Express Math - Mello mathematics - Home

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6.2 Simplifying Expressions Using Distributive Properties 303A

Key Termsf Distributive Property of Multiplication over Addition

f Distributive Property of Multiplication over Subtraction

f Distributive Property of Division over Addition

f Distributive Property of Division over Subtraction

Learning GoalsIn this lesson, you will:

f Write and use the

distributive properties.

f Use distributive properties

to simplify expressions.

Express MathSimplifying Expressions Using Distributive Properties

Common Core State Standards for Mathematics7.NS The Number System

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

3. Solve real-world and mathematical problems involving the four operations with rational numbers.

7.EE Expressions and Equations

Use properties of operations to generate equivalent expressions.

1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

2. Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.

Essential Ideas The Distributive Property provides ways to write numerical and algebraic expressions in equivalent forms.

When the Distributive Property is applied to numerical expressions only, it is a helpful tool in computing math mentally.

The area of a rectangle model is useful in demonstrating the Distributive Property.

There are four versions of the distributive property:

Distributive Property of Multiplication over Addition:If a, b and c are any real numbers, then a ∙ (b 1 c) 5 a ∙ b 1 a ∙ c.

Distributive Property of Multiplication over Subtraction: If a, b and c are any real numbers, then a ∙ (b 2 c) 5 a ∙ b 2 a ∙ c.

Distributive Property of Division over Addition: If a, b, and c are any real numbers and c ! 0, then a 1 b ______ c 5 a __ c 1 b __ c .

Distributive Property of Division over Subtraction: If a, b and c are any real numbers and c ! 0, then a 2 b ______ c 5 a __ c 2 b __ c .

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6.2 Simplifying Expressions Using Distributive Properties 303

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Key Termsf Distributive Property of Multiplication over Addition

f Distributive Property of Multiplication over Subtraction

f Distributive Property of Division over Addition

f Distributive Property of Division over Subtraction

Learning GoalsIn this lesson, you will:

f Write and use the

distributive properties.

f Use distributive properties

to simplify expressions.

Express MathSimplifying Expressions Using Distributive Properties

It once started out with camping out the night before the sale. Then, it evolved

to handing out wrist bands to prevent camping out. Now, it’s all about the

Internet. What do these three activities have in common?

For concerts, movie premieres, and highly-anticipated sporting events, the

distribution and sale of tickets have changed with computer technology.

Generally, hopeful ticket buyers log into a Web site and hope to get a chance to

buy tickets. What are other ways to distribute tickets? What are other things that

routinely get distributed to people?

304 Chapter 6 Numerical and Algebraic Expressions and Equations

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Why was 230 represented as 200 1 30 rather than another sum?

What do the numbers in the boxes represent?

Explain the area model in your own words.

How are the area model and the mental math model similar?

Problem 1This problem addresses all aspects of the distributive property. Students explore the distributive property by calculating the product of two numbers in two ways. They will analyze two models, a method for computing mental math and the area of rectangles diagram, and they connect and use both models. The Distributive Properties of Multiplication over Addition, Multiplication over Subtraction, Division over Addition, and Division over Subtraction are introduced. Students will simplify algebraic expressions using both the area model and symbolic representations. They then reverse the process of the Distributive Property of Multiplication over Addition and Subtraction, and factor numerical and algebraic expressions.

GroupingHave students complete Questions 1 through 3 with a partner. Then share the responses as a class.

Share Phase, Questions 1 and 2

Are you able to calculate the product using mental math? If so, explain your process.

Why is the area model appropriate for this calculation?

304 Chapter 6 Numerical and Algebraic Expressions and Equations©

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Problem 1 Fastest Math in the Wild West

Dominique and Sarah are checking

each other’s math homework. Sarah

tells Dominique that she has a quick

and easy way to multiply a one-digit

number by any other number in her

head. Dominique decides to test

Sarah by asking her to multiply

7 by 230.

1. Calculate the product 7 3 230.

230 3 7 1610

2. Write an expression that shows the mathematical steps Sarah performed to calculate

the product.

7(200) 1 7(30)

Sarah is able to correctly multiply 7 by 230 in her head. She explains her method to Dominique.

Write an expression that shows the mathematical steps Sarah performed to calculate

Sarah 230 3 7First, break 230 into the

sum of 200 and 30. Then,

multiply 7 x 200 to get a

product of 1400 and 7 x 30

to get a product of 210.

Finally, add 1400 and 210

together to get 1610.

Dominique makes the connection between Sarah’s quick mental calculation and calculating the area of a rectangle. She draws the models shown to represent Sarah’s thinking in a different way.

DominiqueCalculating 230 x 7 is the same as

determining the area of a rectangle by

multiplying the length by the width.

But I can also divide the rectangle into

two smaller rectangles and calculate the

area of each rectangle. I can then add

the two areas to get the total. 1400 + 210 = 1610

230

7

200 30

1400 2107

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Share Phase, Questions 3

Why did you choose the particular sum that you did to represent the two- or three-digit number?

Where are the values from your use of Sarah’s method represented in your area model?

Explain your use of parentheses when you used Sarah’s method.

GroupingAsk a student to read the information following Question 3 aloud. Discuss the information as a class.

Discuss Phase, Definitions

Why do you think it is called the Distributive Property of Multiplication over Addition?

Use one of the items from Question 3 to demonstrate the substitution of values for a, b and c in the formal definition.

Use 4(499) to demonstrate using the Distributive Property of Multiplication over Subtraction.

Use 535 ____ 5 to demonstrate

using the Distributive Property of Division over Addition.


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3. First, use Dominique’s method and sketch a model for each. Then, write an

expression that shows Sarah’s method and calculate.

a. 9(48)40 8

9 360 72

9(40 1 8)5 9(40) 1 9(8)5 360 1 72 5 432

b. 6(73)70 3

6 420 18

6(70 1 3)5 6(70) 1 6(3)5 420 1 18 5 438

c. 4(460)400 60

4 1600 240

4(400 1 60)5 4(400) 1 4(60)5 1600 1 240 5 1840

Sarah’s and Dominique’s methods are both examples of the

Distributive Property of Multiplication over Addition, which

states that if a, b, and c are any real numbers, then

a ? ( b 1 c) 5 a ? b 1 a ? c.

Including the Distributive Property of Multiplication over Addition,

there are a total of four different forms of the Distributive Property.

Another Distributive Property is the Distributive Property of Multiplication over Subtraction, which states that if

a, b, and c are any real numbers, then a ? ( b 2 c) 5 a ? b 2 a ? c.

can draw at least two different

models to determine

).

Use 595 ____ 5 to demonstrate

using the Distributive Property of Division over Subtraction.

Explain when it may more appropriate to use subtraction rather than addition.


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A common error is that students will distribute to the first term in the parentheses, but forget to distribute to the second term. Use of the area model helps eliminate this error by having students fill in a rectangle for each term.


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The Distributive Property also holds true for division over addition and division over

subtraction as well.

The Distributive Property of Division over Addition states that if a, b, and c are real

numbers and c fi 0, then a 1 b ______ c 5 a __ c 1 b __ c .

The Distributive Property of Division over Subtraction states that if a, b, and c are real

numbers and c fi 0, then a 2 b ______ c 5 a __ c 2 b __ c .

4. Draw a model for each expression, and then simplify.

a. 6(x 1 9) b. 7(2b 2 5)x 9 2b 25

6 6x 54 7 14b 235

6x 1 54 14b 2 35

c. 22(4a 1 1) d. x 1 15 _______ 5

4a 1 x 15

22 28a 22 1 __ 5

1 __ 5

x 3

28a 2 2 x __ 5 1 3

5. Use one of the Distributive Properties to rewrite each expression in

an equivalent form.

a. 3y(4y 1 2) b. 12( x 1 3)

12y2 1 6y 12x 1 36

c. 24a(3b 2 5) d. 27y(2y 2 3x 1 9)

212ab 1 20a 214y2 1 21xy 2 63y

e. 6m 1 12 ________ 22

f. 22 2 4x ________ 2

23m 2 6 11 2 2x

i iding by is the same as

multiplying by what number

GroupingHave students complete Questions 4 and 5 with a partner. Then share the responses as a class.

Share Phase, Questions 4 and 5

How are these problems in Question 4 different from the previous problems?

Why can’t you simplify your algebraic expressions in Question 4 by adding the areas?

How did you determine the signs of each term?

Is the process any different if there are more than two terms in the parentheses?

Explain the distributive process for the division problems.

MisconceptionsBe sure that students do not over generalize the Distributive Property of Division over Addition or Subtraction. The numerator can be split as a sum for ease in calculations; but the denominator cannot be split as a sum.

For example: a 1 b ______ c 5 a __ c + b __ c

For example: c ______ a 1 b

fi c __ a 1 c __ b

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Problem 2Students simplify five expressions using distributive properties. They will then evaluate two expressions using distributive properties.

GroupingHave students complete Questions 1 and 2 with a partner. Then share the responses as a class.

Share Phase, Question 1

How can a distributive property be used to simplify this expression?

What is the first step?

What is the second step?

Are there any like terms?

What can be combined?


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Problem 2 Simplifying and Evaluating

1. Simplify each expression. Show your work.

a. 26(3x 1 (24y))

26(3x 1 (24y)) 5 (26)(3x) 1 (26)(24y) 5218x 1 24y

b. 24(23x 2 8) 2 34

24(23x 2 8) 2 34 5 (24)(23x) 1 (24)(28) 1 (234) 5 12x 1 32 1 (234) 5 12x 1 (22)

c. 27.2 2 6.4x ____________ 20.8

27.2 1 6.4x ____________ 20.8

5 27.2 _____ 20.8

1 6.4x _____ 20.8

5 9 1 (28x)

d. ( 22 1 __ 2

) ( 3 1 __ 4

) 1 ( 22 1 __ 2

) ( 22 1 __ 4

) ( 22 1 __

2 ) ( 3 1 __

4 ) 1 ( 22 1 __

2 ) ( 22 1 __

4 ) 5 ( 22 1 __

2 ) ( 3 1 __

4 1 ( 22 1 __

4 ) )

5 ( 22 1 __ 2

) (1)

5 22 1 __ 2

e. ( 27 1 __

2 ) 1 5y ___________

2 1 __ 2

( 27 1 __

2 ) 1 5y

___________ 2 1 __

2 5

27 1 __ 2

_____

2 1 __ 2

1

5y ___

2 1 __ 2

5 23 1 2y


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Share Phase, Question 2

Did you use any of the distributive properties to simplify this expression? Which one(s)?

Is the answer in the simplest form? How do you know?


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2. Evaluate each expression for the given value. Then, use properties to simplify the

original expression. Finally, evaluate the simplified expression.

a. 2x (23x 1 7) for x 5 21 2 __ 3

2x(23x 1 7) 5 2 ( 21 2 __ 3

) ( 23 ( 21 2 __ 3

) 1 7 ) 2x(23x 1 7) 5 6x2 1 14x

5 26 ( 21 2 __ 3

) 2

1 14 ( 21 2 __ 3

) 2 5 ( 2 6 __ 1 ) ( 2 5 __ 3 ) ( 2 5 __ 3 ) 1 14 ___ 1 ( 2 5 __ 3 ) 1

5 2 50 ___ 3 1 ( 2 70 ___ 3 ) 5 2 120 ____ 3

5 240

1 5 2 __

1 ( 2 5 __ 3 ) ( 2 3 __ 1 ( 2 5 __ 3 ) 1 7 )

1 5 2 10 ___ 3 (5 1 7) 4 5 2 ( 10 ___ 3 ) ( 12 ___

1 )

1 5 240

b. 4.2x 27 ________ 1.4

for x 5 1.26

4.2x 2 7 ________ 1.4

5 4.2(1.26) 2 7

____________ 1.4 4.2x 2 7 ________ 1.4

5 4.2x ____ 1.4 1 27 ___ 1.4

5 3x 1 (25) 3x 2 5 5 3(1.26) 1 (25) 5 21.22

5 5.292 2 7 _________ 1.4

5 21.708 _______ 1.4

5 21.22

c. Which form—simplified or not simplified—did you prefer to evaluate? Why?

Answers will vary.

Be prepared to share your solutions and methods.

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