CorrectionKey=B DO NOT EDIT--Changes must be made … 6 Sec 1.pdfGuided Practice 1. The manager of a summer camp has 14 baseballs and 23 tennis balls. The manager buys some boxes of

6.1L E S S O N

Algebraic Expressions

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Interactive Whiteboard Interactive example available online

ADDITIONAL EXAMPLE 1Each week, Steven gets an allowance of $8 plus $3 for each chore he does. His younger sister Jill gets an allowance of $6 plus $2 per chore. Write an expression for how much their parents give Steven and Jill each week if they do the same number of chores. 14 + 5c

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ADDITIONAL EXAMPLE 2A group of 5 people go out to eat and buy appetizers and main dishes. They decide to split the bill so each person pays 20% of the total cost. Appetizers are $5 and main dishes are $10. Write an expression to show how much each person pays. 0.2 × (5a + 10m) or a + 2m

EngageESSENTIAL QUESTION

How do you add, subtract, factor, and multiply algebraic expressions? Sample answer: You can use the properties of addition and the Distributive Property to add and subtract algebraic expressions by combining like terms. You can use the Distributive Property to multiply and factor algebraic expressions.

Motivate the LessonAsk: Can you think of some quantities that vary and some quantities that stay the same?

ExploreConnect to Daily LifeDiscuss with students how temperature and prices are examples of quantities that vary, but the length of a day and the year someone was born are examples of quantities that stay the same. Brainstorm other examples of quantities that vary and quantities that stay the same.

ExplainEXAMPLE 1

Connect Multiple Representations Mathematical PracticesMake sure students understand that an algebraic expression is another way of representing the information from a verbal expression.

Questioning Strategies Mathematical Practices • Why is the Commutative Property used to simplify the expression? It allows the order of the addends to be switched so that the like terms are together.

• What property is used to combine like terms? Distributive Property

YOUR TURNAvoid Common ErrorsIn Your Turn 3, students may forget to distribute the negative sign to the second term within the parentheses. Remind students to distribute the negative sign to each term.

EXAMPLE 2Connect Multiple Representations Mathematical PracticesMake sure students understand that a percent can be changed to a decimal or a fraction.

Questioning Strategies Mathematical Practices • How could you write the expression representing what the band gets to keep using a fraction instead of a decimal? 1 __ 4 × (16.60a + 12.20c).

• What expression could you write to represent how much the band does not get to keep? Sample answer: 3 __ 4 × (16.60a + 12.20c).

Florida StandardsThe student is expected to:

Expressions and Equations—7.EE.1.1

Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.Also 7.EE.1.2

Mathematical Practices

MP.4.1 Problem Solving

173 Lesson 6.1

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5. 7(9k + 6m)

6. 0.2(3b - 15c)

7. 2 _ 3 (6e + 9f - 21g)

Simplify each expression.

YOUR TURN

Using the Distributive PropertyYou can use the Distributive Property to remove the parentheses from an algebraic expression like 3(x + 5). Sometimes this is called “simplifying” or “expanding” the expression. Multiply the quantity in front of parentheses by each term within parentheses: 3(x + 5) = 3 · x + 3 · 5 = 3x + 15.

Marc is selling tickets for a high school band concert. The band gets to keep 25% of the money he collects from ticket sales to put toward new uniforms. Write an expression to represent how much the band gets to keep.

Let a represent the number of adult tickets he sells.

Let y represent the number of youth tickets he sells.

The expression 16.60a + 12.20y represents the amount of money Marc collects from ticket sales.

Write 25% as a decimal: 0.25

Write an expression to represent 25% of the money he collects:

0.25 × ( 16.60a + 12.20y )

25% of adult ticket and youth ticket sales sales

Use the Distributive Property to simplify the expression.

0.25(16.60a) + 0.25(12.20y) = 4.15a + 3.05y

Reflect4. Analyze Relationships Instead of using the Distributive Property to

expand 0.25 × (16.60a + 12.20y), could you have first found the sum 16.60a + 12.20y? Explain.

EXAMPLE 2

How much does the band get to keep if Marc sells

20 adult tickets and 40 youth tickets?

Math TalkMathematical Practices

How much does the band

Math Talk

7.EE.1.1, 7.EE.1.2

ADMIT

ONE

TICKET

Northfi

eld

High

School

Wednesday, Nov. 127:00 P.M.Adults $16.60Youth $12.20

$205

No; 16.60a and 12.20y are not like terms, so they

cannot be added together.

63k + 42m 0.6b - 3c 4e + 6f - 14g

Unit 3174

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7_MFLESE056701_U3M06L1.indd 174 4/8/14 7:32 PM

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ESSENTIAL QUESTION

Adding and Subtracting ExpressionsYou can use the properties of addition along with the Distributive Property to add and subtract algebraic expressions.

Jill and Kyle get paid per project. Jill is paid a project fee of $25 plus $10 per hour. Kyle is paid a project fee of $18 plus $14 per hour. Write an expression to represent how much a company will pay to hire both to work the same number of hours on a project.

Write expressions for how much the company will pay each person. Let h represent the number of hours they will work on the project.

Jill: $25 + $10h Kyle: $18 + $14h

Fee + Hourly rate × Hours Fee + Hourly rate × Hours

Add the expressions to represent the amount the company will pay to hire both.

25 + 10h + 18 + 14h

= 25 + 18 + 10h + 14h

= 43 + 24h

The company will pay 43 + 24h dollars to hire both Jill and Kyle.

Reflect1. Critical Thinking What can you read directly from the expression

43 + 24h that you cannot read directly from the equivalent expression 25 + 10h + 18 + 14h?

EXAMPLEXAMPLE 1

STEP 1

STEP 2

How do you add, subtract, factor, and multiply algebraic expressions?

L E S S O N

6.1 Algebraic Expressions


YOUR TURN

2. 3. (-0.25x - 3) - (1.5x + 1.4) ( 3x + 1 _ 2 ) + ( 7x - 4 1 _ 2 ) Math TrainerOnline Assessment

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7.EE.1.1, 7.EE.1.2

7.EE.1.1

Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. Also 7.EE.1.2

Combine their pay.

Use the Commutative Property.

Combine like terms.

The total amount the company pays in fees and per hour

10x - 4 -1.75x - 4.4

173Lesson 6.1

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PROFESSIONAL DEVELOPMENT

Math BackgroundAn algebraic expression is a mathematical statement constructed from at least one variable and possibly one or more operation symbols and one or more numbers. The table shows examples and nonexamples of algebraic expressions.

Note that an algebraic expression must not contain an equal sign. A mathematical statement that contains an equal sign, such as 4x = 16, is an equation.

Algebraic Expressions

Examples x, 3y - 5, -2xy, z 2 + 1

Nonexamples 7, 20 ÷ (4 + 1), 4x = 16

Integrate Mathematical Practices MP.4.1

This lesson provides an opportunity to address this Mathematical Practice standard. It calls for students to model with mathematics. Defining variables links the symbols to their real-world meanings. Substituting values for the variables allows students to interpret the results in the context of the situation. Students use the sum of expressions to model the situation, to solve the problem, and to answer the question in context.

Algebraic Expressions 174


Add Expressions

Subtract Expressions

Multiply Expressions

Factor Expressions

1. Distribute, if necessary.2. Combine like terms.

1. Find a common factor.2. Divide each term by the

common factor.3. Put parentheses around

the simplified expression.4. Put the common factor

outside the parentheses, indicating multiplication.

YOUR TURNAvoid Common Errors When multiplying a constant by a sum or difference, students may only multiply the first term by the constant. Have them draw arrows from the constant to each term in the parentheses to help them remember to distribute fully.

EXPLORE ACTIVITYEngage with the Whiteboard

After finding one rectangular arrangement in Part B, have students try to think of another possible rectangular arrangement of the tiles. Have a volunteer draw it on

the board and use it to write another factored expression: 2(2x + 4).

Questioning Strategies Mathematical Practices • What are the dimensions of each tile? The x-tile is x units long and 1 unit wide. The 1-tiles are 1 unit long and 1 unit wide.

• What do the dimensions of the rectangle represent? the factors

YOUR TURNFocus on Modeling Mathematical PracticesHave students use algebra tiles to model each exercise. Point out that as the coefficients and constants get larger, modeling becomes more unwieldy.

ElaborateTalk About ItSummarize the Lesson

Have students complete a graphic organizer, such as the one shown here, to describe how to add, subtract, multiply, and factor expressions.

GUIDED PRACTICEEngage with the Whiteboard

Have students volunteer to fill in the blanks in Guided Practice 1– 4 and draw algebra tiles to represent the factoring for Guided Practice 5.

Avoid Common Errors Exercise 6 Remind students that there are several ways to factor the expression. The correct answer is found by placing the greatest possible factor outside the parentheses.

175 Lesson 6.1


Guided Practice

1. The manager of a summer camp has 14 baseballs and 23 tennis balls. The manager buys some boxes of baseballs with 12 baseballs to a box and an equal number of boxes of tennis balls with 16 tennis balls to a box. Write an expression to represent the total number of balls. (Example 1)

Write expressions for the total number of baseballs and tennis balls. Let n represent the number of boxes of each type.

baseballs: + ( )n tennis balls: + ( )n

Find an expression for the total number of balls.

+ + +

+ + +

+

So, the total number of baseballs and tennis balls is + .

2. Use the expression you found above to find the total number of baseballs and tennis balls if the manager bought 9 boxes of each type. (Example 1)

Use the Distributive Property to expand each expression. (Example 2)

3. 0.5(12m − 22n)

0.5(12m − 22n) = 0.5( ) − 0.5( )

= −

4. 2 _ 3 (18x + 6z)

2 _ 3 ( ) + 2 _ 3 ( ) = +

Factor each expression. (Example 3)

5. 2x + 12 6. 12x + 24 7. 7x + 35

STEP 1

STEP 2

8. What is the relationship between multiplying and factoring?

ESSENTIAL QUESTION CHECK-IN???

Combine the two expressions.

Use the Commutative Property.

Combine like terms.

Distribute 0.5 to both terms in parentheses.

Multiply.

14 12 23 16

14 12n

12n

23

23

16n

16n

37

289

12m

6m

18x

2(x + 6) 12(x + 2)

You multiply numbers or expressions to produce a product.

You factor a product into the numbers or expressions that were

multiplied to produce it.

7(x + 5)

6z 12x 4z

11n

22n

28n

14

37 28n

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EXPLORE ACTIVITY

× The width is 4 +1-tiles, or 4.

The length is 1 x tile and 2 +1-tiles, or x + 2.

Factoring ExpressionsA factor is a number that is multiplied by another number to get a product. To factor is to write a number or an algebraic expression as a product.

Factor 4x + 8.

Model the expression with algebra tiles.

Use positive x tiles and +1-tiles.

Arrange the tiles to form a rectangle. The total area represents 4x + 8.

Since the length multiplied by the width equals the area, the length and the width of the rectangle are the factors of 4x + 8. Find the length and width.

Use the expressions for the length and width of the rectangle

to write the area of the rectangle, 4x + 8, in factored form.

Reflect8. Communicate Mathematical Ideas How could you use the

Distributive Property to check your factoring?

A

B

C

D

9. 2x + 2

10. 3x + 9

11. 5x + 15 12. 4x + 16

Factor each expression.

YOUR TURN


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7.EE.1.1

4

4(x + 2)

Multiply the factors. The result should be the original expression.

Sample answer:

8 + + + + ++ +

+++

++

+ + + +

++

++

++

++

2(x + 1) 3(x + 3) 5(x + 3) 4(x + 4)

175Lesson 6.1

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ModelingProvide students with algebra tiles. Have students model expressions such as the following.(2x + 8) + (4x - 3)(7x + 5) - (3x + 1)(-4x + 2) - (x + 6)Students should combine tiles of the same type or remove zero pairs to create simplified models of the expressions. Have the students write the simplified expressions.

Visual CuesWhen multiplying a constant by a sum or difference, have students first draw arrows from the constant to each term inside the parentheses. The arrows remind students to also distribute the multiplication to the other terms. For example.

0.4(9x + 5y) = 0.4(9x) + 0.4(5y)

Additional ResourcesDifferentiated Instruction includes: • Reading Strategies • Success for English Learners ELL

• Reteach • Challenge PRE-AP

DIFFERENTIATE INSTRUCTION




and Intervention

Personal

Online homework assignment available

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Lesson Quiz available online

EvaluateGUIDED AND INDEPENDENT PRACTICE

Concepts & Skills Practice

Example 1Adding and Subtracting Expressions

Exercises 1, 2, 9, 10, 14, 15

Example 2Using the Distributive Property

Exercises 3, 4, 13

Explore ActivityFactoring Expressions

Exercises 5–9, 11, 12

Additional ResourcesDifferentiated Instruction includes: • Leveled Practice Worksheets

6.1 LESSON QUIZ

A company sets up a food booth and a game booth at the county fair. The fee for the food booth is $100 plus $5 per day. The fee for the game booth is $50 plus $7 per day.

1. Write an expression for how much both booths will cost for the same number of days.

2. How much does the company pay for both booths for 5 days?


3. (-0.75x + 6) - (2.5x - 1.9)

4. 8(5x - 3y)

Factor each expression.

5. 4x + 20

6. 6x + 54

Exercise Depth of Knowledge (D.O.K.) Mathematical Practices

9–10 2 Skills/Concepts MP.4.1 Modeling

11–12 2 Skills/Concepts MP.5.1 Using tools

13 3 Strategic Thinking MP.3.1 Logic

14–15 2 Skills/Concepts MP.2.1 Reasoning

16 2 Skills/Concepts MP.1.1 Problem Solving

17 2 Skills/Concepts MP.2.1 Reasoning

18 3 Strategic Thinking MP.7.1 Using Structure

19 3 Strategic Thinking MP.3.1 Logic

7.EE.1.1

7.EE.1.1

Exercises 9 and 10 combine concepts from the Florida cluster “Use properties of operations to generate equivalent expressions.”

Answers1. 100 + 5d + 50 + 7d, or 150 + 12d

2. $210

3. -3.25x + 7.9

4. 40x - 24y

5. 4(x + 5)

6. 6(x + 9)

177 Lesson 6.1


Work Area

TennisBasketball

2x + 6

x + 32x

3x – 14

12

x

x

1

11

1 1

16. Persevere in Problem Solving The figures show the dimensions of a tennis court and a basketball court given in terms of the width x in feet of the tennis court.

a. Write an expression for the perimeter of each court.

b. Write an expression that describes how much greater the perimeter of the basketball court is than the perimeter of the

tennis court.

c. Suppose the tennis court is 36 feet wide. Find all dimensions of the

two courts.

17. Draw Conclusions Use the figure to find the product (x + 3)(x + 2). (Hint: Find the area of each small square or rectangle, then add.)

(x + 3)(x + 2) =

18. Communicate Mathematical Ideas Desmond claims that the product shown at the right illustrates the Distributive Property. Do you agree? Explain why or why not.

19. Justify Reasoning Describe two different ways that you could find the product 8 × 997 using mental math. Find the product and explain why your methods work.

FOCUS ON HIGHER ORDER THINKING

58 × 23 174 1160 1,334

T: 6x + 12, B: 7x + 36

x2 + 5x + 6

Agree. To find 58 × 23, let 23 = 3 + 20. Then

find the product 58(3 + 20). First step: 58(3) = 174.

Second step: 58(20) = 1,160. Third step: 174 + 1160 =

1,334. So, 58(23) = 58(3) + 58(20)

(1) Think of 997 as 1,000 − 3. So, 8 × 997 = 8(1,000 − 3).

By the Distributive Property, 8(1,000 − 3) = 8,000 − 24 =

7,976. (2) Think of 997 as 900 + 90 + 7. By the Distributive

Property, 8(900 + 90 + 7) = 7,200 + 720 + 56 = 7,976.

T: 36 ft by 78 ft, B: 50 ft by 94 ft

x + 24

Unit 3178

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++++

++++

++++ + ++ + +

2x + 4x + 3

?

3x - 3

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Name Class Date

Independent Practice6.1

Write and simplify an expression for each situation.

9. A company rents out 15 food booths and 20 game booths at the county fair. The fee for a food booth is $100 plus $5 per day. The fee for a game booth is $50 plus $7 per day. The fair lasts for d days, and all the booths are rented for the entire time. Write and simplify an expression for the amount in dollars that the company is paid.

10. A rug maker is using a pattern that is a rectangle with a length of 96 inches and a width of 60 inches. The rug maker wants to increase each dimension by a different amount. Let ℓ and w be the increases in inches of the length and width. Write and simplify an expression for the perimeter of the new pattern.

In 11–12, identify the two factors that were multiplied together to form the array of tiles. Then identify the product of the two factors.

11.

12.

13. Explain how the figure illustrates that 6(9) = 6(5) + 6(4).

In 14–15, the perimeter of the figure is given. Find the length of the indicated side.

14.

Perimeter = 6x

15.

Perimeter = 10x + 6

7.EE.1.1, 7.EE.1.2

15(100 + 5d) + 20(50 + 7d) = 2,500 + 215d

2(96 + ℓ) + 2(60 + w) = 312 + 2ℓ + 2w

3 and x + 2; 3x + 6

3x − 7 2x + 6

The area is the product of the length and width

(6 × 9). It is also the sum of the areas of the rectangles

separated by the dashed line (6 × 5 and 6 × 4). So,

6(9) = 6(5) + 6(4).

4 and 2x − 1; 8x − 4

177Lesson 6.1

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Activity available online my.hrw.comEXTEND THE MATH PRE-AP

Challenge

A rectangle with a length of x + 5 has a perimeter of 4x + 14.

1. Write the expression for the width of the rectangle in terms of x.

2. Suppose the perimeter of the rectangle is 42 inches. What are the length and width of the rectangle?

3. Write the expression for the area of the rectangle in terms of x.

4. What is the area of the rectangle when x = 7?

1. x + 2

2. length is 12 in., width is 9 in.

3. x 2 + 7x + 10

4. 108 in2



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CorrectionKey=B DO NOT EDIT--Changes must be made … 6 Sec 1.pdfGuided Practice 1. The manager of a summer camp has 14 baseballs and 23 tennis balls. The manager buys some boxes of