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Modeling with mathematics: a bridge to Algebra II PDF

641 Pages·2012·39.387 MB·English
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This page intentionally left blank MODELING with MATHEMATICS A Bridge to Algebra II CCoommaapp22ee__MMooddeelliinngg__FFMM..iinndddd ii 22//33//1122 33::2200 PPMM this page left intentionally blank CCoommaapp22ee__MMooddeelliinngg__FFMM..iinndddd iiii 22//33//1122 33::2200 PPMM 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 CCoommaapp22ee__MMooddeelliinngg__FFMM..iinndddd iiiiii 22//33//1122 33::2200 PPMM 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 CCoommaapp22ee__MMooddeelliinngg__FFMM..iinndddd iivv 22//33//1122 33::2200 PPMM 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. CCoommaapp22ee__MMooddeelliinngg__FFMM..iinndddd vv 22//33//1122 33::2200 PPMM 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 CCoommaapp22ee__MMooddeelliinngg__FFMM..iinndddd vvii 22//33//1122 33::2200 PPMM 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 CCoommaapp22ee__MMooddeelliinngg__FFMM..iinndddd vviiii 22//33//1122 33::2200 PPMM 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 CCoommaapp22ee__MMooddeelliinngg__FFMM..iinndddd vviiiiii 22//33//1122 33::2200 PPMM

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