Taking Apart Numbers and Shapes

Taking Apart

Numbers and

Shapes

Writing Equivalent Expressions Using the

Distributive Property

Lesson Overview

Students divide area models in different ways to see that the sum of the areas of the smaller

regions equals the area of the whole model. They then rewrite the product of two factors

as a factor times the sum of two or more terms, leading to the formalization of the

Distributive Property.

Grade 6

Expressions, Equations, and Relationships

(7) The student applies mathematical process standards to develop concepts of

expressions and equations. The student is expected to:

(D) generate equivalent expressions using the properties of operations: inverse, identity,

commutative, associative, and distributive properties.

ELPS

1.A, 1.C, 1.E, 1.F, 1.G, 2.C, 2.E, 2.I 3.D, 3.E, 4.B, 4.C, 5.B, 5.F, 5.G.

Essential Ideas

• The area of a rectangle is the product of its length and width.

• You can illustrate the Distributive Property using an area model of a rectangle with side

lengths a and (b 1 c).

• The Distributive Property states that for any numbers a, b, and c, a(b 1 c) 5 ab 1 ac.

• You can rewrite equivalent expressions using properties.

LESSON 1: Taking Apart Numbers and Shapes • 1

MATERIALS

None

2 • TOPIC 1: Factors and Multiples

Lesson Structure and Pacing: 1 Day

Engage

Getting Started: Break It Down to Build It Up

Students divide area models for the product 5 3 27 in two different ways. They calculate

the areas of the subdivided parts before determining the area of the whole model.

Develop

Activity 1.1: Connecting Area Models and the Distributive Property

Students rewrite the product of two factors as a factor times the sum of two or more terms,

leading to the formalization of the Distributive Property. They decompose factors and

products into equivalent representations.

Demonstrate

Talk the Talk: The Floor Is Yours

Students design the floor plan in a gymnasium for different after-school activities. They

represent their model using the Distributive Property and then explain their rationale.

LESSON 1: Taking Apart Numbers and Shapes • 3

Getting Started: Break It Down to Build It Up

Facilitation Notes

In this activity, students divide area models for the product 5 3 27 in

two different ways. They calculate the areas of the subdivided parts

before determining the area of the whole model.

Ask a student to read the situation aloud. Have students complete

Question 1 individually. Share responses as a class.

As students work, look for

• Whether students use a vertical, horizontal, or slanted line to

divide the area model.

• Splitting 27 into numbers that make the computation of area

easier.

• Correct dimensions for each of the smaller regions in the area

model.

Questions to ask

• What is an area model?

• Did you split the length to obtain specific values that add up

to 27? If so, explain your thinking.

Misconceptions

Students may decide to make a diagonal line to split the area.

While correct, discuss that their decision makes two trapezoids, or

two triangles, instead of rectangles, and it is much more efficient

to use rectangles. Also, rectangles are required to model the

Distributive Property.

Have students complete Questions 2 and 3 individually. Share

responses as a class.

Questions to ask

• What was the same about each of your area calculations? Why

is that the case?

• Why does everyone get the same total area even though they

split the walkway differently?

Summary

You can divide an area model into smaller regions. The sum of the

areas of each region is the total area of the model.

ENGAGE

4 • TOPIC 1: Factors and Multiples

Activity 1.1

Connecting Area Models and the Distributive

Property

Facilitation Notes

In this activity, students rewrite the product of two factors as a factor

times the sum of two or more terms, leading to the formalization of

the Distributive Property. Students decompose factors and products

into equivalent representations.

Ask a student to read the introduction. Have students complete

Question 1 with a partner or group. Share responses as a class.

Questions to ask

• When is the use of parentheses necessary? What do they

indicate?

• Why is 5 repeated in both parentheses?

• How does this expression relate to your calculations?

• Did anyone split the rectangle up into three regions? How

does that change the way you write the corresponding

equation using the Distributive Property?

• Which combination of values is the most efficient one to use?

Ask a student to read the definition of the Distributive Property

aloud. Ask students to work with a partner or in groups to complete

Question 2. Share responses as a class.

Questions to ask

• What does it mean to distribute something?

• Explain what it is meant by “multiplication over addition.”

• The multiplication symbol is not shown but rather implied.

Where is multiplication implied in a(b 1 c) 5 ab 1 ac?

Read and discuss the Worked Example as a class. Have students

answer Questions 3 through 5 with their partner or group, and share

responses as a class.

Questions to ask

• What is the purpose of showing the arrows in the example?

• Draw a diagram to represent this expression.

• Could you write the expression as (2 1 15)4? Explain

your thinking?

• How did you decide which addend to put in the

parentheses?

• What are the factors in this equation?

DEVELOP

LESSON 1: Taking Apart Numbers and Shapes • 5

• Why did you choose the sum of those two values to

represent the factor?

• Are some sums a better choice than others? Explain

your thinking.

• What was the error in each false statement?

• How do you read each true statement?

• What is important to keep in mind when you are distributing

a factor over a quantity?

Differentiation strategies

• To scaffold support for Question 3, suggest that students use

arrows similar to those used in the Worked Example. Then,

have them work backwards from the partial products.

• For advanced learners, demonstrate how the Distributive

Property can help with mental math by expressing one factor

as the sum of the tens and ones values. For example:

3 3 28 5 3(20 1 8) 5 60 1 24 5 84.

Summary

You can use the Distributive Property to rewrite the product of two

factors as a factor times the sum of two or more terms.

Talk the Talk: The Floor Is Yours

Facilitation Notes

In this activity, students design the floor plan in a gymnasium for

different after-school activities. They represent their model using the

Distributive Property and then explain their rationale.

Have students work with a partner or in groups to complete

Question 1. Share responses as a class.

Questions to ask

• What rationale did you use to split the gym floor to

accommodate the three activities?

• How is the width of the gym floor reflected in your equation?

• How is the area of each activity reflected in your equation?

• A typical volleyball net spans about 36 feet from pole to

pole. Did you leave enough room to set up the net? If not,

how could you change your diagram and the corresponding

equation?

DEMONSTRATE

6 • TOPIC 1: Factors and Multiples

• How does the use of the Distributive Property change when

you are multiplying a single factor by the sum of three

numbers?

• Does the sum of the areas of each of the three regions total

4200square feet?

Summary

The Distributive Property allows you to represent expressions in

different ways.

LESSON 1: Taking Apart Numbers and Shapes • 7

Warm Up Answers

1. 90 square inches

2. 108 square yards

LESSON 1: Taking Apart Numbers and Shapes • 1

LEARNING GOALS

• Write, read, and evaluate equivalent numeric expressions.

• Identify the adjacent side lengths of a rectangle as

factors of the area value.

• Identify parts of an expression, such as the product and

the factors.

• Write equivalent numeric expressions for the area of a

rectangle by decomposing one side length into the sum

of two or more numbers.

• Apply the Distributive Property to rewrite the product of

two factors.

KEY TERMS

• numeric expression

• equation

• Distributive Property

REVIEW

Calculate the area of each

rectangle. Show your work.

6 in.

15 in.

9 yd

12 yd

Taking Apart

Numbers

and Shapes

Writing Equivalent Expressions Using

the Distributive Property

You know how to add, subtract, multiply, and divide numbers using different strategies.

Taking apart numbers before you perform a mathematical operation can highlight important

information or make calculations easier. How can taking apart numbers help you to express

number sentences in different ways?

8 • TOPIC 1: Factors and Multiples

Answers

1. Sample answer shown.

2. Sample answer shown

above.

3. 135 square feet

10 125

100 35

2 • TOPIC 1: Factors and Multiples

Getting Started

Break It Down to Build It Up

Callie is building a rectangular walkway up to her house. The width of

the walkway is 5 feet and the length is 27 feet. She needs to calculate

the area of the walkway to determine the amount of materials needed

to build it.

1. Mark and label 2 different ways you could divide an area

model to determine the area of the walkway.

2. Determine the areas of each of the subdivided parts of

yourmodels.

3. What is the total area of the walkway?

LESSON 1: Taking Apart Numbers and Shapes • 9

Answers

1. Sample answer.

5 3 27 5 5(25 1 2)

5 (5 ∙ 25) 1 (5 ∙ 2)

5 125 1 10

5 135

LESSON 1: Taking Apart Numbers and Shapes • 3

The numeric expression of 5 3 27 represents the area of the walkway

from the Getting Started. A numeric expression is a mathematical

phrase that contains numbers and operations.

The equation 5 3 27 5 135 shows that the expression 5 3 27 is

equal to the expression 135.

An equation is a mathematical sentence that uses an equals sign to

show that two or more quantities are the same as one another.

1. Reect on the different ways you can rewrite the product

of 5 and 27. Select one of your area models to complete the

example.

How did you split the side

length of 27?

5 3 27 5 5(

1 )

What are the factors of each

smaller region?

5 (5 ?

) 1 (5 ? )

What is the area of each

smaller region?

What is the total area?

Connecting Area Models

and the Distributive Property

ACTIVITY

1.1

What are

other ways

you could

split one of

the factors

and write a

corresponding

equation?

What would

the equation

look like if you

split one of

the factors

into more

than two

regions?

10 • TOPIC 1: Factors and Multiples

ELL Tip

Help students

differentiate

between the terms

expression and

equation. Read

the following

statements aloud.

Have students

usethumbs-up or

thumbs-down

to indicate

understanding.

Answers

2. Sample answer.

The area of the whole

rectangle is equal to

a (b+c) , because a is

the width and b+c is

the length. The area of

the smaller rectangle

is a ∙ b and the area of

the larger rectangle is

a ∙ c . The sum of those

areas, ab+ac , is equal

to the area of the whole

rectangle, a(b + c).

3a. 7(3 1 10 ) 5 21 1 70

3b. 3(12 1 15 ) 5 36 1 45

3c. 8(2 1 7 ) 5 16 1 56

3d. 5(6 1 9 ) 5 30 1 45

Write an example or counterexample to illustrate.

• All equations have equal signs in them.

• All expressions are equations.

• All equations include expressions.

4 • TOPIC 1: Factors and Multiples

You just used the Distributive Property!

The Distributive Property, when

applied for multiplication, states

that for any numbers a, b, and c, the

equation a( b 1 c ) 5 ab 1 ac is true.

2. Explain the Distributive Property using the area model shown.

You can read and describe the expression 4 (2 1 15) in different ways.

For example, you can say:

• four times the quantity of two plus fteen,

• four times the sum of two and fteen, or

• the product of four and the sum of two and fteen.

You can describe the expression 4 (2 1 15) as a product of two factors.

The quantity (2 1 15) is both a single factor and a sum of two terms.

3. Fill in the missing addend in each box that makes the

equation true.

a. 7 (

1 10 ) 521 1 70 b. 3 ( 1 15 ) 536 1 45

c. 8 (2 1

) 516 + 56 d. 5 (6 1 ) 530 + 45

WORKED EXAMPLE

Consider this example of the Distributive Property.

4(2 1 15) 5 4 ? 2 1 4 ? 15

You can also use

grouping symbols

to show that you

need to multiply

each set of

factors before

you add them,

? 2) 1 (4 ? 15) .

LESSON 1: Taking Apart Numbers and Shapes • 11

Answers

4a. Sample answer.

4 (10 1 7) 5 68

40 1 28 5 68

68 5 68

4b. Sample answer.

9 3 34 5 306

9 (30 1 4) 5 306

270 1 36 5 306

306 5 306

4c. Sample answer.

3 3 29 5 87

3 (20 1 9) 5 87

60 1 27 5 87

87 5 87

5a. False;

3(2 1 4) 5 3 ? 2 1 3 ? 4

5b. True

5c. False;

7(20 1 8) 5 7 ? 20 1 7 ? 8

5d. False;

4(5 1 10) 5 20 1 40

5e. True

LESSON 1: Taking Apart Numbers and Shapes • 5

4. Rewrite a factor as the sum of two terms in each expression

and use the Distributive Property to verify each product.

a. 4 3 17568 b. 9 3 345306

c. 3 3 29587

5. Identify each statement as true or false. If the statement

is false, show how you could rewrite it to make it a true

statement.

a. True False 3(2 1 4) 5 3 ? 2 1 4

b. True False 6(10 1 5) 5 6 ? 10 1 6 ? 5

c. True False 7(20 1 8) 5 7 1 20 ? 8

d. True False 4(5 1 10) 5 20 1 10

e. True False 2(6 1 11) 5 12 1 22

12 • TOPIC 1: Factors and Multiples

Answers

1. Sample answer.

50(40 1 34 1 10) 5

50 ? 40 1 50 ? 34 1

50 ? 10

5 2000 1 1700 1 500

5 4200

I divided the length of

the gym into three parts

to create three areas of

different sizes for each

activity.

• I made the area for

playing volleyball the

largest, 50 feet by 40

feet.

• I made the area for

playing dodgeball, 50

feet by 34 feet, close

to the same size as the

volleyball area, but a bit

smaller.

• I made the smallest area

of the gym, 50 feet by 10

feet, for playing board

games or reading since

those are activities that

require less movement.

6 • TOPIC 1: Factors and Multiples

TALK th e TALK

The Floor Is Yours

You can apply the Distributive Property to solve

real-world problems.

Consider the situation.

Tyler is setting up the gym oor for an after-school program. He

wants to include a rectangular area for playing volleyball and

another for dodgeball. He also wants to have an area for kids who

like to play board games or just sit and read. The gym oor is

already 50 feet by 84 feet, or 4200 square feet.

1. Create a diagram to show how you would split up the

gym oor. Represent your diagram using the Distributive

Property and write an explanation for the areas assigned to

each activity.

NOTES

40 34 10