Category: Envision Math

Go through the enVision Math Common Core Grade 3 Answer Key Topic 6 Connect Area to Multiplication and Addition regularly and improve your accuracy in solving questions.

enVision Math Common Core 3rd Grade Answers Key Topic 6 Connect Area to Multiplication and Addition

Essential Question:
How does area connect to multiplication and addition?

enVision STEM Project: Design Solutions
Do Research There are different designs that help protect against weather, such as as lightning rods, flood-defense barriers, and wind-resistant roofs. Use the Internet or another source to gather information about these kinds of designs and how they work.

Journal: Write a Report Include what you found. Also in your report:

• Tell how some window or door designs can help protect against weather.
• Use a grid to draw one of the window or door designs. Count the number of unit squares your design measures. Label your drawing to show how the design works to protect against weather.

Review What You Know

Vocabulary
Choose the best term from the box. Write it on the blank.
• addend
• equal groups
• multiply
• array

Question 1.
When you skip count to get the total number, you ________.
Answer:
When you skip count to get the total number, you __add______.

Question 2.
Dividing apples so everyone gets the same number of apples is an example of making __________.
Answer:
Dividing apples so everyone gets the same number of apples is an example of making _equal groups____.

Question 3.
When you display objects in rows and columns, you make a(n). ___________.
Answer:
When you display objects in rows and columns, you make a(n). multiply

Division as Sharing

Question 4.
Chen has 16 model cars. He puts them in 4 rows. Each row has an equal number of cars. How many columns are there?
Answer:

Question 5.
Julie has 24 glass beads to give to 4 friends. Each friend gets an equal share. How many glass beads does each friend get?
Answer:

Arrays

Question 6.
Write an addition equation and a multiplication equation for the array shown at the right.

Answer:
Number of rows = 6
Number of columns = 5
We add columns value as many times as rows.
We add 6 times 5.
Addition equation = 5 + 5 + 5 + 5 + 5 + 5 = 30
Multiplication equation= Rows × Columns = 6 × 5 = 30.

Relating Multiplication and Division

Question 7.
There are 12 team members. They line up in 3 equal rows. Which multiplication equation helps you find how many are in each row?
A. 2 × 6 = 12
B. 1 × 12 = 12
C. 3 × 4 = 12
D. 3 × 12 = 36
Answer:
Total number of team members = 12
Number of rows = 3
Number of members in each row = 12 ÷ 3
12 ÷ 3 = 4
There are 4 members in each row.
3 × 4 = 12.
Option C. 3 × 4 = 12.

Question 8.
There are 20 bottles of juice lined up in 4 equal rows. Explain how you can use a multiplication equation to find out how many bottles of juice are in each row.
Answer:
Number of juice bottles = 20
Number of row the juice bottles lined up = 4
Number of bottles in each row = 20 ÷ 4
20 ÷ 4 = 5
4 × 5 = 20.
There are 5 juice bottles in each row.

Pick a Project

PROJECT 6A
How are cities built?
Project: Plan a Dog Park

PROJECT 6B
What are community gardens?
Project: Design a Community Garden

PROJECT 6C
What are carpenters?
Project: Draw a School Floor Plan

PROJECT 6D
How do you play the game?
Project: Make an Area Game

+

Lesson 6.1 Cover Regions

Solve & Share
Look at Shapes A-C on Area of Shapes Teaching Tool. How many square tiles do you need to cover each shape? Show your answers below. Explain how you decided.
I can … count unit squares to find the area of a shape.

Look Back! Can you be sure you have an accurate answer if there are gaps between the tiles you used? Explain.

Essential Question
How Do You Measure Area?

Visual Learning Bridge
Emily made a collage in art class. She cut shapes to make her design. What is the area of this shape?

Count the unit squares that cover Emily’s shape. The count is the area of the shape.

36 unit squares cover the shape. The area of the shape is 36 square units.

Sometimes you can estimate the area. You can combine partially filled squares to estimate full squares.
Count the unit squares that cover this shape.
About 27 unit squares cover the shape.
The area of the shape is about 27 square units.

Convince Me! Construct Arguments Karen says these shapes each have an area of 12 square units. Do you agree with Karen? Explain.

Answer:
I don’t agree with Karen statement that areas of these shapes is 12 square units each. Because the area of the shape 1
That is circle is approximately 10 square units and the area of the Second shape is 12 square units.
Both the shape have different areas.

Another Example!
Emily wants to cover this octagon.

If she tries to cover it using unit squares, there will be gaps or overlaps.
Emily can break the square into two same-size triangles. She can cover the shape completely using this triangle:

Fourteen triangles cover the octagon. The area of the octagon is 7 square units.

Guided Practice

Do You Understand?
Question 1.
How do you know the area of the octagon is 7 square units?
Answer:
The squares are divided into two equal triangles.
The unit square is 1 unit long

There are 14 triangles filling the Octagon
14 triangles
2 triangles makes 1 square unit
14 ÷ 2 = 7.
The area of the octagon = 7 square unit

Question 2.
Explain how finding the area of a shape is different from finding the length of a shape.
Answer:
Area of a shape
The area of a shape is the number of unit squares that cover the surface of a closed figure. Area is measured in square units.
Length of a shape
Length is a measurement of the distance around something.

Do You Know How?
In 3 and 4, count to find the area. Tell if the area is an estimate.
Question 3.

Answer:

In the above image each square represents 1 unit long
A rectangular shape is represented.
The shape covers 4 rows and 5 columns
Number of square = 5 + 5 + 5 + 5 = 20 units.
Area of the shape  = 5 × 4 = 20 square unit.

Question 4.

Answer:
In the above image each square represents 1  unit long
A Triangle shape is represented in the square table.
The shape covers approximately 5 squares.
Area of the shape  = 5 square units.

Independent Practice

In 5-7, count to find the area. Tell if the area is an estimate.
Question 5.

Answer:

The shape showed in the image  is a T shaped.
Here each square is 1 square unit
Number of squares covered in the horizontal line of T = 4 + 4 = 8 unit
Number of squares covered in the vertical line of T = 2 + 2 = 4
Area of the shape = 8 + 4 = 12 square units.

Question 6.

Answer:

Here each square represents 1 unit long.
An oval shape is represented in the table.
Number of squares covered by the shape = 15 squares.
Area of the shape = 15 square units.

Question 7.

Answer:

Here each square is 1 unit long.
A square is divided into 2 equal triangles.
In the shape above
Number of squares covered = 8 units
Number of triangle covered = 2
2 equal triangles is equal to 1 square.
Area of the shape = 8 units + 1 unit = 9 square units.

Problem Solving

Question 8.
Maggie buys 4 sketch pads. She pays with a 20-dollar bill. How much change does Maggie get back?

Answer:
Number of sketch pads Maggie bought = 4
Cost of each sketch pads = $3
Cost of all the sketch pads = 4 × $3 = $12
$3 + $3 + $3 + $3 = $12
Money she paid = 20 dollar bill.
Amount of change she got back = $20 – $12 = $8.
Maggie got $8 cash back.

Question 9.
Critique Reasoning Janet covers the red square with square tiles. She says, “I covered this shape with 12 unit squares, so I know it has an area of 12 square units.” Do you agree with Janet? Explain.

Answer:
As we can see the tiles covers more area than the red square. I don’t agree with Janet’s saying “I covered this shape with 12 unit squares, so I know it has an area of 12 square units.”

Question 10.
Higher Order Thinking Chester drew this picture of a circle inside a square. What would be a good estimate of the green-shaded area of the square? How did you calculate your answer?

Answer:

Question 11.
Number Sense Arthur puts 18 erasers into equal groups. He says there are more erasers in each group when he puts the erasers in 2 equal groups than when he puts the erasers in 3 equal groups. Is Arthur correct? Explain.
Answer:
Total number of erasers Arthur has = 18 erasers.
Dividing the 18 erasers into 2 equal groups = 18 ÷ 2 = 9 erasers.
Dividing the 18 erasers into 3 equal groups = 18 ÷ 3 = 6 erasers.
Arthur says there are more erasers in each group when he puts the erasers in 2 equal groups than when he puts the erasers in 3 equal groups.
I agree with the Arthur’s sayings that there will be more erasers in 2 equal groups than in 3 equal groups.
There are 9 erasers in 2 groups and 6 erasers in 3 groups
The difference between the 2 groups and 3 groups = 9 – 6 = 3.
There are 3 more erasers in group 2.

Assessment Practice

Question 12.
Daryl draws this shape on grid paper. Estimate the area of the shape Daryl draws.

A. About 4 square units
B. About 5 square units
C. About 6 square units
D. About 7 square units
Answer:
Total number of squares = 6 × 5 = 30 squares.
Number of uncovered squares = 22 squares.
Number of covered squares = total squares – uncovered squares
30 – 22 = 8 squares.
There are 8 covered squares.
But there is white area in the covered shape squares.
So, The area of the Daryl shape is about 7 square units.

Lesson 6.2 Area: Nonstandard Units

Solve & Share
Find the area of the postcard on each grid. What do you notice about the size of the postcard on each grid? What do you notice about the area of the postcard on each grid? Explain.
I can … count unit squares to find the area of a shape.

Look Back! Are the measurements of the areas of the postcard shown above the same? Explain.
Answer:

Explanation:
Image 1
Number of squares = 6 × 10 = 60 squares.
Number of uncovered squares = 36 squares
Number of postcard covered squares = 60 – 36 = 24 squares.
Area of the postcard = 24 square units.

Image 2
Number of squares = 3 × 5 = 15 squares.
Number of uncovered squares = 9 squares
Number of postcard covered squares = 15 – 9 = 6 squares.
Area of the postcard = 6 square units.

1 Square in image 2 is equal to 4 squares in image 1
As shown in the image above.
So, both the postcards have the same area in both the images.

Essential Question
How Can You Measure Area Using Non-Standard Units?

Visual Learning Bridge
Tran designs a bookmark for a book. How can he use unit squares to find the area of the bookmark?

You can count the number of unit squares.

There are 32 unit squares.
Area = 32 square units

You can use a different unit square.

Convince Me! Reasoning How are the areas of these two squares alike and how are they different?

Guided Practice

Do You Understand?
Question 1.
Which of these shapes has an area of 5 square units? How do you know?

Answer:

Do You Know How?
Question 2.
Draw unit squares to cover the figures and then find the area. Use the unit squares shown.

Answer:

Independent Practice

In 3-5, draw unit squares to cover the figures and find the area. Use the unit squares shown.
Question 3.

Answer:

Question 4.

Answer:

Question 5.

Answer:

Problem Solving

Question 6.
Ben finds that the area of this figure is 14 square units. Draw unit squares to cover this figure.

Answer:

Question 7.
Luke eats 6 grapes from the bowl. Then Juan and Luke equally share the grapes that are left. How many grapes does Juan eat? Show how you used reasoning to solve the problem.

Answer:
Total number of grapes = 24 grapes.
Number of grapes Luke ate = 6 grapes.
Number of remaining grapes in the bowl = 24 grapes – 6 grapes = 18 grapes.
Luke and Juan eat the remaining grapes equally.
Number of grapes Juan and Luke eat equally are = 18 ÷ 2 = 9 grapes.
Juan and Luke eat 9 grapes each.

Question 8.
Construct Arguments Riaz estimates that the area of this figure is 45 square units. Martin estimates the area is 48 square units. Whose estimate is closer to the actual measure? Explain.

Answer:

Question 9.
Higher Order Thinking Theo wants to cover the top of a small table with square tiles. The table is 12 square tiles long and 8 square tiles wide. How many tiles will Theo need to cover the table?
Answer:

Assessment Practice

Question 10.
Rick used the smaller unit square and found that the area of this shape is 18 square units. If he used the larger unit square, what would the area of the shape be?

A. 1 square unit
B. 2 square units
C. 3 square units
D. 4 square units
Answer:

Lesson 6.3 Area: Standard Units

Solve & Share
Draw a square to represent 1 unit square. Use your unit square to draw a rectangle that has an area of 8 square units. Compare your shape with a partner’s shape. What is the same? What is different?
I can … measure the area of a shape using standard units.

Look Back! Are the sizes of your and your partner’s shapes something that is the same or something that is different? Explain.

Essential Question
How Can You Measure Area Using Question Standard Units of Length?

Visual Learning Bridge
Meg bought this sticker. What is the area of the sticker in square centimeters?

Count the unit squares.

6 unit squares cover the sticker. The sticker is measured in square centimeters.
So, the area of the sticker is 6 square centimeters.

Convince Me! Be Precise If square inches rather than square centimeters were used for the problem above, would more unit squares or fewer unit squares be needed to cover the shape? Explain.

Guided Practice

Do You Understand?
Question 1.
If Meg’s sticker on the previous page measured 2 inches by 3 inches, what would its area be?
Answer:
The measurement of Meg’s sticker are = 2 inches by 3 inches.
Area = 2 inches × 3 inches = 6 inches.
Area of Meg’s sticker is 6 inches.

Question 2.
Zoey paints a wall that measures 8 feet by 10 feet. What units should Zoey use for the area of the wall? Explain.
Answer:
Measurements of Zoey paint on the wall = 8 feet by 10 feet
Area of the paint on the wall = 8 feet × 10 feet  = 80 square feet.

Do You Know How?
In 3 and 4, each unit square represents a standard unit. Count the shaded unit squares. Then write the area.
Question 3.

Answer:
Number of shaded unit squares = 4 × 3 = 12 squares.
Area of the shaded squares = 12 square ft.
square ft because 1 square is measure is ft.

Question 4.

Answer:
Number of shaded squares = 6 × 2 = 12 squares.
Area of the shaped squares = 12 square m.
square m because 1 square is measure is m.

Independent Practice

In 5-10, each unit square represents a standard unit. Count the shaded unit squares. Then write the area.
Question 5.

Answer:
Number of shaded unit squares = 4 × 4 = 16 unit squares.
Area of shaded unit squares = 16 squares in.
square in because 1 square is measure is in.

Question 6.

Answer:
Number of shaped unit squares = 3 × 3 = 9 unit squares.
Area of the shaded unit squares = 9 square ft.
square ft because 1 square is measure is ft.

Question 7.

Answer:
Number of shaded unit squares = 6 × 3 = 18 unit squares.
Area of shaded unit squares = 18 square in.
square in because 1 square is measure is in.

Question 8.

Answer:
Number of shaded unit squares = 9 × 7 = 63 unit squares.
Area of the shaded unit squares = 63 square m.
square m because 1 square is measure is m.

Question 9.

Answer:
Number of shaded unit squares = 7 × 5 = 35 unit squares.
Area of the shaded unit squares = 35 square cm.
square cm because 1 square is measure is cm.

Question 10.

Answer:
Number of shaded unit squares = 4 ×5 = 20 unit squares.
Area of the shaded unit squares = 20 square ft.
square ft because 1 square is measure is ft.

Problem Solving

Question 11.
Reasoning Mr. Sanchez grows three types of vegetables in his garden. What is the area of the garden that Mr. Sanchez uses to grow lettuce and cucumbers? Explain how to use the units in this problem.

Answer:
Number of squares covered Cucumbers = 4 × 2 = 8 squares
Area of garden used to grow Cucumber= 8 square ft.
Number of square covered to grow Lettuce = 3 × 6 = 18 squares
Area of garden used to grow Lettuce = 18 square ft.
Total area used to grow Lettuce and cucumber in Mr. Sanchez garden = 8 square ft + 18 square ft = 26 square ft.
Area used to grow Lettuce and cucumber is 26 square ft.

Question 12.
Lisa received 34 text messages on Monday and 43 text messages on Tuesday. She received 98 text messages on Wednesday. How many more text messages did Lisa receive on Wednesday than on Monday and Tuesday combined?
Answer:
Number of text messages Lisa received on Monday = 34
Number of text messages Lisa received on Tuesday = 43
Number of text messages Lisa received on Wednesday = 98
Total number of text messages Lisa received on Monday and Tuesday combined = 34 + 43 = 77 text messages.
Number of more text messages did Lisa receive on Wednesday than on Monday and Tuesday combined are = 98 – 77 = 21
Lisa receive 21 more text messages  on Wednesday than on Monday and Tuesday combined.

Question 13.
Monica buys a postage stamp. Is the area of the stamp more likely to be 1 square inch or 1 square meter? Explain.
Answer:
Generally the dimensions of the postage stamp are in inches.
So , the area of the postage stamp Monica bought is 1 square inch.

Question 14.
Algebra Which operation can you use to complete the equation below? 8 = 56 ☐ 7
Answer:
To complete the given equation we use division.
56 ÷ 7 = 8
8 × 7 = 56
So, we use Division.

Question 15.
Higher Order Thinking Brad says a square that has a length of 9 feet has an area of 18 square feet. Is Brad correct? Why or why not?
Answer:
No, Brad is not  correct.
The area of a square is side × side
Here the length of the square = 9 feet
Area of the square = 9 ft  × 9 ft = 81 square ft.
So, what Brad said is wrong the area of the square is 81 square ft not 18 square ft.

Assessment Practice

Question 16.
Each of the unit squares in Shapes A-C represent 1 square foot. Select numbers to tell the area of each shape.

Answer:

Lesson 6.4 Area of Squares and Rectangles

Solve & Share
Jorge is carpeting two rooms. One room is a square with a side that measures 6 meters. The other room is a rectangle with sides that measure 3 meters and 12 meters. How many square meters of carpet does Jorge need?
I can … find the area of squares and rectangles by multiplying.

Look Back! What do you notice about the lengths of the sides and the areas of the two rooms Jorge is carpeting?

Essential Question
How Can You Find the Question Area of a Figure?

Visual Learning Bridge
Mike paints a rectangular wall in his room green. The picture shows the length and width of Mike’s wall. A small can of paint covers 40 square feet. Does Mike need more than one small can to paint the wall of his room?

One Way
Count the unit squares to find area.

There are 48 unit squares. The area of Mike’s wall is 48 square feet.

Another Way
Count the number of rows and multiply by the number of squares in each row. There are 8 rows and 6 squares in each row.

The area of Mike’s wall is 48 square feet. He will need more than one small can of paint.

Convince Me! Model with Math Mike plans to paint a wall in his living room blue. That wall measures 10 feet tall and 8 feet wide. What is the area of the wall Mike plans to paint blue? How many cans of paint will he need?

Another Example!
The area of another wall in Mike’s room is 56 square feet. The wall is 8 feet high. How wide is the wall?

56 = 8 × ?
You can use division: 56 ÷ 8 = ?
56 ÷ 8 = 7
The wall is 7 feet wide.

Guided Practice

Do You Understand?
Question 1.
Suji’s garden is 4 yards long and 4 yards wide. What is the area of Suji’s garden?
Answer:
Length of Suji’s garden = 4 yards
Width of Suji’s garden = 4 yards
Area of Suji’s garden = length × width = 4 yards × 4 yards = 16 square yards.

Question 2.
The area of Michi’s garden is 32 square feet. The garden is 8 feet long. How wide is Michi’s garden?

Answer:
Area of Michi’s garden = 32 square feet
Length of the garden = 8 feet
width of the garden = ? feet
Area = length × width
32 = 8 × width
width = 32 ÷ 8 = 4 feet.
8 ft × 4 ft = 32 square ft
Michi’s garden is 4 ft wide.

Do You Know How?
In 3 and 4, find the area of each figure. Use grid paper to help.
Question 3.

Answer:

Question 4.

Answer:

Independent Practice

In 5 and 6, find the area. In 7, find the missing length. Use grid paper to help.
Question 5.

Answer:
Length of the figure = 3 cm
Width of the figure = 1 cm
Area of the figure = 3 cm × 1 cm = 3 square cm.

Question 6.

Answer:
Length of the figure = 4 ft
Width of the figure = 9 ft
Area of the figure = 4 ft × 9 ft = 36  square ft.

Question 7.

Answer:

Problem Solving

Question 8.
Jen’s garden is 4 feet wide and has an area of 28 square feet. What is the length of Jen’s garden? How do you know?
Answer:
Area of Jen’s garden = 28 square ft
width of Jen’s garden = 4 ft
Length of Jen’s garden = area ÷ width
28 sq ft ÷ 4 ft = 7 ft.
Length of Jen’s garden = 7 ft.

Question 9.
Make Sense and Persevere Briana has 2 grandmothers. She mailed 2 cards to each of them. In each card she put 6 photographs. How many photographs did Briana mail in all?
Answer:
Number of grandmothers Briana has = 2 grandmothers
Number of cards she mailed to them = 2 each
total number of cards she mailed = 2 + 2 = 4
Number of photographs in each card = 6 photographs
Total number of photographs Briana mailed in all = 4 × 6 = 24 photographs.

Question 10.
Kevin thinks he found a shortcut. He says he can find the area of a square by multiplying the length of one side by itself. Is Kevin correct? Why or why not?
Answer:
Area of a square is side × side = side square.
Kevin says he can find the area of a square by multiplying the length of one side by itself.
I agree with Kevin’s saying.
Multiplying the length of one side by itself is same as multiplying side × side.
Because the length of all the side of a square are same.

Question 11.
Higher Order Thinking Ryan measures a rectangle that is 9 feet long and 5 feet wide. Teo measures a rectangle that has an area of 36 square feet. Which rectangle has the greater area? Explain how you found the answer.
Answer:
Length of Ryan rectangle = 9 ft
Width of Ryan rectangle = 5 ft
Area of rectangle = length × width = 9 ft  × 5 ft = 45 ft.
Area of the rectangle Teo measured = 36 square ft.
Ryan rectangle has the greater area.
On comparing the areas of both the rectangles. Ryan rectangle area is 45 square ft and Teo rectangle area is 36 square ft
45 – 36 = 9 square ft greater.
Ryan rectangle area is 9 square ft greater than Teo’s rectangle area.

Assessment Practice

Question 12.
Marla makes maps of state preserves. Two of her maps of the same preserve are shown. Select all the true statements about Marla’s maps.

☐ You can find the area of Map A by counting the unit squares.
☐ You can find the area of Map B by multiplying the side lengths.
☐ The area of Map A is 18 square feet.
☐ The area of Map B is 18 square feet.
☐ The areas of Maps A and B are NOT equivalent.
Answer:
☐ You can find the area of Map A by counting the unit squares.
☐ You can find the area of Map B by multiplying the side lengths.
☐ The area of Map A is 18 square feet.
☐ The area of Map B is 18 square feet.
All the above mentioned statements are true.

Lesson 6.5 Apply Properties: Area and the Distributive Property

Solve & Share
The new reading room floor is a rectangle that is 8 feet wide by 9 feet long. Mrs. Wallace has a rectangular rug that is 8 feet wide by 5 feet long. What area of the reading room floor will not be covered by the rug?
I can … use properties when multiplying to find the area of squares and rectangles.

Look Back! Does your strategy change if the rug is in the corner of the room or in the center? Explain why or why not.

Essential Question
How Can the Area of Rectangles Represent Question the Distributive Property?

Visual Learning Bridge
Gina wants to separate this rectangle into two smaller rectangles. Will the area of the large rectangle equal the sum of the areas of the two small rectangles?

Separate the 8-unit side into two parts.

7 × 8 = 7 × (5 + 3) = (7 × 5) + (7 × 3)

So, the area of the large rectangle is equal to the sum of the areas of the two small rectangles.

Convince Me! Generalize Find another way to separate this rectangle into two smaller parts. Write an equation you can use to find the areas of the two smaller rectangles. Is the area of the large rectangle still the same? What can you generalize?

Guided Practice

Do You Understand?
Question 1.
Describe a way to separate a 6×6 square into two smaller rectangles.
Answer:

6 × 6 = 36
6 × 6 = 6 × (2 + 4 ) = 6 × 2 + 6 × 4 = 12 + 24 = 36.

Question 2.
What multiplication facts describe the areas of the two smaller rectangles you identified in Exercise 1?
Answer:
6 × 6 = 36
6 × 6 = 6 × (2 + 4 ) = 6 × 2 + 6 × 4 = 12 + 24 = 36.

Do You Know How?
Complete the equation that represents the picture.
Question 3.

Answer:

Explanation:
Given 6 × 5
Using distributive property to find the product of 6 × 5.
By breaking one of the facts to get the product.
Here we break 5 in to 2 and 3
Then the equation will be
6 × (2 + 3 )= (6 × 2 ) + (6 × 3)

Independent Practice

In 4 and 5, complete the equation that represents the picture.
Question 4.

Answer:

Explanation:
Given 5 × 7
Using distributive property to find the product of 5 × 7
By breaking one of the facts to get the product.
Here 7 is broken into 4 and 3
Then the equation will be
5 × 7 = 5 × (4 + 3) = (5 ×4) + (5 × 3 ) = 20 + 15 = 35.

Question 5.

Answer:

Explanation:
Given 3 × 8
Using distributive property to find the product of 3 × 8
By breaking one of the facts to get the product.
Here 8 is broken into 4 and 4
Then the equation will be
3 × 8 = 3 × (4 + 4) = (3 ×4) + (3 × 4 ) = 12 + 12 = 24..

In 6, write the equation that represents the picture.
Question 6.

Answer:

Explanation:
Given 4 × 6
Using distributive property to find the product of 4× 6
By breaking one of the facts to get the product.
Here 6 is broken into 2 and 4
Then the equation will be
4 × 6 = 4 × (2 + 4) = (4 ×2) + (4 × 4 ) = 8 + 16 = 24

Problem Solving

Question 7.
Amit sold 3 shells last week for $5 each and 2 more shells this week for $5 each. Show two ways to determine how much money Amit made in the two weeks.
Answer:
Number of shells Amit sold last week = 3 shells
Cost of each shell he sold = $5 each
Total Money Amit made in last week = 3 × $5 = $15
Number of shells he sold this week = 2 shells
Cost of each shell he sold = $5 each.
Total money Amit made this week = 2 × $5 = $10
Total money Amit made in two weeks = $15 + $10 =$25.
Other way
Total number of shells he sold in two weeks = 3 + 2
Cost of each shell = $5
Total money Amit earned in two weeks =
=$5 × (3 + 2)
= ($5 × 3) + ($5 × 2 )
= $15 + $10
= $25.

Question 8.
enVision® STEM Claudia wants to replace the roof of a dog house with a new wind-resistant material. The roof has two rectangular sides that are 6 feet by 4 feet. What is the total area of the roof?
Answer:
Number of rectangular sides = 2
Sizes of the rectangular sides = 6 feet by 4 feet
Area of 1 rectangular sides = 6 ft × 4 ft = 24 square ft
Total area of the roof = 2 × 24 square ft = 48 square ft.

Question 9.
Use Structure Chiya has an 8 x 6 sheet of tiles. Can she separate the sheet into two smaller sheets that are 8 x 4 and 8 x 2? Do the two smaller sheets have the same total area as her original sheet? Explain.

Answer:
The dimensions of Chiya sheet of tiles = 8 × 6
Area of the sheet = 8× 6 = 48.
The sheet is divided into two smaller sheet
Using distributive property and dividing into two smaller parts
8 × 6 = 8 ×(4 + 2) = (8 × 4) + (8 × 2) = 32 + 16 = 48.
In both the cases the area is same.

Question 10.
Higher Order Thinking List all possible ways to divide the rectangle at the right into 2 smaller rectangles.

Answer:
The dimensions of the picture = 4 rows with 5 squares
Total number of squares = 4 × 5 = 20 squares.
Using distributive property and finding all the possible ways.
4 × 5 = 4 × (1 + 4 )= (4 × 1) + (4 × 4) = 4 + 16 = 20
4 × 5 = 4 × (2 + 3 )= (4 × 2) + (4 ×3) = 8  + 12 = 20
4 × 5 = 4 × (3 + 2 )= (4 × 3) + (4 × 2) =12 + 8 = 20
4 × 5 = 4 × (4 + 1 )= (4 × 4) + (4 × 1) = 16 + 4 = 20.

Assessment Practice

Question 11.
Which equation represents the total area of the green shapes?

A. 4 × 8= 4 × (6 + 2) = (4 × 6) + (4 × 2)
B. 4 × 7 = 4 × (3 + 4) = (4 × 3) + (4 × 4)
C. 4 × 7 = 4 × (4 + 3) = (4 × 4) + (4 × 3)
D. 4 × 7 = 4 × (5 + 2) = (4 × 5) + (4 × 2)
Answer:
Number of rows = 4 with 7 square each
Area of the green shapes = 4 × 7 = 28 square units.
Breaking 7 into smaller part
7 = 1 + 6
7 = 2 + 5
7 = 3 + 4
7 = 4 + 3
7 = 5 + 2
7 = 6 + 1
the equations are
B. 4 × 7 = 4 × (3 + 4) = (4 × 3) + (4 × 4)
C. 4 × 7 = 4 × (4 + 3) = (4 × 4) + (4 × 3)
D. 4 × 7 = 4 × (5 + 2) = (4 × 5) + (4 × 2)
which represents the green shapes.

Lesson 6.6 Apply Properties: Area of Irregular Shapes

Solve & Share
Mrs. Marcum’s desk is shaped like the picture below. The length of each side is shown in feet. Find the area of Mrs. Marcum’s desk.
I can … use properties to find the area of irregular shapes by breaking the shape into smaller parts.

Look Back! How can you check your answer? Is there more than one way to solve this problem? Explain.

Essential Question
How Can You Find the Area of an Irregular Shape?

Visual Learning Bridge
Mr. Fox is covering a miniature golf course putting green with artificial grass. Each artificial grass square is 1 square foot. What is the area of the putting green that Mr. Fox needs to cover?

One Way
You can draw the figure on grid paper. Then count the unit squares to find the area.

The area of the putting green is 56 square feet.

Another Way
Divide the putting green into rectangles. Find the area of each rectangle. Then add the areas.

Rectangle A 4 × 3= 12
Rectangle B 4 × 3 = 12
Rectangle C 4 × 8 = 32
12 + 12 + 32 = 56. The area of the putting green is 56 square feet.

Convince Me! Use Structure Find another way to divide the putting green into smaller rectangles. Explain how you can find the area of the putting green using your smaller rectangles.

Answer:

Here dividing the image as per the above image. into 3 rectangles.
Rectangle A = 8 ft × 3 ft = 24 square ft
Rectangle B = 8 ft × 3 ft = 24 square ft
Rectangle C = 4 ft × 2 ft = 8 square ft
Total area of the image = 24 +24 + 8 = 56 square ft.

Guided Practice

Do You Understand?
Question 1.
Explain why you can find the area of the putting green on the previous page using different rectangles.
Answer:

Question 2.
Explain what operation you use to find the total area of the smaller rectangles.
Answer:
I used distributive property.
I divide the shape into smaller shapes and then added all the shapes to find the total shape.
To find the area of the irregular shape find the areas of the smaller area and then add them together.

Do You Know How?
In 3 and 4, find the area of each figure. Use grid paper to help.
Question 3.

Answer:

Question 4.

Answer:

Independent Practice

In 5-8, find the area of each figure. Use grid paper to help.
Question 5.

Answer:

Question 6.

Answer:

Question 7.

Answer:

Question 8.

Answer:

Problem Solving

Question 9.
Reasoning Mrs. Kendel is making a model house. The footprint for the house is shown at the right. What is the total area? Explain your reasoning.

Answer:

Explanation:
Here to find the area of Mrs. Kendel Model House i divided the model into two sections.
Living Section and Sleeping Section.
Finding the area of Living Section and Sleeping Section and then adding them gives the total area of the model house.

Question 10.
Vocabulary Fill in the blanks. Mandy finds the _________ of this shape by dividing it into rectangles. Phil gets the same answer by counting ________.

Answer:
Mandy finds the _Area__ of this shape by dividing it into rectangles. Phil gets the same answer by counting _squares__.

Question 11.
Algebra Write an equation. Use a question mark to represent the unknown quantity for the phrase “six times a number is 24.” Solve your equation.
Answer:
“six times a number is 24.”
4 + 4 + 4 + 4 + 4 + 4 = 24
six times 4 is 24.

Question 12.
Higher Order Thinking Mr. Delancy used 3-inch square tiles to make the design at the right. What is the area of the design he made? Explain how you found it.

Answer:
Dimensions of 1 square = 3 in
Area of 1 square = 3 in × 3 in = 9 square in.
Number of green squares = 4 squares
Number of blue squares = 4 squares.
Total number of squares =4 + 4 = 8 squares.
Total Area of all squares = 8 × 9 square in = 72 square in.

Assessment Practice

Question 13.
Jared drew the figure at the right. Draw lines to show how you can divide the shape to find the area. Then select the correct area for the figure at the right.

A. 6 square inches
B. 24 square inches
C. 30 square inches
D. 33 square inches
Answer:

Lesson 6.7 Problem Solving

Look For and Use Structure
Solve & Share
Mr. Anderson is tiling his kitchen floor. He will not need tiles for the areas covered by the kitchen island or the counter. How many square meters of tiles does Mr. Anderson need?
I can … use the relationships between quantities to break a problem into simpler parts.

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

• What patterns can I see and describe?
• How can I use the patterns to solve the problem?
• Can I see expressions and objects in different ways?
  

Look Back! Use Structure is the tiled area greater than or less than the total area of the kitchen? Explain.

Essential Question
How Can You Use Structure to Solve Problems?

Visual Learning Bridge
Janet is painting a door. She needs to paint the entire door except for the window.
What is the area of the part of the door that needs paint?

What do I need to do to solve this problem?
I need to find the area of the door without the window.

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

• break the problem into simpler parts.
• find equivalent expressions.

I will subtract the area of the window from the total area.
Here’s my thinking…)
Find the area of the whole door.
4 feet × 9 feet = 36 square feet

Find the area of the window.
2 feet × 2 feet = 4 square feet

Subtract to find the area that needs paint.
36 – 4 = 32 square feet

The area of the part of the door that needs paint is 32 square feet.

Convince Me! Use Structure Janet thinks of a different way to solve the problem. She says, “I can divide the area I need to paint into 4 smaller rectangles. Then I will find the area of each smaller rectangle and add the 4 areas.” Does Janet’s strategy make sense? Explain.

Guided Practice

Use Structure Lil glued beads on the border of the frame. What is the area of the part she decorated with beads?

Question 1.
How can you think about the total area of the frame?
Answer:

Question 2.
Use what you know to solve the problem.

Answer:
Given a frame and a image
To find the  decorated area we subtract the image area from  total area.

Independent Practice

Use Structure A keypad has 10 rubber buttons. Each button is 1 centimeter by 2 centimeters. The rest is made out of plastic. Is the area of the plastic greater than the area of the rubber buttons?

Answer:

Question 3.
How can you break the problem into simpler parts? What is the hidden question?
Answer:
Area of the buttons is calculated. Then
Total area of the keypad is calculated.
Finally the area of the plastic is calculated by subtracting buttons area from the total area.

Question 4.
How can you find the area of all the rubber buttons?
Answer:
The area of the rubber = Area of the keypad – Area of the 10 buttons
= 70 square cm – 20 square cm
= 50 square cm.

Question 5.
Use what you know to solve the problem.
Answer:
We know the
Dimensions of the button
Dimensions of the keypad.
Using these dimensions area of buttons and keypad are calculated.
Subtracting the button area from keypad area. Plastic area is calculated.

Problem Solving

Performance Task
Place Mat Genevieve is designing a placemat. The center measures 8 inches by 10 inches. A 2-inch border goes around the center. Genevieve cuts the corners to make the placemat an octagon. She wants to find the area of the placemat.

Question 6.
Use Structure What are the lengths and widths of each rectangular border piece?
Answer:

Question 7.
Use Appropriate Tools How can Genevieve find the area of the 4 corner pieces using grid paper?

Answer:

Area of the corners is 8 square in.

Question 8.
Model with Math What equation can Genevieve use to find the area of the center? Find the area of the center using your equation.

Answer:
Given
The center measures 8 inches by 10 inches.
Area of the center = 8 in × 10 in = 80 square in.

Question 9.
Reasoning How are the quantities in this problem related?
Answer:
They share one side of the other rectangles.

Question 10.
Be Precise Solve the problem. Explain what unit you used for your answer.
Answer:
We used inches. to measure the answer.

Topic 6 Fluency Practice Activity

Follow the Path
Shade a path from START to FINISH. Follow the quotients that are odd numbers. You can only move up, down, right, or left. Once you complete the path, write the fact families for each of the squares you shaded.
I can … multiply and divide within 100.

Topic 6 Vocabulary Review

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

• area
• column
• Distributive Property
• estimate
• multiplication
• product
• row
• square unit
• unit square

Question 1.
A(n) __________ has sides that are each 1 unit long.
Answer:
A(n) _unit square__ has sides that are each 1 unit long.

Question 2.
__________ is the number of unit squares that cover a region or shape.
Answer:
__Area__ is the number of unit squares that cover a region or shape.

Question 3.
You can use the ___________ to break apart facts and find the __________.
Answer:
You can use the __Distributive Property__ to break apart facts and find the ___Area_______.

Question 4.
A unit square has an area of 1 _________.
Answer:
A unit square has an area of 1 __Square unit_______.

Question 5.
When you __________, you give an approximate answer.
Answer:
When you _multiplicate_________, you give an approximate answer.

Write always, sometimes, or never.

Question 6.
Area is __________ measured in square meters.
Answer:
Area is __sometimes_______ measured in square meters.

Question 7.
Multiplication __________ involves joining equal groups.
Answer:
Multiplication __sometimes________ involves joining equal groups.

Question 8.
The area of a shape can __________ be represented as the sum of the areas of smaller rectangles.
Answer:
The area of a shape can __always________ be represented as the sum of the areas of smaller rectangles.

Use Vocabulary in Writing
Question 9.
What is the area of this rectangle? Explain how you solved the problem. Use at least 3 terms from the Word List in your answer.

Answer:
Number of rows = 4
Number of squares in each row = 6 squares.
Area of the rectangle = 4 × 6 = 24 square units.

Topic 6 Reteaching

Set A pages 209-212

A unit square has sides that are 1 unit long.
Count the unit squares that cover the shape. The count is the area of the shape.

Seventeen unit squares cover the shape. The area of the shape is 17 square units.
Sometimes you need to estimate to find the area. First count the full squares. Then estimate the number of partially filled squares.

Remember that area is the number of unit squares needed to cover a region with no gaps or overlaps.

In 1 and 2, count to find the area. Tell if the area is an estimate.
Question 1.

Answer:

Question 2.

Answer:

Set B pages 213-216

Unit squares can be different sizes. The size of a unit square determines the area.

The measurements are different because different sizes of unit squares were used.

Remember that you can use unit squares to measure area.

Draw unit squares to cover the figures and find the area. Use the unit squares shown.
Question 1.

Answer:

Question 2.

Answer:

Set C pages 217-220

The unit squares below represent square inches.
What is the area of the figure below?

Twenty-four unit squares cover the figure. The area of the figure is measured in square inches.
So, the area of the figure is 24 square inches.

Remember that you can measure using standard or metric units of area for unit squares.

In 1 and 2, each unit square represents a standard unit. Count the unit squares. Then write the area.
Question 1.

Answer:

Question 2.

Answer:

Set D pages 221-224

You can find area by counting the number of rows and multiplying by the number of squares in each row.

There are 5 rows.
There are 4 squares in each row.
5 × 4 = 20
The area of the figure is 20 square inches.

Remember that you can multiply the number of rows by the number of squares in each row to find the area.

In 1-3, find the area of each figure. Use grid paper to help.
Question 1.

Answer:

Question 2.

Answer:

Question 3.

Answer:

Set E pages 225-228

Reteaching Continued

You can use the Distributive Property to break apart facts to find the product.
Separate the 5 unit side into two parts.

Area of the large rectangle: 6 × 5 = 30
Areas of the small rectangles:
6 × 2 = 12
6 × 3 = 18
Add the two areas: 12 + 18 = 30
You can write an equation to show that the area of the large rectangle is equal to the sum of the areas of the two small rectangles.
6 × 5 = 6 × (2 + 3) = (6 × 2) + (6 × 3)

Remember that you can separate a rectangle into two smaller rectangles with the same total area.

In 1-3, write the equations that represent the total area of the red shapes. Find the area.
Question 1.

Answer:

Question 2.

Answer:

Question 3.

Answer:

Set F pages 229-232

Find the area of this irregular shape.

You can place the shape on grid paper and count the unit squares. The area of the shape is 21 square inches.

You also can divide the shape into rectangles. Find the area of each rectangle and add.
5 × 3 = 15 square inches
3 × 2 = 6 square inches
15 + 6 = 21 square inches

Remember that you can add smaller areas to find a total area.

In 1 and 2, find the area of each shape.
Question 1.

Answer:

Question 2.

Answer:

Set G pages 233-236

Think about these questions to help you use structure in solving 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?
  

Remember to look for simpler ways of representing an area.

Debra made this design from 1-inch square tiles. What is the area of the blue tiles?

Question 1.
How can you express the area of the blue tiles?
Answer:
Area of the blue tile = area of the outer rectangle – area of the inner rectangle
= 7 in × 7 in – 5 in × 3 in
= 49 square in – 15 square in
= 34 square in.
Area of the blue tile is 34 square in.

Question 2.
Solve the problem. Explain how you solved.
Answer:
Area of the outer rectangle = 7 in × 7 in = 49 square in
Area of the inner rectangle = 5 in × 3 in = 15 square in
Area of the blue tile = area of the outer rectangle – area of the inner rectangle
= 49 square in – 15 square in
= 34 square in.
Area of the blue ti,e is 34 square in.

Topic 6 Assessment Practice

Question 1.
Count to find the area of the shape. Tell if the area is exact or an estimate.

Answer:
There are 10 squares covered by the shape. out of ten 4 are half covered with the shape.
So, area of the shape is around 8 squares unit.

Question 2.
Use the Distributive Property to write the equation that represents the picture. Then give the area of each smaller rectangle and the large rectangle.

Answer:
Area of the larger rectangle = 3 × 5 = 15
Area of the smaller rectangles |
3 × 3 = 9
3 × 2 = 6
adding them
9 + 6 = 15 .
The equation
3 × 5 = 3 × (2 + 3) = (3 × 2) + (3 × 3 ) = 6 + 9 = 15.

Question 3.
Lewis says that the figure below has an area of 4 square meters. Is he correct? Explain.

Answer:
No, Lewis is wrong. As he says the area of the figure is 4 square m. But the  square represent cm.
The area of the figure is = 7 square cm.
There are 4 complete squares and 6 half divided squares.
adding them
4 + (6 ÷ 2 ) = 4 + 3 = 7 square cm.
Area of the figure is 7 square cm.

Question 4.
Select all of the ways to break apart the area of the large rectangle into the sum of the areas of two smaller rectangles. Then give the area of the large rectangle.

☐ 5 × 7 = 5 × (1 + 5) = (5 × 1) + (5 × 5)
☐ 5 × 7 = 5 × (3 + 4) = (5 × 3) + (5 × 4)
☐ 5 × 7 = 5 × (2 + 3) = (5 × 2) + (5 × 3)
☐ 5 × 7 = 5 × (1 + 6) = (5 × 1) + (5 × 6)
☐ 5 × 7 = 5 × (2 + 5) = (5 × 2) + (5 × 5)
Area = ☐ square inches
Answer:
7 is divided into two parts
7 = (1 + 6)
7 = (2 + 5)
7 = (3 + 4)
7 = (4 + 3)
7 = (5 + 2)
7 = (6 + 1)
Area of the larger rectangle = 5 in × 7 in = 35 square in
5 × 7 = 5 × (3 + 4) = (5 × 3) + (5 × 4) = 15 + 20 = 35
5 × 7 = 5 × (1 + 6) = (5 × 1) + (5 × 6) = 5 + 30 = 35
5 × 7 = 5 × (2 + 5) = (5 × 2) + (5 × 5) = 10 + 25 = 35
all these equations are correct.

Question 5.
What is the total area of the design below?

A. 4 × 4 = 16 square inches
B. (4 × 4) + (2 × 2) = 20 square inches
C. (4 × 4) + (2 × 2) + (2 × 2) = 24 square inches
D. 8 × 4 = 32 square inches
Answer:
The design is divided into 3 rectangles
(4 in × 4 in) + (2 in × 2 in) + (2 in × 2 in)
16 square in + 4 square in + 4 square in
24 square in.
So the answer is C. (4 × 4) + (2 × 2) + (2 × 2) = 24 square inches.

Question 6.
Jared draws a rectangle. Explain how to find the area using the Distributive Property.

Answer:
Area of the rectangle using distributive property
Dividing the rectangle into smaller parts
9 ft is divided into 5 ft and 4 ft
Area of the smaller rectangles
5 × 9 = 5 × (5 + 4)
= (5 × 5) + (5 × 4)
= 25 + 20
= 45 square ft.
Area of the rectangle Jared drew is 45 square ft.

Question 7.
Fran has a square flower bed. One side of the flower bed is 3 feet long. How can you find the area of the flower bed?
Answer:
Shape of Fran flower bed is Square.
Square have 4 equal sides.
One side of the flower bed = 3 ft.
Area of a square = side × side
Area of the flower bed = 3 ft × 3 ft = 9 square ft.
Fran square flower bed is 9 square ft.

Question 8.
Find the missing side length. Then find the area and explain how to find it.

Answer:
The opposite side of the missing side has a length of 6 ft.
Here the missing length having 1 ft length above and below it.
6 ft id distributed as
6 = 1 + 1 + ?
? = 6 – 1 – 1
? = 6 – 2
? = 4 ft.
The missing side length is 4 ft.
On adding the lengths 1 ft + 4 ft + 1 ft = 6 ft .
same as the opposite side.

Question 9.
This rectangle has an area of 56 square centimeters. What is the missing length? Use an equation to explain.

Answer:
Given
Area of the rectangle is 56 square cm.
Area of a rectangle = length × width
56 square cm = 7 cm × ? cm
? cm = 56 square cm ÷ 7 cm
? cm = 8 cm.
The missing length is 8 cm.
7 cm × 8 cm = 56 square cm.

Question 10.
What is the area of Ron’s figure? Explain.

Answer:

Question 11.
Maddie makes a mosaic with 1-inch glass squares, as shown below. Which color of glass has the greatest area in Maddie’s mosaic?

Answer:
Number of violet squares = 1 + 3 +5 + 5 + 3 + 1 = 9 + 9 = 18 squares
Area of violet squares = 18 square in
Number of yellow squares = 2 + 6 + 2 = 10 squares.
Area of yellow squares = 10 square in
Number of white squares = 4 + 4 = 8 squares.
Area of white squares = 8 square in

Violet color squares have greater area.

Question 12.
Select the correct side length for each square given its area.

Answer:

Question 13.
Explain how to find the area of each rectangle and the total area of the rectangles.

Answer:
Number of squares Ryan rectangle covers = 6 × 4 = 24 squares.
Area of Ryan rectangle = 24 square cm
Number of squares Jodie rectangle covers = 3 × 2 = 6 squares.
Area of Jodie rectangle = 6 square cm.
Total area of the rectangles = 24 square cm + 6 square c,m = 30 square cm.

Question 14.
Some students in Springfield make a parade float with the letter S on it. Draw lines to divide the shape into rectangles. Then find how many square feet the letter is.

A. 28 square ft
B. 56 square ft
C. 54 square ft
D. 90 square ft
Answer:

Question 15.
Max draws 2 rectangles, each with an area of 24 square centimeters. What could be the side lengths of Max’s rectangles? Show how he could use the Distributive Property to represent the area in each case.
Answer:
Given
Number of rectangles Max draws = 2 rectangles
Area of each rectangles = 24 square cm
The lengths of the rectangle 1 = 4 cm × 6 cm
Using distributive property
to find the area of rectangle 1 =
=  4 cm × 6 cm
= 4 cm × (2 cm + 4 cm )
= (4 cm ×2 cm) + (4 cm × 4 cm)
= 8 square cm + 16 square cm
=24 cm
to find the area of rectangle 2 =
The lengths of the rectangle 2 = 3 cm × 8 cm
= 3 cm × 8 cm
= 3 cm × (3 cm + 5 cm)
= (3 cm × 3 cm ) + (3 cm × 5 cm)
= 9 square cm +15 square cm
= 24 square cm

Question 16.
A community center builds a new activity room in the shape shown below. Explain how to find the area of the room, and solve.

Answer:

Question 17.
Show 2 different unit squares that you can use to measure the area of these rectangles. Find the area with your unit squares.

Answer:

Question 18.
Ethan wants to know the area of the yellow part of this design.

A. Explain how you can break this problem into simpler problems.
Answer:
Finding the area of the outer rectangle.
Finding the area of the inner rectangle .
Then subtracting the area of inner rectangle from outer rectangle gives the area of the yellow region.

B. Find the yellow area. Show your work.
Answer:
Area of the outer rectangle = 6 ft × 4 ft = 24 square ft
Area of the inner rectangle = 3 ft × 2 ft = 6 square ft
Area of the yellow area = Area of outer rectangle – Area of inner rectangle
= 24 square ft – 6 square ft
= 18 square ft.
The yellow area is 18 square ft.

Topic 6 Performance Task

Banner Design
Jessie is designing a banner that has red, blue, and white sections.
The Banner Details list shows the rules for each color.
The Jessie’s Banner diagram shows the different sections of the banner.
Banner Details

• Red sections must have a total area greater than 5 in. 4in. 3 in. 40 square inches.
• Blue sections must have a total area greater than 30 square inches.
• The white section must have an area less than 40 square inches.

Use the Jessie’s Banner diagram to answer Question 1.

Question 1.
To check if his banner fits the rules, Jessie started this table. Complete the table. Use multiplication and addition as needed.

Answer:

Use the table above and the Banner Details list to answer Question 2.

Question 2.
Is Jessie’s banner within the totals in the Banner Details list? Explain.
Answer:

Banner Details

• Red sections must have a total area greater than 5 in. 4in. 3 in. 40 square inches.
• Blue sections must have a total area greater than 30 square inches.
• The white section must have an area less than 40 square inches.

Yes, Jessie banner is not within the totals in the Banner detail list.
Total red color area = 35 square inches + 21 square inches = 56 square inches.
56 square inches is greater than 40 square inches
Total White color area = 30 square inches.
30 square inches less than 40 square inches.
Total blue color area = 9 square inches + 25 square inches = 34 square inches.
34 square inches greater than 30 square inches.

Question 3.
Jessie makes a square patch to go on top of the banner.
Part A
Draw unit squares to cover the patch. How many unit squares cover the patch?

Answer:

Part B
Jessie says if he checks the area by multiplying, the area will be the same as if he counts each unit square. Is he correct? Explain.
Answer:

Yes, Jessie is correct.
Area of the patch by multiplying is same as area by counting the squares.

Question 4.
Jessie uses two colors to make the patch. The colors have different areas.
Part A
Explain how to separate the square into two smaller rectangles with different areas. Use multiplication to find the areas of each of the smaller rectangles.
Answer:

Part B
Is the area of the square equal to the total area of the two smaller rectangles? Use an equation to explain.
Answer:

Go through the enVision Math Common Core Grade 7 Answer Key Topic 4 Generate Equivalent Expressions regularly and improve your accuracy in solving questions.

enVision Math Common Core 7th Grade Answers Key Topic 4 Generate Equivalent Expressions

Topic 4 Essential Question
How can properties of operations help to generate equivalent expressions that can be used in solving problems?

3-ACT MATH

Topic 4 enVision STEM Project

Did You Know?
In 2013, just over 30% of American consumers knew about activity trackers. By 2015, about 82% recognized them.

Continued research and development leads to technological advances and breakthroughs, such as the use of biosensing apparel to track activity.

Your Task: Analyze Activity Tracker Data
The ways that data are communicated and presented to the user are just as important as the types of data collected. You and your classmates will continue your exploration of activity trackers and use data to develop models based on individual fitness goals.

Topic 4 Get Ready

Review What You Know

Vocabulary

Choose the best term from the box to complete each definition.
evaluate
expression
factor
order of operations
substitute
term

Question 1.
When you __________ an expression, you replace each variable with a given value.

Answer:
When you evaluate an expression, you replace each variable with a given value.

Explanation:
In the above-given question,
given that,
when we evaluate an expression, we replace each variable with a given value.
for example:
Evaluate 3a-2b.
for a = 6 and b = 4.
3(6) – 2(4).
18 – 8.
10.

Question 2.
To evaluate a + 3 when a = 7, you can _________ 7 for a in the expression.

Answer:
To evaluate a + 3 when a = 7, you can substitute 7 for a in the expression.

Explanation:
In the above-given question,
given that,
if we evaluate a + 3 when a = 7.
we will substitute 7 for a in the expression.
a + 3.
7 + 3.
10.

Question 3
The set of rules used to determine the order in which operations are performed is called the _________

Answer:
The set of rules used to determine the order in which operations are performed is called the order of operations.

Explanation:
In the above-given question,
given that,
The set of rules used to determine the order in which operations are performed is called the order of operations.
for example:
3 + [6(11 + 1 – 4)]/8 x 2.
3+[6(8)]/8 x 2.
3 + 48 / 8 x 2.
3 + 6 x 2.
3 + 12.
15.

Question 4.
Each part of an expression that is separated by a plus or minus sign is a(n) __________.

Answer:
Each part of an expression that is separated by a plus or minus sign is the term.

Explanation:
In the above-given question,
given that,
Each part of an expression that is separated by a plus or minus sign is the term.
for example:
2x + 4y – 9.
where x and y are variables.
9 is the constant.
2 and 4 are coefficients.
terms are 2x, 4y, and 9.

Question 5.
A(n) __________ is a mathematical phrase that can contain numbers, variables, and operation symbols.

Answer:
An expression is a mathematical phrase that can contain numbers, variables, and operation symbols.

Explanation:
In the above-given question,
given that,
An expression is a mathematical phrase that can contain numbers, variables, and operation symbols.
for example:
n + 7 = 10.
x – 5 = 3.
3p = 15.
y/2 = 5.

Question 6.
When two numbers are multiplied to get a product, each number is called a(n) _________.

Answer:
When two numbers are multiplied to get a product, each number is called a factor.

Explanation:
In the above-given question,
given that,
When two numbers are multiplied to get a product, each number is called a factor.
for example:
3 x 5 = 15.
3 and 5 are the factors.
15 is the product.

Order of Operations

Evaluate each expression using the order of operations.
Question 7.
3(18 – 7) + 2

Answer:
3(18 – 7) + 2 = 35.

Explanation:
In the above-given question,
given that,
3(18 – 7) + 2.
3(11) + 2.
33 + 2.
35.
3(18 – 7) + 2 = 35.

Question 8.
(13 + 2) ÷ (9 – 4)

Answer:
(13 + 2) ÷ (9 – 4) = 3.

Explanation:
In the above-given question,
given that,
(13 + 2) ÷ (9 – 4).
(13 + 2) ÷ (9 – 4).
15 / 5.
3.
(13 + 2) ÷ (9 – 4) = 3.

Question 9.
24 ÷ 4 • 2 – 2

Answer:
24 ÷ 4 • 2 – 2 = 10.

Explanation:
In the above-given question,
given that,
24 ÷ 4 • 2 – 2.
6 . 2 – 2.
12 – 2.
10.
24 ÷ 4 • 2 – 2 = 10.

Equivalent Expressions

Evaluate each expression when a = -4 and b = 3.
Question 10.
ab

Answer:
ab = -12.

Explanation:
In the above-given question,
given that,
a = -4 and b = 3.
– 4 x 3.
-12.
ab = -12.

Question 11.
2a + 3b

Answer:
2a + 3b = 1.

Explanation:
In the above-given question,
given that,
a = -4 and b = 3.
2(-4) + 3(3).
-8 + 9.
1.
2a + 3b = 1.

Question 12.
2(a – b)

Answer:
2(a – b) = -14.

Explanation:
In the above-given question,
given that,
a = -4 and b = 3.
2(-4 – 3).
2(-7).
-14.
2(a – b) = -14.

Question 13.
Explain the difference between evaluating 3 • 7 – 4 ÷ 2 and evaluating 3(7 – 4) ÷ 2.

Answer:
The two expressions are different.

Explanation:
In the above-given question,
given that,
3 . 7 – 4 ÷ 2.
3 . 7 – 2.
21 – 2.
19.
3(7 – 4) ÷ 2.
3(3) / 2.
9 / 2.

Language Development

Complete each math statement using the word bank.

To evaluate an algebraic expression, substitute a __________ for the variable in the expression.

Answer:
To evaluate an algebraic expression, substitute properties of operations for the variable in the expression.

Explanation:
In the above-given question,
given that,
To evaluate an algebraic expression, substitute properties of operations for the variable in the expression.
for example:
n + 7 = 10.
x – 5 = 3.
3p = 15.
y/2 = 5.

In the algebraic expression 3(x – 2), 3 and x – 2 are ___________

Answer:
In the algebraic expression 3(x – 2), 3, and x – 2 are coefficients.

Explanation:
In the above-given question,
given that,
In the algebraic expression 3(x – 2), 3 and x – 2 are coefficients.
for example:
3(x – 2).
3 and x-2 are coefficients.

To generate equivalent expressions, you can use the __________

Answer:
To generate equivalent expressions, you can use the order of operations.

Explanation:
In the above-given question,
given that,
To generate equivalent expressions, you can use the order of operations.
for example:
5(x – 1) + 7.
5(x) + 5(-7) + 7.
5x – 5 + 7.
5x + 2.

In the expression 4x + 2x – 6y, you first need to __________

Answer:
In the expression 4x + 2x – 6y, you first need to add.

Explanation:
In the above-given question,
given that,
In the expression 4x + 2x – 6y, you first need to add.
4x + 2y – 6y.
4x – 4y.

You can use the Distributive Property to __________ the algebraic expression 5(x – 7).

Answer:
You can use the Distributive property to find the algebraic expression.

Explanation:
In the above-given question,
given that,
we can use the distributive property to find the algebraic expression.
for example:
5(x – 7).
5x – 35.

In the algebraic expression, 6x + 10, x is the ________ , 6 is the ________ and 10 is the ___________

Answer:
In the algebraic expression 6x + 10, x is the coefficient, 6 is the variable, and 10 is the constant.

Explanation:
In the above-given question,
given that,
In the algebraic expression 6x + 10, x is the coefficient, 6 is the variable, and 10 is the constant.
for example:
6x + 10.
where 6 is the variable.
x is coefficient.
10 is constant.

Four words that describe operations that can be used with expressions are _________, and ________, _________ and __________.

Answer:
The words that describe the operations are constants, terms, variables, and coefficients.

Explanation:
In the above-given question,
given that,
The words that describe the operations are constants, terms, variables, and coefficients.
for example:
2x + 4y – 9.
where x and y are variables.
9 is the constant.
2 and 4 are coefficients.
terms are 2x, 4y, and 9.

In the algebraic expression 5x + 4 + 6x – 3, you use the Commutative Property to _________ like terms next to each other and the Associative Property to _________ like terms together.

Answer:

Pick A Project

PROJECT 4A
Which emojis would you use to tell the story of your day so far?
PROJECT: WRITE AND ILLUSTRATE A CHILDREN’S BOOK

PROJECT 4B
How many different ways can you represent a dollar?
PROJECT: GENERATE EQUIVALENCE

PROJECT 4C
If you wrote a song, what would it sound like?
PROJECT: COMPOSE A SONG

PROJECT 4D
What was your favorite structure at a playground when you were younger?
PROJECT: BUILD A MODEL PLAYGROUND

Lesson 4.1 Write and Evaluate Algebraic Expressions

Solve & Discuss It!
Mr. Ramirez’s class was playing a game in which students need to match sticky notes that have equivalent expressions. How can you sort the expressions into groups?
I can… write and evaluate algebraic expressions.

Focus on math practices
Reasoning is there more than one way to group the expressions? Give an example.

Answer:
Yes, there are more than one way to group the expressions.

Explanation:
In the above-given question,
given that,
A numerical expression in mathematics can be a combination of numbers, integers combined using.
for example:
16 is an numerical expression.

Essential Question
How can algebraic expressions be used to represent and solve problems?

Answer:
We can use algebra to solve mathematical problems.

Explanation:
In the above-given question,
given that,
we can also interpret the solution in the context of the original problem.
for example:
2x + 5 = 43.
where 43 is the constant.
always has an equal symbol.
2x + 5 = algebraic expression.

Try It!

Misumi started with $217 in her bank account. She deposits $25.50 each week and never withdraws any money. What expression can Misumi use to determine her account balance after w weeks?

Answer:
The expression can Misumi use to determine her account balance after w weeks = 8.5 weeks.

Explanation:
In the above-given question,
given that,
Misumi started with $217 in her bank account.
She deposits $25.50 each week and never withdraws any money.
$217 / 25.50 = 8.5.
so the expression can Misumi use to determine her account balance after w weeks = 8.5.

Convince Me! How did you determine which value to use for the constant and which value to use for the coefficient?

Answer:
x is coefficient and 3 is constant.

Explanation:
In the above-given question,
given that,
2x + 3 is the expression.
x is the coefficient.
3 is the constant.

Try It!

The cost to rent a scooter is $15.50 per hour and the cost to rent a watercraft is $22.80 per hour. Use the expression 15.5s + 22.8w to determine how much it would cost to rent a scooter for 3\(\frac{1}{2}\) hours and a watercraft for 1\(\frac{3}{4}\) hours.

Answer:
The cost would cost to rent a scooter for 3(1/2) hours and watercraft for 1(3/4) hours = $54.25 and $40.

Explanation:
In the above-given question,
given that,
The cost to rent a scooter is $15.50 per hour and the cost to rent a watercraft is $22.80 per hour.
Use the expression 15.5s + 22.8w to determine how much.
15.5s + 22.8w.
3. 1/2 = 7/2.
1. 3/4 = 7/4.
15.5(7/2) + 22.8(7/4).
108.5/2 + 159.6/4.
54.25 + 39.9.

Try It!

Emelia earns $8.74 per hour plus a gas allowance of $3.50 per day at her job. How much does Emelia’s job pay in a day when she works 5\(\frac{1}{2}\) hours? Write an expression and evaluate for 5\(\frac{1}{2}\) hours.

Answer:
Emelia’s job pay in a day when she works 5(1/2) hours = $67.32.

Explanation:
In the above-given question,
given that,
Emelia earns $8.74 per hour plus a gas allowance of $3.50 per day at her job.
$8.74 + $3.50.
$12.24.
5(1/2) = 5.5.
5.5 x $12.24 = $67.32.
so Emelia’s job pay in a day when she works 5(1/2) hours = $67.32.

KEY CONCEPT
Algebraic expressions can be used to represent problems with unknown or variable values. Values can be substituted for variables to evaluate the expression.

Do You Understand?
Question 1.
Essential Question How are algebraic expressions used to represent and solve problems?

Answer:
Algebraic expressions are used to represent problems with unknowns or variable values.

Explanation:
In the above-given question,
given that,
Algebraic expressions are used to represent problems with unknowns or variable values.
Values can be substituted for variables to evaluate the expression.
for example:
2x + 3y = a.
where x = 2 and y = 3.
2 x 2 + 3 x 3 = a.
4 + 9 = a.
13 = a.

Question 2.
Use Structure How is a constant term different than a variable term for an expression that represents a real-world situation?

Answer:
a = 13.

Explanation:
In the above-given question,
given that,
2x + 3y = a.
where x = 2 and y = 3.
2 x 2 + 3 x 3 = a.
4 + 9 = a.
13 = a.

Question 3.
Look for Relationships Explain why you can have different values when evaluating an algebraic expression.

Answer:
To evaluate an algebraic expression we have to substitute a number for each variable and perform the arithmetic operations.

Explanation:
In the above-given question,
given that,
To evaluate an algebraic expression we have to substitute a number for each variable and perform the arithmetic operations.
for example:
x + 6.
where x = 6.
6 + 6 = 12.
if we know the value of our variables, we can replace the variables with their values and then evaluate the expression.

Do You Know How?
Question 4.
A tank containing 35 gallons of water is leaking at a rate of \(\frac{1}{4}\) gallon per minute. Write an expression to determine the number of gallons left in the tank after m minutes.

Answer:
The number of gallons left in the tank after m minutes = 8.75 gallons.

Explanation:
In the above-given question,
given that,
A tank containing 35 gallons of water is leaking at a rate of \(\frac{1}{4}\) gallon per minute.
35 x 1/4 = 35/4.
8.75.
so the number of gallons left in the tank after m minutes = 8.75 gallons.

Question 5.
Write an algebraic expression that Marshall can use to determine the total cost of buying a watermelon that weighs w pounds and some tomatoes that weigh t pounds. How much will it cost to buy a watermelon that weighs 18\(\frac{1}{2}\) pounds and 5 pounds of tomatoes?

Answer:
The much will it cost to buy a watermelon that weighs 18(1/2) pounds and 5 pounds of tomatoes = $29.25 and $3.4.

Explanation:
In the above-given question,
given that,
the cost of tomatoes is $3.25 per lb.
the cost of watermelons is $0.68 per lb.
18/2 = 9.
$3.25 x 9 = $29.25.
0.68 x 5 = $3.4.
so the much will it cost to buy a watermelon that weighs 18(1/2) pounds and 5 pounds of tomatoes = $29.25 and $3.4.

Question 6.
What is the value of \(\frac{3}{8}\)x – 4.5 when x = 0.4?

Answer:
(3/8)x – 4.5 = 4.35.

Explanation:
In the above-given question,
given that,
(3/8)x – 4.5.
where x = 0.4.
(3/8)0.4 – 4.5.
0.375 x 0.4 – 4.5.
0.15 – 4.5.
4.35.
(3/8)x – 4.5 = 4.35.

Question 7.
What is the value of 8.4n – 3.2p when n = 2 and p = 4?

Answer:
8.4n – 3.2p = 4.

Explanation:
In the above-given question,
given that,
8.4n – 3.2p.
8.4 (2) – 3.2 (4).
16.8 – 12.8.
4.
8.4n – 3.2p = 4.

Practice & Problem Solving

Leveled Practice For 8-10, fill in the boxes to complete the problems.
Question 8.
Evaluate 10.2x + 9.4y when x = 2 and y = 3.
10.2 (_______) + 9.4 (_______)
= _______ + 28.2
= _______

Answer:
10.2 x + 9.4 y = 48.6.

Explanation:
In the above-given question,
given that,
10.2 x + 9.4 y.
where x = 2 and y = 3.
10.2 x 2 + 9.4 x 3.
20.4 + 28.2.
48.6.
10.2 x + 9.4 y = 48.6.

Question 9.
Evaluate \(\frac{1}{2}\)t + \(\frac{3}{8}\) when t = \(\frac{1}{4}\)
\(\frac{1}{2}\)(________) + \(\frac{3}{8}\)
= ______ + \(\frac{3}{8}\)
= ______

Answer:
1/2 x 1/4 + 3/8 = 0.5.

Explanation:
In the above-given question,
given that,
1/2 = 0.5.
1/4 = 0.25.
0.5 x 0.25 + 3/8.
0.125 + 0.375.
0.5.
1/2 x 1/4 + 3/8 = 0.5.

Question 10.
Write an expression that represents the height of a tree that began at 6 feet and increases by 2 feet per year. Let y represent the number of years.
_____ + ______ y

Answer:
6x + 2y.

Explanation:
In the above-given question,
given that,
the height of a tree that began at 6 feet and increases by 2 feet per year.
6x + 2y.
where y represents the number of years.
so the expression is 6x + 2y.

For 11-14, evaluate each expression for the given value of the variable(s).
Question 11.
3d – 4
d = 1.2

Answer:
3d – 4 = 0.4.

Explanation:
In the above-given question,
given that,
3d – 4.
where d = 1.2.
3(1.2) – 4.
3.6 – 4.
0.4.
3d – 4 = 0.4.

Question 12.
0.5f – 2.39
f = 12, 9 = 2

Answer:
0.5f – 2.39 = 3.68.

Explanation:
In the above-given question,
given that,
0.5f – 2.39.
where f = 12 and 9 = 2.
0.5 x 12 – 2.32.
6 – 2.32.
3.68.
0.5f – 2.39 = 3.68.

Question 13.
p + 3
p = \(\frac{3}{5}\)

Answer:
p + 3 = 3.6.

Explanation:
In the above-given question,
given that,
p + 3.
where p = 3/5.
3/5 = 0.6.
0.6 + 3.
3.6.
p + 3 = 3.6.

Question 14.
34 + \(\frac{4}{9}\)w
w = –\(\frac{1}{2}\)

Answer:
34 + 4/9x w = 33.9.

Explanation:
In the above-given question,
given that,
34 + 4/9x w.
where w = -1/2.
34 + 4/9(-1/2).
34 + 0.4(-0.5).
34 – 0.1.
33.9.

Question 15.
Model with Math What expression can be used to determine the total cost of buying g pounds of granola for $3.25 per pound and f pounds of flour for $0.74 per pound?

Answer:
$3.25g and $0.74f.

Explanation:
In the above-given question,
given that,
the total cost of buying g pounds of granola for $3.25 per pound.
f pounds of flour for $0.74 per pound.
$3.25g + $0.74f

Question 16.
Model with Math Which expression can be used to determine the total weight of a box that by itself weighs 0.2 kilogram and contains p plaques that weigh 1.3 kilograms each?

A. 1.3p +0.2
B. 0.2p + 1.3
C. 0.2 – 1.3p
D. 1.2p

Answer:
Option A is correct.

Explanation:
In the above-given question,
given that,
total weight of a box that by itself weighs 0.2 kilogram.
and contains p plaques that weigh 1.3 kilograms each.
0.2 + 1.3p.
so option A is correct.

Question 17.
The expression -120 + 13n represents a submarine that began at a depth of 120 feet below sea level and ascended at a rate of 13 feet per minute. What was the depth of the submarine after 6 minutes?

Answer:
The depth of the submarine after 6 minutes = – 42 feet.

Explanation:
In the above-given question,
given that,
The expression -120 + 13n represents a submarine that began at a depth of 120 feet below sea level.
ascended at a rate of 13 feet per minute.
-120 + 13(6).
-120 + 78.
-42.
so the depth of the submarine after 6 minutes = -42 feet.

Question 18.
Be Precise A full grain silo empties at a constant rate. Write an expression to determine the amount of grain left after s seconds.

Answer:
The amount of grain left after 5 seconds = 2982.5 cubic feet.

Explanation:
In the above-given question,
given that,
A full grain silo empties at a constant rate.
the capacity of food grain is 3000 cubic feet.
3000 – 3.5/s.
3000 – 3.5(5).
3000 – 17.5.
2982.5.
so the amount of grain left after 5 seconds = 2982.5 cubic feet.

Question 19.
Higher Order Thinking For the expression 5 – 5x to have a negative value, what must be true about the value of x?

Answer:
The value of x = 4.

Explanation:
In the above-given question,
given that,
the expression is 5 – 5x.
where x = 4.
5 – 5(4).
5 – 20.
-15.

Assessment Practice

Question 20.
Joe bought g gallons of gasoline for $2.85 per gallon and c cans of oil for $3.15 per can.
PART A
What expression can be used to determine the total amount Joe spent on gasoline and oil?

Answer:
The total amount joe spend on gasoline and oil = $2.85g + $3.15c.

Explanation:
In the above-given question,
given that,
Joe bought g gallons of gasoline for $2.85 per gallon.
c cans of oil for $3.15 per can.
$2.85g + $3.15c.
so the total amount joe spends on gasoline and oil = $2.85g + $3.15c.

PART B
Joe spent $15. He bought 2 cans of oil. About how many gallons of gasoline did he buy?
A. 2.5
B. 3
C. 3.5
D. 4

Answer:
The gallons of gasoline did he buy = 3 gallons.

Explanation:
In the above-given question,
given that,
Joe spent $15. He bought 2 cans of oil.
1.5 x 2.
3.
the gallons of gasoline did he buy = 3.
so option B is correct.

Question 21.
The outside temperature was 73°F at 1 P.M. and decreases at a rate of 1.5°F each hour. What expression can be used to determine the temperature h hours after 1 P.M.?

Answer:
The expression can be used to determine the temperature h hours after 1 P.M = 71.5°F.

Explanation:
In the above-given question,
given that,
The outside temperature was 73°F at 1 P.M. and decreases at a rate of 1.5°F each hour.
73 – 1.5.
71.5°F.
so the expression can be used to determine the temperature h hours after 1 P.M = 71.5°F.

Lesson 4.2 Generate Equivalent Expressions

Explore It!
A shipment of eggs contains some cartons with a dozen eggs and some cartons with a half-dozen eggs.
I can… write equivalent expressions for given expressions.

A. How can you represent the total number of eggs in the shipment using diagrams or images? Explain your diagram.

Answer:
1 dozen + 1/2 dozen eggs.

Explanation:
In the above-given question,
given that,
A shipment of eggs contains some cartons with a dozen eggs and some cartons with a half-dozen eggs.
1 dozen = 12.
1/2 doxen = 12/2.
12/2 = 6.
12 + 1/2 eggs.

B. How can you represent the total number of eggs in the shipment using expressions? What variables do you use? What do they represent?

Answer:
1 dozen + 1/2 dozen eggs.

Explanation:
In the above-given question,
given that,
A shipment of eggs contains some cartons with a dozen eggs and some cartons with a half-dozen eggs.
1 dozen = 12.
1/2 doxen = 12/2.
12/2 = 6.
12 + 1/2 eggs.

Focus on math practices
Construct Arguments How do the two representations compare? How are they different?

Essential Question
What are equivalent expressions?

Try It!

Nancy wrote the expression 3x – 12 to represent the relationship in a table of values. Use properties of operations to write two equivalent expressions.
3(x – _____)
_____ + 3x

Answer:
The two equivalent expressions are 36 + 3x.

Explanation:
In the above-given question,
given that,
3x – 12.
3(x – 12).
3x – 36.
36 + 3x.

Convince Me! What property can you use to write an equivalent expression for -5(x – 2)? Explain.

Answer:
-5(x – 2) = -5x – 10.

Explanation:
In the above-given question,
given that,
-5(x – 2).
-5x – 10.
we can use the distributive property.
-5x -10.

Try It!

Use properties of operations to write two expressions that are equivalent to \(\frac{3}{4}\)n + (8 + \(\frac{1}{3}\)z).

Answer:
3/4n + (8 + {1/3}) = 0.75n + 8.3z.

Explanation:
In the above-given question,
given that,
3/4n + (8 + {1/3})z.
3/4n + 8 + 0.3z.
0.75n + 8.3z.
3/4n + (8 + {1/3}) = 0.75n + 8.3z.

Try It!

Write two expressions that are equivalent to –\(\frac{5}{4}\)x – \(\frac{3}{4}\)

Answer:
-5/4 – 3/4 = 2.

Explanation:
In the above-given question,
given that,
-5/4 – 3/4.
-5/4 = 1.25.
3/4 = 0.75.
-1.25 – 0.75.
2.

KEY CONCEPT
You can use properties of operations to write equivalent expressions.

Do You Understand?
What are equivalent
Question 1.
Essential Question expressions?

Answer:
-1/2(x + 8), -1/2x + (-4) and -4 +(-1/2x) are equivalent.

Explanation:
In the above-given question,
given that,
-1/2(x + 8).
-1/2x + (-1/2) . 8.
-1/2x + (-4).
-4 + (-1/2x).
the three expressions are true.

Question 2.
Make Sense and Persevere For which operations is the Commutative Property true?

Answer:
(-4) + -1/2x.

Explanation:
In the above-given question,
given that,
(-4) + -1/2x.
we can use the commutative property for the expression.
-1/2x + (-4).

Question 3.
How can the Associative Property be applied when writing equivalent expressions with variables?

Answer:

Do You Know How?
Question 4.
Write an expression equivalent to -3 + \(\frac{2}{3}\)y – 4 – \(\frac{1}{3}\)y.

Answer:
-3 + (2/3)y – 4 – (1/3)y.

Explanation:
In the above-given question,
given that,
-3 + \(\frac{2}{3}\)y – 4 – \(\frac{1}{3}\)y.
-3 + (2/3)y – 4 – (1/3)y.
-3 – 4 + (2/3)y – (1/3)y.
-7 + 1/3y.
-3 + (2/3)y – 4 – (1/3)y = -7 + 1/3y.

Question 5.
Complete the tables to determine if the expressions are equivalent. If the expressions are equivalent, name the property or properties that make them equivalent.

Answer:
3(x – 5) = 3x – 15.

Explanation:
In the above-given question,
given that,
3(x – 5).
3x – 15.
x = 1.
3 – 15 = -12.
x = 2.
6 – 15 = -9.
x = 3.
9 – 15 = -6.

Question 6.
Use the properties of operations to write an expression equivalent to 4x + \(\frac{1}{2}\) + 2x – 3.

Answer:
4x +[{1/2}] + 2x – 3 = 3x -3.

Explanation:
In the above-given question,
given that,
4x + (1/2) + 2x – 3.
4x + 2x + (1/2) – 3.
6x + (1/2) -3.
3x – 3.
4x +[{1/2}] + 2x – 3 = 3x -3.

Practice & Problem Solving

For 7-9, write an equivalent expression.
Question 7.
-3(7 + 5g)

Answer:
-36g.

Explanation:
In the above-given question,
given that,
-3(7 + 5g).
-3 x 7 = -21.
-3 x 5 = 15.
-21 + (-15g).
-36g.

Question 8.
(x + 7) + 3y

Answer:
24xy.

Explanation:
In the above-given question,
given that,
(x + 7) + 3y.
3y x X + 3y x 7.
3xy + 21y.
24xy.

Question 9.
\(\frac{2}{9}\) – \(\frac{1}{5}\) • x

Answer:
2/9 – 1/5 . X =

Explanation:
In the above-given question,
given that,
2/9 x X – (1/5)x.
2/9 x – 1/5 x.

Question 10.
Which expression is equivalent to t + 4 + 3 – 2t?
A. t + 7
B. -t + 7
C. 6t
D. 10t

Answer:
t + 4 + 3 – 2t = -t + 7.

Explanation:
In the above-given question,
given that,
t + 4 + 3 – 2t.
t + 7 – 2t.
t – 2t + 7.
-t + 7.
t + 4 + 3 – 2t = -t + 7.

Question 11.
The distance in feet that Karina swims in a race is represented by 4d – 4, where d is the distance for each lap. What is an expression equivalent to 4d – 4?

Answer:
The expression equal to 4d – 4 = 4(d – 4).

Explanation:
In the above-given question,
given that,
4d – 4.
4(d – 4).
4d – 16.
4d – 4 = 4d – 16.

Question 12.
Use the Associative Property to write an expression equivalent to (w + 9) + 3.

Answer:
The expression is (3 + w) + 9.

Explanation:
In the above-given question,
given that,
the expression is (w + 9) + 3.
(3 + w) + 3.
6 + w.

Question 13.
Nigel is planning his training schedule for a marathon over a 4-day period. He is uncertain how many miles he will run on two d

Answer:
The number of miles he will run on two days = 14.5 miles.

Explanation:
In the above-given question,
given that,
Nigel is planning his training schedule for a marathon over a 4-day period.
on day 1 he will run 12 miles.
on day 2 he will run 14.5 miles.
on day 3 he will run 17 miles.
12 + 17 = 29.
29/2 = 14.5.
on day 2 he will run 14.5 miles.

Question 14.
Maria said the expression -4n+ 3 + 9n – 4 is equivalent to 4n. What error did Maria likely make?

Answer:
-4n + 3 + 9n – 4 = 5n – 1.

Explanation:
In the above-given question,
given that,
-4n + 3 + 9n – 4.
-4n + 9n = 5n.
5n + 3 – 4.
5n – 1.
-4n + 3 + 9n – 4 = 5n – 1.

Question 15.
Write an expression equivalent to x – 3y + 4.

Answer:
x – 3y + 4 = 4 + x – 3y.

Explanation:
In the above-given question,
given that,
x – 3y + 4.
4 + x – 3y.
x + 4 – 3y.

Question 16.
Andre wrote the expression -2 + 4x = 3 to represent the relationship shown in the table.

Write two other expressions that also represent the relationship shown in the table.

Answer:
-2 + 4x = 3.

Explanation:
In the above-given question,
given that,
-2 + 4x.
x = 0.
-2 + 0.
-2.
x = 6.
-2 + 4(6).
-2 + 24.
22.
x = 12.
-2 + 4(12).
-2 + 48.
46.

Question 17.
Higher Order Thinking to rent a car for a trip, four friends are combining their money. The group chat shows the amount of money that each puts in. One expression for their total amount of money is 189 plus p plus 224 plus q.

a. Use the Commutative Property to write two equivalent expressions.

Answer:
The expressions are 189 + p + 224q.

Explanation:
In the above-given question,
given that,
189 + p + 224q.
p + 189 + 224q.
224q + p + 189.

b. If they need $500 to rent a car, find at least two different pairs of numbers that p and q could be.

Answer:
$500 + p + 224q.

Explanation:
In the above-given question,
given that,
500 – 224 = 276.
276 + p + 224q.
p + 276 + 224q.
224q + 276 + p.

Assessment Practice

Question 18.
Select all expressions equivalent to \(\frac{3}{5}\)x + 3.

Answer:
The expressions equivalent to (3/5)x + 3 = 1 + 2/5x + 3 and 4/5x – 1/5x + 3.

Explanation:
In the above-given question,
given that,
(3/5)x + 3.
1 + 2/5x + 3.
4/5x – 1/5x + 3.
3/5x + 3.
so the expressions equivalent to (3/5)x + 3 = 1 + 2/5x + 3 and 4/5x – 1/5x + 3.

Lesson 4.3 Simplify Expressions

Solve & Discuss It!
How can the tiles below be sorted?
I can… use properties of operations to simplify expressions.

Focus on math practices
Reasoning Would sorting the tiles with positive coefficients together and tiles with negative coefficients together help to simplify an expression that involves all the tiles? Explain.

Answer:
The positive coefficients are 4.25, 3/5y, 3/8, 1/5, 2.1x, and 1/2x.
The negative coefficients are -0.5, -0.5x, -2.1y, and -2.

Explanation:
In the above-given question,
given that,
the coefficients are 4.25, 3/5y, 3/8, 1/5, 2.1x, -0.5, -0.5x, -2.1y, -2, and 1/2x.
the positive coefficients are 4.25, 3/5y, 3/8, 1/5, 2.1x, and 1/2x.
the negative coefficients are -0.5, -0.5x, -2.1y, and -2.

Essential Question
How are properties of operations used to simplify expressions?

Try It!

Simplify the expression – 6 – 6f + 7 – 3f – 9.
______ – 3f – _____ + 7 – ______
_____ – ______

Answer:
-6 – 6f + 7 – 3f – 9 = -9f – 23.

Explanation:
In the above-given question,
given that,
-6 – 6f + 7 – 3f – 9.
-6f  – 3f – 6 + 7 – 9.
-9f – 13 – 9.
-9f – 23.
-6 – 6f + 7 – 3f – 9 = -9f – 23.

Convince Me! How do you decide in what way to reorder the terms of an expression when simplifying it?

Try It!

Simplify each expression.
a. 59.95m – 30 + 7.95m + 45 + 9.49m

Answer:
59.95m – 30 + 7.95m + 45 + 9.49m = 52m – 5.51.

Explanation:
In the above-given question,
given that,
59.95m – 30 + 7.95m + 45 + 9.49.
59.95m + 7.95m – 30 + 45 + 9.49.
52m – 15 + 9.49.
52m – 5.51.

b. -0.5p + \(\frac{1}{2}\)p – 2.75 + \(\frac{2}{3}\)p

Answer:
-0.5p + (1/2)p – 2.75 + (2/3)p = 0.6p – 2.75.

Explanation:
In the above-given question,
given that,
-0.5p + (1/2)p – 2.75 + (2/3)p.
-0.5p + 0.5p – 2.75 + 0.6p.
0.6p – 2.75.
-0.5p + (1/2)p – 2.75 + (2/3)p = 0.6p – 2.75.

Try It!

Simplify the expression -3.7 +59 + 4k + 11.1 – 10g.
(______ – 10g) + 4k + (______ + 11.1)
= _______ + 4k + ______
The simplified expression is ________.

Answer:
-3.7 + 59 + 4k + 11.1 – 10g = 55.3 – 10g.

Explanation:
In the above-given question,
given that,
-3.7 + 59 + 4k + 11.1 – 10g.
-3.7 + 11.1 – 11.1 + 59 – 10g.
-3.7 + 59 – 10g.
55.3 – 10g.

KEY CONCEPT
When simplifying algebraic expressions, use properties of operations to combine like terms.
To simplify the expression below, group like terms.
\(\frac{3}{10}\)y – 3.5x – \(\frac{3}{8}\) +0.53x + 5.25 – 2.75y – 12
(-3.5x + 0.53x) + (\(\frac{3}{10}\)y – 2.75y) + (-\(\frac{3}{8}\) + 5.25 – 12)
Then combine like terms.
-2.97x – 2.45y – 7.125

Do You Understand?
Question 1.
Essential Question How are properties of operations used to simplify expressions?

Answer:
The properties of operations are used to combine like terms.

Explanation:
In the above-given question,
given that,
the properties of operations used to combine like terms.
for example:
\(\frac{3}{10}\)y – 3.5x – \(\frac{3}{8}\) +0.53x + 5.25 – 2.75y – 12
(-3.5x + 0.53x) + (\(\frac{3}{10}\)y – 2.75y) + (-\(\frac{3}{8}\) + 5.25 – 12)
Then combine like terms.
-2.97x – 2.45y – 7.125

Question 2.
Make Sense and Persevere Explain why constant terms expressed as different rational number types can be combined.

Answer:
The constant terms remain the same.

Explanation:
In the above-given question,
given that,
constant terms expressed as different rational number types can be combined.
for example:
\(\frac{3}{10}\)y – 3.5x – \(\frac{3}{8}\) +0.53x + 5.25 – 2.75y – 12
(-3.5x + 0.53x) + (\(\frac{3}{10}\)y – 2.75y) + (-\(\frac{3}{8}\) + 5.25 – 12)
Then combine like terms.
-2.97x – 2.45y – 7.125

Question 3.
Reasoning How do you know when an expression is in its simplest form?

Answer:
The expression is in its simplest form when it has only limited expressions.

Explanation:
In the above-given question,
given that,
for example:
-3.7 + 59 + 4k + 11.1 – 10g.
-3.7 + 11.1 – 11.1 + 59 – 10g.
-3.7 + 59 – 10g.
55.3 – 10g.

Do You Know How?
Question 4.
Simplify -4b + (-9k) – 6 – 3b + 12.

Answer:
-7b – 9k + 6.

Explanation:
In the above-given question,
given that,
-4b + (-9k) – 6 – 3b + 12.
-4b – 3b -9k – 6 + 12.
-7b – 9k + 6.

Question 5.
Simplify -2 + 6.45z – 6+ (-3.25z).

Answer:
-8 + 3.2z.

Explanation:
In the above-given question,
given that,
-2 + 6.45z – 6+ (-3.25z).
-2 + 6.45z – 6 – 3.25z.
-8 + 3.2z.

Question 6.
Simplify –9 + (-\(\frac{1}{3}\)y) +6 – \(\frac{4}{3}\)y.

Answer:
-3 – 5/3y.

Explanation:
In the above-given question,
given that,
–9 + (-\(\frac{1}{3}\)y) +6 – \(\frac{4}{3}\)y.
-9 – 1/3y + 6 – 4/3y.
-9 + 6 – 5/3y.
-3 – 5/3y.

Practice & Problem Solving

In 7-10, simplify each expression.
Question 7.
–2.8f +0.96 – 12 – 4

Answer:
-2.8f + 0.96 – 12 – 4 = -2.8f – 15.04.

Explanation:
In the above-given question,
given that,
-2.8f + 0.96 – 12 – 4.
-2.8f + 0.96 – 16.
-2.8f – 15.04.
-2.8f + 0.96 – 12 – 4 = -2.8f – 15.04.

Question 8.
3.2 – 5.1n – 3n + 5

Answer:
3.2 – 5.1n – 3n + 5 = 8.2 – 8.1n.

Explanation:
In the above-given question,
given that,
3.2 – 5.1n – 3n + 5.
3.2 – 8.1n + 5.
8.2 – 8.1n.
3.2 – 5.1n – 3n + 5 = 8.2 – 8.1n.

Question 9.
2n + 5.5 – 0.9n – 8 + 4.5p

Answer:
2n + 5.5 – 0.9n – 8 + 4.5p = 4.5p – 2.5 + 1.1n.

Explanation:
In the above-given question,
given that,
2n + 5.5 – 0.9n – 8 + 4.5p.
2n – 0.9n + 5.5 – 8 + 4.5p.
1.1n – 2.5 + 4.5p.
4.5p – 2.5 + 1.1n.
2n + 5.5 – 0.9n – 8 + 4.5p = 4.5p – 2.5 + 1.1n.

Question 10.
12 + (-4) – \(\frac{2}{5}\)j – \(\frac{4}{5}\)j + 5

Answer:
12 + (-4) – 2/5j – 4/5j + 5 = 13 – 6/5j.

Explanation:
In the above-given question,
given that,
12 + (-4) – 2/5j – 4/5j + 5.
12 -4 – 6/5j + 5.
8 – 6/5j + 5.
13 – 6/5j.
12 + (-4) – 2/5j – 4/5j + 5 = 13 – 6/5j.

Question 11.
Which expression is equivalent to -5v + (-2) + 1 + (-2v)?
A. -9v
B. -4v
C. -7v – 1
D. -7V + 3

Answer:
Option C is correct.

Explanation:
In the above-given question,
given that,
-5v + (-2) + 1 + (-2v).
-5v -2 + 1 -2v.
-7v -1.
so option C is correct.

Question 12.
Which expression is equivalent to \(\frac{2}{3}\)x + (-3) + (-2) – \(\frac{1}{3}\)x?
A. x + 5
B. –\(\frac{1}{3}\)x + 5
C. \(\frac{1}{3}\)x – 1
D. \(\frac{1}{3}\)x – 5

Answer:
Option D is correct.

Explanation:
In the above-given question,
given that,
\(\frac{2}{3}\)x + (-3) + (-2) – \(\frac{1}{3}\)x.
2/3x -3 -2 -1/3x.
1/3x -5
so option D is correct.

Question 13.
The dimensions of a garden are shown. Write an expression to find the perimeter.

Answer:
The perimeter of the garden = x – 7.

Explanation:
In the above-given question,
given that,
the length of the garden = x.
the width of the garden = 1/2x – 7.
area of the garden = l x b.
x + 1/2x – 7.
2/2x – 7.
x – 7.
so the perimeter of the garden = x – 7.

Question 14.
Simplify the expression 8h + (-7.3d) – 14 + 5d – 3.2h.

Answer:
8h + (-7.3d) – 14 + 5d – 3.2h = 5.2h – 2.3d – 14.

Explanation:
In the above-given question,
given that,
8h + (-7.3d) – 14 + 5d – 3.2h.
8h – 7.3d – 14 + 5d – 3.2h.
5.2h – 2.3d – 14.
8h + (-7.3d) – 14 + 5d – 3.2h = 5.2h – 2.3d – 14.

Question 15.
Simply 4 – 2y + (-8y) + 6.2.

Answer:
4 – 2y + (-8y) + 6.2 = 10.2 – 10y.

Explanation:
In the above-given question,
given that,
4 – 2y + (-8y) + 6.2.
4 – 2y – 8y + 6.2.
4 – 10y + 6.2.
10.2 – 10y.
4 – 2y + (-8y) + 6.2 = 10.2 – 10y.

Question 16.
Simplify \(\frac{4}{9}\)z – \(\frac{3}{9}\)z + 5 – \(\frac{5}{9}\)z – 8.

Answer:
4/9z – 3/9z + 5 – 5/9z – 8 = -4/9z – 3.

Explanation:
In the above-given question,
given that,
4/9z – 3/9z + 5 – 5/9z – 8.
1/9z + 5 – 5/9z – 8.
-4/9z + 5 – 8.
-4/9z – 3.
4/9z – 3/9z + 5 – 5/9z – 8 = -4/9z – 3.

Question 17.
Construct Arguments Explain whether 11t – 4t is equivalent to 4t – 11t. Support your answer by evaluating the expression for t = 2.

Answer:
The values are the same but 11t – 4t is positive and 4t – 11t is negative.

Explanation:
In the above-given question,
given that,
11t – 4t is equivalent to 4t – 11t.
t = 2.
11(2) – 4(2).
22 – 8.
14.
4t – 11t.
4(2) – 11(2).
8 – 22.
-14.

Question 18.
The signs show the costs of different games at a math festival. How much would it cost n people to play Decimal Decisions and Ratio Rage?

Answer:
The cost would take to n people to play Decimal Decisions and Ratio Rage = 6.6n/4 – 3.

Explanation:
In the above-given question,
given that,
the cost of 1 Game is 5.5n – 3.
the cost of 1 Game is n/4.
5.5n – 3 + n/4.
6.6n/4 – 3.
the cost would take to n people to play Decimal Decisions and Ratio Rage = 6.6n/4 – 3.

Question 19.
Higher Order Thinking in the expression ax + bx, a is a decimal and b is a fraction. How do you decide whether to write a as a fraction or b as a decimal?

Answer:
Yes, we can write an as a fraction and b as a decimal.

Explanation:
In the above-given question,
given that,
in the expression ax + bx, a is a decimal and b is a fraction.
for example:
a = 1.1.
b = 1/2.
ax + bx.
1.1x + 1/2x.
so 1.1 is a decimal and 1/2 is a fraction.

Assessment Practice

Question 20.
Select all expressions equivalent to -6z + (-5.5) + 3.5z + 5y – 2.5.
☐ -8 + 5y + 2.52
☐ -2.5z + 5y – 8
☐ -8 + 5y +(-2.5z)
☐ 2.5y + (-2.5z) – 5.5
☐ 5y – 8 – 2.5z

Answer:
Option B and C are correct.

Explanation:
In the above-given question,
given that,
-6z + (-5.5) + 3.5z + 5y – 2.5.
-6z + 3.5z – 5.5 – 2.5 + 5y.
-2.5z -8 + 5y.
so options B and C are correct.

Lesson 4.4 Expand Expressions

Solve & Discuss It!
The school is planning to add a weight room to the gym. If the total area of the gym and weight room should stay under 5,500 square feet, what is one possible length for the new weight room? Show your work. Are there other lengths that would work? Why or why not? -90 ft
I can… expand expressions using the Distributive Property.

Look for Relationships
What is the relationship between the areas of the gym and weight room?

Answer:
The relationship between the areas of the gym and weight room = 550 ft.

Explanation:
In the above-given question,
given that,
The school is planning to add a weight room to the gym.
If the total area of the gym and weight room should stay under 5,500 square feet.
the area of the school is l x b.
where l = 90 ft and b = 55 ft.
area = l x b.
area = 90 x 55.
area = 4950.
5500 – 4950 = 550.
so the relationship between the areas of the gym and weight room = 550 ft.

Focus on math practices
Model with Math What is an expression using x that represents the total area of the gym and the weight room?

Answer:
The relationship between the areas of the gym and weight room = 550 ft.

Explanation:
In the above-given question,
given that,
The school is planning to add a weight room to the gym.
If the total area of the gym and weight room should stay under 5,500 square feet.
the area of the school is l x b.
where l = 90 ft and b = 55 ft.
area = l x b.
area = 90 x 55.
area = 4950.
5500 – 4950 = 550.
so the relationship between the areas of the gym and weight room = 550 ft.

Essential Question
How does the value of an expression change when it is expanded?

Try It!

What is the expanded form of the expression 3.6(t + 5)?

3.6(t + 5)
= ________t + _______ • 5
= _______ + _______
The expanded expression is _______.

Answer:
3.6t + 18.

Explanation:
In the above-given question,
given that,
3.6(t + 5).
3.6 x t = 3.6t.
3.6t + 3.6 x 5.
3.6t + 18.

Convince Me! If you know the value of t, would the evaluated expression be different if you added the known value of t and 5 and then multiplied by 3.6? Explain.

Try It!

Expand the expression t(-1.2w + 3).

Answer:
The expression is -1.2tw + 3t.

Explanation:
In the above-given question,
given that,
the expression is t(-1.2w + 3).
-1.2tw + 3t.
so the expanded expression is -1.2tw + 3t.

Try It!

Simplify the expression –\(\frac{2}{5}\)(10 + 15m – 20n).

Answer:
The expression is -4 -6m – 8n.

Explanation:
In the above-given question,
given that,
–\(\frac{2}{5}\)(10 + 15m – 20n).
-2/5 (10 + 15m – 20n).
-2/5(10 + 15m – 20n).
-20/5 – 30/5m – 40/5n.
-4 – 6m – 8n.

KEY CONCEPT
You can expand an expression using the Distributive Property.
Multiply, or distribute, the factor outside the parentheses with each term inside the parentheses.
-7(3y – 1)
= (-7)(3y) + (-7)(-1)
= -21y + 7
The sign of each term is included in all calculations.

Do You Understand?
Question 1.
Essential Question How does the value of an expression change when it is expanded?

Answer:
The value of an expression change when it is expanded.

Explanation:
In the above-given question,
given that,
the value of an expression change when it is expanded.
for example:
-8(2y – 2).
-8(-2y) + (-8) (-2).
-16y + -16.

Question 2.
Use Structure How does the subtraction part of the expression change when a(b – c) is expanded?

Answer:
The subtraction part of the expression change.

Explanation:
In the above-given question,
given that,
a(b – c).
ax b – a x c.
ab – ac.
the product of in terms is multiplied with outterms.

Question 3.
Make Sense and Persevere When does expanding and simplifying a(b + c) result in a positive value for ac?

Answer:
ab + ac.

Explanation:
In the above-given question,
given that,
the expression is a(b + c).
a x b + a x c.
ab + ac.
the sign is positive.
so the value for ac is also positive.

Do You Know How?
Question 4.
Shoes and hats are on sale. The expression \(\frac{1}{4}\)(s + 24.80) can be used to determine the discount when you buy shoes with a retail price of s dollars and a hat with a retail price of $24.80. Write another expression that can be used to determine the discount.

Answer:
Another expression is $1.55.

Explanation:
In the above-given question,
given that,
Shoes and hats are on sale.
The expression \(\frac{1}{4}\)(s + 24.80).
when you buy shoes with a retail price of s dollars and a hat with a retail price of $24.80.
1/4 (s + 24.80).
s/4 + 24.80/4.
s/4 + 6.2.
s/4 = – 6.2.
s = -6.2/4.
s = 1.55.
so the retail price of the shoes = $1.55.

Question 5.
Expand x(4 – 3.4y).

Answer:
The expression is 4x – 3.4xy.

Explanation:
In the above-given question,
given that,
x(4 – 3.4y).
4x X – 3.4 x X x Y.
4x – 3.4xy.
so the expanded expression is 4x – 3.4xy.

Question 6.
Expand –\(\frac{2}{10}\)(1 – 2x + 2).

Answer:
The expanded expression is -3/5 – 2/5x.

Explanation:
In the above-given question,
given that,
–\(\frac{2}{10}\)(1 – 2x + 2).
-2/10 (1 – 2x + 2).
-1/5 (1 – 2x + 2).
-1/5 – 2/5x – 2/5.
-3/5 – 2/5x.

Practice & Problem Solving

Leveled Practice For 7-8, fill in the boxes to expand each expression.
Question 7.
3(n + 7)
= (3) (_____) + (3) (_____)
= ____ + _____

Answer:
3n + 21.

Explanation:
In the above-given question,
given that,
3(n + 7).
3 x n + 3 x 7.
3n + 21.

Question 8.
4(x – 3)
= ______ x – ______ (3)
= ______ – ______

Answer:
4x – 12.

Explanation:
In the above-given question,
given that,
4(x – 3).
4 x X – 4 x 3.
4x – 12.

For 9-14, write the expanded form of the expression.
Question 9.
y(0.5 + 8)

Answer:
y(0.5 + 8) = 8.5y.

Explanation:
In the above-given question,
given that,
y(0.5 + 8).
0.5y + 8y.
8.5y.
y(0.5 + 8) = 8.5y.

Question 10.
4(3 + 4x – 2)

Answer:
4(3 + 4x – 2) = 4 + 16x.

Explanation:
In the above-given question,
given that,
4(3 + 4x – 2).
4 x 3 + 4x x 4 – 2 x 4.
12 + 16x – 8.
4 + 16x.
4(3 + 4x – 2) = 4 + 16x.

Question 11.
6(y + x)

Answer:
6(y + x) = 6y + 6x.

Explanation:
In the above-given question,
given that,
6(y + x).
6 x y + 6 x x.
6y + 6x.
6(y + x) = 6y + 6x.

Question 12.
-2.5(-3 + 4n + 8)

Answer:
-2.5 (-3 + 4n + 8) = -14.5 – 10n.

Explanation:
In the above-given question,
given that,
-2.5 (-3 + 4n + 8).
-2.5 x -3 – 2.5 x 4n – 2.5 x 8.
-5.5 – 10n – 20.
-14.5 – 10n.
-2.5 (-3 + 4n + 8) = -14.5 – 10n.

Question 13.
–\(\frac{1}{3}\)(y – x)

Answer:
–\(\frac{1}{3}\)(y – x) = -1/3y + x/3.

Explanation:
In the above-given question,
given that,
–\(\frac{1}{3}\)(y – x).
-1/3(y – x).
-1/3y + x/3.
–\(\frac{1}{3}\)(y – x) = -1/3y + x/3.

Question 14.
8(6x – 4)

Answer:
8(6x – 4) = 48x – 32.

Explanation:
In the above-given question,
given that,
8(6x – 4).
8 x 6x – 4 x 8.
48x – 32.
8(6x – 4) = 48x – 32.

Question 15.
Higher Order Thinking A grocery store has a 13%-off sale on all bread. You decide to purchase 6 loaves of bread. Let b be the original price of a loaf of bread. Expand the expression 6(b – 0.13b). Once the expression is expanded, what do the terms represent?

Answer:
6(b – 0.13b) = -5.22b.

Explanation:
In the above-given question,
given that,
A grocery store has a 13%-off sale on all bread.
You decide to purchase 6 loaves of bread.
Let b be the original price of a loaf of bread.
6(b – 0.13b).
6 x b – 0.13b x 6.
6b – 0.78b.
-5.22b.
6(b – 0.13b) = -5.22b.

Question 16.
A gardener plans to extend the length of a rectangular garden. Let x represent the garden’s original length. The expression 4(x + 7) represents the area of the extended garden. When asked for the area of the extended portion, the gardener incorrectly said it was 11 square feet. Describe the error the gardener made.

Answer:
The error the gardener made = 4x + 28.

Explanation:
In the above-given question,
given that,
Let x represent the garden’s original length.
The expression 4(x + 7) represents the area of the extended garden.
4(x + 7).
4 x x + 4 x 7.
4x + 28.

Question 17.
Find a difference equivalent to the product 11(x – y).

Answer:
11(x – y) = 11x – 11y.

Explanation:
In the above-given question,
given that,
11(x – y).
11 x x – 11 x y.
11x – 11y.
11(x – y) = 11x – 11y.

Question 18.
Use the Distributive Property to write an expression equivalent to 0.4(-5 – 7y – 13.8).

Answer:
0.4(-5 – 7y – 13.8) = -10.12 – 2.8y.

Explanation:
In the above-given question,
given that,
the expression is 0.4(-5 – 7y – 13.8).
0.4 x (-5) – 0.4 (7y) – 0.4 (-13.8).
-4.6 – 2.8y – 5.52.
-10.12 – 2.8y.

Question 19.
Make Sense and Persevere Use the Distributive Property to expand 7(7x – 3y) – 6.

Answer:
7(7x – 3y) – 6 = 49x – 21y – 6.

Explanation:
In the above-given question,
given that,
7(7x – 3y) – 6.
7 x 7x – 7 (3y) – 6.
49x – 21y – 6.
7(7x – 3y) – 6 = 49x – 21y – 6.

Question 20.
Use the Distributive Property to write an expression equivalent to y(-3 – 8x).

Answer:
y(-3 – 8x) = -3y – 8xy.

Explanation:
In the above-given question,
given that,
y(-3 – 8x).
-3 x y – 8x (y).
-3y -8xy.

Question 21.
An architect plans to build an extension to Meiling’s rectangular deck. Let x represent the increase, in meters, of her deck’s length. The expression 5(X + 8) represents the area of the deck, where 5 is the width, in meters, and (x + 8) represents the extended length, in meters. Use the Distributive Property to write an expression that represents the total area of Meiling’s new deck.

Answer:
The total area of Meiling’s new deck = 5x + 40.

Explanation:
In the above-given question,
given that,
An architect plans to build an extension to Meiling’s rectangular deck.
Let x represent the increase, in meters, of her deck’s length.
The expression 5(X + 8) represents the area of the deck, where 5 is the width, in meters, and (x + 8) represents the extended length in meters.
5(x + 8).
5 x x + 5 x 8.
5x + 40.

Assessment Practice

Question 22.
Select all expressions equivalent to –\(\frac{1}{2}\)(4 – 2 + 8x).
☐ -4x – 1
☐ 4x – 1
☐ 3x
☐ -2 + 1 – 4x
☐ 2 + 1 – 4x
☐ 4x + 1

Answer:
-4x -1 and -2 + 1 – 4x.

Explanation:
In the above-given question,
given that,
–\(\frac{1}{2}\)(4 – 2 + 8x).
-1/2(4 – 2 + 8x).
-4/2 – 2/2 + 8/2x.
-2 + 1 – 4x.
-1 – 4x.

Question 23.
An expression is shown.
\(\frac{1}{5}\)(5 – 7y + 10) Create an equivalent expression without parentheses.

Answer:
1/5(5 – 7y + 10) = 1 – 7/5y + 2.

Explanation:
In the above-given question,
given that,
1/5(5 – 7y + 10).
5/5 – 7/5y + 10/5.
1 – 7/5y + 2.

Lesson 4.5 Factor Expressions

Explain It!
Tasha is packing gift bags that include the same items. She has 72 glow sticks, 36 markers, and 24 bottles of bubbles. Tasha believes that she can pack no more than 6 bags using all of her supplies.

Answer:
Yes, she can pack 44 bags.

Explanation:
In the above-given question,
given that,
Tasha is packing gift bags that include the same items.
She has 72 glow sticks, 36 markers, and 24 bottles of bubbles.
72 + 36 + 24 = 132.
132/3 = 44.
she can pack 44 bags.

I can… use common factors and the Distributive Property to factor expressions.

Make Sense and Persevere
How can you use what you know about common factors to solve the problem?

A. Critique Reasoning Do you agree with Tasha? Explain.

Answer:
No, Tasha is wrong.

Explanation:
In the above-given question,
given that,
Tasha is packing gift bags that include the same items.
She has 72 glow sticks, 36 markers, and 24 bottles of bubbles.
72 + 36 + 24 = 132.
132/3 = 44.
she can pack 44 bags.

B. If Tasha creates the greatest number of gift bags, how many of each item is in each bag? Explain how you know.

Answer:
Tasha packs 44 bags.

Explanation:
In the above-given question,
given that,
Tasha is packing gift bags that include the same items.
She has 72 glow sticks, 36 markers, and 24 bottles of bubbles.
72 + 36 + 24 = 132.
132/3 = 44.
she can pack 44 bags.

Focus on math practices
Reasoning Tasha added more markers and now has a total of 48 markers. Does this change the possible number of gift bags? Explain.

Essential Question
How does the Distributive Property relate to factoring expressions?

Try It!

Use factoring to write an expression for the length of the pool with the given width.

4x + 20 = _____ (x + _____)
So, the length of the pool is ______ meters.

Answer:
The length of the pool is 4(x + 5) meters.

Explanation:
In the above-given question,
given that,
4x + 20.
4(x + 5).
the length of the pool is 4(x + 5) meters.

Convince Me! How can you use the Distributive Property to check the factored expression? Use the factored expression for Example 1 in your explanation.

Try It!

Show two different ways to factor -(4x – 28).

Answer:
-(4x – 28) = -4(x – 7).

Explanation:
In the above-given question,
given that,
-(4x – 28).
-4x + 28.
-4(x – 7).

Try It!

Write an equivalent expression for the expression above using a negative factor.

Answer:
The equivalent expression for the expression is -4(x – 7).

Explanation:
In the above-given question,
given that,
the equivalent expression for the expression is -4(x – 7).
-(4x – 28).
-4x + 28.
-4(x – 7).

KEY CONCEPT
The greatest common factor (GCF) can be used to factor expressions.
The Distributive Property can be applied to factor an expression. Factoring an expression creates an equivalent expression.

Do You Understand?
Question 1.
Essential Question How does the Distributive Property relate to factoring expressions?

Answer:
2x + 8 = 2(x + 4).

Explanation:
In the above-given question,
given that,
the distributive property can be applied to factor an expression.
factoring an expression creates an equivalent expression.
2x + 8.
2(x + 4).

Question 2.
Susan incorrectly factored the expression below.
12a – 156 + 6
3(4a + 5b + 3)

a. Explain any errors Susan may have made when factoring.

Answer:
12a + 15b + 9.

Explanation:
In the above-given question,
given that,
3(4a + 5b + 3).
(3 x 4a) + (3 x 5b) + (3 x 3).
12a + 15b + 9.

b. Factor the expression correctly.

Answer:
12a + 15b + 9.

Explanation:
In the above-given question,
given that,
3(4a + 5b + 3).
(3 x 4a) + (3 x 5b) + (3 x 3).
12a + 15b + 9.

Do You Know How?
Question 3.
Sahil is putting together supply kits and has 36 packs of x pencils, 12 packs of y crayons, and 24 erasers.
a. Write an expression to show the total number of items.

Answer:
The expression to show the total number of items = 36x + 12y + 24.

Explanation:
In the above-given question,
given that,
Sahil is putting together supply kits and has 36 packs of x pencils, 12 packs of y crayons, and 24 erasers.
36 x X + 12 x Y + 24,
36x + 12y + 24.
so the expression to show the total number of items = 36x + 12y + 24.

b. Use factoring to show many kits Sahil can make while putting every type of item in each kit.

Answer:
The expression to show the total number of items = 36x + 12y + 24.

Explanation:
In the above-given question,
given that,
Sahil is putting together supply kits and has 36 packs of x pencils, 12 packs of y crayons, and 24 erasers.
36 x X + 12 x Y + 24,
36x + 12y + 24.
so the expression to show the total number of items = 36x + 12y + 24.

c. Use the factored expression to find the number of each item in each kit.

Answer:
The expression to show the total number of items = 36x + 12y + 24.

Explanation:
In the above-given question,
given that,
Sahil is putting together supply kits and has 36 packs of x pencils, 12 packs of y crayons, and 24 erasers.
36 x X + 12 x Y + 24,
36x + 12y + 24.
so the expression to show the total number of items = 36x + 12y + 24.

Question 4.
Show two different ways to factor – 12x + 24 – 18y.

Answer:
-12x + 24 – 18y and -2(6x – 12 + 9y).

Explanation:
In the above-given question,
given that,
the expression is – 12x + 24 – 18y.
-2(6x – 12 + 9y).

Question 5.
How can you use the Distributive Property to factor the expression 6x + 15?

Answer:
3(2x + 5).

Explanation:
In the above-given question,
given that,
the given expression is 6x + 15.
3x + 3x +15.
3(2x + 5).

Practice & Problem Solving

Leveled Practice In 6-9, factor the expression.
Question 6.
16a + 10.
The GCF of 16a and 10 is 2.
2 × ______ = 16a
2 × _______ = 10
The factored expression is ________

Answer:
2 x 8a = 16a.
2 x 5 = 10.
The factored expression is 16a + 10.

Explanation:
In the above-given question,
given that,
the expression is 16a + 10.
2(8a + 5).
2 x 8a = 16a.
2 x 5 = 10.

Question 7.
-9y – 3.
The positive GCF of -9y and -3 is 3.
3 × ______ = -9y
3 × ______ = -3
The factored expression is ________

Answer:
3 x -3y = -9y.
3 x 1 = -3.

Explanation:
In the above-given question,
given that,
the expression is -9y – 3.
-3(3y + 1).
-3 x 3y = -9y.
-3 x 1 = -3.

Question 8.
14x + 49

Answer:
7 x 2x = 14x.
7 x 7 = 49.

Explanation:
In the above-given question,
given that,
the expression is 14x + 49.
7(2x + 7).
7 x 2x = 14x.
7 x 7 = 49.

Question 9.
12y – 16

Answer:
2 x 6y = 12y.
2 x 8 = 16.

Explanation:
In the above-given question,
given that,
the expression is 12y – 16.
2(6y – 8).
2 x 6y = 12y.
2 x 8 = 16.

Question 10.
This model shows the area of a garden. Write two expressions that represent the area.

Answer:
The area of the garden = 5x + 10.

Explanation:
In the above-given question,
given that,
the expression is 5x + 10.
5(x + 2).
5 x x = 5x.
5 x 2 = 10.
so the area of the garden = 5x + 10.

Question 11.
Use the GCF to write the factored form of the expression 18x + 24y.

Answer:
The factored form of the expression is 2(9x + 12y).

Explanation:
In the above-given question,
given that,
the expression is 18x + 24y.
2(9x + 12y).
6(3x + 4y).
so the factored form of the expression is 2(9x + 12y).

Question 12.
Find the dimensions of the sports field at the right if the width is at least 60 yards.

Answer:
The dimensions of the sports field = -160 and 60.

Explanation:
In the above-given question,
given that,
the expression is 240 – 400x.
-160x.
-160(60).
-9600.
so the dimensions of the sports field = -160 and 60.

Question 13.
Your friend incorrectly factors the expression 15x – 20xy as 5x( 3 – 4xy).
a. Factor the expression correctly.

Answer:
The expression correctly = 5(3x – 4xy).

Explanation:
In the above-given question,
given that,
the expression is 15x – 20xy.
5(3x – 4xy).
5 x 3x = 15x.
5 x 4xy = 20xy.
so the expression correctly is 5(3x – 4xy).

b. What error did your friend likely make?

Answer:
The expression correctly = 5(3x – 4xy).

Explanation:
In the above-given question,
given that,
the expression is 15x – 20xy.
5(3x – 4xy).
5 x 3x = 15x.
5 x 4xy = 20xy.
so the expression correctly is 5(3x – 4xy).

Question 14.
You are given the expression 12x + 18y + 26.
a. Make Sense and Persevere What is the first step in factoring the expression?

Answer:
The first step in factoring is 2(6x + 9y + 18).

Explanation:
In the above-given question,
given that,
the expression is 12x + 18y + 26.
2(6x + 9y + 13).
2 x 6x = 12x.
2 x 9y = 18y.
2 x 13 = 26.

b. Factor the expression.

Answer:
The expression is 2(6x + 9y + 18).

Explanation:
In the above-given question,
given that,
the expression is 12x + 18y + 26.
2(6x + 9y + 13).
2 x 6x = 12x.
2 x 9y = 18y.
2 x 13 = 26.

Question 15.
A hotel manager is adding a tile border around the hotel’s rectangular pool. Let x represent the width of the pool, in feet. The length is 3 more than 2 times the width, as shown. Write two expressions that give the perimeter of the pool.

Answer:
The perimeter of the pool = 2XxX + 3x.

Explanation:
In the above-given question,
given that,
Let x represent the width of the pool, in feet.
The length is 3 more than 2 times the width, as shown.
the perimeter of the rectangle = length x width.
perimeter = 2x + 3 x X.
perimeter = 2×2 x 3x.

Question 16.
Higher Order Thinking Use the expressions below.
14m + mn
2y + 2x + 4
–\(\frac{3}{4}\)m + 8m + m
4 – 3p
5.75t + 7.75t – t
8xy – 6xy
a. Circle the expressions that have like terms.

Answer:
The expressions that have the like terms = 14m + mn, 5.75t + 7.75t -t, and 8xy – 6xy.

Explanation:
In the above-given question,
given that,
the expressions are 14m + mn
2y + 2x + 4.
–\(\frac{3}{4}\)m + 8m + m.
4 – 3p.
5.75t + 7.75t – t.
8xy – 6xy.
the like terms are 8xy – 6xy = 2xy.
5.75t + 7.75t – t = 13.5t – t.
12.5 t.

b. Explain why the other expressions do not have like terms.

Answer:
The other expressions that do not have like terms are 4 – 3p, 14m + mn.

Explanation:
In the above-given question,
given that,
the expressions are 14m + mn
2y + 2x + 4.
–\(\frac{3}{4}\)m + 8m + m.
4 – 3p.
5.75t + 7.75t – t.
8xy – 6xy.
the unlike terms are 14m + mn.
2y + 2x + 4.

Question 17.
Construct Arguments Ryan says the expression 3 + 5y cannot be factored using a GCF. Is he correct? Explain why or why not.

Answer:
Yes, Ryan was correct.

Explanation:
In the above-given question,
given that,
the expression is 3 + 5y.
we cannot be factored in using a GCF.
so Ryan was correct.

Assessment Practice

Question 18.
Select all the expressions equivalent to 12 + 30y.
☐ 3(4 + 10y)
☐ 4(3 + 10y)
☐ 6(2 + 5y)
☐ 2(6 + 30y)
☐ 6(3 + 10y)

Answer:
The expressions equivaent to 12 + 30y is 3(4 + 10y), 6(2 + 5y).

Explanation:
In the above-given question,
given that,
the expression is 12 + 30y.
3(4 + 10y).
6(2 + 5y).
3 x 4 = 12 + 3 x 10y = 30y.
6 x 2 = 12 + 6 x 5y = 30y.

Question 19.
Write an expression that is the product of two factors and is equivalent to -2x – 10.

Answer:
The expression is -2(x + 5).

Explanation:
In the above-given question,
given that,
the expression is -2x – 10.
-2(x + 5).
-2x + 10.

Topic 4 Mid-Topic Checkpoint

Question 1.
Vocabulary If you write an expression to represent the following situation, how can you determine which is the constant and which is the coefficient of the variable? Lesson 4-1
The zoo charges the Garcia family an admission fee of $5.25 per person and a one-time fee of $3.50 to rent a wagon for their young children.

Answer:
The admission fee is the coefficient and rents a wagon is constant.

Explanation:
In the above-given question,
given that,
The zoo charges the Garcia family an admission fee of $5.25 per person.
a one-time fee of $3.50 to rent a wagon for their young children.
$5.25x + $3.50.
$5.25 is the coefficient of the variable.
$3.50 is the constant.

Question 2.
An online photo service charges $20 to make a photo book with 16 pages. Each extra page costs $1.75. The cost to ship the completed photo book is $5. Write an expression to determine the total cost in dollars to make and ship a photo book with x extra pages. Lesson 4-1

Answer:
The total cost in dollars to make and ship a photo book with X extra pages is $20x + $5 = $1.75.

Explanation:
In the above-given question,
given that,
An online photo service charges $20 to make a photo book with 16 pages.
Each extra page costs $1.75.
The cost to ship the completed photo book is $5.
$20x + $5 = $1.75.
$25x -$1.75 = $5.

Question 3.
Write an expression equivalent to 2a + (\(\frac{3}{4}\)a + \(\frac{1}{5}\)b) by combining like terms. Lesson 4-3

Answer:
The expression equivalent to 2a + 3/4a + 1/5b is 5/4a + 1/5b.

Explanation:
In the above-given question,
given that,
the given expression is 2a + (\(\frac{3}{4}\)a + \(\frac{1}{5}\)b).
2a + 3/4a + 1/5b.
5/4a + 1/5b.

Question 4.
Which expression is equivalent to 3.2y – \(\frac{1}{3}\) + (-7y) + \(\frac{2}{3}\)? Lesson 4-2
A. -10.2y + \(\frac{1}{3}\)
B. -3.8y + \(\frac{1}{3}\)
C. -3\(\frac{7}{15}\)y
D. -3y

Answer:
Option B is the correct.

Explanation:
In the above-given question,
given that,
the expression is 3.2y – \(\frac{1}{3}\) + (-7y) + \(\frac{2}{3}\).
3.2y – 1/3 -7y + 2/3.
-3.8y + 1/3.
so option B is the correct.

Question 5.
Ray wants to buy a hat that costs $10 and some shirts that cost $12 each. The sales tax rate is 6.5%. Write an expression to determine the amount of sales tax that Ray will pay on his entire purchase. Expand to simplify the expression. Lesson 4-4

Answer:
The expression to determine the amount of sales tax that Ray will pay on his entire purchase = $22 + 6.5%.

Explanation:
In the above-given question,
given that,
Ray wants to buy a hat that costs $10 and some shirts that cost $12 each.
The sales tax rate is 6.5%.
$10 + $12 + 6.5%.
$22 + 6.5%.

Question 6.
Factor the expression 28r + 425 – 35. Lesson 4-5

Answer:
The expression is 7(4r + 61 – 5).

Explanation:
In the above-given question,
given that,
the expression is 28r + 425 – 35.
7(4r + 61 – 5).
7 x 4r = 28r.
61 x 7 = 425.
7 x 5 = 35.

Question 7.
Describe two ways the Distributive Property can be used to write equivalent expressions. Lessons 4-4 and 4-5

Answer:
The two ways the distributive property can be used to write equivalent expressions

Explanation:
In the above-given question,
given that,
-1/2(x + 8), -1/2x + (-4) and -4 +(-1/2x) are equivalent.
-1/2(x + 8).
-1/2x + (-1/2) . 8.
-1/2x + (-4).
-4 + (-1/2x).
the three expressions are true.

Topic 4 Mid-Topic Performance Task

Alison is a buyer for a chain of 6 flower shops. This means that she buys flowers in bulk from a supplier and then distributes them to the 6 flower shops in the chain.
PART A
This week Alison bought 108 bunches of carnations and 96 bunches of roses from the supplier. Let c represent the number of carnations in each bunch, and let r represent the number of roses in each bunch. Write an expression to show the total number of carnations and roses that Alison bought.

Answer:
The total number of carnations and roses that Alison bought = 12(9c + 8r).

Explanation:
In the above-given question,
given that,
This week Alison bought 108 bunches of carnations and 96 bunches of roses from the supplier.
Let c represent the number of carnations in each bunch, and let r represent the number of roses in each bunch.
108c + 96r.
12(9c + 8r).
12 x 9c = 108c.
12 x 8r = 96r.

PART B
Alison wants to distribute the carnations and roses equally among the 6 flower shops. Factor the expression from Part A using 6 as the common factor. How does the factored expression help Alison determine how many carnations and how many roses each flower shop should get?

Answer:
The common factor is 3(c + r).

Explanation:
In the above-given question,
given that,
Alison wants to distribute the carnations and roses equally among the 6 flower shops.
3c + 3r.
3(c + r).
1(3c + 3r).
so the common factor is 3(c + r).

PART C
There are 24 carnations in each bunch and 12 roses in each bunch. Use your answer to Part B to determine the total number of carnations and the total number of roses Alison will distribute to each flower shop this week.

Answer:
The total number of carnations and roses = 6(4c + 2r).

Explanation:
In the above-given question,
given that,
There are 24 carnations in each bunch and 12 roses in each bunch.
24 c + 12 r.
2(12c + 6r).
6(4c + 2r).
6 x 4c = 24c.
6 x 2r = 12r.

PART D
Jake manages one of the flower shops. He wants to use the carnations and roses to make bouquets. He wants each bouquet to have the same combination of carnations and roses, with no flowers left over. Determine a way that Jake can divide the flowers to make the bouquets. How many bouquets will there be?

Answer:
The number of bouquets will there be = 6(c + r).

Explanation:
In the above-given question,
given that,
Jake manages one of the flower shops.
He wants to use the carnations and roses to make bouquets.
He wants each bouquet to have the same combination of carnations and roses, with no flowers left over.
6 and 6.
6 x 6 = 36.
6c + 6r.
6(c + r).
so the number of bouquets will there be = 6(c + 1).

3-Act Mathematical Modeling:
I’ve Got You Covered

ACT 1
Question 1.
After watching the video, what is the first question that comes to mind?
Answer:

Question 2.
Write the Main Question you will answer.
Answer:

Question 3.
Construct Arguments Predict an answer to this Main Question. Explain your prediction.

Answer:

Question 4.
On the number line below, write a number that is too small to be the answer. Write a number that is too large.

Answer:
The two numbers are 1 and 10.

Explanation:
In the above-given question,
given that,
the number line is 10 cm long.
the short is 1 cm.
the large is 10 cm.
so the two numbers that are too small and too large is 1 and 10.

Question 5.
Plot your prediction on the same number line.

Answer:
The two numbers are 1 and 10.

Explanation:
In the above-given question,
given that,
the number line is 10 cm long.
the short is 1 cm.
the large is 10 cm.
so the two numbers that are too small and too large is 1 and 10.

ACT 2
Question 6.
What information in this situation would be helpful to know? How would you use that information?

Answer:
The figure contain 7 objects.

Explanation:
In the above-given question,
given that,
the figure contains 7 objects.
3 objects on the left side.
4 objects on the right side.
so the figure contain 7 objects.

Question 7.
Use Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.

Answer:
The figure contain 7 objects.

Explanation:
In the above-given question,
given that,
the figure contains 7 objects.
3 objects on the left side.
4 objects on the right side.
so the figure contain 7 objects.

Question 8.
Model with Math Represent the situation using mathematics. Use your representation to answer the Main Question.
Answer:

Question 9.
What is your answer to the Main Question? Is it higher or lower than your prediction? Explain why.

Answer:

ACT 3
Question 10.
Write the answer you saw in the video.

Answer:

Question 11.
Reasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
Answer:

Question 12.
Make Sense and Persevere Would you change your model now that you know the answer? Explain.

Answer:

ACT 3

Extension
Reflect
Question 13.
Model with Math Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?
Answer:

Question 14.
Generalize What pattern did you notice in your calculations? How did that pattern help you solve the problem?
Answer:

SEQUEL
Question 15.
Reasoning A classmate says that another object needs 512 tiles. What do you know about the dimensions of the object?

Answer:
The dimensions of the object = 22 and 26.

Explanation:
In the above-given question,
given that,
A classmate says that another object needs 512 tiles.
22 x 26 = 512.
the length = 22.
the width = 26.
area = l x b.
22 x 26.
512.

Lesson 4.6 Add Expressions

Solve & Discuss It!
The Smith family took a 2-day road trip. On the second day, they drove the distance they traveled on the first day. What is a possible distance they could have traveled over the 2 days? Is there more than one possible distance? Justify your response.
I can… add expressions that represent real-world problems.

Make Sense and Persevere
How are the quantities in the problem related?

Answer:
The quantities in the problem are the smith family took a 2-day road trip.

Explanation:
In the above-given question,
given that,
The Smith family took a 2-day road trip.
On the second day, they drove the distance they traveled on the first day.
2 + 2 = 4.

Focus on math practices
Use Structure How can two different expressions be used to represent the total distance?

Essential Question
How can properties of operations be used to add expressions?

Try It!

Sophia and Ollie each deposit $120 to open a joint account. They each make monthly deposits as shown. What expression represents the amount in the account after m months?

The amount of money in the joint account after m months is ______ + _______.

Answer:
The amount of money in the joint account after m months is 120 + 120 = $525..

Explanation:
In the above-given question,
given that,
Sophia and Ollie each deposit $120 to open a joint account.
120 + 150 = 270.
120 + 135 = 255.
270 + 255 = 525.
so the amount of money in the joint account after m months is $525.

Convince Me! Explain why the initial deposits and monthly deposits are not combined into one term?

Try It!

Find each sum.
a. (9.740 – 250.50) + (-5.48p + 185.70)

Answer:
Te sum is 60.54.

Explanation:
In the above-given question,
given that,
(9.740 – 250.50) + (-5.48p + 185.70).
(-240.76) + (180.22).
60.54.

b. (\(\frac{2}{11}\)x – 3 – 5y) + (-\(\frac{3}{11}\) + 5y + 5.5)

Answer:
-1/11 x + 2.5 + 5y.

Explanation:
In the above-given question,
given that,
(\(\frac{2}{11}\)x – 3 – 5y) + (-\(\frac{3}{11}\) + 5y + 5.5)
(2/11)x – 3 – 5y + (-3/11) + 5y + 5.5.
-1/11x + 2.5 + 5y.
-1/11 x + 2.5 + 5y.

C. (-14.2b – 97.35) + (6.76d – 118.7 – 3.4d)

Answer:
-14.2b – 216.05 + 3.36d.

Explanation:
In the above-given question,
given that,
(-14.2b – 97.35) + (6.76d – 118.7 – 3.4d).
-14.2b – 97.35 + 3.36d – 118.7.
-14.2b – 216.05 + 3.36d.

d. (\(\frac{3}{8}\) – \(\frac{1}{6}\)m + 5t) + (\(\frac{7}{10}\)m + 9t + \(\frac{1}{4}\))

Answer:
1/3 + 3/2m + 14t.

Explanation:
In the above-given question,
given that,
\(\frac{3}{8}\) – \(\frac{1}{6}\)m + 5t) + (\(\frac{7}{10}\)m + 9t + \(\frac{1}{4}\))
3/8 – 1/6 m + 5t + 7/10 m + 9t + 1/4.
3/8 + 1/4 – 1/6 m + 7/10 m + 5t + 9t.
3/8 + 1/4 + 3/2 m + 14 t.
1/3 + 3/2m + 14t.

KEY CONCEPT

Adding expressions may require combining like terms.
Terms with the same variables are added together and constants are added together.
When adding terms with the same variables, the rules for adding rational numbers apply to their coefficients.

(3.6 + 22.4t) + (2 + 18.9t) = 5.6 + 41.3t

Do You Understand?
Question 1.
Essential Question How can properties of operations be used to add expressions?

Answer:
The properties of operations be used to add expressions with the like terms.

Explanation:
In the above-given question,
given that,
Adding expressions may require combining like terms.
Terms with the same variables are added together and constants are added together.
When adding terms with the same variables, the rules for adding rational numbers apply to their coefficients.
(3.6 + 22.4t) + (2 + 18.9t).
5.6 + 41.3t.

Question 2.
Reasoning Explain whether the coefficients of two terms with different variables can be added to make one new term.

Answer:
Yes the coefficients of two terms with different variables can be added to make the new term.

Explanation:
In the above-given question,
given that,
Adding expressions may require combining like terms.
Terms with the same variables are added together and constants are added together.
When adding terms with the same variables, the rules for adding rational numbers apply to their coefficients.
(3.6 + 22.4t) + (2 + 18.9t).
5.6 + 41.3t.

Question 3.
Be Precise which properties of operations could be used to show that (-5p + 9) + (-2 + p) is equivalent to (-5p) + p + 9 – 2?

Answer:
The equivalent expression is -4p + 7.

Explanation:
In the above-given question,
given that,
the expression is (-5p + 9) + (-2 + p).
-5p + 9 -2 + p.
-4p + 7.

Do You Know How?
Question 4.
Dillon says that 4b and -2b are not like terms because 4b is positive and -2b is negative. Is he correct? Explain.

Answer:
No, they are like terms.

Explanation:
In the above-given question,
given that,
Dillon says that 4b and -2b are not like terms because 4b is positive and -2b is negative.
4b and 2b are same.
4b – 2b = 2b.
so they are like terms.

Question 5.
Joel spent $28 for an Internet data service and pays $14.50 per month. He spent $24.50 to join an online movie streaming site and pays $13.25 per month. Write an expression to represent Joel’s total cost for both memberships after m months.

Answer:
The expression to represent Joel’s total cost for both memberships after m months = $27.75m + $52.50.

Explanation:
In the above-given question,
given that,
Joel spent $28 for an Internet data service and pays $14.50 per month.
He spent $24.50 to join an online movie streaming site and pays $13.25 per month.
$14.50m + $13.25m + $28 + $24.50.
$27.75m + $52.50.

Question 6.
Add \(\frac{1}{3}\)n + \(\frac{2}{3}\) and –\(\frac{1}{6}\)n + \(\frac{1}{6}\)m.

Answer:
-1/3n + 2/3 + 1/6m.

Explanation:
In the above-given question,
given that,
\(\frac{1}{3}\)n + \(\frac{2}{3}\) and –\(\frac{1}{6}\)n + \(\frac{1}{6}\)m.
1/3 n + 2/3 – 1/6 n + 1/6m.
-1/3n + 2/3 + 1/6m.

Question 7.
Find the sum.
(-3.5t – 4s +4.5) + (-7.1 – 0.3s + 4.1t)

Answer:
2.6 + 0.6t – 4.3s.

Explanation:
In the above-given question,
given that,
(-3.5t – 4s +4.5) + (-7.1 – 0.3s + 4.1t).
-3.5t – 4s + 4.5 -7.1 – 0.3s + 4.1t.
0.6 t – 4.3s – 2.6.
2.6 + 0.6t – 4.3s.

Practice & Problem Solving

Leveled Practice For 8-9, fill in the boxes to add the expressions.
Question 8.
(2a + 8) + (4a + 5)
= (2a + ______) + (8 + _____)
= _______ + 13

Answer:
6a + 13.

Explanation:
In the above-given question,
given that,
the expression is (2a + 8) + (4a + 5).
(2a + 4a) + ( 8 + 5).
6a + 13.

Question 9.
(\(\frac{2}{7}\)x – 7) + (\(\frac{1}{7}\)x + 8)
= (_______ + _______) + (-7 + ______)
= ______x + ______

Answer:
3/7 + 1.

Explanation:
In the above-given question,
given that,
(\(\frac{2}{7}\)x – 7) + (\(\frac{1}{7}\)x + 8).
(2/7 + 1/7)x – 7 + 8.
3/7x + 1.

Question 10.
Find the sum.
(8b + 7) + (6x – 4) + (5c + 8)

Answer:
8b + 6x + 11.

Explanation:
In the above-given question,
given that,
(8b + 7) + (6x – 4) + (5c + 8).
8b + 7 + 6x – 4 + 5c + 8.
8b + 3 + 6x + 8.
8b + 6x + 11.

Question 11.
Combine like terms.
(-3y – 5) + (5m + 7y) + (6 + 9m)

Answer:
14m + 4y + 1.

Explanation:
In the above-given question,
given that,
(-3y – 5) + (5m + 7y) + (6 + 9m)
-3y – 5 + 5m + 7y + 6 + 9m.
4y + 1 + 14m.
14m + 4y + 1.

Question 12.
Felipe is going to plant b sunflower seeds in one garden and 5b + 10 sunflower seeds in another. How many seeds is Felipe going to plant altogether?

Answer:
The number of seeds Felipe going to plant altogether = 5bsquare + 10b.

Explanation:
In the above-given question,
given that,
Felipe is going to plant b sunflower seeds in one garden and 5b + 10 sunflower seeds in another.
(5b + 10) b.
5b x b + 10b.
so the number of seeds Felipe goind to plant altogether = 5bsquare + 10b.

Question 13.
An art class is making a mural for the school that has a triangle drawn in the middle. The length of the bottom of the triangle is x. Another side is 1 more than three times the length of the bottom of the triangle. The last side is 2 more than the bottom of the triangle. Write and simplify an expression for the perimeter of the triangle.

Answer:
The perimeter of the triangle =

Explanation:
In the above-given question,
given that,
An art class is making a mural for the school that has a triangle drawn in the middle.
The length of the bottom of the triangle is x.
Another side is 1 more than three times the length of the bottom of the triangle.
The last side is 2 more than the bottom of the triangle.
x + 3x + 2x.
p = x + 2x + 3x.
p = 6x.
so the perimeter of the triangle = 6x.

Question 14.
On a math test, Sarah has to identify all the coefficients and constants of the expression 4 + n + 7m. Sarah identifies the only coefficient as 7 and the only constant as 4.
a. Identify all the coefficients of the expression.

Answer:
The coefficients of the expression is 7.

Explanation:
In the above-given question,
given that,
On a math test, Sarah has to identify all the coefficients and constants of the expression 4 + n + 7m.
Sarah identifies the only coefficient as 7.
so the coefficient is 7.

b. Identify all the constants of the expression.

Answer:
The constants of the expression is n.

Explanation:
In the above-given question,
given that,
On a math test, Sarah has to identify all the coefficients and constants of the expression 4 + n + 7m.
Sarah identifies the only constant as n.
so the constant is n.

c. What error did Sarah likely make?

Answer:
Sarah make the mistake was 4 is the constant.

Explanation:
In the above-given question,
given that,
On a math test, Sarah has to identify all the coefficients and constants of the expression 4 + n + 7m.
Sarah identifies the only constant as 4.
so the constant is 4.

Question 15.
The width of a rectangle is 5x – 2.5 feet and the length is 2.5x + 8 feet. Find the perimeter of the rectangle.

Answer:
The perimeter of the rectangle = 6.25x – 20.

Explanation:
In the above-given question,
given that,
The width of a rectangle is 5x – 2.5 feet and the length is 2.5x + 8 feet.
(5x – 2.5) x (2.5x + 8).
6.25x – 20.
so the perimeter of the rectangle = 6.25x – 20.

Question 16.
Nina has x coins. Clayton has 5 fewer coins than six times the number of coins Nina has. Write an expression for the total number of coins Nina and Clayton have altogether. Then simplify the expression.

Answer:
The expression for the total number of coins Nina and Clayton have altogether = 6x  – 30.

Explanation:
In the above-given question,
given that,
Nina has x coins.
Clayton has 5 fewer coins than six times the number of coins Nina has.
x – 5 x 6.
6x – 30.
so the expression for the total number of coins Nina and Clayton have altogether = 6x – 30.

Question 17.
Higher Order Thinking Use the expression (8x + 2) + (-9x + 7).
a. Find the sum.

Answer:
9 – x.

Explanation:
In the above-given question,
given that
(8x + 2) + (-9x + 7).
8x + 2 -9x + 7.
-x + 9.
9 – x.

b. Reasoning Explain how you know when to combine terms with variables.

Answer:
We have to combine the variables when they have like terms.

Explanation:
In the above-given question,
given that
(8x + 2) + (-9x + 7).
8x + 2 -9x + 7.
-x + 9.
9 – x.

Question 18.
Gabe went to the Florida Mall. He bought k model planes and spent $24 on books. Then he spent another $25 at another store.

a. Write an expression that represents the amount Gabe spent at the mall.

Answer:
The expression that represents the amount Gabe = $49k + $14.99.

Explanation:
In the above-given question,
given that,
Gabe went to the Florida Mall.
He bought k model planes and spent $24k on books.
Then he spent another $25k at another store.
each model panel cost $14.99.
$24k + $25k + $14.99.
$49k + $14.99.

b. How much did Gabe spend in all if he bought 3 model planes?

Answer:
The amount did Gabe spend in all if he bought 3 models planes = $93.97.

Explanation:
In the above-given question,
given that,
Gabe went to the Florida Mall.
He bought k model planes and spent $24k on books.
Then he spent another $25k at another store.
each model panel cost $14.99.
3 x $14.99 = 93.97.
so the amount did Gabe spend in all if he bought 3 models planes = $93.97.

Assessment Practice

Question 19.
A middle school with x students conducted a survey to determine students’ Tuesday afternoon activities.
PART A
Write an expression for each activity.
25 more than one-tenth of the students dance.

20 fewer than three-tenths of the students play soccer.

21 more than one-tenth of the students play baseball

Answer:
25 + 1/10, 20 – 3/10, and 21 + 1/10.

Explanation:
In the above-given question,
given that,
25 more than one-tenth of the students dance.
20 fewer than three-tenths of the students play soccer.
21 more than one-tenth of the students play baseball.
25 + 1/10, 20 – 3/10, and 21/10.

PART B
Write a simplified expression to represent the number of students who either dance or play baseball on Tuesday afternoons.

Answer:
The number of students who either dance or play baseball on  Tuesday afternoons = 25 + 1/10 and 21/10.

Explanation:
In the above-given question,
given that,
25 more than one-tenth of the students dance.
21 more than one-tenth of the students play baseball.
25 + 1/10 and 21/10.

Lesson 4.7 Subtract Expressions

Explore It!
The East Side Bulldogs and the West Side Bears are playing a football game. A fan is keeping score using T for a touchdown plus extra point, worth 7 points total, and F for a field goal, worth 3 points.

I can… subtract expressions using properties of operations.

A. How can you represent the score of each team using expressions?

Answer:
The score of the each team using expressions = 10 points.

Explanation:
In the above-given question,
given that,
East Side Bulldogs and the West Side Bears are playing a football game.
fan is keeping score using T for a touchdown.
plus extra point, worth 7 points total, and F for a field goal, worth 3 points.
7 + 5 = 12.

B. How can you represent the difference of the teams’ scores using an expression?

Answer:
The difference of the teams scores using an expression is 4.

Explanation:
In the above-given question,
given that,
East Side Bulldogs and the West Side Bears are playing a football game.
fan is keeping score using T for a touchdown.
plus extra point, worth 7 points total, and F for a field goal, worth 3 points.
12 – 1 = 11.

C. How can you determine how many more points the winning team had than the losing team?

Answer:
The number of points the winning team had than the losing team = 1.

Explanation:
In the above-given question,
given that,
East Side Bulldogs and the West Side Bears are playing a football game.
fan is keeping score using T for a touchdown.
plus extra point, worth 7 points total, and F for a field goal, worth 3 points.
12 – 1 = 11.
so the number of points the winning team had than the losing team = 1.

Focus on math practices
Look for Relationships How can looking at the coefficients help you determine which team scored the greater number of points?

Essential Question
How can properties of operations be used to subtract expressions?

Try It!

A frame holds a picture that is 15 inches long and x inches wide. The frame border is 3 inches wide around the picture. What expression represents the area of the frame border?
Area of frame border = Area of entire frame – Area of photo = ________ – ________
The area of the frame is _________ in²

Answer:
The expression represents the area of the frame border =

Explanation:
In the above-given question,
given that,
A frame holds a picture that is 15 inches long and x inches wide.
The frame border is 3 inches wide around the picture.
3 = 15 – x.
x = 15/3.
x = 5.
so the expression represents the area of the frame border is 5 square inches.

Convince Me! Why can you choose to add or subtract when subtracting an expression?

Try It!

Subtract (0.95x – 0.04) – (0.99x – 0.13).

Answer:
0.04x + 0.09.

Explanation:
In the above-given question,
given that,
(0.95x – 0.04) – (0.99x – 0.13).
0.95x – 0.04 – 0.99x + 0.13.
0.04x + 0.09.
0.04x + 0.09.

Try It!

Subtract (17 + 4.5m + 8k) – (7.5m – 9 + 4k).

Answer:
4k – 3m + 26.

Explanation:
In the above-given question,
given that,
(17 + 4.5m + 8k) – (7.5m – 9 + 4k).
17 + 4.5m + 8k – 7.5m + 9 – 4k.
26 – 3m + 4k.
4k – 3m + 26.

KEY CONCEPT
To subtract expressions, you can use properties of operations.
Write the subtraction as addition and use the Distributive Property to multiply – 1 to the terms in the expression being subtracted.
5 – (2x – 7)
= 5 – (-2x – 7)
= 5 +(-1)(-2x – 7)
= 5+ (-1)(-2)x + (-1)(-7)
= 5 + 2x + 7
You can use the Distributive Property to distribute the minus sign to the second expression, which changes the signs of the terms.
5 – (-2x – 7)
= 5 + 2x + 7

Do You Understand?
Question 1.
Essential Question How can properties of operations be used to subtract expressions?

Answer:
The properties be used to subtract expressions are Distributive property.

Explanation:
In the above-given question,
given that,
5 – (2x – 7)
= 5 – (-2x – 7)
= 5 +(-1)(-2x – 7)
= 5+ (-1)(-2)x + (-1)(-7)
= 5 + 2x + 7.

Question 2.
Use Structure How is subtracting – 4x from 9x similar to subtracting -4 from 9?

Answer:
The difference is 5x.

Explanation:
In the above-given question,
given that,
-4x and 9x.
9x – 4x.
5x.
so the difference is 5x.

Question 3.
Is adding the quantity – 12 + 8r to an expression the same as subtracting -8r+ 12 from the same expression? Explain your reasoning.

Answer:
No, the expressions are not equal.

Explanation:
In the above-given question,
given that,
-12 + 8r.
8r – 12.
-8r + 12.
12 – 8r.
so both the expressions are not equal.

Do You Know How?
Question 4.
Subtract.
a. (21x) – (-16 + 7x)

Answer:
28x + 16.

Explanation:
In the above-given question,
given that,
subtract the expressions.
21x – (-16 + 7x).
21x + (-1) (-16 + 7x).
21x + (-1)(-16) + (-1)(-7x).
21x + 16 + 7x.
28x + 16.

b. (-13n) – (17 – 5n)

Answer:
18n – 17.

Explanation:
In the above-given question,
given that,
subtract the expressions.
(-13n) – (17 – 5n).
13n – 17 + 5n.
18n – 17.

c. (4y – 7) – (y – 7)

Answer:
3y.

Explanation:
In the above-given question,
given that,
subtract the expressions.
(4y – 7) – (y – 7).
4y – 7 -y + 7.
3y.

d. (-w + 0.4) – (-w – 0.4)

Answer:
0.8.

Explanation:
In the above-given question,
given that,
subtract the expressions.
(-w + 0.4) – (-w – 0.4).
-w + 0.4 + w + 0.4.
0.8.

Question 5.
Jude has 5 pairs of sunglasses that cost the same in his online shopping cart but then decides to get only 2. Each pair of sunglasses is the same price. Let p represent the cost of each pair. Write an expression for the original cost, the updated cost, and the difference in cost.

Answer:
The expression for the original cost, the updated cost, and the difference in cost = $2.02.

Explanation:
In the above-given question,
given that,
Jude has 5 pairs of sunglasses that cost the same in his online shopping cart but then decides to get only 2.
Each pair of sunglasses is the same price. Let p represent the cost of each pair.
5 – 2 = 3p.
3 x $1.49 = 4.47.
$6.49 – $4.47 = $2.02.
so the difference in cost = $2.02.

Question 6.
Subtract and simplify.
\(\frac{1}{6}\)m – (-\(\frac{5}{8}\)m + \(\frac{1}{3}\))

Answer:
53m + 1/3.

Explanation:
In the above-given question,
given that,
\(\frac{1}{6}\)m – (-\(\frac{5}{8}\)m + \(\frac{1}{3}\)).
1/6m + 5/8m + 1/3.
6 x 8 = 48.
48 + 5 = 53.
53m + 1/3.

Practice & Problem Solving

Multimedia Leveled Practice In 7-9, fill in the missing signs or numbers.
Question 7.
Rewrite the expression 14m – (5 + 8m)
14m 5 8m

Answer:
The missing signs are minus.

Explanation:
In the above-given question,
given that,
14m – (5 + 8m).
14m – 5 – 8m.
6m – 5.

Question 8.
Rewrite the expression 13d – (-9d – 4) without parentheses. without parentheses.
13d 9d 4

Answer:
The expression is 22d + 4.

Explanation:
In the above-given question,
given that,
the expression is 13d – (-9d – 4).
13d + 9d + 4.
22d + 4.
so the expression without parentheses is 22d + 4.

Question 9.
Write an equivalent expression to 8k – (5 + 2k) without parentheses. Then simplify.
8k – (5 + 2k) = 8k 5 2k
= 8k 2k 5
= k 5

Answer:
The equivalent expression is 6k – 5.

Explanation:
In the above-given question,
given that,
8k – (5 + 2k).
8k – 5 – 2k.
6k – 5.

Question 10.
A company has two manufacturing plants with daily production levels of 5x + 11 items and 2x – 3 items, respectively, where x represents a minimum quantity. The first plant produces how many more items daily than the second plant?

Answer:
The first plant produces 3 more items daily than the second plant.

Explanation:
In the above-given question,
given that,
A company has two manufacturing plants with daily production levels of 5x + 11 items and 2x – 3 items.
5x + 11 – 2x – 3.
3x + 8.
so the first plant produces 3 more items daily than the second plant.

Question 11.
Two communications companies offer calling plans. With Company X, it costs 35¢ to connect and then 5¢ for each minute. With Company Y, it costs 15¢ to connect and then 4¢ for each minute.
Write and simplify an expression that represents how much more Company X charges than Company Y, in cents, for n minutes.

Answer:
The expression that represents company X charges than company Y, in cents, for n minutes =

Explanation:
In the above-given question,
given that,
Two communications companies offer calling plans. With Company X, it costs 35¢ to connect and then 5¢ for each minute.
With Company Y, it costs 15¢ to connect and then 4¢ for each minute.
35 + 5 = 40¢.
15 + 4 = 19¢.
40 – 19 = 21¢.
so the expression that represents company x charges than company y, in cents, for n minutes = 21¢.

Question 12.
Make Sense and Persevere The base and height of a triangle are each extended 2 cm. What is the area of the shaded region? How do you know?

Answer:
The area of the shaded region = x cm.

Explanation:
In the above-given question,
given that,
The base and height of a triangle are each extended 2 cm.
area = 1/2 x b x h.
area = 1/2 x 2 x x.
area = 2/2x.
area = x cm.
so the area of the shaded region is x cm.

Question 13.
Two friends shop for fresh fruit. Jackson buys a watermelon for $7.65 and 5 pounds of cherries. Tim buys a pineapple for $2.45 and 4 pounds of cherries. Use the variable p to represent the price, in dollars, per pound of cherries. Write and simplify an expression to represent how much more Jackson spent.

Answer:
The expression to represent how much more jackson spent = $6.2p.

Explanation:
In the above-given question,
given that,
Two friends shop for fresh fruit.
Jackson buys a watermelon for $7.65 and 5 pounds of cherries.
Tim buys a pineapple for $2.45 and 4 pounds of cherries.
$2.45 + 4 = $6.45.
$7.65 + 5 = $12.65.
$12.65 – $6.45 = $6.2.
so the expression to represent how much more jacson spent = $6.2p.

Question 14.
Yu’s family wants to rent a car to go on vacation. Envocar charges $50.50 and 8¢ per mile. Freedomride charges $70.50 and 12¢ per mile. How much more does Freedomride charge for driving d miles than EnvoCar?

Answer:
The much does Freedomride charge for driving d miles than Envo Car = $20 and 4¢ .

Explanation:
In the above-given question,
given that,
Yu’s family wants to rent a car to go on vacation.
Envocar charges $50.50 and 8¢ per mile.
Freedomride charges $70.50 and 12¢ per mile.
$50.50 and 8¢ .
$70.50 and 12¢ .
70.50 – 50.50 and 12 – 8.
$20 and 4¢ .

Question 15.
A rectangular garden has a walkway around it. Find the area of the walkway.

Answer:
The area of the walkway = 168x + 136.5 sq ft.

Explanation:
In the above-given question,
given that,
the area of the walkway = l x b.
area = (8x + 6.5 ft ) x 21 ft.
area = 168x + 136.5 square ft.

Question 16.
Critique Reasoning Tim incorrectly rewrote the expression \(\frac{1}{2}\)p – (\(\frac{1}{4}\)p + 4) as \(\frac{1}{2}\)p + \(\frac{1}{4}\)p – 4. Rewrite the expression without parentheses. What was Tim’s error?

Answer:
The Tim’s error = 1/2p – 4.

Explanation:
In the above-given question,
given that,
Tim incorrectly rewrote the expression \(\frac{1}{2}\)p – (\(\frac{1}{4}\)p + 4).
\(\frac{1}{2}\)p + \(\frac{1}{4}\)p – 4.
1/2p – 1/4p – 4.
1/2p – 4.

Question 17.
Higher Order Thinking Find the difference.
(7x – 6\(\frac{2}{3}\)) – (-3x +4\(\frac{3}{4}\))

Answer:
The difference is 10x – 7.

Explanation:
In the above-given question,
given that,
(7x – 6\(\frac{2}{3}\)) – (-3x +4\(\frac{3}{4}\))
7x – 6 x 2/3 – (-3x + 4 (3/4).
7x – 6 (2/3) + 3x – 4 (-3/4).
10x -4 – 3.
10x – 7.

Question 18.
Each month, a shopkeeper spends 5x + 11 dollars on rent and electricity. If he spends 2x – 3 dollars on rent, how much does he spend on electricity?

Answer:
The much he spend on electricity = 3x + 8.

Explanation:
In the above-given question,
given that,
Each month, a shopkeeper spends 5x + 11 dollars on rent and electricity.
If he spends 2x – 3 dollars on rent.
5x + 11 – 2x – 3.
3x + 8.

Question 19.
Use the expression \(\frac{1}{4}\)p – (1 – \(\frac{1}{3}\)p).
a. Rewrite the expression without parentheses. Simplify. Show your work.

Answer:
The expression is 1/7p – 1.

Explanation:
In the above-given question,
given that,
\(\frac{1}{4}\)p – (1 – \(\frac{1}{3}\)p).
1/4 p – (1 – 1/3p).
1/4p – 1 + 1/3p.
1/7p – 1.

b. Use a different method to write the expression without parentheses. Do not simplify.

Answer:
The expression is 1/7p – 1.

Explanation:
In the above-given question,
given that,
\(\frac{1}{4}\)p – (1 – \(\frac{1}{3}\)p).
1/4 p – (1 – 1/3p).
1/4p – 1 + 1/3p.
1/7p – 1.

Assessment Practice

Question 20.
An expression is shown.
(0.25n – 0.3) – (0.8n – 0.25)
Create an equivalent expression without parentheses.

Answer:
The equivalent expression without parantheses = 0.05 – 0.55n.

Explanation:
In the above-given question,
given that,
(0.25n – 0.3) – (0.8n – 0.25).
0.25n – 0.3 – 0.8n + 0.25.
-0.55n + 0.05.

Lesson 4.8 Analyze Equivalent Expressions

Solve & Discuss It!
How many toothpicks make a triangle? Two triangles? Write an expression that represents the number of toothpicks needed to make x triangles that appear side-by-side in a single row, as shown. Explain your reasoning.
I can… use an equivalent expression to find new information.

Look for Relationships
What do you notice about the number of toothpicks needed for more than 1 triangle?

Focus on math practices
Reasoning Can there be more than one expression that represents the total number of toothpicks needed to make x triangles in the arrangement shown? Explain.

Essential Question
How can writing equivalent expressions show how quantities are related?

Try It!

Joe is buying gift cards that are on sale for 15% off. He uses c – 0.15c to determine the sale price of gift cards. What is an equivalent expression that Joe could also use to determine the sale price of a gift card?

Answer:
The equivalent expression that Joe could use to determine the sale price = 0.15 – 0.15c.

Explanation:
In the above-given question,
given that,
Joe is buying gift cards that are on sale for 15% off.
He uses c – 0.15c to determine the sale price of gift cards.
15/100 – 0.15c.
0.15 – 0.15c.

Convince Me! How do you know if an expression is describing a percent increase or a percent decrease?

Try It!

The total area, in square feet, of a rectangular stage that has been widened by x feet is represented by 1,900 + 76x. Use the Distributive Property to factor the expression. What does each factor in the equivalent expression tell you about the stage?

Answer:
The equivalent expression about the stage = 1900x + 76 x square.

Explanation:
In the above-given question,
given that,
The total area, in square feet, of a rectangular stage that has been widened by x feet is represented by 1,900 + 76x.
1900 + 76x x x.
1900x + 76xsquare.

KEY CONCEPT

Rewriting expressions can clarify relationships among quantities or variables. When you rewrite an expression, you are writing an equivalent expression.
4x + 12 is equivalent to 4(x + 3) is equivalent to x + x + x + x + 3 + 3 + 3 + 3

Do You Understand?
Question 1.
Essential Question How can writing equivalent expressions show how quantities are related?

Answer:
The equivalent expressions are 4x + 12.

Explanation:
In the above-given question,
given that,
4x + 12.
4(x + 3).
x + x + x + x + 3 + 3 + 3 + 3.

Question 2.
Use Structure The total area, in square feet, of a rectangular mural that has been extended by x feet is represented by 5.5(7.5 + x). Expand the expression using the Distributive Property. What do each of the terms in the equivalent expression tell you about the mural?

Answer:
The expressions tell you about the mural = 41.25 + 5.5x.

Explanation:
In the above-given question,
given that,
the expression is 5.5(7.5 + x).
5.5 x 7.5 + 5.5 x x.
41.25 + 5.5x.
so the equivalent expression is 41.25 + 5.5x.

Question 3.
The expression (2x + 6) + x represents the perimeter of an isosceles triangle. If x represents the length of one side of the triangle, explain how you can use the Distributive Property to find the length of each of the two equivalent sides?

Answer:
The Distributive property to find the length of each of the two equivalent sides = 2x square + 6x.

Explanation:
In the above-given question,
given that,
The expression (2x + 6) + x represents the perimeter of an isosceles triangle.
(2x + 6) + x.
2x X x + 6x.
so the distributive property to find the length of each of the two equivalent sides = 2x x x + 6x.

Do You Know How?
Question 4.
Rewrite the expression 12x + 8 to find an equivalent expression. Show three possible expressions. What do the rewritten expressions tell you about the relationships among the quantities?

Answer:
The expression are 2(6x + 4) and 4(3x + 2).

Explanation:
In the above-given question,
given that,
Rewrite the expression 12x + 8 to find an equivalent expression.
2 (6x + 4).
4 (3x + 2).

Question 5.
A rope is used to make a fence in the shape of an equilateral triangle around a newly planted tree. The length of the rope is represented with the expression 9x + 15

a. Rewrite the expression to represent the three side lengths of the rope fence.

Answer:
The length of the rope is represented with the expression =

Explanation:
In the above-given question,
given that,
9x + 15.
3(3x + 5).
3 x 3x = 9x.
3 x 5 = 15.
so the length of the rope is 3(3x + 5).

b. What is the length of one side?

Answer:
The length of one side = 3x.

Explanation:
In the above-given question,
given that,
9x + 15.
3(3x + 5).
so the length of one side = 3x.

Question 6.
The expression (x – 0.35x) represents 35% off the cost of an item x. How is this equivalent to multiplying x by 0.65?

Answer:
The expression equivalent to multiplying = 0.4225.

Explanation:
In the above-given question,
given that,
The expression (x – 0.35x) represents 35% off the cost of an item x.
x = 0.65.
0.65 – 0.35(0.65).
0.65 – 0.2275.
0.4225.

Practice & Problem Solving

Question 7.
Reasoning Eric is planning an event at a hotel. Let g stand for the number of Eric’s guests. The two expressions represent the difference between the cost of the rooms. Expression 1: (326 + 37g) – (287 + 23g). Expression 2: 39 + 14g. What can you tell about Expression 2 and Expression 1?

Answer:
The two expressions are same.

Explanation:
In the above-given question,
given that,
Eric is planning an event at a hotel. Let g stand for the number of Eric’s guests.
The two expressions represent the difference between the cost of the rooms.
Expression 1: (326 + 37g) – (287 + 23g). Expression 2: 39 + 14g.
326 + 37g – 287 – 23g.
39 + 14g.
so the expressions 1 and 2 are same.

Question 8.
A student received a coupon for 17% off the total purchase price at a clothing store. Let b be the original price of the purchase. Use the expression b-0.17b for the new price of the purchase. Write an equivalent expression by combining like terms.

Answer:
The equivalent expression is 0.16b.

Explanation:
In the above-given question,
given that,
A student received a coupon for 17% off the total purchase price at a clothing store.
Let b be the original price of the purchase.
Use the expression b-0.17b for the new price of the purchase.
0.17b – b.
0.16b.
so the equivaent expression is 0.16b.

Question 9.
Kirana buys boxes of crackers that each have the same cost, c. She represents the cost of 3 boxes of cheese crackers, 2 boxes of poppy seed crackers, and 2 boxes of plain crackers using the expression 3c + 2c + 2c. What equivalent expression can represent the cost?

Answer:
The equivalent epression that represent the cost = 7c.

Explanation:
In the above-given question,
given that,
Kirana buys boxes of crackers that each have the same cost c.
She represents the cost of 3 boxes of cheese crackers.
2 boxes of poppy seed crackers, and 2 boxes of plain crackers using the expression 3c + 2c + 2c.
3c + 2c + 2c.
5c + 2c.
7c.
so the equivalent expression that represent the cost = 7c.

Question 10.
A student received a coupon for 14% off the total purchase price at a clothing store. Let c be the original price of the purchase. The expression c – 0.14c represents the new price of the purchase. Write an equivalent expression to show another way to represent the new price.

Answer:
The equivalent expression to show another way to represent the new price = 0.13c.

Explanation:
In the above-given question,
given that,
A student received a coupon for 14% off the total purchase price at a clothing store.
Let c be the original price of the purchase.
The expression c – 0.14c represents the new price of the purchase.
c – 0.14c.
0.13c.
so the equivalent expression to show another way to represent the new price = 0.13c.

Question 11.
A farmer recently sold a large plot of land. The sale decreased his total acreage by 8%. Let v be the original acreage.
a. Find two equivalent expressions that will give the new acreage.

Answer:
The two equivalent expressions that will give the new acreage = v – 0.08v and 0.07v.

Explanation:
In the above-given question,
given that,
A farmer recently sold a large plot of land.
The sale decreased his total acreage by 8%.
v – 0.08v.
0.07v.

b. Use the expressions to describe two ways to find the new acreage.

Answer:
The two equivalent expressions that will give the new acreage = v – 0.08v and 0.07v.

Explanation:
In the above-given question,
given that,
A farmer recently sold a large plot of land.
The sale decreased his total acreage by 8%.
v – 0.08v.
0.07v.

Question 12.
An art teacher enlarged the area of a copy of a painting by 49%. Let d represent the area of the original painting. The expression d + 0.49d is one way to represent the area of the new painting. Write two additional expressions that will give the area of the new painting.

Answer:
The two additional expressions that will give the area of the new painting = d + 0.49d and 0.50d.

Explanation:
In the above-given question,
given that,
An art teacher enlarged the area of a copy of a painting by 49%.
Let d represent the area of the original painting.
The expression d + 0.49d is one way to represent the area of the new painting.
so the expressions are d nd d + 0.49d.
0.50d.

Question 13.
Use Structure The area of a rectangular playground has been extended on one side. The total area of the playground, in square meters, can be written as 352 + 22x. Rewrite the expression to give a possible set of dimensions for the playground.

Answer:
The expression to give a possible set of dimensions for the playground = 22(16 + x).

Explanation:
In the above-given question,
given that,
The area of a rectangular playground has been extended on one side.
The total area of the playground, in square meters, can be written as 352 + 22x.
352 + 22x.
22(16 + x).
22 x 16 = 352.
22 x x = 22x.

Question 14.
The manager of a store increases the price of the bathing suits by 7%. Let t be the original price of a bathing suit. The new price is t + 0.07t. a. Find an expression equivalent to t + 0.07t.

Answer:
The expression equivalent to t + 0.07t = 0.08t.

Explanation:
In the above-given question,
given that,
The manager of a store increases the price of the bathing suits by 7%.
Let t be the original price of a bathing suit.
The new price is t + 0.07t.
0.08t.

b. If the original price of a bathing suit was $19.99, estimate the new price by first rounding the original price to the nearest dollar.

Answer:
The original price to the nearest dollar = $20 + $1.4t.

Explanation:
In the above-given question,
given that,
If the original price of a bathing suit was $19.99.
19.99 = 20.
20 + 0.07 x 20.
20 + 1.4t.

Question 15.
Higher Order Thinking A customer at a clothing store is buying a pair of pants and a shirt. The customer can choose between a sale that offers a discount on pants, or a coupon for a discount on the entire purchase. Let n represent the original price of the pants and s represent the price of the shirt

a. Write two expressions that represent the “15% off sale on all pants” option.

Answer:
The expressions that represents is n – 0.015.

Explanation:
In the above-given question,
given that,
A customer at a clothing store is buying a pair of pants and a shirt.
The customer can choose between a sale that offers a discount on pants, or a coupon for a discount on the entire purchase.
n – 0.015.

b. Write two expressions that represent the “10% off her entire purchase” option.

Answer:
The expressions that represents is n – 0.010.

Explanation:
In the above-given question,
given that,
A customer at a clothing store is buying a pair of pants and a shirt.
The customer can choose between a sale that offers a discount on pants, or a coupon for a discount on the entire purchase.
n – 0.010.

c. If the original cost of the pants is $25 and the shirt is $10, which option should the customer choose? Explain.

Answer:
The customer choose the both pants and shirts = $35.

Explanation:
In the above-given question,
given that,
If the original cost of the pants is $25 and the shirt is $10.
$25 + $10.
$35.
so the customer choose the both pants and shirts = $35.

Assessment Practice

Question 16.
At a college, the cost of tuition increased by 10%. Let b represent the former cost of tuition. Use the expression b + 0.10b for the new cost of tuition.
PART A
Write an equivalent expression for the new cost of tuition.

Answer:
The new cost of tuition is 0.11b.

Explanation:
In the above-given question,
given that,
At a college, the cost of tuition increased by 10%.
Let b represent the former cost of tuition.
b + 0.10b.
0.11b.
so the new cost of tution is 0.11b.

PART B
What does your equivalent expression tell you about how to find the new cost of tuition?
Answer:

250
4-8 Analyze Equivalent Expressions

Topic 4 Review

Topic Essential Question
How can properties of operations help to generate equivalent expressions that can be used in solving problems?

Vocabulary Review
Complete each definition and then provide an example of each vocabulary word.

Vocabulary
coefficient
constant
variable
factor
expression

Answer:
A term that contains only a number is constant.
The number part of the term that contains a variable is coefficient.
A variable is a letter that represents an unknown value.

Explanation:
In the above-given question,
given that,
A term that contains only a number is constant.
The number part of the term that contains a variable is coefficient.
A variable is a letter that represents an unknown value.
for example:
2x + 4y – 9.
where 2 and 4 are coefficients.
x and y are variables.
9 is the constant.

Use Vocabulary in Writing
Membership in a digital library has a $5 startup fee and then costs $9.95 per month. Membership in a video streaming service costs $7.99 per month with no startup fee. Use vocabulary words to explain how this information could be used to write an expression for the total cost of both memberships after m months.

Answer:
The total cost of memberships after m months = $49.75 + $7.99m.

Explanation:
In the above-given question,
given that,
Membership in a digital library has a $5 startup fee and then costs $9.95 per month.
Membership in a video streaming service costs $7.99 per month with no startup fee.
5 x $9.95 + $7.99m.
$49.75 + $7.99m.
so the total cost of memberships after m months = $49.75 + $7.99m.

Concepts and Skills Review

Lesson 4.1 Write and Evaluate Algebraic Expressions

Quick Review
You can use an algebraic expression to represent and solve a problem with unknown values. The expression can consist of coefficients, constants, and variables. You can substitute values for variables to evaluate expressions.

Example
A farm charges $1.75 for each pound of strawberries picked and $2 for a basket to hold the strawberries. What is the total cost to pick 5 pounds of strawberries?

Answer:
The total cost to pick 5 pounds of strawberries = $10.75.

Explanation:
In the above-given question,
given that,
A farm charges $1.75 for each pound of strawberries picked and $2 for a basket to hold the strawberries.
($1.75 x $5) + 2.
($8.75) + 2.
$10.75.

Write an expression to represent the total cost in dollars to pick p pounds of strawberries.
1.75p + 2
Substitute 5 for p.
1.75(5) + 2 = 8.75 + 2 = 10.75
It costs $10.75 to pick 5 pounds of strawberries.

Practice
Question 1.
Haddie makes and sells knit scarves. Next week she will pay a $25 fee for the use of a booth at a craft fair. She will charge $12 for each scarf she sells at the fair. Write an expression to determine Haddie’s profit for selling s scarves after paying the fee for the use of the booth.

Answer:
The expression to determine Haddie’s profit for selling scarves = $37.

Explanation:
In the above-given question,
given that,
Haddie makes and sells knit scarves.
Next week she will pay a $25 fee for the use of a booth at a craft fair.
She will charge $12 for each scarf she sells at the fair.
$25 + $12.
$37.
so the expression to determine Haddie’s profit for selling scarves = $37.

Question 2.
The cost to buy p pounds of potatoes at $0.32 per pound and n pounds of onions at $0.48 per pound can be determined by using the expression 0.32p + 0.48n. How much will it cost to buy 4.5 pounds of potatoes and 2.5 pounds of onions?

Answer:
The cost to buy 4.5 pounds of potatoes and 2.5 pounds of onions = 0.4608 + 1.2n.

Explanation:
In the above-given question,
given that,
The cost to buy p pounds of potatoes at $0.32 per pound and
n pounds of onions at $0.48 per pound can be determined by using the expression 0.32p + 0.48n.
0.32 x 4.5p = 0.4608p.
0.48 x 2.5n = 1.2.
0.4608 + 1.2n.

Lessons 4-2 AND 4-3 Generate Equivalent Expressions and Simplify Expressions

Quick Review
You can use properties of operations and combine like terms to simplify expressions. Like terms are terms that have the same variable part.

Example
Simplify the expression below.
-7 + \(\frac{1}{3}\)n – \(\frac{4}{3}\) + 2n
Use the Commutative Property to put like terms together,
\(\frac{1}{3}\)n + 2n – 7 – \(\frac{4}{3}\)
Combine like terms.
2\(\frac{1}{3}\)n – 8\(\frac{1}{3}\)

Practice
Simplify each expression below.
Question 1.
\(\frac{5}{8}\)m + 9 – \(\frac{3}{8}\)m – 15

Answer:
3/8 – 6.

Explanation:
In the above-given question,
given that,
\(\frac{5}{8}\)m + 9 – \(\frac{3}{8}\)m – 15.
Use the Commutative Property to put like terms together,
5/8 – 3/8 + 9 – 15.
combine like terms.
3/8 – 6.

Question 2.
-8w + (-4z) + 2 + 6w + 9z – 7

Answer:
5z – 2w – 5.

Explanation:
In the above-given question,
given that,
-8w + (-4z) + 2 + 6w + 9z – 7.
Use the Commutative Property to put like terms together,
6w – 8w -4z + 9z -7 + 2.
combine like terms.
-2w + 5z -5.
5z – 2w – 5.

Question 3.
-6 + (-2d) + (-4d) + 3d

Answer:
-3(d + 2).

Explanation:
In the above-given question,
given that,
-6 + (-2d) + (-4d) + 3d.
Use the Commutative Property to put like terms together,
-6 – 2d – 4d + 3d.
combine like terms.
3d – 6d -6.
-3d – 6.
-3(d + 2).

Lesson 4.4 Expand Expressions

Quick Review
The Distributive Property allows you to multiply each term inside parentheses by a factor that is outside the parentheses. This means that you can use the Distributive Property to expand expressions.

Example
Expand the expression (6 + 7).
(\(\frac{1}{4}\) × h) + (\(\frac{1}{4}\) × 7) = \(\frac{1}{4}\)h + 1.75

Practice
Question 1.
Expand the expression 3.5(-3n + 4).

Answer:
The expression is -10.5n + 14.

Explanation:
In the above-given question,
given that,
the expression is 3.5(-3n + 4).
3.5 x -3n + 3.5 x 4.
-10.5n + 14.
so the expression is -10.5n + 14.

Question 2.
Simplify the expression –\(\frac{3}{5}\)(-8 + \(\frac{5}{9}\)x – 3).

Answer:
5/9 + 24/5 – 3.

Explanation:
In the above-given question,
given that,
the expression –\(\frac{3}{5}\)(-8 + \(\frac{5}{9}\)x – 3).
-3/5 -8 + 5/9 x – 3.
24/5 + 5/9x – 3.
5/9x + 24/5 – 3.

Lesson 4.5 Factor Expressions

Quick Review
When you factor an expression, you write it as a product of two expressions. The new expression is equivalent to the original expression. The greatest common factor (GCF) and the Distributive Property are tools that you use when you need to factor an expression.

Example
Factor the expression 12x – 9y + 15.
The GCF of 12x, 15, and -9y is 3.
Rewrite each term using 3 as a factor.
12x = 3 • 4x
-9y = 3 • (-3y)
15= 3 • 5
Use the Distributive Property to factor the expression.
3(4x – 3y + 5)

Practice
Factor each expression.
Question 1.
63a – 42b

Answer:
3(21a – 14b).

Explanation:
In the above-given question,
given that,
the expression is 63a – 42b.
3(21a – 14b).
3 x 21a = 63a.
3 x 14b = 42b.

Question 2.
81y + 54

Answer:
9(9y + 6).

Explanation:
In the above-given question,
given that,
the expression is 81y + 54.
9(9y + 6).
9 x 9y = 81y.
9 x 6 = 54.

Question 3.
Which show a way to factor the expression 32t – 48? Select all that apply.
☐ 2(16t – 24)
☐ 4(12t – 48)
☐ 6(26 – 42)
☐ 8(4t – 6)
☐ 16(2t – 3)

Answer:
2(16t – 24) and 8(4t – 6).

Explanation:
In the above-given question,
given that,
the expression is 32t – 48.
2(16t – 24).
2 x 16t = 32t.
2 x 24 = 48.
8(4t – 6).
8 x 4t = 32t.
8 x 6 = 48.

Lessons 4.6 AND 4.7 Add and Subtract Expressions

Quick Review
Adding and subtracting expressions may require combining like terms. This means that you must use the Commutative and Associative Properties to reorder and group terms as needed.

Example
Kerry has n markers. Rachel has 1 marker fewer than twice the number of markers Kerry has. Write and simplify an expression for the total number of markers they have.
Number of markers Kerry has: n
Number of markers Rachel has: 2n – 1
Total number of markers:
n + (2n – 1)
(n + 2n) – 1
3n – 1

Practice
Add the expressions.
Question 1.
5.2C – 7.35) + (-3.9C + 2.65)

Answer:
1.3c – 4.7.

Explanation:
In the above-given question,
given that,
5.2C – 7.35) + (-3.9C + 2.65).
5.2c – 7.35 – 3.9c + 2.65.
5.2c – 3.9c + 2.65 – 7.35.
1.3c – 4.7.

Question 2.
(6x – 2y – 5) – (-5 + 9y – 8x)

Answer:
14x – 11y.

Explanation:
In the above-given question,
given that,
(6x – 2y – 5) – (-5 + 9y – 8x).
6x – 2y – 5 + 5 – 9y + 8x.
14x – 11y.

Question 3.
Last week Jean ran 2 fewer than 4m miles. This week she ran 0.5 miles more than last week. Write and simplify an expression for the total number of miles Jean ran in the two weeks.

Answer:
The total number of miles Jean ran in the two weeks = 2.5m.

Explanation:
In the above-given question,
given that,
Last week Jean ran 2 fewer than 4m miles.
This week she ran 0.5 miles more than last week.
2m + 0.5m.
2.5m.
so the total number of miles Jean ran in the two weeks = 2.5m.

Lesson 4.8 Analyze Equivalent Expressions

Quick Review
Equivalent expressions can help to show new information about a problem. Sometimes the equivalent expression will be an expanded expression. In other cases, it will be a factored expression.

Example
The perimeter of a square is represented with the expression 84 + 44s. What is the length of one side of the square?
A square has 4 sides, so factor 4 out of each term in the expression for the perimeter.
84 + 445 = 4 • 21 + 4 • 11s = 4(21 + 115)
The factor within the parentheses represents the length of one side of the square.
The length of one side is 21 + 11s.

Practice
Question 1.
Hal earns n dollars per hour. Next month he will receive a 2% raise in pay per hour. The expression n + 0.02n is one way to represent Hal’s pay per hour after the raise. Write an equivalent simplified expression that will represent his pay per hour after the raise.

Answer:
The equivalent expression that will represent his pay per hour after the raise = 0.03n.

Explanation:
In the above-given question,
given that,
Hal earns n dollars per hour.
Next month he will receive a 2% raise in pay per hour.
The expression n + 0.02n is one way to represent Hal’s pay per hour after the raise.
n + 0.02n = 0.03n.

Question 2.
The area of a garden plot can be represented by the expression 84z – 54. The garden will be divided into six sections for planting six different vegetables. The sections will be equal in area. Write an expression that represents the area of each section.

Answer:
The expression that represents the area of each section =

Explanati

Topic 4 Fluency Practice

Hidden Clue
For each ordered pair, solve the percent problems to find the coordinates. Then locate and label the corresponding point on the graph. Draw line segments to connect the points in alphabetical order. Use the completed picture to help you answer the riddle below.

I can… represent and solve percent problems.

Go through the enVision Math Common Core Grade 5 Answer Key Topic 2 Use Models and Strategies to Add and Subtract Decimals regularly and improve your accuracy in solving questions.

enVision Math Common Core 5th Grade Answers Key Topic 2 Use Models and Strategies to Add and Subtract Decimals

Use Models and Strategies to Add and Subtract Decimals

Essential Questions: How can sums and differences of decimals be estimated? What are some common procedures for adding and subtracting decimals? How can sums and differences be found mentally?

enVision STEM Project: Producers and Consumers
Do Research Use the Internet or other sources to find information about producers and consumers.
Journal: Write a Report Include what you found. Also in your report:

• What do producers need to survive? What do consumers need to survive?
• Give at least three examples of both producers and consumers.
• Write and solve decimal addition and subtraction problems for the amounts of food the consumers need.

Review What You Know

Vocabulary

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

• addend
• inverse operations
• round
• difference
• equivalent
• sum

Question 1.
The ___ is the result of subtracting one number from another.

Answer:
The difference is the result of subtracting one number from another.

Explanation:
In the above-given question,
given that,
the difference is the result of subtracting one number from another.
for example:
5 – 2 = 3.
3 is the difference.

Question 2.
Two numbers or expressions that have the same value are ___.

Answer:
The two numbers or expressions that have the same value are equivalent.

Explanation:
In the above-given question,
given that,
the two numbers or expressions that have the same value are equivalent.
for example:
2 + 3 = 5.
the two have the same value.

Question 3.
The answer to an addition problem is the ___.

Answer:
The answer to an addition problem is the sum.

Explanation:
In the above-given question,
given that,
the answer to an addition problem is the sum.
for example:
5 + 3 = 8.
8 is the sum.

Question 4.
One way to estimate an answer is to ___ the numbers and then do the calculation.

Answer:
One way to estimate an answer is to inverse the numbers and then do the calculation.

Explanation:
In the above-given question,
given that,
One way to estimate an answer is to inverse the numbers and then do the calculation.
5 + 6 = 11.
11 – 6 = 5.
11 – 5 = 6.

Round Decimals
Round each number to the nearest tenth.

Question 5.
74.362

Answer:
The number equal to the nearest tenth = 74.4.

Explanation:
In the above-given that,
given that,
to round a number to the nearest tenth,
look at the next place value to the right(the hundredths).
if it’s 4 or less, just remove all the digits to the right.
if it’s 5 or greater, add 1 to the digit in the tenths place, and remove all the digits to the right.
74.362 in this the hundredths place is greater than 5.
so add 1 to the tenths place.
3 + 1 = 4.
the number equal to the nearest tenth = 74.4.

Question 6.
28.45

Answer:
The number equal to the nearest tenth = 28.5.

Explanation:
In the above-given that,
given that,
to round a number to the nearest tenth,
look at the next place value to the right(the hundredths).
if it’s 4 or less, just remove all the digits to the right.
if it’s 5 or greater, add 1 to the digit in the tenths place, and remove all the digits to the right.
28.45 in this the hundredths place is greater than 5.
so add 1 to the tenths place.
4 + 1 = 5.
the number equal to the nearest tenth = 28.5.

Question 7.
13.09

Answer:
The number equal to the nearest tenth = 13.1.

Explanation:
In the above-given that,
given that,
to round a number to the nearest tenth,
look at the next place value to the right(the hundredths).
if it’s 4 or less, just remove all the digits to the right.
if it’s 5 or greater, add 1 to the digit in the tenths place, and remove all the digits to the right.
13.09 in this the hundredths place is greater than 5.
so add 1 to the tenths place.
0 + 1 = 1.
the number equal to the nearest tenth = 13.1.

Round each number to the nearest hundredth.

Question 8.
43.017

Answer:
The number equal to the nearest hundredth = 43.02.

Explanation:
In the above-given that,
given that,
to round a number to the nearest hundredth,
look at the next place value to the right(the thousands).
if it’s 4 or less, just remove all the digits to the right.
if it’s 5 or greater, add 1 to the digit in the hundreds place, and remove all the digits to the right.
43.017 in this the thousands place is greater than 5.
so add 1 to the hundredths place.
1 + 1 = 2.
the number equal to the nearest hundredths = 43.02.

Question 9.
186.555

Answer:
The number equal to the nearest hundredth = 186.56.

Explanation:
In the above-given that,
given that,
to round a number to the nearest hundredth,
look at the next place value to the right(the thousands).
if it’s 4 or less, just remove all the digits to the right.
if it’s 5 or greater, add 1 to the digit in the hundreds place, and remove all the digits to the right.
186.555 in this the thousands place is greater than 5.
so add 1 to the hundredths place.
5 + 1 = 6.
the number equal to the nearest hundredths = 186.56.

Question 10.
222.222

Answer:
The number equal to the nearest hundredth = 222.20.

Explanation:
In the above-given that,
given that,
to round a number to the nearest hundredth,
look at the next place value to the right(the thousands).
if it’s 4 or less, just remove all the digits to the right.
if it’s 5 or greater, add 1 to the digit in the hundreds place, and remove all the digits to the right.
222.222 in this the thousands place is less than 5.
so do not add 1 to the hundredths place.
222.20
the number equal to the nearest hundredths = 222.20.

Round each number to the underlined digit.

Question 11.
84.59

Answer:
The number round to the underlined digit is = 85.

Explanation:
In the above-given question,
given that,
the number is 84.59.
the underlined digit is 4.
4 + 1 = 5.
so the number round to the underlined digit = 85.

Question 12.
2.948

Answer:
The number round to the underlined digit is = 2.95.

Explanation:
In the above-given question,
given that,
the number is 2.948.
the underlined digit is 4.
4 + 1 = 5.
so the number round to the underlined digit = 2.95.

Question 13.
30.125

Answer:
The number round to the underlined digit is = 30.13.

Explanation:
In the above-given question,
given that,
the number is 30.125.
the underlined digit is 2.
2 + 1 = 3.
so the number round to the underlined digit = 30.13

Addition and Subtraction of Whole Numbers

Find each sum or difference.

Question 14.
9,536 + 495

Answer:
9536 + 495 = 10031.

Explanation:
In the above-given question,
given that,
the numbers are 9536 and 495.
add the two numbers.
9536 + 495 = 10031.

Question 15.
612 – 357

Answer:
612 – 357 = 255.

Explanation:
In the above-given question,
given that,
the numbers are 612 and 357.
subtract the two numbers.
612 – 357 = 255.

Question 16.
5,052 – 761

Answer:
5052 – 761 = 4291.

Explanation:
In the above-given question,
given that,
the numbers are 5052 and 761.
subtract the two numbers.
5052 – 761 = 4291.

Question 17.
Vivica sees that a printer costs $679 and a computer costs $1,358. What is the total cost of the printer and the computer?

Answer:
The total cost of the printer and the computer = $2037.

Explanation:
In the above-given question,
given that,
Vivica sees that a printer costs $679.
computer costs $1,358.
$679 + $1,358 = $2037.
so the total cost of the printer and the computer = $2037.

Question 18.
The Pecos River is 926 miles long, and the Brazos River is 1,280 miles long. How many miles longer is the Brazos River than the Pecos River?
A. 2,206 miles
B. 1,206 miles
C. 364 miles
D. 354 miles

Answer:
Option D is correct.

Explanation:
In the above-given question,
given that,
The Pecos River is 926 miles long, and the Brazos River is 1,280 miles long.
1280 – 926 = 354.
the longer is the Brazos River than the Pecos River is 354 miles.
so option D is correct.

Pick a Project

PROJECT 2A
How big are alligators and crocodiles?
Project: Compare the Sizes of Reptiles

Answer:
The crocodile weight is 410 – 520 kg approximately.

Explanation:
In the above-given question,
given that,
the water animals are crocodiles and alligators.
the size of the reptiles is very tiny.
so the crocodile weight is 410 – 520 kg approximately.

PROJECT 2B
How much should a theme park ticket cost?
Project: Make a Brochure for a Theme Park

Answer:
The cost of the ticket is $37 approximately.

Explanation:
In the above-given question,
given that,
the theme park is located in the united states of America.
so the cost of the ticket is $37 approximately.

PROJECT 2C
How much and how healthy is your meal in a restaurant?
Project: Plan Your Meal

PROJECT 2D
How far was your trip?
Project: Make a Travel Journal

Answer:
The travel trip is Vizag.

Explanation:
In the above-given question,
given that,
my travel trip is Vizag.
the distance from Hyderabad to Vizag is around 650 km.
so my travel trip is Vizag.

Lesson 2.1 Mental Math

Activity
Solve & Share

Three pieces of software cost $20.75, $10.59, and $18.25. What is the total cost of the software? Use mental math to solve.

You can use reasoning to help you. What do you know about adding three numbers that will make it easier to solve this problem?

Answer:
The total cost of the software is $49.59.

Explanation:
In the above-given question,
given that,
Three pieces of software cost $20.75, $10.59, and $18.25.
20.75 + 10.59 + 18.25 = 49.59.
so the total cost of the software is $49.59.

Look Back! Which two numbers above were easy to add in your head? Why?

Visual Learning Bridge

Essential Question
How Can You Use Mental Math to Add?

A.
Properties of addition can help you find the total cost of these three items.

The Commutative Property and Associative Property make it easy to add $11.45 + $3.39 + $9.55.

The Associative Property lets you change the grouping of addends.
($11.45 + $3.39) + $9.55 = $11.45 + ($3.39 + $9.55)

The Commutative Property lets you add two decimals in any order.

B.
Use the Commutative Property to change the order.
$11.45 + ($3.39 + $9.55) = $11.45 + ($9.55 + $3.39)
Use the Associative Property to change the grouping.
$11.45 + ($9.55 + $3.39) = ($11.45 + $9.55) + $3.39

C.
Add $11.45 and $9.55 first because they are easy to compute mentally.
$11.45 + $9.55 = $21
$21 + $3.39 = $24.39
The three items cost a total of $24.39.
Compatible numbers are numbers that are easy to compute mentally.

Convince Me! Reasoning Use mental math to find the sum. Explain your thinking. Jim earns $22.50, $14.75, and $8.50 on three different days. How much did he earn in all?

Answer:
The much did he earn in all = $45.75.

Explanation:
In the above-given question,
given that,
Jim earns $22.50, $14.75, and $8.50 on three different days.
we can find the answer in three ways.
$22.50 + $14.75 = 37.25.
$37.25 + $8.50 = $45.75.
so the much did he earn in all = $45.75.

Another Example
With compensation, adjust one or both numbers to make the calculation easier. Then adjust the difference or sum to get the final answer.
Use compensation to subtract.

Find 4.25 – 0.08 mentally.
4.25 – 0.10 = 4.15

4.25 – 0.08 = 4.17

Use compensation to add.
Find $3.47 + $4.35 mentally.

$3.47 + $4.35 = $7.82

Guided Practice

Do You Understand?

Question 1.
Show how to use compensation to add $3.18 and $6.50.

Answer:
$3.18 + $6.50 = $9.68.

Explanation:
In the above-given question,
given that,
the two numbers are 3.20 and 6.50.
$3.20 + $6.50 = $9.70.
we have to take 0.02 from $9.70.
$9.70 – 0.02 = $9.68.
so $3.18 + $6.50 = $9.68.

Question 2.
Use properties to rewrite the expression to be easier to solve. Explain. (13 + 4.63) + 7.4

Answer:
We can use two expressions that are associative and commutative properties.

Explanation:
In the above-given question,
given that,
the two properties are associative and commutative.
13 + 4.63 = 17.63.
17.63 + 7.4 = 25.03.
4.63 + 7.4 = 12.03.
12.03 + 13 = 25.03.
so we can use two expressions that are associative and commutative properties.

Do You Know How?

In 3-6, use mental math to add or subtract.

Question 3.
12 + 3.04 + 8.28

Answer:
12 + 3.04 + 8.28 = 23.32.

Explanation:
In the above-given question,
given that,
use mental math to add.
12 + 3.04 = 15.04.
15.04 + 8.28 = 23.32.

Question 4.
6.97 + 4.15

Answer:
6.97 + 4.15 = 11.12.

Explanation:
In the above-given question,
given that,
use mental math to add.
6.97 + 4.15 = 11.12.

Question 5.
9.04 – 6.98

Answer:
9.04 – 6.98 = 2.06.

Explanation:
In the above-given question,
given that,
use mental math to subtract.
9.04 – 6.98 = 2.06.

Question 6.
4.02 +0.19 + 16.48

Answer:
4.02 + 0.19 + 16.48 = 20.69.

Explanation:
In the above-given question,
given that,
use mental math to add.
4.02 + 0.19 = 4.21.
4.21 + 16.48 = 20.69.

Independent Practice

Leveled Practice In 7-12, use mental math to add or subtract.

Question 7.
7.1 + 5.4 + 2.9 =
___ + 5.4 =
_______

Answer:
7.1 + 5.4 + 2.9 = 15.4.

Explanation:
In the above-given question,
given that,
use mental math to add.
7.1 + 5.4 = 12.5.
12.5 + 2.9 = 15.4.

Question 8.
373.4 – 152.9 =
373.4 – __ = 220.4
___ + 0.1 = ___

Answer:
373.4 – 152.9 = 220.5.

Explanation:
In the above-given question,
given that,
use mental math to subtract.
373.4 – 152.9 = 220.5.
373.4 – 152.8 = 220.4.
220.4 + 0.1 = 220.5.

Question 9.
$18.25 + $7.99 + $4.75

Answer:
$18.25 + $7.99 + $4.75 = $30.99.

Explanation:
In the above-given question,
given that,
use mental math to add.
18.25 + 7.99 = 26.24.
26.24 + 4.75 = 30.99.

Question 10.
1.05 + 3 + 4.28 +0.95

Answer:
1.05 + 3 + 4.28 + 0.95 = 6.31.

Explanation:
In the above-given question,
given that,
use mental math to add.
1.05 + 3 = 1.08.
4.28 + 0.95 = 5.23.
5.23 + 1.08 = 6.31.

Question 11.
2,504 + 140 + 160

Answer:
2504 + 140 + 160 = 2804.

Explanation:
In the above-given question,
given that,
use mental math to add.
2504 + 140 = 2644.
2644 + 160 = 2804.

Question 12.
35.7 – 14.8

Answer:
35.7 – 14.8 =20.9.

Explanation:
In the above-given question,
given that,
the two numbers are 35.7 and 14.8.
use mental math to subtract.
35.7 – 14.8 = 20.9.

Problem Solving

Question 13.
Joanne bought three books that cost $3.95, $4.99, and $6.05. How much did she spend in all? Use compensation and mental math to find the sum.

Answer:
The total she spends in all = 14.09.

Explanation:
In the above-given question,
given that,
Joanne bought three books that cost $3.95, $4.99, and $6.05.
$3.95 + $4.100 = 8.05.
8.05 – 0.01 = 8.04.
8.04 + 6.05 = 14.09.
so the total she spends in all = 14.09.

Question 14.
Construct Arguments Use compensation to find each difference mentally. Explain how you found each difference.
A. 67.9 – 29.9
B. 456 – 198

Answer:
A. 67.9 – 29.9 = 38.
B. 456 – 198 = 258.

Explanation:
In the above-given question,
given that,
67.9 + 0.01 = 68.0.
68.0 – 0.01 = 67.9.
67.9 – 29.9 = 38.
456 – 200 = 256.
256 + 2 = 258.

Question 15.
Number Sense The table shows how many points Eduardo scored during each game. Use mental math to find how many points he scored in the first three games.

Answer:
The number of points Eduardo scored in the first three games = 99.

Explanation:
In the above-given question,
given that,
The table shows how many points Eduardo scored during each game.
in the first game, he scored 54 points.
in the second game, he scored 19 points.
in the third game, he scored 26 points.
54 + 19 + 26 = 99.
so the number of points Eduardo scored in the first three games = 99.

Question 16.
On three different days at her job, Sue earned $27, $33, and $49. She needs to earn $100 to buy a desk for her computer. If she buys the desk, how much money will she have left over?

Answer:
The money will she have leftover = 9.

Explanation:
In the above-given question,
given that,
On three different days at her job.
Sue earned $27, $33, and $49.
She needs to earn $100 to buy a desk for her computer.
$27 + $33 + $49 = 109.
109 – 100 = 9.
so the money will she have leftover = 9.

Question 17.
A shelf can hold 50 DVDs. Jill has 27 DVDs. She plans to buy 5 new ones. Each DVD costs $9. After she buys the new ones, how many more DVDs can the shelf hold?

Answer:
The number of DVDs can the shelf hold = 32.

Explanation:
In the above-given question,
given that,
A shelf can hold 50 DVDs. Jill has 27 DVDs.
She plans to buy 5 new ones.
Each DVD costs $9.
27 + 5 = 32.
so the number of DVDs can the shelf hold = 32.

Question 18.
When finding the difference of two numbers mentally, can you use the Commutative Property? Explain.

Answer:
No, we cannot use the commutative property.

Explanation:
In the above-given question,
given that,
the difference between the two numbers we cannot use the commutative property.
for example:
the two numbers are 5 and 3.
5 – 3 = 2.
so the difference is 2.

Question 19.
Higher Order Thinking Daria bought a skein of alpaca yarn for $47.50, a skein of angora yarn for $32.14, a skein of wool yarn for $16.50, and a pair of knitting needles for $3.86. How much did she spend in all? Describe how you calculated your answer.

Answer:
The more did she spend in all = $100.

Explanation:
In the above-given question,
given that,
Daria bought a skein of alpaca yarn for $47.50, a skein of angora yarn for $32.14.
a skein of wool yarn for $16.50, and a pair of knitting needles for $3.86.
47.50 + 32.14 + 16.50 + 3.86.
47.50 + 32.14 = 79.64.
16.50 + 3.86 = 20.36.
79.64 + 20.36 =
so the more did she spend in all = $100.

Assessment Practice

Question 20.
Mrs. Healer’s class took a field trip to a park 12.3 miles away. Mr. Dean’s class drove 4.9 miles to the public library. How much farther did Mrs. Healer’s class travel than Mr. Dean’s class? Explain how you used mental math to determine the difference.

Answer:
The much farther did Mrs. Healer’s class travel than Mr. Dean’s class = 7.4.

Explanation:
In the above-given question,
given that,
Mrs. Healer’s class took a field trip to a park 12.3 miles away.
Mr. Dean’s class drove 4.9 miles to the public library.
12.3 – 4.9 = 7.4.
so much farther did Mrs. Healer’s class travel than Mr.Dean’s class = 7.4.

Lesson 2.2 Estimate Sums and Differences of Decimals

Activity

Solve & Share

An amusement park has two roller coasters. One is 628 feet long, and the other is 485 feet long. If you ride both roller coasters, about how many feet will you travel in all? Use estimation to solve.

You can use reasoning to decide what you are asked to find. Is the problem looking for an exact answer? How can you tell?

Look Back! About how much longer is the one coaster than the other? Show your work.

Answer:
The number of feet will you travel in all = 143 feet.

Explanation:
In the above-given question,
given that,
An amusement park has two roller coasters.
One is 628 feet long, and the other is 485 feet long.
628 – 485 = 143.
so the number of feet will you travel in all = 143 feet.

Visual Learning Bridge

Essential Question How Can You Estimate Sums?

A.
Students are collecting dog food to give to an animal shelter. Estimate how many pounds were collected in Weeks 3 and 4.

There is more than one way to find an estimate.

B. One Way
Round each addend to the nearest hundred.

237.5 + 345.1 is about 500.
The students collected about 500 pounds of dog food in Weeks 3 and 4.

C.
Another Way

Substitute compatible numbers.

Compatible numbers are easy to add!

237.5 + 345.1 is about 600.
The students collected about 600 pounds of dog food in Weeks 3 and 4.

Convince Me! Critique Reasoning Tomás said, “We did great in Week 4! We collected just about twice as many pounds as in Week 1!”
Use estimation to decide if he is right. Explain your thinking.

Another Example
You can estimate differences.
Estimate 22.84 – 13.97.

One Way
Round each number to the nearest whole number.

22.84 – 13.97 is about 9.

Another Way
Substitute compatible numbers.

22.84 – 13.97 is about 10.

Guided Practice

Do You Understand?

Question 1.
In the example above, which estimate is closer to the actual difference? How can you tell without subtracting?

Answer:
22.84 is near to the 23.

Explanation:
In the above-given question,
given that,
for example:
22.84 we can write it as 23.
22.84 we can write it as 25.
so 23 is near to 22.84.

Question 2.
In the example on the previous page, students collected more pounds of dog food in Week 4 than in Week 3. Estimate about how many more.

Answer:
The many more they have collected = 100 pounds.

Explanation:
In the above-given question,
given that,
the two numbers are 237.5 and 345.1.
250 + 350 = 600.
so the many more they have collected = 100 pounds.

Do You Know How?

In 3-10, estimate each sum or difference.

Question 3.
49 + 22.88

Answer:
49 + 23 = 72.

Explanation:
In the above-given question,
given that,
the two numbers are 49 and 23.
add the two numbers.
49 + 23 = 72.

Question 4.
86.9 – 18

Answer:
86.9 – 18 = 68.9

Explanation:
In the above-given question,
given that,
the two numbers are 86.9 and 18.
subtract the two numbers.
86.9 – 18 = 68.9.

Question 5.
179 + 277.1

Answer:
179 + 277.1 = 456.1

Explanation:
In the above-given question,
given that,
the two numbers are 179 and 277.1.
add the two numbers.
179 + 277.1 = 456.1.

Question 6.
23.2 – 9.71

Answer:
23.2 – 9.71 =13.49

Explanation:
In the above-given question,
given that,
the two numbers are 23.2 and 9.71.
subtract the two numbers.
23.2 – 9.71 = 13.49.

Question 7.
23.8 – 4.7

Answer:
23.8 – 4.7 = 19.1.

Explanation:
In the above-given question,
given that,
the two numbers are 23.8 and 4.7.
subtract the two numbers.
23.8 – 4.7 = 19.1.

Question 8.
87.2 + 3.9

Answer:
87.2 + 3.9 = 91.1.

Explanation:
In the above-given question,
given that,
the two numbers are 87.2 and 3.9.
add the two numbers.
87.2 – 3.9 = 91.1.

Question 9.
38.9 – 21.4

Answer:
38.9 – 21.4 = 17.5.

Explanation:
In the above-given question,
given that,
the two numbers are 38.9 and 21.4.
subtract the two numbers.
38.9 – 21.4 = 17.5.

Question 10.
576 +94.6

Answer:
576 + 94.6 = 670.6.

Explanation:
In the above-given question,
given that,
the two numbers are 576 and 94.6.
add the two numbers.
576 + 94.6 = 670.6.

Independent Practice

In 11-18, estimate each sum or difference.

Question 11.
79.1 + 32.4

Answer:
79.1 + 32.4 = 111.5.

Explanation:
In the above-given question,
given that,
the two numbers are 79.1 and 32.4.
add the two numbers.
79.1 + 32.4 = 111.5.

Question 12.
788.9 – 572

Answer:
788.9 – 572 = 216.9.

Explanation:
In the above-given question,
given that,
the two numbers are 788.9 and 572.
subtract the two numbers.
788.9 – 572 = 216.9.

Question 13.
837 + 488.12

Answer:
837 + 488.12 = 1325.12

Explanation:
In the above-given question,
given that,
the two numbers are 837 and 488.12.
add the two numbers.
837 + 488.12 = 1325.12.

Question 14.
418.5 – 23.7

Answer:
418.5 – 23.7 = 394.8.

Explanation:
In the above-given question,
given that,
the two numbers are 418.5 and 23.7.
subtract the two numbers.
418.5 – 23.7 = 394.8.

Question 15.
2.9 + 3.9

Answer:
2.9 + 3.9 = 6.8.

Explanation:
In the above-given question,
given that,
the two numbers are 2.9 and 3.9.
add the two numbers.
2.9 – 3.9 = 6.8.

Question 16.
$12.99 – $3.95

Answer:
$12.99 – $3.95 = 9.04.

Explanation:
In the above-given question,
given that,
the two numbers are 12.99 and 3.95.
subtract the two numbers.
12.99 – 3.95 = 9.04.

Question 17.
8.1 + 3.7 + 7.9.

Answer:
8.1 + 3.7 + 7.9 = 19.7.

Explanation:
In the above-given question,
given that,
the three numbers are 8.1, 3.7, and 7.9.
add the three numbers.
8.1 + 3.7 = 11.8.
7.9 + 11.8 = 19.7.

Question 18.
3.8 + 4.1 + 3.3

Answer:
3.8 + 4.1 + 3.3 = 11.2.

Explanation:
In the above-given question,
given that,
the three numbers are 3.8, 4.1, and 3.3.
add the three numbers.
3.8 + 4.1 = 7.9.
3.3 + 7.9 = 11.2.

Problem Solving

Question 19.
Construct Arguments The cost of one DVD is $16.98, and the cost of another DVD is $9.29. Ed estimated the cost of the two DVDs to be about $27. Is his estimate higher or lower than the actual cost? Explain.

Answer:
The estimate is equal.

Explanation:
In the above-given question,
given that,
The cost of one DVD is $16.98, and the cost of another DVD is $9.29.
Ed estimated the cost of the two DVDs to be about $27.
16.98 + 9.29 = 26.27.
the estimate is equal to the actual cost.

Question 20.
Higher Order Thinking A teacher is organizing a field trip. Each bus can seat up to 46 people. Is it better to estimate a greater or lesser number than the actual number of people going on the field trip? Why?

Answer:
The estimation is lesser than the actual number of people going on the field trip.

Explanation:
In the above-given question,
given that,
A teacher is organizing a field trip.
Each bus can seat up to 46 people.
the estimate a lesser number than the actual number of people going on the field trip.

Question 21.
The size and shape of Golden Gate Park are often compared to the size and shape of Central Park. About how many more acres does Golden Gate Park cover than Central Park?

Answer:
The number of acres does Golden Gate Park covers than central park = 174 acres.

Explanation:
In the above-given question,
given that,
The size and shape of Golden Gate Park are often compared to the size and shape of the central park.
the central park in New York City has an area of 843 acres.
Golden Gate Park in San Francisco, California, has an area of 1,017 acres.
1017 – 843 = 174.
so the number of acres does Golden Gate Park covers than central park = 174.

Assessment Practice

Question 22.
Three rock samples have masses of 74.05 grams, 9.72 grams, and 45.49 grams. A scientist estimates the total mass of the samples by rounding each mass to the nearest whole number. Which lists the numbers he will add?
A. 75, 10, and 46
B. 74.1, 9.7, and 45.5
C. 74, 10, and 45
D. 75, 10, and 50

Answer:
Option A is the correct answer.

Explanation:
In the above-given question,
given that,
Three rock samples have masses of 74.05 grams, 9.72 grams, and 45.49 grams.
A scientist estimates the total mass of the samples by rounding each mass to the nearest whole number.
74.05 = 75.
9.72 = 10.
45.49 = 46.
so option A is the correct answer.

Question 23.
Umberto buys a game for $7.89 and some batteries for $5.49. He pays with a $20 bill. Which is the best estimate of how much change he should get?
A. $5.00
B. $7.00
C. $13.00
D. $17.00

Answer:
The better estimate of change he should get = $7.00.

Explanation:
In the above-given question,
given that,
Umberto buys a game for $7.89 and some batteries for $5.49.
He pays with a $20 bill.
7.89 + 5.49 = 13.38.
20 – 13.38 = 6.62.
so option B is correct.

Lesson 2.3 Use Models to Add and Subtract Decimals

Solve & Share
Gloria rode her bicycle 0.75 mile in the morning and 1.40 miles in the afternoon. How many miles did Gloria ride in all? Solve this problem any way you choose.

You can use tools, such as place-value blocks, to help determine how many miles Gloria rode.

Answer:
The number of miles did Gloria ride in all = 2.15.

Explanation:
In the above-given question,
given that,
Gloria rode her bicycle 0.75 miles in the morning and 1.40 miles in the afternoon.
0.75 + 1.40 = 2.15.
so the number of miles did Gloria ride in all = 2.15.

Look Back! Make Sense and Persevere How can you check that your answer is reasonable?

Visual Learning Bridge

Essential Question
How Can You Use Models to Add Decimals?

A.
Use the table at the right to find the total monthly cost of using the dishwasher and the DVD player.

Place-value blocks can be used to add decimals.

Use compatible numbers to estimate. $0.85 + $0.40 is about $0.80 + $0.40, so the sum is about $1.20.

B.
Use place-value blocks to model $0.85 + $0.40.

C.
Combine the blocks. Regroup when you can.

The monthly cost of using the dishwasher and DVD player is $1.25.

Convince Me! Critique Reasoning For the example above, Jesse said, “The total monthly cost of using the ceiling light and the dishwasher was $0.74.” Is Jesse correct? Explain.

Answer:
No, Jesse was not correct.

Explanation:
In the above-given question,
given that,
the total monthly cost of using the ceiling light and the dishwasher was $0.74.
the total monthly cost of using the ceiling light = $0.89.
the total monthly cost of using the dishwasher = $0.85.
0.89 + 0.85 = 1.74.
so Jesse was not correct.

Another Example
You can subtract decimals with place-value blocks. Find 1.57 – 0.89.

Step 1
Show 1.57 with place-value blocks.

Step 2
To subtract 0.89, regroup the blocks and remove 8 tenths and 9 hundredths. The blocks remaining are the difference.

Guided Practice

Do You Understand?

Question 1.
Explain how to use place-value blocks to find the difference between the monthly cost of using the DVD player and the dishwasher. Then find the difference.

Answer:
The difference between the DVD player and the dishwasher = $0.45.

Explanation:
In the above-given question,
given that,
the cost of a DVD player is $0.40.
the cost of the dishwasher is $0.85.
0.85-0.40 = 0.45.
so the difference between DVD player and the dishwasher = $0.45.

Do You Know How?

In 2-7, use place-value blocks to add or subtract.

Question 2.
1.22 +0.34

Answer:
1.22 + 0.34 = 1.56.

Explanation:
In the above-given question,
given that,
the two numbers are 1.22 and 0.34.
add the two numbers.
1.22 + 0.34 = 1.56.

Question 3.
0.63 +0.41

Answer:
0.63 + 0.41 = 1.04.

Explanation:
In the above-given question,
given that,
the two numbers are 0.63 and 0.41.
add the two numbers.
0.63 + 0.41 = 1.04.

Question 4.
2.73 – 0.94

Answer:
2.73 – 0.94 = 1.99.

Explanation:
In the above-given question,
given that,
the two numbers are 2.73 and 0.94.
subtract the two numbers.
2.73 – 0.94 = 1.99.

Question 5.
$1.38 – $0.73

Answer:
$1.38 – $0.73 = $0.65.

Explanation:
In the above-given question,
given that,
the two numbers are 1.38 and 0.73.
subtract the two numbers.
1.38 – 0.73 = $0.65.

Question 6.
0.47 – 0.21

Answer:
0.47 – 0.21 = 0.26.

Explanation:
In the above-given question,
given that,
the two numbers are 0.47 and 0.21.
subtract the two numbers.
0.47 – 0.21 = 0.26.

Question 7.
2.02 + 0.8

Answer:
2.02 + 0.8 = 2.82.

Explanation:
In the above-given question,
given that,
the two numbers are 2.02 and 0.8.
add the two numbers.
2.02 + 0.8 = 2.82.

Independent Practice

In 8-11, add or subtract. Use place-value blocks to help.

Question 8.
0.1 + 0.73

Answer:
0.1 + 0.73 = 0.83.

Explanation:
In the above-given question,
given that,
the two numbers are 0.1 and 0.73.
add the two numbers.
0.1 + 0.73 = 0.83.

Question 9.
$1.33 – $0.35

Answer:
$1.33 – $0.35 = 0.98.

Explanation:
In the above-given question,
given that,
the two numbers are 1.33 and 0.35.
subtract the two numbers.
1.33 – 0.35 = $0.98.

Question 10.
$0.37 + $0.47

Answer:
$0.37 + $0.47 = $0.84.

Explanation:
In the above-given question,
given that,
the two numbers are $0.37 and 0.47.
add the two numbers.
0.37 + 0.47 = 0.84.

Question 11.
1.11 +0.89

Answer:
1.11 + 0.89 = 2.00.

Explanation:
In the above-given question,
given that,
the two numbers are 1.11 and 0.89.
add the two numbers.
1.11 + 0.89 = 2.

Problem Solving

Question 12.
Construct Arguments How is adding 4.56 + 2.31 similar to adding $2.31 + $4.56?

Answer:
4.56 + 2.31 is equal to 2.31 + 4.56.

Explanation:
In the above-given question,
given that,
the two numbers are 4.56 and 2.31.
add the two numbers.
2.31 + 4.56 = 6.87.

Question 13.
Write an expression that is represented by the model below.

Answer:
The number is 1.45.

Explanation:
In the above-given question,
given that,
there is one hundred blocks.
1.
there are 4 tens blocks.
0.4.
there are 5 ones blocks.
1 + 0.4 + 0.05 = 1.45.

Question 14.
Without adding the decimals, tell if the sum of 0.46 + 0.25 is less than or greater than 1? Explain.

Answer:
The sum is greater than 1.

Explanation:
In the above-given question,
given that,
0.46 + 0.25 = 0.71.
46 + 25 = 71.
71 is greater than 1.
so the sum is greater than 1.

Question 15.
Number Sense Estimate to decide if the sum of 314 + 175 is more or less than 600.

Answer:
The sum is less than 600.

Explanation:
In the above-given question,
given that,
the two numbers are 314 + 175 = 489.
489 is less than 600.
so the sum is less than 600.

Question 16.
Higher Order Thinking Do you think the difference of 1.4 – 0.95 is less than or greater than 1? Explain.

Answer:
The difference of 1.4 and 0.95 = 0.45.

Explanation:
In the above-given question,
given that,
the two numbers are 1.4 and 0.95 = 489.
0.45 is less than 1.
so the difference is less than 1.

Question 17.
Vocabulary Estimate 53.8 – 27.6. Circle the compatible numbers to substitute.
54 – 28 53 – 28 55 – 27 55 – 25

Answer:
The numbers are 54 – 28 = 26.

Explanation:
In the above-given question,
given that,
the numbers are 53.8 and 27.6.
the numbers are 54-28, 53-28, 55-27, and 55-26.
53.8 is equal to 54.
27.6 is equal to 28.
54 – 28 = 26.

Question 18.
Algebra Write an expression that can be used to find the perimeter of the pool shown to the right. Remember, perimeter is the distance around a figure.

Answer:
The perimeter of the figure = 1250 sq m.

Explanation:
In the above-given question,
given that,
the length and breadth of the rectangle.
l = 50 m.
b = 25 m.
area = l x b.
area of the rectangle = l x b.
50 x 25 = 1250.
so the perimeter of the figure = 1250 sq m.

Assessment Practice

Question 19.
Each set of place-value blocks below represents a decimal.

Part A
What is the sum of the decimals?
Part B
Explain how you found your answer.

Answer:
0.3 + 0.01 = 0.31.
0.7 + 0.05 = 0.75.

Explanation:
In the above-given question,
given that,
there are 3 tens blocks.
there are 2 one’s blocks.
0.3 + 0.01 = 0.31.
there are 7 tens blocks.
there are 5 ones blocks.
0.7 + 0.05 = 0.75.

Lesson 2.4 Use Strategies to Add Decimals

Activity

Solve & Share

Mr. Davidson has two sacks of potatoes. The first sack weighs 11.39 pounds. The second sack weighs 14.27 pounds. How many pounds of potatoes does Mr. Davidson have in all? Solve this problem any way you choose.

You can generalize what you know about whole number addition to decimal addition

Look Back! How is adding decimals similar to adding whole numbers?

Answer:
The number of pounds of potatoes does Mr. Davidson has in all = 25.66.

Explanation:
In the above-given question,
given that,
Mr. Davidson has two sacks of potatoes.
The first sack weighs 11.39 pounds.
The second sack weighs 14.27 pounds.
11.39 + 14.27 = 25.66.
so the number of pounds of potatoes does Mr. Davidson has in all = 25.66.

Visual Learning Bridge

Essential Question How Can You Add Decimals?

A.
A swim team participated in a relay race. The swimmers’ times for each leg of the race were recorded in a table. What was the combined time for Caleb and Bradley’s legs of the relay race?

You can find 21.39 + 21.59, but estimate first: 21 + 22 = 43.

B.
Step 1
Just like with whole numbers, line up the addends by place value. The decimal points will also line up.

C.
Step 2
Use what you know about partial sums to add the hundredths, tenths, ones, and tens.

The combined time for Caleb and Bradley was 42.98 seconds. The sum is close to the estimate.

Adding decimals is just like adding whole numbers!

Convince Me! Critique Reasoning André said the last two legs of the race took 3,938 seconds. What mistake did he make?

Another Example
Carson ran 7.81 miles last week. He ran 14 miles this week. How many miles did he run in the two weeks?

Use properties and a number line to find the sum.
7.81 + 14 = 14 + 7.81 (Commutative Property)
= 14 + (7 + 0.81)
= (14 + 7) + 0.81 (Associative Property)
= 21 + 0.81
= 21.81

Carson ran 21.81 miles in all.
A number line can help you visualize adding decimals.

Guided Practice

Do You Know How?

Question 1.
How is adding 21.39 + 21.59 like adding 2,139 + 2,159?

Answer:
The sum is the same.

Explanation:
In the above-given question,
given that,
the two numbers are 21.39 and 21.59.
add two numbers.
21.39 + 21.59 = 42.98.
2139 + 2159 = 4298.
so both are the same.

Do You Understand?

In 2-5, use place value and properties to find each sum.

Question 2.
0.82 + 4.21

Answer:
0.82 + 4.21 = 5.03.

Explanation:
In the above-given question,
given that,
the two numbers are 0.82 and 4.21.
add the two numbers.
0.82 + (4 + 0.21).
(0.82 + 0.21) + 4.
1.03 + 4.
5.03.

Question 3.
9.1 + 7.21

Answer:
9.1 + 7.21 = 16.31.

Explanation:
In the above-given question,
given that,
the two numbers are 9.1 and 7.21.
add the two numbers.
9.1 + (7 + 0.21).
(9.1 + 0.21) + 7.
9.31 + 7.
16.31.

Question 4.
0.26 + 8.3

Answer:
0.26 + 8.3 = 8.56.

Explanation:
In the above-given question,
given that,
the two numbers are 0.26 and 8.3.
add the two numbers.
0.26 + (8 + 0.3).
(0.26 + 0.3) + 8.
0.56 + 8.
8.56.

Question 5.
4.98 + 3.02

Answer:
4.98 + 3.02 = 8.00.

Explanation:
In the above-given question,
given that,
the two numbers are 4.98 and 3.02.
add the two numbers.
4.98 + (3 + 0.2).
(4.98 + 0.2) + 3.
5 + 3.
8.

Independent Practice

You can estimate first to be sure your answers are reasonable.

Leveled Practice In 6-12, use place value and properties of operations to find each sum.

Question 6.

Answer:
1.03 + 0.36 = 1.39.

Explanation:
In the above-given question,
given that,
the two numbers are 1.03 and 0.36.
add the two numbers.
0.36 + (1 + 0.3).
(0.36 + 0.3) + 1.
0.96 + 1.
1.03 + 0.36 = 1.39.

Question 7.

Answer:
6.9 + 2.8 = 9.7.

Explanation:
In the above-given question,
given that,
the two numbers are 6.9 and 2.8.
add the two numbers.
6.9 + (2 + 0.8).
(6.9 + 0.8) + 2.
7.7 + 2.
6.9 + 2.8 = 9.7

Question 8.

Answer:
45.08 + 2.01 = 47.09.

Explanation:
In the above-given question,
given that,
the two numbers are 45.08 and 2.01.
add the two numbers.
45.08 + (2 + 0.1).
(45.08 + 0.1) + 2.
45.09 + 2.
45.08 + 2.01 = 47.09.

Question 9.

Answer:
2.00 + 0.78 = 2.78.

Explanation:
In the above-given question,
given that,
the two numbers are 2.00 and 0.78.
add the two numbers.
0.78 + (1 + 1.0).
(0.78 + 1.0) + 1.
1.78 + 1.
2.00 + 0.78 = 2.78.

Question 10.
$271.90 + $34.22

Answer:
$271.90 + $34.22 = $316.12.

Explanation:
In the above-given question,
given that,
the two numbers are $271.90 and $34.22.
add the two numbers.
271.90 + (34 + 0.22).
(271.90 + 0.22) + 34.
272.12 + 34.
$271.90 + $34.22 = $316.12.

Question 11.
7.2 + 3.96 + 8.8

Answer:
7.2 + 3.96 + 8.8 = 19.96.

Explanation:
In the above-given question,
given that,
the two numbers are 7.2, 88, and 3.96.
add the three numbers.
7.2 + (88 + 3.96).
(7.2 + 3.96) + 88.
1.16 + 88.
7.2 + 3.96 + 8.8 = 19.96.

Question 12.
16.62 + 4 + 2.38

Answer:
16.62 + 4 + 2.38 = 23.

Explanation:
In the above-given question,
given that,
the two numbers are 16.62,4 and 2.38.
add the two numbers.
16.62 + (4 + 2.38).
(16.62 + 4) + 2.38.
16.66 + 2.38.
16.62 + 4 + 2.38  = 23.

Problem Solving

Question 13.
A farmer sold 53.2 pounds of carrots and 29.4 pounds of asparagus to a restaurant. How many pounds of these two vegetables did the restaurant buy?

Answer:
The number of pounds of these two vegetables did the restaurant buy = 82.6.

Explanation:
In the above-given question,
given that,
A farmer sold 53.2 pounds of carrots and 29.4 pounds of asparagus to a restaurant.
the two numbers are 53.2 and 29.4.
53.2 + 29.4 = 82.6.
so the number of pounds of these two vegetables did the restaurant buy = 82.6.

For 14 and 15, use the table.

Question 14.
enVision® STEM Which two cities have the greatest combined rainfall for a typical year?

Answer:
The two cities have the greatest combined rainfall for a typical year =

Explanation:
In the above-given question,
given that,
the cities that have the rainfall are:
Macon 45 inches.
Boise 12.19 inches.
Caribou 37.44 inches.
Springfield 44.97 inches.
the two cities that have the greater rainfall are Caribou and Springfield.

Question 15.
Number Sense Which location had less than 45 inches of rain but more than 40 inches of rain?

Answer:
The location that had less than 45 inches of rain but more than 40 inches of rain = Springfield, MO.

Explanation:
In the above-given question,
given that,
the cities that have the rainfall are:
Macon 45 inches.
Boise 12.19 inches.
Caribou 37.44 inches.
Springfield 44.97 inches.
44.97 < 45.
44.97 > 40.
so the location that had less than 45 inches of rain but more than 40 inches of rain = Springfield, MO.

Question 16.
Higher Order Thinking Tim earned $16 babysitting and $17.50 mowing a lawn. He paid $8.50 for a movie and bought a small popcorn for $1.95. Write an expression to show how much money he has left.

Answer:
The expression to show how much money he has left = $43.95.

Explanation:
In the above-given question,
given that,
Tim earned $16 babysitting and $17.50 mowing a lawn.
He paid $8.50 for a movie and bought a small popcorn for $1.95.
$16 + $17.50 + $8.50 + $1.95.
$16 + $17.50 = 33.5.
$8.50 + $1.95 = 10.45.
$33.5 + 10.45 = $43.95.

Question 17.
Critique Reasoning Juan adds 3.8 + 4.6 and gets a sum of 84. Is his answer correct? Tell how you know.

Answer:
No, he was not correct.

Explanation:
In the above-given question,
given that,
Juan adds 3.8 + 4.6 and gets a sum of 84.
3.8 + 4.6 = 8.4.
the sum is the same but the values are not the same.
so he was not correct.

Assessment Practice

Question 18.
Choose all expressions that are equal to 12.9.
0.02 + 12 + 0.88
0.06 + 12.03
11.9 + 1
6.2 + 3.4 + 2.3
3.01 + 2.01 + 7.7

Answer:
Options A and B are correct.

Explanation:
In the above-given question,
given that,
the expressions are 0.02 + 12 + 0.88.
0.02 + 12 = 12.02.
12.02 + 0.88 = 12.90.
11.9 + 1 = 12.9.
so the options A and B are correct.

Question 19.
Choose all expressions that are equal to 16.02.
16 + 0.02
3.42 + 8 + 4.6
16.01 + 1
12.06 + 3.14
7.36 +8.66

Answer:
Options A and B are correct.

Explanation:
In the above-given question,
given that,
the expressions are 16 + 0.02.
16.02.
3.42 + 8 + 4.6 = 16.02.
so options A and B are correct.

Lesson 2.5 Use Strategies to Subtract Decimals

Solve & Share
Ms. Garcia is an electrician and has a length of wire that is 32.7 meters long. She has another length of wire that is 15.33 meters long. How much longer is one wire than the other? Solve this problem any way you choose.

You can use what you know about whole number subtraction to subtract decimals.

Answer:
The longer is one wire than the other = 17.37 meters.

Explanation:
In the above-given question,
given that,
Ms. Garcia is an electrician and has a length of wire that is 32.7 meters long.
She has another length of wire that is 15.33 meters long.
15.33 + 32.7 = 48.03.
32.7 – 15.33 = 17.37.
so the longer is one wire than the other = 17.37 meters.

Look Back! Use Structure How can you use the relationship between addition and subtraction to check your work above?

Visual Learning Bridge

Essential Question
How Can You Subtract Decimals?

A.
What is the difference in the wingspans of the two butterflies?

Estimate before you find the exact answer.
6 – 4 = 2

B.
You can use a number line to subtract.

C.
Here’s one way to subtract using partial differences.
Find 5.92 – 4.37

5.92 – 4.37 = 1.55

Convince Me! Be Precise in the solution above, what strategies can you use to subtract the 7 hundredths?

Guided Practice

Do You Understand?

Question 1.
How can you use addition to check that 1.55 cm is the difference in the wingspans of the two butterflies?

Answer:
The answers are 3.00 and 1.45.

Explanation:
In the above-given question,
given that,
the two numbers are 3.00 and 1.45.
3.00 – 1.45 = 1.55.
so the difference is 1.55.
the answers are 3.00 and 1.45.

Question 2.
Maria rewrote 45.59 – 7.9 as 45.59 – 7.90. Did she change the value of 7.9 by placing the zero after 7.9? Why or why not?

Answer:
Yes, she changed the value of 7.9 by placing the zero.

Explanation:
In the above-given question,
given that,
Maria rewrote 45.59 – 7.9 as 45.59 – 7.90.
45.59 – 7.9 = 37.69.
yes, she changed the value of 7.9 by placing the zer0.

Do You Know How?

In 3-10, subtract the decimals.

Question 3.
16.82 – 5.21

Answer:
16.82 – 5.21 = 11.61.

Explanation:
In the above-given question,
given that
the two numbers are 16.82 and 5.21.
subtract the numbers.
16.82 – 5.21 = 11.61.

Question 4.
7.21 – 6.1

Answer:
7.21 – 6.1 = 1.1.

Explanation:
In the above-given question,
given that
the two numbers are 7.21 and 6.1.
subtract the numbers.
7.21 – 6.1 = 1.1.

Question 5.
23.06 – 8.24

Answer:
23.06 – 8.24 = 14.82

Explanation:
In the above-given question,
given that
the two numbers are 23.06 and 8.24.
subtract the numbers.
23.06 – 8.24 = 14.82.

Question 6.
$4.08 – $2.12

Answer:
$4.08 – $2.12 = $1.96.

Explanation:
In the above-given question,
given that
the two numbers are $4.08 and $2.12.
subtract the numbers.
$4.08 – $2.12 = $1.96.

Question 7.
56.8 – 2.76

Answer:
56.8 – 2.76 = 54.04.

Explanation:
In the above-given question,
given that
the two numbers are 56.8 and 2.76.
subtract the numbers.
56.8 – 2.76 = 54.04.

Question 8.
$43.80 – $16.00

Answer:
$43.80 – $16.00 = $27.8.

Explanation:
In the above-given question,
given that
the two numbers are $43.80 and $16.00.
subtract the numbers.
$43.80 – $16.00 = $27.8.

Question 9.
22.4 – 10.7

Answer:
22.4 – 10.7 = 11.7.

Explanation:
In the above-given question,
given that
the two numbers are 22.4 and 10.7.
subtract the numbers.
22.4 – 10.7 = 11.7.

Question 10.
$36.40 – $21.16

Answer:
$36.40 – $21.16 = $15.24.

Explanation:
In the above-given question,
given that
the two numbers are $36.40 and $21.16.
subtract the numbers.
$36.40 – $21.16 = $15.24.

Independent Practice

Leveled Practice In 11-26, find the difference.

Question 11.
7.8 – 4.9

Answer:
7.8 – 4.9 = 2.9.

Explanation:
In the above-given question,
given that
the two numbers are 7.8 and 4.9.
subtract the numbers.
7.8 – 4.9 = 2.9.

Question 12.
$20.60 – $14.35

Answer:
$20.60 – $14.35 = $6.25.

Explanation:
In the above-given question,
given that
the two numbers are $20.60 and $14.35.
subtract the numbers.
$20.60 – $14.35 = $6.25.

Question 13.
43.90 – 7.52

Answer:
43.90 – 7.52 = 36.38.

Explanation:
In the above-given question,
given that
the two numbers are 43.90 and 7.52.
subtract the numbers.
43.90 – 7.52 = 36.38.

Question 14.
65.90 – 28.38

Answer:
65.90 – 28.38 = 37.52.

Explanation:
In the above-given question,
given that
the two numbers are 65.90 and 28.38.
subtract the numbers.
65.90 – 28.38 = 37.52.

Question 15.
15.03 – 4.12

Answer:
15.03 – 4.12 = 10.91.

Explanation:
In the above-given question,
given that
the two numbers are 15.03 and 4.12.
subtract the numbers.
15.03 – 4.12 = 10.91.

Question 16.
13.9 – 3.8

Answer:
13.9 – 3.8 = 10.1.

Explanation:
In the above-given question,
given that
the two numbers are 13.9 and 3.8.
subtract the numbers.
13.9 – 3.8 = 10.1.

Question 17.
65.18 – 12.05

Answer:
65.18 – 12.05 = 53.13.

Explanation:
In the above-given question,
given that
the two numbers are 65.18 and 12.05.
subtract the numbers.
65.18 – 12.05 = 53.13.

Question 18.
$52.02 – $0.83

Answer:
$52.02 – $0.83 = $51.19.

Explanation:
In the above-given question,
given that
the two numbers are $52.02 and $0.83.
subtract the numbers.
$52.02 – $0.83 = $51.19.

Question 19.
7.09 – 3.65

Answer:
7.09 – 3.65 = 3.44.

Explanation:
In the above-given question,
given that
the two numbers are 7.09 and 3.65.
subtract the numbers.
7.09 – 3.65 = 3.44.

Question 20.
34.49 – 12.61

Answer:
34.49 – 12.61 = 21.88

Explanation:
In the above-given question,
given that
the two numbers are 34.49 and 12.61.
subtract the numbers.
34.49 – 12.61 = 21.88.

Question 21.
85.22 – 43.5

Answer:
85.22 – 43.5 = 41.72.

Explanation:
In the above-given question,
given that
the two numbers are 85.22 and 43.5.
subtract the numbers.
85.22 – 43.5 = 41.72.

Question 22.
$10.05 – $4.50

Answer:
$10.05 – $4.50 = $5.55.

Explanation:
In the above-given question,
given that
the two numbers are $10.05 and $4.50.
subtract the numbers.
$10.05 – $4.50 = $5.55.

Question 23.
5.27 – 3.4

Answer:
5.27 – 3.4 = 1.87.

Explanation:
In the above-given question,
given that
the two numbers are 5.27 and 3.4.
subtract the numbers.
5.27 – 3.4 = 1.87

Question 24.
23.6 – 8.27

Answer:
23.6 – 8.27 = 15.33.

Explanation:
In the above-given question,
given that
the two numbers are 23.6 and 8.27.
subtract the numbers.
23.6 – 8.27 = 15.33.

Question 25.
8.04 – 0.3

Answer:
8.04 – 0.3 = 7.74.

Explanation:
In the above-given question,
given that
the two numbers are 8.04 and 0.3.
subtract the numbers.
8.04 – 0.3 = 7.74.

Question 26.
$21.37 – $10.95

Answer:
$21.37 – $10.95 = $10.42.

Explanation:
In the above-given question,
given that
the two numbers are $21.37 and $10.95.
subtract the numbers.
$21.37 – $10.95 = $10.42.

Problem Solving

Question 27.
Algebra The Pyramid of Khafre measured 143.5 meters high. The Pyramid of Menkaure measured 65.5 meters high. Write and solve an equation to find d, the difference in the heights of these two pyramids.

Answer:
The difference in the heights of these two pyramids = 143.5.

Explanation:
In the above-given question,
given that,
The Pyramid of Khafre measured 143.5 meters high.
The Pyramid of Menkaure measured 65.5 meters high.
143.5 – 65.5 = 78.
78 + 65.5 = 143.5.
so the difference in the heights of these two pyramids = 143.5.

Question 28.
Higher Order Thinking Jonah bought a 1.5-liter bottle of seltzer. He used 0.8 liter of seltzer in some punch. Which is greater, the amount he used or the amount he has left? Explain how you decided

Answer:
The amount he used is greater.

Explanation:
In the above-given question,
given that,
Johan bought a 1.5-liter bottle of seltzer.
He used 0.8 liters of seltzer in some punch.
1.5 – 0.8 = 0.7.
the amount he used is greater.

Question 29.
Sue subtracted 2.9 from 20.9 and got 1.8. Explain why this is not reasonable.

Answer:
2.9 – 20.9 = 18.

Explanation:
In the above-given question,
given that,
Sue subtracted 2.9 from 20.9 and got 1.8.
2.9 and 2.9 is the difference and got 1.8.
2.9 – 20.9 = 18.
so this is not reasonable.

Question 30.
Make Sense and Persevere Abe had $156.43 in his bank account at the beginning of the month. He made the two withdrawals shown in his check register. How much money does he have left in his bank account? He must have at least $100 in his account by the end of the month or he will be charged a fee. How much money does he need to deposit to avoid being charged a fee?

Answer:
The money does he need to deposit to avoid being charged a fee = $60.59.

Explanation:
In the above-given question,
given that,
Abe had $156.43 in his bank account at the beginning of the month.
He made the two withdrawals shown in his check register.
$156.43 + $17.85 = $174.28.
100 – 39.41 = 60.59.
so the money does he need to deposit to avoid being charged a fee = $60.59.

Assessment Practice

Question 31.
Circle all of the subtraction problems with a difference of 1.65.
27.30 – 16.65
11.23 – 9.58
40.4 – 23.9
12.68 – 2.03
21.74 – 20.09

Answer:
The subtraction problems with a difference of 1.65 is second and fifth expression.

Explanation:
In the above-given question,
given that,
27.30 – 16.65 = 10.65.
11.23 – 9.58 = 1.65.
40.4 – 23.9 = 16.5.
12.68 – 2.03 = 10.65.
21.74 – 20.09 = 1.65.

Lesson 2.6 Model with Math

Activity

Problem Solving

Solve & Share
At a baseball game, Sheena bought a sandwich for $6.95 and two pretzels for $2.75 each. She paid with a $20 bill. How much change did she receive? Solve this problem any way you choose. Use bar diagrams to help.

Answer:
The change did she receive = $10.3.

Explanation:
In the above-given question,
given that,
At a baseball game, Sheena bought a sandwich for $6.95 and two pretzels for $2.75 each.
She paid with a $20 bill.
$6.95 + $2.75 = $9.7.
$20 – $9.7 = $10.3.
so the change did she recieve = $10.3.

Thinking Habits

Be a good thinker!
• These questions can help you. How can I use math I know to help solve the 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 other way can you represent this problem situation?

Visual Learning Bridge

Essential Question How Can You Represent a Problem with Bar Diagrams?

A.
Monica wants to buy all of the art supplies shown on this sign. She has a coupon for $5.50 off the cost of her purchases. What will Monica’s total cost be after the discount?

Model with math means you apply math you have learned to solve problems.

What do I need to do to solve the problem?
I need to find Monica’s cost for the art supplies.

Here’s my thinking

B.
How can I model with math?
I can
• use the math I know to help solve the problem.
• find and answer any hidden questions.
• use bar diagrams and equations to represent and solve this problem.

C.
I will use bar diagrams to represent this situation.

$59.95 + $24.95 + $9.75 + $13.50 = $108.15
The total cost before the discount is $108.15.

$108.15 – $5.50 = $102.65
Monica’s cost after the discount is $102.65.

Convince Me! Model with Math How could you decide if your answer makes sense?

Guided Practice
Model with Math Nate has $30.50. He wants to buy his dog a sweater that costs $15, a toy that costs $3.79, and a leash that costs $14.79. How much more money does he need?

Answer:
The more money does he need = $3.08.

Explanation:
In the above-given question,
given that,
Nate has $30.50.
He wants to buy his dog a sweater that costs $15.
a toy that costs $3.79, and a leash that costs $14.79.
$15 + $3.79 + $14.79 = 33.58.
33.58 – 30.50 = 3.08.
so the more money does he need = $3.08.

When you model with math you use the math you already know to solve new problems!

Question 1.
What do you need to find before you can solve the problem?

Answer:
The more money does he need = $3.08.

Explanation:
In the above-given question,
given that,
Nate has $30.50.
He wants to buy his dog a sweater that costs $15.
a toy that costs $3.79, and a leash that costs $14.79.
$15 + $3.79 + $14.79 = 33.58.
33.58 – 30.50 = 3.08.
so the more money does he need = $3.08.

Question 2.
Draw bar diagrams to represent the problem and then solve the problem. Show the equations you used to solve the problem.

Answer:
The more money does he need = $3.08.

Explanation:
In the above-given question,
given that,
Nate has $30.50.
He wants to buy his dog a sweater that costs $15.
a toy that costs $3.79, and a leash that costs $14.79.
$15 + $3.79 + $14.79 = 33.58.
33.58 – 30.50 = 3.08.
so the more money does he need = $3.08.

Independent Practice

Model with Math
Luz Maria has $15. She buys a ticket to a movie and a smoothie. How much money does she have left?

Question 3.
What do you need to find before you can solve the problem?

Answer:
The money does she have left = 412.35.

Explanation:
In the above-given question,
given that,
Luz Maria has $15.
She buys a ticket to a movie and a smoothie.
$9.50 + $2.85 = $12.35.
so the money does she have left = $12.35.

Question 4.
Draw two bar diagrams to represent the problem.

Answer:
The money does she have left = 412.35.

Explanation:
In the above-given question,
given that,
Luz Maria has $15.
She buys a ticket to a movie and a smoothie.
$9.50 + $2.85 = $12.35.
so the money does she have left = $12.35.

Question 5.
What is the solution to the problem? Show the equations you used to solve the problem.

Answer:
The money does she have left = 412.35.

Explanation:
In the above-given question,
given that,
Luz Maria has $15.
She buys a ticket to a movie and a smoothie.
$9.50 + $2.85 = $12.35.
so the money does she have left = $12.35.

Problem Solving

Performance Task

School Trip
Audrey is saving for a school trip. She needs $180 for the bus tickets, $215 for the hotel, and $80 for meals. The table shows how much money she and her sister, Kelsey, have saved over a 4-month period. How much more money does Audrey need for the trip?

Question 6.
Make Sense and Persevere What are you trying to find?

Answer:
The more money does Audrey needs for the trip = $143.

Explanation:
In the above-given question,
given that,
Audrey is saving for a school trip.
She needs $180 for the bus tickets, $215 for the hotel, and $80 for meals.
$68 + $31.50 + $158 + $74.75 = 332.25.
180 + 215 + 80 = 475.
475 – 332 = 143.
so the more money does Audrey needs for the trip = $143.

Question 7.
Construct Arguments Should you multiply Audrey’s savings for September by 4 since there are 4 months? Explain.

Answer:
No, we cannot multiply by 4.

Explanation:
In the above-given question,
given that,
Audrey’s savings for September is $68.
$68 x 4 = $272.
so we cannot multiply by 4.

You can model with math by using what you know about adding and subtracting whole numbers to add and
subtract decimals.

Question 8.
Model with Math Draw bar diagrams to represent the total cost of Audrey’s trip and the total she has saved. Then find the total cost and total savings.

Answer:
The total cost of Audrey’s trip is 332.25.

Explanation:
In the above-given question,
given that,
The total she has saved = 332.25.
the total cost of Audrey’s trip = 332.25.

Question 9.
Model with Math Write and solve an equation to determine how much more money Audrey needs for the trip.

Answer:
The more money Audrey needs for the trip = 475.

Explanation:
In the above-given question,
given that,
Audrey is saving for a school trip.
She needs $180 for the bus tickets, $215 for the hotel, and $80 for meals.
$68 + $31.50 + $158 + $74.75 = 332.25.
180 + 215 + 80 = 475.
475 – 332 = 143.
so the more money does Audrey needs for the trip = $143.

Topic 2 Fluency Review Activity

Point & Tally

Find a partner. Get paper and a pencil. Each partner chooses either light blue or dark blue.
Partner 1 and Partner 2 each point to a black number at the same time. Each partner subtracts the lesser number from the greater number.
If the answer is on your color, you get a tally mark. Work until one partner has twelve tally marks.

Topic 2 Vocabulary Review

Glossary

Understand Vocabulary

Word List
• Associative Property of Addition
• Commutative Property of Addition
• compatible numbers
• compensation
• equivalent decimals
• inverse operations

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

Question 1.
When you adjust one number and change another number in the problem to make a computation easier, you use ____.

Answer:
We will use the compensation.

Explanation:
In the above-given question,
given that,
when you adjust one number and change another number in the problem to make computation easier.
for example:
2 + 3 + 4.
2 + (3 + 4).
(2 + 3) + 4.
5 + 4 = 9.
so we will use the compensation.

Question 2.
You can replace the values in a problem with _____ so that it’s easier to use mental math to complete the computation.

Answer:
We can replace the values in a problem with compatible numbers.

Explanation:
In the above-given question,
given that,
you can replace the values in a problem with compatible numbers.
for example:
3.9 = 4.
4.8 = 5.
4 + 5 = 9.
so we can replace the values in a problem with compatible numbers.

Question 3.
To align decimal points in a decimal addition problem, annex zeros to write ____ so that all addends have the same number of decimal places.

Answer:
To align decimal points in a decimal addition problem, annex zeros to write compensation so that all addends have the same number of decimal places.

Explanation:
In the above-given question,
given that,
for example:
2.99 + 3.79.
3 + 4 = 7.

Question 4.
Because of the ____, I know that 477.75 + (76.89 + 196.25) = (76.89 + 196.25) + 477.75 without adding.

Answer:
We can use associative and commutative properties.

Explanation:
In the above-given question,
given that,
477.75 + (76.89 + 196.25).
(76.89 + 196.25) + 477.75.
so we can use associative and commutative properties.

Question 5.
Cross out the numbers below that are NOT equivalent to 500.0.
500.00 5 × 10 5 × 10² 50.05 500.500

Answer:
500, 5 x 10 x 10.

Explanation:
In the above-given question,
given that,
the numbers are 500, 5 x 10, 5 x 10 x 10, 50.05.
5 x 10 x 10 = 500.
so the numbers that are not equivalent to 500.

Question 6.
Cross out the numbers below that are NOT equivalent to 53.2 + 16.8.
7 × 10¹ 0.070 7.0 7 × 7 (7 × 10) + (0 × 1)

Circle the problem that uses compensation.

Question 7.
32.7 + 15.6 = 32.6 + 15.7 45.7 + 26.2 = 45.7 + 26.3 – 0.1

Answer:
Yes, this problem uses the compensation.

Explanation:
In the above-given question,
given that,
the expressions are 32.7 + 15.6 = 32.6 + 15.7.
45.7 + 26.2 = 45.7 + 26.3 – 0.1.
so this uses a compensation.

Question 8.
14.24 – 11.8 = 14.24 – 12 + 0.2 168.3 – 53.8 = 168.3 – 53.4 – 0.4

Answer:
Yes this problem also uses the compensation.

Explanation:
In the above-given question,
given that,
the expressions are 14.24 – 11.8.
14.24 – 12 + 0.2.
168.3 – 53.8 = 168.3 – 53.4 – 0.4.
so this problem also uses the compensation method.

Use Vocabulary in Writing

Question 9.
Explain how the Commutative Property of Addition, the Associative Property of Addition and mental math can help you find 75.2 + (57.376 + 24.8). What is the sum?

Answer:
The sum is 157.376.

Explanation:
In the above-given question,
given that,
by using the Associative property of addition and Commutative property of addition.
75.2 + (57.376 + 24.8).
(75.2 + 57.376) + 24.8.
132.576 + 24.8 = 157.376.
so the sum is 157.376.

Topic 2 Reteaching

Set A
pages 45-48
Add 15.3 + 1.1 + 1.7 using mental math.

15.3 and 1.7 are compatible numbers because they are easy to calculate mentally.
The Commutative Property of Addition enables us to add in any order.
15.3 + 1.1 + 1.7 = 15.3 + 1.7 + 1.1
= 17.0 + 1.1
= 18.1

Remember that you can use compatible numbers or compensation to find sums and differences.
Use mental math to add or subtract.

Question 1.
8.6 + 23.4 + 1.4

Answer:
8.6 + 23.4 + 1.4 = 33.4.

Explanation:
In the above-given question,
given that,
by using the Associative property of addition and Commutative property of addition.
8.6 + (23.4 + 1.4).
(8.6 + 23.4) + 1.4.
32 + 1.4 = 33.4.

Question 2.
27 – 9.9

Answer:
27 – 9.9 = 17.1.

Explanation:
In the above-given question,
given that,
by using the Associative property of addition and Commutative property of subtraction.
27 – 9.9 = 17.1.

Question 3.
13.5 + 5.7 + 36.5

Answer:
13.5 + 5.7 + 36.5 = 55.7.

Explanation:
In the above-given question,
given that,
by using the Associative property of addition and Commutative property of addition.
13.5 + (5.7 + 36.5).
(13.5 + 5.7) + 36.5.
19.2 + 36.5 = 55.7.

Question 4.
205.4 – 99.7

Answer:
205.4 – 99.7 = 105.7.

Explanation:
In the above-given question,
given that,
the two numbers are 205.4 and 99.7.
subtract the numbers.
205.4 – 99.7 = 105.7.

Set B
pages 49-52

Estimate 22.4 – 16.2.

22.4 – 16.2 is about 5.

Remember that compatible numbers can give a different estimate than rounding. Estimate each sum or difference.

Question 1.
358 + 293

Answer:
358 + 293 = 651.

Explanation:
In the above-given question,
given that,
the two numbers are 358 + 293.
add the two numbers.
358 + 293 = 651.

Question 2.
15.01 – 4.4

Answer:
15.01 – 4.4 = 10.61.

Explanation:
In the above-given question,
given that,
the two numbers are 15.01 and 4.4.
subtract the two numbers.
15.01 – 4.4 = 10.61.

Question 3.
80.01 + 2.89

Answer:
80.01 + 2.89 = 82.9.

Explanation:
In the above-given question,
given that,
the two numbers are 80.01 and 2.89.
add the two numbers.
80.01 + 2.89 = 82.9.

Question 4.
25,003 – 12,900

Answer:
25003 – 12900 = 12103.

Explanation:
In the above-given question,
given that,
the two numbers are 25003 and 12900.
subtract the two numbers.
25003 – 12900 = 12103.

Set C
pages 53-56

Use place-value blocks to subtract 1.86 – 0.95.

Show 1 flat, 8 longs, and 6 small squares to represent 1.86.

Remove 9 tenths and 5 hundredths, regrouping as needed.

Count what is left.
1.86 – 0.95 = 0.91

Add or subtract. Use place-value blocks for help.

Question 1.
0.02 + 0.89

Answer:
0.02 + 0.89 = 0.91.

Explanation:
In the above-given question,
given that,
the two numbers are 0.02 and 0.89.
add the two numbers.
0.02 + 0.89 = 0.91.

Question 2.
0.67 – 0.31

Answer:
0.67 – 0.31 = 0.36.

Explanation:
In the above-given question,
given that,
the two numbers are 15.01 and 4.4.
subtract the two numbers.
0.67 – 0.31 = 0.36.

Question 3.
0.34 + 0.34

Answer:
0.34 + 0.34 = 0.68.

Explanation:
In the above-given question,
given that,
the two numbers are 0.34 and 0.34.
add the two numbers.
0.34 + 0.34 = 0.68.

Question 4.
0.81 – 0.78

Answer:
0.81 – 0.78 = 0.03.

Explanation:
In the above-given question,
given that,
the two numbers are 0.81 and 0.78.
subtract the two numbers.
0.81 – 0.78 = 0.03.

Set D
pages 57-60, 61-64

Lucy bought 2.12 pounds of pears and 3 pounds of apples. Find how many more pounds of apples than pears Lucy bought.
Use place-value blocks to help.

Add or subtract. Use place-value blocks to help.

Question 1.
7.06 + 0.85

Answer:
7.06 + 0.85 = 7.91.

Explanation:
In the above-given question,
given that,
the two numbers are 7.06 and 0.85.
add the two numbers.
7.06 + 0.85 = 7.91.

Question 2.
24.07 – 5.31

Answer:
24.07 – 5.31 = 18.76.

Explanation:
In the above-given question,
given that,
the two numbers are 24.07 and 5.31.
subtract the two numbers.
24.07 – 5.31 = 18.76.

Question 3.
51.92 – 28.03

Answer:
51.92 – 28.03 = 23.89.

Explanation:
In the above-given question,
given that,
the two numbers are 51.92 and 28.03.
subtract the two numbers.
51.92 – 28.03 = 23.89.

Question 4.
8.71 – 0.4

Answer:
8.71 – 0.4 = 8.31.

Explanation:
In the above-given question,
given that,
the two numbers are 8.71 and 0.4.
subtract the two numbers.
8.71 – 0.4 = 8.31.

Question 5.
98 + 3.79

Answer:
98 + 3.79 = 101.79.

Explanation:
In the above-given question,
given that,
the two numbers are 98 and 3.79.
add the two numbers.
98 + 3.79 = 101.79.

Question 6.
Talia measured two strings. The green string was 2.37 cm long. The blue string was 4 cm long. How many centimeters longer was the blue string than the green string?

Answer:
The number of centimeters longer was the blue string than the green string = 1.63 cm long.

Explanation:
In the above-given question,
given that,
Talia measured two strings.
The green string was 2.37 cm long.
The blue string was 4 cm long.
4 – 2.37 = 1.63.
so the number of centimeters longer was the blue string than the green string = 1.63 cm long.

Set E
pages 65-68
Think about these questions to help you model with math.

Thinking Habits
• How can I use math I know to help solve the 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 a good model clearly shows how the quantities in the problem are related. Alberto ran 15.6 km on Monday, 12.8 km on Tuesday, and 6.5 km on Wednesday. Dennis ran 11.25 km on Monday, 14.6 km on Tuesday, and 8 km on Wednesday. Who ran farther? How much farther?

Question 1.
What do you need to find before you can solve the problem?

Answer:
Alberto ran farther than Dennis.

Explanation:
In the above-given question,
given that,
Alberto ran 15.6 km on Monday, 12.8 km on Tuesday, and 6.5 km on Wednesday.
Dennis ran 11.25 km on Monday, 14.6 km on Tuesday, and 8 km on Wednesday.
15.6 + 12.8 + 6.5 = 34.9.
11.25 + 14.6 + 8 = 33.85.
Alberto ran farther than Dennis.
34.9 – 33.85 = 1.05.

Question 2.
Write equations to model this problem. Then solve the problem.

Answer:
Alberto ran farther than Dennis.

Explanation:
In the above-given question,
given that,
Alberto ran 15.6 km on Monday, 12.8 km on Tuesday, and 6.5 km on Wednesday.
Dennis ran 11.25 km on Monday, 14.6 km on Tuesday, and 8 km on Wednesday.
15.6 + 12.8 + 6.5 = 34.9.
11.25 + 14.6 + 8 = 33.85.
Alberto ran farther than Dennis.
34.9 – 33.85 = 1.05.

Topic 2 Assessment Practice

Question 1.
Kayla’s dollhouse has 15.15 square feet downstairs and 6.45 square feet upstairs. What is the estimated total area if you round the decimals to the nearest tenth?
A. 21.0
B. 21.6
C. 21.7
D. 22.0

Answer:
Option C is correct.

Explanation:
In the above-given question,
given that,
Kayla’s dollhouse has 15.15 square feet downstairs and 6.45 square feet upstairs.
15.15 = 15.2.
6.45 = 6.5.
15.2 + 6.5 = 21.7.
so option C is correct.

Question 2.
Estimate the sum of $12.15, $16.85, and $1.74 by rounding each number to the nearest tenth.
A. $30.70
B. $30.80
C. $31.00
D. $30.00

Answer:
Option B was correct.

Explanation:
In the above-given question,
given that,
estimate the sum by rounding each number to the nearest tenth.
$12.15 + $16.85 + $1.74 = $30.80.
so option B was correct.

Question 3.
What is the sum of 2.65 +3.78?
A. 5.33
B. 5.43
C. 6.33
D. 6.43

Answer:
Option D is correct.

Explanation:
In the above-given question,
given that,
the two numbers are 2.65 and 3.78.
add the two numbers.
2.65 + 3.78 = 6.43.
so option D is correct.

Question 4.
Which decimal makes this equation true? 4.95 + = 12.1
A. 7.15
B. 7.85
C. 8.15
D. 8.85

Answer:
Option  A is correct.

Explanation:
In the above-given question,
given that,
the two numbers are 4.95 and 12.1.
4.95 – 12.1 = 7.15.
so option A is correct.

Question 5.
Lawrence spent $1.89 on a bottle of paint and $0.45 on a brush.
A. What was the total amount he spent? Use the model to help you.

B. Explain how the model helps you find the sum.

Answer:
The total amount he spent = $2.34.

Explanation:
In the above-given question,
given that,
Lawrence spent $1.89 on a bottle of paint and $0.45 on a brush.
$1.89 + $0.45 =
so the total amount he spent = $2.34.

Question 6.
Match each expression on the left with the equivalent decimal.

Answer:
3.05 + 1.65 = 4.7.
8.5 – 4.8 = 3.7.
4.25 + 1.55 = 5.8.
11.4 – 6.6 = 4.8.

Explanation:
In the above-given question,
given that,
match the expressions on the left with an equivalent symbol.
3.05 + 1.65 = 4.7.
8.5 – 4.8 = 3.7.
4.25 + 1.55 = 5.8.
11.4 – 6.6 = 4.8.

Question 7.
Ed is training for a race. He ran 12.56 miles on one day, 12.98 miles the second, and 13.04 miles the third day.
A. What is his combined distance for the first three days?
B. How much farther did he run the second day than the first day?

Answer:
A. The combined distance for the first three days = 38.58 miles.
B. The farther he runs the second day than the first day = 0.42 miles.

Explanation:
In the above-given question,
given that,
Ed is training for a race.
He ran 12.56 miles on one day, 12.98 miles on the second, and 13.04 miles on the third day.
12.56 + 12.98 + 13.04 = 38.58.
12.98 – 12.56 = 0.42.
the combined distance for the first three days = 38.58 miles.
the farther he runs the second day than the first day = 0.42 miles.

Question 8.
The Thomas Jefferson Memorial is on 18.36 acres of land, the Franklin Delano Roosevelt Memorial is on 7.5 acres of land, the Vietnam Veterans Memorial is on 8.9 acres of land, and the World War I Memorial is on 9.6 acres of land. Which two memorials have the greatest difference in area?
What is the difference between the areas of these two memorials? Explain.

Answer:
The difference between the areas of these two memorials = 10.86.

Explanation:
In the above-given question,
given that,
The Thomas Jefferson Memorial is on 18.36 acres of land,
the Franklin Delano Roosevelt Memorial is on 7.5 acres of land,
the Vietnam Veterans Memorial is on 8.9 acres of land,
and the World War I Memorial is on 9.6 acres of land.
18.36 – 7.5 = 10.86.
8.9 – 9.6 = 0.7.
so the difference between the areas of these two memorials = 10.86.

Question 9.
Amber bought a hardcover book for $23.70 and a paperback for $6.91. How much did she spend in all? If she paid with 2 twenty-dollar bills, how much change did she get?

Answer:
The much change did she get = $10.61.

Explanation:
In the above-given question,
given that,
Amber bought a hardcover book for $23.70 and a paperback for $6.91.
23.70 + 6.91 = 30.61.
If she paid with 2 twenty-dollar bills.
2 x 20 = 40.
$40 – $30.61 = $10.61.

Question 10.
Kassandra has a rectangular patio in her backyard. The patio is 12.74 meters long and 5.45 meters wide.

A. Round the length and width to the nearest whole number. Then estimate the perimeter of Kassandra’s patio. Write an equation to model your work.
B. Round the length and width to the nearest tenth. Then estimate the perimeter of Kassandra’s patio. Write an equation to model your work.
C. Find the exact perimeter. Which estimate is closer? Explain why you think that estimate is closer.

Answer:
The length and width to the nearest whole number = 5.5 and 13.
the length and width to the nearest tenth = 5.5 and 12.7.
the perimeter is 69.433.

Explanation:
In the above-given question,
given that,
Kassandra has a rectangular patio in her backyard.
The patio is 12.74 meters long and 5.45 meters wide.
the nearest whole number is 5.5 and 13.
the nearest tenth is 5.5 and 12.7.
the perimeter is 69.433.

Topic 2 Performance Task

Video Games
Four students are playing the same video game. Their scores for the first three levels are added together to see if the student has enough points to move on to Round 2.

Question 1.
The students’ scores are shown in the table below.

Part A
A student must have at least 18 points to advance to Round 2. Use estimation to decide if any of the students did not get 18 points.
Part B
Use estimation to decide which student had the greatest number of points. Explain your reasoning.

Answer:
The student has the greater number of points = Zoey.

Explanation:
In the above-given question,
given that,
kim had 7.18 + 6.55 + 6.45 = 20.18.
Sally had 5.49 + 6.18 + 5.72 = 17.39.
Tina had 8.02 + 7.94 + 8.38 = 24.34.
zoey had 8.64 + 8.32 + 8.13 = 25.09.

Question 2.
Complete the table to find the total number of points for each student.

Answer:
The total number of points of Kim = 20.18.
the total number of points of Sally = 17.39.
the total number of points Tina = 24.34.
the total number of points Zoey = 25.09.

Explanation:
In the above-given question,
given that,
kim had 7.18 + 6.55 + 6.45 = 20.18.
Sally had 5.49 + 6.18 + 5.72 = 17.39.
Tina had 8.02 + 7.94 + 8.38 = 24.34.
zoey had 8.64 + 8.32 + 8.13 = 25.09.

Question 3.
How many more points did Zoey score than Sally? Write an equation to model your work.

Answer:
The more points did Zoey score than Sally = 7.7.

Explanation:
In the above-given question,
given that,
kim had 7.18 + 6.55 + 6.45 = 20.18.
Sally had 5.49 + 6.18 + 5.72 = 17.39.
Tina had 8.02 + 7.94 + 8.38 = 24.34.
zoey had 8.64 + 8.32 + 8.13 = 25.09.
25.09 – 17.39 = 7.7.

Question 4.
Use the total points scored for each student.
Part A
About how many points did the four students score in Round 1? Estimate by rounding each point total to the nearest whole number.
Part B
Complete the bar diagram to show the exact total number of points the students scored.

Answer:
The total number of points the students scored = 87.

Explanation:
In the above-given question,
given that,
kim had 7.18 + 6.55 + 6.45 = 20.18.
Sally had 5.49 + 6.18 + 5.72 = 17.39.
Tina had 8.02 + 7.94 + 8.38 = 24.34.
zoey had 8.64 + 8.32 + 8.13 = 25.09
20.18 + 17.39 + 24.34 + 25.09 = 87.

Question 5.
In Round 2, Zoey had a total of 23.43 points. She got a score of 7.96 in Level 2 and a score of 8.03 in Level 3.
Part A
What score did she receive in Level 1?
Part B
Explain how you found your answer.

Answer:
In level 1 Zoey scored 20.18 points.

Explanation:
In the above-given question,
given that,
Zoey had a total of 23.43 points.
She got a score of 7.96 in Level 2.
the score of 8.03 in Level 3.
23.43 + 7.96 + 8.03 = 20.18.
so the total score is 20.18 points.

Question 6.
Kim recorded her scores for Round 2. To estimate her total, she rounds to the nearest whole number and says, “7 + 9 + 7 = 23, so my total is at least 23 points.” Do you agree? Explain your reasoning.

Answer:
No, it was not correct.

Explanation:
In the above-given question,
given that,
Kim recorded her scores for Round 2.
7 + 9 + 7 = 23.
but the scores are 6.77, 8.48, and 7.13.
6.77 + 8.48 + 7.13 = 22.38.
so it was not correct.