manipulatives

Teaching Mathematics with Manipulatives

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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 translucentmanipulatives designed for use with an overhead projector.

• The Glencoe Mathematics Classroom Manipulative Kit contains classroom sets offrequently used manipulatives in algebra, geometry, measurement, probability, and statistics.

• The Glencoe Mathematics Student Manipulative Kit contains an individual set ofmanipulatives 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.

ISBN13 ISBN10Glencoe Mathematics Overhead Manipulative Kit 978-0-07-830593-1 0-07-830593-4Glencoe Mathematics Classroom Manipulative Kit 978-0-02-833116-4 0-02-833116-8Glencoe Mathematics Student Manipulative Kit 978-0-02-833654-1 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 Math Connects, Course 1. Any other reproduction, forsale or other use, is expressly prohibited.

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

ISBN: 978-0-07-889763-4MHID: 0-07-889763-7 Teaching Mathematics with Manipulatives

Course 1

Printed in the United States of America.

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Contents

Easy-to-Make Manipulatives Page Base-Ten Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1Decimal Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2Fraction Models: Bars . . . . . . . . . . . . . . . . . . . . . . . . .3Fraction Models: Circles . . . . . . . . . . . . . . . . . . . . . . .4Fraction Wheel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5Counters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6Integer Counters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7Pattern for Cup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8Integer Mat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9Algebra Tiles—1 tiles . . . . . . . . . . . . . . . . . . . . . . . .10Algebra Tiles—Variables . . . . . . . . . . . . . . . . . . . . .11Product Mats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12Equation Mat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13Quarter-Inch Grid . . . . . . . . . . . . . . . . . . . . . . . . . . .14Centimeter Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . .15Square Dot Paper . . . . . . . . . . . . . . . . . . . . . . . . . . .16Isometric Dot Paper . . . . . . . . . . . . . . . . . . . . . . . . .17Number Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18First Quadrant Grid . . . . . . . . . . . . . . . . . . . . . . . . . .19Coordinate Planes . . . . . . . . . . . . . . . . . . . . . . . . . . .20Percent Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21Spinners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22Number Cube Patterns . . . . . . . . . . . . . . . . . . . . . . .23Protractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24Rectangular Prism Pattern . . . . . . . . . . . . . . . . . . . .25Cube Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26Cylinder Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . .27Cone Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28Pyramid Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . .29Pattern Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30Tangram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31Circle Graph Template . . . . . . . . . . . . . . . . . . . . . . .32Problem-Solving Guide . . . . . . . . . . . . . . . . . . . . . . .33

Activities PageCHAPTER 1

Teaching Notes and Overview . . . . . . . . . . .341-2 Mini-Project: Divisibility Patterns . . . . . . . .361-2 Using Overhead Manipulatives:

Rectangular Arrays . . . . . . . . . . . . . . . . . . .371-5 Using Overhead Manipulatives:

Variables and Expressions . . . . . . . . . . . . . .391-8 Mini-Project: Solving Equations . . . . . . . . .41Explore 1-9

Algebra Lab Recording Sheet . . . . . . . . . . .42

CHAPTER 2Teaching Notes and Overview . . . . . . . . . . .44

2-2 Mini-Project: Bar Graphs and Line Graphs . . . . . . . . . . . . . . . . . . . . . . . . .45

Extend 2-8Statistics Lab Recording Sheet . . . . . . . . . .46


Explore 3-1Using Overhead Manipulatives: Decimals Through Hundredths . . . . . . . . . .52

3-2 Mini-Project: Comparing and Ordering Decimals . . . . . . . . . . . . . . . . . . . .54

Explore 3-5Math Lab Recording Sheet . . . . . . . . . . . . .55






4-3 Using Overhead Manipulatives: Modeling Improper Fractions . . . . . . . . . . .62

4-5 Using Overhead Manipulatives: Least Common Multiple . . . . . . . . . . . . . . . . . . . .63



5-3 Mini-Project: Adding and SubtractingFractions . . . . . . . . . . . . . . . . . . . . . . . . . . . .68


5-4 Using Overhead Manipulatives: Renaming Sums . . . . . . . . . . . . . . . . . . . . . .71




Extend 6-1Math Lab Recording Sheet . . . . . . . . . . . . .75

6-3 Using Overhead Manipulatives: Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .77



7-1 Using Overhead Manipulatives: Fractions and Percents . . . . . . . . . . . . . . . . .83

7-4 Using Overhead Manipulatives: Fair and Unfair Games . . . . . . . . . . . . . . . . .84

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Extend 7-4Probability Lab Recording Sheet . . . . . . . . .86

Extend 7-4Using Overhead Manipulatives:Experimental Probability . . . . . . . . . . . . . . .87

7-8 Mini-Project: Percent and Estimation . . . . . . . . . . . . . . . . . . . . . . . . . .88


8-2 Using Overhead Manipulatives: Measurement . . . . . . . . . . . . . . . . . . . . . . . .91

Explore 8-3Measurement Lab Recording Sheet . . . . . . .93

8-6 Mini-Project: Using the Metric System . . . . . . . . . . . . . . . . . . . . . . .95

Extend 8-8Measurement Lab Recording Sheet . . . . . . .96


9-2 Using Overhead Manipulatives: Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . .100

Explore 9-4Geometry Lab Recording Sheet . . . . . . . .102

Explore 9-5Geometry Lab Recording Sheet . . . . . . . .103

Extend 9-7Geometry Lab Recording Sheet . . . . . . . .104

CHAPTER 10Teaching Notes and Overview . . . . . . . . . .105

Explore 10-1Measurement Lab Recording Sheet . . . . .108


10-2 Mini-Project: Circumference . . . . . . . . . . .111

10-2 Using Overhead Manipulatives: Diameter and Circumference . . . . . . . . . . .112

10-3 Using Overhead Manipulatives: Area of Irregular Shapes . . . . . . . . . . . . . .113


10-4 Mini-Project: Areas of Triangles . . . . . . . .116Explore 10-7

Measurement Lab Recording Sheet . . . . . . . . . . . . . . . . . . . . .117

Extend 10-7Measurement Lab Recording Sheet . . . . . . . . . . . . . . . . . . . . .118


Explore 11-2Algebra Lab Recording Sheet . . . . . . . . . .120

11-7 Mini-Project: The Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . .121




12-3 Mini-Project: Solving Addition Equations . . . . . . . . . . . . . . . . . . . . . . . . . .128


Extend 12-4Algebra Lab Recording Sheet . . . . . . . . . .130

12-5 Using Overhead Manipulatives: Solving Multiplication Equations . . . . . . .131

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Teacher’s Guide to Using Teaching Mathematics with Manipulatives

The book contains two sections of masters—Easy-to-Make Manipulatives and activities forMiddle School Mathematics. Tabs help you

locate the activities for each chapter. A complete list ofmanipulatives available in each of the three types ofGlencoe Mathematics Manipulative Kits appears on thenext page.

Easy-to-Make ManipulativesThe first section of this book contains masters formaking your own manipulatives. To make moredurable manipulatives, consider using card stock. Tomake algebra tiles similar to those shown in theStudent Edition, have students use markers to color thetiles appropriately or use colored card stock.

You can also make transparencies of frequently useditems such as grid paper and number lines.

Activity MastersEach chapter begins with Teaching Notes andOverview that summarizes the activities for the chapterand includes sample answers. There are three types ofmasters.

Mini-Projects are short projects that enable students toindependently investigate mathematical concepts.

Using Overhead Manipulatives provides instructionsfor the teacher to demonstrate an alternate approach tothe concepts of the lesson by using manipulatives onthe overhead projector.

Student Recording Sheets accompany the LabActivities found in the Student Edition. Students caneasily record the results of the activity on preparedgrids, charts, and figures.

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Glencoe Mathematics Manipulatives

Glencoe Mathematics Overhead Manipulative ResourcesISBN: 0-07-830593-4

Transparencies Overhead Manipulatives

integer mat centimeter grid algebra tilesequation mat number lines spinnersproduct mat lined paper two-dimensional cupsinequality mat regular polygons red and yellow countersdot paper polynomial models decimal models (base-ten blocks)isometric dot paper integer models compasscoordinate grids equation models protractor

geoboard/geobandsgeometric shapestransparency pens in 4 colors

Glencoe Mathematics Classroom Manipulative KitISBN: 0-02-833116-8

Measurement, Probability,Algebra and Statistics Geometry

algebra tiles base-ten models compassescounters marbles geoboardscups measuring cups geobandscentimeter cubes number cubes geomirrorsequation mat/product mat protractors isometric dot grid stampcoordinate grid stamp and rulers pattern blocks

ink pad scissors tangramsspinnersstopwatchestape measures

Glencoe Mathematics Student Manipulative KitISBN: 0-02-833654-2

algebra tiles protractorred and yellow counters scissorscups geoboardequation/product mat geobandscompass/ruler tape measure

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Easy-to-Make Manipulatives 1 Teaching Mathematics with Manipulatives, Course 1

NAME ______________________________________________ Base-Ten ModelsC

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NAME ______________________________________________ Decimal ModelsC

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NAME ______________________________________________ Fraction Models: Bars

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NAME ______________________________________________ Fraction Models: Circles

1 112

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NAME ______________________________________________ Fraction Wheel

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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



NAME ______________________________________________ Algebra Tiles—1 Tiles

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NAME ______________________________________________ Algebra Tiles—Variables

�x �x �x �x �x �x �x �x �x �x

�x �x �x �x �x �x �x �x �x �x

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NAME ______________________________________________ Product MatC

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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 ______________________________________________ Number Lines

–11–10

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NAME ______________________________________________ First-Quadrant Grids

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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

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NAME ______________________________________________ Protractors

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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 ______________________________________________ Tangram


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NAME ______________________________________________ Circle Graph Template

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NAME ______________________________________________ Problem-Solving Guide

Problem: Understand

CheckThese stepscan helpyou solveproblems.

Plan

Solve


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Chapter 1 34 Teaching Mathematics with Manipulatives, Course 1

Algebra: Number Patterns and FunctionsTeaching Notes and Overview

Mini-ProjectDivisibility Patterns(p. 36 of this booklet)

Use With Lesson 1-2.

Objective Apply the rules for divisibility toidentify patterns.

Materialsbeansjar

Students use the rules for divisibility toidentify divisibility patterns. Students willalso count and divide beans to determine allthe numbers that will divide a given number.

Answers1. 1, 2, 3, 6, 9, 18

2. See students’ work.

3. 6; 18 � 1 � 18; 18 � 2 � 9; 18 � 3 � 6;18 � 6 � 3; 18 � 9 � 2; 18 � 18 � 1

4. Sample answer: 15

5. Sample answer: 1, 3, 5, 15

6. See students’ work.

7. Sample answers: 4; 15 � 1 � 15; 15 �3 � 5; 15 � 5 � 3; 15 � 15 � 1

Using OverheadManipulativesRectangular Arrays(pp. 37–38 of this booklet)


Objective Identify prime and compositenumbers by using rectangular arrays.

Materialsalgebra tiles*blank transparencytransparency pens** � available in Overhead Manipulative Resources Kit

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

prime and composite numbers by usingrectangular arrays.

• Demonstration 2 asks students how to builda rectangular array and determine whether agiven number is prime or composite.

• Students will build rectangular arrays forseveral numbers independently and tellwhether each number is prime or composite.

AnswersAnswers appear on the teacher demonstrationinstructions on pages 37–38.

Using OverheadManipulativesVariables and Expressions(pp. 39–40 of this booklet)


Objective Model algebraic expressions.

Materialsabout 20 counters*three paper cupsblank transparenciestransparency pen** � available in Overhead Manipulative Resources Kit

This demonstration contains four activities.• Demonstrations 1 and 2 show how to model

and find the values of addition expressions.

• Demonstrations 3 and 4 show how to modeland find the values of multiplicationexpressions.

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Chapter 1 Algebra: Number Patterns and Functions

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Mini-ProjectSolving Equations(p. 41 of this booklet)


Objective Model and solve algebraicequations.

Materialscupscounters

Students use cups and counters to model andsolve addition, subtraction, and multiplicationequations.

Answers1. 8 2. 4

3. 3 4. 1

5. 11 6. 3

Algebra Lab RecordingSheetWriting Formulas(pp. 42–43 of this booklet)

Use With Explore 1-9. This corresponds tothe activity on pages 61–62 in the StudentEdition.

Objective Use tables of data to generateformulas.

Materialscentimeter grid paper

Students will explore how area and sidelengths of rectangles and squares are related.Students will use a formula to express therelationship as an equation.

AnswersSee Teacher Edition pp. 61–62.


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Mini-Project(Use with Lesson 1-2)

NAME ______________________________________________ DATE ____________ PERIOD __________

Divisibility Patterns1. List the numbers that divide 18 evenly. Use the rules for divisibility.

2. Take 18 beans. Separate them into groups evenly. The numbers you found in Exercise 1 will work as group numbers. Try to find all of the possible arrangements. Sketch each arrangement you find.

3. How many different numbers divide 18 evenly? List each way.

4. Put your 18 beans into a jar with everyone else’s beans. Grab some morebeans without counting. When you have the beans you want, count them.How many beans do you have?

5. What numbers will divide the number of beans you have evenly? Use your

rules of divisibility.

6. Separate your beans into groups evenly. Find all of the arrangements you can.Sketch each one.

7. How many different ways can you divide your beans?

List the numbers represented by each grouping.


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Using Overhead Manipulatives(Use with Lesson 1-2)

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Rectangular Arrays

Teacher Demonstration for Activity 1• Tell students that a composite number has more than two factors. For example,

4 is a composite number because it has factors of 1, 2, and 4. Place 4 smallalgebra tiles on the screen. Show that there is more than one way to make arectangle (2-by-2 and 1-by-4 or 4-by-1).

• Tell students that a prime number has exactly two factors. Point out that thesefactors are the number and 1. Place 3 tiles on the screen. Show that there isonly one way to make a rectangle (3-by-1 or 1-by-3). Tell them that this showsthat 3 is prime; its only factors are 1 and 3.

• Tell students that you are going to find outwhether 8 is a prime or a composite number. Place 8 tiles on the screen to form a 1-by-8rectangle. Record the arrangement on the blanktransparency. Ask students to suggest any otherarrangements. (2-by-4 rectangle) If studentssuggest 8-by-1, point out that it is the same shape as 1-by-8, only rotated.

• Tell students that these arrangements show that the factors of 8 are 1, 2, 4, and8. Point out that since 8 has more than two factors, it is a composite number.

Teacher Demonstration for Activity 2• Tell students that you want to determine whether 13 is a prime number or a

composite number.

• Place 13 squares on the screen. Ask students how you can arrange them tomake a rectangle. (1-by-13) Build the rectangle. Record the arrangement onthe transparency.

1 � 13

1 � 8

2 � 4

Objective Identify prime and composite numbers by using rectangulararrays.

Materials• algebra tiles*

• blank transparency

• transparency pens*

* � available in Overhead Manipulative Resources Kit


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Using Overhead Manipulatives

2. Which numbers of square tiles had only one arrangement? (2, 3, 5, 7, 11)

3. Which numbers of square tiles had more than one arrangement?(4, 6, 8, 9, 10, 12)

4. Is there a relationship between a multiplication table and the number ofarrangements you have? If so, describe it. (The greater the number ofarrangements, the more times the number appears on themultiplication table.)

5. Make a guess about which numbers between 12 and 25 square units can havemore than one rectangular shape. Explain why you selected those numbers.(12, 14, 15, 16, 18, 20, 21, 22, 24, 25; these numbers have severalfactors.)

6. Write a sentence describing the characteristics of prime and composite numbers.(Sample answer: A prime number has exactly 2 factors, 1 and thenumber itself. A composite number has more than 2 factors.)

• Point out that the drawing shows that the factors of 13 are 1 and 13. Since 13has exactly two factors, it is a prime number.

Have students complete Exercises 1–6 below. If tiles are notavailable for all students, they can draw squares on their paper tomake the arrangements.1. Repeat the process with areas of 2 square units through 12 square units.

2 3 4 5 6

7 8 9

10 11 12


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Variables and Expressions

Teacher Demonstration for Activities 1 and 2• Before turning on the projector, place 4 counters on the left side of the screen.

Place 5 counters inside a paper cup and place it on the right side. Then turn onthe projector.

• Tell students that this model represents the phrase the sum of four and somenumber. Write the phrase on a blank transparency. Tell them that the somenumber is an unknown value, and that when you assign a value to “somenumber,” then you can find the value of the expression.

• Ask students how many counters they can see. (4) Tell them that 4 representsthe known value and that the cup can contain any number of counters.

• Tell students that you have assigned a value to “some number” by placingsome counters in the cup.

• Empty the cup onto the screen and count the counters that were in the cup.Tell students that by adding this to the 4 other counters, you can find the valueof the expression.

• Ask the following questions. In the expression, what is the “some number?”(the unknown value) In the expression the sum of four and some number,when we replace “some number” with its known value, what is the value ofthe expression? (9)

• Repeat using different numbers of counters in the cup.

Objective Model algebraic expressions.

Materials• about 20 counters*

• three paper cups

• blank transparencies

• transparency pen*



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• Clear the screen. Ask students how to model the expression some number plussix. (Place a cup and 6 counters on the screen.) Write the expressionon a transparency. Ask, “What is the value of this expression if “somenumber” is 9?” (15) You may want to place 9 counters in the cup, empty thecup, and find the total number of counters.

• Write a � 8 on a transparency. Ask, “What is the value of a � 8 if a equals14?” (22)

Teacher Demonstration for Activities 3 and 4• Place 4 counters in each of 3 cups. Place the cups on the screen.

• Tell students that each cup has the same number of counters in it and that thismodel represents the phrase 3 times some number.

• Ask students how many cups they see. (3) Tell them that 3 represents theknown value and that each cup contains an unknown number of counters.Empty the cups to show all the counters. Ask students how many countersthere are in all. (12)

• Ask the following questions. What is the unknown value? (4) What is theproduct of 3 and some number in this case? (12)

• Repeat using a different number of counters under the cups.

• Ask, “How many cups would you need to model the expression six times anumber?” (6 cups)

• Write 5 • x on a blank transparency. Ask, “What is the value of 5 • x if x equals 4?” (20)

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NAME ______________________________________________ DATE ____________ PERIOD __________

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Solving Equations

Materialscups, counters

Model each equation using cups and counters. Let a cup represent the variable and let counters represent thenumbers. Use the guess-and-check strategy to solve eachequation.

1. c � 7 � 15 2. n � 5 � 9

3. 6s � 18 4. d � 3 � 4

5. 9 � e � 20 6. 5y � 15


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Algebra Lab Recording Sheet(Use with the activity on pages 61–62 in Explore 1-9 of the Student Edition)

NAME ______________________________________________ DATE ____________ PERIOD __________

Writing Formulas


Activity 1Record your data in the table.

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RectangleLength Width Area

(cm) (cm) (sq cm)

A 2 3

B 2 4

C 2 5

D 3 4

E 4 4

F 5 4

Analyze the Results1. Describe one or more patterns in the table.

2. Describe the relationship between the area of a rectangle and its length andwidth in words.

3. MAKE A CONJECTURE What would be the area of a rectangle witheach of the following dimensions? Test your conjecture by modeling eachrectangle and counting the number of shaded squares.

a. length, 2 cm; width, 8 cm b. length, 9 cm; width, 4 cm

4. WRITE A FORMULA If A represents the area of a rectangle, write anequation that describes the relationship between the rectangle’s area A, length �, and width w.

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Activity 2For each step on in Activity 2 on p. 62, draw new rectangles on grid paperand find the areas. Then organize the information into a table.

Analyze the ResultsCompare the areas you found in each step to the original areas. Write asentence describing how the area changed. Explain.

5. Step 1

6. Step 2

7. Step 3

Activity 3Record your data in the table.

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Square Side Length (cm) Area (sq cm)

A 2

B 3

C 4

Analyze the Results8. Describe a pattern in the rows of the table.

9. MAKE A CONJECTURE What would be the area of a square with sidelengths of 8 centimeters? Test your conjecture.

10. WRITE A FORMULA If A represents the area of a square, write anequation that describes the relationship between the square’s area A andside length s.


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Statistics and GraphsTeaching Notes and Overview

Mini-Project Bar Graphs and Line Graphs (p. 45 of this booklet)


Objective Collect data to create bar graphsand line graphs.

Materials paper

Students survey their classmates to find thenumber of people living in each person’shousehold. They create a bar graph to displaytheir data. Students keep track of the amountof time they spend doing a certain activityeach day. They create a line graph to displaythis data.

AnswersAnswers will vary. See students’ data andgraphs.

Statistics LabRecording SheetCollecting Data to Solve a Problem(pp. 46–48 of this booklet)

Use With Extend 2-8. This corresponds tothe activity on page 119 in the StudentEdition.

Objective Solve a problem by collecting,organizing, displaying, and interpreting data.

Using rating scales, frequency charts, surveyquestions, and a log, students will collect,organize, display, and interpret data in orderto solve a problem.


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NAME ______________________________________________ DATE ____________ PERIOD __________

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Bar Graphs and Line Graphs

Survey your classmates. Ask: “Including yourself, how manypersons live in your household?”

Make a bar graph of the frequency table on a separate sheet of paper.

Keep track of the number of minutes you spend doing someactivity each day for one week. Then make a line graph of the data on a separate sheet of paper.

Some examples that you may want to choose include the following: doinghomework, reading for pleasure, watching TV, practicing music, exercising,talking on the phone, playing video games, working on a hobby.

Day of the Week Time Spent (minutes)

Sun. _________________________________________

Mon. _________________________________________

Tues. _________________________________________

Wed. _________________________________________

Thurs. _________________________________________

Fri. _________________________________________

Sat. _________________________________________

Number in Tally FrequencyHousehold

2

3

4

5

6

7

8

9

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NAME ______________________________________________ DATE ____________ PERIOD __________C

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Statistics Lab Recording Sheet(Use with the activity on pages 119–120 in Extend 2-8 of the Student Edition)

Collecting Data to Solve a Problem

Activity Rating Scales

Step 1 Make a data collection plan.

• Choose on of the topics on page 119 of the Student Edition and agroup to survey.

Topic ___________________________________________________

Survey Group ____________________________________________

• Write one or more survey questions that include the rating scaleshown above to determine student opinion on this topic.

Question#1 ______________________________________________

Question#2 ______________________________________________

Question#3 ______________________________________________

• Identify an audience for your results.

Audience to present results _________________________________

Step 2 Collect the data.

• Conduct your survey and record the results.

• Collect responses from at least 10 people in the population you chose.

• Record your results in the frequency table.

Strongly Agree (5)

Agree (4)

No Opinion (3)

Disagree (2)

Strongly Disagree(1)



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Step 3 Create a display of the data.

• Choose an appropriate type of display and scale for your data. Thencreate an accurate diplay.

Analyze the Results1. What are the mean, median, mode, and range of your data?

Mean ____________________

Median ____________________

Mode ____________________

Range ____________________

2. Use your display to describe the distribution of your data.

3. How would you summarize the opinions of those you surveyed?

4. Based on your analysis, what course of action would you recommend to thegroup interested in your data?

5. Present your findings and recommendation to the whole class. Includeposter-size versions of both your displays and a written report of your dataanalysis.

6. MAKE A CONJECTURE What other factors might influence the results ofyour survey.



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Activity Log

Step 1 Make a data collection plan.

• Choose one of the topics on page 120 of the Student Edition and areasonable period of time over which to collect the data.

Topic ___________________________________________________

• Create a log that you can use to collect the data.

Step 2 Collect the data and create an appropriate display.

• Record the necessary data in your log. Then choose and create anappropriate display for the data.

Analyze the Results

7. What type of display did you choose?

8. Describe the change in your data over the time period you chose.

9. If possible, use your display to make a prediction about future data. Explainyour reasoning. If not possible, explain why not.

Day Time Spent Watching TVTime Temperature

OR



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.Operations with DecimalsTeaching Notes and Overview

Using OverheadManipulativesDecimals Through Hundredths(pp. 48–49 of this booklet)


Objective Model decimals throughhundredths.

Materials decimal models*blank transparencytransparency pen** � available in Overhead Manipulative Resources Kit

This demonstration contains two activities.• Demonstration 1 shows how to use base-ten

models to represent decimals, and how totrade tenths for hundredths.

• Demonstration 2 asks students to identifythe decimal modeled with both base-tenmodels and hundredths blocks.

• Students will perform the same kind ofactivities independently.

• An Extension activity shows how to tradeones for tenths when modeling a decimal.


Mini-ProjectComparing and Ordering Decimals(p. 50 of this booklet)


Objective Compare decimals using models.

Materials none

Students draw decimal models for givendecimals in order to determine which of twodecimals is greater.

Answers1. 0.87

2. 0.65

3. 0.41

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Chapter 3 Operations with DecimalsC

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4. 0.98

5. 0.5

6. 0.63

Math Lab Recording Sheet Modeling Decimals (p. 55 of this booklet)

Use With Explore 3-5. This corresponds tothe activity on page 155 in the StudentEdition.

Objective Use models to add and subtractdecimals.

Materials centimeter grid paper

Using models, students add and subtractdecimals. Space is provided for students toexplain how to compute with decimals bothwith and without models.

AnswersSee Teacher Edition p. 155.



Math Lab Recording Sheet Multiplying Decimals by WholeNumbers(p. 56 of this booklet)


Objective Use models to multiply a decimalby a whole number.

Materialsgrid papercolored pencilsscissors

Students use decimal models to find theproduct of a decimal and a whole number.Space is also given for students to make andtest a conjecture about the product of a wholenumber and a decimal.


Math Lab Recording Sheet Multiplying Decimals(p. 57 of this booklet)


Objective Use decimal models to multiplydecimals.


Students use decimal models to find theproduct of two decimals. Students willidentify and explain a pattern in the numberof decimal places in each factor and in eachproduct.


Math Lab Recording Sheet Dividing by Decimals(p. 58 of this booklet)


Objective Use models to divide a decimalby a decimal.

Materialsbase-ten blocks

Using base-ten blocks, students will find thequotient of two decimals. Space is alsoprovided for students to explain their workand make conclusions.


Chapter 3 Operations with Decimals

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Decimals Through Hundredths

Teacher Demonstration for Activity 1• Tell students that you can use base-ten models to represent decimals. At the

top of a blank transparency, label the models as shown below.

• Place two tenths strips on the screen. Point out that this is a model for two tenths.

• Tell students that you can also model two tenths by trading the tenths forhundredths. Trade each tenth for 10 hundredths. You will need to drawadditional hundredths on the transparency. Then ask how many hundredthsthere are. (20)

Teacher Demonstration for Activity 2• Clear the screen. Show three tenths and five hundredths with base-ten models

on a blank transparency. Ask students what decimal you have modeled. (0.35)• Show students as you trade the tenths for hundredths. (You will need to draw

additional tenths on the transparency.) Ask students how many hundredthsblocks there are. (35)

• Point out that since there are 35 hundredth blocks, the decimal modeled is 35hundredths.

1 0.1 0.01

Objective Model decimals through hundredths.

Materials• decimal models*






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Have students complete Exercises 1–4 below.If enough base-ten blocks are not available for modeling, have studentssketch the blocks on their paper. Tell them to erase or cross out the blocksthey trade.

1. Three tenths is the same as how many hundredths? (thirty hundredths)

2. Show six tenths and four hundredths with base-ten blocks. Trade the tenthsfor hundredths. How many hundredths do you have now? (See students’work; 64 hundredths)

3. How many tenths are the same as ninety hundredths? Model using base-tenblocks. (9 tenths)

4. If you separated a hundredth block into 100 equal parts, what decimal wouldbe modeled by seventeen of the new parts? (17 ten-thousandths)

ExtensionShow 2 ones and 4 tenths. Trade the ones for tenths. Ask students how manytenths you now have. (24)



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NAME ______________________________________________ DATE ____________ PERIOD __________

Comparing and Ordering Decimals

Shade a hundred grid to show each decimal. Which decimal is greater?1. 0.78 0.87 2. 0.65 0.49

is greater. is greater.

3. 0.3 0.41 4. 0.98 0.9


5. 0.5 0.47 6. 0.09 0.63


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Math Lab Recording Sheet(Use with the activity on page 155 in Explore 3-5 of the Student Edition)

NAME ______________________________________________ DATE ____________ PERIOD __________

Adding and Subtracting Decimals Using Models


Check Your ProgressFind each sum or difference using decimal models.

a. 0.14 � 0.67 b. 0.35 � 0.42 c. 0.03 � 0.07

d. 0.75 � 0.36 e. 0.68 � 0.27 f. 0.88 � 0.49

Analyze the Results1. Explain how you can use grid paper to model 0.8 � 0.37.

2. MAKE A CONJECTURE Write a rule you can use to add or subtractdecimals without using models.

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Multiplying Decimals by Whole Numbers

Materialsgrid paper, colored pencils, scissors

Check Your ProgressUse decimal models to show each product.

a. 3 � 0.5 b. 2 � 0.7 c. 0.8 � 4

Writing Math 1. MAKE A CONJECTURE Is the product of a whole number and a decimal

greater than the whole number or less than the whole number? Explain yourreasoning.

2. Test your conjecture on 7 � 0.3. Check your answer by making a model orby using a calculator.



NAME ______________________________________________ DATE ____________ PERIOD __________

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Multiplying Decimals

Materialsgrid paper, colored pencils, scissors

Check Your ProgressUse decimal models to show each product on grid paper.

a. 0.3 � 0.3 b. 0.4 � 0.9 c. 0.9 � 0.5

Writing Math1. Tell how many decimal places are in each factor and in each product of

Exercises a–c above.

2. MAKE A CONJECTURE Use the pattern you discovered in Exercise 1 to find 0.6 � 0.2. Check your conjecture with a model or a calculator.

3. Find two decimals whose product is 0.24.

Check Your ProgressUse decimal models to show each product on grid paper.

d. 1.5 � 0.7 e. 0.8 � 2.4 f. 1.3 � 0.3

Analyze the Results4. MAKE A CONJECTURE How does the number of decimal places in the

product relate to the number of decimal places in the factors?

5. Analyze each product.

a. Explain why the first product is less than 0.6.

b. Explain why the second product is equal to 0.6.

c. Explain why the third product is greater than 0.6

Math Lab Recording Sheet(Use with the activity on pages 167–168 in Explore 3-7 of the Student Edition)

First SecondProductFactor Factor

0.9 � 0.6 � 0.54

1.0 � 0.6 � 0.60

1.5 � 0.6 � 0.90




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Dividing by Decimals

Materialsbase-ten blocks

Check Your ProgressUse base-ten blocks to find each quotient.

a. 2.4 � 0.6 b. 1.2 � 0.4 c. 1.8 � 0.6

d. 0.9 � 0.09 e. 0.8 � 0.04 f. 0.6 � 0.05

Analyze the Results1. Explain why the base-ten blocks representing the dividend must be replaced

or separated into the smallest place value of the divisor.

2. Tell why the quotient 0.4 � 0.05 is a whole number. What does the quotientrepresent?

3. Determine the missing divisor in the sentence 0.8 � � 20. Explain.

4. MAKE A CONJECTURE Tell whether 1.2 � 0.03 is less than, equal to, orgreater than 1.2. Explain your reasoning.



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Fractions and DecimalsTeaching Notes and Overview

Math Lab Recording Sheet Equivalent Fractions(pp. 60–61 of this booklet)


Objective Use models to determine aprocedure for generating equivalent fractions.

Materials counters

Students use models to generate equivalentfractions. Space is also provided for studentsto explain a given model and make aconjecture about a rule for writing fractions in simplest form.


Using OverheadManipulativesModeling Improper Fractions(p. 62 of this booklet)


Objective Use models to express mixednumbers as improper fractions.

Materials ruler*transparency pens** � available in Overhead Manipulative Resources Kit

• This demonstration shows how to usemodels to express mixed numbers asimproper fractions.

• Students will also learn to use models tofind equivalent fractions.

AnswersAnswers appear on the teacher demonstrationinstructions on page 62.

Using OverheadManipulativesLeast Common Multiple(p. 63 of this booklet)


Objective Use models to find the leastcommon multiple.

Materials rectangular dot paper transparency*transparency pens*centimeter ruler** � available in Overhead Manipulative Resources Kit

• This demonstration shows how to usemodels to find the least common multiple.

• Students will learn how to make rectangleson rectangular dot paper for their models.


Chapter

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Simplifying Fractions

Materialscounters

Check Your ProgressUse counters to generate three fractions equivalent to each fraction.

a. �34

� b. �31

�

c. �25

� d. �65

�

Check Your ProgressUse counters to generate a simpler fraction equivalent to each fraction.

e. �11

06� f. �

261�

g. �284� h. �

324

0�



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Analyze the Results1. In Activity 1, an equivalent fraction is created by combining equal groups

that have the same number of red counters and the same number of totalcounters. What operation does this model?

2. MAKE A CONJECTURE Use the operation you found in Exercise 1 to

generate a fraction equivalent to �78

�. Justify your answer.

3. In Activity 2, an equivalent fraction is created by separating a group ofcounters into equal groups of that have the same number of red counters and the same number of total counters. What operation does this model?

4. MAKE A CONJECTURE Use the operation you found in Exercise 2 to

generate a fraction equivalent to �34

00�. Justify your answer.



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panies,Inc.Using Overhead Manipulatives(Use with Lesson 4-3)

Modeling Improper Fractions

Teacher Demonstration • Tell students that you are going to draw a model for a mixed number.

• Draw two rectangles. Shade the rectangles. Tell students that this represents 2.

• Draw an identical rectangle beside the first two. Ask students how to show �23

�.

(Separate the rectangle into 3 equal parts and shade 2 parts.)

Ask students to name the number shown by the rectangles. �2�23

��

• Separate the whole number rectangles into thirds. Ask students how manythirds are shaded in all three rectangles. (8) Ask students to name the

fraction that is equivalent to 2�23

�. ��83

��

Objective Use models to express mixed numbers as improper fractions.

Materials• ruler*





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.Using Overhead Manipulatives(Use with Lesson 4-5)

Least Common Multiple

Teacher Demonstration • Place the dot paper transparency horizontally on the screen. Tell students that

you are going to model multiples of 2 and 3. As you draw 2-by-1 rectanglesend-to-end on the dot paper, tell them that each is 2 centimeters long and thatyou are making a 2-centimeter rectangle train.

• Next, draw 3-by-1 rectangles end-to-end to make a train below the 2-by-1rectangles as shown. Ask students to look for places where the ends of therectangles line up.

• Place the ruler below the trains and ask students at what lengths the ends ofthe rectangles line up. (0 cm, 6 cm, and 12 cm) Ask if 0, 6, and 12 aremultiples of both 2 and 3. (yes)

• Ask students, “At what measurement, other than zero, do the rectangles lineup for the first time?” (6 cm) Point out that 6 is called the least commonmultiple of 2 and 3.

Objective Use models to find the least common multiple.

Materials• rectangular dot paper transparency*


• centimeter ruler*




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panies,Inc.Operations with FractionsTeaching Notes and Overview

Math Lab Recording SheetRounding Fractions(p. 67 of this booklet)


Objective Round fractions to the nearesthalf.

Materialsgrid papercolored pencils

Students will round several fractions to thenearest half. Then they will sort the fractionsinto three groups: those that round to 0, thosethat round to �

12

�, and those that round to 1. Bycomparing the numerators and denominators,students make a conjecture about how toround any fraction to the nearest half.


Mini-ProjectAdding and Subtracting Fractions(p. 68 of this booklet)


Objective Use models to add and subtractfractions.

Materialsnone

Students shade grids to model fractions.Using these models, students find sums anddifferences of fractions.

Answers1. �

196�

2. �24

�

3. �280�

4. �2126� or 1�

166�

��

��

��

� �

Chapter

5



Chapter 5 Operations with Fractions

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Math Lab Recording SheetUnlike Denominators(pp. 69–70 of this booklet)


Objective Use models to add and subtractfractions with unlike denominators.

Materialsfraction strips

Students use fraction strips to add andsubtract fractions with unlike denominators.Students will make a conjecture about therelationship between common multiples andadding and subtracting unlike fractions.


Using OverheadManipulativesRenaming Sums (p. 71 of this booklet)


Objective Find common unit names foradding different objects.

Materials3 congruent squares (2 inches-by-2 inches)5 congruent trapezoids (2 inches-by-1 inch)4 pencils2 pensblank transparencytransparency pen** � available in Overhead Manipulative Resources Kit

• Using a group of squares and trapezoidsand a group of pens and pencils, studentswill find common unit names for addingthese different objects.

• Students will choose the best commonname and explain their reasoning.

• Students will find common unit names forgroups of objects that they list themselves.

• An Extension activity asks students to lookfor objects in their homes that can bedescribed using a common unit name.




Chapter 5 Operations with FractionsC

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Math Lab Recording SheetMultiplying Fractions(p. 72 of this booklet)


Objective Multiply fractions using models.

Materialspapermarkers

Students will use models to find the productsof fractions. Then they will explain therelationship between the numerators of theproblem and the numerator of the product.They will also compare the denominators ofthe problem and the denominator of theproduct. Finally, students will write a rulethey can use to multiply fractions.


Math Lab Recording SheetDividing Fractions(p. 73 of this booklet)


Objective Divide fractions using models.

Materialspapercolored pencilsscissors

Students will divide fractions using models.They will also determine when the quotient oftwo fractions is greater than, less than, orequal to 1 and give examples to support theirreasoning. Finally, students will explainwhether division is commutative and giveexamples to support their answer.



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NAME ______________________________________________ DATE ____________ PERIOD __________

Rounding Fractions

Materialsgrid paper, colored pencils

Check Your ProgressDraw and shade a model to represent each fraction. Then use the model toround each fraction to the nearest half.

a. �12

30� b. �

87

�

c. �190� d. �

51

�

e. �11

15� f. �

225�

g. �160� h. �

2107�

i. �18

� j. �176�

Analyze the Results

1. Sort the fractions in Exercises a–j into three groups: those that round to 0,

those that round to �12

�, and those that round to 1.

2. MAKE A CONJECTURE Compare the numerators and denominators ofthe fractions in each group. Explain how to round any fraction to the nearesthalf without using a model.

3. Test your conjecture by repeating the activity and Exercise 1 using the

fractions �35

�, �137�, �

12

60�, �

123�, �

254�, �

175�, �

79

�, and �191�.

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NAME ______________________________________________ DATE ____________ PERIOD __________

Adding and Subtracting Fractions

Shade each figure to model each fraction. Use the models to find the sum.1. �

156� � �

146� �

2. �34

� � �14

� �

3. �1210� � �

230� �

4. �196� � �

1136� �

��

��

��

� �


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NAME ______________________________________________ DATE ____________ PERIOD __________

Unlike Denominators

Materialsfraction strips

Check Your ProgressUse fraction strips to add.

a. �110� � �

52

�

b. �16

� � �21

�

c. �12

� � �43

�

Check Your ProgressUse fraction strips to subtract.

d. �38

� � �41

�

e. �89

� � �31

�

f. �23

� � �14

�

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Analyze the ResultsUse the models from Activities 1 and 2 to complete the following.

1. �12

� � �15

� � ��10

� � ��10

� 2. �78

� � �34

� � ��8� � �

�8�

Write an addition or subtraction expression for each model. Then add or subtract.

3.

4.

5. MAKE A CONJECTURE What is the relationship between the number ofseparations on the answer fraction strip and the denominators of the fractionsadded or subtracted?


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Ch

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Renaming Sums

Teacher Demonstration• Place the squares and trapezoids on the screen. Ask students to think of unit

names to describe the sum of the squares and trapezoids. Record theirresponses. [Sample answers: 8 things, 8 shapes, 8 four-sidedshapes (or quadrilaterals)]

• Have students look at the list and choose the best unit name for the squaresand trapezoids. (four-sided shapes, quadrilaterals)

• Hold up the pencils and pens. Ask students to think of unit names to describetheir sum. Record their responses. (6 things; 6 objects; 6 long, thinobjects; 6 writing tools)

• Have students choose the best unit name for the pens and pencils. (writingtools)

• Ask students why a common unit name is needed to find the sum. (Theaddends and sum must have the same unit name.)

• Ask students what helped them select the “best” unit name for the objects.(Sample answer: The name “writing tools” is more representative;it describes the objects in the group better.)

Have students complete Exercises 1–2 below.1. Make a list of different objects that could have a common unit name. (Sample

answer: paper clips, staples, and rubber bands—paper holders)

2. What do you think you need to do to find the sum of �12

� and �34

�? (Sampleanswer: Find a common denominator.)

ExtensionAsk students to look for groups of objects in their homes that can be describedusing a common unit name. For example, sheets and pillowcases can be calledbed linens.

Objective Find common unit names for adding different objects.

Materials• 3 congruent squares (about 2 inches by 2 inches)

• 5 congruent trapezoids (about 2 inches by 1 inch)

• 4 pencils

• 2 pens







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Multiplying Fractions

Materialspaper, markers

Check Your ProgressFind each product using a model.

a. �14

� � �12

� b. �13

� � �14

� c. �12

� � �51

�

Analyze the Results1. Describe how you would change the model in Activity 1 to find �

12

� � �13

�.

Is the product the same as �13

� � �12

�? Explain.

Check Your ProgressFind each product using a model. Then write in simplest form.

d. �34

� � �23

� e. �25

� � �56

� f. �45

� � �83

�

Analyze the Results2. Draw a model to show that �

23

� � �56

� � �11

08�. Then explain how the model shows

that �1108� simplifies to �

59

�.

3. Explain the relationship between the numerators of the problem and thenumerator of the product. What do you notice about the denominators of theproblem and the denominator of the product?

4. MAKE A CONJECTURE Write a rule you can use to multiply fractions.



NAME ______________________________________________ DATE ____________ PERIOD __________

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Dividing Fractions

Materialspaper, colored pencils, scissors

There are 8 small prizes that are given away 2 at a time. How many peoplewill get prizes?

1. How many 2s are in 8? Write as a division expression.

Suppose there are two granola bars divided equally among 8 people. Whatpart of a granola bar will each person get?

2. What part of 8 is in 2? Write as a division expression.

Check Your ProgressFind each quotient using a model.

a. 2 � �15

� b. 3 � �13

� c. 3 � �23

� d. 2 � �43

�

e. �140� � �

15

� f. �34

� � �12

� g. �45

� � �15

� h. �16

� � �31

�

Analyze the ResultsUse greater than, less than, or equal to to complete each sentence. Then givean example to support your answer.

1. When the dividend is equal to the divisor, the quotient is ? 1.

2. When the dividend is greater than the divisor, the quotient is ? 1.

3. When the dividend is less than the divisor, the quotient is ? 1.

4. You know that multiplication is commutative because the product of 3 � 4 isthe same as 4 � 3. Is division commutative? Give examples to explain youranswer.


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Ratio, Proportion, and FunctionsTeaching Notes and Overview

Math Lab Recording SheetRatios and Tangrams(pp. 75–76 of this booklet)

Use With Extend 6-1. This corresponds tothe activity on pp. 320–321 in the StudentEdition.

Objective Explore ratios and therelationship between ratio and area.

Materials2 sheets of patty paperscissors

Students use a tangram to explore ratios andthe relationship between ratio and area. Theywill compare the areas of triangles usingratios to discover the relationship betweenratio and area.


Using OverheadManipulativesRatios(pp. 77–78 of this booklet)


Objective Explore ratios that are equivalentand ratios that are not equivalent.

Materialspaper or colored transparency, if availablescissorsblank transparencytransparency pen** � available in Overhead Manipulative Resources Kit

• Using a triangle as the unit of measure,students create shapes and use them to showratios that are equivalent and ratios that arenot equivalent.

• Students then explain how to findequivalent ratios and how to determinewhen two ratios are not equivalent.

AnswersAnswers appear on the teacher demonstrationinstructions on pp. 77–78.

Chapter

6


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Math Lab Recording Sheet(Use with the activity on pages 320–321 in Extend 6-1 of the Student Edition)

NAME ______________________________________________ DATE ____________ PERIOD __________

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Ratios and Tangrams

Materials2 sheets of patty paper, scissors

Writing MathWork with a partner. Refer to pp. 320–321.

1. Suppose the area of triangle B is 1 square unit. Find the area of each trianglebelow.

a. triangle C b. triangle F

2. Explain how the area of each of these triangles compares to the area oftriangle B.

3. Explain why the ratio of the area of triangle C to the original large square is1 to 8.


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4. Tell why the area of square E is equal to the area of figure G.

5. Find the ratio of the area of triangle F to the original large square. Explainyour reasoning.

6. Complete the table. Write the fraction that compares the area of each figureto the original square. What do you notice about the denominators?

Figure A B C D E F G

FractionalPart of the

LargeSquare


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Ratios

Teacher Demonstration • Create the shapes shown below. Use a 2-inch square cut along a diagonal as

the triangular unit of measure. The dashed lines on each shape are to be usedonly as a help in preparing the shapes, and should not appear on the finishedpieces. You will need a minimum of 10 small triangles, 3 parallelograms, 2 trapezoids, 4 large triangles, and 2 pentagons.

• Show students the small triangular shape. Say, “This triangle will be our unitof measure for this activity.”

• Copy the following ratio using the shapes and a transparency pen. Askstudents what shape or shapes can be used to make an equal ratio.

• Repeat the previous step for the following ratios. Be prepared to build anyshape with the small triangles if needed to show students how the shapescompare.

a.

b.

or )(�?

or )(�?

or )(�?

Objective Explore ratios that are equivalent and ratios that are notequivalent.

Materials• paper or colored transparency, if available

• scissors


• transparency pen** � available in Overhead Manipulative Resources Kit


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c.

d.

• Ask students to suggest a shape that has a ratio not equal to each ratio shownbelow.

e.

f.

• Ask, “How did you find the equivalent ratios?” (Sample answer:Compare the known parts; keep the same comparison to find theunknown shape.)

• Ask, “How did you know when two ratios were not equivalent?” (Sampleanswer: The comparison between corresponding parts was notthe same.)

)(�?

not

)(� ? not

)(� ?

or )(�?


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Ch

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Teaching Notes and Overview

Math Lab Recording SheetModeling Percents(p. 82 of this booklet)


Objective Use models to illustrate themeaning of percents.

Materialsone hundred square gridsStudents will use one hundred square grids tomodel percents.


Using OverheadManipulativesFractions and Percents(p. 83 of this booklet)


Objective Express fractions as percents.

Materialscompass*paper clip12-inch rulerblank transparencytransparency pen** � available in Overhead Manipulative Resources Kit

• This demonstration illustrates dropping apaper clip onto a piece of paper andrecording the number of times it landswithin a circle drawn on the paper.

• Students will write a ratio of the number oftimes the clip lands within the circle to thenumber of drops that are made.

• Using this ratio, students will find thepercent of drops that land within the circle.

AnswersAnswers appear on the teacher demonstrationinstructions on p. 83.

Using OverheadManipulativesFair and Unfair Games (pp. 84–85 of this booklet)


Objective Explore fair and unfair games.

Materials 2 spinners*blank transparency, prepared as describedtransparency pens** � available in Overhead Manipulative Resources Kit

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

play an addition game using spinners anddetermine whether the game is fair.

• An Extension activity asks students tocompare what results should occur most oftenin the game to the actual results of the game.

• Demonstration 2 shows students how to playa multiplication game using the spinners anddetermine whether the game is fair.


Chapter

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Chapter 7 Percent and Probability

Probability Lab Recording Sheet Experimental and TheoreticalProbability(p. 86 of this booklet)

Use With Extend 7-4. This corresponds to theactivity on page 387 in the Student Edition.

Objective Compare experimentalprobability with theoretical probability.

Materials blue cubesred cubes

Students will conduct a simulation by rollingcubes. They will use the results from severaltrials to compare the experimental andtheoretical probabilities of given events.Students will examine the results within theirgroup, as well as the results of the other groupsin the class.


Using OverheadManipulativesExperimental Probability(p. 87 of this booklet)

Use With Extend 7-4.

Objective Determine the experimentalprobability for a given set of data.

Materials 6-section spinner*blank transparencytransparency pens** � available in Overhead Manipulative Resources Kit

• Students conduct a simulation by spinning aspinner 60 times to find the experimentalprobability of given events.

• Students will also use a given set of resultsfrom a different experiment to find theexperimental probability of given events.




Mini-ProjectPercent and Estimation(p. 88 of this booklet)


Objective Estimate the percent of a figurethat is shaded.

Materialsnone

Students estimate the percent of each figurethat is shaded by just looking. Then theycount the grid squares to find the actualpercent that is shaded. Students compare theirestimates to the actual percents.

AnswersAll estimates are sample answers.

1. 75%; 60%

2. 50%; 50%

3. 35%; 35%

4. 50%; 51%

5. Sample answer: Estimates are close to theactual percents, but not exact.

6. Answers will vary. See students’ work.

Chapter 7 Percent and Probability

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NAME ______________________________________________ DATE ____________ PERIOD __________

Modeling Percents

Activity

Check Your ProgressModel each percent.

a. b.

c. d.

e. __________________ f. __________________ g. __________________

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Analyze the Results1. Identify the fraction of each model in Exercises a-g that is shaded.

2. MAKE A CONJECTURE How can you write a percent as a fraction?How can you write a fraction with a denominator of 100 as a percent?


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Fractions and Percents

Teacher Demonstration • Draw a circle with a 3-inch diameter

in the center of a blank transparency. Place this transparency on the screen.

• Hold the ruler on its end so that it shows a height of 12 inches. Drop the paper clip 20 times from a height of 12 inches. Record the number of times the clip lands completely within the circle. (Sample answer:9 times)

• Ask students to name the ratio of the number of times the clip landed within the circle to the number of drops made. (Sample answer: �

290�)

• Write �290� � �

10?0

� at the bottom of the transparency. Ask students how to

calculate the percent of drops that landed within the circle. (Set up aproportion. Find the cross products. Solve for the unknown value.)

32

14

56

78

91

01

11

2

Objective Express fractions as percents.

Materials• compass*

• paper clip

• 12-inch ruler


• transparency pen** � available in Overhead Manipulative Resources Kit

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Fair and Unfair Games

Teacher Demonstration for Activity 1 • Tell students that in this activity you will explore fair and unfair games. Ask

students to define a fair game. (A fair game is a game in which playershave an equal chance of winning.) Then ask them to define an unfairgame. (In an unfair game, players do not have an equal chance ofwinning.)

• Prepare two identical spinners with four equal sections. Number the sectionson each spinner 1–4. (Hint: You could also use just one spinner and spin ittwice for each round of the game.)

• On the blank transparency, prepare a chart like the one shown below.

Also prepare the addition table shown in the Extension for Activity 1, and thechart shown in Activity 2. Place the transparency on the screen so only thefirst table is showing. Use tape to hold it in place if necessary.

• Divide the class into two sections, for example, left side and right side. Callthem Team One and Team Two. Tell them the rules for the game as follows.“We will spin the spinners. If the sum of the two numbers is even, Team Onegets a point. If the sum of the two numbers is odd, Team Two gets a point.”

• Spin both spinners. Keep track of each sum by marking an X above thenumber in the chart. Repeat until you have recorded 30 sums.

• Have a member from each team count and report the number of points fortheir team. (Answers will vary; there will be about 15 of each.)

• Ask students whether they think the game is fair and why or why not. (Thisis a fair game. There are 8 ways to get an odd-numbered sum and

4 1

3 2

4 1

3 2

Objective Explore fair and unfair games.

Materials• 2 spinners*

• blank transparency, prepared as described below



Sum 2 3 4 5 6 7 8


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8 ways to get an even-numbered sum. However, this particulartrial might not look fair. Encourage students to express anyreasonable answers.)

• The chart should resemble a bar graph. Ask students to describe the shape ofthe graph. (Sample answer: The bars are lower on the ends andhigher in the middle.)

• Ask students which sums occurred most often and least often. (Sampleanswer: 5; 2 and 8)

Extension for Activity 1Show students the addition table at the right. Ask thefollowing questions. Which sum or sums from theaddition table occur most often? (5) Which sum orsums occur least often? (2 and 8) Ask students tocompare these answers to the results of the game.(They should be very similar.)

Teacher Demonstration for Activity 2• Place the chart transparency so that the following chart is showing.

• Tell students that for this activity, Team One gets a point if the product of thetwo numbers is even, and Team Two gets a point if the product is odd.

• Spin both spinners. Keep track of each product by marking an X above thenumber in the chart. Repeat until you have recorded 30 products.

• Have a member from each team count and report the number of points fortheir team. (Answers will vary; there will be about 3 times as manyeven products as odd products.)

• Ask students whether they think the game is fair and why or why not.(Sample answer: The game is not fair because the even productsoccur much more often. Some students may suggest that this isbecause, although there are the same number of even and oddfactors, the only products that are odd are the products of twoodd numbers, while odd times even, even times odd, and eventimes even give even products.)

• Ask them to describe the shape of the graph. (Sample answer: It goes upand down more than once.)

• Ask which product(s) occurred most often and least often. (Sampleanswer: 4 occurred most often; 1, 9, and 16 occurred least often.)

1 1 2 3 4

1 2 3 4 5

2 3 4 5 6

3 4 5 6 7

4 5 6 7 8

Product 1 2 3 4 6 8 9 12 16



Probability Lab Recording Sheet(Use with the activity on page 387 in Extend 7-4 of the Student Edition)

NAME ______________________________________________ DATE ____________ PERIOD __________

Experimental and Theoretical Probability

Materialsblue cubesred cubes

Analyze the Results1. To find the experimental probability of selecting a blue cube, write the ratio

of the number of times a blue cube was selected to the number of trials. Whatis the experimental probability of selecting a blue cube?

2. What is the theoretical probability of selecting a blue cube? How does thisprobability compare to the experimental probability found in Exercise 1?Explain any differences.

3. Compare your results to the results of other groups in your class. Why do youthink the experimental probabilities usually vary when an experiment isrepeated?

4. Find the experimental probability for the entire class’s trials. How do theexperimental and theoretical probability compare?

5. MAKE A CONJECTURE Explain why the experimental probabilityobtained in Exercise 4 may be closer in value to the theoretical probabilitythat the experimental probability in Exercise 1.

6. COLLECT THE DATA Work with a partner. Have your partner place adifferent number of red and blue cubes totaling 10 into the bag. Useexperimental probability to guess the correct number of red and blue cubes inthe bag. Explain your reasoning.

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Using Overhead Manipulatives(Use with Extend 7-4)

Ch

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Experimental Probability

Teacher Demonstration • Prepare the spinner by writing the numbers from 1 to

6 in the sections of the spinner. Prepare the blanktransparency with a chart like the one shown.

• Show students the chart and the spinner. Point out that the spinner has 6equal-size numbered sections. Tell them that you are going to spin the spinner.Ask students to estimate the number of times the spinner will show 1, 3, and 5if you spin it 60 times. (Estimates may vary; a reasonable estimate isabout 10 times for each.)

• Have students help spin the spinner 60 times. Have another student make atally of each spin in the chart.

• Compare the estimates to the actual results. (Answers will vary.)• Tell students that experimental probability is a ratio that compares the number

of ways a certain outcome occurs to the total number of outcomes. Since youspun the spinner 60 times, there were 60 outcomes in this experiment. Write afraction comparing the number of times the spinner showed a 3 to 60. Tellstudents that this ratio represents the experimental probability of spinning a 3.Have students tell you how to use the results in the chart to write ratiosrepresenting the experimental probability of spinning a 1 and of spinning a 5.(Answers will vary.)

Have students complete Exercises 1–4 below. Six index cards are labeled L, O, C, K, E, and R. Without looking, Jackiechooses a card, records its letter, and replaces it. She repeats the activity 48 times. The chart shows the results of her experiment.

1. What is the experimental probability of choosing an O?

2. What is the experimental probability of choosing an E?

3. What is the experimental probability of choosing an O or a C?

4. Find the experimental probability of choosing a vowel. Write the answer in simplest form.

�1. �478� 2. �

458� 3. �

1478� 4. �

14

��

Objective Determine the experimental probability for a given set of data.

Materials• 6-section spinner*




1 3 5

estimate

actual

L O C K E R

8 7 10 10 5 8


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NAME ______________________________________________ DATE ____________ PERIOD __________

Percent and Estimation

Estimate the percent of each figure that is shaded by just looking. Thencount the 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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Ch

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Systems of MeasurementTeaching Notes and Overview

Using OverheadManipulativesMeasurement (pp. 91–92 of this booklet)


Objective Compare weights among the Sunand the planets.

Materialstransparency showing weight relative to Earth6 measuring cups10 containers (having a 3-cup capacity)waterblank transparencytransparency pen** = available in Overhead Manipulative Resources Kit

• Using containers of water, students compareweights among the Sun and the planets.

• Students find the weight of various objectson certain planets.

• An Extension activity asks students to findtheir relative Earth weight for severalplanets.

AnswersAnswers appear on the teacher demonstrationinstructions on pp. 91–92.

Measurement LabRecording Sheet The Metric System (pp. 93–94 of this booklet)

Use With Explore 8-3. This corresponds tothe activity on pp. 430–431 in the StudentEdition.

Objective Measure in metric units.

Materials tape measure

Students use metric units to measure severalitems. They explain which metric unit is mostappropriate for each item. Then they examinetheir measurements to identify a relationshipbetween the numbers. Students also selectthree objects around the classroom that wouldbe best measured in each of three metricunits. Finally, students will measure the sidesof rectangles and find the area and perimeterof each one.


Chapter

8


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Chapter 8 Systems of Measurement

Mini-ProjectUsing the Metric System(p. 95 of this booklet)


Objective Measure in metric units.

Materials centimeter ruler

Students measure several items in centimeters.They write the measurement of each item inmillimeters, meters, and kilometers.

Answers1. 5 cm; 50 mm; 0.05 m; 0.005 km

2. 3 cm; 30 mm; 0.03 m; 0.003 km

3. 7.5 cm; 75 mm; 0.075 m; 0.0075 km

4. 1.5 cm; 15 mm; 0.015 m; 0.0015 km

5–8. Answers will vary. See students’ work.

Measurement LabRecording SheetUsing Appropriate Units and Tools(pp. 96–97 of this booklet)

Use With Extend 8-8. This corresponds tothe activities on pages 459–460 in the StudentEdition.

Objective Select appropriate units and toolsto measure objects or activities.

Materialsa variety of measuring tools such as tapemeasures; inch and metric rulers, yardsticks;balance scales; measuring cups and spoons;pint, quart, and gallon containers;thermometers; and clocks and stopwatches.

Students measure common attributes ofclassroom objects and activities.


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Ch

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Measurement

Teacher Demonstration• Prepare a blank transparency with the diagram shown below.

• Fill a measuring cup with one cup of water to represent the weight of water on Earth. Pour this amount into a container. Label the container Earth.

• Show the transparency of Weight relative to Earth. Separate the class into 5 groups and assign each group a planet or the Moon. Each group should have two containers, a measuring cup, and water.

• Instruct the groups to pour one cup of water into a container and label itEarth, as you have shown.

• Instruct each group to fill as many measuring cups with water as needed torepresent the weight of water of their planet. For example, the Moon’s

weight is �16

� that of Earth, so that group would empty �16

� cup of water into a

container and label it Moon.

• Ask each group which of their two containers weighs more and why. On theblank transparency, record the name of each planet and whether the waterweighs more or less than on Earth. (Mercury, less; Venus, less; Moon,less; Mars, less; Jupiter, more)

Sun28

Mercury Venus Earth

Planets’ Weight Factors

Moon Mars

109

61

831

31

Jupiter3

Objective Compare weights among the Sun and the planets.

Materials Needed• transparency showing weight relative to Earth

• 6 measuring cups

• 10 containers (having a 3-cup capacity)

• water



* = available in Overhead Manipulative Resources Kit


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• Then ask whether the water on each planet weighs more or less than theweight of one cup of water on Jupiter. (less)

Have students complete Exercises 1–4 below.1. Which planet’s container weighs the most? Explain your reasoning.

(The container labeled Jupiter. It has the most water.)

2. How much would a 22-pound dog weigh on Jupiter? (66 pounds)

3. How much would you weigh on the Moon? (Sample answer: A 126-pound student would weigh 21 pounds.)

4. On Earth, a certain object weighs 1 pound. How many ounces would thesame object weigh on Mars? (6 ounces)

ExtensionHave students do research to find their relative Earth weight for the planetsSaturn, Neptune, Uranus, and Pluto.


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Measurement Lab Recording Sheet(Use with the activity on pages 430–431 in Explore 8-3 of the Student Edition)

NAME ______________________________________________ DATE ____________ PERIOD __________

The Metric System

Materialstape measure

ActivityRecord your data in the table below.

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ObjectMeasure

m cm mm

length of pencil

length of sheet of paper

length of your hand

width of your little finger

length of table or desk

length of chalkboard eraser

width of door

height of door

distance from doorknob to the floor

length of classroom

Analyze the Results1. Tell which unit of measure is most appropriate for each item. How did you

decide which unit was most appropriate?

2. LOOK FOR A PATTERN Examine the pattern between the numbers ineach column. How are the numbers in the first and second columns related?In the first and third columns? In the second and third columns?


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3. MAKE A CONJECTURE If you know the length of an object measured inmillimeters, explain how you could find its length measured in centimeters.

4. MAKE A CONJECTURE If you know the length of an object measured inmeters, explain how you could find its length measured in centimeters.

5. Select three objects around your classroom that would be best measured inmeters, three objects that would be best measured in centimeters, and threeobjects that would be best measured in millimeters. Explain your choices.

6. Write the name of a common object that you think has a length thatcorresponds to each length. Explain your choices.

a. 5 centimeters

b. 3 meters

c. 1 meter

d. 75 centimeters


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NAME ______________________________________________ DATE ____________ PERIOD __________

Ch

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Using the Metric System

Use a ruler to measure the lines in centimeters. Then write your measurement in millimeters, meters, and kilometers.1. __________ centimeters

__________ millimeters __________ meters __________ kilometers

2. __________ centimeters






Find four small objects in your classroom to measure with yourruler. Record what you are measuring. Record its length incentimeters. Write the measurement in millimeters, meters, andkilometers.5. Object: ________________________________ __________ centimeters


6. Object: ________________________________ __________ centimeters








Measurement Lab Recording Sheet(Use with the activity on pages 459–460 in Extend 8-8 of the Student Edition)

NAME ______________________________________________ DATE ____________ PERIOD __________

Using Appropriate Units and Tools

Materialsa variety of measuring tools such as tape measures; inch and metric rulers,yardsticks; balance scales; measuring cups and spoons; pint, quart, and galloncontainers; thermometers; and clocks and stopwatches.

Activity 1Record your results in the table below.

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Object Attribute(s) Tool Measurement

Analyze the Results1. Express each attribute of the object you measured using different units.

For example, if you measured the length of the object in centimeters,write this length in meters.

2. Write a real-world problem in which one of your measurements isneeded to solve the problem. For example, if you measured the time ittakes to sharpen one new pencil, a problem might be to estimate the timeit would take for each student in your class to sharpen a new pencilbefore a test.


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Activity 2Record your results in the table below.

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Activity Attribute(s) Tool Measurement

Analyze the Results3. Express each attribute of the activity you measured using different

units. For example, if you measured the time it took to do an activityin minutes, write this time using seconds.

4. Write a real-world problem in which one of your measurements isneeded to solve the problem. For example, if you measured the time ittook to sharpen one new pencil, a problem might be to estimate thetime it would take for each student in your class to sharpen a newpencil before taking a test.

Suppose you were going to organize the following events for a fieldday at your school. What tools would you need to set up the event anddetermine a winner?

5. 50-meter dash: Who can run the fastest?

6. Softball throw: Who can throw the farthest?

7. Water relay: Which team can fill up their bucket the fastest using aleaking cup?

8. Indicate an ideal outdoor temperature, in Fahrenheit and Celsiusdegrees, for the field day event described in Exercise 7. Explain yourreasoning.


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Geometry: Angles and PolygonsTeaching Notes and Overview

Using OverheadManipulativesAngles(pp. 100–101 of this booklet)


Objective Estimate measures of angles.

Materialswax paper, or colored transparencycompass*scissorsprotractor*blank transparenciestransparency pens** � available in Overhead Manipulative Resources Kit

• Students estimate the measures of anglesusing sections cut out of a circle.

• Students discuss how they could estimatethe measure of an angle without using aprotractor or the sections from the circle.


Geometry LabRecording Sheet Angles in Triangles(p. 102 of this booklet)

Use With Explore 9-4. This corresponds tothe activities on page 485 in the StudentEdition.

Objective Explore the relationship amongthe angles of a triangle.

Materialsnotebook paper or construction paper

Triangles have three angles. Students willexplore how the three angles of a triangle arerelated.


Geometry LabRecording SheetAngles in Quadrilaterals(p. 103 of this booklet)


Objective Explore the relationship amongthe angles of different quadrilaterals.

Materialsgrid paper

Students explore how the angles of differentquadrilaterals are related.


Chapter

9


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Chapter 9 Geometry: Angles and Polygons

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Geometry LabRecording Sheet Tessellations(p. 104 of this booklet)

Use With Extend 9-7. This corresponds tothe activity on page 508 in the StudentEdition.

Objective Create tessellations using patternblocks.

Materials pattern blocks

Students create tessellations using patternblocks shown. Students also identify twofigures that cannot be used to create atessellation and use a drawing to justify their answers.


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Angles

Teacher Demonstration • Use the compass to draw a circle with a radius of 4 inches on wax paper or a

colored transparency. Cut out the circle.

• Place the circle on the screen. Show students as you fold it in half and then inhalf again. Unfold the circle and tell students that the point where the foldsintersect is the center of the circle. Mark the center.

• Draw a line from the center to the edge of the circle along one of the folds.Ask students how you could draw an angle of 180° using this line as one sideof the angle. (Since 180° is a straight angle, you can extend the lineto the other side of the circle. Another solution is to position aprotractor with the center mark at the center of the circle and the0° mark on the line. Find 180° on the opposite side. Make a mark.Draw a line from the vertex through the mark.) Draw the angle andwrite the measure on the paper.

• On the same circle, draw angles of 90°and 45°, as shown. Label each section. Cut out the labeled angles. Make sure the edges of each section have pen lines showing. (If you are using a colored transparency and do not wish to cut it, you can use the angles without cutting the circle.)

• Ask a student to draw an angle on a blank transparency. Then ask studentshow they could use the labeled sections to estimate the measure of this angle.(Choose a section to compare. Place the side of the angle on theside of that section. See if the angle is greater or less than thelabeled angle. Compare to the next larger or smaller section.

180˚90˚ 45˚

Objective Estimate measures of angles.

Materials• wax paper, or colored transparency

• compass*

• scissors

• protractor*





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Continue until you find the 2 measures that the angle is between.Then estimate.) Measure the angle with a protractor to verify the accuracyof the estimate.

• Repeat for several different angles.

• Ask students to discuss how they could estimate the measure of an anglewithout using a protractor or the sections above. [Sample answer:Compare the angle to 0°, 90° (like a square corner), and 180°(straight line). Then estimate how far it is from the known angleand add or subtract that amount. For example, if it is abouthalfway between 0° and 90°, then the angle is about 45°.]

• Show students the remaining circle wedge. Ask what the measure of thisremaining wedge is. (45°)

Ch

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Geometry Lab Recording Sheet(Use with the activity on page 485 in Explore 9-4 of the Student Edition)

NAME ______________________________________________ DATE ____________ PERIOD __________

Angles in Triangles

Materialsnotebook paper or construction paper

Analyze the Results1. What does each torn corner represent?

2. The point where these three corners meet is the vertex of another angle as shown. Classify this angle as right, acute, obtuse, or straight. Explain.

3. What is the measure of this angle?

4. MAKE A CONJECTURE What is the sum of the measures of angles 1, 2, and 3 for each of your triangles? Verify your conjecture by measuring each angle using a protractor. Then find the sum of these measures for each triangle.

5. MAKE A CONJECTURE What is the sum of the measures of the angles of any triangle?

1 23

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NAME ______________________________________________ DATE ____________ PERIOD __________

Angles in Quadrilaterals

Materialsgrid paper

Analyze the ResultsRecord your results in the table below.

Ch

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Quadrilateral m∠A m∠B m∠C m∠D Sum of Angles

1

2

3

4

5

Analyze the Results1. Describe any patterns you see in the angle measurements of

Quadrilateral 1 and Quadrilateral 2.

2. Describe any patterns you see in the angle measurements ofQuadrilaterals 1–4.

3. MAKE A CONJECTURE Are any of the patterns found inQuadrilaterals 1–4 present in Quadrilateral 5? If not, make aconjecture as to what makes Quadrilateral 5 different fromQuadrilaterals 1–4.


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Geometry Lab Recording Sheet(Use with the activity on page 508 in Extend 9-7 of the Student Edition)

NAME ______________________________________________ DATE ____________ PERIOD __________

Tessellations

Materialspattern blocks

Your TurnCreate a tessellation using the pattern blocks shown.

a. b. c.

Analyze the Results1. Tell if a tessellation can be created using a square and an equilateral triangle.

Justify your answer with a drawing.

2. MAKE A CONJECTURE What is the sum of the measures of the angleswhere the vertices of the figures meet? Is this true for all tessellations?

3. Name two figures that cannot be used to create a tessellation. Use a drawingto justify your answer.


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Ch

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Measurement: Perimeter, Area, and VolumeTeaching Notes and Overview

Measurement LabRecording SheetArea and Perimeter(pp. 108–109 of this booklet)

Use With Explore 10-1. This corresponds tothe activities on page 520–521 in the StudentEdition.

Objective Explore changes in area andperimeter of rectangles.


Students will find the area and perimeter ofdifferent rectangles to investigate relationshipsbetween areas and perimeters of originalfigures and those of the new figures. Theywill discover how changing the dimensions ofa figure affects its area and perimeter.


Measurement LabRecording SheetCircumference(p. 110 of this booklet)


Objective Describe the relationship betweenthe diameter and circumference of a circle.

Materialsstringcircular object: jar lidcentimeter ruler

Students will use string and a circular objectto find the relationship between the distancearound a circle and the distance across a circlethrough its center.


Mini-Project Circumference(p. 111 of this booklet)


Objective Find the circumference of a circle.

Materialsstringruler

Using a piece of string and a ruler, studentsfind the circumferences of several circles.They will also measure the diameter of eachcircle and find its circumference using theformula for the circumference of a circle.Students then compare the two methods forfinding the circumference of a circle.

AnswersSample answers are given.

1. 4 in.; 1�14

� in.; 3�1134� in.

2. 4�34

� in.; 1�12

� in.; 4�57

� in.

3. 1�38

� in.; �12

� in.; 1�47

� in.

4. 2�78

� in.; �78

� in.; 2�34

� in.

5. The formula method; it is easier tomeasure the diameter than thecircumference.

Chapter

10


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Chapter 10 Measurement: Perimeter, Area, and Volume

Using OverheadManipulativesDiameter and Circumference(p. 112 of this booklet)


Objective Discover the relationship betweenthe diameter and circumference of a circle.

Materialscompass*stringscissorsruler*calculatorblank transparenciestransparency pen** � available in Overhead Manipulative Resources Kit

• This demonstration shows how to find thecircumference of a circle using a piece ofstring.

• Students will measure the circumferenceand diameter of several circles, then dividethe circumference by the diameter todiscover the relationship between the twomeasurements.


Using OverheadManipulativesArea of Irregular Shapes (pp. 113–114 of this booklet)


Objective Find the area of irregular shapes.

Materials centimeter grid transparency*transparency pens**� available in Overhead Manipulative Resources Kit

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

area of an irregular figure on a centimetergrid.

• Demonstration 2 shows how to estimate thearea of an irregular figure by drawing arectangle around the figure.

• Students explain which estimation methodthey think is more accurate.

• An Extension activity asks students tochoose a method and estimate the area of acircle and a triangle on a centimeter grid.


Measurement LabRecording Sheet Area of Triangles (p. 115 of this booklet)


Objective Discover the formula for the areaof a triangle using the properties ofparallelograms and a table of values.

Materials grid paper colored pencilsscissors


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Chapter 10 Measurement: Perimeter, Area, and Volume

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Students use the properties of parallelogramsto find the area of triangles. Students discoverthe formula for the area of a triangle.


Mini-Project Areas of Triangles (p. 116 of this booklet)


Objective Find the areas of triangles.

Materials metric ruler

Students use a metric ruler to measure thebase and height of several triangles. Usingthese measurements, students find the area ofeach triangle to the nearest square millimeter.

AnswersSample answers are given.

1. 450 mm2

2. 672 mm2

3. 308 mm2

4. 263 mm2

5. 463 mm2

6. 442 mm2

Geometry LabRecording Sheet Using a Net to Build a Cube (p. 117 of this booklet)

Use With Explore 10-7. This corresponds to the activity on page 554 in the StudentEdition.

Objective Make a two-dimensional patternfor a cube and use it to build another cube.

Materials cubescissorspaper

Students identify and draw nets that will forma cube, as well as nets that will not form acube. They draw a net for a rectangular prismand compare it to the nets that form a cube.Finally, students identify the figures formedby given nets.


Measurement LabRecording SheetSelecting Formulas and Units(p. 118 of this booklet)

Use With Extend 10-7. This corresponds tothe activities on page 560 in the StudentEdition.

Objective select appropriate units, tools, andformulas to measure objects.

Materialsmeasuring tools

Students use direct measures of one or moreattributes of an object to calculate measuressuch as circumference, area, or volume.




Measurement Lab Recording Sheet(Use with the activities on pages 520–521 in Explore 10-1 of the Student Edition)

NAME ______________________________________________ DATE ____________ PERIOD __________

Area and Perimeter


Activity 1Record your data in the table below.

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Rectangle Length (cm) Width (cm) Area (sq cm) Perimeter (cm)

original

2

3

4

5

Analyze the Results1. Describe how the dimensions of rectangles A, B, and C are different than the

original rectangle.

2. Describe how the area perimeter of the original rectangle changed when thelength and width were both doubled.

3. Describe how the perimeter of the original rectangle changed when thelength and width were both doubled.

4. Describe how the area and the perimeter of the original rectangle changedwhen the length and width were both tripled.

5. Describe how the area and the perimeter of the original rectangle changedwhen the length and width were both quadrupled.

6. Draw a rectangle whose length and width are half those of the originalrectangle. Describe how the area and perimeter changes.


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7. MAKE A CONJECTURE How are the perimeter and area of a rectangleaffected if the length and the width are changed proportionally?

Activity 2Record your data in the table below.

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Square Side Length (cm) Area (sq cm) Perimeter (cm)

original

2

3

4

5

Analyze the Results8. Describe how the dimensions of squares A, B, and C are different from the

original square.

9. Describe how the perimeter of the original square changed when the sidelengths increased by one centimeter.

10. Describe any ratios in the table above.

11. Suppose the perimeter of a square is 60 centimeters. Explain how you canfind the length of its side. Then find its side length.

12. WRITE A FORMULA If P represents the perimeter of a square, write anequation that describes the relationship between the square’s side length sand perimeter P.

13. MAKE A CONJECTURE Suppose you double the side lengths of theoriginal square. Use what you learned in Activity 1 to predict the area andperimeter of the new square. Explain your reasoning.



Measurement Lab Recording Sheet(Use with the activity on page 527 in Explore 10-2 of the Student Edition)

NAME ______________________________________________ DATE ____________ PERIOD __________

Circumference

Materialsstringcircular object such as a jar lidcentimeter rules

Analyze the ResultsRecord your results in the table below.

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Object C d �dC

�

Analyze the Results1. MAKE A CONJECTURE If you know the diameter of a circle, how can

you find the distance around the circle?

2. MAKE A PREDICTION What would be the approximate distance arounda circle that is 4 inches across?

3. MAKE A CONJECTURE How can you find the distance around a circleif you know the distance from the center of the circle to the edge of thecircle?


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NAME ______________________________________________ DATE ____________ PERIOD __________

Circumference

Place a piece of string around the circumference of each circle.Measure the string to the nearest eighth of an inch. Record themeasurement. Then draw the diameter and measure it to thenearest eighth of an inch. Use the formula C � �d to calculate the circumference. Use �

272� as an approximation of �.

1. 2.

Circumference � ____________ Circumference � ______________

Diameter � ________________ Diameter � __________________

�d � ______________________ �d � ________________________

3. 4.

Circumference � ____________ Circumference � ______________

Diameter � ________________ Diameter �____________________

�d � ______________________ �d � ________________________

Compare the two circumference measures for each circle.5. Which method is better for determining the circumference? ______________

________________________________________________________

Why? ________________________________________________________

______________________________________________________________

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Teacher Demonstration • With the compass, draw a circle with a radius of 2 cm on a blank transparency.

Show students the circle. Remind them that the circumference of the circle isthe distance around the circle. Tell students that you can use string to help findthe measure of the circumference of this circle. Place the end of the string on apoint of the circle. (You may want to tape it in place.) Show students as youtrace the circumference of the circle with the string. Cut the string at the pointwhere it meets the other end.

• Place the string on the screen and use the ruler to find the length of the string.(about 12.6 cm) Ask students what measurement of the circle this lengthrepresents. (the circumference)

• Use the ruler to measure the diameter of the circle. (4 cm)• Have students use their calculators to divide the circumference, 12.6, by the

diameter, 4, of the circle. (3.15) Record the result on the transparency.

• Repeat the steps for circles with other diameters.

• Ask students what they notice about the quotients you obtained when youdivided each circle’s circumference by its diameter. (The quotients areclose to 3.)

• Ask students how the circumference and diameter of a circle appear to berelated. (The circumference of a circle is always a little more than 3times the diameter.)

Diameter and Circumference

Objective Discover the relationship between the diameter and circumferenceof a circle.

Materials• compass*

• string

• scissors

• ruler*

• calculator



* � available in Overhead Manipulative Resources Kit.


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Area of Irregular Shapes

Teacher Demonstration for Activity 1 • On the centimeter grid transparency,

draw a figure similar to the one shown at the right.

• Shade the whole squares within the outlineof the figure. Have students count thenumber of shaded squares. Record thisnumber on the grid.

• Using a different colored transparency pen,shade the squares that touch the outline anywhere. Ask students to count the number of these squares. Add this to thenumber of whole squares within the outline. Record this number on the grid.

• Ask students to find the mean of the two recorded numbers. Ask students howthis number represents an estimate of the area within the outline. (The areais at least the number of whole squares and at most the numberof whole and partial squares. The mean is halfway between thesetwo numbers.)

• Ask students to think of another way to estimate the area of the figure.(Sample answer: Draw a rectangle around the shape and 2 smallrectangles in the area not in the shape. Subtract the area of thesmall rectangles from the area of the large rectangle.)

Objective Find the area of irregular shapes.

Materials• centimeter grid transparency*


*� available in Overhead Manipulative Resources Kit


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Teacher Demonstration for Activity 2• On the centimeter grid transparency, draw a figure similar to

the one shown at the right.

• Draw a rectangle that encloses most of the figure.

• Have students count squares to find the length and the width of the rectangle. Record the numbers on the grid. (Answerswill vary.)

• Ask students to estimate the area of the figure by finding the area of therectangle. (The estimate is the length times the width.)

• Next, use the method in Activity 1 to estimate the area of the figure.(Answers will vary.)

• Ask students which estimation method they think is more accurate. Have themexplain their reasoning. (Answers will vary. In general, the methodusing the rectangle that encloses most of the figure is moreaccurate when the parts of the figure inside the rectangle andoutside the rectangle are about the same.)

ExtensionOn the centimeter grid, draw the following figures. Ask students to describe away to estimate the area of each figure. Then estimate each area using thosemethods. (Answers will vary.)a. circle b. triangle


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Parallelogram Base, b Height, hArea of Area of

Parallelogram Each Triangle

A 4 6

B 2 5

C 3 4

D 5 3

E 7 5


Measurement Lab Recording Sheet(Use with the activity on page 539 in Explore 10-4 of the Student Edition)

NAME ______________________________________________ DATE ____________ PERIOD __________

Area of Triangles


ActivityRecord your results in the table below.

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Analyze the Results1. Compare the base and height of each triangle to the base and height of

the original parallelogram. What do you notice?

2. Compare the two triangles formed. How are they related?

3. What is the area of each triangle? Record your answer in the table.

4. Repeat Steps 2 through 5 for Parallelograms B through E. Calculatethe area of each triangle formed and record your results in the table.

5. LOOK FOR A PATTERN What patterns do you notice in the rows ofthe table?

6. MAKE A CONJECTURE Write a formula that relates the area A ofa triangle to the length of its base b and height h.


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NAME ______________________________________________ DATE ____________ PERIOD __________

Areas of Triangles

Use a metric ruler to measure the base and height of each triangle.If the height is not shown, sketch it. Label these segments withtheir measurements to the nearest millimeter. Use yourmeasurements to calculate the area to the nearest squaremillimeter.1. 2.

Area � _________ Area � _________

3. 4.

Area � _________ Area � _________

5. 6.

Area � _________ Area � _________


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NAME ______________________________________________ DATE ____________ PERIOD __________

Using a Net to Build a Cube

Materialscubepaperscissors

Analyze the Results1. Explain whether both nets formed a cube. If not, describe why the net or nets

did not cover the cube.

2. Draw three other nets that will form a cube and three other nets that will notform a cube. Describe a pattern in the nets that do form a cube.

3. Measure the edges of the cube in the activity above. Use this measure to findthe area of one side of the cube.

4. MAKE A CONJECTURE Write an expression for the total of all thesurfaces of a cube with edge length s.

5. Draw a net for a rectangular prism. Explain the difference between this netand the nets that formed a cube.

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Measurement Lab Recording Sheet(Use with the activity on page 560 in Extend 10-7 of the Student Edition)

NAME ______________________________________________ DATE ____________ PERIOD __________

Selecting Formulas and Units

Materialsmeasuring tools

ActivityRecord your results in the table below.

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Object AttributeFormula Direct CalculatedNeeded Measure(s) Measure(s)

shoebox

chalkboard

desktop

cereal box

clock face

bulletin board

basketball

Analyze the Results1. Which object did you find most difficult to measure directly? How did you

solve this problem?

2. WRITING IN MATH Write a real-world problem that could be solved usingone of the objects and the measure you calculated.



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.Integers and Transformations Teaching Notes and Overview

Algebra LabRecording Sheet Zero Pairs (p. 120 of this booklet)


Objective Use models to understand zeropairs.

Materialscountersinteger mat

Students use counters to model pairs ofintegers, then form zero pairs to find the sumof the integers.


Mini-ProjectThe Coordinate System(p. 121 of this booklet)


Objective Graph and label points.

Materials none

Students graph and label sets of points. Theyconnect the points and identify each figure.

Answers1. square ABCD 2. rectangle JKLM

3. trapezoid PQRS 4. parallelogram EFGH

5. rectangle STUV 6. octagon PQRSTUVW

x

y

O-5 -4 -3 -2 -1 1 2 3 4 5

-5-4

-3-2-1

12345

P Q

R

S

W

V

TU

x

y

O-5 -4 -3 -2 -1 1 2 3 4 5

-5-4-3-2-1

12345

S T

UV

x

y

O-5 -4 -3 -2 -1 1 2 3 4 5

-5-4-3-2-1

12345

E

F

H

G

x

y

O-5 -4 -3 -2 -1 1 2 3 4 5

-5-4-3-2-1

12345

P Q

S R

x

y

O-5 -4 -3 -2 -1 1 2 3 4 5

-5-4-3-2-1

12345

J K

M L

x

y

O-5 -4 -3 -2 -1 1 2 3 4

-5-4-3-2-1

1234

5

5

A B

D C

Chapter

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Algebra Lab Recording Sheet(Use with the activity on page 576 in Explore 11-2 of the Student Edition)

Zero Pairs

Materials: counters, integer mat

Check Your Progress Use counters to model each pair of integers. Then form zero pairs to findthe sum of the integers.

a. �3, �3 b. �5, �5 c. �7, �7

Analyze the Results 1. What is the value of a zero pair? Explain your reasoning.

2. Suppose there are 5 zero pairs on an integer mat. What is the value of thesezero pairs? Explain.

3. Explain the effect of removing a zero pair from the mat. What effect doesthis have on the remaining counters?

4. Integers like 14 and 24 are called opposites. What is the sum of any pair ofopposites?

5. Write a sentence describing how zero pairs are used to find the sum of anypair of opposites.

6. MAKE A CONJECTURE How do you think you could find �5 � (�2)using counters?



NAME ______________________________________________ DATE ____________ PERIOD __________

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The Coordinate Systema. Graph and label each point.

b. Connect the points in order, including the last and the first points.

c. Name the figure.

1. A(�3, 3), B(1, 3), C(1, �1), 2. J(�5, 2), K(3, 2), L(3, �3), D(�3, �1) M(�5, �3)

3. P(�3, 3), Q(2, 3), R(5, �2), 4. E(�3, 4), F(4, 2), G(4, �2), S(�5, �2) H(�3, 0)

5. S(�4, 5), T(0, 5), U(0, �2), 6. P(�2, 4), Q(1, 4), R(3, 1), S(3, �1),V(�4, �2) T(1, �4), U(�2, �4), V(�4, �1),

W(�4, 1)

x

y

O-5 -4 -3 -2 -1 1 2 3 4

-5-4-3-2-1

1234

5

5

x

y

O-5 -4 -3 -2 -1 1 2 3 4

-5-4-3-2-1

1234

5

5

x

y

O-5 -4 -3 -2 -1 1 2 3 4

-5-4-3-2-1

1234

5

5

x

y

O-5 -4 -3 -2 -1 1 2 3 4

-5-4-3-2-1

1234

5

5

x

y

O-5 -4 -3 -2 -1 1 2 3 4

-5-4-3-2-1

1234

5

5

x

y

O-5 -4 -3 -2 -1 1 2 3 4

-5-4-3-2-1

1234

5

5


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Algebra: Properties and EquationsTeaching Notes and Overview

Algebra LabRecording SheetDistributive Property(pp. 125–126 of this booklet)


Objective Solve equations using theDistributive Property.

Students will use and draw models toillustrate the Distributive Property. They willuse these models and the DistributiveProperty to solve equations.


Algebra Lab Recording Sheet Solving Addition Equations UsingModels(p. 127 of this booklet)


Objective Solve addition equations usingcups and counters.

Materials cupscountersequation mat

Using cups and counters, students will solveaddition equations using models. Studentswill explain how to solve equations usingmodels and discover a rule they can use tosolve addition equations without usingmodels.


Mini-Project Solving Addition Equations(p. 128 of this booklet)


Objective Solve addition equations usingmodels.


Students write equations represented bymodels. Using cups and counters, they solveaddition equations with models. Space isprovided for students to sketch their models.

Answers1. 4 � x � 6

2. x � 3 � 9 or x � (�3) � 9

3. x � 2 � 6

4. x � (�3) � 4 or x � 3 � 4

5. x � 4

++

++

+ ++

++

+ ++ +

+ ++ +

++ +

Chapter

12


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Chapter 12 Algebra: Properties and Equations

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6. x � �5

7. x � �4

8. �7

Algebra LabRecording Sheet Solving Subtraction EquationsUsing Models(p. 129 of this booklet)


Objective Solve subtraction equations withmodels.


Using cups and counters, students will solvesubtraction equations using models. Students

will explain how to solve equations usingmodels and discover a rule they can use tosolve subtraction equations without usingmodels.


Algebra Lab Recording Sheet Solving Inequalities Using Models(p. 130 of this booklet)

Use With Extend 12-4. This corresponds tothe activity on pages 655–656 in the StudentEdition.

Objective Use models to solve simpleaddition and subtraction inequalities.

Materials

cups and counters

Students will use cups and counters to modeland solve inequalities.

AnswersSee Teacher Edition p. 655–656.

—++ —— — ————

————

– ––– –– –––– –

– ––––

––

– ––– –


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Chapter 12 Algebra: Properties and Equations

Using OverheadManipulativesSolving Multiplication Equations(pp. 131–132 of this booklet)


Objective Solve multiplication equations byusing models.

Materials cups*counters*equation mat transparency*transparency pens** = available in Overhead Manipulative Resources Kit

• This demonstration shows how to solvemultiplication equations by using models.

• Students then solve equations independentlyusing models.

• Students explain how they would solvemultiplication equations without usingmodels.


122_138_TMWMC1_889763 12/3/07 5:35 PM Page 124



NAME ______________________________________________ DATE ____________ PERIOD __________

The Distributive Property

CHOOSE Your MethodDraw a model showing that each equation is true.

a. 2(4 � 6) � (2 � 4) � (2 � 6) b. 4(3 � 2) � (4 � 3) � (4 � 2)

c. 7(10 � 8) � (7 � 10) � (7 � 8) d. 6(20 � 3) � (6 � 20) � (6 � 3)

Analyze the Results

Activity 11. Refer to Check Your Progress c above. How could you use the Distributive

Property to evaluate 7(18) mentally?

2. Use the Distributive Property to evaluate 9(33) mentally.

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Activity 2Tell whether each statement is true or false. Justify your answer with tilesor a drawing.

1. 3(x � 1) � 3x � 3 2. 4(x � 1) � 4x � 1

3. 3(2x � 1) � 6x � 2 4. 2(3x � 2) � 6x � 4

5. MAKE A CONJECTURE Use what you learned in this lab to make aconjecture about the expressions 5(2x � 3) and 10x � 15.

6. REASONING Use what you learned in this lab to rewrite the expressionsbelow without parentheses.

2(x � 1) � ________________________________

6(x � 4) � ________________________________

3(5x � 6) �

7. WRITING IN MATH A friend decides that 4(x � 3) � 4x � 3. Howwould you explain to your friend that 4(x � 3) � 4x � 12? Include drawingsin your explanation.

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NAME ______________________________________________ DATE ____________ PERIOD __________

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Solving Addition Equations Using Models

Materialscups, counters, equation mat

Check Your ProgressSolve each equation using models.

a. 1 � x � 8 b. x � 2 � 7 c. 9 � x � 3

d. x � 3 � �7 e. 2 � x � �5 f. �3 � x � 3

Analyze the Results 1. Explain how you decide how many counters to add or subtract from

each side.

2. Write an equation in which you need to remove zero pairs in order to solve it.

3. Model the equation some number plus 5 is equal to �2. Then solve theequation.

4. MAKE A CONJECTURE Write a rule that you can use to solve anequation like x � 3 � 6 without using models.


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NAME ______________________________________________ DATE ____________ PERIOD __________


Solving Addition Equations

Write the equation that is represented by each model.1. 2.

_____________________________ _____________________________

3. 4.

_____________________________ _____________________________

Solve each equation using cups and counters. Sketch thearrangement in the boxes.5. x � 2 � 6

6. x � (�2) � �7

7. x � 1 � �3

8. Solve x � 4 � �3 without using models. x � ____________

+ ++

++ +

+ ++ + –––

+ + ++ + ++ + +

+ ++

++ +

+ + ––– + ++ +


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NAME ______________________________________________ DATE ____________ PERIOD __________

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Algebra Lab Recording Sheet(Use with the activity on page 650 in Explore 12-4 of the Student Edition)

Solving Subtraction Equations Using Models

Materialscups, counters, equation mat

Check Your ProgressSolve each equation using models.

a. x � 4 � 2 b. �3 � x � 1 c. x � 5 � �1

Analyze the Results1. Explain why it is helpful to rewrite a subtraction problem as an addition

problem when solving equations using models.

2. MAKE A CONJECTURE Write a rule for solving equations like x � 7 � �5 without using models.



Algebra Lab Recording Sheet(Use with the activity on page 655–656 in Extend 12-4 of the Student Edition)

NAME ______________________________________________ DATE ____________ PERIOD __________

Solving Inequalities Using Models

Check Your ProgressSolve each of the following inequalities algebraically and using a model.Draw a picture of your model below.

a. x � 4 � 7 b. x � 2 � 3 c. x � 5 � 1

Analyze the Results1. Write two different inequalities, one involving addition and one involving

subtraction, both of whose solutions are x � 4.

2. WRITING IN MATH Explain how you could solve the inequality x � 7 � 12 algebraically.

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Solving Multiplication Equations

Teacher Demonstration • Place 2 cups on the left side of the equation mat. Place 6 red counteres on the

right side of the mat. Remind students that each cup represents an unknownvalue, x. Ask them to state the equation represented by the models. (2x � �6)Write the equation at the bottom of the mat.

• Tell students that each cup must contain the same number of counters.Arrange the counters into two equal groups to correspond to the two cups.

Objective Solve multiplication equations by using models.

Materials• cups*

• counters*

• equation mat transparency*


* = available in Overhead Manipulative Resources Kit

2x � �6

–––

–––

2x � �6

–

–

–

–

–

–

• Ask students what the solution of the equation is. (�3)



Using Overhead ManipulativesC

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2. What operation did you use to find each solution? (division)

3. The coefficient of an expression like 3x is the numerical part, 3. How canyou use the coefficient to solve the equation 3x � 12? (Divide each sideby the coefficient, 3.)

4. How would you solve 2x � 5 without using models? (Divide 5 by 2.)

Have students complete Exercises 1–4 below.1. Solve each equation using models.

a. 5x � 20 (4) b. �12 � 4x (�3) c. 3x � 3 (1)

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