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The Distributive PropertyThe Distributive Property

Lesson 7-1

Vocab

Essential

Question

Why are algebraic rules useful?

Common Core

State Standards

Content Standards 7.NS.2, 7.NS.2c, 7.EE.1, 7.EE.2

Mathematical Practices 1, 3, 4, 5

Vocabulary

equivalent expressionsDistributive Property

Interactive

Study Guide

See pages 149–150 for:• Getting Started• Real-World Link• Notes

What You’ll Learn

• Use the Distributive Property to write equivalent numerical expressions.

• Use the Distributive Property to write equivalent algebraic expressions.

Real-World Link

Entertainment The Newport Aquarium in Kentucky has acrylic tunnels that allow visitors to walk underneath aquatic life. The cost of admission is $23 per person. The aquarium also offers Behind-the-Scenes Tours for $15 per person and Penguin Encounters for $25 per person.

Use the Distributive Property to write each expression as an equivalent numeric expression. Then evaluate the expression.

a. 5(12 + 4)

5(12 + 4) = 5 · 12 + 5 · 4

= 60 + 20 Multiply.

= 80 Add.

b. (20 - 3)8.2

(20 - 3)8.2 = 20 · 8.2 - 3 · 8.2

= 164 - 24.6 Multiply.

= 139.4 Subtract.

1a. (6 + 3)4 36 1b. 3 _ 4 (9 - 2) 5 1

_ 4

Tutor

Example 1

Do these problems to find out. Got It ?Got It ?

Expressions are equivalent expressions when they have the same value. Example 1 shows how the Distributive Property relates to equivalent expressions.

Numerical ExpressionsNumerical Expressions

Words To multiply a sum or difference by a number, multiply each term inside the parentheses by the number outside the parentheses.

Symbols a(b + c) = ab + ac a(b - c) = ab - ac

Examples 5(6 + 7) = 5 · 6 + 5 · 7 (9 - 3)8 = 9 · 8 - 3 · 8

Distributive Property Key ConceptWatch

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Objective use the Distributive Property to write equivalent numerical and algebraic expressions

Building on the Essential Question

At the end of the lesson, students should be able to answer “How can you use the Distributive Property and mental math to simplify 2 1 _ 2 · 4 1 _ 2 ?”

Example 1

What’s the Math? evaluate numerical expressions• In Example 1a, what number must be multiplied by

both addends? 5

• In Example 1b, why does it not matter which side of the parentheses, left or right, the number is on? The Commutative Property says you can multiply the factors in either order.

Need Another Example?

Use the Distributive Property to write each expression as an equivalent numeric expression. Then evaluate the expression.a. 4(5 + 8) 4 · 5 + 4 · 8; 52

b. (6 - 9)2 6 · 2 - 9 · 2; -6

2 Teach the Concept

292 Chapter 7 Algebraic Expressions

Proportionality and Linear Relationships

The Distributive Property allows you to find some products mentally. For example, you can find 7 · 34 mentally by evaluating 7 · (30 + 4).

7 · (30 + 4) = 7 · 30 + 7 · 4

= 210 + 28 Think: 7 · 30 = 210

= 238 Think: 210 + 28 = 238

Financial Literacy On a school visit to Washington, D.C., Dichali and his class visited the Smithsonian National Air and Space Museum. Tickets to the IMAX movie cost $8.99. Find the total cost for 20 students to see the IMAX movie.

You can use the Distributive Property and mental math to find the total cost for the movie. To find the total cost mentally, find 20($9.00 - $0.01).

20($9.00 - $0.01) = 20($9.00) - 20($0.01) Distributive Property

= $180 - $0.20 Multiply.

= $179.80 Subtract.

So, the total cost is $179.80.

You can check your result by multiplying 20 · $9 to get $180. Since $180 is close to $179.80, the answer is reasonable.

2a. A spaghetti dinner at the Italian Village restaurant costs $10.25. Use the Distributive Property and mental math to find the total cost of the dinner for Sherita, her brother, and her parents. 4($10 + $0.25); 4 ($10) + 4 ($0.25); $41

2b. After dinner, they each order gelato for $1.50. What is the new total?

4($1 + $0.50); 4 ($1) + 4 ($0.50); $6; $47 total

You can model the Distributive Property by using algebra tiles and variables.

The model shows 2(x + 4).There are 2 groups (x + 4).

Separate the tiles into 2 groupsof x and 2 groups of 4.

x + 4 4

2 2

x

2xx

xx

1 1 1 11 1 1 1

1 1 1 11 1 1 11 1 1 1

+

2(x + 4) = 2x + 2 · 4

= 2x + 8

The expressions 2(x + 4) and 2x + 8 are equivalent expressions because no matter what the value of x is, these expressions have the same value.

Tutor

Example 2

Do these problems to find out. Got It ?Got It ?

Algebraic ExpressionsAlgebraic Expressions

Vocabulary Link

Distribute

Everyday Use to deliver to each member of a group

Math Use a property that allows you to multiply each member of a sum by a number

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Example 2

What’s the Math? use the Distributive Property• Why is it helpful to break $8.99 into ($9.00 - $0.01)?

It is easier to multiply ($9.00 - $0.01) by 20 mentally than to multiply $8.99 by 20.

Need Another Example?

A community center offers a canoeing day trip. The canoeing fee is $99 per person. Find the total cost for a family of four. 4(100 - 1); 4 · 100 - 4 · 1; $396

Lesson 7-1 The Distributive Property 293

Use the Distributive Property to write each expression as an equivalent algebraic expression.

a. 4(x + 5)

4(x + 5) = 4x + 4 · 5

= 4x + 20 Simplify.

b. (y + 10)6

(y + 10)6 = y · 6 + 10 · 6

= 6y + 60 Simplify.

3a. 2.4(a + 5) 2.4a + 12 3b. (b + 6)3 3b + 18

Use the Distributive Property to write each expression as an equivalent algebraic expression. a. 3(m - 4)

3(m - 4) = 3[m + (-4)] Rewrite m - 4 as m + (- 4).

= 3 · m + 3 · (-4) Distributive Property

= 3m + (-12) Simplify.

= 3m - 12 Definition of subtraction

b. -9.5(n - 7 )

-9.5(n - 7 ) = -9.5[n + (-7)] Rewrite n - 7 as n + (-7).

= -9.5 · n + (-9.5)(-7) Distributive Property

= -9.5n + 66.5 Simplify.

4a. 2 _ 3 (d - 3) 2 _ 3

d - 2 4b. -7(e - 4) -7e + 28

Example 3Tools Tutor

Do these problems to find out. Got It ?Got It ?

Example 4Tools Tutor

Do these problems to find out. Got It ?Got It ?

Watch Out!

Distributive Property In Example 3a, remember to distribute the 4 to both values inside the parentheses.

Use the Distributive Property to write each expression as an equivalent numeric expression. Then evaluate the expression. (Example 1)

1. 7(9 + 3) 7 · 9 + 7 · 3; 84 2. 2 _ 5 (3 + 5) 2 _ 5

· 3 + 2 _ 5

· 5; 3 1 _ 5

3. (7 + 8)2.2 7 · 2.2 + 8 · 2.2; 33 4. (5 + 6)8 5 · 8 + 6 · 8; 88

5. You purchase 3 blue notebooks and 2 red notebooks. Each notebook costs $1.30. Use mental math to find the total cost of the notebooks. Justify your answer by using the Distributive Property. (Example 2) $6.50; 5($1 + $0.30) = 5($1) + 5($0.30)

Use the Distributive Property to write each expression as an equivalent algebraic expression. (Examples 3 and 4)

6. 3 _ 4 (m + 4) 3 _ 4

m + 3 7. ( p + 4)5 5p + 20

8. -6(b - 5) -6b + 30 9. 9.5(a - 10) 9.5a - 95

Guided PracticeGuided PracticeCheck

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Example 3

What’s the Math? simplify algebraic expressions• In Example 3a, what number must be multiplied by

both addends? 4

• In Example 3b, why can you not simplify 6y + 60 any further? There are no like terms.

Need Another Example?Use the Distributive Property to write each expression as an equivalent algebraic expression. a. 2(x+ 4) 2x + 8 b. (y+ 3)6 6y + 18

Example 4

What’s the Math? simplify algebraic expressions• In Example 4a, could you leave this as a subtraction

problem instead of adding the inverse of 4? yes

• In Example 4b, what is the best way to subtract an integer? Add its additive inverse.

Need Another Example?Use the Distributive Property to write each expression as an equivalent algebraic expression. a. 4(x- 2) 4x - 8 b. -2(n- 3) -2n + 6

Formative AssessmentGuided Practice Use these exercises to assess students’ understanding of the concept of the lesson. If they need more help, use the Personal Tutors available online.

Ask students to draw a model to explain the Distributive Property. Students should label their drawings and make certain to use the correct names of the mathematical procedures. See students’ work.

Ask students to dra

TICKETTICKETOut tut the hthhhe DooDooDD orOut the DoorOut thhe DooDD orrDD

294 Chapter 7 Algebraic Expressions

Proportionality and Linear Relationships

3 Practice and ApplyHomework

The Independent Practice pages are meant to be used as the homework assignment. If you do not wish to assign the entire exercise set, you can use the table below to select appropriate exercises for your students’ needs.

Differentiated Homework Options

ALApproaching Level

10–23, 37, 39–53

OL On Level11–23 odd, 24, 25–33 odd, 35–37, 39–53

BL Beyond Level 24–53

Create Your Own Homework Online

can be used to create worksheets for the suggested assignments above, or create your own worksheets for differentiated homework or review.

MATHEMATICAL PRACTICES

Emphasis On Exercise(s)

1 Make sense of problems and persevere in solving them.

38

3 Construct viable arguments and critique the reasoning of others.

24, 37

5 Use appropriate tools. 25–30

Mathematical Practices 1, 3, and 4 are aspects of mathematical thinking that are emphasized in every lesson. Students are given opportunities to be persistent in their problem solving, to express their reasoning, and to apply mathematics to real-world situations.

Use the Distributive Property to write each expression as an equivalent numeric expression. Then evaluate the expression. (Example 1)

10. -4(8 - 5) 11. 4.5(16 + 8) 12. (23 - 7)6 23 · 6 - 7 · 6; 96

13. 12.3(9 + 4) 14. (6 + 18) 2 _ 3 6 · 2

_ 3

+ 18 · 2

_ 3

; 16 15. 5 _ 8 (20 - 4) 5

_ 8

· 20 - 5

_ 8

· 4; 10

16. Martine bought two pairs of jeans that are on sale for $32.85 each. Use mental math to find the total cost of the jeans. Justify your answer by using the Distributive Property. (Example 2) $65.70; 2($32.85) = 2($33 - $0.15)

17. Financial Literacy Sarah charges $6.50 per hour to babysit. She babysat for 3 hours on Friday and 5 hours on Saturday. Write two equivalent expressions for her total wages. Then find her total wages. (Example 2) $6.50(3 + 5), $6.50(3) + $6.50(5); $52

Use the Distributive Property to write each expression as an equivalent algebraic expression. (Examples 3 and 4)

18. -7.4(10 + a) -74 - 7.4a 4 _ 5 (t - 15) 4

_ 5

t - 12 20. 3.7(r - 1) 3.7r - 3.7

21. (b + 4)12 12b + 48 22. 5(x - 9) 5x - 45 23. - 1 _ 2 (n + 4) -

1

_ 2

n - 2

24. Multiple Representations The volume of seed in a bird feeder is represented by the equation V = 12(24 - h).

a. Symbols Use the Distributive Property to write 12(24 - h) as an equivalent algebraic expression. 288 - 12h

b. Table Make a table of ordered pairs (h, V ).

c. Graph Graph the ordered pairs on the coordinate plane.

d. Words Explain what happens to the volume as the height increases. The volume decreases.

Use Math Tools Find each product mentally. Justify your answer.

25. 8 · 22 176 26. 13 · 39 507 27. 19 · 41 779

28. 29 · 13 377 29. 75 · 40 3000 30. 95 · 38 3610

Use the Distributive Property to write each expression as an equivalent numeric expression. Then evaluate the expression.

(Hint: 3 1 _ 4 can be written as the sum 3 + 1 _ 4 .

)

31. 4 1 _ 5 · 5 5 · 4 + 5 · 1

_ 5

; 21 32. 10 · 5 1 _ 2 10 · 5 + 10 · 1

_ 2

; 55

33. 6 · 4 2 _ 3 6 · 4 + 6 · 2

_ 3

; 28 34. 2 2 _ 7 · 14 14 · 2 + 14 · 2

_ 7

; 32

Aiko uses 2 1 _ 3 yards of fabric to make costumes for a play. Use the Distributive Property to find how much fabric she will need if she makes 9 costumes. 9 · 2 + 9 ·

1

_ 3

; 21 yd

36. Admission to Hersheypark Theme Park in Hershey, Pennsylvania, is $52.95 for an adult and $31.95 for children. The Diego family has a coupon for $10 off each admission ticket. Write an expression to find the cost for x adults and y children.

Independent PracticeIndependent Practice Go online for Step-by-Step SolutionseHelp

-4 · 8 - (-4)5; -12 4.5 · 16 + 4.5 · 8; 108

12.3 · 9 + 12.3 · 4; 159.9

19

h

24 in.

B

b–c. See Answer Appendix.

25–30. See students’ work for justifications.

35

x ($52.95 - $10) + y ($31.95 - $10)

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Lesson 7-1 The Distributive Property 295

Watch Out!Watch Out!Find the Error If students have difficulty finding the error in Exercise 37, have them replace x with 8 in 3(x + 2) and 3x + 2 to determine the error.

H.O.T. Problems H.O.T. Problems Higher Order Thinking

37. Find the Error Julia is using the Distributive Property to simplify 3(x + 2). Find her mistake and correct it.

3(x + 2) = 3x + 2

38. Persevere with Problems Is 3 + (x · y) = (3 + x) · (3 + y) a true statement? If so, explain your reasoning. If not, give a counterexample. No; 3 + (4 · 5) = 23, (3 + 4)(3 + 5) = 56

39. Building on the Essential Question Explain how you can use the Distributive Property and mental math to simplify 2 1 _ 2 · 4 1 _ 2 . See margin.

C

Sample answer: Julia did not distribute 3 to the second term inside the parentheses; the correct answer is 3x + 6.

Find each sum or difference. Write in simplest form. 7.NS.1

44. - 5 _ 8 + 3 _ 4 1 _

8 45. -2 1 _ 2 - 2 _ 3 -3 1 _

6 46. 2 _ 5 + 1 _ 6 17 _

30 47. -5 6 _ 7 + 1 _ 9

48. Jessica needs 5 5 _ 8 yards of fabric to make a skirt and 14 1 _ 2 yards to make a coat. How much fabric does she need in all? 7.NS.3 20 1 _

8 yd

49. Tate’s flower garden has a perimeter of 25 feet. He plans to add 2 feet 9 inches to the width and 3 feet 9 inches to the length. What is the new perimeter in feet? 7.NS.3 38 ft

Evaluate each expression. 7.NS.2

50. -6h, if h = -20 120 51. -4st, if s = -9 and t = 3 108

52. x _ -5 , if x = -85 17 53. 108 _ m , if m = -9 -12

- 362 _ 63

or - 5 47 _ 63

Common Core Review Common Core Review

40. Admission to a science museum is d dollars and a ticket for the 3-D movie is t dollars. Which expression represents the total cost of admission and a movie for p people? C

A dtp C p(d + t) B p + (dt) D d( p + t)

41. Which expression represents the total area of the rectangles? G

2

7x

F 2 + x + 7 + 2 + x + 7 G 2 x + 14 H 2 x + 7 J 14 x

42. Which expression can be written as 7(c + d )? C

A 7c · 7d B (7 + c) · (7 + d) C 7c + 7d D (7 + c) + (7 + d)

43. Short Response A car rental company charges $45 per day to rent a car. If you rent the car for more than seven days, the cost will be reduced by $10 per day.

Write an equation to show the total cost c of renting a car for d days if you rent the car for more than seven days. c = d(45 - 10)

Standardized Test PracticeStandardized Test Practice

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Standardized Test PracticeStandardized Test Practice

Diagnose Student ErrorsSurvey student responses for each item. Class trends may indicate common errors and misconceptions.

40. A does not understand the problem

B does not understand the problem

C correct

D interchanged the d and p variables

41. F found the perimeter

G correct

H did not distribute the 2 times the 7

J multiplied all values

42. A does not understand the Distributive Property

B interchanged the multiplication and the addition signs

C correct

D used all addition signs instead of multiplication and addition

Additional Answer

39. Sample answer: You can break up 2 1 _ 2

to be

2 + 1 _ 2

and 4 1 _ 2

to be 4 + 1 _ 2

. Then set up the

multiplication expression 2 ( 4 + 1 _ 2

) + 1 _ 2

( 4 + 1 _ 2

) .

Distribute: 2 · 4 + 2 · 1 _ 2

+ 1 _ 2

· 4 + 1 _ 2

· 1 _ 2

. Add: 8 + 1

+ 2 + 1 _ 4

= 11 1 _ 4

.

296 Chapter 7 Algebraic Expressions