lesson 14 introduction multiply fractions using an area model · introduction 126 ©curriculum...

Introduction

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Lesson 14 Multiply Fractions Using an Area Model

Lesson 14Multiply Fractions Using an Area Model

a. Write a multiplication equation for the area of the square with side lengths of 1 meter.

b. What expression can you write for the area of the smaller square with side lengths

of 5 ·· 10 meter?

c. The square with the darker shading is 5 units wide and 5 units long. How many square

units is this?

d. What is the total number of small squares in the large square?

e. What fraction of the whole square is the part with the darker shading?

f. The area of the square with the darker shading is square meter.

Mr. Thompson has a 1-meter-square whiteboard. He creates a square with 5 ·· 10 -meter sides on his whiteboard to post a weekly puzzle. How many square meters of whiteboard space does he use for the weekly puzzle?

1 meter

1 meter

meter510

meter510

In Lesson 13, you learned about multiplying fractions. Now you will use area models to multiply fractions. Take a look at this problem.

5.NF.B.4b

©Curriculum Associates, LLC Copying is not permitted. 127Lesson 14 Multiply Fractions Using an Area Model

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The area of a square with sides 5 ·· 10 meter represents

the product 5 ·· 10 3 5 ·· 10 .

• The length of each unit is 1 ·· 10 meter. The length and the

width of the blue square are both 5 units, or 5 ·· 10 meter.

• The area of one square unit is 1 ··· 100 square meter. The area

of the blue square is 25 square units, or 25 ··· 100 square meter.

5 ·· 10 meter 3 5 ·· 10 meter 5 25 ··· 100 square meter

You can also think about multiplying 5 ·· 10 3 5 ·· 10 another way.

5 ·· 10 is 5 3 1 ·· 10 . Rewrite. 5 ·· 10 3 5 ·· 10 5 1 5 3 1 ·· 10 2 3 1 5 3 1 ·· 10 2 Change the order of the factors. 5 5 3 5 3 1 ·· 10 3 1 ·· 10

Then multiply. 5 25 3 1 ··· 100 , or 25 ··· 100

Reflect1 Look at the models on these pages. Explain why multiplying the numerators tells you

the number of parts in the product.

510 meter

510 meter

Modeled and Guided Instruction

Learn About


Lesson 14


Multiplying Unit Fractions to Find Areas

Read the problem below. Then explore different ways to understand multiplying two unit fractions.

Titus has a square sheet of paper measuring 1 foot on each side. He folds the paper in half vertically and then folds it into fourths horizontally. Titus unfolds the paper as shown below. What is the area of each part?

1 foot

1 foot

Picture It You can understand the problem by picturing parts of the unfolded paper.

The folding created 8 equal sections.

foot14

foot12

Each section is

of the whole.18

1 foot

1 foot

Each section is 1 ·· 2 foot long and 1 ·· 4 foot wide.

Model It You can model the problem with an equation.

Each equal part of the paper has a length of 1 ·· 2 foot and width of 1 ·· 4 foot.

area 5 1 ·· 2 foot 3 1 ·· 4 foot

1 ·· 2 foot 3 1 ·· 4 foot 5 1 3 1 ····· 2 3 4 square foot


Connect It Now you will solve the problem from the previous page by connecting the model to the equation.

2 Explain why each part of the paper is 1 ·· 2 foot long.

3 Explain why each part of the paper is 1 ·· 4 foot wide.

4 What expression do you use to find the area of 1 part of the paper?

5 Multiply the denominators of these fractions. How does the product relate to the

size of the units compared to the size of the whole model?

6 Multiply the numerators of these fractions. How does the product relate to the

number of outlined parts of the model?

7 Explain how to find the area of one part of the sheet of paper.

Try It Use what you just learned about multiplying unit fractions to solve this problem. Show your work on a separate sheet of paper.

8 What is the area of a paper strip with a width of 1 ·· 3 yard and length of 1 ·· 6 yard?

Modeled and Guided Instruction

Learn About


Lesson 14


Multiplying Fractions Greater than One

Read the problem below. Then explore different ways to understand multiplying fractions greater than 1.

A postage stamp has a width of 3 ·· 4 inch and length of 3 ·· 2 inches. What is the area of

the stamp in square inches?

Picture It You can picture the problem using the area of a rectangle.

The first area model shows 1 ·· 4 inch 3 1 ·· 2 inch 5 1 ·· 8 square inch. The second model uses the

same 1 ·· 8 -square-inch parts to show an area that is 3 ·· 4 inch 3 3 ·· 2 inches.

in.12

in.12

in.14

in.14

in.14

in.14

in.12

in.12

in.12

in.14

in.14

in.14

in.14

1 ·· 4 3 1 ·· 2 3 ·· 4 3 3 ·· 2

Model It You can model the problem with an equation.

The dimensions of the stamp are 3 ·· 4 inch and 3 ·· 2 inches, so multiply the fractions to find

the area.

area 5 3 ·· 4 3 3 ·· 2 5 3 3 3 ····· 4 3 2


Connect It Now you will solve the problem from the previous page by connecting the model to the equation.

9 Look at the area model for 3 ·· 4 3 3 ·· 2 . Explain why each part shows 1 ·· 8 square inch.

10 What is the total width of the three 1 ·· 2 -inch columns?

What is the total height of the three 1 ·· 4 -inch rows?

11 How many 1 ·· 8 -square-inch parts are shaded purple? parts

Write the area of the purple section as a fraction greater than 1. square inches

12 Now look at the equation in Model It. Multiply the numerators, multiply the denominators, and write the fraction. How does this product compare with the one shown by the area model?

13 How do you multiply fractions greater than 1?

Try It Use what you just learned about multiplying fractions to solve this problem. Show your work on a separate sheet of paper.

14 Bernice’s math workbook is 2 ·· 3 foot wide and 5 ·· 6 foot long. What is the area of a

page in the workbook?

Guided Practice

Practice


Lesson 14

Multiplying Fractions to Find Area


Example

Pair/Share

How can you write

5 ·· 8 3 1 ·· 2

as a product of

unit fractions and

whole numbers?

Pair/ShareFind the area of a rectangle with side lengths of 3 ·· 4 yard and 6 ·· 5 yards. How is the model different?

How can I represent a fractional side length with an area model?

1 ·· 8 inch 3 1 ·· 2 inch is

1 ·· 16 square inch. How

many sixteenth square

inches are shown in the

model?

15 What is the area of a rectangle with a length of 1 ·· 2 yard and a width

of 11 ·· 6 yards? Write an equation to represent your solution.

Show your work.

Solution

Rachel is designing a newspaper ad. The ad will include a piece of

art whose dimensions are 5 ·· 8 inch long and 1 ·· 2 inch wide. How many

square inches of space will the art cover?

Look at how you could show your work using an area model.

58

12

5 ·· 8 3 1 ·· 2 5 5 3 1 ····· 8 3 2 5 5 ··· 16

Solution

Study the example below. Then solve problems 15–17.

5 ··· 16 square inch


Pair/ShareDoes Ollie’s answer make sense?

Pair/ShareWrite an equation to represent your model. Explain the meaning of the numerators.

If I draw a square to represent a square foot, how can I represent thirds and fifths on the square?

Think about the size of the two fractions. Will the product of the fractions be greater than 1 or less than 1?

16 Brent is designing a poster that has an area of 1 square foot. He is

going to paste a photo collage on a section of the poster that is

1 ·· 3 foot wide and 3 ·· 5 foot long. What part of a square foot will the

photo collage cover?

Show your work.

Solution

17 What is the area of the square? yd67

yd67

Circle the letter of the correct answer.

A 36 ·· 49 square yard

B 12 ·· 14 square yard

C 49 ·· 36 square yards

D 12 ·· 7 square yards

Ollie chose D as the correct answer. How did he get that answer?

Independent Practice

Practice


Lesson 14

Multiplying Fractions to Find Area


Solve the problems.

1 The square below represents 1 square unit.

Which expression represents the area of the dark blue section?

A 7 ·· 3 3 3 ·· 1 square units

B 3 ·· 7 3 1 ·· 3 square units

C 1 ·· 7 3 1 ·· 3 square units

D 7 ·· 3 3 1 ·· 3 square units

2 Fill in the missing numbers to make the equation true. Then complete the area model to check your answer.

1 ·· 6 3 5 1 ·· 24

3 Which products could you find by shading the model below? Circle the letter for all that apply.

A 3 ·· 4 3 1 ·· 3

B 1 ·· 3 3 1 ·· 6

C 2 ·· 3 3 1 ·· 4

D 5 ·· 3 3 1 ·· 4

E 3 ·· 4 3 3 ·· 4

Self Check


Go back and see what you can check off on the Self Check on page 93.

4 Draw an area model to represent the expression 5 ·· 4 inches 3 4 ·· 5 inch.

5 Explain how to find the area of the model you drew in problem 4, then find the area .

6 Write the dimensions of a different rectangle that has the same area as the rectangle you drew in problem 4. Show how you know the area is the same.

lesson 14 introduction multiply fractions using an area model · introduction 126 ©curriculum...

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