Teaching Mathematics With Manipulative

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Course 2

Teaching Mathematics withManipulatives


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Manipulatives

Glencoe offers three types of kits to enhance the use of manipulatives in your

Middle School Mathematics classroom.

• The Glencoe Mathematics Overhead Manipulative Resources contains translucent

manipulatives designed for use with an overhead projector.

• The Glencoe Mathematics Classroom Manipulative Kit contains classroom sets of

frequently used manipulatives in algebra, geometry, measurement, probability, and

statistics.

• The Glencoe Mathematics Student Manipulative Kit contains an individual set of

manipulatives often used in Student Edition activities.

The manipulatives contained in each of these kits are listed on page vi of this

booklet.

Each of these kits can be ordered from Glencoe by calling (800) 334-7344.

Glencoe Mathematics Overhead Manipulative Kit 0-07-830593-4

Glencoe Mathematics Classroom Manipulative Kit 0-02-833116-8

Glencoe Mathematics Student Manipulative Kit 0-02-833654-2

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Permission is granted toreproduce the material contained herein on the condition that such materials be reproduced onlyfor classroom use; be provided to students, teachers, and families without charge; and be usedsolely in conjunction with the Glencoe Mathematics: Applications and Concepts, Course 2,program. Any other reproduction, for sale or other use, is expressly prohibited.

Send all inquiries to:Glencoe/McGraw-Hill8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-07-860125-8 Teaching Mathematics with Manipulatives

Printed in the United States of America.

1 2 3 4 5 6 7 8 9 10 045 11 10 09 08 07 06 05 04 03


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iii

Contents

Easy-to-Make Manipulatives PageBase-Ten Models . . . . . . . . . . . . . . . . . . . . . . .1

Decimal Models . . . . . . . . . . . . . . . . . . . . . . . .2

Fraction Models: Bars . . . . . . . . . . . . . . . . . . .3

Fraction Models: Circles . . . . . . . . . . . . . . . . .4

Counters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Integer Counters . . . . . . . . . . . . . . . . . . . . . . . . 6

Pattern for Cup . . . . . . . . . . . . . . . . . . . . . . . . . 7

Integer Mat . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Equation Mat . . . . . . . . . . . . . . . . . . . . . . . . . .9

Quarter-Inch Grid . . . . . . . . . . . . . . . . . . . . . . 10

Centimeter Grid . . . . . . . . . . . . . . . . . . . . . . .11

Square Dot Paper . . . . . . . . . . . . . . . . . . . . . .12

Isometric Dot Paper . . . . . . . . . . . . . . . . . . . .13

Tangram . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14

Number Lines . . . . . . . . . . . . . . . . . . . . . . . . .15First Quadrant Grids . . . . . . . . . . . . . . . . . . . . 16

Coordinate Planes . . . . . . . . . . . . . . . . . . . . . . 17

Percent Models . . . . . . . . . . . . . . . . . . . . . . . . 18

Spinners . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19

Number Cube Patterns . . . . . . . . . . . . . . . . . .20

Protractors . . . . . . . . . . . . . . . . . . . . . . . . . . .21

Rectangular Prism Pattern . . . . . . . . . . . . . . .22

Cube Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Cylinder Pattern . . . . . . . . . . . . . . . . . . . . . . .24

Cone Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Pyramid Pattern . . . . . . . . . . . . . . . . . . . . . . .26

Pattern Blocks . . . . . . . . . . . . . . . . . . . . . . . .27

Circle Graph Template . . . . . . . . . . . . . . . . . .28

Problem-Solving Guide . . . . . . . . . . . . . . . . .29

Activities PageCHAPTER 1

Teaching Notes and Overview . . . . . .30

1-2 Mini-Project: Modeling Powers and

Exponents . . . . . . . . . . . . . . . . . . .311-4 Using Overhead Manipulatives:

Variables and Expressions . . . . . . .32

1-7b Hands-On Lab Recording Sheet . . . . .34

CHAPTER 2Teaching Notes and Overview . . . . . .35

2-2 Mini-Project: Story Graph . . . . . . . . .37


2-6 Using Overhead Manipulatives:

Quartiles . . . . . . . . . . . . . . . . . . . .39


How Much is a Handful? . . . . . . . .41

CHAPTER 3


3-3 Mini Project: Coordinate Plane

Puzzle . . . . . . . . . . . . . . . . . . . . . .44

3-4a Hands-On Lab Recording Sheet . . . . .45



Multiplying Integers . . . . . . . . . . . .47


4-2a Hands-On Lab Recording Sheet . . . . .514-3 Mini-Project: Solving Multiplication

Equations . . . . . . . . . . . . . . . . . . . .52


Solving Two-Step Equations . . . . . .53



A Function of Time . . . . . . . . . . . .56

CHAPTER 5


5-1a Hands-On Lab Recording Sheet . . . . .595-5 Using Overhead Manipulatives:

Percent . . . . . . . . . . . . . . . . . . . . . . 60



Multiplying Fractions and Mixed

Numbers . . . . . . . . . . . . . . . . . . . .62

6-8 Mini-Project: Perimeter . . . . . . . . . . .64




Equal Ratios . . . . . . . . . . . . . . . . . . 68



7-4 Mini-Project: Scale Drawings . . . . . . .72



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iv


8-1 Mini-Project: Percent and

Estimation . . . . . . . . . . . . . . . . . . .75



Percent of Change . . . . . . . . . . . . .77



Exploring Permutations . . . . . . . . .81


9-6 Mini-Project: Theoretical and

Experimental Probability . . . . . . . .84






Jelly Bean Statistics . . . . . . . . . . . .92




Investigating Triangles and

Quadrilaterals . . . . . . . . . . . . . . . . .95

10-7 Mini-Project: Quadrilateral

Tesselations . . . . . . . . . . . . . . . . . .97


Angles of a Polygon . . . . . . . . . . . .98


Inscribed Polygons . . . . . . . . . . . .100

10-8b Using Overhead Manipulatives:

Dilations . . . . . . . . . . . . . . . . . . .102

10-9b Hands-On Lab Recording Sheet . . . .104

CHAPTER 11Teaching Notes and Overview . . . . .105

11-3a Hands-On Lab Recording Sheet . . . .107


Area . . . . . . . . . . . . . . . . . . . . . . . 108


11-6 Mini-Project: Areas of Circles,

Rectangles, and Squares . . . . . . . .111


Probability and Area Models . . . .112

CHAPTER 12Teaching Notes and Overview . . . . .114



Volume of Pyramids . . . . . . . . . . .117

12-3 Mini-Project: Volume of Solids . . . . .119


12-4b Hands-On Lab Recording Sheet . . . .121


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v

Teacher’s Guide to UsingTeaching Mathematics with Manipulatives

The book contains two sections of masters—

Easy-to-Make Manipulatives and activities for

Middle School Mathematics. Tabs help you

locate the activities for each chapter. A complete list of

manipulatives available in each of the three types of

Glencoe Mathematics Manipulative Kits appears on the

next page.

Easy-to-Make Manipulatives

The first section of this book contains masters for

making your own manipulatives. To make more

durable manipulatives, consider using card stock. To

make algebra tiles similar to those shown in the

Student Edition, have students use markers to color the

tiles appropriately or use colored card stock.

You can also make transparencies of frequently used

items such as grid paper and number lines.

Activity Masters

Each chapter begins with Teaching Notes and

Overview that summarizes the activities for the chapter

and includes sample answers. There are three types of

masters.

Mini-Projects are short projects that enable students to

independently investigate mathematical concepts.

Using Overhead Manipulatives provides instructions

for the teacher to demonstrate an alternate approach to

the concepts of the lesson by using manipulatives on

the overhead projector.

Student Recording Sheets accompany the Hands-On

Lab Activities found in the Student Edition. Students

can easily record the results of the activity on prepared

grids, charts, and figures.


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vi

Glencoe Mathematics Manipulatives

Glencoe Mathematics Overhead Manipulative Resources

ISBN: 0-07-830593-4

Transparencies Overhead Manipulatives

integer mat centimeter grid algebra tilesequation mat number lines spinners

product mat lined paper two-dimensional cups

inequality mat regular polygons red and yellow counters

dot paper polynomial models decimal models (base-ten blocks)

isometric dot paper integer models compass

coordinate grids equation models protractor

geoboard/geobands

geometric shapes

transparency pens in 4 colors

Glencoe Mathematics Classroom Manipulative Kit

ISBN: 0-02-833116-8

Measurement, Probability,

Algebra and Statistics Geometry

algebra tiles base-ten models compasses

counters marbles geoboards

cups measuring cups geobands

centimeter cubes number cubes geomirrors

equation mat/product mat protractors isometric dot grid stamp

coordinate grid stamp and rulers pattern blocks

ink pad scissors tangrams

spinners

stopwatches

tape measures

Glencoe Mathematics Student Manipulative Kit

ISBN: 0-02-833654-2

algebra tiles protractor

red and yellow counters scissorscups geoboard

equation /product mat geobands

compass/ruler tape measure


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© Glencoe/McGraw-Hill 1 Mathematics: Applications and Concepts, Course 2

NAME ______________________________________________ Base-Ten Models


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NAME ______________________________________________ Decimal Models


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NAME ______________________________________________ Fraction Models: Bars

1

1

2

1

2

1

12

1

12

1

12

1

12

1

12

1

12

1

12

1

12

1

12

1

12

1

12

1

12

110

110

110

110

110

110

110

110

110

110

1

9

1

9

1

9

1

9

1

9

1

9

1

9

1

9

1

9

1

8

1

8

1

8

1

8

1

8

1

8

1

8

1

8

1

7

1

7

1

7

1

7

1

7

1

7

1

7

1

6

1

6

1

6

1

6

1

6

1

6

1

5

1

5

1

5

1

5

1

5

1

4

1

4

1

4

1

4

1

3

1

3

1

3


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NAME ______________________________________________ Fraction Models: Circles

1 1

1

2

1

2

15

15

15

15

15

14

14

14

13

13

13

16

16

16

16

17

17

17

17 1

7

17

17

1

8

1

818

18

18

19

19

1

9

1

9

110

110

110

110

112

112

112

112

112

1

12112

1

121

12

112

112

112

110

110 1

10110

110

1101

919 1

9

19

19

18

18

18

16

16

14


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NAME ______________________________________________ Counters


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NAME ______________________________________________ Integer Counters

+++++

+++++

+++++

++––––––––

–––––

–––––


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NAME ______________________________________________ Pattern for Cup


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NAME ______________________________________________ Integer Mat


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NAME ______________________________________________ Equation Mat

=


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NAME ______________________________________________ Quarter-Inch Grid


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NAME ______________________________________________ Centimeter Grid


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NAME ______________________________________________ Square Dot Paper


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NAME ______________________________________________ Isometric Dot Paper


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NAME ______________________________________________ Tangram


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NAME ______________________________________________ Number Lines

–1 1

–1 0

– 9

– 8

–7

– 6

– 5

–4

– 3

–2

–1

0

1

2

3

4

5

6

7

8

9

1 0

1 1

0

1

2

3

4

5

6

7

8

9

1 0

1 1

1 2

1

3

1 4

1 5

1 6

1 7

1 8

1 9

2 0

2 1

2 2

0

1

2

3

4

5

6

7

8

9


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NAME ______________________________________________ First-Quadrant Grids

10

9

8

7

6

5

4

3

2

1

O 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

O 1 2 3 4 5 6 7 8 9 10


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NAME ______________________________________________ Coordinate Planes


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NAME ______________________________________________ Percent Models


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NAME ______________________________________________ Spinners


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NAME ______________________________________________ Number Cube Patterns

2

13 4

9 2 1

4

3

5


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NAME ______________________________________________ Protractors

90 8 0

1 0 0

10 0

8 0

1 1 0

7 0

1 2 0

6 0

1 3 0

5 0 1 4 0

4 0 1

5 0

3 0

1 6 0 2

0 1 7 0

1 0

1 8 0

0

7 0

1 1 0 6 0

1 2 0 5 0

1 3 0

4 0

1 4 0

3 0

1 5 0

2 0

1 6 0

1 0

1 7

0

0 1 8 0

90 8 0

1 0 0

10 0

8 0

1 1 0

7 0

1 2 0

6 0

1 3 0

5 0 1 4 0

4 0 1

5 0

3 0

1 6 0 2

0 1 7 0

1 0

1 8 0

0

7 0

1 1 0 6 0

1 2 0 5 0

1 3 0

4 0

1 4 0

3 0

1 5 0

2 0

1 0

1 7

0

0 1 8 0

90 8 0

1 0 0

10 0

8 0

1 1 0

7 0

1 2 0

6 0

1 3 0

5 0 1 4 0

4 0 1

5 0

3 0

1 6 0 2

0

1 7 0

1 0

1 8 0

0

7 0

1 1 0 6 0

1 2 0 5 0

1 3 0

4 0

1 4 0

3 0

1 5 0

2 0

1 0

1 7

0

0 1 8 0

90 8 0

1 0 0

10 0

8 0

1 1 0

7 0

1 2 0

6 0

1 3 0

5 0 1 4 0

4 0 1

5 0

3 0

1 6 0 2

0

1 7 0

1 0

1 8 0

0

7 0

1 1 0 6 0

1 2 0 5 0

1 3 0

4 0

1 4 0

3 0

1 5 0

2 0

1 6 0

1 0

1 7

0

0 1 8 0

90 8 0

1 0 0

10 0

8 0

1 1 0

7 0

1 2 0

6 0

1 3 0

5 0 1 4 0

4 0 1

5 0

3 0

1 6 0 2

0

1 7 0

1 0

1 8 0

0

7 0

1 1 0 6 0

1 2 0 5 0

1 3 0

4 0

1 4 0

3 0

1

5 0

2 0

1 0

1 7

0

0 1 8 0

90 8 0

1 0 0

10 0

8 0

1 1 0

7 0

1 2 0

6 0

1 3 0

5 0 1 4 0

4 0 1

5 0

3 0

1 6 0 2

0

1 7 0

1 0

1 8 0

0

7 0

1 1 0 6 0

1 2 0 5 0

1 3 0

4 0

1 4 0

3 0

1

5 0

2 0

1 0

1 7

0

0 1 8 0

1 6 0

1 6 0

1 6 0

1 6 0


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NAME ______________________________________________ Rectangular Prism Pattern


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NAME ______________________________________________ Cube Pattern


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NAME ______________________________________________ Cylinder Pattern


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NAME ______________________________________________ Cone Pattern


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NAME ______________________________________________ Pyramid Pattern


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NAME ______________________________________________ Pattern Blocks


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NAME ______________________________________________ Circle Graph Template

0%95%

90%

85%

80%

75%

70%

65%

60%

55% 50% 45%

40%

35%

30%

25%

20%

15%

10%

5%

1

4

2 3

1 3

1 5

1 6

1 8

1 1 0


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NAME ______________________________________________ Problem Solving Guide

Problem: Explore

ExamineThese steps

can helpyou solveproblems.

Plan

Solve


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Decimal Patterns and AlgebraTeaching Notes and Overview

Mini-Project Modeling Powers and Exponents

(p. 31 of this booklet)

Use With Lesson 1-2.

Objective Use centimeter cubes to model powers and exponents.

Materials

centimeter cubesgrid paper

Students use centimeter cubes to model pow-ers and exponents. On grid paper, they sketchthe top and side view of each model.

Answers

1. top side

2. top side

3. top side

Using Overhead

Manipulatives Variables and Expressions(pp. 32–33 of this booklet)


Objective Use cups and counters to modelalgebraic expressions.

Materials

counters*

cups*integer mat transparency** = available in Overhead Manipulative Resources Kit

• Students use cups and counters to modeland solve algebraic expressions.

• An Extension activity asks students tomodel and solve an algebraic expression

using different values for the variable.

Answers

Answers appear on the teacher demonstrationinstructions on pages 32–33.

Hands-On Lab Recording SheetExploring Sequences(p. 34 of this booklet)

Use With Lesson 1-7b. This corresponds tothe activity on page 37 in the Student Edition.

Objective Explore patterns in sequencesusing paper folding.

Materials

calculator piece of paper colored pencils

A table is provided for students to record their data. Space is also given for students to write

and explain their answers.

Answers

See Teacher Wraparound Edition p. 37.


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Mini-Project(Use with Lesson 1-2)

NAME ______________________________________________ DATE ____________ PERIOD __________

Use centimeter cubes to build each square or cube. On grid paper,

sketch the top view and the side view.1. 23 2. 53 3. 62

top view top view top view

side view side view side view

Modeling Powers and Exponents

Materials

centimeter cubes, grid paper

You can build a square or a cube using centimeter cubes. 32 is a three-by-threesquare and is 1 centimeter cube high. You need 9 centimeter cubes to build 3 2.

The top view of 32 is shown at the left below. The side view of 32 is shown atthe right below.

You need 64 centimeter cubes to build 43. The top view is shown at the left below. The side view is shown at the right below.


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Using Overhead Manipulatives(Use with Lesson 1-4)

Variables and Expressions

Teacher Demonstration

• Tell students that the phrase the sum of two and some number is an algebraicexpression. The number 2 in this phrase is a constant, a number that youknow, and “some number” is an unknown value. The phrase can be repre-sented by a cup for the unknown value and 2 counters for the number 2.

Place a cup and 2 counters on the integer mat.

• Place 6 counters in the cup. The cup now has a value of 6. Remove the coun-ters from the cup and count all the counters. There are a total of 8 counters.

Thus, the expression has a value of 8.

• Ask students what the value of the expression would be if 4 counters were placed in the cup and if no counters were placed in the cup. Model the correctanswers if necessary. (6, 2)

• Clear the mat.

Objective Use cups and counters to model algebraic expressions.

Materials

• counters*

• cups*

• integer mat transparency*

* = available in Overhead Manipulative Resources Kit


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• Tell students that the phrase three times some number is also an algebraicexpression. Ask students how this phrase can be expressed mathematically.(Use three cups.) Stress that the same number of counters must be placed in

each cup. Place 2 counters in each cup. Empty the cups and count all thecounters. There are 6. The expression has a value of 6.

• Ask students what the value of the expression is if 1 counter is placed in each

cup; if 5 counters are placed in each cup. Demonstrate the correct answers if necessary. (3, 15)

Have students complete Exercises 1–5.

Model each phrase with cups and counters. Then put four counters in each

cup. How many counters are there in all? Record your answers by drawing

pictures of your models.

1. the sum of 3 and a number (7) 2. 3 times a number (12)

3. 5 more than a number (9) 4. twice a number plus 1 (9)

5. Write a sentence to describe what the cup represents.

(Sample answer:The cup represents the variable or unknownquantity.)

Extension

Model the phrase four more than three times some number as a third example.

Use 3 cups to represent “three times some number”, and use 4 counters to repre-sent the number 4. Place 5 counters in each cup. Empty the cups and count allthe counters. There are 19. The expression has a value of 19. Ask students whatthe value of the expression is if 1 counter is placed in each cup: if 3 counters arein each cup. (7, 13)

Using Overhead Manipulatives


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Hands-On Lab Recording Sheet(Use with the activity on page 37 in Lesson 1-7b of the Student Edition)

NAME ______________________________________________ DATE ____________ PERIOD __________

Exploring Sequences

Materials

calculator, piece of paper, colored pencils

Investigate

Work with a partner. Complete the table.

Writing Math

Work with a partner.

1. Examine the sequence of numbers in the “Layers” column of your table. Isthe sequence arithmetic or geometric? Then write a rule to find the next threeterms.

2. Examine the sequence of numbers in the “Fraction” column of your table. Is

the sequence arithmetic or geometric? Then write a rule to find the next threeterms. ( Hint: Write each fraction as a decimal.)

3. Assume you could continue the paper-folding process indefinitely. Supposeyour unfolded piece of paper is 0.002 inch thick. Add a column to your tableand find the thickness of the paper for the first five folds.

4. How many folds would it take until the paper is as tall as you?

5. Explain the relationship between the number of layers and the fraction of theshaded region.

Number Layers Fraction of Paperof Folds of Paper Not Shaded

1 2 12

2 4 14

3 8

4

5


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Statistics: Analyzing DataTeaching Notes and Overview

Mini-Project Story Graph



Objective Interpret and describe a linegraph.

Materials

none

Students interpret a given line graph and

describe it by writing a story that fits the datashown.

Sample Answer

I left my house at 4:15 P.M. to go to the storefor Mom. She asked me to buy some milk and tortillas for dinner. I walked 5 minutes beforemeeting my friend, Lina. We talked for 5 min-utes. Then I continued walking to the store atthe same pace I had been walking. I arrived at

the store 10 minutes later. It took me 10 min-

utes to buy milk and tortillas. On the wayhome, I walked a little faster because Momwanted the groceries by 5 P.M. I made it homein 10 minutes. I was early.

Hands-On LabRecording SheetAre You Average?(p. 38 of this booklet)

Use With Lesson 2-4b. This corresponds tothe activity on page 73 in the Student Edition.

Objective Use mean, median, mode, and range to describe a set of data.

Materials

markers

ruler

Students work in groups to collect data, create

a frequency table, and create an appropriategraph to display their data. Students are asked to explain why they chose the type of graphthey did for their data.

Answers


Using Overhead Manipulatives Quartiles(pp. 39–40 of this booklet)


Objective Graph quartiles and determine theinterquartile range.

Materials blank transparencies, prepared as described below

transparency pens*


• Students find and graph the upper and lower quartiles and the interquartile rangeof a given set of data.

• An Extension activity asks students to find and graph the upper and lower quartiles and

the interquartile range of a second set of data, then compare the two sets of data.

Answers



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NAME ______________________________________________ DATE ____________ PERIOD __________

Story Graph Make up a story to fit this graph. The graph describes the distance you are fromyour house with respect to time. The story should include where you begin, end,

and stop along the way. Make sure you also mention how long it takes you totravel from place to place and how long you stay at each place.

To help you understand what is happening, you may want to act out the graph.Use seconds instead of minutes and ignore the distance scale.

100

5 10 15 20 25 30 35 40

200

300

400

500

600

700

800

900

0


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Quartiles


• Prepare blank transparencies with copies of the tables shown below and in theExtension.

• Show students the first table. Tell them that the table shows various suspen-sion bridges in the United States and the length of their main span.

• Tell students that in a large set of data, it is helpful to separate the data intofour equal parts called quartiles. Tell them you are going to find and graph

the quartiles for these data.

• Ask students to list the data in order from least to greatest. (533, 549, 564,610, 655, 655, 701, 704, 853, 1,067, 1,158, 1,280, 1,298) Write the datain order below the table.

• Ask students how to find the median of the data. (Find the data numberthat is in the middle when the data are arranged in order from

least to greatest.) Then have them find the median. (701) Circle themedian and point out that it separates the data into two equal groups.

• On the list of data, place anarrow between 564 and 610 and

between 1,158 and 1,067 asshown. Find their means and tellstudents that these numbers rep-resent the median of the lower half, the lower quartile, and themedian of the upper half, the

upper quartile, of the data.

Objective Graph quartiles and determine the interquartile range.

Materials

• blank transparencies, prepared as described below

• transparency pens*


Length ofName, Location

Main Span (m)

Ambassador International, Detroit 564

Benjamin Franklin, Philadelphia 533

Bronx-Whitestone, New York City 701

Delaware Memorial, Wilmington, Delaware 655

George Washington, New York City 1,067

Golden Gate, San Francisco 1,280

Mackinac Straits, Michigan 1,158

Tacoma Narrows II, Tacoma, Washington 853

San Francisco-Oakland Bay, San Francisco 704

Seaway Skyway, Ogdensburg, New York 655

Throgs Neck, New York City 549

Varrazano-Narrows, New York 1,298

Walt Whitman, Philadelphia 610

Source: Federal Highway Administration



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533 549 564 610 655 655 701 704 853 1,067 1,158 1,280 1,298

564

2

610

587 median

1,067

2

1,158

1,112.5

• Point out that half of the data numbers, the middle half, lie between the lower and upper quartiles, 587 and 1,112.5. Tell them that the range of the middlehalf of the data is called the interquartile range. Ask students to find theinterquartile range of these data. (525.5)

• Below the data, draw a number line from 500 to 1,300. Graph the median of the

data above the number line. Then show students as you graph the least value, thegreatest value, the upper quartile, and the lower quartile above the number line.

• Point out that the numbers graphed separate the data into four groups. Ask how many of the numbers in the data set fall in each group. (3)

Extension

Show students the table below. Tell them that the table shows various suspension

bridges internationally and the lengths of their main spans.

Have students find the median,

upper quartile, lower quartile, leastvalue, and greatest value for the

data. (1,043.5; 1,377; 988; 668;1,990) Then have them graphthose values. Finally, compare thenumber lines for the two sets of data. Ask students what conclu-

sions they can make from the data.(Sample answer: The middlehalf of the data for interna-tional bridges is less spread

out. International bridgesgenerally have longer spansthan those in the UnitedStates.)

500 1,3001,100900700

LQ587533

M701

UQ1,112.5 1,298

Length ofName, Location

Main Span (m)Akashi Kaiko, Japan 1,990

First Bosporus, Istanbul, Turkey 1,074

Forth Road, Queensferry, Scotland 1,006

Humber, Hull, Britain 1,410

Kita Bisan-Seto, Japan 990

Minami Bisan-Seto, Japan 1,100

Ohnaruto, Japan 876

Pierre Laporte, Quebec, Canada 668

Ponte 25 de Abril, Lisbon, Portugal 1,013

Second Bosporus, Istanbul, Turkey 1,090Severn, Beachley, England 988

Shimotsui Straits, Japan 940

Storebelt, Denmark 1,624

Tsing Ma Bridge, Hong Kong 1,377

Source: Federal Highway Administration


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How Much is a Handful?


• Ask four volunteers to trace the outline of their hand on a transparency.

• Place one of the outlines on the overhead and cover the shape with counters.Count the counters used and record on a fifth transparency. Repeat the proce-

dure for each of the three outlines.

• Find the mode, median, and mean of the data. Ask the students which of these“averages” is most representative of the data.

• Discuss the most effective way to present the data. Alternatives include thefrequency table, bar graph, line plot, or stem-and-leaf plot. Use the sixth blank

transparency to prepare the data in the agreed-upon manner.

• Ask students to predict how many counters would be needed to cover a hand outline for a student in the same grade but not in the class.

• Have students randomly choose four students who are not in your class and trace the outline of their hands on a transparency. Record the counters needed

to cover the shape. Have students compare the number of counters with their prediction.

• Trace your own hand on a transparency and find the number of countersneeded to cover the shape. Ask how this number compares with the other data.(In most cases, it will be greater than the average of the studentdata.)

• Ask whether you could use the information about the teacher’s hand to predictabout how many counters would cover other adults’ hands. (Data from one

adult would not be sufficient to predict for other adults.)

Objective Use data to make predictions.

Materials


• 11 blank transparencies

• 40 counters** = available in Overhead Manipulative Resources Kit


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Algebra: IntegersTeaching Notes and Overview

Mini-Project Coordinate Plane Puzzle



Objective Locate points for ordered pairs ona coordinate plane

Materials

none

Given ordered pairs, students locate the

appropriate points labeled on a coordinate plane. Students solve the puzzle by matchingeach point with the correct ordered pair.

Answer

WE ARE COORDINATED!

Hands-On Lab Recording SheetAdding Integers


Use With Lesson 3-4a. This corresponds to

the activity on pages 118–119 in the StudentEdition.

Objective Use counters to model the addi-tion of integers.

Materials

counters

integer mat

Students use counters to add integers. Spaceis provided for students to explain their work.

They will write their own addition sentencesand draw conclusions on how to add integers.

Answers

See Teacher Wraparound Edition pp. 118–119.

Hands-On Lab Recording SheetSubtracting Integers


Use With Lesson 3-5a. This corresponds tothe activity on pages 126–127 in the StudentEdition.

Objective Use counters to model the sub-traction of integers.

Materials

counters

integer mat

Students use counters to subtract integers.

Space is provided for students to explain their work. They will write their own subtractionsentences and draw conclusions on how tosubtract integers.

Answers

See Teacher Wraparound Edition pp. 126–127.


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Multiplying Integers(pp. 47–48 of this booklet)


Objective Multiply integers by usingmodels.

Materials

counters*integer mat transparency*

transparency pen** = available in Overhead Manipulative Resources Kit

This demonstration contains two activities.• Demonstration 1 shows how to model the

multiplication of two positive integers or a positive and a negative integer.

• Demonstration 2 shows how to model the

multiplication of two negative integers.

• Extension questions ask students to modeland solve integer multiplication problems

independently.

Answers


Chapter 3 Algebra: Integers


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NAME ______________________________________________ DATE ____________ PERIOD __________

(4, 5) (5, 6) (5, 6) (4, 2) (2, 4)

(0, 2) (2, 2) (7, 7) (6, 0) (4, 6) (5, 1) (1, 0) (2, 4) (2, 7) (8, 4) (8, 6)

Coordinate Plane Puzzle Use the coordinate plane to find the point for each ordered pair below.

Then write the letter for each point under its ordered pair at the bottom of

the page. When you have filled in all the letters, you will know what two

points on a grid said to each other.

x

y

O -8 -7 -6 -5 -4 -3 -1-2 1 2 3 4 5 6 7 8

-7

-8

-6

-5

-4

-3

-1

-2

5

6

7

8

1

2

3

4

W

A

I

E

R

A

H

J

J

B

G

T

E

D

C

O

O

R

D

N

E

-9

9

9

-9


51/127


Hands-On Lab Recording Sheet(Use with the activity on pages 118–119 in Lesson 3-4a of the Student Edition)

NAME ______________________________________________ DATE ____________ PERIOD __________

Adding Integers

Materials

counters, integer mat

Your Turn

Use counters to find each sum.

a. 5 + 6 b. 3 (5) c. 5 (4)

d. 7 3 e. 2 (5) f. 8 (6)

g. 6 5 h. 3 (6) i. 2 7

j. 8 (3) k. 9 1 l. 4 10

Writing Math

1. Write two addition sentences where the sum is positive. In each sentence,one addend should be positive and the other negative.

2. Write two addition sentences where the sum is negative. In each sentence,one addend should be positive and the other negative.

3. MAKE A CONJECTURE Write a rule that will help you determine thesign when finding the sum of integers.


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Hands-On Lab Recording Sheet(Use with the activity on pages 126–127 in Lesson 3-5a of the Student Edition)

NAME ______________________________________________ DATE ____________ PERIOD __________

Subtracting Integers

Materials

counters, integer mat

Your Turn

Use counters to find each difference.

a. 7 6 b. 5 (3) c. 6 (3)

d. 5 8 e. 6 (3) f. 7 3

g. 5 (7)

Writing Math


1. Write two subtraction sentences where the difference is positive. Make sureyou use a combination of positive and negative integers.

2. Write two subtraction sentences where the difference is negative. Make sure

you use a combination of positive and negative integers.

3. MAKE A CONJECTURE Write a rule that will help you determine thesign of the difference of two integers.


53/127



Multiplying Integers

Teacher Demonstration for Activity 1

• Review with students the meaning of the two colors of counters. If necessary,mark the yellow counters to show they are “positive” counters and the red

counters to show they are “negative” counters.

• Remind students that 3 2 means three sets of two items. Tell students thatmodeling 3 2 means to place 3 sets of 2 positive counters on the mat.Model 3 2. Ask students to complete: 3 2 ____ ? . (6)

• Place 3 sets of 2 negative counters on the mat. Ask students to state the

multiplication sentence that has been modeled. (3 (2) 6)


• Clear the mat. Write 3 2 at the base of the mat. Tell students thatsince 3 is the opposite of 3, 3 2 means to remove 3 sets of 2 positive

counters.

• Place 6 zero pairs on the mat. Ask students to state the value of the mat. (0)

–+ –+ –+

–+ –+ –+

––

––

––

––

––

––

++

++

++

++

++

++

Objective Multiply integers by using models.

Materials

counters*

integer mat transparency*

transparency pen*



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• Remove 3 sets of 2 positive counters. Ask students to state the valueof the mat. (6)

• Ask students to complete the sentence 3 2 = ____ ? . (6)

• Clear the mat. Write 3 (2) at the base of the mat. Ask students howmany zero pairs must be placed on the mat to model 3 (2). (6)

• Place 6 zero pairs on the mat. Ask students whether positive or negativecounters should be removed to model 3 (2). (negative) Remove3 sets of 2 red counter. Ask students the value of the mat. (6)

• Ask students to complete the sentence 3 (2) = ____ ? . (6)


Use counters to find each product. Use your result to write a multiplication

sentence.

1. 2 3 (2 3 6) 2. 2 (4) (2 (4) 8)

3. 2 (3) (2 (3) 6) 4. 3 (4) (3 (4) 12)

5. 4 0 (4 0 0) 6. 1 (5) (1 (5) 5)

7. Find 9 (3) without using models. (27)

Extension

Ask students to model each multiplication and then write a multiplication

sentence.4 3 (4 3 12)4 (3) (4 (3) 12)

4

(

3) (4 (3) 12)4 3 (4 3 12)

+

+

–

–

+

+

–

–

+

+

+

+

+

+

+

+

–

–

+

+

–

–

+

+

–

–

+

+

–

–

–

–

–

–

–

–


55/127


56/127


Hands-On Lab Recording Sheet

Functions and Graphs(p. 55 of this booklet)

Use With Lesson 4-6a. This corresponds tothe activity on page 176 in the StudentEdition.

Objective Graph a function on a scatter plot.

Materials

stop watchuncooked spaghetti

Students will perform the “wave” with variousnumbers of students in the class. They are provided with a graph on which they canrecord their data. They use the graph to find a pattern, make predictions, and explain their

reasoning.

Answers



A Function of Time(pp. 56–57 of this booklet)


Objective Use a function rule to find theoutput of a function.

Materials

coordinate grid transparency*transparency pen*

transparency prepared with the two tablesshown in the demonstration


• Students use given data for the input valuesand the function rule to find the output val-ues of a function.

• Students then use the graph of the function

to make predictions.

• An Extension activity asks students to giveexamples of other functions of time.

Answers


Chapter 4 Algebra: Linear Equations and Functions


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Hands-On Lab Recording Sheet(Use with the activity on pages 154–155 in Lesson 4-2a of the Student

Edition)

NAME ______________________________________________ DATE ____________ PERIOD __________

Solving Equations Using Models

Materials

cups and counters, equation mat

Investigate

The scale at the right is balanced, and the bag contains a certainnumber of blocks.

1. Suppose you cannot look in the bag. How can you find thenumber of blocks in the bag?

2. In what way is a balanced scale like an equation?

3. What does it mean to solve an equation?

Your Turn

Solve each equation using models.

a. x 1 3 b. x 3 7 c. x 4 4

d. x 3 2 e. x 4 1 f. 2 x 1

g. x 3 2 h. x 1 3 i. 4 x 2

Writing Math

1. How is solving an equation similar to keeping a scale in balance.2. For any equation, how can you determine how many counters to add or sub-

tract from each side?

3. Identify the property of numbers that is illustrated by a zero pair.

4. Identify the property of numbers that allows you to add or subtract zerowithout changing the value of a number.

5. MAKE A CONJECTURE Write a rule that you can use to solve an equa-

tion like x 3 2 without using models.


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NAME ______________________________________________ DATE ____________ PERIOD __________

Solving Multiplication Equations

Write the equation modeled by the cups and counters.

1. 2.

________________________________ ________________________________

Solve each equation using cups and counters. Sketch the arrangement in the boxes.

3. 4 x 12 x __________________

4. 2 x 14 x __________________

5. 3 x 15 x __________________

Solve without using models.

6. 5 x 10 x __________________


59/127



Solving Two-Step Equations


• Remind students that, in modeling, red counters represent negative integers,yellow counters represent positive integers, and a cup represents x.

• Review the use of zero pairs. Ask students what happens to an equation if azero pair is added to or subtracted from a side. (An equivalent equationresults.)

• Place 3 cups and 2 yellow counters on the left side of an equation mat and

place 4 red counters on the right side. Ask students what equation is modeled.(3x 2 4)

• Add 2 red counters to each side of the equation mat. Point out that you do thisto create zero pairs on the left side. Remove the zero pairs. Ask students whyyou can remove the zero pairs. (Their value is 0.)

+ – – –

– –

– –+ –

+

+

–

–

–

–

Objective Solve two-step equations by using models.

Materials

• cups*

• counters*

• equation mat transparency*



60/127



• Ask students what equation is represented on the mat. (3x 6)

• Pair up an equal number of counters with each cup.

• Ask students what the solution of the equation 3 x 2 4 is. (2)

Extension

Have students write a few sentences explaining how the modeling demonstrated in this lab uses the work backward strategy presented in Lesson 4-4a. (Sampleanswer: Modeling the solution to two-step equations uses thereverse of the order of operations.)

– –

– –

– –


61/127


Hands-On Lab Recording Sheet(Use with the activity on page 176 in Lesson 4-6a of the Student Edition)

NAME ______________________________________________ DATE ____________ PERIOD __________

Functions and Graphs

Materials

stop watch, uncooked spaghetti

Writing Math


1. Graph the ordered pairs (number of students,

time) on the coordinate grid at the right.

2. Describe how the points appear on your graph.

3. Place one piece of uncooked spaghetti on

your graph so that it covers as many of the points as possible. Predict how long it would take 30 students to complete the “wave.”Make a prediction for 50 students.

4. Find a pattern in the data and use the pattern to predict how long it would takethe students in your school to complete the “wave.” Explain your reasoning.

5. A function describes the relationship between two quantities. In a function, onequantity depends on the other. Complete the sentence: The time it takes to do

the “wave” depends on ____ ? .

T i m e

( s e c o n d s )

Students

50

5

10

15

20

25

30

35

40

10 15 20 25 30 35 40


62/127



A Function of Time


• Tell students that a certain secretary can type an average of 55 words per minute. Ask students to use this information to complete the table.

• Tell students that in this example the minutes are called the input values, thetotal words processed are called output values, and the middle column con-tains the function rule.

• Graph the ordered pairs (minutes, words) on the coordinate grid transparency.Draw a line passing through the points.

• Ask students how they could use the graph to estimate the number of wordstyped in 8 minutes. (Extend the line; 440 words.)

• Say, “Let t be the time in minutes. What is the expression for the total number of words typed in t minutes?” (55t )

Objective Use a function rule to find the output of a function.

Materials

• coordinate grid transparency*

• transparency pen*

• transparency prepared with the two tables shown below


Minutes AverageTotalMinutes Number of Words

Wordsper Minute

1 1 ? (55)

2 2 ? (110)

3 3 ? (165)

4 4 ? (220)

5 5 ? (275)


63/127


ExtensionHave students give examples of other functions of time. (Sample answer:revolutions per minute)

• Tell students that another secretary can type an average of 70 words per minute. Repeat the activity using the information below.


Minutes AverageTotalMinutes Number of Words

Wordsper Minute

1 1 ? (70)

2 2 ? (140)

3 3 ? (210)

4 4 ? (280)

5 5 ? (350)


64/127


Fractions, Decimals, and PercentsTeaching Notes and Overview

Hands-On Lab Recording SheetExploring Factors(p. 59 of this booklet)


Objective Discover factors of wholenumbers.

Materials

15 index cards cut in half markers

Students work as a class to determine factorsof whole numbers by identifying patterns.Space is provided for students to explain their answers and make predictions.

Answers


Using Overhead Manipulatives Percent(p. 60 of this booklet)


Objective Illustrate the meaning of percentusing models.

Materials

centimeter grid transparency*transparency pens*


• Using a model on the centimeter grid trans-

parency, students write a ratio of the number of shaded squares to the total number of squares.

• Students make conclusions about the mean-ing of percent based on the model.

Answers

Answers appear on the teacher demonstration

instructions on page 60.


65/127



NAME ______________________________________________ DATE ____________ PERIOD __________

Exploring Factors

Materials

15 index cards cut in half, markers

Writing Math

Work as a class.

1. How many students are standing at the end of the activity? Which cards are

they holding?

2. LOOK FOR A PATTERN Suppose there were 100 students holding indexcards. Extend the pattern in Exercise 1 to predict the numbers that would beheld by students standing at the end of the activity.

3. Explain the relationship between the numbers on the front and the back of the cards.

4. Separate the cards into two groups: one group with exactly two numbers on

the back of the card and one group with more than two numbers. Describe

any special characteristics of each group.


66/127



Percent


• On the centimeter grid transparency, outline a 10-by-10 square. Using a

different colored pen, shade 25 of the squares as shown.

• Ask, “How many small squares are in the model?” (100)

• Ask, “How many small squares are shaded?” (25)

• Ask students to write a ratio of shaded squares to squares in the model. 12050• Tell students that the model represents 25 percent. Ask them to make a

conjecture about the meaning of the word percent . (Sample answer:Percent is a ratio comparing a number to 100.)

Objective Illustrate the meaning of percent using models.

Materials

• centimeter grid transparency*




67/127


Applying FractionsTeaching Notes and Overview

Using Overhead Manipulatives Multiplying Fractions andMixed Numbers(pp. 62–63 of this booklet)


Objective Use models to multiply fractionsand mixed numbers.

Materials

blank transparency

ruler*transparency pens*


This demonstration contains three activities.• Demonstration 1 shows the multiplication

of two fractions.

• Demonstration 2 shows the multiplication

of a whole number and a fraction.

• Demonstration 3 shows the multiplication

of a fraction and a mixed number.

• Students are asked to find products of frac-tions and mixed numbers independently.

• An Extension activity asks students to modelthe multiplication of two mixed numbers.

Answers


Mini-Project

Perimeter(p. 64 of this booklet)


Objective Measure the sides of figures and find the perimeter.

Materials

ruler

Given several figures, students measure thesides, label them, and then find the perimeter.

Answers

1. P 11

4 1

1

4 1

1

4 1

1

4 5 in.

2. P 34 1

38 2 1

78 6 in.

3. P 11

8 1

1

8 1

7

8 1

1

8 5

1

4 in.

4. P 1 1 11

4 3

1

4 in.

5. P 11

8

5

8 1 1

1

8 2

3

8 6

1

4 in.

6. P 58 1 1 58 78 114 78 614 in.

Hands-On Lab Recording SheetCircumference(p. 65 of this booklet)


the activity on page 274 in the Student

Edition.

Objective Find a relationship between

circumference and diameter.

Materials

ruler measuring tape

circular objects

Students find the diameter and circumferenceof various circular objects and record their

measurements in a table. A graph is provided for students to graph their data, find the slopeof the line, and discover a relationship between circumference and diameter.

Answers



68/127



Multiplying Fractions and Mixed Numbers

Objective Use models to multiply fractions and mixed numbers.

Materials

• blank transparency

• ruler*




• Use the ruler and a black transparency pen todraw a square like the one shown. Divide the

square vertically into halves and horizontallyinto fourths.

• Tell students you are going to model 1

2

1

4.

• Color one half of the square blue.

• Color one fourth of the square red.

• Count the small rectangles that are shaded bothcolors. Ask students what number is represented.

18 Write 1

2

1

4

1

8 below the model.


• Use the ruler and a black transparency pen todraw 2 squares side-by-side. Divide them

horizontally into fifths.

• Tell students you are going to model 2 3

5.

• Color the 2 squares blue.

• Color three-fifths of the squares red.

• Count the small rectangles that are shaded bothcolors. Point out that 6 small rectangles are

shaded and each has an area of 1

5 of a unitsquare. Ask students to add the areas.

65 or 115 Write 2 35

65 or 1

15 below the

model.

Purple Red

Blue

Purple

Blue

Red


69/127



• Draw 2 unit squares side-by-side. Divide them vertically into fourths and horizontally into thirds.

• Tell students you are going to model 11

4

2

3.

• Color 11

4 of the squares blue.

• Color 2

3 of the squares red.

• Count the small rectangles that are shaded both colors. Point out that 10 of the smallrectangles are shaded and each has an area

of 1

1

2 of a unit square. Ask students to add

the areas. 1102 or

56 Write 114

23

1102

or 5

6 below the model.


Use area models to find each product.

1. 1

2

1

3

1

6

2.

2

3

3

4

1

6

2

or 1

2

3. 3

1

2

3

2

or 11

2

4. 2 34 64 or 1

12 5. 113

14 1

42 or

13 6. 312

12 74 or 1

34

7. Find 212 1

13. Use area models if necessary. 26

0 or 3

13

Extension

• Ask students how to suggest a model to

show the product 11

2 1

1

3. (Make a

drawing 2 squares across and

2 squares down. Shade 112 of the

squares vertically and 113 horizon-tally.) Draw the model. Have students find

the product. 63 or 2


Purple

Blue

Red

113

112


70/127



NAME ______________________________________________ DATE ____________ PERIOD __________

Perimeter

Measure the segments in each figure to the nearest eighth of an inch. Label the

segments with their measurements. Then find the perimeter of each figure.

1. 2.

Perimeter __________ Perimeter __________

3. 4.


5. 6.



71/127



NAME ______________________________________________ DATE ____________ PERIOD __________

Circumference

Materials

ruler, measuring tape, circular objects

Investigate

Work with a partner. Record your data in the table below.

Write About It


1. For each object, divide the circumference by the diameter.Record the results in the table above. Round to the nearesttenth if necessary.

2. What do you notice about the ratios?

3. Graph the ordered pair (diameter, circumference) on thecoordinate plane for each object. What do you find?

4. Select two points on the graph and find the slope betweenthem. Select two different points and find the slope. Whatdo you observe about the slopes?

5. Use the graph to predict the circumference of a circular object that has a diameter of 18 centimeters.

6. Write a rule describing how you would find the circumference C of a circle

if you know the diameter d .

ObjectDiameter Circumference

Circumference Diameter(cm) (cm)

y

x O


72/127


Ratios and ProportionsTeaching Notes and Overview

Using Overhead Manipulatives Equal Ratios(pp. 68–69 of this booklet)


Objective Explore the meaning of ratio and proportion.

Materials

counters* blank transparency

transparency pen** = available in Overhead Manipulative Resources Kit

• Students use counters to model ratios byfinding equivalent fractions.

• An Extension activity asks students to work in pairs to find equivalent ratios.

Answers

Answers appear on the teacher demonstrationinstructions on pages 68-69.

Hands-On Lab Recording SheetRate of Change(p. 70 of this booklet)

Use With Lesson 7-2b. This corresponds tothe activity on page 296 in the StudentEdition.

Objective Investigate rate of change.

Materials

square tiles

Students use tiles to build models and find the perimeter of each model. A table and coordi-

nate plane is provided for students to record and graph their data. Students explore rate of change by finding the slope of the graph.

Answers


Hands-On Lab Recording SheetWildlife Sampling


Use With Lesson 7-3b. This corresponds tothe activity on page 301 in the StudentEdition.

Objective Use proportions to estimate.

Materials

small bowl

dried beans

paper cupmarkers

Using dried beans in a bowl to representdeer in a forest, students model the capture-recapture technique used to estimate animal populations. A table is provided for studentsto record their data, as well as space for themto explain their results.

Answers



73/127


Mini-Project Scale Drawings



Objective Create a scale drawing.

Materials

ruler

Using the grid provided, students create a

scale drawing of a room they would like to build. The drawing includes everything they

wish to include in the room drawn to scale.Students must label their drawing and list theobjects in the room and their actual size.

Answers

1. See students’ work. Make sure drawingsare the correct scale based on this informa-tion and the scale of the grid.

2–3. Answers will vary.

Hands-On Lab Recording Sheet

Using a Percent Model(p. 73 of this booklet)


Objective Use a percent model to find a part.

Materials

none

Grids are provided for students to drawmodels to find the percent of a number.

Answers


Chapter 7 Ratios and Proportions


74/127



Equal Ratios


• Define ratio as the comparison of two numbers.

• Place 6 counters in a pile on a blank transparency. Label this pile X. Place 2

counters in a second pile. Label this pile Y. Divide pile X into two groups of 3and pile Y into two groups of 1. Point out that for every three counters in pileX, there is one counter in pile Y. Tell students that piles X and Y have a ratioof 3 to 1.

• Clear the screen. Divide a blank transparency into 2 rows with 4 columns ineach row. Label the sections as shown below. Place 3 counters in section A

and 4 counters in section B. Ask students to name the ratio of the counters inA and B. (3 to 4)

• Place 6 counters in section J. Ask students how many counters must be placed in K so that the ratio of J to K is 3 to 4. (8)

• Place 12 counters in section M. Ask students how many counters must be placed in L so that the ratio of L to M is 3 to 4. (9)

A J L N

B K M P

X Y

Objective Explore the meaning of ratio and proportion.

Materials

• counters*

• blank transparency

• transparency pen*



75/127


• Place 15 counters in section N. Ask students how many counters must be placed in P so that the ratio of N to P is 3 to 4. (20)

• Ask students how they decided how many counters to place in sections K,

L, and P. (Sample answer: Write equivalent fractions. 34 6

8

192

1250)

• Ask, “If there were 72 counters in section M, how many counters would be

needed in section L?” (54)

Extension

In pairs, have students work together asking how many counters should be

placed in sections K, L, and P as they change the number of counters in sectionsJ, M, and N.



76/127


77/127


Hands-On Lab Recording Sheet(Use with the activity on page 301 in Lesson 7-3b of the Student Edition)

NAME ______________________________________________ DATE ____________ PERIOD __________

Wildlife Sampling

Materials

small bowl, dried beans, paper cup, markers

Investigate

Work in groups of three. Record your data in the table below.

Original Number Captured ______

Trial Sample Recaptured P

A

B

C

D

E

F

G

H

I

J

Total

Write About It

Work in groups of three.

1. Find the average of the estimates in column P . Is this a good estimate of thenumber of beans in the bowl? Explain your reasoning.

2. Count the actual number of beans in the bowl. How does this number compare to your estimate?


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NAME ______________________________________________ DATE ____________ PERIOD __________

1. List the objects in your room and their actual size.

2. Why did you want these things in your room?

3. Why did you choose these sizes?

Scale Drawings

The Room of Your Dreams

What if you could build a room of any size and put anything you

wanted in it?

Use the grid as a blueprint. Title your drawing and choose a scale.

Then sketch the things you would put in your room, such as a bed,

a desk, closet, TV, windows, a door. You can make the furnishings

larger or smaller than usual. Make sure you draw everything to

scale. Label everything in the room.


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NAME ______________________________________________ DATE ____________ PERIOD __________

Using a Percent Model

Materials:

none

Your Turn Draw a model to find each part.

a. 70% of 50 b. 45% of 20 c. 20% of 75

Writing Math

1. Suppose your whole family had dinner with you and the total bill was $50.How could you change your model to find the tip?

2. Write a sentence that describes what is represented by

the model at the right.

3. Write a percent problem that can be represented and solved using a model.

4. Make a conjecture about how you could use a model to estimate the percent represented by the ratio 8 out of 50.

0 5025 10075 125 175 225

10

150 200 250

60 70 80 90 10050 4030200

Part

Percent


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Mini-Project Percent and Estimation



Objective Create a scale drawing.

Materials

ruler

Using the grid provided, students create a

scale drawing of a room they would like to

build. The drawing includes everything theywish to include in the room drawn to scale.Students must label their drawing and list theobjects in the room and their actual size.

Answers

1. See students’ work. Make sure drawingsare the correct scale based on this informa-tion and the scale of the grid.

2–3. Answers will vary.

Hands-On LabRecording SheetSampling(p. 76 of this booklet)


the activity on page 344 in the StudentEdition.

Objective Investigate rate of change.

Materials

square tiles

Students use tiles to build models and find the perimeter of each model. A table and coordi-

nate plane is provided for students to record and graph their data. Students explore rate of change by finding the slope of the graph.

Answers


Using Overhead Manipulatives Percent of Change

(pp. 77–78 of this booklet)


Objective Use dot paper to show percent of increase and percent of decrease.

Materials

rectangular dot paper transparency*transparency pens*

* available in Overhead Manipulative Resources Kit

• Students use models on dot paper to showand find percent of increase and percent of

decrease.

• An Extension activity asks students to use agiven figure to show an increase and

decrease of 331

3%.

Answers



Applying PercentTeaching Notes and Overview


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Mini-Project(Lesson 8-1)

NAME ______________________________________________ DATE ____________ PERIOD __________

Percent and Estimation

Estimate the percent of the shaded portion for each figure.

Then count grid squares to find the actual percent shaded.

1. 2.

Estimate: __________________ Estimate: ___________________

Actual: ____________________ Actual: _____________________

3. 4.

Estimate: __________________ Estimate: ___________________

Actual: ____________________ Actual: _____________________

5. How did your estimates compare with the actual percents?

6. Shade your own grid. Estimate the percent shaded and count to find the exact percent.

Estimate: ___________________

Actual: _____________________


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NAME ______________________________________________ DATE ____________ PERIOD __________

Sampling

Materials:

none

Writing Math

Work in groups of three.

State whether each sample is random. Explain.

1. To determine the favorite spectator sport for women over 25 years old,350 women over 25 are surveyed at a professional basketball game.

2. To collect data about the study habits of middle school students in the

Franklin School District, the name of every middle school student in thedistrict is placed in a bag and 250 names are randomly selected.

For Exercises 3-7, refer to the information below.

Suppose you are to survey students in your school.

3. Formulate a hypothesis about students’ activities.

4. Design and conduct a survey. Describe the technique that you used to get arandom sample.

5. Use the back of this sheet to organize and display the results of your surveyin a table or graph.

6. Anal

Teaching Mathematics With Manipulative

Documents

Transcript of Teaching Mathematics With Manipulative