# enVision Math Common Core Grade 3 Answer Key Topic 6 Connect Area to Multiplication and Addition

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## 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.
• equal groups
• multiply
• array

Question 1.
When you skip count to get the total number, you ________.
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 __________.
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). ___________.
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?

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?

Arrays

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

Number of rows = 6
Number of columns = 5
We add columns value as many times as rows.
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
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.
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.

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?
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.
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.

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.

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.

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.

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.

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?

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.

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?

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.
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.

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.

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?

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

Independent Practice

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

Question 4.

Question 5.

Problem Solving

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

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.

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.

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?

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

### 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?
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.
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.

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.

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.

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.

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.

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.

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.

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.

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.

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?
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.
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
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?
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.

### 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?
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?

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.

Question 4.

Independent Practice

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

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.

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.

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?
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?
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?
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.
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.
☐ 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.

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?
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.

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.

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.

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.

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.
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?
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.

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.

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)
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.

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.

Question 2.
Explain what operation you use to find the total area of the smaller rectangles.
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.

Question 4.

Independent Practice

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

Question 6.

Question 7.

Question 8.

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.

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 ________.

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.
“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.

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

### 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

• 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?

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

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?

Question 3.
How can you break the problem into simpler parts? What is the hidden question?
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?
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.
We know the
Dimensions of the button
Using these dimensions area of buttons and keypad are calculated.
Subtracting the button area from keypad area. Plastic area is calculated.

Problem Solving

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?

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

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.

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?
They share one side of the other rectangles.

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

### Topic 6 Fluency Practice Activity

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.
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.
__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 __________.
You can use the __Distributive Property__ to break apart facts and find the ___Area_______.

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

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

Write always, sometimes, or never.

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

Question 7.
Multiplication __________ involves joining equal groups.
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.
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.

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.

Question 2.

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.

Question 2.

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.

Question 2.

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.

Question 2.

Question 3.

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.

Question 2.

Question 3.

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.

Question 2.

Set G pages 233-236

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?
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.
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.

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.

Area of the larger rectangle = 3 × 5 = 15
Area of the smaller rectangles |
3 × 3 = 9
3 × 2 = 6
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.

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.
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
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
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.

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?
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.

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.

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.

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?

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.

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

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

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.
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.

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.

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.
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.
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.

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.

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.

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?

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.