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Grade 6 Lesson 1-14: Properties of Equality Everything you need to review and teach a Grade 6 lesson. Print Teacher's Edition pages, Student Edition pages, and ancillaries for Lesson 1-14, Properties of Equality--all at once! Lesson Components: · Teacher's Edition, Chapter 1 Introduction and Lesson 1-14 · · · · · · · · · · · · · · · · · · · Student Edition, pages 44­47 "Equation," "Inverse Operations," and "Properties of Equality" Math Vocabulary Kit cards Home-School Connection, Chapter 1 Family Letters Every Student Learns, pages 11 and 12 Teaching Tools 7 and 9 Problem of the Day 1-14 Problem Solving Masters/Workbook, page 14 Discovery Channel School Masters Introduction Benchmark Tests Introduction Overview of the Math Diagnosis and Intervention System Math Diagnosis and Intervention System, "Equality and Inequality" activity Spiral Review and Test Prep, page 14 Assessment Sourcebook Overview Test-Taking Practice Transparencies, page 14 Practice Masters/Workbook, page 14 Reteaching Masters/Workbook, page 14 Enrichment Masters/Workbook, page 14 Homework Workbook, page 14 Review from Last Year Masters Overview and Table of Contents

Lesson 1-14

Properties of Equality

Key Idea

Properties of equality are used to keep equations balanced.

1. 12 2. 15

12 15 6 18

19 8 7 3

19 8 7 3

How can you keep an equation balanced?

An equation is a sentence stating that two expressions are equal. You can think of a pan balance as a model of an equation. To keep the pans balanced, you do the same thing to both sides. To keep an equation balanced, you do the same operation to both sides by using properties of equality. Properties of Equality

Addition Property of Equality Adding the same number to both sides of an equation does not change the equality. Subtraction Property of Equality Subtracting the same number from both sides of an equation does not change the equality.

3. 6

4. 18

Vocabulary

· equation · properties of equality · inverse operations

4

2

6

You know: Therefore: 9

9 8

8 3 7 5 3 2 8 4

17. 17 3. 17. 17 5. 12. 12 2. 20. 20 4.

You know: 10 Therefore: 10 7 4 3

Multiplication Property of Equality Multiplying You know: both sides of an equation by the same Therefore: 4 nonzero number does not change the equality. Division Property of Equality Dividing both sides of an equation by the same nonzero number does not change the equality.

You know: 12 Therefore: (12 8)

Properties of equality can be used with numerical equations. Example A

You know 11 12 23. Does 11 12 8 23 8? Why or why not?

Yes; the same number, 8, was added to both sides of the equation.

Example B

You know 10 3 30. Does 10 3 5 10 3 Why or why not?

No; 5 was subtracted from the left side of the equation, but the right side was divided by 5.

5?

Both 11 12 8 and 23 8 equal 31.

10 10

3 3

5 5

25, but 6

44

Properties of equality can be used with equations that contain variables. Example C

You know 3x 15. Does 3x 3 15 3? Why or why not?

Yes; both sides of the equation were divided by the same number, 3.

Example D

You know n 6 8. Does n 6 6 8 6? Why or why not?

Yes; the same number, 6, was subtracted from both sides of the equation.

Talk About It

1. Number Sense What could be done to the equation in Example B to keep the sides equal to each other?

Sample answer: From the start, do the same operation to both sides of the equation.

How can you get the variable alone in an equation?

You can get the variable alone by using inverse operations to "undo" the operation applied to the variable. Addition and subtraction are inverse operations, as are multiplication and division. Example E

Explain how to get the variable alone. x x 12 12 12 19 19

The operation is subtraction.

Think It Through

I can think of a pan balance to model equations.

Example F

Explain how to get the variable alone.

n 15 n 15

45 45

The operation is division.

12 Add 12 to both sides

to get x alone and to keep the equation balanced.

15

15 Multiply both sides

by 15 to get n alone.

Talk About It

2. In Example E, why was 12 added to both sides of the equation?

To isolate the variable on one side of the equation.

For another example, see Set 1-14 on p. 69.

Explain how to get the variable alone in each equation. 3. k 13 25 Add 13 to

y

Property of Equality has been followed.

1. You know 35 22 57. Does 35 22 63 57 63? Why or why not? Yes; The Addition

2. You know 10m 120. Does 10m 18 120 18? Why or why not? Yes; The Subtraction 64 sides of the

Divide both

Property of Equality has been followed.

4. 7 14 Multiply both sides 5. 8n both sides of the equation. of the equation by 7. 6. Number Sense A balanced scale shows 8 6 14. If 5 weights are removed from one side, what needs to be done to the other side to keep the scale balanced? 5 weights need to be removed.

equation by 8.

Section C Lesson 1-14

45

For more practice, see Set 1-14 on p. 73.

Skills and Understanding

7. You know 20 4 80. Does 20 4 6 80 6? Why or why not? Yes; The Addition 8. You know 5n 350 5n 350 Does 5 5 ? Why or why not? Yes; The Division

Property of Equality has been followed. Property of Equality has been followed. c c 9. You know 12 11. Does 12 14 11 4? Explain. No; The same number must be added to both sides of the equation for it to remain true.

10. You know 25

Explain how to get the variable alone in each equation.

See margin for Exercises 11­22 . w 11. k 13 29 12. 15 45

Yes; The Subtraction Property of Equality has been followed.

a

13. Does 25

13

a

13

13? Explain.

y

13. 6n 0 60

d 17. 10

42 17 12 t

14. 14 18. 5c 22. 30

7 65 x 14

15. 21 19. 72

m 9y

32

16. a 20. b

48 43

21. 63

23. Reasoning A level pan balance shows 4a should divide to get the variable alone.

20. Explain why you

Division will undo the multiplication of the variable.

Reasoning and Problem Solving

26. Add 8 to both sides of the equation.

24. Keisha makes stained-glass windows using 5 different pieces of glass for each window. She has 125 pieces of glass altogether. If w is the number of windows she can make, the equation 5w 125 represents this situation. How can you get the variable alone? Divide both sides 25. The cost for materials and time is $140 per window. The selling price of $225 includes the materials, time, and profit p for each window. The equation 140 p 225 represents this situation. How can you get the variable alone? Subtract 140 from both

sides of the equation. of the equation by 5.

26. The cost to insure and ship one window is $8. Since Keisha uses part of her profit to pay the shipping cost, the equation p 8 85 represents this situation. How can you get the variable alone? See above. 27. Is the answer below correct? If not, tell why and write the correct expression. A level pan balance shows b to get the variable alone. 3 13. Explain how

Think It Through

I should check if my answer is reasonable.

I would take 3 away from both sides of the balance.

No; To undo the subtraction he must add 3 to both sides of the equation.

46

Extensions

Evaluate each expression for x 28. 2y 2 18 29. y 4

x

12 and y

3. 30. xy 36 31. x y 9

Write each word phrase as an algebraic expression. 32. 23 less than a number m m Evaluate each expression for x 34. 8x

19, 35, 59 23

33. a number s divided by 8 36. 15 24? x 37. x 2

s 8

3, 5, and 8. 3

18, 20, 23 12, 28, 67

5

35. x

38. Which is the best estimate for 39 A. 600 B. 800

0, 2, 5

3

C. 1,000

D. 1,200

Enrichment

The Reflexive, Symmetric, and Transitive Properties

The equality properties are the rules you use when solving equations. Here are three more properties that are used.

Property

Reflexive Symmetric

Statement

a a

Meaning

A number is equal to itself. If numbers are equal, they will remain equal if their order is changed. c, If numbers are equal to the same number, they are equal to each other.

Example

8 If x 8 5, then 5 x

If a b, then b a. If a b and b then a c.

Transitive

If 9 3 6, and 6 4 2, then 9

3

4

2

For 1­3, name the property shown. 1. x In 4­6, name a property that describes each situation.

Symmetric Property Transitive Property

4

y and y

7, then x

4

7

Reflexive Property

2. 4g

4g

Symmetric Property

3. x

11

11

x

4. Kyle is the same height as Derek, so Derek is the same height as Kyle. 5. Ana is the same age as Emilio, and Emilio is the same age as Eva. So, Ana and Eva are the same age. Transitive Property 6. One million equals one million. Reflexive Property

All text pages available online and on CD-ROM. Section C Lesson 1-14 47

Name

TEACHING TOOL

7

© Pearson Education, Inc. 6

Grade 6, Place-Value Blocks

7

Name

TEACHING TOOL

9

© Pearson Education, Inc. 6

Grade 6, Tiles and Counters

9

1­14

Lexie has 6 cats and 2 carriers that can each hold 1 cat. How many different ways can she take 2 cats to the vet for yearly shots? (Hint: Make an organized list.)

Grade 6 Chapter 1 Lesson 14 © Scott Foresman

Name

Properties of Equality

1. Amanda earns $4 an hour baby-sitting, and she made $36 last weekend. If h is the number of hours she baby-sat, the equation 4h 36 represents this situation. How can you get the variable alone?

PS 1-14

2. In Middle School A there were 37 more students in 2002 than in 2000. There were 489 students in the school in 2000. If b represents the number of students in 2002, the equation b 489 37 represents this situation. How can you get the variable alone?

3. Dave and his three friends picked peaches for their mothers today. They each went home with 22 lb of peaches. If x is the x number of peaches they picked altogether, the equation 4 22 represents this situation. How can you get the variable alone?

4. Jennifer's sister is 3 years older than Jennifer. This year Jennifer's sister is 21. If y is Jennifer's age, then y 3 21 represents this situation. How can you get the variable alone?

5. A total of $108 is received for booklets that are sold for $2 each. If k is the number of booklets sold, the equation 2k 108 represents this situation. How can you get the variable alone?

6. Writing in Math You know that 4 Why or why not?

9

36. Does 4

9

12

36

12?

© Pearson Education, Inc. 6

14

Use with Lesson 1-14.

Math Diagnosis and Intervention System

Name

Intervention Lesson J9

Equality and Inequality

More, less, or equal?

Example 1 Compare 4 4 7 4 3 12 3 12 7 5 3 and 12 5 5.

Example 2 Compare 6 6 13 6 7 15 7 15 6 9 7 and 15 9 9.

Compare. Write

1. 10

, 9

, or

for each .

2. 14

49 7 13 6 13

96 13 17 3 15

11

3. 7

2 16 59

7

4. 20

4

5. 8

4

6. 12

5

7. 6

1

8. 18

1

© Pearson Education, Inc.

Replace the with a number to make the sentence true.

9. 9

19

10. 17

9

11. 14

9

12.

6

8

13. 7

14

14. 12

8

17

Math Diagnosis and Intervention System

Name

Intervention Lesson J9

Equality and Inequality (continued)

Compare. Write

15. 23 8

,

, or

for each .

16. 15

8

9 24

1

17. 30

6 12

12

18. 6

9 10

20

19. 22

9 37

2

20. 5

8 42

32

21. 30

3 24

3

22. 25

1 22

1

23. 40

2 54

2

24. 22

68

3

25. 34

5 26

3

Find two whole numbers that make each number sentence true.

26. 16

13

27. 23

26

Test Prep Circle the correct letter for the answer.

28. Compare. 29. Compare.

Choose the correct symbol. 7

A B

Choose the correct symbol. 40

A B

5 12

6

C D

22 15

C D

3

© Pearson Education, Inc.

30. Which number goes in the to make the number sentence true?

14

A 20 18

20

B 14 C 6 D 5

Name

Spiral Review and Test Prep 1-14

Circle the correct answer. 1. Which ratio is equal to 22:55? A. 3:15 B. 2 to 5 C. 33 to 66 D. 88:66

6 mm

5 mm

10 mm

2. How would this word phrase be written as an algebraic expression? 6 less than 7 times the number r A. 6 7r B. 7r 6

x 1 6 2 12

4. What is the volume of the rectangular prism?

C. 6 D. 7r

3 18

7r 6

4 24

5. Darla is 3 times as old as her sister Julia. Write an algebraic expression for Darla's age. Let Julia's age x.

6. 8

8. ( 25)

( 5)

20

14

Use with Lesson 1-14.

© Pearson Education, Inc. 6

3. Which of the following is the rule for the table? Write true or false for each. A. x 5 C. 3x 7. ( 6) ( 4) 10 B. 2x D. 6x

3 4

Overview of the

Assessment Sourcebook

Assessment and instruction are interwoven strands in the fabric of mathematics education. The primary purpose of assessment is to promote learning, so assessment may be referred to as the glue that holds curriculum and instruction together. As a result, the various instructional methods used in Scott Foresman­Addison Wesley Mathematics are supported by different assessment methods. This overview is a brief introduction to the kinds of assessment available in this Assessment Sourcebook, including both formal and informal types of assessment.

Formal Written Tests

A variety of formal written tests are provided to assess students' mastery of important mathematics concepts and skills. Materials Provided Blackline masters (starting on page 1) · Diagnosing Readiness in Grades 1­6 to assess students' understanding of mathematical concepts developed in the previous grade level. · Chapter Tests for use with all individual chapters in the student text. In Grades K­2 there are two forms of the Free Response and the Multiple Choice chapter tests. In Grades 3­6 the chapter tests are called Mixed Formats because they contain free-response, multiple-choice, and writing in math questions. There are two forms of each chapter test. · Cumulative Tests provided for use after Chapters 3, 6, 9, and 12. · A bubble-form Answer Sheet to allow students to practice answering test questions on a separate response sheet.

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vi

Overview

Journal Writing

Journal Writing encourages students to use mathematical language as they reflect on what they are learning. It also provides an opportunity for you, the teacher, to gain insight as to how students approach problem-solving. Materials Provided (starting on page viii) · Tips for assessing and responding to journal entries · Ideas for Journal Prompts

Performance Assessment

Performance tests give a way to assess students' qualities of imagination, creativity, and perseverance. By using performance assessment, you can evaluate how students · reason through problems, · make and test conjectures, · use number sense to predict reasonable answers, and · utilize alternative strategies. Materials Provided (starting on page xviii)

Portfolio Assessment

Portfolio Assessment provides a way of tracking a student's growth and progress over time. A portfolio should include many types of assessment. Materials Provided (starting on page xiii) · Tips and ideas for compiling and managing mathematics portfolios · Inside My Mathematics Portfolio (blackline master) serves as a table of contents for the portfolios · A Mathematics Portfolio Assessment Sheet (blackline master) to record how student portfolios track growth in various areas

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· Performance Assessment tasks to be used after each chapter · Notes that identify the mathematical concepts and skills needed · A four-point Scoring Rubric

Basic-Facts Timed Tests

Basic-Facts Timed Tests provide students with the opportunity to review and practice basic facts. Materials Provided (starting on page 25) · Tips for administering the tests · Tips on adjusting time limits · Additional materials · Basic-Facts Timed Tests to be used before each chapter

Overview

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Name

Test-Taking Practice 1-14

1. You know that g Does g 14 3 14 36 36. 3? Explain.

2. How would you get the variable alone in the equation 34d 177? A. Multiply both sides of the equation by 34. B. Add 177 to both sides of the equation. C. Divide both sides of the equation by 34. D. Subtract 177 from both sides of the equation.

© Pearson Education, Inc. 6

14

Use with Lesson 1-14.

Name

Properties of Equality

1. You know 16 4 20. Does 16 4 4 20 4? Why or why not?

P 1-14

Explain how to get the variable alone in each equation. 2.

f 25

3 t 8 70 6v 70 32 2. 132 24

3. 54 4. m 5. 7t 6. 42 7.

y 20

8. Reasoning Explain why multiplication gets the variable alone in a

Test Prep

9. How would you get the variable alone in the following equation: g A. multiply both sides of the equation by 9

© Pearson Education, Inc. 6

9

7?

B. multiply both sides of the equation by 7 C. divide both sides of the equation by 9 D. divide both sides of the equation by 7 10. Writing in Math To get the variable alone, Ranier added 29 to both sides of b 18 29. Was he correct? Explain.

14

Use with Lesson 1-14.

Name

Properties of Equality

Balanced Equation Unbalanced Equation

R 1-14

To keep both sides of an equation equal, you must do the same thing to both sides.

1

Add 2

1

Add 2 Add 2

1

1

Add 4

3

3

3

5

The equation is balanced because both sides are equal, or have the same amount. We added the same amount to each side of the equation.

The equation is not balanced. 3 does not equal 5. We did not add the same amount to both sides of the equation.

You can use inverse operations to get the variable alone in an equation. Example: x x x 5 5 16 21 5 21 5 The operation is addition, so use its inverse, subtraction. Subtract 5 from both sides of the equation to get x alone and keep the equation balanced.

Explain how to get the variable alone in each equation. 1. 3p 2. 10 27

© Pearson Education, Inc. 6

h

54 15. Explain

3. Reasoning A level pan balance shows g 47 why you should add to get the variable alone.

14

Use with Lesson 1-14.

Name

Equation Tables

1. Solve the equation x 3 y for the given values of x or y to complete the table. 2. Examine the completed table from Exercise 1. What would happen to the value of y if the value of x was increased by five? x y 3 0

E 1-14

REASONING

6 6

3. How would the value of y change if the equation was rewritten as x y 3, the value of x was increased by five, and the equation was still true?

4. How would the value of y change if the equation was rewritten as y + 3 x, the value of x was increased by five, and the equation was still true?

5. Compare the equations in Exercises 1­4. How are they alike? How are they different?

7. How would the value of v change if the value of n was doubled and the equation was still true?

v

2

8

14

Use with Lesson 1-14.

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6. Solve the equation n v 2 for the given values of n or v to complete the table.

n

4

10

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