Second Verse, 3 Same as the MATERIALS First 2€¦ · Activity 3.2: Algebra Tiles and the Distributive Property Students use algebra tiles to multiply expressions. The example provided

3

Lesson OverviewStudents consider a situation about packing two suitcases for a camping trip and then combining the contents of the suitcases to motivate the need to combine like terms in algebraic expressions. Students model and simplify algebraic expressions first by using algebra tiles to make sense of combining like terms and then by using the rules and properties. Algebra tiles are then used as a method to make sense of the Distributive Property. Students rewrite expressions using the Distributive Property, the Order of Operation Rules, and combining like terms. Then students use algebra tiles to apply the distributive property to division problems. Finally, students rewrite expressions as a product of two factors.

Grade 6 Expressions and Equations

Apply and extend previous understandings of arithmetic to algebraic expressions.

2. Write, read, and evaluate expressions in which letters stand for numbers.a. Write expressions that record operations with numbers and with letters standing for numbers.

3. Apply the properties of operations to generate equivalent expressions.

Essential Ideas• Algebra tiles are a helpful tool to make sense of rewriting algebraic expressions.• Like terms are two or more terms that have the same variable raised to the same power.• The Distributive Property states that if a, b, and c are any real numbers, then a(b 1 c) 5 ab 1 ac.

Because subtraction is a special form of addition and division is a special form of multiplication,

the Distributive Property can also be expressed as a(b 2 c) 5 ab 2 ac, a 1 b _____ c 5 a __ c 1 b __ c ,

and a 2 b ______ c 5 a __ c 2 b __ c .• An algebraic expression can be written as the product of two factors by applying the

Distributive Property.

Second Verse, Same as the FirstEquivalent Expressions

LESSON 3: Second Verse, Same as the First • M3-35A

MATERIALSAl geblocks/algebra tiles

(must have x, y, x2, 1 tiles/blocks)

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M3-35B • TOPIC 1: Expressions

Lesson Structure and Pacing: 2 Days

Day 1

EngageGetting Started: Packing for a Camping TripStudents consider a situation of two brothers packing for a camping trip. Students model the situation with algebra tiles for different parts of the situation.

DevelopActivity 3.1: Algebra Tiles and Combining Like TermsStudents use algebra tiles to model algebraic expressions. The models are then used to make sense of combining like terms. They simplify algebraic expressions by combining like terms using algebra tiles. Students simplify simple algebraic expressions without algebra tiles.

Activity 3.2: Algebra Tiles and the Distributive PropertyStudents use algebra tiles to multiply expressions. The example provided simplifies the expression 5(x 1 1). Students create similar models as a method to make sense of the Distributive Property. They then simplify expressions using the Distributive Property, Order of Operations, and combining like terms. They also use the Distributive Property to rewrite a division expression: (4x 1 8) 4 4.

Day 2

Activity 3.3: Factoring Algebraic ExpressionsStudents use the Distributive Property to factor algebraic expressions. Students rewrite expressions as a product of two factors, including expressions where the coefficients of the original terms do not have common factors.

Activity 3.4: Simplifying Algebraic ExpressionsStudents simplify algebraic expressions by combining like terms or by both using the Distributive Property and combining like terms without using algebra tiles.

DemonstrateTalk the Talk: Write RightStudents examine a list of expressions and determine whether each is a correct rewriting of a given expression. For each expression that is not rewritten correctly, students will make corrections using the Distributive Property and combining like terms.

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LESSON 3: Second Verse, Same as the First • M3-35C

Getting Started: Packing for a Camping Trip

Facilitation NotesIn this activity, students consider a situation of two brothers packing for a camping trip. Students model the situation with algeblocks/algebra tiles and write expressions for different parts of the situation.

Read the scenario once through and direct the students to model the story as you read. Discuss the different types of clothing that they are packing and how they can be grouped. We can combine the long-sleeved shirts and the short-sleeved shirts for “Shirts” and the shorts and long pants for “Pants”.

Provide students time to draw the models in Question 1 before reading Question 2. Provide each student with algebra tiles for this activity. Students may draw circles for each brother’s suitcase and place the blocks inside.

SummaryAlgebra tiles and models can be used to model real-life situations that can be represented with numeric and algebraic expressions.

Activity 3.1Algebra Tiles and Combining Like Terms

Facilitation NotesIn this activity, students use algebra tiles to model algebraic expressions. The models are then used to make sense of combining like terms. Students simplify algebraic expressions by combining like terms using algebra tiles. Equivalent algebraic expressions and equivalent models are made explicit to students. Students simplify simple algebraic expressions without algebra tiles.

Provide each group of students with a set of algeblocks/tiles. Students need the following tiles: x, y, x2, y2, 1.

For Question 1, hold up each tile in turn so that students can record the tile and write the appropriate value on their drawings. Then have students complete Question 2.

Differentiation strategyFor students who struggle, you may want to practice more with the x, x2, and 1 tiles before you introduce the y and y2 tiles.

ENGAGE

DEVELOP

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M3-35D • TOPIC 1: Expressions

Questions to ask• Explain why the smallest tile can be called a unit square. • What are the length and width of the x tile? • What are the length and width of the y tile? What are the length and

width of the unit tiles? • Explain why the unit tiles match up with the sides of the x tiles.• What tiles have a side length in common with the unit square? Explain.• What tiles have a side length in common with the larger squares?

Explain.Select a student to read the paragraphs after Question 2. Have students work with a partner or in groups to complete Questions 3 through 4. Share responses as a class.

Questions to ask• What are like terms? How are like terms represented with the

algebra tiles? • If more than one tile is needed to represent a term, does it matter if

those tiles are connected to make one large rectangle or are placed with spaces between them?

Have students work with a partner or in groups to complete Questions 5 through 7. Share responses as a class.

Questions to ask• What properties allow you to move the tiles and regroup them with

similar tiles together? • What operation are you using when you combine like terms? • Compare the algebraic expression with the original expression. What

do you notice?• Which expressions contain like terms: 4x 1 3x, 3x 1 3y, 5x2 1 8x,

6 1 9, 10m 2 3m?• Why can you not model all algebraic expressions with the algebra tiles?

SummaryGrouping algebra tiles is similar to combining like terms in an algebraic expression. Like terms are terms of an expression that have the same variable part.

Activity 3.2Algebra Tiles and the Distributive Property

Facilitation NotesIn this activity, students use algebra tiles to multiply expressions. The example provided simplifies the expression 5(x 1 1). Students create similar

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LESSON 3: Second Verse, Same as the First • M3-35E

models as a method to make sense of the Distributive Property. They then simplify expressions using the Distributive Property, Order of Operations, and combining like terms. They also use the Distributive Property to rewrite a division expression: (4x 1 8) 4 4.

Ask a student to read the information introducing the lesson and explaining the worked example. Discuss the worked example as a class and answer Question 1 as a class.

Questions to ask• In what ways is this expression and algebra tile model different from

the previous ones? • How is the Distributive Property modeled using the algebra tiles? • Where are the factors represented in the model? • How do you know which term should be placed horizontally across

the top row of the algebra tile model? • How do you know which term should be placed vertically down the

leftmost column of the algebra tile model? • What does the dot in the uppermost left corner of the algebra tile

model represent? • Which part of the algebra tile model is the product? How is the

product determined? • How do you use algebra tiles to multiply expressions? • Show that the algebra tile model and the process of using the

Distributive Property yield the same results.

Have students work with a partner or in groups to complete Question 2. Share responses as a class.

Questions to ask• How can I combine a two x-tiles and one unit tile to make

a rectangle?• How is the 4 represented with your algebra tiles?• What are the dimensions of the new rectangle?• How is Question 2, part (b) different from the other questions? • What property can be used to rewrite Question 2, part (b) so that it

looks like the other questions? • Was your process of solving Question 2, part (b) any different from

solving the previous questions? If so, explain.


Questions to ask • Explain your process of solving this problem. • What is different about this problem? How did this difference affect

how you solved the problem?

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M3-35F • TOPIC 1: Expressions

• How can you tell if the resulting algebraic expression is simplified completely?

• Why is it okay to rearrange the numbers in Question 3 as you are simplifying? What properties are you using?

Misconceptions• Students often wonder why they are using Distributive Property

rather than Order of Operation Rule stating to solve what is inside parentheses first. This is a good time to make explicit that the in a numeric expression, such as 4(10 1 2), both Order of Operation Rules and Distributive Property are valid procedures and yield the same result. The problems in Question 3 cannot be solved using Order of Operation Rules because the algebraic expression inside parentheses cannot be combined because the terms are not like terms; in this case, Distributive Property is another method they now have available to clear the parentheses.

• A common mistake occurs when students encounter problems such as 5 1 2(x 1 3), where like terms are written in front of the parentheses. In this case, students often combine the like terms first, resulting in the erroneous expression 7(x 1 3). If this occurs, remind students to use the Order of Operation rules; this will result in students completing the Distributive Property first.

• A common mistake occurs when students encounter problems with fractions in front of the parentheses, such as in Question 3, part (b): 2 __ 3 (6x 1 12). Students will often multiply the fraction by the first term, and in the process reduce portions of the fraction when using the dividing out process. Then, when multiplying the fraction by the second term, they erroneously continue with the reduced fraction instead of the correct initial fraction.

Differentiation strategies• For students who struggle, allow them to use the algebra tiles until

they feel comfortable solving these problems without them. • To extend the activity, have students verify their solution using

algebra tiles.

Have students work with a partner or in groups to complete Questions 4 and 5. Share responses as a class.

Questions to ask about Questions 4 and 5• Does it matter how you arrange the algebra tiles? Explain. • How do you know you created equal groups? • Does the operation of division always result in equal parts? • How is the process of distribution shown in the notation?

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LESSON 3: Second Verse, Same as the First • M3-35G

• How is the number of terms in the dividend related to the number of terms in the quotient?

• Is it always possible to rearrange equal groups to make a rectangle? Explain.

Differentiation strategyFor students who struggle using the Distributive Property with division, have then circle each individual division problem to make sure they calculate division with each term.


Questions to ask• What is the difference between the two methods? • Which method uses the Distributive Property before dividing?• In Method 1, why do you not have to divide 3 into the algebraic

expression inside the parentheses.

SummaryAlgebra tiles and models can be used to show how the Distributive Property applies to algebraic expressions.

Activity 3.3Factoring Algebraic Expressions

Facilitation NotesIn this activity, students use the Distributive Property to factor algebraic expressions. Students rewrite expressions as a product of two factors, including expressions where the coefficients of the original terms do not have common factors.

Ask a student to read the introduction to the activity aloud. Complete Question 1 as a class.

Questions to ask• Provide an example of the Distributive Property. • Use your example to explain what is meant by the phrase, “multiply

or divide an expression by a given value” • Use your example to explain what is meant by the phrase, “product

of 2 factors”.• Use your example to explain what is meant by the phrase, “a constant

and a sum of terms”.• What would you have to divide by to make the coefficient of 3

become a 1?

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M3-35H • TOPIC 1: Expressions

• What would the constant become when you divide it by 3?• Why is your new algebraic expression considered a “product of

2 factors”?• Is the value of the original expression changed when you complete

this process?Differentiation strategyTo assist all students, you may want to provide a few more examples of your own so that students begin to recognize the process as “undoing” the Distributive Property. Practice with whole number values.

Ask a student to read the introduction to the worked example aloud. Discuss the worked example and answer Question 2 as a class.

Questions to ask• Why do you think the process is called “factoring”?• How is this worked example different from Question 1?• How do you decide what number to use as the factor outside of

the parentheses?• Why is a 4 placed outside of the parentheses and as a divisor of

each term?• How is this process the same as multiplying by 1 (like you completed

in conversions)?• How do you know whether the constant term inside the parentheses

will be a fraction or not?

SummaryAlgebraic expressions can be factored into a product of two or more algebraic expressions.

Activity 3.4Simplifying Algebraic Expressions

Facilitation NotesIn this activity, students simplify algebraic expressions by combining like terms or by applying the Distributive Property and then combining like terms without using algebra tiles.

Have students work with a partner or in groups to complete the entire activity. Share responses as a class.

Differentiation strategiesFor students who struggle,

• Allow them to use the algebra tiles until they feel comfortable solving these problems without them.

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LESSON 3: Second Verse, Same as the First • M3-35I

• Modify the assignment by having them complete half of the problems in each section.

• Have students mark like terms by underlining or circling them before combining them; make sure that they include the sign in front of each term.

Misconceptions• Outside of the mathematics classroom, “combine” generally means

to add. It may need to be made explicit to students that combining like terms means to add or subtract based upon the signs in the problem.

• Student sometimes refer to the incorrect operation symbol when combining like terms. For example, in the expression 4x 2 2 1 3x, students will calculate 4x 2 3x, looking after the 4x instead of in front of the 3x for the correct operation symbol.

• How can you tell when you need to use the Distributive Property by multiplying or dividing and when you need to use the Distributive Property to factor?

• Did you rewrite the problem with like terms placed together to solve this problem?

• If so, how did you know what sign to place between the terms when you moved them?

• If not, explain how you knew which terms were to be combined and how you knew whether to add or subtract them.

• How do you know when the algebraic expression is simplified as much as possible?

• What is the coefficient of x? What is the coefficient of x2? • If a problem requires both the use of the Distributive Property and

combining like terms, does it matter which operation is completed first? Why or why not.

SummaryAlgebraic expressions can be simplified by combining like terms and using number properties such as the Distributive Property.

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Talk the Talk: Write Right

Facilitation NotesIn this activity, students examine a list of expressions and determine whether each is a correct rewriting of a given expression. For each expression that is not rewritten correctly, students will make corrections using the Distributive Property and combining like terms.

Have students complete Questions 1 and 2 individually. Have them share their answers and reasoning with a partner or in their group before discussing as a whole class.

SummaryKnowing about number properties and combining like terms is helpful when checking work on simplifying algebraic expressions.

DEMONSTRATE

M3-35J • TOPIC 1: Expressions

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Warm Up Answers1. 6 2 __ 3

2. 0.4

3. 11

4. 11

5. The answers are the same.

LESSON 3: Second Verse, Same as the First • M3-35


LEARNING GOALS• Model algebraic expressions with algebra tiles.• Simplify algebraic expressions using algebra tiles.• Simplify algebraic expressions using the associative,

commutative, and distributive properties.• Apply properties of operations to create

equivalent expressions.• Rewrite expressions as the product of two factors.

KEY TERMS• like terms• Distributive Property• equivalent expressions

You have evaluated numeric expressions and written and evaluated algebraic expressions. How do you combine algebraic expressions, like you did with numeric expressions, into as few terms as possible?

WARM UPEvaluate each expression.

1. 5 4 3 __ 4

2. 0.24 4 0.6

3. (14 1 8)

________ 2

4. 14 ___ 2

1 8 __ 2

5. What do you notice about the answers to Questions 3 and 4?

3Second Verse, Same as the FirstEquivalent Expressions

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ELL TipInitiate an Anticipation Chat using the key terms listed for this lesson. Have students write each term in a learning journal, read it aloud, and anticipate the meaning. In their journals, have students self-assess on a scale of 1–10:

• I am comfortable saying this key term aloud.

• I understand what this key term means.

As you work through the lesson, have students refer to their learning journal, repeat the process, and assign new ranks. Check to see that they are becoming more comfortable speaking and understanding each term.

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M3-36 • TOPIC 1: Expressions


Getting Started

Packing for a Camping Trip

Jaden and Jerome, twin brothers, are packing for a weekend camping trip. They lay out the following items to go in the suitcase.

Jaden: Jerome:

1. How many shirts and pairs of pants is each brother packing? Together, how many shirts and pairs of pants are they packing?

Shirts Pants

Jaden

Jerome

Together

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Answers1. Shirts Pants

Jaden 5 3

Jerome 4 3

Together 9 6



Answers2. See below.

Answers1.

2a. 1 1 1 1 1

2b. x 1 x 1 x

2c. x2 1 x2 1 x2

x

x

1 1 111

x

x2 y2

y

y

y

x y

1 1 1

1 1 1

1 1 1

xxx

111

x

x2 x2 x2

x x x

x


As you may have seen in the previous activity, when using algebra tiles to model situations and expressions, it is important to have a shared meaning for each differently-sized algebra tile.

Your teacher will hold up each differently-sized algebra tile and tell you the conventional value of each.

1. Sketch each tile and record its value.

2. Represent each numeric or algebraic expression using algebra tiles. Write an addition expression that highlights the different tiles used in the model. Then, sketch the model below the expression.

a. 3 b. 3x c. 3x2

Algebra Tiles and Combining

Like Terms

ACTIVIT Y

3.1

Your teacher has provided you with algebra tiles.

2. How can you use algebra tiles to model the number of shirts packed by each brother and the number of shirts they packed together?

Your addition expressions should each have 3 terms. Why?

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2. Jaden Jerome Jaden and

Jerome




In an algebraic expression, like terms are two or more terms that have the same variable raised to the same power. The coefficients of like terms can be different. Let’s start our exploration of combining like terms with a review of the properties of arithmetic and algebra that you will use to combine terms.

The expression you wrote in each part of Question 2 was made up of like terms. All tiles that are the same size and have the same value represent like terms.

3. Given the algebra tile model, write an addition expression that highlights the different tiles in the model. Then, if necessary, combine like terms and write the expression using as few terms as possible.

x2

x2

x 1

1

1

1

x

x

4. Analyze the last expression you wrote in Question 3.

a. How many terms are in your expression with the fewest terms? How does this relate to the algebra tile model?

b. What is the greatest exponent in the expression?

c. What is the coefficient of x in the expression? How does this relate to your algebra tile model?

When I combine like terms using models, I just group all the same tiles together.

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Answers3. x2 1 x2 1 x 1 x 1 x 1 1 1 1

1 1 1 1 5 2x2 1 3x 1 4

4a. 3 terms; there are 3 different sized tiles in the model.

4b. The greatest exponent is 2.

4c. The coefficient of x is 3; there are 3 of the x tiles.




5. Consider the model.

xy2 y2

y2

x

x

x

x

a. Write an addition expression that highlights the different tiles in the model.

b. Rearrange the tiles to combine all of the like tiles. How many terms does your expression have now?

c. Write the new algebraic expression represented.

6. Represent the algebraic expression 3x2 1 x 1 2 using algebra tiles. How many types of tiles are needed?

Algebra tiles are helpful tools for combining like terms in algebraic expressions. However, because they only represent whole number tiles, they cannot be used to model all algebraic expressions.

7. Use what you have learned about combining like terms to rewrite each algebraic expression with as few terms as possible.

a. 2x 1 3x 2 4.5x b. 3 1 __ 2

y 1 2 1 4y 1 1 1 __ 4

c. 4.5x 1 6y 2 3.5x 1 7 d. 3 __ 4

x 1 2 1 3 __ 8

x

e. 5x 1 2y 1 1 __ 3

x2 2 3x

So, combining like terms means to add or subtract terms with the same variables. Like 3x 1 5x. That’s 8x.

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Answers5a. 2x 1 y2 1 3x 1 2y2

5b. 2 terms

5c. 5x 1 3y2

6.

7a. 0.5x

7b. 7 1 __ 2 y 1 3 1 __ 4

7c. 1x 1 6y 1 7 5 x 1 6y 1 7

7d. 9 __ 8 x 1 2

7e. 1 __ 3 x2 1 2x 1 2y

y2

y2

y2

x

x

x

x

x

x2

1

1

x2

x2

x




Algebra Tiles and the

Distributive Property

ACTIVIT Y

3.2

Let’s use algebra tiles to explore rewriting algebraic expressions with the Distributive Property.

1

1

11 x

x

WORKED EXAMPLE

Consider the expression 5(x 1 1). This expression has two factors: 5 and the quantity (x 1 1). You can use the Distributive Property to rewrite this expression. In this case, multiply the 5 by each term of the quantity (x 1 1). The model using algebra tiles is shown.

1x

x + 1

1x

1x

1x

1x

1x

1

1

15

1

1

5(x 1 1) 5 5x 1 5

This model is just adding the quantity x 1 1 five times!

When you are

speaking about an

algebraic expression

that is grouped

together with

parentheses, use the

words “the quantity.”

For example 2(x 1 3)

in words would

be “two times the

quantity x plus three.”

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1. Analyze the parts of the mathematical expressions in the worked example. Explain each response.

a. Which expression, 5(x 1 1) or 5x 1 5, shows a product of two factors?

b. How many terms are in 5x 1 5?

c. The number 5 is a coefficient in which expression?

2. Create a model of each expression using your algebra tiles. Then, sketch the model and rewrite the expression using the Distributive Property.

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

NOTES

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Answers1a. 5(x 1 1)

1b. 2

1c. 5x 1 5. It is a coefficient of x.

2a. See below.

2b. 6x 1 2

• 1 1

x x x

x x x

x x x

1 1 1

2a. 8x 1 4

• x x 1

1 x x 1

1 x x 1

1 x x 1

1 x x 1




3. Rewrite each expression using the Distributive Property. Then, combine like terms if possible.

a. 2(x 1 4)

b. 2 __ 3

(6x 1 12)

c. 2(x 1 5) 1 4(x 1 7)

d. 5x 1 2(3x 2 7)

e. 2(y 1 5) 1 2(x 1 5)

f. 1 __ 2

(4x 1 2) 1 8x

So far in this activity, you have multiplied expressions together using the Distributive Property. Now let’s think about how to divide expressions.

Ah, so I distribute like this: 2(x 1 2) 2 ? x 1 2 ? 22x 1 4

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Answers3a. 2x 1 8

3b. 4x 1 8

3c. 6x 1 38

3d. 11x 2 14

3e. 2x 1 2y 1 20

3f. 10x 1 1




How do you think the Distributive Property will play a part in dividing expressions? Let’s find out.

4. Consider the expression (4x 1 8) 4 4, which can also be rewritten as 4x 1 8 ______ 4 .

a. First, represent 4x 1 8 using your algebra tiles. Sketch the model you create.

b. Next, divide your algebra tile model into four equal groups. Then, sketch the model you created with your algebra tiles.

c. Write an expression to represent each group from your sketches in part (b).

d. Verify you created equal groups by multiplying your expression from part (c) by 4. The product you calculate should equal 4x 1 8.

I know that multiplication and division are inverse operations. So, I should start thinking in reverse.

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Answers4a.

4b.

4c. x 1 2

4d. 4(x 1 2) 5 4x 1 8

x 1 1

x 1 1

x 1 1

x 1 1

x 1 1

x 1 1

x 1 1

x 1 1




The model you created in Question 4 is an example that shows that the Distributive Property can be used with division as well as with multiplication.

5. Consider the expression 2x 1 6y 1 4

____________ 2

.

a. Use algebra tiles to represent the division expression.

b. Rewrite the division expression using the Distributive Property. Then, simplify the expression.

2x 1 6y 1 4

____________ 2

5 2x ____ 1

6y ____ 1

4 ____

c. Verify that your answer is correct.

5

Let’s consider the division expression from Question 4.

WORKED EXAMPLE

You can rewrite an expression of the form 4x 1 8 ______ 4 using the Distributive Property.

4x 1 8 _______ 4

5 4x ___ 4

1 8 __ 4

5 1x 1 2

5 x 1 2

So, 4x 1 8 _______ 4

5 x 1 2

To rewrite the

expression, divide

the denominator into

both terms in the

numerator.

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Answers5a.

5b. 2, 2, 2

5c. x 1 3y 1 2; This expression is the same as the one modeled by the algebra tiles.

x y 1

x y 1

y 1

y 1

y

y

x 1

y 1

y

y

x 1

y 1

y

y




Zachary thinks he can simplify algebraic expressions that use the Distributive Property with division without using algebra tiles. He

wants to rewrite 6 1 3(x 1 1)

__________ 3 in as few terms as possible and proposes two different methods.

6. Analyze each correct method.

a. Explain the reasoning used in each method.

b. Which method do you prefer. Why?

Method 1

6 + 3(x + 1) ________ 3 = 6 __ 3 + 3(x + 1)

______ 3

= 2 + (x + 1)

= x + 3

Method 2

6 + 3(x + 1) _________ 3 = 6 + 3x + 3

________ 3

= 3x + 9 _____ 3

= 3x __ 3 + 9 __ 3

= x + 3

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Answers6a. Method 1 divided the

two terms, 6 and 3(x 1 1), each by 3, and then simplified each term. Method 2 distributed 3 to the x and 1, simplified the numerator, and then divided each remaining term in the numerator, 3x and 9, by 3, and simplified.

6b. Answers will vary.




You have used the Distributive Property to multiply and divide algebraic expressions by a given value. The Distributive Property can also be used to rewrite an algebraic expression as a product of two factors: a constant and a sum of terms.

You can write any expression as a product of two factors. In many types of math problems, you often need the coefficient of a variable to be 1. Let’s explore how to use the Distributive Property — without algebra tiles—to rewrite expressions so that the coefficient of the variable is 1.

1. Consider the expression 3x 1 6.

a. Identify the coefficient of the variable term.

b. Use the Distributive Property to rewrite the expression as the product of two factors: the coefficient and a sum of terms.

c. How can you check your work?

Factoring Algebraic

Expressions

ACTIVIT Y

3.3

Using the Distributive

Property to write

an expression as

a product of two

factors is also known

as factoring.

C01_SE_M03_T01_L03.indd 46 3/29/17 11:07 AM

Answers1a. 3

1b. 3(x 1 2)

1c. I divided 3 by 3 to get 1, and I divided 6 by 3 to get 2.




Using the Distributive Property to rewrite the sum of two terms as the product of two factors is also referred to as factoring expressions. In the expression 3x 1 6, you factored out the common factor of 3 from each term and rewrote the expression as 3(x 1 2). In other words, you divided 3 from each term and wrote the expression as the product of 3 and the sum of the remaining factors, (x 1 2).

You can use the same strategy to rewrite an algebraic expression so that the coefficient of the variable is 1 even if the terms do not have common factors.

2. Use the Distributive Property to check that the new expression is equivalent to the original expression in the Worked Example.

3. Rewrite each expression as the product of two factors. Check your answers.

a. 4x 1 5 b. 8x 2 3

c. 1 __ 2 x 2 4 d. 1.1x 1 1.21

WORKED EXAMPLE

Let’s rewrite the expression 4x 2 7 so the coefficient of the variable is 1.

To rewrite the expression, factor out the coefficient 4 from each term. The equivalent expression is the product of the coefficient and the difference of the remaining factors.

4x 2 7 5 4 ( 4x ___ 4

2 7 __ 4

) 5 4 (x 2 7 __

4 )

Remember, you can

multiply or divide any

expression by 1 and

not change its value.

C01_SE_M03_T01_L03.indd 47 4/7/17 9:49 AM

Answers

2. 4(x 2 7 __ 4 ) 5 4 ? x 2 4 ? 7 __ 4 5 4x 2 7

3a. 4(x 1 5 __ 4 )

3b. 8 (x 2 3 __ 8 )

3c. 1 __ 2 (x 2 8)

3d. 1.1(x 1 1.1)


Answers1. x 1 8

2. 5(3x 2 2)

3. x 1 5

4. 1 __ 4 (10 1 x)

5. 7y

6. 4x2 1 4y 1 3x 1 2y2

7. x 1 7

8. 5x2 1 2y


Simplifying Algebraic

Expressions

ACTIVIT Y

3.4

Rewrite each expression using the Distributive Property.

1. 32 1 4x ________ 4

2. 15x 2 10

3. 3(x 1 1) 1 12

_____________ 3

4. 2 1 __ 2

1 1 __ 4

x

Rewrite each algebraic expression in as few terms as possible.

5. 3x 1 5y 2 3x 1 2y 6. 4x2 1 4y 1 3x 1 2y2

7. 7x 1 5 2 6x 1 2 8. x2 1 5y 1 4x2 2 3y

C01_SE_M03_T01_L03.indd 48 3/29/17 11:07 AM



Answers9. x 1 20y

10. 7x 1 19y

11. 13x 1 35

12. 2x

13. 4x 1 3y

14. 3x 1 26y


Rewrite each algebraic expression by applying the Distributive Property and then combining like terms.

9. 4(x 1 5y) 2 3x

11. 3x 1 5(2x 1 7)

13. 3(x 1 2y) 1 3x 2 9y

_______ 3

10. 2(2x 1 5y) 1 3(x 1 3y)

12. 4x 1 6y

________ 2 2 3y

14. 2(x 1 3y) 1 4(x 1 5y) 2 3x

C01_SE_M03_T01_L03.indd 49 3/29/17 11:07 AM



Answers1a. Incorrect; they added 7

to each term instead of multiplying by 7.

1b. Correct

1c. Incorrect; they only multiplied the first term by 7 instead of both.

1d. Correct

1e. Incorrect; they incorrectly combined 3a and 5b to get 8ab.

2a. Correct

2 b. Correct

2c. Incorrect; they added 8 1 3 before they multiplied 3 by (2x 1 5).

2d. Correct

2e. Incorrect; they didn’t multiply 3 by 5.


NOTESTALK the TALK

Write Right

Mr. Martin asked his class to write expressions equivalent to 7(3a 1 5b) and 8 1 3(2x 1 5) and got 5 different responses for each. For each response, determine if the original expression was rewritten correctly. For those not rewritten correctly, describe the mistake that was made in rewriting the expression.

1. 7(3a 1 5b)

a. 10a 1 12b

b. 7(3a) 1 7(5b)

c. 21a 1 5b

d. 21a 1 35b

e. 7(8ab)

2. 8 1 3(2x 1 5)

a. 8 1 3 ∙ 2x 1 3 ∙ 5

b. 23 1 6x

c. 11(2x 1 5)

d. 8 1 6x 1 15

e. 13 1 6x

C01_SE_M03_T01_L03.indd 50 3/29/17 11:07 AM



Assignment AnswersWrite

Sample answer.

You can use the Distributive Property to multiply an expression by a term: 4(2x 1 1) 5 8x + 4.

You can use the Distributive Property to divide an expression by a term:

8x 1 4 ______ 4 5 2x 1 1.

You can use the Distributive Property to factor an expression: 8x 1 4 5 4(2x 1 1).

Practice

1a. x2 1 2y2 1 5

1b. y2 1 4y 1 1

2a. x 1 6y 1 7

2b. 7 __ 8 x 1 2 __ 3 y 1

3 __ 4

3a. Let x represent the number of hours the fire burns. He will use 3x 1 6 logs.

3b. 2(3x 1 6)

3c. See below.3d. 6x 1 12

3e. 3x 1 6 ______ 3 5 3x __ 3 1 6 __ 3 5 x 1 2

3f. 6x 1 2y 23, if x is the number of cars and y is the number of visitors

Assignment

Practice1. Represent each algebraic expression by sketching algebra tiles. Rewrite the expression in a fewer

number of terms, if possible.

a. x2 1 2y2 1 5 b. y2 1 3y 1 1 1 y

2. Rewrite each expression by combining like terms.

a. 4.5x 1 (6y 2 3.5x) 1 7 b. ( 2 __ 3 y 1 5 __ 8 x 1 1 __ 4 ) 1 ( 1 __ 4 x 1 1 __ 2 ) 3. Nelson is going on an overnight family reunion camping trip. He is in charge of bringing the wood for

the campfire. He will start the fire with 6 logs and then plans to add 3 logs for each hour the fire burns.

a. Represent the number of logs he will use as an algebraic expression.

b. Suppose the family decides to stay for 2 nights next year. Write the expression for the number of logs

they would need for 2 nights.

c. Create a model of the situation in part (b) using your algebra tiles, and then sketch the model.

d. Rewrite the expression in part (c) using as few terms as possible.

e. Nelson’s cousin believes they will only need one-third of the firewood Nelson brings for one night.

Represent this as an expression and then use the Distributive Property to rewrite the expression.

f. There are several family members who will be visiting for the day only. The campground charges $6

per car, plus $2 per visitor. One of the families brings a coupon for $3 off their total fee. Write the

expression that represents their total cost for the day. Define the variables.

g. The two oldest uncles at the reunion insist on paying the bill for the daily visitors. They will split

the bill equally. Represent the amount of money each uncle will pay as an expression. Then use the

Distributive Property to rewrite the expression.

4. Rewrite each expression by applying the Distributive Property and combining like terms.

a. 7(2x 1 y) 1 5(x 1 4y) b. 9x 1 6y 1 12y 1 16x

________ 4

c. 6(x 1 1) 1 30

___________ 6

5. Rewrite each expression as a product of two factors, so that the coefficient of the variable is 1.

a. 6x 1 7 b. 2 __ 3 x 1 8

WriteDescribe 3 different ways that

you can use the Distributive

Property to rewrite expressions.

Provide an example for each.

RememberTo rewrite an algebraic expression with as few terms as possible,

use the properties of arithmetic and the Order of Operations.

An algebraic expression containing terms can be written as the

product of two factors by applying the Distributive Property.


C01_SE_M03_T01_L03.indd 51 4/7/17 9:49 AM

3g. 6x 1 2y 23

__________ 2

5 3x + y 2 3 __ 2

4a. 19x 1 27y

4b. 13x 1 9y

4c. x 1 6

5a. 6(x 1 7 __ 6 )

5b. 2 __ 3 (x 1 12)

x2 y2

y2

+ + 1

1

11

1

3c. • x x x 1 1 1 1 1 1

1 x x x 1 1 1 1 1 1

1 x x x 1 1 1 1 1 1

3x 1 6

1

y

y2

+ +y

y

y



Stretch1. Simplify the algebraic expression to include as few terms as possible.

3 [2x 1 4(5y 1 1)] 1 1 __ 4 [8y 1 12 ( 2 __ 3 x 1 1 __ 6 ) ] 2. Rewrite each algebraic expression as the product of two factors, such that the coefficient of the term with

the highest exponent is 1.

a. 2x2 1 5x 1 1

b. 3 __ 4 x3 2 9x2 1 2 __ 3 x 1 10

c. 2.6y2 1 3.9y 2 12.48

Review1. Sheldon Elementary School has a school store that sells many items including folders, pencils, erasers,

and novelty items. The parent association is in charge of buying items for the store.

a. One popular item at the store is scented pencils that come in packs of 24 from the retailer. Write an

algebraic expression that represents the total number of scented pencils they will have available to sell.

Let p represent the number of packs of scented pencils.

b. Another popular item at the store is animal-themed folders. Each pack of folders contains 6 folders.

The store currently has 4 packs in the store and would like to order more. Write an algebraic expression

for the total number of folders they will have after they order more folders. Let x represent the number

of packs of folders they buy.

c. The latest fad is animal-shaped rubber bracelets. The bracelets come in a pack of 24. Write an

algebraic expression that represents the cost of each bracelet. Let c represent the cost of a pack

of 24 bracelets.

2. Determine which rate is faster.

a. 185 miles in 3 hours or 490 miles in 8 hours

b. 70 miles per hour or 100 kilometers per hour

3. Calculate the volume of each solid formed by rectangular prisms.

a. 2 cm 2 cm

2 cm

6 cm

6 cm

2 cm

1 cm 1 cm

b. 2 yd

2 yd

2 yd

4 yd

1 yd

6 yd

C01_SE_M03_T01_L03.indd 52 4/7/17 9:49 AM

Assignment AnswersStretch

1. 8x 1 62y 1 25 ___ 2

2a. 2 (x2 1 5 __ 2 x 1 1 __ 2 )

2b. 3 __ 4 (x3 212x2 1 8 __ 9 x 1 40 ___ 3 )

2c. 2.6 (y2 1 1.5y 24.8)

Review

1a. 24p

1b. 24 1 6x or 4(6) 1 6x

1c. c ___ 24

2a. 185 miles in 3 hours is the fastest rate.

2b. 70 miles per hour is the faster rate.

3a. 36 cubic cm

3b. 20 cubic yd

Documents

Second Verse, 3 Same as the MATERIALS First 2€¦ · Activity 3.2: Algebra Tiles and the Distributive Property Students use algebra tiles to multiply expressions. The example provided