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810 Unit 9 More about Variables, Formulas, and Graphs

Advance PreparationFor the optional Readiness activity in Part 3, fill in problems on Math Masters, page 296 before making copies. See Part 3 for suggested problems.

Teacher’s Reference Manual, Grades 4–6 pp. 289–294

Key Concepts and Skills• Convert between fractions and decimals. 

[Number and Numeration Goal 5]

• Add and subtract decimals and signed numbers. [Operations and Computation Goal 1]

• Add and subtract fractions. [Operations and Computation Goal 3]

• Use a method to solve equations. [Patterns, Functions, and Algebra Goal 2]

• Use distributive strategies to simplify algebraic expressions. [Patterns, Functions, and Algebra Goal 4]

• Use inverse operations and properties of equality to find equivalent equations. [Patterns, Functions, and Algebra Goal 4]

Key ActivitiesStudents simplify equations by eliminating parentheses and combining like terms. They solve the simplified equations using the equivalent-equations method learned in Lesson 6 �11.

Ongoing Assessment: Recognizing Student Achievement Use journal page 339. [Patterns, Functions, and Algebra Goal 2]

Key Vocabularyequivalent equations � simplify an equation

MaterialsMath Journal 2, pp. 338 and 339Student Reference Book, pp. 251 and 252Study Link 9�4

Applying the Distributive PropertyMath Journal 2, p. 337Students use the distributive property to solve number stories.

Math Boxes 9�5Math Journal 2, p. 340Geometry Template Students practice and maintain skillsthrough Math Box problems.

Study Link 9�5Math Masters, p. 295 Students practice and maintain skillsthrough Study Link activities.

READINESS

Revisiting Pan-Balance ProblemsMath Masters, p. 296Students review a systematic method for solving equations.

ENRICHMENTGenerating Unsimplified Equations to Find Equivalent NamesMath Masters, pp. 297A and 297BStudents use the distributive property to generate equivalent names for numbers.

ENRICHMENTWriting and Solving EquationsMath Masters, p. 297Students translate word sentences into equations and solve them.

EXTRA PRACTICE Solving EquationsMath Masters, p. 298Students simplify and solve equations.

Teaching the Lesson Ongoing Learning & Practice Differentiation Options

Simplifying andSolving Equations

Objective To simplify and solve equations.

������

Common Core State Standards

810_EMCS_T_TLG2_G6_U09_L05_576922.indd 810 2/22/11 11:36 AM

Algebra

Solve 3y + 10 = 7y - 6

Check: Substitute the solution, 4, for y in the original equation:

3y + 10 = 7y - 6 3 ∗ 4 + 10 = 7 ∗ 4 - 6 12 + 10 = 28 - 6 22 = 22

Since 2 2 = 2 2 is true, the solution, 4, is correct.So, y = 4.

Note

A constant is just a number, such as 3 or 7.5 or π. Constants don’t change, or vary, the way variables do.

A Systematic Method for Solving Equations

Many equations with just one unknown can be solved using only addition, subtraction, multiplication, and division. If the unknown appears on both sides of the equals sign, you must change the equation to an equation with the unknown appearing on one side only. You may also have to change the equation to one with all the constants on the other side of the equals sign.

The operations you use to solve an equation are similar to the operations you use to solve a pan-balance problem. Remember—you must always perform the same operation on both sides of the equals sign.

Each step in the above example produced a new equation that looks different from the original equation. But even though these equations look different, they all have the same solution (which is 4). Equations that have the same solution are called equivalent equations.

Step

1. Remove the unknown term (the variable term) from the left side of the equation.

2. Remove the constant term from the right side of the equation.

3. Change the 4 y term to a 1y term. (Remember, 1 y and 1 ∗ y and y all mean the same thing.)

Operation

Subtract 3y from each side.(S 3y)

Add 6 to both sides.(A 6)

Divide both sides by 4.(D 4)

Equation

3 y + 10 = 7 y -6 -3 y -3 y 10 = 4 y -6

10 = 4 y - 6 + 6 + 6 16 = 4 y

16 = 4 y 16 � 4 = 4 y � 4 4 = y

237_254_EMCS_S_SRB_G6_ALG_576523.indd 251 3/15/11 11:05 AM

Student Reference Book, p. 251

Student Page

Lesson 9�5 811

Getting Started

Math MessageRead pages 251 and 252 in your Student Reference Book. Explain why equations a–c are equivalent.

a. 2 _ 3 y + 4 = 10b. 2y - (-12) = 30c. 150 = 60 + 10y

Study Link 9�4 Follow-UpGo over the answers as a class. Resolve disagreements by asking students to substitute at least two different values for each of the variables.

Mental Math and Reflexes Remind students that one way to check if two expressions are equivalent is to substitute a value for the variable. Write pairs of expressions. Students show thumbs-up if the expressions are equivalent and thumbs-down if they are not. Suggestions:

16x ; 4x + 12x thumbs-up 3y ; y + y + y thumbs-up

2(z - 2); 4z - 2 thumbs-down 8n + 12n; 4(2n + 3n) thumbs-up

12g ; 120g ÷ 10g thumbs-down 4(m - 2); -2(4 - 2m) thumbs-up

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

(Student Reference Book, pp. 251 and 252)

Algebraic Thinking Use the same steps outlined in the example on page 251 of the Student Reference Book to solve each equation. Because the solution (y = 9) is the same for all three equations, they are equivalent equations.

Go over the example on page 252. Discuss the general outline of the solution strategy. The equation is first transformed into an equivalent equation that looks like the equations students have been solving.

Review the procedure used in Lesson 9-4 to simplify an equation:

1. Eliminate all parentheses.

2. Combine like terms on each side of the equal sign.

3. Solve. Check the solution using substitution.

Demonstrate how to use the distributive property to remove the parentheses in the first step of the procedure above.

5(b + 3) = (5 ∗ b) + (5 ∗ 3) = 5b + 15

4(b - 1) = (4 ∗ b) - (4 ∗ 1) = 4b - 4

SOLVINGGGGGGLLLLLLLLLLLLVINVINVINNVINNVINVINVINVINVVINVINVINVINVINV GGGGGGGGGGGGOLOOOLOLOLOLOOLOLOOOSOSOSOOSOSOSOSOSOSOOSOSOSOOOOSOOO GGGGGGGGGGGGGLLLLLLLLVINVINVINVINVINVINVINVINVINVINVINVININV GGGGGGGGGGLLLLLLLLLL GGGGGGGGGGOOOOOOOOOOOLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSOLVING

Mathematical PracticesSMP1, SMP2, SMP3, SMP6, SMP8Content Standards6.NS.4, 6.EE.4, 6.EE.5

811-815_EMCS_T_TLG2_G6_U09_L05_576922.indd 811 3/20/12 12:44 PM

Like terms are terms that have exactly the same unknown orunknowns. The terms 4x and 2x are like terms because theyboth contain x. The terms 6 and 15 are like terms because theyboth contain no variables; 6 and 15 are both constants.

If an equation has parentheses, or if the unknown or constantsappear on both sides of the equals sign, here is how you cansimplify it.

♦ If an equation has parentheses, use the distributive propertyor other properties to write an equation without parentheses.

♦ If an equation has two or more like terms on one side of theequals sign, combine the like terms. To combine like termsmeans to rewrite the sum or difference of like terms as asingle term. For example, 4y � 7y � 11y and 4y � 7y � �3y.

Algebra

Solve 5(b � 3) � 3b � 5 � 4(b � 1).Reminder:5(b � 3) means the sameas 5 * (b � 3).

Equations that include thesquare of a variable, like3x2 � 2x � 1, can bedifficult to solve.

In 1145, Abraham barHiyya Ha-Nasi gave acomplete solution to anyequation that can be writtenas ax2 � bx � c � 0.

The solutions are:

x�[�b��(b2�4�ac)� ]/2a

x�[�b��(b2�4�ac)� ]/2a

Operation

1. Use the distributive propertyto remove the parentheses.

2. Combine like terms.

3. Subtract 2b from both sides. (S 2b)

4. Add 4 to both sides. (A 4)

5. Divide both sides by 2. (D 2)

Equation

5b � 15 � 3b � 5 � 4b � 4

2b � 20 � 4b � 4

2b � 20 � 4b � 4� 2b ��2b

20 � 2b � 4

20 � 2b � 4� 4 � � 4

24 � 2b

24 � 2b24 / 2 � 2b / 2

12 � b

1. Check that 12 is the solution of the equation in the example above.

Solve.2. 5x � 7 � 1 � 3x 3. 5 * (s � 12) � 10 * (3 � s) 4. 3(9 � b) � 6(b � 3)

Check your answers on page 423.

Student Reference Book, p. 252

Student Page

Student Page

Simplifying and Solving EquationsLESSON

9�5

Date Time

250–252

Simplify each equation. Then solve it. Record the operations you used for each step.

1. 6y � 2y � 40 2. 5p � 28 � 88 � p

Solution Solution

3. 8d � 3d � 65 4. 12e � 19 � 7 � e

Solution Solution

5. 3n � �12� n � 42 6. 3m � 1 � m � 6 � 2 � 9

Solution Solution

7. 3(1 � 2y ) � y � 2y � 4y 8. 8 � 12x � 6 º (1 � x )

Solution Solution

9. �4.8 � b � 0.6b � 1.8 � 3.6b 10. 4t � 5 � t � 7

Solution Solution

x � �19�

m � �3

e � 2d � 13

n � 12

t � 4b � �3.3

p � 10y � 10

y � 3

Math Journal 2, p. 338

Student Page

�

�

11. 8v � 25 � v � 80 12. 3z � 6z � 60 � z

Solution Solution

13. g � 3g � 32 � 27 � 5g � 2 14. 16 � 3s � 2s � 24 � 2s � 20

Solution Solution

15. Are the following 2 equations equivalent?

5y � 3 � �6y � 4 � 12y 5y � 3 � �6y � 4(1 � 3y)

Explain your answer.

16. Are the following 2 equations equivalent?

5(f � 2) � 6 � 16 f � 1 � 3

Explain your answer.

17. Solve �2z

5� 4� � z � 1 Solution

(Hint: Multiply both sides by 5.)

z � 3

Yes

Yess � 12

z � 6v � 15

Simplifying and Solving Equations continuedLESSON

9�5

Date Time

250–252

Try This

Sample answer: They havethe same solution, y � �1.

Sample answer: They havethe same solution, f � 4.

g � 3

Math Journal 2, p. 339

812 Unit 9 More about Variables, Formulas, and Graphs

▶ Simplifying and

INDEPENDENT ACTIVITY

Solving Equations(Math Journal 2, pp. 338 and 339)

Algebraic Thinking Work through Problem 2 on journal page 338 as a class. Write all steps and operations for solving the problem on the board.

Problem 2:

=

=

= 10p

606p

886p + 28

=5p + 28 88 – p

Operation

Add 1p(A 1p)

Subtract 28.(S 28)

Divide by 6.(D 6)

Remind students to check the solution by substituting it for the variable in the original equation.

5(10) + 28 = 88 - (10)

78 = 78

Assign the remaining problems on journal pages 338 and 339. Allow enough time to go over the answers as a class.

PROBLEMBBBBBBBBBBBOOOOOOOOOOOBBBBBBBBBBBBBBBBBBBBBBBBBBBB MMMMMEEEEMMMLEBLELBLEBLELLLBLEBLEBLEBLEBLEBLEBLEBLEEEMMMMMMMMMMMMMOOOOOOOOOOOBBBBBLBLBLBLBLBLBLLLLPROPROPROPROPROPROPROPROPROPROPROPPRPROPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPROROROROROROOOPPPPPPP MMMMMMMMMMMMMMMMMMMMMMEEEEEEEEEEEEELELELELEEEEEEEELLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRRRRRPROBLEMSOLVING

BBBBBBBBBBBBBBBBBBBLELEELEMMMMMMMMMOOOOOOOOOBLBLBLBLBLBLBLBBLLOOOOROROROROROROROROROO LELELELEEEEEELEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRGGGGLLLLLLLLLLLLLVINVINVINVINNNNVINVINVINNVINVINVINVINVV GGGGGGGGGGGOLOOOLOOLOLOLOO VINVINVVLLLLLLLLVINVINVINVINVINVINVINVINVINVINVINVINNGGGGGGGGGGOOOLOLOLOLOLOLOOOO VVVLLLLLLLLLLVVVVVVVVVOOSOSOOSOSOSOSOSOSOSOOSOSOSOSOOOOSOSOSOSOSOSOSSOOSOSOSOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVVLLLLLLLVVVVVVVVVLVVVVVVVLLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSSS GGGGGGGGGGGGGGGGGOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIIISOLVING

811-815_EMCS_T_TLG2_G6_U09_L05_576922.indd 812 2/22/11 11:36 AM

Number Stories and the Distributive PropertyLESSON

9�5

Date Time

Solve each problem mentally. Then record the number model you used.

1. A carton of milk costs $0.60. John bought 3 cartons of milk one day and 4 cartons the next day.

How much did he spend in all?

Number model

2. During a typical week, Karen runs 16 miles and Jacob runs 14 miles.

About how many miles in all do Karen and Jacob run in 8 weeks?

Number model

3. Mark bought 6 CDs that cost $12 each. He returned 2 of them.

How much did he spend in all?

Number model

4. Max collects stamps. He had 9 envelopes, each containing 25 stamps. He sold 3 envelopes to another collector.

How many stamps did he have left?

Number model

5. Jean is sending party invitations to her friends. She has 8 boxes with 12 invitationsin each box. She has already mailed 5 boxes of invitations.

How many invitations are left?

Number model

36 invitations

150 stamps

$48

About 240 miles

$4.20

248 249

0.60(3 � 4) � n, or (0.60 º 3) � (0.60 º 4) � n

(8 º 16) � (8 º 14) � m, or 8(16 � 14) � m

(6 º 12) � (2 º 12) � c, or (6 � 2) º 12 � c

(9 º 25) � (3 º 25) � s, or (9 � 3) º 25 � s

(8 º 12) � (5 º 12) � p, or (8 � 5) º 12 � p

Math Journal 2, p. 337

Student Page

Math Boxes LESSON

9�5

Date Time

1. The area of the shaded part of therectangle is 20 units2.

Write a number sentence to find the value of h.

Number sentence:

Solve for h.h � units

5. The polygon below is a regular polygon.Find the measure of angle X without using a protractor.

m�X ��120

2. Solve.

a. �13�f � 6 � �8

Solution

b. �30 � b � 6 � 11b

Solution

c. 4g � 4 � 2g � 36

Solution g � 16

b � �2

f � �6

3. Triangles THG and TIN are similar.

a. Length of I�N� � m

b. Length of H�I� � m

c. The size-change factor: �ttrriaianngglele

TTHING

� � : 411512

12

16

h

5

20 � h(16 � 12)

248 249

179

233169

250 251

T N

I

H

G4 m 12 m

3 m5 m

C O

E V

NX

4. I am a quadrangle with 2 pairs ofcongruent adjacent sides. One of my diagonals is also my only line of symmetry.How many sides do I have?

Use your Geometry Template to draw this polygon in the space provided at the right.

4

Math Journal 2, p. 340

Student Page

Lesson 9�5 813

Ongoing Assessment: Journal page 339Problems 15 and 16 �Recognizing Student Achievement

Use journal page 339, Problems 15 and 16 to assess students’ abilities to simplify equations and recognize equivalent equations. Students are making adequate progress if their explanations for Problems 15 and 16 indicate that each equation has the same solution (y = -1); (f = 4). [Patterns, Functions, and Algebra Goal 2]

Point out that the fraction bar in the Try This problem acts as a grouping symbol. The numerator and the denominator in the expression 2z + 4 _ 5 can each be treated as if there were parentheses around them. The expression may be written as the division expression (2z + 4) ÷ 5 or as the multiplication expression 1 _ 5 ∗ (2z + 4). One way to start is to multiply both sides of the equation by 5.

2z + 4 _ 5 = z - 1

5 ∗ ( 2z + 4 _ 5 ) = 5 ∗ (z - 1)

2z + 4 = 5z - 5

2 Ongoing Learning & Practice

▶ Applying the

INDEPENDENT ACTIVITY

Distributive Property(Math Journal 2, p. 337)

Algebraic Thinking Students use the distributive property to solve number stories.

▶ Math Boxes 9�5

INDEPENDENT ACTIVITY

(Math Journal 2, p. 340)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lessons 9-1 and 9-3. The skills in Problems 4 and 5 preview Unit 10 content.

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LESSON

9 � 5

Name Date Time

“Complexifying” Equations to Find Equivalent Names

When you solve an equation, you simplify it first. You can also apply the steps backward to “unsimplify” or “complexify” equations. This process can be used to generate interesting equivalent names for numbers.

Example: “Complexify” an equation to generate an equivalent name for 92.

Solution: Step 1: Start by writing an equation stating that 92 is equal to itself.

92 = 92

Step 2: Next, write 92 as a sum of two whole numbers.

64 + 28 = 92

Step 3: Find a common factor of the two addends on the left side, and use the distributive property to factor it out. For this example, we will use the GCF of 64 and 28, which is 4.

4 ∗ (16 + 7) = 92

This equation shows that 4 ∗ (16 + 7) is another name for 92.

Answer the questions below to generate more names for 92.

1. a. Name another common factor of 64 and 92. 2 b. Repeat Step 3 above using this common factor to generate a different

name for 92.

2 ∗ (32 + 14) = 92

2. a. Repeat Step 2 above by writing 92 as the sum of two whole numbers other than 64 and 28.

Sample answer: 38 + 54 = 92 b. Find the greatest common factor of your two addends and factor it out to

generate another name for 92.

GCF: Sample answer: 2 Equation: Sample answer: 2 ∗ (19 + 27) = 92

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Math Masters, p. 297A

Teaching Master Teaching Master

LESSON

9 � 5

Name Date Time

“Complexifying” Equations to Find Equivalent Names continued

pyg

gp

The equivalent name for 92 that was generated in the Example on page 297A is written in the name-collection box below. Write the two other names for 92 you found in Problems 1 and 2. Then use the same procedure to find at least three equivalent names for the other numbers and write them in the name-collection boxes.

Sample answers are given.

2 ∗ (32 + 14)2 ∗ (19 + 27)

5 ∗ (3 + 6)15 ∗ (1 + 2)5 ∗ (5 + 4)

3 ∗ (6 + 20)6 ∗ (3 + 10)2 ∗ (35 + 4)

4 ∗ (25 + 4)2 ∗ (23 + 35)4 ∗ (15 + 14)

78 116

92

4 º (16 + 7)

45

297A-297B_EMCS_B_G6_MM_U09_576981.indd 297B 3/9/11 12:11 PM

Math Masters, p. 297B

STUDY LINK

9�5

Name Date Time

Equivalent Equations

Each equation in Column 2 is equivalent to an equation in Column 1.Solve each equation in Column 1. Write Any number if all numbers are solutions of the equation.

Match each equation in Column 1 with an equivalent equation in Column 2. Write the letter label of the equation in Column 1 next to the equivalent equation in Column 2.

Column 1 Column 2

A 4x - 2 = 6

Solution

B 3s = -6

Solution

C 3y - 2y = y

Solution

D 5a = 7a

Solution

ABCA

AD

CA

BBAD

x = 2

s = -2

a = 0

Any number

6j + 8 = 8 + 6j

2c - 1 = 3

6w = -12

2h _ 2h = 1

3q _ 3 - 6 = -4

3(r + 4) = 18

2(5x + 1) = 10x + 2

-5x - 5(2 - x) = 2(x - 7)

s = 0

5b - 3 - 2b = 6b + 3

t _ 4 + 3 = 2 1 _ 2

6z = 12

2a = (4 + 7)a

C

Write each product or quotient in exponential notation.

1. 22 ∗ 23 2. 3. 52 ∗ 52 4. 43 _ 42 104

_ 102 415425

Practice

251 252

102

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Math Masters, p. 295

Study Link Master

814 Unit 9 More about Variables, Formulas, and Graphs

▶ Study Link 9�5

INDEPENDENT ACTIVITY

(Math Masters, p. 295)

Home Connection Students solve equations and find equivalent equations.

3 Differentiation Options

READINESS

INDEPENDENT ACTIVITY

▶ Revisiting 5–15 Min

Pan-Balance Problems(Math Masters, p. 296)

To review solving equations, have students generate equivalent equations and record the operations they used on Math Masters, page 296. Remind students that they have solved problems like these as pan-balance problems in the past. Use the suggested problems below or generate problems according to students’ needs. Suggestions:

� 8y + (-5) = 5y + 13 y = 6

� 16f - 24 = 8f f = 3

� 11 + 9k = 71 - 3k k = 5

� 4r + 37 = 100 - 5r r = 7

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LESSON

9�5

Name Date Time

Writing and Solving Equations

Sometimes you need to translate words into algebraic expressions to solve problems.

Example: The second of two numbers is 4 times the first. Their sum is 50. Find the numbers.

If n � the first number, then

4n � the second number, and n � 4n � 50.

Because 5n � 50, n � 10.

The first number is 10 and the second number is 4(10), or 40.

For each problem, translate the words into algebraic expressions. Then write an equationand solve it.

1. The larger of two numbers is 12 more than the smaller. Their sum is 84. Find the numbers.

Equation

Smaller number Larger number

2. Mr. Zock’s sixth-grade class of 29 students has 9 more boys than girls. How many girls are in the class?

Equation Number of girls

Sometimes it helps to label a diagram when you are translating words intoalgebraic expressions.

3. The base (b) of a parallelogram is 3 times as long as anadjacent side (s). The perimeter of the parallelogramis 64 m. What is the length of the base?

Label the diagram at the right. Then write an equationand solve it.

Equation

Length of the base units24

10

4836

x � (x � 12) � 842x � 12 � 84; 2x � 72

g � (g � 9) � 292g � 9 � 29; 2g � 20

3s

s

b

2(3s) � 2s � 646s � 2s � 64; 8s � 64

Math Masters, p. 297

Teaching Master

LESSON

9 �5

Name Date Time

More Simplifying and Solving of Equations

pyg

gp

Simplify each equation. Then solve it. Show your work.

1. 4(5t � 7) � 10t � 2 2. 18(m � 6) � 15m � 6

Solution Solution

3. 4(12 � 8w) � w � 18 4. 3g � 8(2g � 6) � 2 � 14g

Solution Solution

5. �7(1 � 4y) � 13(2y � 3) 6. 4n � 5(7n � 3) � 9(n � 5)

Solution Solution

7. 2(6v � 3) � 18 � 3(16 � 3v) 8. �5 � (�15d � 1) � 2(7d � 16) � d

Solution Solution d � 1v � �12

n � �1y � �16

g � 10w � 2

m ��38t � 3

Math Masters, p. 298

Teaching Master

Lesson 9�5 815

ENRICHMENT PARTNER ACTIVITY

▶ Generating Unsimplified 5–15 Min

Equations to Find Equivalent Names(Math Masters, pp. 297A and 297B)

To extend their work with simplifying equations, remind students that they can take two steps to simplify an equation: first, use the distributive property to eliminate parentheses; then combine like terms.

For example, to simplify this equation: 4 ∗ (9 + 2) = 44

They can distribute the 4: 36 + 8 = 44

Then, they can combine like terms: 44 = 44

Explain that students can also apply these steps backward, starting with a simple equation like 44 = 44 and obtaining a more complex equation like 4 ∗ (9 + 2) = 44. Tell students that using these steps to “complexify” a simple equation can be a way to generate interesting names for numbers.

Have students read the example on Math Masters, page 297A. Then have them work in pairs to complete the problems on Math Masters, pages 297A and 297B.

ENRICHMENT PARTNER ACTIVITY

▶ Writing and Solving Equations 15–30 Min

(Math Masters, p. 297)

Students translate word sentences into equations and then solve the equations.

EXTRA PRACTICE

INDEPENDENT ACTIVITY

▶ Solving Equations 5–15 Min

(Math Masters, p. 298)

To provide extra practice with equations, have students simplify and solve equations involving variable terms on both sides of the equal sign.

Planning Ahead

Gather small #1 standard (1 3 _ 16 in. long) and jumbo size (1 13 _ 16 in. long) paper clips for the Readiness activity in Part 3 of Lesson 9-6.

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LESSON

9 � 5

Name Date Time

“Complexifying” Equations to Find Equivalent Names

297A

Copyright ©

Wright G

roup/McG

raw-H

ill

When you solve an equation, you simplify it first. You can also apply the steps backward to “unsimplify” or “complexify” equations. This process can be used to generate interesting equivalent names for numbers.

Example: “Complexify” an equation to generate an equivalent name for 92.

Solution: Step 1: Start by writing an equation stating that 92 is equal to itself.

92 = 92

Step 2: Next, write 92 as a sum of two whole numbers.

64 + 28 = 92

Step 3: Find a common factor of the two addends on the left side, and use the distributive property to factor it out. For this example, we will use the GCF of 64 and 28, which is 4.

4 ∗ (16 + 7) = 92

This equation shows that 4 ∗ (16 + 7) is another name for 92.

Answer the questions below to generate more names for 92.

1. a. Name another common factor of 64 and 92.

b. Repeat Step 3 above using this common factor to generate a different name for 92.

2. a. Repeat Step 2 above by writing 92 as the sum of two whole numbers other than 64 and 28.

b. Find the greatest common factor of your two addends and factor it out to generate another name for 92.

GCF:

Equation:

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LESSON

9 � 5

Name Date Time

“Complexifying” Equations to Find Equivalent Names continued

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The equivalent name for 92 that was generated in the Example on page 297A is written in the name-collection box below. Write the two other names for 92 you found in Problems 1 and 2. Then use the same procedure to find at least three equivalent names for the other numbers and write them in the name-collection boxes.

297B

78 116

92

4 º (16 + 7)

45

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