3}{5}$$ = $$\frac{1+3}{5}$$= $$\frac{4}{5}$$, 2. $$\frac{2}{5}$$ and $$\frac{2}{5}$$ = $$\frac{2+2}{5}$$= $$\frac{4}{5}$$, 3. $$\frac{3}{5}$$ and $$\frac{1}{5}$$ = $$\frac{3+1}{5}$$= $$\frac{4}{5}$$. Question 12. Use Appropriate Tools Draw an area model to show $$\frac{4}{10}+\frac{3}{10}+\frac{2}{10}=\frac{9}{10}$$ Answer: Question 13. In a class of 12 students, 8 students are boys. Write two equivalent fractions that tell which part of the class is boys. Answer: Question 14. Use Appropriate Tools Find three different ways to decompose 1$$\frac{5}{6}$$. Use number lines to justify your answer. Answer: Question 15. Higher Order Thinking Jason wrote 1$$\frac{1}{3}$$ as the sum of three fractions. None of the fractions had a denominator of 3. What fractions might Jason have used? Answer: Assessment Practice Question 16. A teacher distributes a stack of paper to 3 groups. Each group receives a different amount of paper. Select all the ways the teacher can distribute the paper by decomposing 1$$\frac{2}{3}$$ inches. Use a fraction model if needed. Answer: ### Lesson 9.3 Add Fractions with Like Denominators Solve & Share Jonas is making nachos and tacos for a family party. He uses $$\frac{2}{5}$$ bag of shredded cheese for the nachos and $$\frac{1}{5}$$ bag for the tacos. How much of the bag of shredded cheese does Jonas use? Solve this problem any way you choose. I can … use my understanding of addition as joining parts of the same whole to add fractions with like denominators. Look Back! Look for Relationships What do you notice about the denominators in your equation? Answer: Denominators in the equation are same. $$\frac{2}{5}$$ + $$\frac{1}{5}$$ Total $$\frac{3}{5}$$ bag of shredded cheese used for the nachos and tacos. Explanation: Essential Question How Can You Add Fractions with Like Denominators? Answer: Build each fraction (if needed) so that both denominators are equal. Add the numerators of the fractions. The new denominator will be the denominator of the built-up fractions. Reduce or simplify the answer, if needed. Explanation: Visual Learning Bridge The table shows the results of a fourth-grade Pets Club survey. What fraction of the club members chose a hamster or a dog as their favorite pet? Find $$\frac{2}{12}+\frac{4}{12}$$ using a model. Find $$\frac{2}{12}+\frac{4}{12}$$ by joining parts. Add the numerators. Write the sum over the like denominator. $$\frac{2}{12}+\frac{4}{12}=\frac{2+4}{12}=\frac{6}{12}$$ $$\frac{6}{12}$$ is equivalent to $$\frac{1}{2}$$ One half of the club members chose a hamster or a dog as their favorite pet. Convince Me! Critique Reasoning Frank solved the problem above and found $$\frac{2}{12}+\frac{4}{12}=\frac{6}{24}$$. What error did Frank make? Explain. Answer: After solving the problem Frank wrote 24 in denominator, which is not correct. Explanation: A it was discussed above in addition when the given fraction with same denominators then only numerators to be added. But Frank added denominators also, which is wrong. $$\frac{2}{12}+\frac{4}{12}=\frac{2+4}{12}=\frac{6}{24}$$ Another Example! Guided Practice Do You Understand? Question 1. Using the survey on the previous page, what fraction of the club members chose either a bird or a cat? Answer: $$\frac{5}{12}+\frac{1}{12}=\frac{6}{12}$$ Explanation: Question 2. Greg found $$\frac{1}{3}+\frac{2}{3}=\frac{3}{6}$$. What error did Greg make? Answer: Adding of denominators is the error did by Greg. Explanation: Greg added both the numerator and denominators For adding fractions Numerators should be added and the denominators should be same. $$\frac{1}{3}+\frac{2}{3}=\frac{3}{3}$$ = 1 Do You Know How? For 3-6, find each sum. Use drawings or fraction strips as needed. Question 3. $$\frac{2}{4}+\frac{1}{4}$$ Answer: $$\frac{3}{4}$$ Explanation: $$\frac{2}{4}+\frac{1}{4}$$ = $$\frac{3}{4}$$ Question 4. $$\frac{1}{3}+\frac{2}{3}$$ Answer: $$\frac{3}{3}$$ Explanation: $$\frac{1}{3}+\frac{2}{3}$$ = $$\frac{3}{3}$$ Question 5. $$\frac{2}{12}+\frac{11}{12}$$ Answer: 1$$\frac{1}{12}$$ Explanation: $$\frac{2}{12}+\frac{11}{12}$$ = $$\frac{13}{12}$$ = 1$$\frac{1}{12}$$ Question 6. $$\frac{1}{10}+\frac{4}{10}$$ Answer: $$\frac{5}{10}$$ Explanation: $$\frac{1}{10}+\frac{4}{10}$$ = $$\frac{5}{10}$$ Independent Practice For 7-18, find each sum. Use drawings or fraction strips as needed. Question 7. $$\frac{2}{8}+\frac{1}{8}$$ Answer: $$\frac{3}{8}$$ Explanation: $$\frac{2}{8}+\frac{1}{8}$$ = $$\frac{3}{8}$$ Question 8. $$\frac{3}{6}+\frac{2}{6}$$ Answer: $$\frac{5}{6}$$ Explanation: $$\frac{3}{6}+\frac{2}{6}$$ = $$\frac{5}{6}$$ Question 9. $$\frac{1}{8}+\frac{4}{8}$$ Answer: $$\frac{5}{8}$$ Explanation: $$\frac{1}{8}+\frac{4}{8}$$ = $$\frac{5}{8}$$ Question 10. $$\frac{3}{10}+\frac{2}{10}$$ Answer: $$\frac{5}{10}$$ Explanation: $$\frac{3}{10}+\frac{2}{10}$$ = $$\frac{5}{10}$$ Question 11. $$\frac{3}{10}+\frac{5}{10}$$ Answer: $$\frac{8}{10}$$ Explanation: $$\frac{3}{10}+\frac{5}{10}$$ = $$\frac{8}{10}$$ Question 12. $$\frac{5}{12}+\frac{4}{12}$$ Answer: $$\frac{9}{12}$$ Explanation: $$\frac{5}{12}+\frac{4}{12}$$ = $$\frac{9}{12}$$ Question 13. $$\frac{4}{5}+\frac{3}{5}+\frac{2}{5}$$ Answer: 1$$\frac{4}{5}$$ Explanation: $$\frac{4}{5}+\frac{3}{5}+\frac{2}{5}$$ =1$$\frac{4}{5}$$ Question 14. $$\frac{3}{10}+\frac{2}{10}+\frac{6}{10}$$ Answer: 1$$\frac{1}{10}$$ Explanation: $$\frac{3}{10}+\frac{2}{10}+\frac{6}{10}$$ = $$\frac{10}{10}$$+$$\frac{1}{10}$$=1$$\frac{1}{10}$$ Question 15. $$\frac{2}{6}+\frac{5}{6}$$ Answer: $$\frac{7}{6}$$ Explanation: $$\frac{2}{6}+\frac{5}{6}$$ = $$\frac{6}{6}+\frac{1}{6}$$ = 1$$\frac{1}{6}$$ Question 16. $$\frac{3}{6}+\frac{9}{6}$$ Answer: 2 $$\frac{12}{6}$$ = $$\frac{6}{6}$$ + $$\frac{6}{6}$$ = 2 Explanation: $$\frac{3}{6}+\frac{9}{6}$$ = $$\frac{6}{6}+\frac{6}{6}$$ = 1+1 = 2 Question 17. $$\frac{11}{10}+\frac{11}{10}$$ Answer: 2$$\frac{2}{10}$$ Explanation: $$\frac{11}{10}+\frac{11}{10}$$ $$\frac{10}{10}$$+ $$\frac{10}{10}$$+$$\frac{2}{10}$$ 2$$\frac{2}{10}$$ Question 18. $$\frac{7}{8}+\frac{1}{8}$$ Answer: 1 $$\frac{8}{8}$$ Explanation: $$\frac{7}{8}+\frac{1}{8}$$ = $$\frac{8}{8}$$ = 1 Problem Solving For 19-21, use the table at the right. Question 19. What fraction of the set is either triangles or rectangles? Answer: Triangles are $$\frac{2}{10}$$ Rectangles are $$\frac{4}{10}$$ Explanation: Question 20. Model with Math Write and solve an equation to find what fraction, f, of the set is either circles or rectangles. Answer: f = 1 – $$\frac{x}{10}$$ Rectangles = $$\frac{4}{10}$$ Circles = $$\frac{3}{10}$$ Question 21. Which two shapes make up half of the set? Find two possible answers. Answer: Circles and Triangles $$\frac{3}{10}+\frac{2}{10}$$ = $$\frac{5}{10}$$ = $$\frac{1}{2}$$ Rectangle and Hexogon $$\frac{4}{10}+\frac{1}{10}$$ = $$\frac{5}{10}$$ = $$\frac{1}{2}$$ Question 22. There are 64 crayons in each box. A school bought 25 boxes of crayons for the art classes. If the crayons are shared equally among 5 classes, how many crayons will each class receive? Explain. Answer: Each class receives 320 crayons Explanation: 64 crayons X 25 boxes = 1600 crayons 1600 crayons / 5 classes = 320 crayons for each class Question 23. Higher Order Thinking Three-tenths of Ken’s buttons are blue, $$\frac{4}{10}$$ are green, and the rest are black. What fraction of Ken’s buttons are black? Answer: $$\frac{3}{10}$$ Explanation f=1 – ($$\frac{3}{10}$$ + $$\frac{4}{10}$$), = 1 – ($$\frac{3 + 4 }{10}$$), = 1 – ($$\frac{7}{10}$$), =($$\frac{1}{1}$$ – $$\frac{7}{10}$$), =$$\frac{3}{10}$$. Assessment Practice Question 24. Match each expression with its sum. Answer: Question 25. Jayla did some chores in the morning. She did $$\frac{3}{12}$$ of her chores in the evening. By the end of the day, she had completed $$\frac{7}{12}$$ of her chores. What fraction of the chores c, did Jayla do in the morning? A. c = $$\frac{1}{12}$$ B. c = $$\frac{2}{12}$$ C. c = $$\frac{3}{12}$$ D. c = $$\frac{4}{12}$$ Answer: D. $$\frac{4}{12}$$ Explanation: ### Lesson 9.4 Model Subtraction of Fractions Solve & Share Mr. Yetkin uses $$\frac{4}{6}$$ of a sheet of plywood to board up a window. How much of the plywood is left? Solve this problem any way you choose. I can … use tools such as fraction strips or area models to subtract fractions with like denominators. Look Back! Be Precise Explain why $$\frac{4}{6}$$ is subtracted from $$\frac{6}{6}$$ to find how much of the plywood is left. Answer $$\frac{2}{6}$$ Explanation $$\frac{6}{6}$$ – $$\frac{4}{6}$$ = $$\frac{2}{6}$$ Essential Question How Can You Use Tools to Subtract Fractions? Answer: To subtract fractions with like denominators, subtract the numerator and write the difference over the denominator. We can use tools such as Fraction Strips or Number line. For example, Number line. Fraction Strip Visual Learning Bridge A flower garden is divided into eighths. If $$\frac{2}{8}$$ of the garden is used to grow yellow roses, what fraction is left to grow other flowers? You can use tools such as fraction strips to represent subtraction. Convince Me! Use Appropriate Tools In the problem above, suppose six sections of the garden are used for yellow roses and two other sections are used for petunias. How much more of the garden is used for yellow roses than is used for petunias? Use fraction strips or another tool to help. Write your answer as a fraction. f = $$\frac{6}{8}$$ + $$\frac{2}{8}$$ =$$\frac{6 + 2}{8}$$ = $$\frac{8}{8}$$ = 1 Another Example! Guided Practice Do You Understand? Question 1. In the problem at the top of the previous page, suppose one other section was used to grow peonies. What fraction of the garden is now available for flowers? Answer: f = $$\frac{8}{8}$$ – ( $$\frac{2}{8}$$ + $$\frac{1}{8}$$) = $$\frac{8}{8}$$ – $$\frac{2 + 1}{8}$$ = $$\frac{8}{8}$$ – $$\frac{3}{8}$$ =$$\frac{5}{8}$$ Do You Know How? For 2-5, use fraction strips or other tools to subtract. Question 2. $$\frac{1}{3}-\frac{1}{3}$$ Answer: 0 Explanation: By using fraction strip $$\frac{1}{3}-\frac{1}{3}$$ = 0 Question 3. $$\frac{5}{5}-\frac{2}{5}$$ Answer: $$\frac{3}{5}$$ Explanation: Question 4. $$\frac{7}{12}-\frac{3}{12}$$ Answer: $$\frac{4}{12}$$ Explanation: Question 5. $$\frac{7}{8}-\frac{1}{8}$$ Answer: $$\frac{6}{8}$$ Explanation: Independent Practice Leveled Practice For 6-14, find each difference. Use fraction strips or other tools as needed. Question 6. Answer: $$\frac{6}{12}$$ $$\frac{1}{2}$$ Question 7. Answer: $$\frac{1}{2}$$ Question 8. Answer: $$\frac{1}{3}$$ Question 9. $$\frac{4}{5}-\frac{2}{5}$$ Answer: $$\frac{2}{5}$$ Explanation: Question 10. $$\frac{17}{10}-\frac{3}{10}$$ Answer: 1$$\frac{4}{10}$$ Explanation: Question 11. $$\frac{8}{6}-\frac{2}{6}$$ Answer: $$\frac{8}{6}-\frac{2}{6}$$ =1 Explanation: Question 12. $$\frac{9}{6}-\frac{1}{6}$$ Answer: 1$$\frac{2}{6}$$ = 1$$\frac{1}{3}$$ Explanation: Question 13. $$\frac{21}{10}-\frac{1}{10}$$ Answer: $$\frac{20}{10}$$ = 2 Explanation: Question 14. $$\frac{1}{5}-\frac{1}{5}$$ Answer: $$\frac{1}{5}-\frac{1}{5}$$ = 0 Explanation: Problem Solving Question 15. Model with Math Leesa has $$\frac{7}{8}$$ gallon of juice. She shares $$\frac{3}{8}$$ gallon. Write and solve an equation to find j, how much juice Leesa has left. Answer: $$\frac{4}{8}$$ Explanation: $$\frac{7}{8}$$ – $$\frac{3}{8}$$ = $$\frac{4}{8}$$ = $$\frac{1}{2}$$ Question 16. Higher Order Thinking Using only odd numbers for numerators, write two different subtraction problems that have a difference of $$\frac{1}{2}$$. Remember, you can find equivalent fractions for $$\frac{1}{2}$$. Answer: $$\frac{1}{2}$$ Explanation: $$\frac{9}{8}$$ – $$\frac{5}{8}$$ = $$\frac{4}{8}$$ = $$\frac{1}{2}$$ Question 17. In Kayla’s class, some of the students are wearing blue shirts. $$\frac{6}{8}$$ of the students are NOT wearing blue shirts. What fraction of the students are wearing blue shirts? Show your work. Answer: $$\frac{2}{8}$$ = $$\frac{1}{4}$$ Explanation: f= 1- $$\frac{6}{8}$$ = $$\frac{1}{1}$$ – $$\frac{6}{8}$$ = $$\frac{2}{8}$$ = $$\frac{1}{4}$$ Question 18. In Exercise 17, what number represents the whole class? How do you know what fraction to use to represent this number? Answer: 1 number represents the whole class $$\frac{1}{1}$$ Explanation: Question 19. Rick shared his bag of grapes with friends. He gave $$\frac{2}{10}$$ of the bag to Melissa and $$\frac{4}{10}$$ of the bag to Ryan. What fraction of the bag of grapes does Rick have left? Show your work. Answer: $$\frac{4}{10}$$ Explanation: f=1-{$$\frac{4}{10}$$ + $$\frac{2}{10}$$}=$$\frac{1}{1}$$ – $$\frac{6}{10}$$ = $$\frac{4}{10}$$ Question 20. Teresa gave away 8 baseball cards and has 4 baseball cards left. Write a subtraction problem to show the fraction of the baseball cards Teresa has left. Answer: $$\frac{12}{12}$$ – $$\frac{8}{12}$$ = $$\frac{4}{12}$$ Explanation: Assessment Practice Question 21. Which subtraction problem has a difference of $$\frac{1}{3}$$? A. $$\frac{2}{2}-\frac{1}{2}$$ B. $$\frac{5}{3}-\frac{3}{3}$$ C. $$\frac{4}{3}-\frac{3}{3}$$ D. $$\frac{5}{3}-\frac{1}{3}$$ Answer: Option C $$\frac{4}{3}-\frac{3}{3}$$ = $$\frac{1}{3}$$ Question 22. Which subtraction problem has a difference of $$\frac{10}{8}$$? A. $$\frac{20}{8}-\frac{10}{8}$$ B. $$\frac{8}{10}+\frac{2}{10}$$ C. $$\frac{10}{8}-\frac{4}{8}$$ D. $$\frac{6}{8}-\frac{1}{4}$$ Answer: Option A, $$\frac{20}{8}-\frac{10}{8}$$ = $$\frac{10}{8}$$ ### Lesson 9.5 Subtract Fractions with Like Denominators Solve & Share Leah and Josh live the same direction from school and on the same side of Forest Road. Leah’s house is $$\frac{8}{10}$$ mile from school. Josh’s house is $$\frac{5}{10}$$ mile from school. How much farther does Leah have to walk home when she reaches Josh’s house? Solve this problem any way you choose. I can … use my understanding of subtraction as separating parts of the same whole to subtract fractions with like denominators. Look Back! Model with Math How could you represent the problem above with a bar diagram and an equation? Tell what your variable means. Answer $$\frac{3}{10}$$ Explanation Leah’s house is $$\frac{8}{10}$$ mile and Josh’s house is $$\frac{5}{10}$$ mile from school. $$\frac{8}{10}$$ – $$\frac{5}{10}$$ = $$\frac{3}{10}$$ Essential Question How Can You Subtract Fractions with Like Denominators? Answer: To subtract fractions with like denominators, just subtract the numerators and write the difference over the denominator. Always reduce your final answer to its lowest term. Explanation: For Example, $$\frac{4}{10}$$ + $$\frac{5}{10}$$ = $$\frac{9}{10}$$ Visual Learning Bridge Tania is squeezing lemons to make lemonade. The recipe calls for $$\frac{5}{8}$$ cup of lemon juice. The amount Tania has squeezed is shown at the right. What fraction of a cup of lemon juice does Tania still need to squeeze? Subtract the fractions to find the difference. Convince Me! Reasoning in the problem above, suppose Tania decided to double the amount of lemonade she wants to make. Then how much more lemon juice would Tania need to squeeze? Guided Practice Do You Understand? Question 1. Jesse has a bottle that contains $$\frac{7}{10}$$ liter of water. He drinks $$\frac{2}{10}$$ liter. Jesse says he has $$\frac{1}{2}$$ liter left. Is he correct? Explain. Answer: YES, JESSE IS CORRECT Explanation: Question 2. What addition sentence can you use to subtract $$\frac{4}{10}$$ from $$\frac{9}{10}$$? Answer: $$\frac{4}{10}$$ + $$\frac{5}{10}$$= $$\frac{9}{10}$$ $$\frac{9}{10}$$ – $$\frac{4}{10}$$ = $$\frac{5}{10}$$ Explanation: Do You Know How? For 3-10, subtract the fractions. Question 3. $$\frac{2}{3}-\frac{1}{3}$$ Answer: $$\frac{1}{3}$$ Explanation: Question 4. $$\frac{3}{4}-\frac{2}{4}$$ Answer: $$\frac{1}{4}$$ Explanation: Question 5. $$\frac{5}{6}-\frac{2}{6}$$ Answer: $$\frac{3}{6}$$ Explanation: Question 6. $$\frac{9}{12}-\frac{3}{12}$$ Answer: $$\frac{6}{12}$$ Explanation: Question 7. $$\frac{9}{8}-\frac{3}{8}$$ Answer: $$\frac{6}{8}$$ Explanation: Question 8. $$\frac{17}{10}-\frac{9}{10}$$ Answer: $$\frac{8}{10}$$ Explanation: Question 9. $$\frac{4}{8}-\frac{1}{8}$$ Answer: $$\frac{3}{8}$$ Explanation: Question 10. $$\frac{1}{2}-\frac{1}{2}$$ Answer: 0 Independent Practice Leveled Practice For 11-18, subtract the fractions. Question 11. Answer: n = $$\frac{4}{6}$$ Explanation: Question 12. Answer: n = $$\frac{5}{100}$$ = $$\frac{1}{20}$$ Explanation: Question 13. Answer: n = $$\frac{2}{4}$$ Explanation: Question 14. Answer: n = $$\frac{2}{8}$$ Explanation: Question 15. $$\frac{5}{6}-\frac{4}{6}$$ Answer: $$\frac{1}{6}$$ Explanation: Question 16. $$\frac{40}{10}-\frac{20}{10}$$ Answer: $$\frac{20}{10}$$ = 2 Explanation: Question 17. $$\frac{80}{100}-\frac{40}{100}$$ Answer: $$\frac{40}{100}$$ Explanation: Question 18. $$\frac{19}{10}-\frac{8}{10}$$ Answer: 1$$\frac{1}{10}$$ Explanation: Problem Solving Question 19. Joey ran $$\frac{1}{4}$$ mile in the morning and $$\frac{1}{4}$$ mile farther than in the morning in the afternoon. If he wants to run a full mile, how much more does Joey have to run? Write equations to explain. Answer: $$\frac{1}{4}$$ Explanation: Question 20. Reasoning Explain how subtracting $$\frac{4}{5}-\frac{3}{5}$$ involves subtracting 4 – 3. Answer: n = $$\frac{1}{5}$$ n = $$\frac{4}{5}$$ – $$\frac{3}{5}$$ n = $$\frac{4 – 3}{5}$$ n = $$\frac{1}{5}$$ Question 21. Higher Order Thinking The flags of all 5 Nordic countries are displayed. What fraction describes how many more of the flags displayed are 2-color flags than are 3-color flags? Answer: $$\frac{1}{5}$$ Explanation: Assessment Practice Question 22. Brian had a piece of chalk $$\frac{9}{10}$$ centimeter long. A piece cracked off as he was drawing on the sidewalk. Then Brian’s chalk was only $$\frac{6}{10}$$ centimeter long. How long was the piece of chalk that cracked off? Answer: $$\frac{3}{10}$$ Explanation: n = $$\frac{9}{10}$$ – $$\frac{6}{10}$$ n = $$\frac{3}{10}$$ Question 23. Marietta baked a chicken pot pie. She serves $$\frac{2}{3}$$ of the pie at dinner. How much of the pie remains? Answer: $$\frac{1}{3}$$ Explanation: ### Lesson 9.6 Add and Subtract Fractions with Like Denominators Solve & Share The dirt bike track shown is $$\frac{7}{8}$$ of a mile long from start to finish. The track is divided into four sections. What is the length of the longest section? Solve this problem any way you choose. I can … use a number line to add and subtract fractions when the fractions refer to the same whole. Answer: $$\frac{2}{8}$$ Explanation: Number line Look Back! How did you decide which section of the track was the longest? Answer: The length of the longest section = $$\frac{3}{8}$$ Essential Question How Do You Add and Subtract Fractions on a Number Line? Answer: In math, a number line is a straight line with numbers placed at equal intervals or segments along its length. A number line can be extended infinitely in any direction, and we usually represent it horizontally. The numbers on the number line in addition increase as we move from left to right and in subtraction decrease moving from right to left. Explanation: For Example, Visual Learning Bridge Mary rides her bike $$\frac{2}{10}$$ mile to pick up her friend Marcy for soccer practice. Together, they ride $$\frac{5}{10}$$ mile to the soccer field. What is the distance from Mary’s house to the soccer field? You can use jumps on the number line to add or subtract fractions. Use a number line to show $$\frac{2}{10}+\frac{5}{10}$$. Draw a number line for tenths. Locate $$\frac{2}{10}$$ on the number line. To add, move $$\frac{5}{10}$$ to the right. When you add, you move to the right on the number line. Write the addition equation. Add the numerators. Write the sum over the like denominator. $$\frac{2}{10}+\frac{5}{10}=\frac{2+5}{10}=\frac{7}{10}$$ The distance from Mary’s house to the soccer field is $$\frac{7}{10}$$ mile. Convince Me! Use Appropriate Tools Use the number line below to find $$\frac{5}{8}+\frac{2}{8}$$. Can you also use the number line to find $$\frac{5}{8}-\frac{2}{8}$$? Explain. Answer: $$\frac{5}{8}+\frac{2}{8}$$ = $$\frac{7}{8}$$ $$\frac{5}{8}-\frac{2}{8}$$ = $$\frac{3}{8}$$ Explanation: Another Example! Find $$\frac{6}{8}-\frac{4}{8}$$ Start at $$\frac{6}{8}$$. To subtract, move $$\frac{4}{8}$$ to the left. The ending point is $$\frac{2}{8}$$ So, $$\frac{6}{8}-\frac{4}{8}=\frac{2}{8}$$ Guided Practice Do You Understand? Question 1. In the example above, how is the denominator illustrated on the number line? Answer: The denominator illustrates the number of parts of the whole, In the above example 8 is the denominator, that is 8 parts. $$\frac{1}{8}$$ + $$\frac{1}{8}$$ + $$\frac{1}{8}$$ + $$\frac{1}{8}$$ + $$\frac{1}{8}$$ + $$\frac{1}{8}$$ + $$\frac{1}{8}$$ + $$\frac{1}{8}$$ Question 2. Draw a number line to represent $$\frac{3}{12}+\frac{5}{12}$$. Answer: $$\frac{3}{12}+\frac{5}{12}$$ = $$\frac{8}{12}$$ Explanation: Do You Know How? For 3-4, write the equation shown by each number line. Question 3. Answer: $$\frac{3}{5}$$ $$\frac{1}{5}+\frac{2}{5}$$ = $$\frac{3}{5}$$ Question 4. Answer: $$\frac{2}{6}$$ $$\frac{5}{6} – \frac{3}{6}$$ = $$\frac{2}{6}$$ Independent Practice For 5-8, write the equation shown by each number line. Question 5. Answer: $$\frac{6}{10}$$ $$\frac{2}{10} + \frac{4}{10}$$ = $$\frac{6}{10}$$ Question 6. Answer: $$\frac{1}{5}$$ $$\frac{4}{5} – \frac{3}{5}$$ = $$\frac{1}{5}$$ Question 7. Answer: $$\frac{3}{4}$$ $$\frac{2}{4} + \frac{1}{4}$$ = $$\frac{3}{4}$$ Question 8. Answer: $$\frac{2}{6}$$ $$\frac{4}{6} – \frac{2}{6}$$ = $$\frac{2}{6}$$ Problem Solving Question 9. Number Sense How do you know the quotient 639 ÷ 6 is greater than 100 before you actually divide? Answer: 106 $$\frac{3}{6}$$ Explanation: = $$\frac{600}{6}$$ + $$\frac{36}{6}$$ + $$\frac{3}{6}$$ = 100 + 6 + $$\frac{3}{6}$$ =106 $$\frac{3}{6}$$ Question 10. On average, a largemouth bass weighs about 12 pounds. Two fishermen weighed the largemouth bass they both caught and found their catch weighed 82 pounds. What is the greatest number of largemouth bass caught that are average weight? Answer: 22 pounds 1 $$\frac{10}{12}$$ Explanation: = $$\frac{82}{12}$$ = $$\frac{60}{12}$$ + $$\frac{22}{12}$$ = 5 + 1$$\frac{10}{12}$$ Question 11. Isaac started his bike ride at the trailhead. He reached the picnic area and continued to the lookout tower. If Isaac rode his bike for a total of $$\frac{10}{4}$$ miles, how much farther did he ride beyond the lookout tower? Answer: $$\frac{5}{4}$$ Explanation: n = $$\frac{10}{4}$$ – [$$\frac{2}{4}$$+ $$\frac{3}{4}$$] n = $$\frac{10}{4}$$ – $$\frac{5}{4}$$ n = $$\frac{5}{4}$$ n = 1 $$\frac{1}{4}$$ Question 12. Model with Math Ricky completely filled a bucket to wash his car. After he finished washing the car, $$\frac{5}{8}$$ of the water remained in the bucket. Write and solve an equation to find n, the fraction of the water Ricky used. Answer: n = $$\frac{3}{8}$$ water used by Ricky. Explanation: n = $$\frac{1}{1}$$ – $$\frac{5}{8}$$ n = $$\frac{3}{8}$$ Question 13. Higher Order Thinking Sarah and Jenny are running an hour-long endurance race. Sarah ran $$\frac{2}{6}$$ hour before passing the baton to Jenny. Jenny ran $$\frac{3}{6}$$ hour, then passed the baton back to Sarah. What fraction of the hour does Sarah still need to run to complete the race? Answer: $$\frac{1}{6}$$ to complete the race. Explanation: n = 1 – $$\frac{2}{6}$$ + $$\frac{3}{6}$$ n = 1 – $$\frac{2 + 3}{6}$$ n = 1 – $$\frac{5}{6}$$ n = $$\frac{1}{6}$$ Assessment Practice Question 14. Choose numbers from the box to fill in the missing numbers in each equation. Use each number once. Answer: Question 15. Choose numbers from the box to fill in the missing numbers in each equation. Use each number once. Answer: ### Lesson 9.7 Model Addition and Subtraction of Mixed Numbers Solve & Share Tory is cutting loaves of bread into fourths. She needs to wrap 3$$\frac{3}{4}$$ loaves to take to a luncheon and 1$$\frac{2}{4}$$ loaves for a bake sale. How many loaves does Tory need to wrap for the luncheon and the bake sale? Solve this problem any way you choose. I can ….. use models and equivalent fractions to help add and subtract mixed numbers. Look Back! Reasoning How can you estimate the sum above? 3$$\frac{3}{4}$$ + 1$$\frac{2}{4}$$ = (3 + 1)($$\frac{3}{4}$$ + $$\frac{2}{4}$$) = (3 + 1)($$\frac{3 + 2}{4}$$) = (4)($$\frac{3 + 2}{4}$$) = 4($$\frac{5}{4}$$) = (4 + 1)($$\frac{1}{4}$$) = $$5\frac{1}{4}$$ Essential Question How Can You Add or Subtract Mixed Numbers? Answer: To add or subtract mixed numbers, firstly convert the mixed numbers to improper fractions and find the least common denominator, then add or subtract the whole numbers. Finally write the lowest terms. For example, 3$$\frac{3}{4}$$ + 1$$\frac{2}{4}$$ = $$\frac{15}{4}$$ + $$\frac{6}{4}$$ = $$\frac{15 + 6}{4}$$ = $$\frac{21}{4}$$ = $$5\frac{1}{4}$$ Visual Learning Bridge Bill has 2 boards to use to make picture frames. What is the total length of the two boards? How much longer is one board than the other? You can use addition to find the total length of the two boards. You can use subtraction to find how much longer one board is than the other. Convince Me! Use Appropriate Tools Suppose Bill’s boards were 2$$\frac{11}{12}$$ feet and 1$$\frac{5}{12}$$ feet. What would be the total length of the two boards? How much longer is one board than the other? Use fraction strips or draw number lines to show your work. Answer 4$$\frac{4}{12}$$ Explanation Another Example! Guided Practice Do You Understand? Question 1. In the problem on the previous page, why does $$\frac{16}{12}$$ = 1$$\frac{4}{12}$$? Use decomposing to explain. Answer: $$\frac{16}{12}$$ = 1$$\frac{4}{12}$$ Explanation: Do You Know How? For 2-3, use a tool to find each sum or difference. Question 2. 1$$\frac{2}{5}$$ + 2$$\frac{4}{5}$$ Answer: 4$$\frac{1}{5}$$ Explanation: Question 3. 1$$\frac{1}{4}$$ + 2$$\frac{3}{4}$$ Answer: 4 Explanation: Independent Practice For 4-11, use a tool to find the sum or difference. Question 4. Answer: $$\frac{2}{4}$$ $$\frac{1}{2}$$ Question 5. Answer: 4 $$\frac{13}{3}$$ Explanation: 1$$\frac{2}{3}$$ + 2$$\frac{2}{3}$$ $$\frac{5}{3}$$ + $$\frac{8}{3}$$ $$\frac{13}{3}$$ 4 $$\frac{13}{3}$$ Question 6. Answer: 1 Explanation: 2$$\frac{3}{4}$$ – 1$$\frac{3}{4}$$ $$\frac{11}{4}$$ – $$\frac{7}{4}$$ $$\frac{4}{4}$$ = 1 Question 7. Answer: 3, Explanation: 1$$\frac{3}{6}$$ + 1$$\frac{3}{6}$$, $$\frac{9}{6}$$ + $$\frac{9}{6}$$, $$\frac{18}{6}$$, 3 Question 8. $$2 \frac{3}{5}+1 \frac{3}{5}$$ Answer: 4 $$\frac{1}{5}$$, Explanation: Question 9. $$4 \frac{5}{12}+1 \frac{7}{12}$$ Answer: 6 Explanation: Question 10. $$4 \frac{9}{10}+3 \frac{7}{10}$$ Answer: 8 $$\frac{2}{10}$$ Explanation: Question 11. $$5 \frac{3}{4}+2 \frac{3}{4}$$ Answer: 8 $$\frac{2}{4}$$, Explanation: Problem Solving Question 12. Use Appropriate Tools Kit said, “On summer vacation, I spent 1$$\frac{1}{2}$$ weeks with my grandma and one week more with my aunt than with my grandma.” How many weeks did she spend visiting family? Use a tool to find the sum. Answer: 2$$\frac{1}{2}$$ Explanation: 1$$\frac{1}{2}$$ + 1 week $$\frac{3}{2}$$ + 1 = $$\frac{3 + 2}{2}$$ = $$\frac{5}{2}$$ = $$\frac{4}{2}$$ + $$\frac{1}{2}$$ = 2$$\frac{1}{2}$$ Question 13. Use Appropriate Tools If Kit spent 3$$\frac{1}{2}$$ weeks in swimming lessons, how much more time did Kit spend visiting family than in swimming lessons? Use a tool to find the difference. Answer: 6 weeks Explanation: 1$$\frac{1}{2}$$ + 1 week + 3$$\frac{1}{2}$$ $$\frac{3}{2}$$ + 1 + $$\frac{7}{2}$$ = $$\frac{3 + 2 + 7}{2}$$ = $$\frac{12}{2}$$ = 6 Question 14. Hannah used 1$$\frac{5}{8}$$ gallons of paint for theceiling. Hannah used 6 gallons of paint for the walls and ceiling combined. How much paint did Hannah use for the walls? Answer: 4$$\frac{3}{8}$$ gallons of Paint Hannah use for the walls Explanation: Total paint (Wall + Ceiling) = Paint for Ceiling + Paint for Wall 6 gallons = 1$$\frac{5}{8}$$ + Paint for Wall Paint for Wall = 6 – 1$$\frac{5}{8}$$ = 6 – $$\frac{13}{8}$$= $$\frac{35}{8}$$ Paint for Wall = 4$$\frac{3}{8}$$ Paint for Wall = Total paint (Wall + Ceiling) – Total paint Ceiling Question 15. A furlong is a unit of length still used today in racing and agriculture. A race that is 8 furlongs is 1 mile. A mile is 5,280 feet. How many feet are in a furlong? Answer: 660 feet is one furlong Explanation: 8 furlongs is 1 mile. A mile is 5,280 feet one furlong $$\frac{5280}{8}$$ = 660 feet Question 16. Higher Order Thinking A recipe calls for 1$$\frac{2}{3}$$ cups of brown sugar for the granola bars and 1$$\frac{1}{3}$$ cups of brown sugar for the topping. Dara has 3$$\frac{1}{4}$$ cups of brown sugar. Does she have enough brown sugar to make the granola bars and the topping? Explain. You can use fraction strips or a number line to compare amounts. Answer: YES Dara has enough sugar She need 3 cups of brown sugar, she has 3$$\frac{1}{4}$$ cups 1$$\frac{2}{3}$$ + 1$$\frac{1}{3}$$ = $$\frac{5 + 4}{3}$$=$$\frac{9}{3}$$ = 3 3$$\frac{1}{4}$$ >3 Assessment Practice Question 18. Megan finishes the scarf. It is 5$$\frac{6}{12}$$ feet in length. She finds a mistake in her knitting and unravels 2$$\frac{4}{12}$$ feet to correct the mistake. How long is the scarf now? A. s = $$2 \frac{7}{12}+2 \frac{11}{12}$$ B. s = $$2 \frac{5}{12}+2 \frac{7}{12}$$ C. s = $$2 \frac{11}{12}-2 \frac{7}{12}$$ D. s = $$4 \frac{11}{12}-2 \frac{7}{12}$$ Answer: $$3 \frac{4}{12}$$ Explanation: $$5 \frac{6}{12} – 2 \frac{4}{12}$$ $$\frac{66}{12} – \frac{28}{12}$$ = 3$$\frac{4}{12}$$ D. s = $$4 \frac{11}{12}-1\frac{9}{12}$$ {some correction may required in option D} Question 17. Megan is knitting a scarf. She has knitted 2$$\frac{7}{12}$$ feet so far. She needs to knit another 2$$\frac{11}{12}$$ feet. Which of the following equations can Megan use to find s, the length of the completed scarf? A. 8 $$\frac{10}{12}$$ feet B. 5 $$\frac{4}{12}$$ feet C. 3 $$\frac{2}{12}$$ feet D. 1 $$\frac{4}{12}$$ feet Answer: Option B. 5 $$\frac{6}{12}$$ feet {some correction may required in option B} Explanation: s = $$2 \frac{7}{12} + 2 \frac{11}{12}$$ s = $$\frac{31}{12} + \frac{35}{12}$$ = $$\frac{66}{12}$$ s = 5$$\frac{6}{12}$$ ### Lesson 9.8 Add Mixed Numbers Solve & Share Joaquin used 1$$\frac{3}{6}$$ cups of apple juice and 1$$\frac{4}{6}$$ cups of orange juice in a recipe for punch. How much juice did Joaquin use? Solve this problem any way you choose. I can… use equivalent fractions and properties of operations to add mixed numbers with like denominators. Generalize. You can use what you know about adding fractions to solve this problem. Look Back! Could you find the sum of $$1 \frac{3}{6}+1 \frac{4}{6}$$ by adding (1 + 1) + $$\left(\frac{3}{6}+\frac{4}{6}\right)$$? Explain.. Answer 3$$\frac{1}{6}$$ Explanation Essential Question How Can You Add Mixed Numbers? Answer: As you may recall, a mixed number consists of an integer and a proper fraction. Any mixed number can also be written as an improper fraction, in which the numerator is larger than the denominator. For example, 3$$\frac{1}{8}$$ = $$\frac{25}{8}$$ The whole numbers, 3 and 1 sum to 4. The fractions$$\frac{2}{5}$$ and $$\frac{3}{5}$$ add upto $$\frac{5}{5}$$ or 1. Add the 1 to 4 to get the answer which is 5. Visual Learning Bridge Brenda mixes sand with 2$$\frac{7}{8}$$ cups of potting mixture to prepare soil for her plant. After mixing them together, how many cups of soil does Brenda have? You can use properties of operations to add mixed numbers. When you break apart a mixed number to add, you are using the Commutative and the Associative Properties. Convince Me! Reasoning How is adding mixed numbers like adding fractions and whole numbers? Guided Practice Do You Understand? Question 1. How do the Commutative and Associative Properties allow you to add the fraction parts, add the whole number parts, and then add them together? Use $$2 \frac{5}{10}+1 \frac{9}{10}=\left(2+\frac{5}{10}\right)+\left(1+\frac{9}{10}\right)$$ as an example. Answer: $$4 \frac{4}{10}$$ Explanation: Question 2. How can you use equivalent fractions to find $$4 \frac{2}{8}+1 \frac{1}{8}$$? Answer: $$5 \frac{3}{8}$$ Explanation: Do You Know How? For 3-8, find each sum. Question 3. Answer: $$3 \frac{1}{8}$$ Explanation: Question 4. Answer: $$7 \frac{9}{10}$$ Explanation: Question 5. $$4 \frac{2}{3}+1 \frac{2}{3}$$ Answer: $$6 \frac{1}{3}$$ Explanation: Question 6. $$6 \frac{5}{12}+4 \frac{11}{12}$$ Answer: $$11 \frac{4}{12}$$ Explanation: Question 7. $$2 \frac{1}{3}+2 \frac{1}{3}$$ Answer: $$4 \frac{2}{3}$$ Explanation: Question 8. $$1 \frac{9}{12}+2 \frac{5}{12}$$ Answer: $$4 \frac{2}{12}$$ Explanation: Independent Practice Leveled Practice For 9-22, find each sum by adding mixed numbers or by adding equivalent fractions. Question 9. Answer: $$4 \frac{1}{6}$$ Explanation: Question 10. Answer: $$5 \frac{3}{4}$$ Explanation: Question 11. Answer: $$8 \frac{3}{6}$$ Explanation: Question 12. Answer: $$4 \frac{6}{10}$$ Explanation: Question 13. Answer: $$5 \frac{4}{8}$$ Explanation: Question 14. Answer: $$15 \frac{2}{8}$$ Explanation: Question 15. $$4 \frac{1}{10}+6 \frac{5}{10}$$ Answer: $$10 \frac{6}{10}$$ Explanation: Question 16. $$1 \frac{7}{12}+4 \frac{9}{12}$$ Answer: $$6 \frac{4}{12}$$ Explanation: Question 17. $$5+3 \frac{1}{8}$$ Answer: $$8 \frac{1}{8}$$ Explanation: Question 18. $$3 \frac{3}{4}+2 \frac{3}{4}$$ Answer: $$6 \frac{2}{4}$$ Explanation: Question 19. $$2 \frac{4}{5}+2 \frac{3}{5}$$ Answer: $$5 \frac{2}{5}$$ Explanation: Question 20. $$3 \frac{2}{6}+2 \frac{5}{6}$$ Answer: $$6 \frac{1}{6}$$ Explanation: Question 21. $$1 \frac{7}{12}+2 \frac{10}{12}$$ Answer: $$4 \frac{5}{12}$$ Explanation: Question 22. $$3 \frac{6}{8}+1 \frac{3}{8}$$ Answer: $$5 \frac{1}{8}$$ Explanation: Problem Solving For 23, use the map at the right. Question 23. a. Find the distance from the start of the trail to the end of the trail. Answer: 6 $$\frac{6}{8}$$ Explanation: b. Linda walked from the start of the trail to the bird lookout and back. Did Linda walk more or less than if she had walked from the start of the trail to the end? Answer: Linda neither walked more nor less, when it comes to start of the trail to the end. Explanation: Linda walked from the start of the trail to the bird lookout and back is 1+1 = 2 she had walked from the start of the trail to the end is one way. its only 1. Question 24. Joe biked 1$$\frac{9}{12}$$ miles from home to the lake, then went some miles around the lake, and then back home. Joe biked a total of 4$$\frac{9}{12}$$ miles. How many miles did Joe bike around the lake? Answer: 3 miles – Joe bike around the lake Explanation: Let f is miles around the lake 4$$\frac{9}{12}$$ miles = 1$$\frac{9}{12}$$ miles + f f = 4$$\frac{9}{12}$$ – 1$$\frac{9}{12}$$ f = $$\frac{57}{12}$$ – $$\frac{21}{12}$$ f = $$\frac{57 – 21}{12}$$ f = $$\frac{36}{12}$$ f = 3 miles Question 25. Reasoning The bus took 4$$\frac{3}{5}$$ hours to get from Jim’s home station to Portland and 3$$\frac{4}{5}$$ hours to get from Portland to Seattle. How long did the bus take to get from Jim’s home station to Seattle? Answer: 8$$\frac{2}{5}$$ hours Explanation: f = 4$$\frac{3}{5}$$ + 3$$\frac{4}{5}$$ f = $$\frac{23}{5}$$ + $$\frac{19}{5}$$ f = $$\frac{23 + 19}{5}$$ f = $$\frac{42}{5}$$ f = 8$$\frac{2}{5}$$ Question 26. Higher Order Thinking A male Parson’s chameleon is 23$$\frac{3}{4}$$ inches long. It can extend its tongue up to 35$$\frac{1}{4}$$ inches. What are 3 possible lengths for the chameleon when its tongue is extended? Answer: 3 possible lengths for the chameleon when its tongue is extended are ass follows, i) 23$$\frac{3}{4}$$ + 35$$\frac{1}{4}$$ inches ii) $$\frac{95}{4}$$ + $$\frac{141}{4}$$ iii) 23 + $$\frac{3}{4}$$ + 35 + $$\frac{1}{4}$$ Assessment Practice Question 27. Julie attaches an extension cord that is 2$$\frac{6}{8}$$ yards long to a cord that is 2$$\frac{3}{8}$$ yards long. How long are the two cords together? Select all the correct ways to find the sum. Answer: Question 28. Select all the correct sums. Answer: ### Lesson 9.9 Subtract Mixed Numbers Solve & Share Evan is walking 2 miles to his aunt’s house. He has already walked mile. How much farther does Evan have to go? Solve this problem any way you choose. Generalize. You can use what you know about subtracting fractions to solve this problem. I can … use equivalent fractions, properties of operations, and the relationship between addition and subtraction to subtract mixed numbers with like denominators. Look Back! You found 2$$\frac{1}{8}$$ – $$\frac{6}{8}$$ = m. Write a related addition equation. Essential Question How Can You Subtract Mixed Numbers? Change the mixed number (with 1 as the whole number) into an improper fraction. Subtract the whole number of the smaller mixed number from the whole number of the new larger mixed number. If the proper fractions have similar denominators, subtract the numerators directly and remain the denominator as it is. For example, Visual Learning Bridge A golf ball measures about 1$$\frac{4}{6}$$ inches across the center. What is the difference between the distances across the centers of a tennis ball and a golf ball? You can use properties of operations and the relationship between addition and subtraction to help subtract mixed numbers. Convince Me! Reasoning Explain why you rename 4$$\frac{1}{4}$$ to find 4$$\frac{1}{4}$$ – $$\frac{3}{4}$$. Guided Practice Do You Understand? Question 1. A hole at the golf course is 3$$\frac{3}{6}$$ inches wide. How much wider is the hole than the golf ball? Answer: 1$$\frac{5}{6}$$ Explanation: Question 2. How could you use the relationship between addition and subtraction with counting up to find 3$$\frac{1}{4}$$ – 1$$\frac{3}{4}$$? Answer: 3$$\frac{1}{4}$$ – 1$$\frac{3}{4}$$ = 1 $$\frac{2}{4}$$ 3$$\frac{1}{4}$$ = 1$$\frac{3}{4}$$ + 1 $$\frac{2}{4}$$ Explanation: 3$$\frac{1}{4}$$ – 1$$\frac{3}{4}$$ = $$\frac{13}{4}$$ – $$\frac{7}{4}$$ = $$\frac{13 – 7}{4}$$ = $$\frac{6}{4}$$ = 1 $$\frac{2}{4}$$ Do You Know How? For 3-8, find each difference. Question 3. Answer: 5 $$\frac{1}{8}$$ Explanation: Question 4. Answer: $$2 \frac{1}{4}$$ Explanation: Question 5. $$6 \frac{3}{10}-1 \frac{8}{10}$$ Answer: $$4 \frac{5}{10}$$ Explanation: Question 6. $$9 \frac{4}{12}-4 \frac{9}{12}$$ Answer: $$4 \frac{7}{12}$$ Explanation: Question 7. $$4 \frac{5}{6}-2 \frac{1}{6}$$ Answer: $$2\frac{4}{6}$$ Explanation: Question 8. $$1 \frac{9}{12}-\frac{10}{12}$$ Answer: $$\frac{11}{12}$$ Explanation: Independent Practice For 9-24, find each difference. Question 9. Answer: $$6 \frac{3}{8}$$ Explanation: $$8 \frac{7}{8}-2\frac{4}{8}$$ = $$\frac{71}{8}-\frac{20}{8}$$ = $$\frac{71 – 20}{8}$$ = $$\frac{51}{8}$$ = $$6 \frac{3}{8}$$ Question 10. Answer: $$2 \frac{6}{10}$$ Explanation: $$4 \frac{5}{10}-1\frac{9}{10}$$ = $$\frac{45}{10}-\frac{19}{10}$$ = $$\frac{45 – 19}{10}$$ = $$\frac{26}{10}$$ = $$2 \frac{6}{10}$$ Question 11. Answer: $$2 \frac{5}{8}$$ Explanation: $$4 \frac{1}{8}-1\frac{4}{8}$$ = $$\frac{33}{8}-\frac{12}{8}$$ = $$\frac{33 – 12}{8}$$ = $$\frac{21}{8}$$ = $$2 \frac{5}{8}$$ Question 12. Answer: $$3 \frac{1}{5}$$ Explanation: 6 – $$2\frac{4}{5}$$ = $$\frac{6}{1}-2\frac{4}{5}$$ = $$\frac{6}{1}-\frac{14}{5}$$ = $$\frac{30 – 14}{5}$$ = $$\frac{16}{5}$$ = $$3 \frac{1}{5}$$ Question 13. $$6 \frac{1}{3}-5 \frac{2}{3}$$ Answer: $$\frac{2}{3}$$ Explanation: $$6 \frac{1}{3}-5 \frac{2}{3}$$ = $$\frac{19}{3}-\frac{17}{3}$$ = $$\frac{19 – 17}{3}$$ = $$\frac{2}{3}$$ Question 14. $$9 \frac{2}{4}-6 \frac{3}{4}$$ Answer: $$2\frac{3}{4}$$ Explanation: $$9 \frac{2}{4}-6 \frac{3}{4}$$ = $$\frac{38}{4}-\frac{27}{4}$$ = $$\frac{38 – 27}{4}$$ = $$\frac{11}{4}$$ = $$2 \frac{3}{4}$$ Question 15. $$8 \frac{3}{8}-3 \frac{5}{8}$$ Answer: $$4\frac{6}{8}$$ Explanation: $$8 \frac{3}{8}-3 \frac{5}{8}$$ = $$\frac{67}{8}-\frac{29}{8}$$ = $$\frac{67 – 29}{8}$$ = $$\frac{38}{8}$$ = $$4 \frac{6}{8}$$ Question 16. $$7-3 \frac{1}{2}$$ Answer: $$3 \frac{1}{2}$$ Explanation: 7 – $$3\frac{1}{2}$$ = $$\frac{7}{1}-3\frac{1}{2}$$ = $$\frac{7}{1}-\frac{7}{2}$$ = $$\frac{14 – 7}{2}$$ = $$\frac{7}{2}$$ = $$3 \frac{1}{2}$$ Question 17. $$6 \frac{1}{6}-4 \frac{5}{6}$$ Answer: $$1\frac{2}{6}$$ Explanation: $$6 \frac{1}{6}-4 \frac{5}{6}$$ = $$\frac{37}{6}-\frac{29}{6}$$ = $$\frac{37 – 29}{6}$$ = $$\frac{8}{6}$$ = $$1 \frac{2}{6}$$ Question 18. $$3 \frac{1}{12}-1 \frac{3}{12}$$ Answer: $$1\frac{10}{12}$$ Explanation: $$3 \frac{1}{12}-1 \frac{3}{12}$$ = $$\frac{37}{12}-\frac{15}{12}$$ = $$\frac{37 – 15}{12}$$ = $$\frac{22}{12}$$ = $$1 \frac{10}{12}$$ Question 19. $$6 \frac{2}{5}-2 \frac{3}{5}$$ Answer: $$3\frac{4}{5}$$ Explanation: $$6 \frac{2}{5}-2 \frac{3}{5}$$ = $$\frac{32}{5}-\frac{13}{5}$$ = $$\frac{32 – 13}{5}$$ = $$\frac{19}{5}$$ = $$3 \frac{4}{5}$$ Question 20. $$4 \frac{5}{10}-1 \frac{7}{10}$$ Answer: $$2\frac{8}{10}$$ Explanation: $$4 \frac{5}{10}-1 \frac{7}{10}$$ = $$\frac{45}{10}-\frac{17}{10}$$ = $$\frac{45 – 17}{10}$$ = $$\frac{28}{10}$$ = $$2 \frac{8}{10}$$ Question 21. $$12 \frac{9}{12}-10 \frac{7}{12}$$ Answer: $$2\frac{2}{12}$$ Explanation: $$12 \frac{9}{12}-10 \frac{7}{12}$$ = $$\frac{153}{12}-\frac{127}{12}$$ = $$\frac{153 – 127}{12}$$ = $$\frac{26}{12}$$ = $$2 \frac{2}{12}$$ Question 22. $$25 \frac{1}{4}-20$$ Answer: = $$5 \frac{1}{4}$$ Explanation: $$25\frac{1}{4} – 20$$ = $$25 \frac{1}{4}-\frac{20}{1}$$ = $$\frac{101}{4}-\frac{20}{1}$$ = $$\frac{101 – 80}{4}$$ = $$\frac{21}{4}$$ = $$5 \frac{1}{4}$$ Question 23. $$7-2 \frac{1}{8}$$ Answer: = $$4 \frac{7}{8}$$ Explanation: 7 – $$2\frac{1}{8}$$ = $$\frac{7}{1}-2\frac{1}{8}$$ = $$\frac{7}{1}-\frac{17}{8}$$ = $$\frac{56 – 17}{8}$$ = $$\frac{39}{8}$$ = $$4 \frac{7}{8}$$ Question 24. $$6 \frac{3}{5}-3 \frac{4}{5}$$ Answer: $$2\frac{4}{5}$$ Explanation: $$6 \frac{3}{5}-3 \frac{4}{5}$$ = $$\frac{33}{5}-\frac{19}{5}$$ = $$\frac{33 – 19}{5}$$ = $$\frac{14}{5}$$ = $$2 \frac{4}{5}$$ Problem Solving Question 25. The average weight of a basketball is 21$$\frac{1}{8}$$ ounces. The average weight of a baseball is 5$$\frac{2}{8}$$ ounces. How many more ounces does the basketball weigh? Answer: 15$$\frac{7}{8}$$ ounces Explanation 21$$\frac{1}{8}$$ – 5$$\frac{2}{8}$$ = $$\frac{169}{8}$$ – $$\frac{42}{8}$$ = $$\frac{169 – 42}{8}$$ = $$\frac{127}{8}$$ = 15$$\frac{7}{8}$$ Question 26. What is the value of the 4 in 284,612? Answer: place value : 4000 face value : 4 Explanation : the value of 4 in 284612 is 4000, as its in thousands place and value of four is 4 Question 27. Two of the smallest mammals on Earth are the bumblebee bat and the Etruscan pygmy shrew. How much shorter is the bat than the shrew? Answer: $$\frac{1}{5}$$ inches Explanation: 1$$\frac{2}{5}$$ – 1$$\frac{1}{5}$$ = $$\frac{7}{5}$$ – $$\frac{6}{5}$$ = $$\frac{7 – 6}{5}$$ = $$\frac{1}{5}$$ Question 28. Make Sense and Persevere The average length of an adult female hand is about 6 $$\frac{3}{5}$$ inches. About how much longer is the hand than the lengths of the bat and shrew combined? Answer: 4 inches Explanation: 6 $$\frac{3}{5}$$ – (1$$\frac{2}{5}$$ + 1$$\frac{1}{5}$$) = 6 $$\frac{3}{5}$$-($$\frac{7}{5}$$ + $$\frac{6}{5}$$) = $$\frac{33}{5}$$-($$\frac{7 + 6}{5}$$) = $$\frac{33}{5}$$–$$\frac{13}{5}$$ = $$\frac{33 – 13}{5}$$ = $$\frac{20}{5}$$ = 4 Question 29. Jack made 5$$\frac{1}{4}$$ dozen cookies for the bake sale, and his sister made 3$$\frac{3}{4}$$ dozen cookies. How many more dozen cookies did Jack make than his sister? Answer: 1$$\frac{2}{4}$$ Explanation: 5$$\frac{1}{4}$$ – 3$$\frac{3}{4}$$ = $$\frac{21}{4}$$ – $$\frac{15}{4}$$ = $$\frac{21 – 15}{4}$$ = $$\frac{6}{4}$$ = 1$$\frac{2}{4}$$ Question 30. Higher Order Thinking Jenna has a spool that contains 5$$\frac{3}{4}$$ meters of ribbon. She uses 3$$\frac{2}{4}$$ meters for a school project and 1$$\frac{1}{4}$$ meters for a bow. How much ribbon remains on the spool? Answer: 1 meter Explanation: 5$$\frac{3}{4}$$ – (3$$\frac{2}{4}$$ + 1$$\frac{1}{4}$$) = $$\frac{23}{4}$$ – ($$\frac{14}{4}$$ + $$\frac{5}{4}$$) = $$\frac{23}{4}$$ – ($$\frac{14 + 5}{4}$$) = $$\frac{23}{4}$$ – $$\frac{19}{4}$$ = $$\frac{23 – 19}{4}$$ = $$\frac{4}{4}$$ =1 Assessment Practice Question 31. Last week, the office used 5$$\frac{1}{12}$$ boxes of paper. This week, they used 1$$\frac{5}{12}$$ boxes of paper. How many more boxes did they use last week than this week? A. 10 $$\frac{6}{12}$$ boxes B. 4 $$\frac{8}{12}$$ boxes C. 4 $$\frac{4}{12}$$ boxes D. 3 $$\frac{8}{12}$$ boxes Answer: Option D. 3 $$\frac{8}{12}$$ boxes Explanation: 5$$\frac{1}{12}$$ – 1$$\frac{5}{12}$$ = 5$$\frac{1}{12}$$ – 1$$\frac{5}{12}$$ = $$\frac{61}{12}$$ – $$\frac{17}{12}$$ = $$\frac{61 – 17}{12}$$ = $$\frac{44}{12}$$ = 3$$\frac{8}{12}$$ Question 32. A store sold 6$$\frac{1}{5}$$ cases of juice on Friday and 4$$\frac{4}{5}$$ cases of juice on Saturday. How many more cases of juice did the store sell on Friday than on Saturday? A. 11 cases B. 3 $$\frac{1}{5}$$ cases C. 2 $$\frac{2}{5}$$ cases D. 1 $$\frac{2}{5}$$ cases Answer: D. 1 $$\frac{2}{5}$$ cases Explanation: 6$$\frac{1}{5}$$ – 4$$\frac{4}{5}$$ =6$$\frac{1}{5}$$ – 4$$\frac{4}{5}$$ =$$\frac{31}{5}$$ – $$\frac{24}{5}$$ =$$\frac{31 – 24}{5}$$ =$$\frac{7}{5}$$ =1$$\frac{2}{5}$$ ### Lesson 9.10 Problem Solving Model with Math Solve & Share The table shows how long Jamie studied for a math test over 3 days. How much more time did Jamie spend studying on Tuesday and Wednesday than on Thursday? I can … use math I know to represent and solve problems. Answer: =$$\frac{2}{4}$$ Explanation: (1$$\frac{3}{4}$$ + $$\frac{3}{4}$$) – ($$\frac{2}{4}$$) =(1$$\frac{3}{4}$$ + $$\frac{3}{4}$$) – ($$\frac{2}{4}$$) =($$\frac{7}{4}$$ + $$\frac{3}{4}$$) – ($$\frac{2}{4}$$) =($$\frac{7 – 3}{4}$$ – $$\frac{2}{4}$$) =($$\frac{4}{4}$$ – $$\frac{2}{4}$$) =$$\frac{4-2}{4}$$ =$$\frac{2}{4}$$ Thinking Habits Be a good thinker! These questions can help you. • 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? Look Back! Model with Math What representations can you use to help solve this problem? Essential Question How Can You Use Math to Model Problems? Visual Learning Bridge Brad and his father hiked the Gadsen Trail and the Rosebriar Trail on Saturday. They hiked the Eureka Trail on Sunday. How much farther did they hike on Saturday than on Sunday? What do you need to find? I need to find how far Brad and his father hiked on Saturday and how much farther they hiked on Saturday than on Sunday. How can I model with math? I can • use previously learned concepts and skills. • use bar diagrams and equations to represent and solve this problem. • decide if my results make sense. Here’s my thinking. Find 2$$\frac{4}{10}$$ – $$\frac{6}{10}$$. Use a bar diagram and write an equation to solve. Brad and his father hiked 1$$\frac{8}{10}$$ miles farther on Saturday than on Sunday. Convince Me! Model with Math How do the bar diagrams help you decide if your answer makes sense? Answer : Bar diagram is the pictorial representation of math model as per the above picture on Saturday, Brad and his father hiked 1$$\frac{9}{10}$$ + $$\frac{5}{10}$$ = 2$$\frac{4}{10}$$ and on Sunday $$\frac{6}{10}$$ the difference is denoted by d and is calculated as below d = 2$$\frac{4}{10}$$ – $$\frac{6}{10}$$ = 1$$\frac{8}{10}$$ Guided Practice Model with Math Alisa hiked a trail that was $$\frac{9}{10}$$ mile and Joseph hiked a trail that was $$\frac{5}{10}$$ mile. How much farther, d, did Alisa hike than Joseph? Question 1. Draw a bar diagram to represent the problem and show the relationships among the quantities. Answer: Question 2. What equation can you write to represent the problem? Answer: d = $$\frac{9}{10}$$ – $$\frac{5}{10}$$ Explanation: d = $$\frac{9}{10}$$ – $$\frac{5}{10}$$ d = $$\frac{9 – 5}{10}$$ d = $$\frac{4}{10}$$ Question 3. How much farther did Alisa hike than Joseph? Answer: $$\frac{4}{10}$$ Explanation: d = $$\frac{9}{10}$$ – $$\frac{5}{10}$$ d = $$\frac{9 – 5}{10}$$ d = $$\frac{4}{10}$$ Independent Practice Model with Math The smallest female spider measures about $$\frac{3}{5}$$ millimeter in length. The smallest male spider measures about $$\frac{1}{5}$$ millimeter in length. How much longer, n, is the smallest female spider than the smallest male spider? Use Exercises 4-6 to answer the question. Question 4. Draw a picture and write an equation to represent the problem. Answer: Question 5. What previously learned math can you use to solve the problem? Answer: YES Explanation: n = $$\frac{3}{5}$$ – $$\frac{1}{5}$$ n = $$\frac{3 – 1}{5}$$ n = $$\frac{2}{5}$$ Question 6. How much longer is the smallest female spider than the smallest male spider? Answer: $$\frac{2}{5}$$ Explanation: n = $$\frac{3}{5}$$ – $$\frac{1}{5}$$ n = $$\frac{3 – 1}{5}$$ n = $$\frac{2}{5}$$ Problem Solving Performance Task On Safari Sandra and Ron traveled in a safari car while they were in Tanzania. The diagram shows the distances in miles they traveled from start to finish. How far did Sandra and Ron travel from the leopards to the elephants? Question 7. Reasoning What quantities are given in the problem and what do they mean? Answer: Sandra and Ron traveled in a safari car in Tanzania Quantities are given in the problem from starting point $$\frac{3}{8}$$ mile to see zebras 2$$\frac{4}{8}$$ miles to see Leopards 1$$\frac{5}{8}$$ miles to see Elephants 1$$\frac{7}{8}$$ miles to Finish Question 8. Make Sense and Persevere What is a good plan for solving the problem?. Answer: First add the distances from start point to zebra and Leopards fractions $$\frac{3}{8}$$+$$\frac{4}{8}$$ = $$\frac{7}{8}$$ then subtract the above fraction from distance from start to Elephant 1$$\frac{5}{8}$$ – $$\frac{7}{8}$$ = $$\frac{13 – 7}{8}$$ = $$\frac{6}{8}$$ Question 9. Model with Math Draw pictures and write and solve equations to find how far Sandra and Ron travel from the leopards to the elephants. Answer: Sandra and Ron travel from the leopards to the elephants n = 1$$\frac{5}{8}$$ – $$\frac{7}{8}$$ n = $$\frac{13}{8}$$ – $$\frac{7}{8}$$ n = $$\frac{13 – 7}{8}$$ n = $$\frac{6}{8}$$ ### Topic 9 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 sum is exactly 1,000. B. The sum is exactly 1,001. C. The difference is exactly 371. D. The difference is between 40 and 45. E. The difference is exactly 437. F. The difference is between 150 and 160. G. The sum is between 995 and 1,000. H. The sum is exactly 1,899. ### Answer: ### Topic 9 Vocabulary Review Understand Vocabulary Word List • decompose • denominator • equivalent fractions • fraction • like denominators • mixed number • numerator • whole number Question 1. Circle the label that best describes $$\frac{1}{2}$$. fraction mixed number whole number Answer: Question 2. Circle the label that best describes 1$$\frac{1}{3}$$. fraction mixed number whole number Answer: Question 3. Circle the label that best describes 4. fraction mixed number whole number Answer: Question 4. Draw a line from each term to its example. Answer: Use Vocabulary in Writing Question 5. Find 1$$\frac{1}{3}$$ + 2$$\frac{2}{3}$$. Use at least 3 terms from the Word List to describe how to find the sum. Answer: To add mixed numbers, firstly convert the mixed numbers to improper fractions and find the least common denominator, then add or subtract the whole numbers. Finally write the lowest terms. Sum = 1$$\frac{1}{3}$$ + 2$$\frac{2}{3}$$ = 4 ### Topic 9 Reteaching Set A pages 333-336, 341-344 Remember you can use tools or add the numerators and write the sum over the like denoninator. Question 1. $$\frac{2}{5}+\frac{2}{5}$$ Answer: $$\frac{4}{5}$$ Explanation: $$\frac{2}{5}+\frac{2}{5}$$ =$$\frac{2 + 2}{5}$$ =$$\frac{4}{5}$$ Question 2. $$\frac{2}{4}+\frac{1}{4}+\frac{1}{4}$$ Answer: $$\frac{4}{4}$$ = 1 Explanation: $$\frac{2}{4}+\frac{1}{4}+\frac{1}{4}$$ = $$\frac{2 + 1 + 1}{4}$$ = $$\frac{4}{4}$$ = 1 Question 3. $$\frac{3}{8}+\frac{4}{8}$$ Answer: $$\frac{7}{8}$$ Explanation: $$\frac{3}{8}+\frac{4}{8}$$ = $$\frac{3 + 4}{8}$$ = $$\frac{7}{8}$$ Question 4. $$\frac{4}{10}+\frac{2}{10}+\frac{3}{10}$$ Answer: $$\frac{9}{10}$$ Explanation: $$\frac{4}{10}+\frac{2}{10}+\frac{3}{10}$$ = $$\frac{4 + 2 + 3}{10}$$ = $$\frac{9}{10}$$ Question 5. $$\frac{4}{10}+\frac{3}{10}$$ Answer: $$\frac{7}{10}$$ Explanation: $$\frac{4}{10}+\frac{3}{10}$$ = $$\frac{4 + 3}{10}$$ = $$\frac{7}{10}$$ Question 6. $$\frac{7}{12}+\frac{2}{12}$$ Answer: $$\frac{9}{12}$$ Explanation: $$\frac{7}{12}+\frac{2}{12}$$ = $$\frac{7 + 2}{12}$$ = $$\frac{9}{12}$$ Set B pages 337-340 Decompose 1$$\frac{5}{6}$$ two different ways. Remember you can decompose fractions in more than one way. Decompose each fraction or mixed number in two different ways. Question 1. $$\frac{3}{5}$$ Answer: $$\frac{1}{5}$$ + $$\frac{1}{5}$$ + $$\frac{1}{5}$$ $$\frac{2}{5}$$ + $$\frac{1}{5}$$ Explanation: Question 2. $$\frac{9}{12}$$ Answer: $$\frac{6}{12}$$ + $$\frac{3}{12}$$ $$\frac{3}{12}$$ +$$\frac{3}{12}$$ + $$\frac{3}{12}$$ Explanation: Question 3. 1$$\frac{1}{2}$$ Answer: $$\frac{1}{2}$$ + 1$$\frac{2}{2}$$ $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$ Explanation: Question 4. 2$$\frac{2}{3}$$ Answer: $$\frac{3}{3}$$ + $$\frac{3}{3}$$ +$$\frac{2}{3}$$ $$\frac{2}{3}$$ + $$\frac{2}{3}$$ +$$\frac{2}{3}$$ + $$\frac{2}{3}$$ Explanation: Set C pages 345-352 Find $$\frac{5}{8}-\frac{2}{8}$$ $$\frac{5}{8}-\frac{2}{8}=\frac{3}{8}$$ Subtract the numerators. Keep the like denominator. Remember you can use different tools to show how to subtract fractions. Question 1. $$\frac{3}{3}-\frac{1}{3}$$ Answer: $$\frac{2}{3}$$ Explanation: $$\frac{3}{3}-\frac{1}{3}$$ =$$\frac{3 – 1}{3}$$ =$$\frac{2}{3}$$ Question 2. $$\frac{5}{6}-\frac{2}{6}$$ Answer: $$\frac{3}{6}$$ =$$\frac{1}{2}$$ Explanation: $$\frac{5}{6}-\frac{2}{6}$$ =$$\frac{5 – 2}{6}$$ =$$\frac{3}{6}$$ =$$\frac{1}{2}$$ Question 3. $$\frac{6}{8}-\frac{3}{8}$$ Answer: $$\frac{3}{8}$$ Explanation: $$\frac{6}{8}-\frac{3}{8}$$ =$$\frac{6 – 3}{8}$$ =$$\frac{3}{8}$$ Question 4. $$\frac{4}{10}-\frac{3}{10}$$ Answer: $$\frac{1}{10}$$ Explanation: $$\frac{4}{10}-\frac{3}{10}$$ =$$\frac{4 – 3}{10}$$ =$$\frac{1}{10}$$ Question 5. $$\frac{5}{5}-\frac{3}{5}$$ Answer: $$\frac{2}{5}$$ Explanation: $$\frac{5}{5}-\frac{3}{5}$$ =$$\frac{5 – 3}{5}$$ =$$\frac{2}{5}$$ Question 6. $$\frac{4}{6}-\frac{2}{6}$$ Answer: $$\frac{2}{6}$$ = $$\frac{1}{3}$$ Explanation: $$\frac{4}{6}-\frac{2}{6}$$ =$$\frac{4-2}{6}$$ =$$\frac{2}{6}$$ = $$\frac{1}{3}$$ Set D pages 353-356 Find the sum or difference shown on each number line. Remember you can show adding or subtracting fractions on a number line. Write each equation shown. Question 1. Answer: $$\frac{3}{4}$$ Explanation: $$\frac{1}{4}+\frac{2}{4}$$ = $$\frac{3}{4}$$ Question 2. Answer: $$\frac{4}{6}$$ Explanation: $$\frac{5}{6}+\frac{1}{6}$$ = $$\frac{4}{6}$$ Set E pages 357-368 Remember you can use different tools to add and subtract mixed numbers. Question 1. $$5 \frac{4}{8}+2 \frac{1}{8}$$ Answer: 7$$\frac{5}{8}$$ Explanation: Question 2. $$3 \frac{3}{6}+1 \frac{5}{6}$$ Answer: 5$$\frac{3}{8}$$ Explanation: Question 3. $$5 \frac{7}{10}+4 \frac{4}{10}$$ Answer: 10$$\frac{1}{10}$$ Explanation: Question 4. $$9-3 \frac{3}{8}$$ Answer: 5$$\frac{3}{8}$$ Explanation: Set F pages 369-372 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? Remember to draw a bar diagram and write an equation to help solve a problem. Bonnie ran $$\frac{1}{4}$$ mile, Olga ran $$\frac{3}{4}$$ mile, Gracie ran $$\frac{5}{4}$$ miles, and Maria ran $$\frac{2}{4}$$ mile. How much farther, f, did Gracie run than Bonnie and Maria combined, c? Answer: $$\frac{2}{4}$$ Explanation: $$\frac{5}{4}$$ – ( $$\frac{2}{4}$$ + $$\frac{1}{4}$$) = $$\frac{5}{4}$$ – $$\frac{2+1}{4}$$ = $$\frac{5 }{4}$$ – $$\frac{3}{4}$$ = $$\frac{5 – 3 }{4}$$ = $$\frac{2}{4}$$ ### Topic 9 Assessment Practice Question 1. Match each expression on the left to an equivalent expression. Answer: Question 2. On Monday, $$\frac{3}{12}$$ of the students went on a field trip. What fraction of the students did NOT go on the field trip? Explain. Answer: $$\frac{9}{12}$$ Explanation: 1 – $$\frac{3}{12}$$ = $$\frac{12}{12}$$ – $$\frac{3}{12}$$ =$$\frac{9}{12}$$ Question 3. Riley planted flowers in some of her garden. Then, she planted vegetables in $$\frac{2}{8}$$ of her garden. Now, $$\frac{7}{8}$$ of Riley’s garden is planted. What fraction of Riley’s garden is planted with flowers? How much of her garden is not planted? A. $$\frac{2}{8}$$ of her garden; $$\frac{1}{8}$$ of her garden is not planted B. $$\frac{3}{8}$$ of her garden; $$\frac{2}{8}$$ of her garden is not planted C. $$\frac{4}{8}$$ of her garden; $$\frac{5}{8}$$ of her garden is not planted D. $$\frac{5}{8}$$ of her garden; $$\frac{1}{8}$$ of her garden is not planted Answer: Option (D) $$\frac{5}{8}$$ of her garden; $$\frac{1}{8}$$ of her garden is not planted Explanation: f + $$\frac{2}{8}$$ = $$\frac{7}{8}$$ f = $$\frac{7}{8}$$ – $$\frac{2}{8}$$ f = $$\frac{7 – 2}{8}$$ f = $$\frac{5}{8}$$ (flowers) 1- $$\frac{7}{8}$$ = $$\frac{1}{8}$$ (garden is not planted) Question 4. Select all the expressions that show a way to decompose $$\frac{7}{8}$$. Answer: Question 5. Which equation is NOT true when $$\frac{4}{12}$$is the missing number? Answer: Option(C) equation is NOT true Explanation: Question 6. Zoe had 3$$\frac{1}{8}$$ feet of orange ribbon. She used some ribbon to make a bow for a gift. Now she has 1$$\frac{1}{8}$$ feet of ribbon left. How much orange ribbon did Zoe use? Use the model to write an equation, and solve. Answer: 1$$\frac{3}{8}$$ Explanation: 3$$\frac{1}{8}$$ – 2$$\frac{1}{8}$$ = 3$$\frac{1}{8}$$ – 1 = $$\frac{25}{8}$$ – 1 = $$\frac{25 – 8}{8}$$ = $$\frac{17}{8}$$ = 2$$\frac{1}{8}$$ 2$$\frac{1}{8}$$ – $$\frac{6}{8}$$ =$$\frac{17}{8}$$ – $$\frac{6}{8}$$ =$$\frac{17 – 6}{8}$$ =$$\frac{11}{8}$$ =1$$\frac{3}{8}$$ Question 7. Roger and Sulee each decomposed 1$$\frac{1}{6}$$. Roger wrote $$\frac{1}{6}+\frac{1}{6}+\frac{2}{6}+\frac{3}{6}$$. Sulee wrote $$\frac{3}{6}+\frac{4}{6}$$ Who was correct? Explain. Answer: Both are correct, as no change is answer Explanation: 1$$\frac{1}{6}$$ =$$\frac{7}{6}$$ Roger $$\frac{1}{6}+\frac{1}{6}+\frac{2}{6}+\frac{3}{6}$$ = $$\frac{1+1+2+3}{6}$$ = $$\frac{7}{6}$$ Sulee $$\frac{3}{6}+\frac{4}{6}$$ = $$\frac{3+4}{6}$$ = $$\frac{7}{6}$$ Question 8. The number line shows which of the following equations? Answer: Option (B) Explanation: $$\frac{6}{10}$$+$$\frac{2}{10}$$ = $$\frac{6+2}{10}$$= $$\frac{8}{10}$$ Question 9. Ryan kayaks 1$$\frac{7}{8}$$ miles before lunch and 2$$\frac{3}{8}$$ miles after lunch. Select all of the equations you would use to find how far Ryan kayacked. Answer: Question 10. The Jacobys kept track of the time they spent driving on their trip. A. Find how many hours the Jacobys drove on Monday and Tuesday. Draw a bar diagram to represent the problem. Answer: 10$$\frac{2}{4}$$ Explanation: 5$$\frac{3}{4}$$+4$$\frac{3}{4}$$ = $$\frac{23}{4}$$+$$\frac{19}{4}$$ = $$\frac{23 + 19}{4}$$ = $$\frac{42}{4}$$ = 10$$\frac{2}{4}$$ B. Find how many hours the Jacobys drove in all. Explain your work. Answer: 19$$\frac{2}{4}$$ Explanation: 5$$\frac{3}{4}$$+4$$\frac{3}{4}$$ +2$$\frac{1}{4}$$+6$$\frac{3}{4}$$ = $$\frac{23}{4}$$+$$\frac{19}{4}$$ +$$\frac{9}{4}$$+$$\frac{27}{4}$$ = $$\frac{23 + 19 + 9 + 27}{4}$$ = $$\frac{23 + 19 + 9 + 27}{4}$$ = $$\frac{78}{4}$$ = 19$$\frac{2}{4}$$ ### Topic 9 Performance Task Water Race In one of the games at the class picnic, students balanced containers filled with water on their heads. The goal was to carry the most water to the finish line. The teams are listed in the Water Race Teams table. The amount of water each student carried is listed in the Water Race Results table. Question 1. Mia will hand out the prize to the winning team. Part A Draw a bar diagram and write an equation to find c, the cups of water Team 1 carried. Answer: c = 1$$\frac{5}{8}$$ Explanation: c = $$\frac{6}{8}$$ + $$\frac{7}{8}$$ c =$$\frac{6 + 7}{8}$$ c =1$$\frac{5}{8}$$ Part B How many cups of water did Team 2 carry? Use fraction strips to show the sum. Answer: 2$$\frac{4}{8}$$ Explanation: n = $$\frac{5}{8}$$ + 1$$\frac{7}{8}$$ n = $$\frac{5}{8}$$ + $$\frac{15}{8}$$ n =$$\frac{5 + 15}{8}$$ n =$$\frac{20}{8}$$ n =2$$\frac{4}{8}$$ Part C How many cups of water did Team 3 carry? Use the number line to show the sum. Answer: 2$$\frac{3}{8}$$ Explanation: f = 1$$\frac{6}{8}$$ + $$\frac{5}{8}$$ f = $$\frac{14}{8}$$ + $$\frac{5}{8}$$ f =$$\frac{14 + 5}{8}$$ f =$$\frac{19}{8}$$ f =2$$\frac{3}{8}$$ Part D Which team carried the most water? Answer: Team 2 2$$\frac{4}{8}$$ Explanation: n = $$\frac{5}{8}$$ + 1$$\frac{7}{8}$$ n = $$\frac{5}{8}$$ + $$\frac{15}{8}$$ n =$$\frac{5 + 15}{8}$$ n =$$\frac{20}{8}$$ n =2$$\frac{4}{8}$$ Question 2. Team 1 wanted to know how they did compared to Team 2. Part A Draw a bar diagram and write an equation that could be used to find m, how much more water Team 2 carried than Team 1. Answer: m = Team 2 – Team 1 = $$\frac{7}{8}$$ Explanation: Part B How much more water did Team 2 carry than Team 1? Explain how to solve the problem using your equation from Part A. Show your work. Answer: $$\frac{7}{8}$$ Explanation: Team 2 = $$\frac{5}{8}$$ + 1$$\frac{7}{8}$$ = $$\frac{5}{8}$$ + $$\frac{15}{8}$$ = $$\frac{5 + 15}{8}$$ =$$\frac{20}{8}$$ =2$$\frac{4}{8}$$ Team 1 = $$\frac{6}{8}$$ + $$\frac{7}{8}$$ =$$\frac{6 + 7}{8}$$ =1$$\frac{5}{8}$$ m = 2$$\frac{4}{8}$$ – 1$$\frac{5}{8}$$ = $$\frac{20}{8}$$ – $$\frac{13}{8}$$ = $$\frac{20 – 13}{8}$$ = $$\frac{7}{8}$$ ## Envision Math Grade 1 Answer Key Topic 2 Reteaching ## Envision Math 1st Grade Textbook Answer Key Topic 2 Reteaching Reteaching Set A You can stack cubes to find which group has more. Write each number. Circle is less than or is greater than. Question 1. Answer: Question 2. Answer: Set B You can use cube towers to put numbers in order. Use cubes. Write the numbers in order from least to greatest. Question 3. Answer: Question 4. Answer: Set C You can use a number line to find missing numbers. Write the missing number. Use the number line. Write the missing numbers. Question 5. 7 _____ 9 Answer: Question 6. 4 ______ 6 Answer: Set D You can use cubes to act out a story. Ed is 7. Ann is 4. Reza is 8. Who is oldest? 8 is the greatest number. So Reza is oldest. Use cubes to act out the story. Write the numbers in order from least to greatest. Question 7. Mary made 5 cards. Jim made 9 cards. Saul made 6 cards. Who made the most cards? ________ Answer: ## Envision Math Grade 1 Answer Key Topic 1.1 0 to 5 ## Envision Math 1st Grade Textbook Answer Key Topic 1.1 0 to 5 Review What You Know Question 1. Circle the number that tells how many. Answer: Question 2. Circle the group that has more Answer: Question 3. Circle the group that has 4 Answer: Home-School Connection Dear Family, Today my class started Topic 1, Numbers to 12. I will learn ways to count and model numbers using real objects. Here are some things we can do to help me with my math. Love, ___________ Book to Read Reading math stories reinforces concepts. Look for this title in your local library: Numbers by Henry Pluckrose (Children’s Press, 1995) Home Activity Have your child make a number collage. Ask him or her to write a number between 1 and 12 on a piece of paper. Then ask him or her to glue or tape that number of items to the paper. Use common items such as buttons, toothpicks, or paper clips. Counting Safari 0 to 5 Home Connection Your child showed the numbers 0 through 5 with counters. Home Activity Have your child use pennies or other small items to show the numbers 0 through 5 Question 1. Answer: Question 2. Answer: Question 3. Answer: Question 4. Answer: Guided Practice Draw the counters to show the number. Question 1. Answer: Question 2. Answer: Question 3. Answer: Question 4. Answer: Question 5. Answer: Question 6. Answer: Do you understand? How is 3 the same as ? Answer: Independent Practice Write the number that tells how many. Question 7. Answer: Question 8. Answer: Question 9. Answer: Question 10. Answer: Question 11. Answer: Question 12. Answer: Number Sense Question 13. Use counters. Show a number from 1 to 5. Draw the counters. Write the number. Answer: Problem Solving Solve the problems below. Question 14. Ann has 3 dogs. Each dog has a toy. Draw their toys. Answer: Question 15. Bob likes acorns. How many does he have? Answer: Question 16. Journal Draw a picture of some friends. How many did you draw? Write the number and the number word. Answer: ## enVision Math Common Core Grade 4 Answer Key Topic 14 Algebra Generate and Analyze Patterns Go through the enVision Math Common Core Grade 4 Answer Key Topic 14 Algebra: Generate and Analyze Patterns regularly and improve your accuracy in solving questions. ## enVision Math Common Core 4th Grade Answers Key Topic 14 Algebra Generate and Analyze Patterns Essential Questions: How can you use a rule to continue a pattern? How can you use a table to extend a pattern? How can you use a repeating pattern to predict a shape? enVision STEM Project: Patterns and Waves Journal: Write a Report Include what you found. Also in your report: Do Research Use the Internet or other sources to learn about 2 industries where oscilloscopes can be used. Name the industry and what can be observed using the oscilloscope. Oscilloscopes are used to observe patterns in waves. Suppose a scientist created a pattern with three levels of sounds: quiet, loud, medium. If the scientist repeats the pattern of sounds, what would be the 41st sound in the pattern? Explain. Review What You Know Vocabulary Choose the best term from the box. Write it on the blank. • even number • odd number • inverse operations • variable Question 1. A(n) ________ can be divided into groups of 2 without a remainder. Answer: Even number Explanation: Even numbers are those numbers that can be divided into two equal groups or pairs and are exactly divisible by 2. Question 2. A symbol or letter that stands for a number is called a(n) ___________. Answer: Variable Explanation: A variable is a letter or symbol used as a placeholder for an unknown value Question 3. Operations that undo each other are called _________ Answer: Inverse Operation. Explanation: Inverse operations are operations that are opposite or “undo” each other. For example, addition undoes subtraction and division undoes multiplication. Inverse operations are useful when solving equations Addition and Subtraction Patterns Add or subtract to find the missing number in each pattern. Question 4. 3, 6, 9, 12, ____,18 Answer: 15 Explanation: Pattern : Add 3 Question 5. 4,8, 12, ____, 20, 24 Answer: 16 Explanation: Pattern : Add 4 Question 6. 8, 7, 6, ____,4, 3 Answer: 5 Explanation: Pattern : Add 1 then less 1 Question 7. 30, 25, 20, 15, ____,5 Answer: 10 Explanation: Pattern : Subtract 5 Question 8. 1, 5, 9, ____, 17, 21 Answer: 13 Explanation: Pattern : Add 4 Question 9. 12, 10, 8, 6, ___, 2 Answer: 4 Explanation: Pattern : Subtract 2 Multiplication and Division Patterns Multiply or divide to find the missing number in each pattern. Question 10. 1, 3, 9, 27, ___, 243 Answer: 81 Explanation: Pattern : Multiply 3 Question 11. 64, 32, 16, ____, 4, 2 Answer: 8 Explanation: Pattern : Divide by 2 Question 12. 1,5, 25, ____, 625 Answer: 125 Explanation: Pattern : Multiply 5 Question 13. 1, 2, 4, 8, ____, 32 Answer: Explanation: Pattern : Multiply by 2 Question 14. 1, 4, 16, ____, 256 Answer: 64 Explanation: Pattern : Multiply by 4 Question 15. 729, 243, 81,27, 9, ____ Answer: 3 Explanation: Pattern : Divided by 3 Problem Solving Question 16. Look for Relationships James places 1 counter in the first box. He places 2 counters in the second box, 4 counters in the third box, 8 counters in the fourth box, and continues the pattern until he gets to the tenth box. How many counters did James place in the tenth box? Answer: 512 Explanation: Pattern is multiply by 2 Pick a Project PROJECT 14A How have roller coasters changed through the years? Project: Make a Model Roller-Coaster Car PROJECT 14B How can you use currency from different countries? Project: Make Your Own Currency PROJECT 14C How can patterns be used in sidewalks? Project: Design Your Own Sidewalk PROJECT 14D How many stadiums in the United States have retractable roofs? Project: Make a Seating Diagram ### Lesson 14.1 Number Sequences Solve & Share Look at the rules and starting numbers below. What are the next 6 numbers in each pattern? Tell how you decided. Describe features of the patterns. Solve these problems any way you choose. I can … use a rule to create and extend a number pattern and identify features of the number pattern not described by the rule. Look Back! Look for Relationships Create two patterns that use the same rule but start with different numbers. Identify a feature of each pattern. For example, identify whether the numbers are all even, all odd, or alternate between even and odd. Essential Question How Can You Use a Rule to Continue a Pattern? Visual Learning Bridge The house numbers on a street follow the rule “Add 4.” If the pattern continues, what are the next three house numbers? Describe a feature of the pattern. You can use a number line to help make sense of the problem and find the next three house numbers. Use a number line to continue the pattern. A rule is a mathematical phrase that tells how numbers or shapes in a pattern are related. The rule for the house numbers is “Add 4.” The next three house numbers are 20, 24, and 28. Describe features of the pattern. Some patterns have features that are not given in the rule. 16, 20, 24, 28 One of the features of this pattern is all of the house numbers are even numbers. Another feature is all of the house numbers are multiples of 4. Convince Me! Generalize Can you use the rule “Add 4” to create a different pattern with all odd numbers? Explain. Explanation: No, we cannot apply this rule on odd numbers when u add 4 to a odd number the answers all will be in odd numbers but not the multiples. Another Example! On another street, the house numbers follow the rule “Subtract 5.” What are the next three house numbers after 825? Describe a feature of the pattern. The next three house numbers are 820, 815, and 810. All of the house numbers are multiples of 5. Guided Practice Do You Understand? Question 1. Rudy’s rule is “Add 2.” He started with 4 and wrote the numbers below. Which number does NOT belong to Rudy’s pattern? Explain. 4,6, 8, 9, 10, 12 Answer: 9 Explanation: 9 number does NOT belong to Rudy’s pattern Because it is not the multiple of 2 Do You Know How? Continue the pattern. Describe a feature of the pattern. Question 2. Subtract 6 48, 42, 36, 30, 24, _____, _____, ______ Answer: 18,12,6. Explanation: Pattern = multiples of 6 Independent Practice For 3-6, continue each pattern. Describe a feature of each pattern. Question 3. Subtract 3: 63, 60, 57, _____, _____ Answer: 54,51,48 Explanation: Pattern is multiples of 3 Question 4. Add 7: 444, 451, 458, _____, _____ Answer: 465,472. Explanation: pattern = Multiples of 7 Question 5. Add 25: 85, 110, 135, _____, _____ Answer: 160,185 Explanation: The sequence is completed by adding 25 Question 6. Subtract 4: 75, 71, 67, _____, _____ Answer: 63,59 Explanation: The sequence is completed by subtracting 4 For 7-12, use the rule to generate each pattern. Question 7. Rule: Subtract 10 90, _____, _____ Answer: 80,70 Explanation: The sequence is completed by subtracting 10 Question 8. Rule: Add 51 16, _____, _____ Answer: 67,83 Explanation: The sequence is completed by adding 51 Question 9. Rule: Add 5 96, _____, _____ Answer: 101, 106 Explanation: The sequence is completed by adding 5 Question 10. Rule: Add 107 43, _____, _____ Answer: 150, 257 Explanation: The sequence is completed by adding 107 Question 11. Rule: Subtract 15 120, _____, _____ Answer: 105, 90 Explanation: The sequence is completed by Subtracting 15 Question 12. Rule: Subtract 19 99, _____, _____ Answer: 80, 61 Explanation: The sequence is completed by Subtracting 19 Problem Solving Question 13. Reasoning Orlando delivers mail. He sees one mailbox that does not have a number. If the numbers are in a pattern, what is the missing number? Answer: 31 Explanation: Pattern : Add 2 The missing number is 31 Question 14. A bus tour runs 9 times a day, 6 days a week. The bus can carry 30 passengers. Find the greatest number of passengers who can ride the tour bus each week. Answer: 1620 Explanation: 6 x 9 = 54 trips 54 x 30 = 1620 The greatest number of passengers who can ride the tour bus each week is 1620 passengers. Question 15. The year 2017 was the year of the Rooster on the Chinese calendar. The next year of the Rooster will be 2029. The rule is “Add 12.” What are the next five years of the Rooster? Answer: 2041, 2053, 2065, 2077, 2089 Explanation: The next five years of the Rooster are 2041, 2053, 2065, 2077, 2089 Question 16. Describe a feature of the year of the Rooster pattern. Answer: Multiples of 12 Explanation: Feature of the year of the Rooster pattern is multiples of 12 Question 17. Vocabulary Define rule. Create a number pattern using the rule “Subtract 7.” Answer: Rule = multiples of 7 Explanation: 70, 63, 56, 49 Is the pattern of Subtract 7 Question 18. Higher Order Thinking Some patterns use both addition and subtraction in their rules. The rule is “Add 3, Subtract 2.” Find the next three numbers in the pattern. 1, 4, 2, 5, 3, 6, 4, 7, _____, _____, _____ Answer: 5, 8 , 6 Explanation: The next three numbers in the pattern is 5, 8 , 6 Assessment Practice Question 19. Rima used “Subtract 3” as the rule to make a pattern. She started with 60, and wrote the next six numbers in her pattern. Which number does NOT belong in Rima’s pattern? A. 57 B. 54 C. 45 D. 26 Answer: D Explanation: 26 number does NOT belong in Rima’s pattern Question 20. Ivan counted all the beans in a jar. If he counted the beans in groups of 7, which list shows the numbers Ivan could have named? A. 77, 84, 91, 99 B. 301, 308, 324, 331 C. 574, 581, 588, 595 D. 14, 24, 34, 44 Answer: B Explanation: 301, 308, 324, 331 list shows the numbers Ivan could have named ### Lesson 14.2 Patterns: Number Rules Solve & Share There are 6 juice boxes in 1 pack, 12 in 2 packs, and 18 in 3 packs. How many juice boxes are in 4 packs? in 5 packs? in 6 packs? Use the rule to complete the table. Describe features of the pattern. Then find how many juice boxes are in 10 packs and 100 packs. I can … use a rule to extend a number pattern, identify features of the number pattern, and use the number pattern to solve a problem. Look Back! Reasoning Create a table showing the relationship between the number of bicycles and the number of bicycle wheels. Start with 1 bicycle. Complete 5 rows of the table using the rule “Multiply by 2.” Describe features of the pattern. gf Essential Question What is the Pattern? Visual Learning Bridge There are 3 leaflets on 1 cloverleaf. There are 6 leaflets on 2 cloverleaves. There are 9 leaflets on 3 cloverleaves. How many leaflets are on 4 cloverleaves? How many cloverleaves will have 12 leaflets? You can use a table to create, extend, and identify features of a pattern. How many leaflets are on 4 cloverleaves? There are 12 leaflets on 4 cloverleaves. The number of leaflets is a multiple of the number of cloverleaves. How many cloverleaves for 12 leaflets? There are 4 cloverleaves for 12 leaflets. The number of cloverleaves is a factor of the number of leaflets. Convince Me! Model with Math If you know the number of leaflets, l, what expression can you use to find the number of cloverleaves, c? If you know the number of cloverleaves, what expression can you use to find the number of leaflets? Guided Practice Do You Understand? Question 1. The rule for this table is “Multiply by 4.” What number does not belong? Answer: 15 Explanation: 4 x 4 = 16 15 does not belong to the group Do You Know How? Complete the table. Describe a feature of the pattern. Question 2. Rule: Divide by 4 Answer: 5 Explanation: 5 x 4 = 20 Multiply by 4 is the pattern Independent Practice For 3-6, use the rule to complete each table. Describe a feature of each pattern. You can multiply or divide to find the patterns in these tables. Question 3. Rule: Multiply by 8 Answer: Explanation: 2 x 8 = 16 5 x 8 = 40 The sequence is completed by the multiplies of 8 Question 4. Rule: Divide by 5 Answer: Explanation: The sequence is completed by Divide by 5 Question 5. Rule: Multiply by 16 Answer: Explanation: The sequence is completed by the multiples of 16 Question 6. Rule: Divide by 2 Answer: Explanation: The sequence is completed by dividing with 2 500 divided by 2 is 250 730 divided by 2 is 365 Problem Solving Question 7. The table shows how much money Joe makes painting. How much money will Joe make when he paints for 6 hours? Rule: Multiply by 45 Answer: 270 Explanation:$270 money will Joe make when he paints for 6 hours

Question 8.
The table shows the total number of pounds of potatoes for different numbers of bags. How many bags does it take to hold 96 pounds of potatoes?
Rule: Divide by 8

Explanation:
12 bags it take to hold 96 pounds of potatoes

Question 9.
Number Sense What is the greatest number you can make using each of the digits 1,7,0, and 6 once?
Explanation:
7106  is the greatest number you can make using each of the digits 1,7,0, and 6 once

Question 10.
Algebra A penguin can swim 11 miles per hour. At this speed, how far can it swim in 13 hours? Use s as a variable. Write and solve an equation.
Explanation:
s = 11 x 13
s = 143 miles

For 11-12, the rule is “Multiply by 3.”

Question 11.
Reasoning Using the rule, how many batteries do 8 flashlights need?
Explanation:
8 x 3 = 24
24 batteries has 8 flashlights needed.

Question 12.
Higher Order Thinking How many more batteries do 20 flashlights need than 15 flashlights? Explain.
15 x 3 = 45
20 x 3 = 60
60 – 45 = 15
Explanation:
15 more batteries do 20 flashlights need than 15 flashlights

Assessment Practice

Question 13.
There are 6 rolls in each package. Use the rule “Divide by 6” to show the relationship between the number of rolls and the number of packages. Use each digit from the box once to complete the table.

Explanation:
Used each digit from the box once to complete the table.

### Lesson 14.3 Patterns: Repeating Shapes

Solve & Share
The rule for the repeating pattern below is “Square, Triangle.” What will be the 37th shape in the pattern? Explain. Solve this problem any way you choose.
I can … use a rule to predict a number or shape in a pattern.

Look Back! When the pattern has 37 shapes, how many are triangles?
37 divided by 2 is 18 times triangles
37 is the triangle

Essential Question
How Can You Use a Repeating Pattern to Predict a Shape?

Visual Learning Bridge
Rashad is making a repeating pattern for the rule “Triangle, Square, Trapezoid.” What will be the 49th shape in the pattern?
A repeating pattern is made up of shapes or numbers that form a part that repeats.

Explanation:
49th shape is triangle
48 divided by 3 is 16
so, the next is triangle

Look for Features of the Repeating Pattern

The trapezoid is the 3rd, 6th, and 9th shape in the pattern. The positions of the trapezoids are multiples of 3.

The triangle is the 15th, 4th, and 7th shape in the pattern. The positions of the triangles are 1 more than a multiple of 3.

The square is the 2nd, 5th, and 8th shape in the pattern. The positions of the squares are 1 less than a multiple

Use the Repeating Pattern to Solve
When you divide 49 by 3, the quotient is 16 R1. The pattern repeats 16 times. The 1st shape in the repeating pattern, a triangle, then appears.

49 is one more than a multiple of 3.
The 49th shape is a triangle.

Convince Me! Be Precise Suppose the rule is “Square, Triangle, Square, Trapezoid” in a repeating pattern. What is the 26th shape in the pattern? Describe features of the repeating pattern. Be precise in your description.

Another Example!
Write the next three numbers in the repeating pattern. Then name the 100th number in the pattern.

Rule: 1, 3, 5, 7 1, 3, 5, 7, 1, 3, 5, 7, 1, 3, 5, 7, 1, 3, 5 …
There are 4 items in the repeating pattern. To find the 100th number, divide by 4. The pattern repeats 25 times. The 100th number is 7.

Guided Practice

Do You Understand?
Question 1.
In the “Triangle, Square, Trapezoid” example on the previous page, what will be the 48th shape? the 50th shape? Explain.
Answer: 48th is trapezoid, 50th is square
Explanation:
When you divide 49 by 3, the quotient is 16 R1. The pattern repeats 16 times. The 1st shape in the repeating pattern, a triangle, then appears.

Do You Know How?
Question 2.
What is the 20th shape? The rule is “Triangle, Circle, Circle.”

Explanation:
When you divide 20 by 3, the quotient is 6 R2. The pattern repeats 6 times. The 1st shape in the repeating pattern, a circle, then appears.

Question 3.
Write the next three numbers. The rule is “9, 2, 7, 6.”
9, 2, 7, 6, 9, 2, 7, 6, ____, _____, ______
Explanation:
The next three numbers 9, 2, 7

Independent Practice

For 4-7, draw or write the next three items to continue each repeating pattern.
Question 4.
The rule is “Square, Triangle, Square.”

Explanation:
The pattern is completed with the help of rule

Question 5.
The rule is “Up, Down, Left, Right.”

Explanation:
The pattern is completed with the help of rule

Question 6.
The rule is “1, 1, 2.”
1, 1, 2, 1, 1, 2, ____ , ____, ____ ……
Explanation:
The pattern is completed with the help of rule

Question 7.
The rule is “5, 7, 4, 8.”
5, 7, 4, 8, 5, 7, 4, 8, 5, 7, ____, ____, ____ ……
Explanation:
The pattern is completed with the help of rule

For 8-9, determine the given shape or number in each repeating pattern.
Question 8.
The rule is “Tree, Apple, Apple.” What is the 19th shape?

Explanation:
When you divide 19 by 3, the quotient is 6 R1. The pattern repeats 6 times. The 1st shape in the repeating pattern, a tree, then appears.

Question 9.
The rule is “1, 2.” What is the 42nd number?
1, 2, 1, 2, 1, 2, …
Explanation:
When you divide 42 by 2, the quotient is 21 R0. The pattern repeats 21 times.
The pattern is closed so the pattern end with 2.

Problem Solving

Question 10.
Create a repeating pattern using the rule “Triangle, Square, Square.”

Explanation:
Created a repeating pattern using the rule “Triangle, Square, Square.”
Ends with the triangle.

Question 11.
enVision® STEM Margot measured the distance for 6 wavelengths of visible light as 2,400 nanometers. What is the distance for 1 wavelength?
2400 divided by 6 = 400
Explanation:
400 is the distance for 1 wavelength

Question 12.
Look for Relationships Hilda is making a repeating pattern with the shapes below. The rule is “Heart, Square, Triangle.” If Hilda continues the pattern, what will be the 11th shape?

Explanation:
When you divide 11 by 3, the quotient is 3 R2. The pattern repeats 3 times.
The 2nd shape in the repeating pattern, a square, then appears.

Question 13.
Look for Relationships Josie puts beads on a string in a repeating pattern. The rule is “Blue, Green, Yellow, Orange.” There are 88 beads on her string. How many times did Josie repeat her pattern?
Explanation:
When you divide 88 by 4, the quotient is 22 R0. The pattern repeats 22 times.
The pattern is closed so the pattern end with yellow

Question 14.
How many more years passed between the first steam locomotive and the gasoline-powered automobile than between the gasoline-powered automobile and the first diesel locomotive?

Explanation:
steam locomotive and the gasoline-powered automobile
1885 – 1804 = 81
81 years
gasoline-powered automobile and the first diesel locomotive
1912 – 1885= 27

Question 15.
Louisa used the rule “Blue, Green, Green, Green” to make a bracelet with a repeating pattern. She used 18 green beads. How many beads did Louisa use to make the bracelet? How many beads were NOT green?
18 divided by 3 is 6
so, pattern repeated 6 times
6 x 4 = 24
24 – 18 = 6
Explanation:

Question 16.
Higher Order Thinking Marcus is using shapes to make a repeating pattern. He has twice as many circles as squares. Make a repeating pattern that follows this rule.
Answer: The rule is circle , square , square

Explanation:
The pattern is made according to the rule.

Assessment Practice

Question 17.
Which rules give a repeating pattern that has a square as the 15th shape? Select all that apply.
☐ Square, Circle
☐Circle, Square, Triangle
☐ Square, Circle, Triangle
☐ Circle, Triangle, Square
☐ Trapezoid, Circle, Square

Explanation:
Selected all that apply
where the 15th shape is a square.

Question 18.
Which rules give a repeating pattern that has a 7 as the 15th number? Select all that apply.
☐ 1, 7
☐ 1, 7, 9
☐ 1, 9, 7
☐ 1, 7, 7
☐ 7, 1, 9

Explanation:
selected pattern rules give a repeating pattern that has a 7 as the 15th number

### Lesson 14.4 Problem Solving

Look For and Use Structure
Solve & Share
Evan’s baby brother is stacking blocks. Using the rule “Add 1 block to the number of blocks in the previous stack,” how many blocks will be in the 6th stack? Explain. Justify your answer.
I can … use patterns to help solve problems.

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?

Look Back! Look For Relationships How many blocks are in the 10th stack? Explain.

Essential Question
How Can I Look For and Make Use of Structure?

Visual Learning Bridge
Alisa made three walls with cubes. She recorded her pattern. If she continues the pattern, how many cubes will be in a 10-layer wall? a 100-layer wall?

What do you need to do to find the number of cubes in a 10-layer and 100-layer wall?
I need to continue the pattern using the rule and analyze the pattern to find features not stated in the rule itself.

How can I make use of structure to solve this problem?
I can

• look for and describe patterns in three dimensional shapes.
• use the rule that describes how objects or values in a pattern are related.
• use features of the pattern not stated in the rule to generate or extend the pattern.

Here’s my thinking.
Make a table and look for patterns.

There are 4 cubes in each layer. Multiply the number of layers by 4 to calculate the number of cubes.
A 10-layer wall contains 10 × 4 = 40 cubes.
A 100-layer wall contains 100 × 4 = 400 cubes.

Convince Me! Look for Relationships How could you use multiples to describe Alisa’s pattern?

Guided Practice

Use Structure
Leah arranged triangular tiles in a pattern like 1 row the one shown. She used the rule “Multiply the number of rows by itself to get the number of small triangles.” How many small triangles would be in the pattern if there were 10 rows?

Question 1.
Complete the table to help describe the pattern.

Explanation:
Multiplied the number by itself

Question 2.
Describe the pattern another way.
4 x 4 = 12
5 x 5 = 25
Explanation:
used the rule “Multiply the number of rows by itself to get the number of small triangles

Question 3.
How many triangles would be in 10 rows?
Explanation:
10 x 10= 100
used the rule “Multiply the number of rows by itself to get the number of small triangles

Independent Practice

Look for Relationships
Alan built the towers shown using the rule “Each story has 2 blocks.” How many blocks will a 10-story tower have? Use Exercises 4-6 to answer the question.

Question 4.
Complete the table to help describe the pattern.

Answer: 10 x 2 = 20

Explanation:

Question 5.
What is another way to describe the pattern that is not described by the rule?
Explanation:

Question 6.
How many blocks are in a 10-story tower? Explain.
Answer: 10 x 2 = 20
Explanation:
used the rule “Multiply the number with 2″

Problem Solving

Glass Stairs
An art gallery staircase is built using glass cubes. The diagram below shows 4 steps are 4 cubes high and 4 cubes across. Five steps are 5 cubes high and 5 cubes across. How many glass cubes are used to make 7 steps? Use Exercises 7-10 to answer the question.

Question 7.
Make Sense and Persevere What do you know, and what do you need to find?
4 steps are 4 cubes high and 4 cubes across. Five steps are 5 cubes high and 5 cubes across
is what we know
7 steps is to find
seven steps are 7 cubes high and 7 cubes across

Question 8.
Reasoning Complete the table.

Explanation:
Pattern = multiply by 3

Question 9.
Look For Relationships What pattern can you determine from the table?

Answer: Pattern = multiply by 3
Explanation:
multiply by 3 pattern can you determine from the table

Question 10.
Reasoning How many cubes are needed for 7 steps? Write and solve an equation.
Explanation:
7 x 7 = 49
seven steps are 7 cubes high and 7 cubes across
49 cubes are needed for 7 steps

### Topic 14 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 1 and Partner 2 each point to a black number at the same time. Each partner adds the two numbers. If the answer is on your color, you get a tally mark. Work until one partner has twelve tally marks.
I can … add multi-digit whole numbers.

### Topic 14 Vocabulary Review

Understand Vocabulary
Word List

• equation
• even number
• factor
• multiple
• odd number
• repeating pattern
• rule
• unknown

Question 1.
Circle the term that best describes 28.
even
odd
equation
unknown
Explanation:
Even is the word which is divisible by 2

Question 2.
Circle the term that best completes this sentence: 4 is a _________ of 16.
even
odd
factor
multiple
Explanation:
2, 4 , 8 are the factors of 16

Question 3.
Circle the term that best describes 17.
even
odd
equation
unknown
Explanation:
Odd numbers are the numbers that cannot be divided by 2 evenly.

Question 4.
Circle the term that best completes this sentence: 9 is a _________ of 3.
even
odd
factor
multiple
Explanation:
3 x 3 = 9
9 is the multiple of 3

Question 5.
Draw a line from each term to its example.

Explanation:
The preview  words are matched with the correct answers.

Use Vocabulary in Writing
Question 6.
Use at least 3 terms from the Word List to describe the pattern.
50, 48, 46, 44, 42 …
Even numbers
Multiple
Rule
Explanation:
The above all numbers are even numbers
They are multiples of 2
Rule= less 2

### Topic 14 Reteaching

Set A pages 521-524

You can use the rule “Subtract 3” to continue the pattern.

The next three numbers in the pattern are 9,6, and 3
A feature of the pattern is all the numbers are multiples of 3.
Another feature is all the numbers in the pattern alternate even, odd.

Remember to check that the numbers in your pattern follow the rule.

Use the rule to continue each pattern. Describe a feature of the pattern.
Question 1.
771, 791, 811, _____, _____, _____
Explanation:
A feature of the pattern is all the numbers are odd numbers

Question 2.
Rule: Subtract 12
122, 110, 98, _____, _____, ______
Explanation:
A feature of the pattern is all the numbers are even numbers

Set B pages 525-528

The regular price is twice the sale price. You can use the rule “Divide by 2” to continue the pattern.

Remember to look for features of the pattern not described by the rule.

Use the rule to continue each pattern. Describe a feature of the pattern.
Question 1.
Rule: Multiply by 18

Explanation:
A feature of the pattern is all the numbers are even numbers

Question 2.
Rule: Divide by 9

Answer:  $100 Explanation: A feature of the pattern is all the numbers are even and odd numbers Question 3. Rule: Multiply by 24 Answer: Explanation: A feature of the pattern is all the numbers are even Set C pages 529-532 You can use the rule “Circle, Triangle, Square” to continue the repeating pattern. You can use the rule to find the 25th shape in the pattern. 25 ÷ 3 = 8 R1. The pattern will repeat 8 times, then the 1st shape will appear. The circle is the 25th shape in the pattern. Remember to use the rule to continue the pattern. Question 1. a. Draw the next three shapes in the repeating pattern. The rule is “Right, Up, Up.” Answer: Explanation: Up, Up, Right is the next three shapes in the repeating pattern b. Draw the 50th shape in the pattern. Answer: Explanation: You can use the rule to find the 50th shape in the pattern. 50 ÷ 3 = 16 R2. The pattern will repeat 16 times, then the 1st shape will appear. The up is the 50th shape in the pattern. Question 2. a. Write the next three numbers in the repeating pattern. The rule is “3, 5, 7, 9.” 3, 5, 7, 9, 3, 5, 7, ____, _____, ______ Answer: 9, 3, 5, 7 Explanation: 9, 3, 5, 7 the next three numbers in the repeating pattern b. What will be the 100th number in the pattern? Answer: Explanation: You can use the rule to find the 100th number in the pattern. 100 ÷ 4 = 25 R0. The pattern will repeat 25 times, There the pattern ends so, the 100th number is 7 Set D pages 533-536 Think about these questions to help you Look For and Use Structure. 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? Remember to use the rule that describes how objects or values in a pattern are related. Sam creates a pattern using the rule “Each story has 3 blocks.” Question 1. Draw the next shape in Sam’s pattern. Answer: Explanation: 4 x 4 is the Sam’s next shape Question 2. Use the rule to continue Sam’s pattern. Answer: Explanation: pattern = multiple of 3 Question 3. How many blocks are in the 10th shape in Sam’s pattern? Answer: 10 x 10 = 100 Explanation: 100 blocks are in the 10th shape in Sam’s pattern. ### Topic 14 Assessment Practice Question 1. Football players come out of the tunnel, and their jerseys have the number pattern shown below. They follow the rule “Add 4.” A. What number belongs on the front of the blank jersey? Explain. Answer: 24 Explanation: Number 24 belongs on the front of the blank jersey B. Describe two features of the pattern. Answer: Explanation: A feature of the pattern is all the numbers are even A feature of the pattern is all the numbers are multiples of 4 Question 2. One dozen eggs is 12 eggs. Two dozen eggs is 24 eggs. Match the number of dozens to the number of eggs. The rule is “Multiply by 12.” Answer: Explanation: Matched the eggs with the dozen Question 3. Use the rule “Multiply by 6” to continue the pattern. Then describe a feature of the pattern. Answer: Explanation: A feature of the pattern is all the numbers are even Question 4. Use the rule “Divide by 3” to continue the pattern. Then write 4 terms of a different pattern that follows the same rule. 729, 243, 81, ____, _____ Answer: 27, 9 , 3, 1 Explanation: 4 terms of a different pattern that follows the same rule is 27, 9 , 3, 1 Question 5. Nicole arranges her shopping purchases by price. Each item costs$6 more than the last. The first item costs $13. The last costs$61. Her brother John says that the price of each item is an odd number. Is John correct? Find the cost of each item to explain.
Answer: 19 , 25 , 31, 37 , 43, 49, 55, 61.
Explanation:
John says that the price of each item is an odd number. yes, John correct

Question 6.
The rule for the repeating pattern is “5, 7, 2, 8.” Write the next three numbers in the pattern. Then tell what will be the 25th number in the pattern. Explain.
5, 7, 2, 8, 5, 7, 2, 8, 5, _____, _____, _____
5, 7, 2, 8, 5, 7, 2, 8, 5, 7, 2, 8
Explanation:
You can use the rule to find the 25th number in the pattern.
25 ÷ 4 = 6 R1.
The pattern will repeat 6 times,
then the 1st number will appear.
The 5 is the 25th number in the pattern

Question 7.
Jackson wrote different patterns for the rule “Subtract 5.” Select all of the patterns that he could have written. Then write 4 terms of a different pattern that follows the same rule.
☐ 27, 22, 17, 12,7
☐ 5, 10, 15, 20, 25
☐ 55, 50, 35, 30, 25
☐ 100, 95, 90, 85, 80
☐ 75, 65, 55, 45, 35

Explanation:
different patterns for the rule “Subtract 5.”
Selected all of the patterns that he could have written

Question 8.
The rule is “Subtract 7.” What are the next 3 numbers in the pattern? Describe two features of the pattern.
70,63, 56, 49, 42,35
Explanation:
28, 21, 14, 7 the next 3 numbers in the pattern

Question 9.
The table shows the different number of teams formed by different numbers of players. The rule is “Divide by 8.”

A. How many teams can be formed with 40 players?
_______ teams
Explanation:
5 teams can be formed with 40 players

B. How many players are there on 13 teams? How do you know?
Explanation:
104 players are there on 13 teams

Question 10.
A. Select all the true statements for the repeating pattern. The rule is “Circle, Heart, Triangle.”

☐ The next shape is the circle.
☐ The circle only repeats twice.
☐ The 10th shape is the heart.
☐ The 12th shape is the triangle.
☐ The circle is the 1st, 4th, 7th, etc. shape.

Explanation:
Selected all the true statements for the repeating pattern.

B. How many triangles are there among the first 22 shapes?
Explanation:
7 triangles are there among the first 22 shapes

Wall Hangings Michael uses knots to make wall hangings to sell.

Question 1.
The Michael’s Basic Wall Hanging figure shows a simple wall hanging Michael makes by repeating the shapes shown. What is the 16th shape in the repeating pattern? The rule is “Circle, Triangle, Square.” Explain.
Explanation:
You can use the rule to find the 25th number in the pattern.
16 ÷ 3 =  R1.
The pattern will repeat 5 times,
then the 1st shape will appear.
The circle is the 16th shape in the pattern

Question 2.
The Snowflake Design figure shows a knot Michael likes to use.

Part A
List the number of knots that Michael uses to form 1 to 6 snowflake designs. The rule is “Add 11.”
Answer: 11 , 22, 33, 44, 55, 66
Explanation:
The number of knots that Michael uses to form 1 to 6 snowflake designs 11 , 22, 33, 44, 55, 66

Part B
Describe a feature of the pattern you listed in Part A that is not part of the rule. Explain why it works.
Explanation:
A feature of the pattern is all the numbers are even and odd

Question 3.
The Michael’s Wall Hanging figure shows the design of a wall hanging Michael makes using the Snowflake Design. Answer the following to find how many knots Michael ties to make a wall hanging with 28 snowflakes.

Part A
Each column of 4 snowflakes has 4 connectors. There are also 4 connectors between columns. Complete the Connectors table using the rule “Add 8 connectors for each column.” Describe a feature of the pattern.

Explanation:
Completed the table by following the rule

Part B
Complete the Total Knots table using the following rules.
Snowflake Knots rule: Multiply the number of snowflakes by 11.
Connector Knots rule: Multiply the number of connectors from the Connectors table by 3.
Total Knots rule: Add the number of snowflake knots and the number of connector knots.

Explanation:
Snowflake Knots rule: Multiply the number of snowflakes by 11.
Connector Knots rule: Multiply the number of connectors from the Connectors table by 3.
Total Knots rule: Add the number of snowflake knots and the number of connector knots.

## enVision Math Common Core Grade 2 Answer Key Topic 11 Subtract Within 1,000 Using Models and Strategies

Go through the enVision Math Common Core Grade 2 Answer Key Topic 11 Subtract Within 1,000 Using Models and Strategies regularly and improve your accuracy in solving questions.

## enVision Math Common Core 2nd Grade Answers Key Topic 11 Subtract Within 1,000 Using Models and Strategies

enVision STEM Project: Making Models

Find Out Use a paintbrush as a model of a bee’s leg. Dip the brush in a bowl of sugar. Then dip the brush in a bowl of pepper. Take turns. What happens to the sugar? What happens to the pepper?
Journal: Make a Book Show what you learn in a book. In your book, also:

• Tell how bees help move pollen between plants.
• Show how to use a model to help subtract three-digit numbers.

Review What You Know

Vocabulary

Question 1.
Circle each number that is less than 607.
598
608
706

Explanation:
I circled 598 as it is lessthan 607.

Question 2.
Circle each number that is greater than 299.
352
300
298

Explanation:
I circles 352,300 as they are greater than 299.

Question 3.
Circle the group of numbers that decrease by 100 from left to right.
650, 550, 450, 350
320, 420, 520, 620
570, 560, 550, 540

Explanation:
I circled the first group of numbers 650, 550, 450, 350 as the numbers in this group are in drecreasing order from left to right by 100.

Subtraction Facts

Question 4.
Write each difference.
14 – 7 = ___
11 – 4 = ____
16 – 9 = ___

14 – 7 = 7
11 – 4 = 7
16 – 9 = 7

Explanation:
The differences of all the three questions above is 7, by thinking of the addition facts 7+7=14, 7+4=11, 7+9=16 i wrote the differences.

Partial Differences

Question 5.
Use partial differences to find 54 – 29.

Explanation:
First i parted the number 29 as 20 and 9, then i parted the number 9 as 4 and 5.So, i subtracted easily.

Math Story
Question 6.
Ben has 64 comic books. He gives 36 comic books to friends. How many comic books does Ben have left?
____ comic books

Explanation:
First i parted the number 36 as 30 and 6, then i parted the number 6 as 4 and 2.So, i subtracted easily.

Pick a Project

PROJECT 11A
How tall is the world’s tallest redwood free?
Project: Create a Redwood Trees Booklet

PROJECT 11B
Where does a lot of snow fall?
Project: Build a Snow Sculpture

PROJECT 11C
How high are Florida’s mountains?
Project: Make a Poster About Florida Mountains

Math Modeling

The Chemistry Set

3-ACT MATH PREVIEW

Before watching the video, talk to a classmate: When was the last time you mixed something together? What happened to the items you mixed? How did they change?

### Lesson 11.1 Subtract 10 and 100

Solve & Share

Jill’s Pumpkin Patch sells straw men.
A large straw man costs $134. A medium straw man costs$10 less than a large straw man. A small straw man costs $100 less than a large straw man. What is the cost of a medium straw man? A small straw man? Use dollar bills, place-value blocks, or mental math to solve. Be ready to explain how you solved the problem. Answer: Explanation: A large straw man costs$134, A medium straw man costs $10 less than a large straw man. A small straw man costs$100 less than a large straw man
By using place value blocks i found the value of medium straw man and a small straw man
Cost of medium straw man is $124, cost of small straw man is$34.

Visual Learning Bridge

Convince Me!
Use mental math to find 457 – 10 and 457 – 100. Explain your reasoning.
447 and 357

Explanation:
The number 457 has 4 hundreds 5 tens and 7 ones, if we remove 1 tens from it the answer is 4 hundreds 4 tens and 5 ones that is 447.
The number 457 has 4 hundreds 5 tens and 7 ones, if we remove 1 hundreds from it the answer is 3 hundreds 5 tens and 5 ones that is 357.

Guided Practice
Subtract. Use place-value blocks or mental math. Then write an equation to show the subtraction.

Question 1.

Question 2.

___ – ___ = ___
525 – 10 = 515

Explanation:
In the above questions there are 5 hundreds place value blocks, 2 ten blocks, 5 one blocks and from this 1 ten block is to be subracted.So, if we subtract one ten from it, their will be 4 hundreds, 1 tens and 5 ones that is 515.

Question 3.

___ – ___ = ___
738 – 100 = 638

Explanation:
In the above questions there are 7 hundreds place value blocks, 3 tens blocks, 8 ones blocks and from this 1 hundred block is to be subracted.So, if we subtract one ten from it, their will be 6 hundreds, 3 tens and 8 ones that is 638.

Question 4.

___ – ___= ___
100 – 100 = 0

Explanation:
In the above questions there is 1 hundred block. If we subtract 1 hundred, their will be 0 blocks.So, 100 minus 100 is 0.

Independent Practice

Subtract. Use place-value blocks or mental math.

Question 5.
250 – 10 = ___
250- 10 = 240

Explanation:
The difference between 250 and 10 is 240.

Question 6.
604 – 10 = ___
604 – 10 = 594

Explanation:
The difference between 604 and 10 is 594.

Question 7.
$102 –$100 = ___
102 – 100 = 2

Explanation:
The difference between 102 and 100 is 2.

Question 8.
719 – 10 = ___
719 – 10 = 709

Explanation:
The difference between 719 and 10 is 709.

Question 9.
$400 –$100 = ___
400 – 100 = 300

Explanation:
The difference between 400 and 100 is 300.

Question 10.
308 – 10 = ___

Question 11.
520 – 100 = ___
520 – 100 = 420

Explanation:
The difference between 520 and 100 is 420.

Question 12.
975 – 10 = ___
975 – 10 = 965

Explanation:
The difference between 975 and 10 is 965.

Question 13.
143 – 100 = ___
143 – 100 = 43

Explanation:
The difference between 143 and 100 is 43.

Question 14.
$825 –$10 = ___
$825 –$10 = $815 Explanation: The difference between 825 and 10 is 815. Question 15. 409 – 10 = ___ Answer: 409 – 10 = 399 Explanation: The difference between 409 and 10 is 399. Question 16.$200 – $100 = ___ Answer:$200 – $100 =$100

Explanation:
The difference between 200 and 10 is 100.

Algebra Find the missing numbers. Use mental math to solve.

Question 17.
362 – = 352
362 – 10 = 352

Explanation:
6 tens minus 1 tens is 5tens.So, 362 – 10 = 352.

Question 18.
801 – = 701
801 – 100 = 701

Explanation:
8 hundreds minus 1 hundred is 7 hundreds.So, 801 – 100 = 701.

Question 19.
449 = 549 –
449 = 549 – 100

Explanation:
5 hundreds minus 1 hundred is 4 hundreds.So, 549 – 100 = 449.

Question 20.
657 – = 647
657 – 10 = 647

Explanation:
5 tens minus 1 tens is 4tens.So, 362 – 10 = 352.

Question 21.
215 – = 205
215 – 10 = 205

Explanation:
1 tens minus 1 tens is 0tens.So, 215 – 10 = 205.

Question 22.
700 – = 690
700 – 10 = 690

Explanation:
10 tens minus 1 tens is 9tens.So, 700 – 10 = 690.

Problem Solving

Solve each problem. Show your work.

Question 23.
enVision® STEM Marni is studying facts about bees. She finds that one type of bee can pollinate 955 plants each day. A different type of bee pollinates 100 fewer plants. How many plants does it pollinate?

Explanation:
Marni tells that one Bee can pollinate 995 plants and another Bee caan pollinate 100 fewer than the first Bee.
To find the number of plats the second Bee pollinates, subtract 100 from 955.So, 955 – 100 = 855.
Therefore the second Bee can pollinate 855 plants.

Question 24.
Model Alex has five $100 bills, three$10 bills, and four $1 bills. He buys a pair of shoes for$100. How much money does he have left?
___ plants

Explanation:
Model Alex has five $100 bills, three$10 bills, and four $1 bills that means he has 534.If he buys a pair of shoes for$100, he will have$434(534-100=434) left with him. Question 25. Higher Order Thinking Think of a 3-digit number. Write a story about subtracting 100 from your number. Then complete the equation to show your subtraction. Answer: Explanation: 6 hundreds minus 1 hundred is 5 hundreds.So, 559 – 100 = 559. Question 26. Assessment Practice Which equations are true? Choose all that apply. 303 – 10 = 293 493 – 100 = 393 563 – 10 = 453 309 – 100 = 299 Answer: Explanation: In the above subtraction equations the marked equation are true. ### Lesson 11.2 Subtract on an Open Number Line Solve & Share There are 224 girls and some boys in a parade. There are 471 children in the parade. How many boys are in the parade? Use the open number line to solve the problem. Show your work. Answer: Explanation: Add up from 224, the number you are subtracting.Stop at 471. 100+100+10+10+10+10+1+1+1+1+1+1+1=247. The difference between 471 and 224 is 247. Visual Learning Bridge Convince Me! What is another way you could add up to find 382 – 247? Explain. Answer: Guided Practice Use the open number line to subtract. Question 1. 573 – 459 = ___ Answer: Add up from 459, the number you are subtracting.Stop at 573. 100+10+1+1+1+1 = 114. The difference between 573 and 459 is 114. Question 2. 672 – 547 = ___ Answer: Explanation: Add up from 547, the number you are subtracting.Stop at 672. 100+10+10+1+1+1+1+1=125 The difference between 672 and 547 is 125. Independent Practice Use the open number line to subtract. Question 3. 530 – 318 = Answer: Explanation: Add up from 318, the number you are subtracting.Stop at 530. 100+100+10+1+1=212 The difference between 530 and 318 is 212. Question 4. 735 – 429 = ___ Answer: Explanation: Add up from 429, the number you are subtracting.Stop at 735. 100+100+100+1+1+1+1+1+1+1=306. The difference between 735 and 429 is 306. Question 5. 802 – 688 = Answer: Explanation: Add up from 688, the number you are subtracting.Stop at 802. 100+10+1+1+1+1=114 The difference between 802 and 688 is 114. Question 6. Vocabulary Complete the sentences using each word below once. ones add number You can add up to subtract on an open number line. Start at the ___ you are subtracting ___ up hundreds, tens, and ______ Stop at the number you subtract from. Answer: Problem Solving Solve each problem. Check your work. Question 7. Reasoning Yun has 780 blocks. Marsha has 545 fewer blocks than Yun. How many blocks does Marsha have? ___ blocks Answer: Explanation: Add up from 545, the number you are subtracting.Stop at 780. 100+100+10+10+10+1+1+1+1+1=235 The difference between 780 and 545 is 235. Question 8. Higher Order Thinking Use open number lines to find 463 – 258 in two different ways. 463 – 258 = __ Answer: Explanation: First way: Add up from 258, the number you are subtracting.Stop at 463. 100+100+1+1+1+1+1=205. The difference between 463 and 258 is 205. Second way: Count back from 463,the number from which you have to subtract.Stop at 258. 1+1+1+1+1+100+100=205. The difference between 463 and 258 is Question 9. Assessment Practice Write a math story for 653 – 529. Then solve your story. Answer: Explanation: Add up from 529, the number you are subtracting.Stop at 653. 100+10+10+1+1+1+1=124. The difference between 653 and 529 is 124. ### Lesson 11.3 Subtract Using Models Activity Solve & Share Use place-value blocks to find 482 – 127. Tell which place value you subtracted first and why. Then draw a picture to show your work. Answer: Explanation: Take away 1 hundreds Take away 2 tens Regroup 1 tens as 10 ones, take away 7 ones So, 482 – 127 = 355. Visual Learning Bridge Convince Me! Explain why regrouping works in the problem above. Guided Practice Use and draw blocks to find each difference. Show your work. Question 1. Answer: Take away 1 hundreds Regroup 1 hundreds as 10 tens, take away 4 tens Take away 3 ones So, 326 – 143 = 183. Question 2. 363 – 127 = ___ Answer: Explanation: Take away 1 hundreds Take away 2 tens Regroup 1 tens as 10 ones, take away 7 ones So, 363 – 127 = 236. Question 3. 546 – 271 = ___ Answer: Explanation: Take away 2 hundreds Regroup 1 hundreds as 10 tens, take away 7 tens Take away 1 ones So, 546 – 271 = 275. Independent Practice Use and draw blocks to find each difference. Show your work. Question 4. 314 – 152 = ___ Answer: Explanation: Take away 1 hundreds Regroup 1 hundreds as 10 tens, take away 5 tens Take away 2 ones So, 314 – 152 = 162. Question 5. 653 – 419 = ___ Answer: Explanation: Take away 4 hundreds Take away 1 tens Regroup 1 tens as 10 ones, take away 9 ones So, 653 – 419 = 234. Question 6. 438 – 162 = __ Answer: Explanation: Take away 1 hundreds Regroup 1 hundreds as 10 tens, take away 6 tens Take away 2 ones So, 438 – 162 = 276. Question 7. 662 – 480 = ___ Answer: Explanation: Take away 4 hundreds Regroup 1 hundreds as 10 tens, take away 8 tens Take away 0 ones So, 662 – 480 = 182. Question 8. 999 – 834 = ___ Answer: Explanation: Take away 8 hundreds Take away 3 tens Take away 4 ones So, 999 – 834 = 165. Question 9. 599 – 209 = ___ Answer: Explanation: Take away 2 hundreds Take away 0 tens Take away 9 ones So, 599 – 209 = 390. Question 10. 954 – 738 = ___ Answer: Explanation: Take away 7 hundreds Take away 3 tens Take away 8 ones So, 954 – 738 = 216. Question 11. Number Sense Tyler says that the difference of 676 – 367 is greater than 200. Is what Tyler says reasonable? Why or why not? Answer: Yes, the diference of 676-367 is Explanation: Take away 3 hundreds Take away 6 tens Regroup 1 tens as 10 ones, take away 7 ones So, 676 – 367 = 309. Problem Solving Solve each problem below. Show your work. Question 12. Tia collected cans to raise money for school. She collected 569 cans on Monday. Tia collected some more cans on Tuesday. Now she has 789 cans. How many cans did Tia collect on Tuesday? Answer: Explanation: Take away 5 hundreds Take away 6 tens Take away 9 ones So, 789 – 569 = 220. Question 13. Make Sense Josh has these bills. How much money does he have? Answer: Question 14. Higher Order Thinking Write a subtraction problem about recycling. Use 3-digit numbers. Then solve the problem. Answer: Explanation: Take away 3 hundreds Regroup 1 hundreds as 10 tens, take away 5 tens Take away 2 ones So, 314 – 152 = 162. Question 15. Assessment Practice Use the place value-blocks to find 864 – 319. Which is the difference? A. 454 B. 535 C. 545 D. 555 Answer: Explanation: Take away 3 hundreds Take away 1 tens Regroup 1 tens as 10 ones, take away 9 ones So, 864 – 319 = 545. ### Lesson 11.4 Subtract Using Models and Place Value Activity Solve & Share Larissa has$353. She buys a pair of basketball shoes for $117. How much money does she have left over? Use or draw place-value blocks to solve. Be ready to explain what you did and why it works. Answer: Explanation:$117 + $117 =$234.
Subtract 2 hundreds
Subtract 3 tens
First subtract 3 ones
Regroup 1 tens as 10 ones. then subtract 1 ones
So, $353 –$234 = \$119.

Visual Learning Bridge

Convince Me!
Find 254 – 174. Jason says he can subtract 100, and then 4, and then 70 to find the difference. Do you agree? Explain.

Explanation:
Subtract 1 hundreds
First subtract 4 tens
Regroup 1 hundreds as 10 tens. then subtract 3 tens
Subtract 4 ones
So, 254 – 174 = 80.

Guided Practice
Draw blocks to find the partial differences. Record the partial differences to find the difference.

Question 1.

Independent Practice

Draw blocks to find the partial differences. Record the partial differences to find the difference.

Question 2.
598 – 319 = __

Explanation:
Subtract 3 hundreds
Subtract 1 tens
First subtract 8 ones
Regroup 1 tens as 10 ones. then subtract 1 ones
So, 598 – 319 = 279.

Question 3.
794 – 452 = ___

Explanation:
Subtract 4 hundreds
Subtract 5 tens
Subtract 2 ones
So, 7+4 – 452 = 342

Question 4.
871 – 355 = ___

Explanation:
Subtract 3 hundreds
Subtract 5 tens
First subtract 1 ones
Regroup 1 tens as 10 ones. then subtract 4 ones
So, 871 – 355 = 516.

Solve. Draw blocks to help.

Question 5.
Higher Order Thinking There were 642 people at the beach. There were 271 adults at the beach. The rest were children. How many children were at the beach?

___ children

Explanation:
Subtract 2 hundreds
First subtract 4 tens
Regroup 1 hundreds as 10 tens. then subtract 3 tens
Subtract 1 ones
So, 642 – 271 = 371.

Problem Solving

Solve each problem. You can use models to help. Show your work.

Question 6.
Model Jeff has 517 baseball cards. He has 263 football cards. How many more baseball cards than football cards does he have?
___ more baseball cards

Explanation:
Subtract 2 hundreds
First subtract 1 tens
Regroup 1 hundreds as 10 tens. then subtract 5 tens
Subtract 3 ones
So, 517 – 263 = 254.

Question 7.
Reasoning Felipe has 453 stamps in his collection. Emily has 762 stamps in her collection. How many more stamps does Emily have?

___ more stamps

Explanation:
Subtract 4 hundreds
Subtract 5 tens
First subtract 2 ones
Regroup 1 tens as 10 ones. then subtract 1 ones
So, 762 – 453 = 309.

Question 8.
Assessment Practice Which numbers complete this partial difference problem for 423 – 219? Choose all that apply.

Explanation:
Subtract 2 hundreds
Subtract 1 tens
First subtract 3 ones
Regroup 1 tens as 10 ones. then subtract 6 ones
So, 423 – 219 = 204.

### Lesson 11.5 Explain Subtraction Strategies

Activity

Solve & Share

Find 532 – 215. Use any strategy. Then explain why your strategy works.

Visual Learning Bridge

Convince Me!
Show how to count up on the open number line to find 437 – 245. Explain why your way works.

Explanation:
I counted up from 245 to 437.
I started at 245, I added 192, i ended at 437.So, 245+192=437.
Therefore, 437-245= 192.

Guided Practice
Subtract any way you choose. Show your work. Then explain why the strategy works.

Question 1.

Subtract 1 hundreds
Subtract 1 tens
First subtract 5 ones
Regroup 1 tens as 10 ones. then subtract 1 ones
So, 345-116=229.

Independent Practice

Choose any strategy to solve each subtraction problem. Show your work. Then explain why the strategy works.

Question 2.
312 – 179 = __

Explanation:
Subtract 1 hundreds
Regroup 1 hundreds as 10 tens, subtract 7 tens
First subtract 2 ones
Regroup 1 tens as 10 ones. then subtract 7 ones
So, 312 – 179 =133.

Question 3.
464 – 155 = ___

Explanation:
Subtract 1 hundreds
Subtract 5 tens
First subtract 5 ones
Regroup 1 tens as 10 ones. then subtract 1 ones
So, 464-155=309.

Question 4.
612 – 478 = ___

Explanation:
Subtract 2 hundreds
Regroup 1 hundreds as 10 tens, subtract 7 tens
First subtract 2 ones
Regroup 1 tens as 10 ones. then subtract 6 ones
So, 612-478=134.

Question 5.
Number Sense Use place value to find 748 – 319. Complete the equations.
319 = 300 + ___ + 9
Hundreds: 748 – __ = ___
Tens: ___ – 10 = __
Ones: ___ – __ = ___
319 = 300 + 10 + 9
Hundreds: 748 – 300 = 448
Tens: 30 – 10 = 20
Ones: 18 – 9 = 9

Explanation:
Subtract 3 hundreds
Subtract 1 tens
First subtract 8 ones
Regroup 1 tens as 10 ones. then subtract 1 ones
So, 748-319=428

Problem Solving

Solve each problem. Show your work.

Question 6.
Explain Ava wants to use mental math to find 352 – 149. Show how she could find the difference. Is this a good strategy for Ava to use? Explain why or why not.

Question 7.
Higher Order Thinking Kristin found 562 – 399 = 163 using an open number line. She added up to subtract. First she added 1, then 100, and then 62.
Draw Kristin’s number line. Do you think Kristin’s strategy was helpful? Explain.

Explanation:
I counted up from 399 to 562.
I started at 399, I added 163, i ended at 562.So, 399+163=562.
Therefore, 562-399=163.

Question 8.
Assessment Practice Jeff counted back on this open number line to find 812 – 125.
Use the numbers on the cards to find the missing numbers in the open number line. Write the missing numbers.

Explanation:
Count back from 692, 5 lessthan 692 is 687
Count forward  from 692, 10 morethan 692 is 702
10 morethan 702 is 712
100 morethan 712 is 812.

### Lesson 11.6 Persevere

Solve & Share

Jody wants to bake 350 muffins. She bakes one batch of 160 muffins and one batch of 145 muffins. How many more muffins does Jody need to bake?
Solve any way you choose. Show your work. Be prepared to explain why your way works.

Thinking Habits
What do I know? What do I need to find? How can I check that my solution makes sense?
Jody wants to bake 350 muffins. She bakes one batch of 160 muffins and one batch of 145 muffins
I know the total number of muffins baked 160+145=305
I need to find the number of muffins to be baked
350-305=45.
I checked my answer by drawing place value blocks.

Visual Learning Bridge

Convince Me!
What questions can you ask yourself when you get stuck? Be ready to explain how questions can help.

Guided Practice
Solve the problem. Remember to ask yourself questions to help. Show your work.

Question 1.
Kim had 455 shells. First, she gives 134 of the shells to a friend. Then she finds 54 more shells. How many shells does Kim have now?

Kim will have 375 shells.

Explanation:
Kim had 455 shells. First, she gives 134 of the shells to a friend
455-134=?
Count up to subtract
134+66=200
200+255=455
455-134=6+60+255=321

Then she finds 54 more shells
321+54=?
321+4=325
325+50=375

The number of shells kim have now is Kim have now 321+54=375.

Independent Practice

Use the table to solve each problem. Show your work.

Question 2.
How much heavier is a grizzly bear than an arctic wolf and a black bear together?
544 pounds

Explanation:
Arctic wolf and black bear together weigh
176+270=446

Grizzly bear weigh 990pounds, subtract weight of arctic wolf and a black bear together from it.
990-446=?
Count up to subtract
446+54=500
500+490=990
990-446=54+490=544
Therefore a Grizzly bear weighs 544pounds more than arctic wolf and a black bear together.

Question 3.
How much less does a black bear weigh than the weight of 2 mule deer?
126 pounds

Explanation:
Weight of mule deer is 198 pounds
Weight of 2 mule deers is 396

Weight of black bear is 270

396-270=?
Count up to subtract
270+30=300
300+96=396
396-270=30+96=126
Therefore the black bear is 126pounds lessthan weight of 2 mule deers.

Question 4.
How much more does a polar bear weigh than an arctic wolf, a black bear, and a mule deer together?

301 pounds

Explanation:
Weight of polar bear is 945
Weight of an arctic wolf, a black bear, and a mule deer together is 176+270+198=644
176+270=
176+200=376
376+24+46=446
446+198=
446+100=546
546+54+44=644

Subrtact 945 and 644
945-644=?
Count up to subtract
644+56=700
700+200=900
900+45=945
945-644=56+200+45=301
Therefore a polarbear weighs 301pounds morethan the weight of an arctic wolf, a black bear, and a mule deer together.

Problem Solving

Big Truck
The picture at the right shows the height of a truck and the height of a smokestack on top of the truck. The height of a bridge is 144 inches. Use the information at the right. Can the truck travel under the bridge?

Question 5.
Make Sense What do you know? What are you trying to find?
I know
Height of the bridge is 144inches
I am trying to find the height of the truck.

Question 6.
Make Sense What hidden question do you need to answer first? Find the answer to the hidden question.
I need to answer the hidden question that is the toatal height of the truck.

Explanation:
The height of the truck is 112+27=?inches
112+7=119
119+20=139inches

Question 7.
Explain Can the truck travel under the bridge? Show your work. Why does your solution make sense?
Yes, the truck can travel under the bridge.

Explanation:
The height of the truck is 112+27=?inches
112+7=119
119+20=139inches
Height of the bridge is 144inches
As the height of the truck is lessthan the height of the bridge the truck can travel under the bridge.

### Topic 11 Vocabulary Review

Understand Vocabulary
Draw a line from each term to its example.
Word List
• difference
• hundreds
• mental math
• open number line
• partial differences
• regroup

Explanation:
I matched the words with the example given.

Question 4.
This open number line is incomplete. It needs to show counting back to find 538 – 115. Write in the missing numbers and labels.

Explanation:
The misssing numbers are 115, 118, 158 and 358
The labels are open number line, hundreds and difference.

Use Vocabulary in Writing

Question 5.
Find 235 – 127. Use terms from the Word List to explain your work.

Explanation:
I used open number line to find the difference between 235 and 127,
The difference is 108.

### Topic 11 Reteaching

Set A

You can subtract 10 or 100 mentally.

Subtract using place-value blocks or mental math.

Question 1.
426 – 10 = __

Explanation:
Ten minus 426 is 416.

Question 2.
345 – 100 = ___

Explanation:
Hundred minus 345 is 245.

Question 3.
287 – 100 = ___

Explanation:
Hundred minus 287 is 187.

Question 4.
309 – 10 = ___

Explanation:
Ten minus 309 is 299.

Question 5.
800 – 10 = ___

Explanation:
Ten minus 800 is 790.

Question 6.
140 – 100 = __

Explanation:
Hundred minus 140 is 40.

Set B
Find 213 – 108.
One Way
Start at 108 on an open number line. Add up to 213.

Count back from 213 of an open number line.

So, 213 – 108 = 105

Use the open number line to subtract.

Question 7.
449 – 217 = ___

Explanation:
I counted up from 217 to 449 to find the difference 449-217.
449-217=232.

Question 8.
903 – 678 = ___

Explanation:
I counted up from 678 to 903 to find the difference 903-678.
903-678=225.

Set C
You can draw place-value blocks to show subtraction. Find 327 – 219.

Subtract 2 hundreds
Subtract 1 tens
First subrtact 7 ones
Regroup 1 tens as 10 ones, subtract 2 ones
327-219=108.

Draw blocks to find the partial differences. Record the partial differences to find the difference.

Question 9.
653 – 427 = ___

Explanation:
I subrtacted 4 hundreds 2 tens and 7 ones.
653-427=226.

Set D
Thinking Habits
Persevere
What do I know?
What do I need to find?
How can I check that my solution makes sense?

I knew how many peenies Marni have, how many she gave to her sister and how many pennies she got from her mother.
I need to find the number of pennies Marni will have after she gets pennies from her mother.
I can check my solution by drawing place value blocks to subtract.

Solve the problem. Ask yourself questions to help.

Question 10.
Marni has 354 pennies. First, she gives 149 pennies to her sister. Then, she gets 210 more pennies from her mother. How many pennies does Marni have now?
415 pennies

Explanation:
Marni has 354 pennies. First, she gives 149 pennies to her sister
354-149=?
149+1=150
150+204=354
354-149=1+204=205

she gets 210 more pennies from her mother
205+210=?
205+10=215
215+200=415
Therefore, there are 415pennies with Marni now.

### Topic 11 Assessment Practice

Question 1.
Which equals 100 less than 763? Choose all that apply.
663
600 + 60 + 3
863
800 + 60 + 3
600 + 100 + 60

Explanation:
100 minus 763 is 663, it can also be written as 600+60+3.

Question 2.
The open number line below shows subtraction.

Complete the equation. Write the numbers being subtracted and the difference.
___ – ___ = ___
587 – 464 = 123

Explanation:
The numbers being subtracted are 100, 10, 10, 1, 1, 1.
100+10+10+1+1+1=123
Therefore, 587-464=123.

Question 3.
Nico collected 235 coins. Amber collected 120 fewer coins than Nico. How many coins did they collect in all?
A. 350
B. 115
C. 500
D. 550
A.350

Explanation:
Nico collected 235 coins
Amber collected 120 fewer coins than Nico
Number of coins Nico collected 235-120=?
120+15=135
135+100=235
235-120=15+100=115.

235+115=?
235+15=250
250+100=350.
Therefore, they collected 350coins in altogether.

Question 4.
There are 537 boys and 438 girls at the concert. How many more boys than girls are at the concert?
A. 89
B. 99
C. 101
D. 109
B.99

Explanation:
There are 537 boys and 438 girls at the concert
537-438=?
438+12=450
450+87=537
537-438=12+87=99
Therefore, there are 99 boys more than girls in the concert.

Question 5.
Show how to add up on an open number line to find 740 – 490. Then write the difference below.

740 – 490 = ___

Question 6.
Look at your work in Item 5. Explain how you used the number line to find the difference.
I added up from 490 to 740, the difference between 740 and 490 is 250.

Question 7.
Use place value and partial differences to find 374 – 157. Show your work.
374 – 157 = ___

Explanation:
Subtract 1 hundred
Subtract 15 tens
First subtract 4 ones
Regroup 1 tens as 10 ones, then subtract 3 ones
So, 374-157=217.

Question 8.
Draw place-value blocks to find the difference of 643 – 418.

643 – 418 = ___

Explanation:
The difference between 643 and 418 is 225.

The chart shows the number of beads sold at Betty’s craft store for 4 weeks.

Question 1.
How many more beads did Betty sell in Week 2 than in Week 1? Write the missing numbers in the equation. Then use any strategy to solve.
__ – ___ = ___
536 – 400 = 136

Explanation:
536-400=136
So, Betty sold 136 beads more in week2 than in week1.

Question 2.
458 glass beads were sold in Week 3. The other beads sold in Week 3 were plastic. How many plastic beads were sold in Week 3?
Use the open number line to solve.

___ – __ = ___
Explain how you solved the problem. Tell how you know your answer is correct.
675 – 458 = 217.

Explanation:
I used open number line to solve this problem, i counted and added up from 458 to 675.
So, the difference between 675 and 458 is 217.We can check the answer by counting back from 675 to 458.

Question 3.
Dex buys 243 beads at Betty’s store. He uses 118 of them to make a bracelet. How many beads does Dex have left?
Solve the problem. Show your work. Explain which strategy you used.

___ – ___ = ___

243 – 118 = 125

Explanation:
I used open number line to solve this problem, i counted and added up from 118 to 243.
So, the difference between 243 and 118 is 125.

Question 4.
Part A
What is the hidden question in the problem?
The hidden question is the sum of blue, orange and white beads.
Part B
Solve the problem. Show your work. Explain which strategy you used.