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Course 2
Teaching Mathematics withManipulatives
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Manipulatives
Glencoe offers three types of kits to enhance the use of manipulatives in your
Middle School Mathematics classroom.
• The Glencoe Mathematics Overhead Manipulative Resources contains translucent
manipulatives designed for use with an overhead projector.
• The Glencoe Mathematics Classroom Manipulative Kit contains classroom sets of
frequently used manipulatives in algebra, geometry, measurement, probability, and
statistics.
• The Glencoe Mathematics Student Manipulative Kit contains an individual set of
manipulatives often used in Student Edition activities.
The manipulatives contained in each of these kits are listed on page vi of this
booklet.
Each of these kits can be ordered from Glencoe by calling (800) 334-7344.
Glencoe Mathematics Overhead Manipulative Kit 0-07-830593-4
Glencoe Mathematics Classroom Manipulative Kit 0-02-833116-8
Glencoe Mathematics Student Manipulative Kit 0-02-833654-2
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Permission is granted toreproduce the material contained herein on the condition that such materials be reproduced onlyfor classroom use; be provided to students, teachers, and families without charge; and be usedsolely in conjunction with the Glencoe Mathematics: Applications and Concepts, Course 2,program. Any other reproduction, for sale or other use, is expressly prohibited.
Send all inquiries to:Glencoe/McGraw-Hill8787 Orion PlaceColumbus, OH 43240-4027
ISBN: 0-07-860125-8 Teaching Mathematics with Manipulatives
Printed in the United States of America.
1 2 3 4 5 6 7 8 9 10 045 11 10 09 08 07 06 05 04 03
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iii
Contents
Easy-to-Make Manipulatives PageBase-Ten Models . . . . . . . . . . . . . . . . . . . . . . .1
Decimal Models . . . . . . . . . . . . . . . . . . . . . . . .2
Fraction Models: Bars . . . . . . . . . . . . . . . . . . .3
Fraction Models: Circles . . . . . . . . . . . . . . . . .4
Counters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Integer Counters . . . . . . . . . . . . . . . . . . . . . . . . 6
Pattern for Cup . . . . . . . . . . . . . . . . . . . . . . . . . 7
Integer Mat . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Equation Mat . . . . . . . . . . . . . . . . . . . . . . . . . .9
Quarter-Inch Grid . . . . . . . . . . . . . . . . . . . . . . 10
Centimeter Grid . . . . . . . . . . . . . . . . . . . . . . .11
Square Dot Paper . . . . . . . . . . . . . . . . . . . . . .12
Isometric Dot Paper . . . . . . . . . . . . . . . . . . . .13
Tangram . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14
Number Lines . . . . . . . . . . . . . . . . . . . . . . . . .15First Quadrant Grids . . . . . . . . . . . . . . . . . . . . 16
Coordinate Planes . . . . . . . . . . . . . . . . . . . . . . 17
Percent Models . . . . . . . . . . . . . . . . . . . . . . . . 18
Spinners . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19
Number Cube Patterns . . . . . . . . . . . . . . . . . .20
Protractors . . . . . . . . . . . . . . . . . . . . . . . . . . .21
Rectangular Prism Pattern . . . . . . . . . . . . . . .22
Cube Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Cylinder Pattern . . . . . . . . . . . . . . . . . . . . . . .24
Cone Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Pyramid Pattern . . . . . . . . . . . . . . . . . . . . . . .26
Pattern Blocks . . . . . . . . . . . . . . . . . . . . . . . .27
Circle Graph Template . . . . . . . . . . . . . . . . . .28
Problem-Solving Guide . . . . . . . . . . . . . . . . .29
Activities PageCHAPTER 1
Teaching Notes and Overview . . . . . .30
1-2 Mini-Project: Modeling Powers and
Exponents . . . . . . . . . . . . . . . . . . .311-4 Using Overhead Manipulatives:
Variables and Expressions . . . . . . .32
1-7b Hands-On Lab Recording Sheet . . . . .34
CHAPTER 2Teaching Notes and Overview . . . . . .35
2-2 Mini-Project: Story Graph . . . . . . . . .37
2-4b Hands-On Lab Recording Sheet . . . . .38
2-6 Using Overhead Manipulatives:
Quartiles . . . . . . . . . . . . . . . . . . . .39
2-8 Using Overhead Manipulatives:
How Much is a Handful? . . . . . . . .41
CHAPTER 3
Teaching Notes and Overview . . . . . .42
3-3 Mini Project: Coordinate Plane
Puzzle . . . . . . . . . . . . . . . . . . . . . .44
3-4a Hands-On Lab Recording Sheet . . . . .45
3-5a Hands-On Lab Recording Sheet . . . . .46
3-6 Using Overhead Manipulatives:
Multiplying Integers . . . . . . . . . . . .47
CHAPTER 4Teaching Notes and Overview . . . . . .49
4-2a Hands-On Lab Recording Sheet . . . . .514-3 Mini-Project: Solving Multiplication
Equations . . . . . . . . . . . . . . . . . . . .52
4-4 Using Overhead Manipulatives:
Solving Two-Step Equations . . . . . .53
4-6a Hands-On Lab Recording Sheet . . . . .55
4-6 Using Overhead Manipulatives:
A Function of Time . . . . . . . . . . . .56
CHAPTER 5
Teaching Notes and Overview . . . . . .58
5-1a Hands-On Lab Recording Sheet . . . . .595-5 Using Overhead Manipulatives:
Percent . . . . . . . . . . . . . . . . . . . . . . 60
CHAPTER 6Teaching Notes and Overview . . . . . .61
6-4 Using Overhead Manipulatives:
Multiplying Fractions and Mixed
Numbers . . . . . . . . . . . . . . . . . . . .62
6-8 Mini-Project: Perimeter . . . . . . . . . . .64
6-9a Hands-On Lab Recording Sheet . . . . .65
CHAPTER 7Teaching Notes and Overview . . . . . .66
7-1 Using Overhead Manipulatives:
Equal Ratios . . . . . . . . . . . . . . . . . . 68
7-2b Hands-On Lab Recording Sheet . . . . .70
7-3b Hands-On Lab Recording Sheet . . . . .71
7-4 Mini-Project: Scale Drawings . . . . . . .72
7-8a Hands-On Lab Recording Sheet . . . . .73
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iv
CHAPTER 8Teaching Notes and Overview . . . . . .74
8-1 Mini-Project: Percent and
Estimation . . . . . . . . . . . . . . . . . . .75
8-3a Hands-On Lab Recording Sheet . . . . .76
8-4 Using Overhead Manipulatives:
Percent of Change . . . . . . . . . . . . .77
CHAPTER 9Teaching Notes and Overview . . . . . .79
9-4 Using Overhead Manipulatives:
Exploring Permutations . . . . . . . . .81
9-5a Hands-On Lab Recording Sheet . . . . .83
9-6 Mini-Project: Theoretical and
Experimental Probability . . . . . . . .84
9-6b Hands-On Lab Recording Sheet . . . . .85
CHAPTER 10Teaching Notes and Overview . . . . . .86
10-1a Hands-On Lab Recording Sheet . . . . .90
10-1b Hands-On Lab Recording Sheet . . . . .91
10-2 Using Overhead Manipulatives:
Jelly Bean Statistics . . . . . . . . . . . .92
10-3b Hands-On Lab Recording Sheet . . . . .93
10-4b Hands-On Lab Recording Sheet . . . . .94
10-5 Using Overhead Manipulatives:
Investigating Triangles and
Quadrilaterals . . . . . . . . . . . . . . . . .95
10-7 Mini-Project: Quadrilateral
Tesselations . . . . . . . . . . . . . . . . . .97
10-7 Using Overhead Manipulatives:
Angles of a Polygon . . . . . . . . . . . .98
10-7 Using Overhead Manipulatives:
Inscribed Polygons . . . . . . . . . . . .100
10-8b Using Overhead Manipulatives:
Dilations . . . . . . . . . . . . . . . . . . .102
10-9b Hands-On Lab Recording Sheet . . . .104
CHAPTER 11Teaching Notes and Overview . . . . .105
11-3a Hands-On Lab Recording Sheet . . . .107
11-4 Using Overhead Manipulatives:
Area . . . . . . . . . . . . . . . . . . . . . . . 108
11-5a Hands-On Lab Recording Sheet . . . .110
11-6 Mini-Project: Areas of Circles,
Rectangles, and Squares . . . . . . . .111
11-8 Using Overhead Manipulatives:
Probability and Area Models . . . .112
CHAPTER 12Teaching Notes and Overview . . . . .114
12-1a Hands-On Lab Recording Sheet . . . .116
12-2 Using Overhead Manipulatives:
Volume of Pyramids . . . . . . . . . . .117
12-3 Mini-Project: Volume of Solids . . . . .119
12-4a Hands-On Lab Recording Sheet . . . .120
12-4b Hands-On Lab Recording Sheet . . . .121
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v
Teacher’s Guide to UsingTeaching Mathematics with Manipulatives
The book contains two sections of masters—
Easy-to-Make Manipulatives and activities for
Middle School Mathematics. Tabs help you
locate the activities for each chapter. A complete list of
manipulatives available in each of the three types of
Glencoe Mathematics Manipulative Kits appears on the
next page.
Easy-to-Make Manipulatives
The first section of this book contains masters for
making your own manipulatives. To make more
durable manipulatives, consider using card stock. To
make algebra tiles similar to those shown in the
Student Edition, have students use markers to color the
tiles appropriately or use colored card stock.
You can also make transparencies of frequently used
items such as grid paper and number lines.
Activity Masters
Each chapter begins with Teaching Notes and
Overview that summarizes the activities for the chapter
and includes sample answers. There are three types of
masters.
Mini-Projects are short projects that enable students to
independently investigate mathematical concepts.
Using Overhead Manipulatives provides instructions
for the teacher to demonstrate an alternate approach to
the concepts of the lesson by using manipulatives on
the overhead projector.
Student Recording Sheets accompany the Hands-On
Lab Activities found in the Student Edition. Students
can easily record the results of the activity on prepared
grids, charts, and figures.
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vi
Glencoe Mathematics Manipulatives
Glencoe Mathematics Overhead Manipulative Resources
ISBN: 0-07-830593-4
Transparencies Overhead Manipulatives
integer mat centimeter grid algebra tilesequation mat number lines spinners
product mat lined paper two-dimensional cups
inequality mat regular polygons red and yellow counters
dot paper polynomial models decimal models (base-ten blocks)
isometric dot paper integer models compass
coordinate grids equation models protractor
geoboard/geobands
geometric shapes
transparency pens in 4 colors
Glencoe Mathematics Classroom Manipulative Kit
ISBN: 0-02-833116-8
Measurement, Probability,
Algebra and Statistics Geometry
algebra tiles base-ten models compasses
counters marbles geoboards
cups measuring cups geobands
centimeter cubes number cubes geomirrors
equation mat/product mat protractors isometric dot grid stamp
coordinate grid stamp and rulers pattern blocks
ink pad scissors tangrams
spinners
stopwatches
tape measures
Glencoe Mathematics Student Manipulative Kit
ISBN: 0-02-833654-2
algebra tiles protractor
red and yellow counters scissorscups geoboard
equation /product mat geobands
compass/ruler tape measure
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© Glencoe/McGraw-Hill 1 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Base-Ten Models
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© Glencoe/McGraw-Hill 2 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Decimal Models
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© Glencoe/McGraw-Hill 3 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Fraction Models: Bars
1
1
2
1
2
1
12
1
12
1
12
1
12
1
12
1
12
1
12
1
12
1
12
1
12
1
12
1
12
110
110
110
110
110
110
110
110
110
110
1
9
1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
5
1
5
1
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1
4
1
4
1
4
1
3
1
3
1
3
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© Glencoe/McGraw-Hill 4 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Fraction Models: Circles
1 1
1
2
1
2
15
15
15
15
15
14
14
14
13
13
13
16
16
16
16
17
17
17
17 1
7
17
17
1
8
1
818
18
18
19
19
1
9
1
9
110
110
110
110
112
112
112
112
112
1
12112
1
121
12
112
112
112
110
110 1
10110
110
1101
919 1
9
19
19
18
18
18
16
16
14
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© Glencoe/McGraw-Hill 5 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Counters
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© Glencoe/McGraw-Hill 6 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Integer Counters
+++++
+++++
+++++
++––––––––
–––––
–––––
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© Glencoe/McGraw-Hill 7 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Pattern for Cup
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© Glencoe/McGraw-Hill 8 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Integer Mat
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© Glencoe/McGraw-Hill 9 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Equation Mat
=
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© Glencoe/McGraw-Hill 10 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Quarter-Inch Grid
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© Glencoe/McGraw-Hill 11 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Centimeter Grid
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© Glencoe/McGraw-Hill 12 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Square Dot Paper
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© Glencoe/McGraw-Hill 13 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Isometric Dot Paper
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© Glencoe/McGraw-Hill 14 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Tangram
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© Glencoe/McGraw-Hill 15 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Number Lines
–1 1
–1 0
– 9
– 8
–7
– 6
– 5
–4
– 3
–2
–1
0
1
2
3
4
5
6
7
8
9
1 0
1 1
0
1
2
3
4
5
6
7
8
9
1 0
1 1
1 2
1
3
1 4
1 5
1 6
1 7
1 8
1 9
2 0
2 1
2 2
0
1
2
3
4
5
6
7
8
9
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© Glencoe/McGraw-Hill 16 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ First-Quadrant Grids
10
9
8
7
6
5
4
3
2
1
O 1 2 3 4 5 6 7 8 9 10
10
9
8
7
6
5
4
3
2
1
O 1 2 3 4 5 6 7 8 9 10
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© Glencoe/McGraw-Hill 17 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Coordinate Planes
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© Glencoe/McGraw-Hill 18 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Percent Models
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© Glencoe/McGraw-Hill 19 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Spinners
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© Glencoe/McGraw-Hill 20 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Number Cube Patterns
2
13 4
9 2 1
4
3
5
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© Glencoe/McGraw-Hill 21 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Protractors
90 8 0
1 0 0
10 0
8 0
1 1 0
7 0
1 2 0
6 0
1 3 0
5 0 1 4 0
4 0 1
5 0
3 0
1 6 0 2
0 1 7 0
1 0
1 8 0
0
7 0
1 1 0 6 0
1 2 0 5 0
1 3 0
4 0
1 4 0
3 0
1 5 0
2 0
1 6 0
1 0
1 7
0
0 1 8 0
90 8 0
1 0 0
10 0
8 0
1 1 0
7 0
1 2 0
6 0
1 3 0
5 0 1 4 0
4 0 1
5 0
3 0
1 6 0 2
0 1 7 0
1 0
1 8 0
0
7 0
1 1 0 6 0
1 2 0 5 0
1 3 0
4 0
1 4 0
3 0
1 5 0
2 0
1 0
1 7
0
0 1 8 0
90 8 0
1 0 0
10 0
8 0
1 1 0
7 0
1 2 0
6 0
1 3 0
5 0 1 4 0
4 0 1
5 0
3 0
1 6 0 2
0
1 7 0
1 0
1 8 0
0
7 0
1 1 0 6 0
1 2 0 5 0
1 3 0
4 0
1 4 0
3 0
1 5 0
2 0
1 0
1 7
0
0 1 8 0
90 8 0
1 0 0
10 0
8 0
1 1 0
7 0
1 2 0
6 0
1 3 0
5 0 1 4 0
4 0 1
5 0
3 0
1 6 0 2
0
1 7 0
1 0
1 8 0
0
7 0
1 1 0 6 0
1 2 0 5 0
1 3 0
4 0
1 4 0
3 0
1 5 0
2 0
1 6 0
1 0
1 7
0
0 1 8 0
90 8 0
1 0 0
10 0
8 0
1 1 0
7 0
1 2 0
6 0
1 3 0
5 0 1 4 0
4 0 1
5 0
3 0
1 6 0 2
0
1 7 0
1 0
1 8 0
0
7 0
1 1 0 6 0
1 2 0 5 0
1 3 0
4 0
1 4 0
3 0
1
5 0
2 0
1 0
1 7
0
0 1 8 0
90 8 0
1 0 0
10 0
8 0
1 1 0
7 0
1 2 0
6 0
1 3 0
5 0 1 4 0
4 0 1
5 0
3 0
1 6 0 2
0
1 7 0
1 0
1 8 0
0
7 0
1 1 0 6 0
1 2 0 5 0
1 3 0
4 0
1 4 0
3 0
1
5 0
2 0
1 0
1 7
0
0 1 8 0
1 6 0
1 6 0
1 6 0
1 6 0
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© Glencoe/McGraw-Hill 22 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Rectangular Prism Pattern
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© Glencoe/McGraw-Hill 23 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Cube Pattern
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© Glencoe/McGraw-Hill 24 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Cylinder Pattern
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© Glencoe/McGraw-Hill 25 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Cone Pattern
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© Glencoe/McGraw-Hill 26 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Pyramid Pattern
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© Glencoe/McGraw-Hill 27 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Pattern Blocks
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© Glencoe/McGraw-Hill 28 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Circle Graph Template
0%95%
90%
85%
80%
75%
70%
65%
60%
55% 50% 45%
40%
35%
30%
25%
20%
15%
10%
5%
1
4
2 3
1 3
1 5
1 6
1 8
1 1 0
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© Glencoe/McGraw-Hill 29 Mathematics: Applications and Concepts, Course 2
NAME ______________________________________________ Problem Solving Guide
Problem: Explore
ExamineThese steps
can helpyou solveproblems.
Plan
Solve
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© Glencoe/McGraw-Hill 30 Mathematics: Applications and Concepts, Course 2
Decimal Patterns and AlgebraTeaching Notes and Overview
Mini-Project Modeling Powers and Exponents
(p. 31 of this booklet)
Use With Lesson 1-2.
Objective Use centimeter cubes to model powers and exponents.
Materials
centimeter cubesgrid paper
Students use centimeter cubes to model pow-ers and exponents. On grid paper, they sketchthe top and side view of each model.
Answers
1. top side
2. top side
3. top side
Using Overhead
Manipulatives Variables and Expressions(pp. 32–33 of this booklet)
Use With Lesson 1-4.
Objective Use cups and counters to modelalgebraic expressions.
Materials
counters*
cups*integer mat transparency** = available in Overhead Manipulative Resources Kit
• Students use cups and counters to modeland solve algebraic expressions.
• An Extension activity asks students tomodel and solve an algebraic expression
using different values for the variable.
Answers
Answers appear on the teacher demonstrationinstructions on pages 32–33.
Hands-On Lab Recording SheetExploring Sequences(p. 34 of this booklet)
Use With Lesson 1-7b. This corresponds tothe activity on page 37 in the Student Edition.
Objective Explore patterns in sequencesusing paper folding.
Materials
calculator piece of paper colored pencils
A table is provided for students to record their data. Space is also given for students to write
and explain their answers.
Answers
See Teacher Wraparound Edition p. 37.
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© Glencoe/McGraw-Hill 31 Mathematics: Applications and Concepts, Course 2
Mini-Project(Use with Lesson 1-2)
NAME ______________________________________________ DATE ____________ PERIOD __________
Use centimeter cubes to build each square or cube. On grid paper,
sketch the top view and the side view.1. 23 2. 53 3. 62
top view top view top view
side view side view side view
Modeling Powers and Exponents
Materials
centimeter cubes, grid paper
You can build a square or a cube using centimeter cubes. 32 is a three-by-threesquare and is 1 centimeter cube high. You need 9 centimeter cubes to build 3 2.
The top view of 32 is shown at the left below. The side view of 32 is shown atthe right below.
You need 64 centimeter cubes to build 43. The top view is shown at the left below. The side view is shown at the right below.
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© Glencoe/McGraw-Hill 32 Mathematics: Applications and Concepts, Course 2
Using Overhead Manipulatives(Use with Lesson 1-4)
Variables and Expressions
Teacher Demonstration
• Tell students that the phrase the sum of two and some number is an algebraicexpression. The number 2 in this phrase is a constant, a number that youknow, and “some number” is an unknown value. The phrase can be repre-sented by a cup for the unknown value and 2 counters for the number 2.
Place a cup and 2 counters on the integer mat.
• Place 6 counters in the cup. The cup now has a value of 6. Remove the coun-ters from the cup and count all the counters. There are a total of 8 counters.
Thus, the expression has a value of 8.
• Ask students what the value of the expression would be if 4 counters were placed in the cup and if no counters were placed in the cup. Model the correctanswers if necessary. (6, 2)
• Clear the mat.
Objective Use cups and counters to model algebraic expressions.
Materials
• counters*
• cups*
• integer mat transparency*
* = available in Overhead Manipulative Resources Kit
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© Glencoe/McGraw-Hill 33 Mathematics: Applications and Concepts, Course 2
• Tell students that the phrase three times some number is also an algebraicexpression. Ask students how this phrase can be expressed mathematically.(Use three cups.) Stress that the same number of counters must be placed in
each cup. Place 2 counters in each cup. Empty the cups and count all thecounters. There are 6. The expression has a value of 6.
• Ask students what the value of the expression is if 1 counter is placed in each
cup; if 5 counters are placed in each cup. Demonstrate the correct answers if necessary. (3, 15)
Have students complete Exercises 1–5.
Model each phrase with cups and counters. Then put four counters in each
cup. How many counters are there in all? Record your answers by drawing
pictures of your models.
1. the sum of 3 and a number (7) 2. 3 times a number (12)
3. 5 more than a number (9) 4. twice a number plus 1 (9)
5. Write a sentence to describe what the cup represents.
(Sample answer:The cup represents the variable or unknownquantity.)
Extension
Model the phrase four more than three times some number as a third example.
Use 3 cups to represent “three times some number”, and use 4 counters to repre-sent the number 4. Place 5 counters in each cup. Empty the cups and count allthe counters. There are 19. The expression has a value of 19. Ask students whatthe value of the expression is if 1 counter is placed in each cup: if 3 counters arein each cup. (7, 13)
Using Overhead Manipulatives
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© Glencoe/McGraw-Hill 34 Mathematics: Applications and Concepts, Course 2
Hands-On Lab Recording Sheet(Use with the activity on page 37 in Lesson 1-7b of the Student Edition)
NAME ______________________________________________ DATE ____________ PERIOD __________
Exploring Sequences
Materials
calculator, piece of paper, colored pencils
Investigate
Work with a partner. Complete the table.
Writing Math
Work with a partner.
1. Examine the sequence of numbers in the “Layers” column of your table. Isthe sequence arithmetic or geometric? Then write a rule to find the next threeterms.
2. Examine the sequence of numbers in the “Fraction” column of your table. Is
the sequence arithmetic or geometric? Then write a rule to find the next threeterms. ( Hint: Write each fraction as a decimal.)
3. Assume you could continue the paper-folding process indefinitely. Supposeyour unfolded piece of paper is 0.002 inch thick. Add a column to your tableand find the thickness of the paper for the first five folds.
4. How many folds would it take until the paper is as tall as you?
5. Explain the relationship between the number of layers and the fraction of theshaded region.
Number Layers Fraction of Paperof Folds of Paper Not Shaded
1 2 12
2 4 14
3 8
4
5
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© Glencoe/McGraw-Hill 35 Mathematics: Applications and Concepts, Course 2
Statistics: Analyzing DataTeaching Notes and Overview
Mini-Project Story Graph
(p. 37 of this booklet)
Use With Lesson 2-2.
Objective Interpret and describe a linegraph.
Materials
none
Students interpret a given line graph and
describe it by writing a story that fits the datashown.
Sample Answer
I left my house at 4:15 P.M. to go to the storefor Mom. She asked me to buy some milk and tortillas for dinner. I walked 5 minutes beforemeeting my friend, Lina. We talked for 5 min-utes. Then I continued walking to the store atthe same pace I had been walking. I arrived at
the store 10 minutes later. It took me 10 min-
utes to buy milk and tortillas. On the wayhome, I walked a little faster because Momwanted the groceries by 5 P.M. I made it homein 10 minutes. I was early.
Hands-On LabRecording SheetAre You Average?(p. 38 of this booklet)
Use With Lesson 2-4b. This corresponds tothe activity on page 73 in the Student Edition.
Objective Use mean, median, mode, and range to describe a set of data.
Materials
markers
ruler
Students work in groups to collect data, create
a frequency table, and create an appropriategraph to display their data. Students are asked to explain why they chose the type of graphthey did for their data.
Answers
See Teacher Wraparound Edition p. 73.
Using Overhead Manipulatives Quartiles(pp. 39–40 of this booklet)
Use With Lesson 2-6.
Objective Graph quartiles and determine theinterquartile range.
Materials blank transparencies, prepared as described below
transparency pens*
* = available in Overhead Manipulative Resources Kit
• Students find and graph the upper and lower quartiles and the interquartile rangeof a given set of data.
• An Extension activity asks students to find and graph the upper and lower quartiles and
the interquartile range of a second set of data, then compare the two sets of data.
Answers
Answers appear on the teacher demonstrationinstructions on pages 39–40.
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© Glencoe/McGraw-Hill 37 Mathematics: Applications and Concepts, Course 2
Mini-Project(Use with Lesson 2-2)
NAME ______________________________________________ DATE ____________ PERIOD __________
Story Graph Make up a story to fit this graph. The graph describes the distance you are fromyour house with respect to time. The story should include where you begin, end,
and stop along the way. Make sure you also mention how long it takes you totravel from place to place and how long you stay at each place.
To help you understand what is happening, you may want to act out the graph.Use seconds instead of minutes and ignore the distance scale.
100
5 10 15 20 25 30 35 40
200
300
400
500
600
700
800
900
0
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Using Overhead Manipulatives(Use with Lesson 2-6)
Quartiles
Teacher Demonstration
• Prepare blank transparencies with copies of the tables shown below and in theExtension.
• Show students the first table. Tell them that the table shows various suspen-sion bridges in the United States and the length of their main span.
• Tell students that in a large set of data, it is helpful to separate the data intofour equal parts called quartiles. Tell them you are going to find and graph
the quartiles for these data.
• Ask students to list the data in order from least to greatest. (533, 549, 564,610, 655, 655, 701, 704, 853, 1,067, 1,158, 1,280, 1,298) Write the datain order below the table.
• Ask students how to find the median of the data. (Find the data numberthat is in the middle when the data are arranged in order from
least to greatest.) Then have them find the median. (701) Circle themedian and point out that it separates the data into two equal groups.
• On the list of data, place anarrow between 564 and 610 and
between 1,158 and 1,067 asshown. Find their means and tellstudents that these numbers rep-resent the median of the lower half, the lower quartile, and themedian of the upper half, the
upper quartile, of the data.
Objective Graph quartiles and determine the interquartile range.
Materials
• blank transparencies, prepared as described below
• transparency pens*
* = available in Overhead Manipulative Resources Kit
Length ofName, Location
Main Span (m)
Ambassador International, Detroit 564
Benjamin Franklin, Philadelphia 533
Bronx-Whitestone, New York City 701
Delaware Memorial, Wilmington, Delaware 655
George Washington, New York City 1,067
Golden Gate, San Francisco 1,280
Mackinac Straits, Michigan 1,158
Tacoma Narrows II, Tacoma, Washington 853
San Francisco-Oakland Bay, San Francisco 704
Seaway Skyway, Ogdensburg, New York 655
Throgs Neck, New York City 549
Varrazano-Narrows, New York 1,298
Walt Whitman, Philadelphia 610
Source: Federal Highway Administration
© Glencoe/McGraw-Hill 39 Mathematics: Applications and Concepts, Course 2
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© Glencoe/McGraw-Hill 40 Mathematics: Applications and Concepts, Course 2
Using Overhead Manipulatives
533 549 564 610 655 655 701 704 853 1,067 1,158 1,280 1,298
564
2
610
587 median
1,067
2
1,158
1,112.5
• Point out that half of the data numbers, the middle half, lie between the lower and upper quartiles, 587 and 1,112.5. Tell them that the range of the middlehalf of the data is called the interquartile range. Ask students to find theinterquartile range of these data. (525.5)
• Below the data, draw a number line from 500 to 1,300. Graph the median of the
data above the number line. Then show students as you graph the least value, thegreatest value, the upper quartile, and the lower quartile above the number line.
• Point out that the numbers graphed separate the data into four groups. Ask how many of the numbers in the data set fall in each group. (3)
Extension
Show students the table below. Tell them that the table shows various suspension
bridges internationally and the lengths of their main spans.
Have students find the median,
upper quartile, lower quartile, leastvalue, and greatest value for the
data. (1,043.5; 1,377; 988; 668;1,990) Then have them graphthose values. Finally, compare thenumber lines for the two sets of data. Ask students what conclu-
sions they can make from the data.(Sample answer: The middlehalf of the data for interna-tional bridges is less spread
out. International bridgesgenerally have longer spansthan those in the UnitedStates.)
500 1,3001,100900700
LQ587533
M701
UQ1,112.5 1,298
Length ofName, Location
Main Span (m)Akashi Kaiko, Japan 1,990
First Bosporus, Istanbul, Turkey 1,074
Forth Road, Queensferry, Scotland 1,006
Humber, Hull, Britain 1,410
Kita Bisan-Seto, Japan 990
Minami Bisan-Seto, Japan 1,100
Ohnaruto, Japan 876
Pierre Laporte, Quebec, Canada 668
Ponte 25 de Abril, Lisbon, Portugal 1,013
Second Bosporus, Istanbul, Turkey 1,090Severn, Beachley, England 988
Shimotsui Straits, Japan 940
Storebelt, Denmark 1,624
Tsing Ma Bridge, Hong Kong 1,377
Source: Federal Highway Administration
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© Glencoe/McGraw-Hill 41 Mathematics: Applications and Concepts, Course 2
Using Overhead Manipulatives(Use with Lesson 2-8)
How Much is a Handful?
Teacher Demonstration
• Ask four volunteers to trace the outline of their hand on a transparency.
• Place one of the outlines on the overhead and cover the shape with counters.Count the counters used and record on a fifth transparency. Repeat the proce-
dure for each of the three outlines.
• Find the mode, median, and mean of the data. Ask the students which of these“averages” is most representative of the data.
• Discuss the most effective way to present the data. Alternatives include thefrequency table, bar graph, line plot, or stem-and-leaf plot. Use the sixth blank
transparency to prepare the data in the agreed-upon manner.
• Ask students to predict how many counters would be needed to cover a hand outline for a student in the same grade but not in the class.
• Have students randomly choose four students who are not in your class and trace the outline of their hands on a transparency. Record the counters needed
to cover the shape. Have students compare the number of counters with their prediction.
• Trace your own hand on a transparency and find the number of countersneeded to cover the shape. Ask how this number compares with the other data.(In most cases, it will be greater than the average of the studentdata.)
• Ask whether you could use the information about the teacher’s hand to predictabout how many counters would cover other adults’ hands. (Data from one
adult would not be sufficient to predict for other adults.)
Objective Use data to make predictions.
Materials
• transparency pens*
• 11 blank transparencies
• 40 counters** = available in Overhead Manipulative Resources Kit
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© Glencoe/McGraw-Hill 42 Mathematics: Applications and Concepts, Course 2
Algebra: IntegersTeaching Notes and Overview
Mini-Project Coordinate Plane Puzzle
(p. 44 of this booklet)
Use With Lesson 3-3.
Objective Locate points for ordered pairs ona coordinate plane
Materials
none
Given ordered pairs, students locate the
appropriate points labeled on a coordinate plane. Students solve the puzzle by matchingeach point with the correct ordered pair.
Answer
WE ARE COORDINATED!
Hands-On Lab Recording SheetAdding Integers
(p. 45 of this booklet)
Use With Lesson 3-4a. This corresponds to
the activity on pages 118–119 in the StudentEdition.
Objective Use counters to model the addi-tion of integers.
Materials
counters
integer mat
Students use counters to add integers. Spaceis provided for students to explain their work.
They will write their own addition sentencesand draw conclusions on how to add integers.
Answers
See Teacher Wraparound Edition pp. 118–119.
Hands-On Lab Recording SheetSubtracting Integers
(p. 46 of this booklet)
Use With Lesson 3-5a. This corresponds tothe activity on pages 126–127 in the StudentEdition.
Objective Use counters to model the sub-traction of integers.
Materials
counters
integer mat
Students use counters to subtract integers.
Space is provided for students to explain their work. They will write their own subtractionsentences and draw conclusions on how tosubtract integers.
Answers
See Teacher Wraparound Edition pp. 126–127.
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© Glencoe/McGraw-Hill 43 Mathematics: Applications and Concepts, Course 2
Using Overhead Manipulatives
Multiplying Integers(pp. 47–48 of this booklet)
Use With Lesson 3-6.
Objective Multiply integers by usingmodels.
Materials
counters*integer mat transparency*
transparency pen** = available in Overhead Manipulative Resources Kit
This demonstration contains two activities.• Demonstration 1 shows how to model the
multiplication of two positive integers or a positive and a negative integer.
• Demonstration 2 shows how to model the
multiplication of two negative integers.
• Extension questions ask students to modeland solve integer multiplication problems
independently.
Answers
Answers appear on the teacher demonstrationinstructions on pages 47–48.
Chapter 3 Algebra: Integers
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© Glencoe/McGraw-Hill 44 Mathematics: Applications and Concepts, Course 2
Mini-Project(Use with Lesson 3-3)
NAME ______________________________________________ DATE ____________ PERIOD __________
(4, 5) (5, 6) (5, 6) (4, 2) (2, 4)
(0, 2) (2, 2) (7, 7) (6, 0) (4, 6) (5, 1) (1, 0) (2, 4) (2, 7) (8, 4) (8, 6)
Coordinate Plane Puzzle Use the coordinate plane to find the point for each ordered pair below.
Then write the letter for each point under its ordered pair at the bottom of
the page. When you have filled in all the letters, you will know what two
points on a grid said to each other.
x
y
O -8 -7 -6 -5 -4 -3 -1-2 1 2 3 4 5 6 7 8
-7
-8
-6
-5
-4
-3
-1
-2
5
6
7
8
1
2
3
4
W
A
I
E
R
A
H
J
J
B
G
T
E
D
C
O
O
R
D
N
E
-9
9
9
-9
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© Glencoe/McGraw-Hill 45 Mathematics: Applications and Concepts, Course 2
Hands-On Lab Recording Sheet(Use with the activity on pages 118–119 in Lesson 3-4a of the Student Edition)
NAME ______________________________________________ DATE ____________ PERIOD __________
Adding Integers
Materials
counters, integer mat
Your Turn
Use counters to find each sum.
a. 5 + 6 b. 3 (5) c. 5 (4)
d. 7 3 e. 2 (5) f. 8 (6)
g. 6 5 h. 3 (6) i. 2 7
j. 8 (3) k. 9 1 l. 4 10
Writing Math
1. Write two addition sentences where the sum is positive. In each sentence,one addend should be positive and the other negative.
2. Write two addition sentences where the sum is negative. In each sentence,one addend should be positive and the other negative.
3. MAKE A CONJECTURE Write a rule that will help you determine thesign when finding the sum of integers.
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© Glencoe/McGraw-Hill 46 Mathematics: Applications and Concepts, Course 2
Hands-On Lab Recording Sheet(Use with the activity on pages 126–127 in Lesson 3-5a of the Student Edition)
NAME ______________________________________________ DATE ____________ PERIOD __________
Subtracting Integers
Materials
counters, integer mat
Your Turn
Use counters to find each difference.
a. 7 6 b. 5 (3) c. 6 (3)
d. 5 8 e. 6 (3) f. 7 3
g. 5 (7)
Writing Math
Work with a partner.
1. Write two subtraction sentences where the difference is positive. Make sureyou use a combination of positive and negative integers.
2. Write two subtraction sentences where the difference is negative. Make sure
you use a combination of positive and negative integers.
3. MAKE A CONJECTURE Write a rule that will help you determine thesign of the difference of two integers.
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© Glencoe/McGraw-Hill 47 Mathematics: Applications and Concepts, Course 2
Using Overhead Manipulatives(Use with Lesson 3-6)
Multiplying Integers
Teacher Demonstration for Activity 1
• Review with students the meaning of the two colors of counters. If necessary,mark the yellow counters to show they are “positive” counters and the red
counters to show they are “negative” counters.
• Remind students that 3 2 means three sets of two items. Tell students thatmodeling 3 2 means to place 3 sets of 2 positive counters on the mat.Model 3 2. Ask students to complete: 3 2 ____ ? . (6)
• Place 3 sets of 2 negative counters on the mat. Ask students to state the
multiplication sentence that has been modeled. (3 (2) 6)
Teacher Demonstration for Activity 2
• Clear the mat. Write 3 2 at the base of the mat. Tell students thatsince 3 is the opposite of 3, 3 2 means to remove 3 sets of 2 positive
counters.
• Place 6 zero pairs on the mat. Ask students to state the value of the mat. (0)
–+ –+ –+
–+ –+ –+
––
––
––
––
––
––
++
++
++
++
++
++
Objective Multiply integers by using models.
Materials
counters*
integer mat transparency*
transparency pen*
* = available in Overhead Manipulative Resources Kit
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© Glencoe/McGraw-Hill 48 Mathematics: Applications and Concepts, Course 2
Using Overhead Manipulatives
• Remove 3 sets of 2 positive counters. Ask students to state the valueof the mat. (6)
• Ask students to complete the sentence 3 2 = ____ ? . (6)
• Clear the mat. Write 3 (2) at the base of the mat. Ask students howmany zero pairs must be placed on the mat to model 3 (2). (6)
• Place 6 zero pairs on the mat. Ask students whether positive or negativecounters should be removed to model 3 (2). (negative) Remove3 sets of 2 red counter. Ask students the value of the mat. (6)
• Ask students to complete the sentence 3 (2) = ____ ? . (6)
Have students complete Exercises 1–7.
Use counters to find each product. Use your result to write a multiplication
sentence.
1. 2 3 (2 3 6) 2. 2 (4) (2 (4) 8)
3. 2 (3) (2 (3) 6) 4. 3 (4) (3 (4) 12)
5. 4 0 (4 0 0) 6. 1 (5) (1 (5) 5)
7. Find 9 (3) without using models. (27)
Extension
Ask students to model each multiplication and then write a multiplication
sentence.4 3 (4 3 12)4 (3) (4 (3) 12)
4
(
3) (4 (3) 12)4 3 (4 3 12)
+
+
–
–
+
+
–
–
+
+
+
+
+
+
+
+
–
–
+
+
–
–
+
+
–
–
+
+
–
–
–
–
–
–
–
–
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© Glencoe/McGraw-Hill 50 Mathematics: Applications and Concepts, Course 2
Hands-On Lab Recording Sheet
Functions and Graphs(p. 55 of this booklet)
Use With Lesson 4-6a. This corresponds tothe activity on page 176 in the StudentEdition.
Objective Graph a function on a scatter plot.
Materials
stop watchuncooked spaghetti
Students will perform the “wave” with variousnumbers of students in the class. They are provided with a graph on which they canrecord their data. They use the graph to find a pattern, make predictions, and explain their
reasoning.
Answers
See Teacher Wraparound Edition p. 176.
Using Overhead Manipulatives
A Function of Time(pp. 56–57 of this booklet)
Use With Lesson 4-6.
Objective Use a function rule to find theoutput of a function.
Materials
coordinate grid transparency*transparency pen*
transparency prepared with the two tablesshown in the demonstration
* = available in Overhead Manipulative Resources Kit
• Students use given data for the input valuesand the function rule to find the output val-ues of a function.
• Students then use the graph of the function
to make predictions.
• An Extension activity asks students to giveexamples of other functions of time.
Answers
Answers appear on the teacher demonstrationinstructions on pages 56–57.
Chapter 4 Algebra: Linear Equations and Functions
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© Glencoe/McGraw-Hill 51 Mathematics: Applications and Concepts, Course 2
Hands-On Lab Recording Sheet(Use with the activity on pages 154–155 in Lesson 4-2a of the Student
Edition)
NAME ______________________________________________ DATE ____________ PERIOD __________
Solving Equations Using Models
Materials
cups and counters, equation mat
Investigate
The scale at the right is balanced, and the bag contains a certainnumber of blocks.
1. Suppose you cannot look in the bag. How can you find thenumber of blocks in the bag?
2. In what way is a balanced scale like an equation?
3. What does it mean to solve an equation?
Your Turn
Solve each equation using models.
a. x 1 3 b. x 3 7 c. x 4 4
d. x 3 2 e. x 4 1 f. 2 x 1
g. x 3 2 h. x 1 3 i. 4 x 2
Writing Math
1. How is solving an equation similar to keeping a scale in balance.2. For any equation, how can you determine how many counters to add or sub-
tract from each side?
3. Identify the property of numbers that is illustrated by a zero pair.
4. Identify the property of numbers that allows you to add or subtract zerowithout changing the value of a number.
5. MAKE A CONJECTURE Write a rule that you can use to solve an equa-
tion like x 3 2 without using models.
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© Glencoe/McGraw-Hill 52 Mathematics: Applications and Concepts, Course 2
Mini-Project(Use with Lesson 4-3)
NAME ______________________________________________ DATE ____________ PERIOD __________
Solving Multiplication Equations
Write the equation modeled by the cups and counters.
1. 2.
________________________________ ________________________________
Solve each equation using cups and counters. Sketch the arrangement in the boxes.
3. 4 x 12 x __________________
4. 2 x 14 x __________________
5. 3 x 15 x __________________
Solve without using models.
6. 5 x 10 x __________________
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© Glencoe/McGraw-Hill 53 Mathematics: Applications and Concepts, Course 2
Using Overhead Manipulatives(Use with Lesson 4-4)
Solving Two-Step Equations
Teacher Demonstration
• Remind students that, in modeling, red counters represent negative integers,yellow counters represent positive integers, and a cup represents x.
• Review the use of zero pairs. Ask students what happens to an equation if azero pair is added to or subtracted from a side. (An equivalent equationresults.)
• Place 3 cups and 2 yellow counters on the left side of an equation mat and
place 4 red counters on the right side. Ask students what equation is modeled.(3x 2 4)
• Add 2 red counters to each side of the equation mat. Point out that you do thisto create zero pairs on the left side. Remove the zero pairs. Ask students whyyou can remove the zero pairs. (Their value is 0.)
+ – – –
– –
– –+ –
+
+
–
–
–
–
Objective Solve two-step equations by using models.
Materials
• cups*
• counters*
• equation mat transparency*
* = available in Overhead Manipulative Resources Kit
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© Glencoe/McGraw-Hill 54 Mathematics: Applications and Concepts, Course 2
Using Overhead Manipulatives
• Ask students what equation is represented on the mat. (3x 6)
• Pair up an equal number of counters with each cup.
• Ask students what the solution of the equation 3 x 2 4 is. (2)
Extension
Have students write a few sentences explaining how the modeling demonstrated in this lab uses the work backward strategy presented in Lesson 4-4a. (Sampleanswer: Modeling the solution to two-step equations uses thereverse of the order of operations.)
– –
– –
– –
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© Glencoe/McGraw-Hill 55 Mathematics: Applications and Concepts, Course 2
Hands-On Lab Recording Sheet(Use with the activity on page 176 in Lesson 4-6a of the Student Edition)
NAME ______________________________________________ DATE ____________ PERIOD __________
Functions and Graphs
Materials
stop watch, uncooked spaghetti
Writing Math
Work with a partner.
1. Graph the ordered pairs (number of students,
time) on the coordinate grid at the right.
2. Describe how the points appear on your graph.
3. Place one piece of uncooked spaghetti on
your graph so that it covers as many of the points as possible. Predict how long it would take 30 students to complete the “wave.”Make a prediction for 50 students.
4. Find a pattern in the data and use the pattern to predict how long it would takethe students in your school to complete the “wave.” Explain your reasoning.
5. A function describes the relationship between two quantities. In a function, onequantity depends on the other. Complete the sentence: The time it takes to do
the “wave” depends on ____ ? .
T i m e
( s e c o n d s )
Students
50
5
10
15
20
25
30
35
40
10 15 20 25 30 35 40
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© Glencoe/McGraw-Hill 56 Mathematics: Applications and Concepts, Course 2
Using Overhead Manipulatives(Use with Lesson 4-6)
A Function of Time
Teacher Demonstration
• Tell students that a certain secretary can type an average of 55 words per minute. Ask students to use this information to complete the table.
• Tell students that in this example the minutes are called the input values, thetotal words processed are called output values, and the middle column con-tains the function rule.
• Graph the ordered pairs (minutes, words) on the coordinate grid transparency.Draw a line passing through the points.
• Ask students how they could use the graph to estimate the number of wordstyped in 8 minutes. (Extend the line; 440 words.)
• Say, “Let t be the time in minutes. What is the expression for the total number of words typed in t minutes?” (55t )
Objective Use a function rule to find the output of a function.
Materials
• coordinate grid transparency*
• transparency pen*
• transparency prepared with the two tables shown below
* = available in Overhead Manipulative Resources Kit
Minutes AverageTotalMinutes Number of Words
Wordsper Minute
1 1 ? (55)
2 2 ? (110)
3 3 ? (165)
4 4 ? (220)
5 5 ? (275)
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© Glencoe/McGraw-Hill 57 Mathematics: Applications and Concepts, Course 2
ExtensionHave students give examples of other functions of time. (Sample answer:revolutions per minute)
• Tell students that another secretary can type an average of 70 words per minute. Repeat the activity using the information below.
Using Overhead Manipulatives
Minutes AverageTotalMinutes Number of Words
Wordsper Minute
1 1 ? (70)
2 2 ? (140)
3 3 ? (210)
4 4 ? (280)
5 5 ? (350)
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© Glencoe/McGraw-Hill 58 Mathematics: Applications and Concepts, Course 2
Fractions, Decimals, and PercentsTeaching Notes and Overview
Hands-On Lab Recording SheetExploring Factors(p. 59 of this booklet)
Use With Lesson 5-1a. This corresponds tothe activity on page 196 in the StudentEdition.
Objective Discover factors of wholenumbers.
Materials
15 index cards cut in half markers
Students work as a class to determine factorsof whole numbers by identifying patterns.Space is provided for students to explain their answers and make predictions.
Answers
See Teacher Wraparound Edition p. 196.
Using Overhead Manipulatives Percent(p. 60 of this booklet)
Use With Lesson 5-5.
Objective Illustrate the meaning of percentusing models.
Materials
centimeter grid transparency*transparency pens*
* = available in Overhead Manipulative Resources Kit
• Using a model on the centimeter grid trans-
parency, students write a ratio of the num- ber of shaded squares to the total number of squares.
• Students make conclusions about the mean-ing of percent based on the model.
Answers
Answers appear on the teacher demonstration
instructions on page 60.
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© Glencoe/McGraw-Hill 59 Mathematics: Applications and Concepts, Course 2
Hands-On Lab Recording Sheet(Use with the activity on page 196 in Lesson 5-1a of the Student Edition)
NAME ______________________________________________ DATE ____________ PERIOD __________
Exploring Factors
Materials
15 index cards cut in half, markers
Writing Math
Work as a class.
1. How many students are standing at the end of the activity? Which cards are
they holding?
2. LOOK FOR A PATTERN Suppose there were 100 students holding indexcards. Extend the pattern in Exercise 1 to predict the numbers that would beheld by students standing at the end of the activity.
3. Explain the relationship between the numbers on the front and the back of the cards.
4. Separate the cards into two groups: one group with exactly two numbers on
the back of the card and one group with more than two numbers. Describe
any special characteristics of each group.
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© Glencoe/McGraw-Hill 60 Mathematics: Applications and Concepts, Course 2
Using Overhead Manipulatives(Use with Lesson 5-5)
Percent
Teacher Demonstration
• On the centimeter grid transparency, outline a 10-by-10 square. Using a
different colored pen, shade 25 of the squares as shown.
• Ask, “How many small squares are in the model?” (100)
• Ask, “How many small squares are shaded?” (25)
• Ask students to write a ratio of shaded squares to squares in the model. 12050• Tell students that the model represents 25 percent. Ask them to make a
conjecture about the meaning of the word percent . (Sample answer:Percent is a ratio comparing a number to 100.)
Objective Illustrate the meaning of percent using models.
Materials
• centimeter grid transparency*
• transparency pens*
* = available in Overhead Manipulative Resources Kit
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© Glencoe/McGraw-Hill 61 Mathematics: Applications and Concepts, Course 2
Applying FractionsTeaching Notes and Overview
Using Overhead Manipulatives Multiplying Fractions andMixed Numbers(pp. 62–63 of this booklet)
Use With Lesson 6-4.
Objective Use models to multiply fractionsand mixed numbers.
Materials
blank transparency
ruler*transparency pens*
* = available in Overhead Manipulative Resources Kit
This demonstration contains three activities.• Demonstration 1 shows the multiplication
of two fractions.
• Demonstration 2 shows the multiplication
of a whole number and a fraction.
• Demonstration 3 shows the multiplication
of a fraction and a mixed number.
• Students are asked to find products of frac-tions and mixed numbers independently.
• An Extension activity asks students to modelthe multiplication of two mixed numbers.
Answers
Answers appear on the teacher demonstrationinstructions on pages 62–63.
Mini-Project
Perimeter(p. 64 of this booklet)
Use With Lesson 6-8.
Objective Measure the sides of figures and find the perimeter.
Materials
ruler
Given several figures, students measure thesides, label them, and then find the perimeter.
Answers
1. P 11
4 1
1
4 1
1
4 1
1
4 5 in.
2. P 34 1
38 2 1
78 6 in.
3. P 11
8 1
1
8 1
7
8 1
1
8 5
1
4 in.
4. P 1 1 11
4 3
1
4 in.
5. P 11
8
5
8 1 1
1
8 2
3
8 6
1
4 in.
6. P 58 1 1 58 78 114 78 614 in.
Hands-On Lab Recording SheetCircumference(p. 65 of this booklet)
Use With Lesson 6-9a. This corresponds to
the activity on page 274 in the Student
Edition.
Objective Find a relationship between
circumference and diameter.
Materials
ruler measuring tape
circular objects
Students find the diameter and circumferenceof various circular objects and record their
measurements in a table. A graph is provided for students to graph their data, find the slopeof the line, and discover a relationship between circumference and diameter.
Answers
See Teacher Wraparound Edition p. 274.
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© Glencoe/McGraw-Hill 62 Mathematics: Applications and Concepts, Course 2
Using Overhead Manipulatives(Use with Lesson 6-4)
Multiplying Fractions and Mixed Numbers
Objective Use models to multiply fractions and mixed numbers.
Materials
• blank transparency
• ruler*
• transparency pens*
* = available in Overhead Manipulative Resources Kit
Teacher Demonstration for Activity 1
• Use the ruler and a black transparency pen todraw a square like the one shown. Divide the
square vertically into halves and horizontallyinto fourths.
• Tell students you are going to model 1
2
1
4.
• Color one half of the square blue.
• Color one fourth of the square red.
• Count the small rectangles that are shaded bothcolors. Ask students what number is represented.
18 Write 1
2
1
4
1
8 below the model.
Teacher Demonstration for Activity 2
• Use the ruler and a black transparency pen todraw 2 squares side-by-side. Divide them
horizontally into fifths.
• Tell students you are going to model 2 3
5.
• Color the 2 squares blue.
• Color three-fifths of the squares red.
• Count the small rectangles that are shaded bothcolors. Point out that 6 small rectangles are
shaded and each has an area of 1
5 of a unitsquare. Ask students to add the areas.
65 or 115 Write 2 35
65 or 1
15 below the
model.
Purple Red
Blue
Purple
Blue
Red
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© Glencoe/McGraw-Hill 63 Mathematics: Applications and Concepts, Course 2
Teacher Demonstration for Activity 3
• Draw 2 unit squares side-by-side. Divide them vertically into fourths and horizontally into thirds.
• Tell students you are going to model 11
4
2
3.
• Color 11
4 of the squares blue.
• Color 2
3 of the squares red.
• Count the small rectangles that are shaded both colors. Point out that 10 of the smallrectangles are shaded and each has an area
of 1
1
2 of a unit square. Ask students to add
the areas. 1102 or
56 Write 114
23
1102
or 5
6 below the model.
Have students complete Exercises 1–7.
Use area models to find each product.
1. 1
2
1
3
1
6
2.
2
3
3
4
1
6
2
or 1
2
3. 3
1
2
3
2
or 11
2
4. 2 34 64 or 1
12 5. 113
14 1
42 or
13 6. 312
12 74 or 1
34
7. Find 212 1
13. Use area models if necessary. 26
0 or 3
13
Extension
• Ask students how to suggest a model to
show the product 11
2 1
1
3. (Make a
drawing 2 squares across and
2 squares down. Shade 112 of the
squares vertically and 113 horizon-tally.) Draw the model. Have students find
the product. 63 or 2
Using Overhead Manipulatives
Purple
Blue
Red
113
112
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© Glencoe/McGraw-Hill 64 Mathematics: Applications and Concepts, Course 2
Mini-Project(Use with Lesson 6-8)
NAME ______________________________________________ DATE ____________ PERIOD __________
Perimeter
Measure the segments in each figure to the nearest eighth of an inch. Label the
segments with their measurements. Then find the perimeter of each figure.
1. 2.
Perimeter __________ Perimeter __________
3. 4.
Perimeter __________ Perimeter __________
5. 6.
Perimeter __________ Perimeter __________
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© Glencoe/McGraw-Hill 65 Mathematics: Applications and Concepts, Course 2
Hands-On Lab Recording Sheet(Use with the activity on page 274 in Lesson 6-9a of the Student Edition)
NAME ______________________________________________ DATE ____________ PERIOD __________
Circumference
Materials
ruler, measuring tape, circular objects
Investigate
Work with a partner. Record your data in the table below.
Write About It
Work with a partner.
1. For each object, divide the circumference by the diameter.Record the results in the table above. Round to the nearesttenth if necessary.
2. What do you notice about the ratios?
3. Graph the ordered pair (diameter, circumference) on thecoordinate plane for each object. What do you find?
4. Select two points on the graph and find the slope betweenthem. Select two different points and find the slope. Whatdo you observe about the slopes?
5. Use the graph to predict the circumference of a circular object that has a diameter of 18 centimeters.
6. Write a rule describing how you would find the circumference C of a circle
if you know the diameter d .
ObjectDiameter Circumference
Circumference Diameter(cm) (cm)
y
x O
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© Glencoe/McGraw-Hill 66 Mathematics: Applications and Concepts, Course 2
Ratios and ProportionsTeaching Notes and Overview
Using Overhead Manipulatives Equal Ratios(pp. 68–69 of this booklet)
Use With Lesson 7-1.
Objective Explore the meaning of ratio and proportion.
Materials
counters* blank transparency
transparency pen** = available in Overhead Manipulative Resources Kit
• Students use counters to model ratios byfinding equivalent fractions.
• An Extension activity asks students to work in pairs to find equivalent ratios.
Answers
Answers appear on the teacher demonstrationinstructions on pages 68-69.
Hands-On Lab Recording SheetRate of Change(p. 70 of this booklet)
Use With Lesson 7-2b. This corresponds tothe activity on page 296 in the StudentEdition.
Objective Investigate rate of change.
Materials
square tiles
Students use tiles to build models and find the perimeter of each model. A table and coordi-
nate plane is provided for students to record and graph their data. Students explore rate of change by finding the slope of the graph.
Answers
See Teacher Wraparound Edition p. 296.
Hands-On Lab Recording SheetWildlife Sampling
(p. 71 of this booklet)
Use With Lesson 7-3b. This corresponds tothe activity on page 301 in the StudentEdition.
Objective Use proportions to estimate.
Materials
small bowl
dried beans
paper cupmarkers
Using dried beans in a bowl to representdeer in a forest, students model the capture-recapture technique used to estimate animal populations. A table is provided for studentsto record their data, as well as space for themto explain their results.
Answers
See Teacher Wraparound Edition p. 301.
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© Glencoe/McGraw-Hill 67 Mathematics: Applications and Concepts, Course 2
Mini-Project Scale Drawings
(p. 72 of this booklet)
Use With Lesson 7-4.
Objective Create a scale drawing.
Materials
ruler
Using the grid provided, students create a
scale drawing of a room they would like to build. The drawing includes everything they
wish to include in the room drawn to scale.Students must label their drawing and list theobjects in the room and their actual size.
Answers
1. See students’ work. Make sure drawingsare the correct scale based on this informa-tion and the scale of the grid.
2–3. Answers will vary.
Hands-On Lab Recording Sheet
Using a Percent Model(p. 73 of this booklet)
Use With Lesson 7-8a. This corresponds tothe activity on page 322 in the StudentEdition.
Objective Use a percent model to find a part.
Materials
none
Grids are provided for students to drawmodels to find the percent of a number.
Answers
See Teacher Wraparound Edition p. 322.
Chapter 7 Ratios and Proportions
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© Glencoe/McGraw-Hill 68 Mathematics: Applications and Concepts, Course 2
Using Overhead Manipulatives(Use with Lesson 7-1)
Equal Ratios
Teacher Demonstration
• Define ratio as the comparison of two numbers.
• Place 6 counters in a pile on a blank transparency. Label this pile X. Place 2
counters in a second pile. Label this pile Y. Divide pile X into two groups of 3and pile Y into two groups of 1. Point out that for every three counters in pileX, there is one counter in pile Y. Tell students that piles X and Y have a ratioof 3 to 1.
• Clear the screen. Divide a blank transparency into 2 rows with 4 columns ineach row. Label the sections as shown below. Place 3 counters in section A
and 4 counters in section B. Ask students to name the ratio of the counters inA and B. (3 to 4)
• Place 6 counters in section J. Ask students how many counters must be placed in K so that the ratio of J to K is 3 to 4. (8)
• Place 12 counters in section M. Ask students how many counters must be placed in L so that the ratio of L to M is 3 to 4. (9)
A J L N
B K M P
X Y
Objective Explore the meaning of ratio and proportion.
Materials
• counters*
• blank transparency
• transparency pen*
* = available in Overhead Manipulative Resources Kit
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© Glencoe/McGraw-Hill 69 Mathematics: Applications and Concepts, Course 2
• Place 15 counters in section N. Ask students how many counters must be placed in P so that the ratio of N to P is 3 to 4. (20)
• Ask students how they decided how many counters to place in sections K,
L, and P. (Sample answer: Write equivalent fractions. 34 6
8
192
1250)
• Ask, “If there were 72 counters in section M, how many counters would be
needed in section L?” (54)
Extension
In pairs, have students work together asking how many counters should be
placed in sections K, L, and P as they change the number of counters in sectionsJ, M, and N.
Using Overhead Manipulatives
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© Glencoe/McGraw-Hill 71 Mathematics: Applications and Concepts, Course 2
Hands-On Lab Recording Sheet(Use with the activity on page 301 in Lesson 7-3b of the Student Edition)
NAME ______________________________________________ DATE ____________ PERIOD __________
Wildlife Sampling
Materials
small bowl, dried beans, paper cup, markers
Investigate
Work in groups of three. Record your data in the table below.
Original Number Captured ______
Trial Sample Recaptured P
A
B
C
D
E
F
G
H
I
J
Total
Write About It
Work in groups of three.
1. Find the average of the estimates in column P . Is this a good estimate of thenumber of beans in the bowl? Explain your reasoning.
2. Count the actual number of beans in the bowl. How does this number com- pare to your estimate?
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© Glencoe/McGraw-Hill 72 Mathematics: Applications and Concepts, Course 2
Mini-Project(Use with Lesson 7-4)
NAME ______________________________________________ DATE ____________ PERIOD __________
1. List the objects in your room and their actual size.
2. Why did you want these things in your room?
3. Why did you choose these sizes?
Scale Drawings
The Room of Your Dreams
What if you could build a room of any size and put anything you
wanted in it?
Use the grid as a blueprint. Title your drawing and choose a scale.
Then sketch the things you would put in your room, such as a bed,
a desk, closet, TV, windows, a door. You can make the furnishings
larger or smaller than usual. Make sure you draw everything to
scale. Label everything in the room.
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© Glencoe/McGraw-Hill 73 Mathematics: Applications and Concepts, Course 2
Hands-On Lab Recording Sheet(Use with the activity on page 322 in Lesson 7-8a of the Student Edition)
NAME ______________________________________________ DATE ____________ PERIOD __________
Using a Percent Model
Materials:
none
Your Turn Draw a model to find each part.
a. 70% of 50 b. 45% of 20 c. 20% of 75
Writing Math
1. Suppose your whole family had dinner with you and the total bill was $50.How could you change your model to find the tip?
2. Write a sentence that describes what is represented by
the model at the right.
3. Write a percent problem that can be represented and solved using a model.
4. Make a conjecture about how you could use a model to estimate the percent represented by the ratio 8 out of 50.
0 5025 10075 125 175 225
10
150 200 250
60 70 80 90 10050 4030200
Part
Percent
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Mini-Project Percent and Estimation
(p. 75 of this booklet)
Use With Lesson 8-1.
Objective Create a scale drawing.
Materials
ruler
Using the grid provided, students create a
scale drawing of a room they would like to
build. The drawing includes everything theywish to include in the room drawn to scale.Students must label their drawing and list theobjects in the room and their actual size.
Answers
1. See students’ work. Make sure drawingsare the correct scale based on this informa-tion and the scale of the grid.
2–3. Answers will vary.
Hands-On LabRecording SheetSampling(p. 76 of this booklet)
Use With Lesson 8-3a. This corresponds to
the activity on page 344 in the StudentEdition.
Objective Investigate rate of change.
Materials
square tiles
Students use tiles to build models and find the perimeter of each model. A table and coordi-
nate plane is provided for students to record and graph their data. Students explore rate of change by finding the slope of the graph.
Answers
See Teacher Wraparound Edition p. 344.
Using Overhead Manipulatives Percent of Change
(pp. 77–78 of this booklet)
Use With Lesson 8-4.
Objective Use dot paper to show percent of increase and percent of decrease.
Materials
rectangular dot paper transparency*transparency pens*
* available in Overhead Manipulative Resources Kit
• Students use models on dot paper to showand find percent of increase and percent of
decrease.
• An Extension activity asks students to use agiven figure to show an increase and
decrease of 331
3%.
Answers
Answers appear on the teacher demonstrationinstructions on pages 77–78.
© Glencoe/McGraw-Hill 74 Mathematics: Applications and Concepts, Course 2
Applying PercentTeaching Notes and Overview
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© Glencoe/McGraw-Hill 75 Mathematics: Applications and Concepts, Course 2
Mini-Project(Lesson 8-1)
NAME ______________________________________________ DATE ____________ PERIOD __________
Percent and Estimation
Estimate the percent of the shaded portion for each figure.
Then count grid squares to find the actual percent shaded.
1. 2.
Estimate: __________________ Estimate: ___________________
Actual: ____________________ Actual: _____________________
3. 4.
Estimate: __________________ Estimate: ___________________
Actual: ____________________ Actual: _____________________
5. How did your estimates compare with the actual percents?
6. Shade your own grid. Estimate the percent shaded and count to find the exact percent.
Estimate: ___________________
Actual: _____________________
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© Glencoe/McGraw-Hill 76 Mathematics: Applications and Concepts, Course 2
Hands-On Lab Recording Sheet(Use with the activity on page 344 in Lesson 8-3a of the Student Edition)
NAME ______________________________________________ DATE ____________ PERIOD __________
Sampling
Materials:
none
Writing Math
Work in groups of three.
State whether each sample is random. Explain.
1. To determine the favorite spectator sport for women over 25 years old,350 women over 25 are surveyed at a professional basketball game.
2. To collect data about the study habits of middle school students in the
Franklin School District, the name of every middle school student in thedistrict is placed in a bag and 250 names are randomly selected.
For Exercises 3-7, refer to the information below.
Suppose you are to survey students in your school.
3. Formulate a hypothesis about students’ activities.
4. Design and conduct a survey. Describe the technique that you used to get arandom sample.
5. Use the back of this sheet to organize and display the results of your surveyin a table or graph.
6. Anal