Discovering Advanced Algebra I

Discovering Advanced Algebra i1 de ingCourses/DAA/fr. 2004 Key Curriculum Press31-03-2009 18:42

Discovering Advanced Algebra i1 de ingCourses/DAA/c. 2004 Key Curriculum Press31-03-2009 18:43

Contribute1 de ingCourses/DAA/c.Project EditorLadie MalekProject AdministratorShannon MillerEditorsChristian Aviles-Scott, Josephine NoahEditorial AssistantsErin Gray, Eric Martin, Susan Minarcin, Sigi Nacson,Jason TaylorEditorial ConsultantsLarry Copes, Fred Decovsky, Heather Dever, DanDitty, David RasmussenMathematical Content ReviewerJames Wilson, Ph.D., University of Georgia, Athens,GeorgiaMulticultural and Equity ReviewerSwapna Mukhopadhyay, Ph.D., Portland StateUniversity, Portland, OregonTeacher ReviewersJennifer North Morris, Santa Rosa High School, SantaRosa, CaliforniaWilliam Putnam, John Marshall High School,Rochester, MinnesotaScientific Content ReviewerArt Fortgang, Mills High School, Millbrae, CaliforniaAccuracy CheckersDudley Brooks, Cavan Fang, Monica Johnston,Abby TanenbaumEditorial Production ManagerDeborah CoganProduction EditorChristine OsborneCopyeditorMargaret MooreProduction DirectorDiana Jean RayProduction CoordinatorAnn RothenbuhlerCover DesignerJill KongabelText DesignerMarilyn PerryArt EditorJason LuzPhoto EditorMargee RobinsonArt and Design CoordinatorCaroline AyresIllustratorsRobert Arnow, Pamela Hobbs, William Pasini, Sue ToddTechnical ArtMatt Perry, Jason LuzCompositor and PrepressTSI GraphicsPrinterVon Hoffmann PressExecutive EditorCasey FitzSimonsPublisherSteven Rasmussen 2004 by Key Curriculum Press. All rights reserved.No part of this publication may be reproduced, stored in aretrieval system, or transmitted, in any form or by any means,electronic, photocopying, recording, or otherwise, withoutthe prior written permission of the publisher. The Geometer’s Sketchpad, Dynamic Geometry, and KeyCurriculum Press are registered trademarks of KeyCurriculum Press. The Discovering Mathematics logo andSketchpad are trademarks of Key Curriculum Press. Fathom Dynamic Statistics is a trademark ofKCP Technologies.All other trademarks are held by their respective owners.This material is based upon work supported by the NationalScience Foundation under award number MDR9154410.Any opinions, findings, and conclusions or recommendationsexpressed in this publication are those of the authors and donot necessarily reflect the views of the National ScienceFoundation.Key Curriculum Press1150 65th StreetEmeryville, CA 94608editorial@keypress.comwww.keypress.comPrinted in the United States of America10 9 8 7 6 5 4 3 2 1 08 07 06 05 04 03ISBN 1-55953-606-3 2004 Key Curriculum Press31-03-2009 18:43

Acknowledgments iii1 de ingCourses/DAA/a.a textbook and its supplementary materials is a team effort involvingCreatingmany individuals and groups. We are especially grateful to thousands ofAdvanced Algebra Through Data Exploration and Discovering Algebra teachers andstudents, to teachers who participated in the summer institutes and workshops,and to manuscript readers, all of whom provided suggestions, reviewed material,located errors, and most of all, encouraged us to continue with the project.Our students, their parents, and our administrators at Interlochen Arts Academyhave played an important part in the development of this book. Most importantly,we wish to thank Carol Murdock, our parents, and our children for their love,encouragement, and support.As authors we are grateful to the National Science Foundation for supportingour initial technology-and-writing project that led to the 1998 publication ofAdvanced Algebra Through Data Exploration. Discovering Advanced Algebra hasbeen developed and shaped by what we learned during the writing and publicationof both Advanced Algebra Through Data Exploration and Discovering Algebra, andour work with so many students, parents, and teachers who were searching for amore meaningful algebra curriculum.Over the course of our careers, many individuals and groups have beeninstrumental in our development as teachers and authors. The Woodrow WilsonNational Fellowship Foundation provided the initial impetus for involvement inleading workshops. Publications and conferences produced by the NationalCouncil of Teachers of Mathematics and Teachers Teaching with Technology haveguided the development of this curriculum. Individuals such as Ron Carlson,Helen Compton, Frank Demana, Arne Engebretsen, Paul Foerster, Christian Hirsch,Glenda Lappan, Richard Odell, Heinz-Otto Peitgen, James Sandefur, James Schultz,Dan Teague, Charles VonderEmbse, Bert Waits, and Mary Jean Winter haveinspired us.The development and production of Discovering Advanced Algebra has been acollaborative effort between the authors and the staff at Key Curriculum Press.We truly appreciate the cooperation and valuable contributions offered by theEditorial and Production Departments at Key Curriculum Press. Finally, a specialthanks to Key’s president, Steven Rasmussen, for encouraging and publishing atechnology-enhanced Discovering Mathematics series that offers groundbreakingcontent and learning opportunities.Jerald MurdockEllen KamischkeEric Kamischke 2004 Key Curriculum Pressiii31-03-2009 18:44

A Note from the Publisher iv1 de ingCourses/DAA/a.you find in this book won’t look quite like the algebra you may haveTheseenalgebrain older textbooks. The mathematics we learn and teach in school has tochange continually to reflect changes in our world. Our workplaces are changing,and technology is present everywhere, fundamentally changing the work we do.There are some new topics that are now possible to explore with technology, andsome standard topics that can be approached in new ways. As the NationalCouncil of Teachers of Mathematics (NCTM) Technology Principle says, “Whentechnological tools are available, students can focus on decision making, reflection,reasoning, and problem solving.” This has been the focus of the authors and theKey Curriculum Press editorial team in the creation of Discovering AdvancedAlgebra: An Investigative Approach. As you progress through this book, you’ll seethat graphing calculators and other technologies are used to explore patterns andto make, test, and generalize conjectures.When Key Curriculum Press published the first version of this text, AdvancedAlgebra Through Data Exploration: A Graphing Calculator Approach, in 1998, fewbooks were available that had a similar foundation in technology. In this revision,you’ll see that that foundation has been enriched with projects, explorations,and exercises that utilize not only graphing calculators, but also the powerfulanalysis tools The Geometer’s Sketchpad and Fathom Dynamic Statistics .Based on feedback from users and reviewers, this revision is reorganized andeasier to read. Discovering Advanced Algebra also completes the fully updatedDiscovering Mathematics series. All of the features that make Discovering Algebraand Discovering Geometry innovative and exciting are now incorporated into thisbook as well, to make a coherent and streamlined series.Investigations are at the heart of each book. Through the investigations, you’llexplore interesting problems and generalize concepts. And if you, as a student,forget a concept, formula, or procedure, you can always re-create it—because youdeveloped it yourself the first time! You’ll find that this approach allows you toform a deep and conceptual understanding of advanced algebra topics.As Glenda Lappan, mathematics professor at Michigan State University and formerNCTM president, said about the first edition of this book, “Students coming outof a year with this text . . . will know the mathematics they know in deeper, moreflexible ways. They will have developed a set of mathematical habits of mind thatwill serve them very well as students or users of mathematics. They will emergewith a sense of mathematics as a search for regularity that allows prediction.”If you are a student, we hope that what you learn this year will serve you well inlife. If you are a parent, we hope you will enjoy watching your student developmathematical confidence. And if you are a teacher, we hope Discovering AdvancedAlgebra greatly enriches your classroom. The professional team at Key CurriculumPress wishes you success and joy in the lifetime of mathematics ahead of you.We look forward to hearing about your experiences.Steven Rasmussen, PresidentKey Curriculum Pressiv 2004 Key Curriculum Press31-03-2009 18:44

Contents Page V1 de ingCourses/DAA/co.A Note to Students from the AuthorsCHAPTERProblem Solving0.1 Pictures, Graphs, and DiagramsInvestigation: Camel Crossing the Desert0.2 Symbolic RepresentationInvestigation: Problems, Problems, ProblemsProject: Create Your Own Computer Icon0.3 Organizing InformationInvestigation: Who Owns the Zebra?Chapter 0 ReviewTake Another LookAssessing What You've LearnedCHAPTERxiiiPatterns and Recursion1.1 Recursively Defined SequencesInvestigation: Monitoring Inventory1.2 Modeling Growth and DecayInvestigation: Looking for the Rebound1.3 A First Look at LimitsInvestigation: Doses of Medicine1.4 Graphing SequencesInvestigation: Match Them UpExploration:Recursion in GeometryActivity: Two Spirals1.5 Loans and InvestmentsInvestigation: Life’s Big ExpendituresProject: The Pyramid Investment PlanExploration:Refining the Growth ModelActivity: Cornering the MarketChapter 1 ReviewTake Another LookAssessing What You’ve Learned 2004 Key Curriculum 66768717475v31-03-2009 18:44

Contents Page VI1 de ingCourses/DAA/co.CHAPTERCHAPTERviDescribing Data2.1 Measures of Central Tendency and Box PlotsInvestigation: Pulse Rates2.2 Measures of SpreadInvestigation: A Good Design2.3 Histograms and Percentile RanksInvestigation: Eating on the RunProject: Stem-and-Leaf PlotsExploration:Census MicrodataActivity: Different Ways to Analyze DataChapter 2 ReviewTake Another LookAssessing What You’ve Learned76778085859498104105106108111112Linear Models and Systems3.1 Linear Equations and Arithmetic SequencesInvestigation: Match Point3.2 Revisiting SlopeInvestigation: Balloon Blastoff3.3 Fitting a Line to DataInvestigation: The WaveProject: Talkin’ Trash3.4 The Median-Median LineInvestigation: Spring ExperimentProject: Counting Forever3.5 ResidualsInvestigation: Airline SchedulesExploration:Residual Plots and Least SquaresActivity: A Good Fit?3.6 Linear SystemsInvestigation: Population Trends3.7 Substitution and EliminationInvestigation: It All Adds UpChapter 3 ReviewMixed ReviewTake Another LookAssessing What You’ve 49151152157159164166169170 2004 Key Curriculum Press31-03-2009 18:44