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MODELING with
MATHEMATICS
A Bridge to Algebra II
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MODELING with
MATHEMATICS
A Bridge to Algebra II
SECOND EDITION
Nancy Crisler
Gary Simundza
Wentworth Institute of Technology
Houston, TX
COMAP Inc.
Lexington, VA
WH Freeman and Company/BFW
New York
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National Sales Manager, Director High School: Craig Bleyer
Publisher: Ann Heath
Associate Editor: Dora Figueiredo
Freelance Development Editor: Ann Stollenwerk
Media Editor: Laura Capuano
Assistant Media Editor: Catriona Kaplan
Executive Marketing Manager: Cindi Weiss
Photo Editor: Bianca Moscatelli
Photo Researcher: Deborah Anderson
Cover Designer: Vicki Tomaselli
Text Designer: Marsha Cohen
Cover Image: S uzanne DeChillo/The New York Times/
Redux
Project Editor: Teresa Christie
Illustrations: Precision Graphics, Network Graphics
Illustration Coordinator: Bill Page
Production Manager: Paul Rohloff
Composition: MPS Limited
Printing and Binding: RR Donnelley and Sons
TI-83TM, TI-84TM, screen shots are used with permission of the publisher:
© 1996, Texas Instruments Incorporated.
TI-83TM, TI-84TM, Graphic Calulators are registered trademarks of Texas
Instruments Incorporated.
Library of Congress Control Number: 2011942628
ISBN-13: 978-1-4292-6255-2
ISBN-10: 1-4292-6255-9
© 2012, 2006 by W. H. Freeman and Company
First printing
All rights reserved
Printed in the United States of America
W. H. Freeman and Company
41 Madison Avenue
New York, NY 10010
Houndmills, Basingstoke RG21 6XS, England
www.whfreeman.com
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About the Authors
NANCY CRISLER, a Presidential Awardee for Excellence in Mathematics
Teaching, has taught mathematics at many levels including middle school,
high school, and college. She has also worked as the Curriculum Specialist
for the Pattonville School District in the St. Louis area for several years. In
addition to full-time teaching, Nancy has been a lead teacher in numerous
summer institutes dealing with topics of discrete mathematics for high school
students and developmental mathematics for college-level students. She has
also coauthored several articles and books that focus on topics in mathematical
modeling, activity-based learning, and discrete mathematics.
GARY SIMUNDZA is Professor of Mathematics at Wentworth Institute of
Technology in Boston, MA, where he teaches both mathematics and physics
after having previously been a high school science teacher. He has directed
several faculty workshops for the Mathematics Association of America aimed
at helping teachers create activity-based, applied developmental mathematics
courses. He has also coauthored modeling-oriented algebra textbooks and
other curriculum materials for the high school and college levels.
Nancy and Gary are the authors of Developing Mathematics through
Applications: Elementary and Intermediate (2003), Activities for Introductory
College Mathematics (2009), and A New Beginning: Introductory College
Mathematics through Applications (2010).
REGION 4 Education Service Center (ESC) is the largest Texas educational
service center. It provides services to 53 school districts, representing over
1,000,000 students and more than 83,000 professional educators. Region 4
provides nationally recognized professional development, technical assistance
services, and quality, economical products to districts and businesses.
COMAP, the Consortium for Mathematics and Its Applications, is an award-
winning non-profi t organization whose mission is to improve mathematics
education for students of all ages. Since 1980, COMAP has worked with
teachers, students, and business people to create learning environments
where mathematics is used to investigate and model real issues in our world.
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Contents
Chapter 1 MATHEMATICAL Chapter 4 MEASUREMENT:
MODELING 1 PERIMETER, AREA, AND
VOLUME 95
What Is a Mathematical Model? 2
1.1 Activity: Animal Populations 3 Why Is Package Design Important? 96
1.2 Proportions as Models 6 4.1 Polygons 97
1.3 R.A.P. (Review And Practice) 12 4.2 Investigation: Formulas and Literal Equations 103
1.4 Investigation: Patterns and Explanations 14 4.3 Activity: Area and Package Design 107
Modeling Project 18 4.4 Investigation: Volumes of Solid Figures 112
A Picture Is Worth a Thousand Words 4.5 R.A.P. 117
Chapter Review 19 4.6 Activity: Surface Area 119
4.7 Investigation: Similar Figures—Perimeter, Area,
and Volume 122
Chapter 2 DIRECT VARIATION 21 Modeling Project 126
Building a Better Box
How Is Mathematics Related to Bungee
Chapter Review 127
Jumping? 22
Extension 132
2.1 Activity: Bungee Jumping 23
Constructing the Net of a Cone
2.2 Investigation: Proportional Relationships 26
2.3 Direct Variation Functions 30
2.4 R.A.P. 38
Chapter 5 FUNCTIONS AND THEIR
2.5 Slope 40
REPRESENTATIONS 135
Modeling Project 48
It’s Only Water Weight How Can You Find the Value of an Unknown
Chapter Review 49 Quantity? 136
Extension 53 5.1 Investigation: Representing Functions 137
Inverse Variation 5.2 Activity: Linear Relationships 144
5.3 Investigation: Identifying Linear Functions 148
5.4 R.A.P. 153
Chapter 3 SOLVING EQUATIONS 5.5 Graphing Linear Equations 155
AND INEQUALITIES 55 5.6 Investigation: Writing Equations of Lines 159
5.7 Slopes of Parallel and Perpendicular Lines 163
How Can We Care for Our Forests? 56
Modeling Project 167
3.1 Activity: Tree Growth 57 It’s in the News
3.2 Investigation: From Expression to Equation 61
Chapter Review 168
3.3 Using Arrow Diagrams and Tables to Solve
Extension 172
Equations 67
Special Functions
3.4 R.A.P. 73
3.5 Investigation: Solving Multi-Step Equations 75
3.6 Investigation: Solving Inequalities 82
3.7 Compound Inequalities 87
Modeling Project 91
Algebra’s Next Top Model
Chapter Review 92
vi Contents
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Chapter 6 DESCRIBING DATA 175 Chapter 8 SYSTEMS OF EQUATIONS
AND INEQUALITIES 261
How Does the Census Help Us Understand
Our Country? 176 How Are Business Decisions Made? 262
6.1 Investigation: Graphs of Data 177 8.1 Investigation: Using Tables and Graphs to
6.2 Activity: The Center of a Data Set 187 Compare Linear Models 263
6.3 Stem-and-Leaf Diagrams and Box Plots 192 8.2 Solving Systems of Two Equations Using
6.4 R.A.P. 203 Substitution 266
6.5 Activity: Sampling 205 8.3 Investigation: Solving Systems of Two Equations
Modeling Project 209 Using Elimination 272
What a State You’re In! 8.4 R.A.P. 278
Chapter Review 210 8.5 Graphing Inequalities in Two Variables 280
Extension 214 8.6 Investigation: Systems of Linear Inequalities 288
Frequency Tables and Histograms Modeling Project 294
Sizing Up the Mail
Chapter Review 295
Chapter 7 LINEAR MODELS
Extension 298
FROM DATA 219 Linear Programming
What Happened to Amelia Earhart? 220
7.1 Activity: Bones and Height 221 Chapter 9 SYMMETRY AND
7.2 Investigation: Fitting Lines to Data 225
TRANSFORMATIONS 301
7.3 Investigation: Residuals and Judging the
Quality of Fit 229 How Do Archaeologists Use Mathematics
7.4 Investigation: Predicting from Linear Models 234 When Studying Past Civilizations? 302
7.5 R.A.P. 238 9.1 Activity: Symmetry 303
7.6 Activity: Linear Regression 240 9.2 Transformations and Rigid Motions 307
7.7 Comparing Models 244 9.3 Investigation: Rigid Motions and Patterns 311
Modeling Project 251 9.4 R.A.P. 315
Who Am I? 9.5 Activity: Rigid Motions in the Coordinate Plane 317
Chapter Review 253 Modeling Project 323
Extension 257 It’s a Masterpiece!
Correlation Chapter Review 324
Extension 328
Perspective in Art
Contents vii
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Chapter 10 EXPONENTIAL Chapter 12 INDIRECT
FUNCTIONS 333 MEASUREMENT 425
Will the World Have Enough Water? 334 How Can You Determine the Height of a Model
Rocket? 426
10.1 Activity: Nonlinear Change 335
10.2 Properties of Exponents 340 12.1 Investigation: Right Triangles 427
10.3 Investigation: Negative Exponents 345 12.2 Investigation: Distance and Midpoint Formulas 434
10.4 R.A.P. 353 12.3 Activity: Similar Triangles as Models 439
10.5 Investigation: Modeling Exponential Growth 355 12.4 Investigation: Trigonometric Ratios 443
10.6 Investigation: Modeling Exponential Decay 362 12.5 Using Trigonometric Ratios 448
Modeling Project 367 12.6 R.A.P. 452
My, How You’ve Grown! 12.7 Investigation: Finding Acute Angles 454
Chapter Review 368 Modeling Project 460
What’s Your Angle?
Extension 371
Newton’s Law of Cooling Chapter Review 461
Extension 465
Measuring Angles of Elevation
Chapter 11 QUADRATIC
FUNCTIONS 373
Chapter 13 PERIODIC FUNCTIONS
Is It a Parabola? 374
AND SOUND 467
11.1 Investigation: Parabolas–A Geometric Look 375
How Are Music and Repeated Motion
11.2 Activity: Parabolas–An Algebraic Look 382
Related? 468
11.3 Investigation: Graphing Quadratic Functions 386
11.4 R.A.P. 390 13.1 Activity: Periodic Motion 469
11.5 Polynomial Arithmetic 392 13.2 Activity: Frequency and Pitch 474
11.6 Investigation: Solving Quadratic Equations 398 13.3 Investigation: Modeling with Sound Waves 479
11.7 Investigation: The Quadratic Formula 404 13.4 R.A.P. 481
11.8 Modeling with Quadratic Functions 409 Modeling Project 483
Modeling Project 416 Bottle Harmony
Designing a Hero’s Fall Chapter Review 484
Chapter Review 417 Extension 488
Extension 420 Sinusoidal Functions
Exploring Quadratic Data
viii Contents
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