IInntteerraaccttiivvee MMaatthheemmaattiiccss PPrrooggrraamm 33 ®® Fireworks Integrated High School Mathematics Pennant Fever R A E Y Dan Fendel Diane Resek Lynne Alper Meadows or Malls? Sherry Fraser Key Curriculum Press® Small World, Isn’t It? Orchard Hideout 3 Interactive Mathematics Program R ® A E Y Integrated High School Mathematics Dan Fendel and Diane Resek with Lynne Alper and Sherry Fraser Project Editor This material is based upon work supported by the National Science Casey FitzSimons Foundation under award number Editorial Assistant ESI-9255262.Any opinions,findings, Jeff Gammon and conclusions or recommendations Additional Editorial Development expressed in this publication are those of the authors and do not necessarily Masha Albrecht,Mary Jo Cittadino reflect the views of the National Art Developer Science Foundation. Ellen Silva Production Editor © 1999 by the Interactive Mathematics Program. Caroline Ayres All rights reserved. Cover and Interior Design Unauthorized copying of the Terry Lockman Interactive Mathematics Program: Lumina Designworks Year 3 is a violation of federal law. Production Management ®Interactive Mathematics Program Diana Jean Parks,Steve Rogers and Key Curriculum Press Art Editor are registered trademarks of Laura Murray Key Curriculum Press. Technical Graphics ™IMP and the IMP logo are Laurel Technical Services trademarks of Key Curriculum Press. Illustration Key Curriculum Press Taylor Bruce,Deborah Drummond,Tom Fowler,Evangelia Philippidis, 1150 65th Street Sara Swan,Diane Varner,Martha Weston,April Goodman Willy Emeryville,California 94608 Publisher 10 9 8 7 6 5 06 05 04 03 ISBN 1-55953-293-9 Steven Rasmussen Printed in the Editorial Director United States of America John Bergez MATHEMATICSREVIEW Rick Marks,Ph.D.,Sonoma State University, Rohnert Park,California MULTICULTURALREVIEWS Genevieve Lau,Ph.D.,Skyline College, San Bruno,California Arthur Ramirez,Ph.D.,Sonoma State University, Rohnert Park,California Marilyn Strutchens,Ph.D.,University of Maryland, College Park,Maryland TEACHERREVIEWS Daniel R.Bennett,Hoolehua,Hawaii Maureen Burkhart,North Hollywood,California Dwight Fuller,Shingle Springs,California Daniel Johnson,San Jose,California Brian Lawler,Aurora,Colorado Brent McClain,Hillsboro,Oregon Susan Miller,Philadelphia,Pennsylvania Amy C.Roszak,Cottage Grove,Oregon Carmen Rubino,Aurora,Colorado Barbara Schallau,San Jose,California Kathleen H.Spivack,New Haven,Connecticut Wendy Tokumine,Honolulu,Hawaii ACKNOWLEDGMENTS Many people have contributed to the development of the IMP™ curriculum,including the hundreds of teachers and many thousands of students who used preliminary versions of the materials.Of course,there is no way to thank all of them individually,but the IMP directors want to give some special acknowledgments. We want to give extraordinary thanks to these people who played unique roles in the development of the curriculum. • Bill Finzer was one of the original • Lori Green left the classroom as a directors of IMP and helped develop the regular teacher after piloting Year 1 and concept of a problem-based unit. became a traveling resource for IMP classroom teachers.She has compiled • Nitsa Movshovitz-Hadar suggested the central problem for Orchard Hideout many of her classroom insights in the and wrote the first draft of that unit. Teaching Handbook for the Interactive Mathematics Program. • Matt Bremer pilot-taught the entire curriculum,did the initial revision of • Celia Stevenson developed the every unit after its pilot testing,and did charming and witty graphics that graced major work on subsequent revisions. the prepublication versions of the IMP units. • Mary Jo Cittadino became a high school student once again during the piloting of the curriculum,which gave her a unique perspective on the curriculum. In creating this program,we needed help in many areas other than writing curriculum and giving support to teachers. The National Science Foundation (NSF) has been the primary sponsor of the Interactive Mathematics Program®.We want to thank NSF for its ongoing support,and we especially want to extend our personal thanks to Dr.Margaret Cozzens,Director of NSF’s Division of Elementary, Secondary,and Informal Education,for her encouragement and her faith in our efforts. We also want to acknowledge here the initial support for curriculum development from the California Postsecondary Education Commission and the San Francisco Foundation, and the major support for dissemination from the Noyce Foundation and the David and Lucile Packard Foundation. Keeping all of our work going required the help of a first-rate office staff.This group of talented and hard-working individuals worked tirelessly on many tasks,such as sending out Interactive Mathematics Program iii Acknowledgments Year 3 units,keeping the books balanced,helping us get our message out to the public,and handling communications with schools,teachers,and administrators.We greatly appreciate their dedication. • Barbara Ford—Secretary • Marianne Smith—Communications Manager • Tony Gillies—Project Manager • Linda Witnov—Outreach Coordinator IMP National Advisory Board We have been further supported in this work by our National Advisory Board—a group of very busy people who found time in their schedules to give us more than a piece of their minds every year.We thank them for their ideas and their forthrightness. David Blackwell Milton A.Gordon Diane Ravitch (1994 only) Professor of Mathematics and President and Professor of Senior Research Scholar, Statistics Mathematics Brookings Institution University of California,Berkeley California State University, Fullerton Roy Romer (1992–1994 only) Constance Clayton Governor Professor of Pediatrics Shirley Hill State of Colorado Chief,Division of Community Curator’s Professor of Education Health Care and Mathematics Karen Sheingold Medical College of Pennsylvania School of Education Research Director University of Missouri Educational Testing Service Tom Ferrio Manager,Professional Calculators Steven Leinwand Theodore R.Sizer Texas Instruments Mathematics Consultant Chairman Connecticut Department of Coalition of Essential Schools Andrew M.Gleason Education Hollis Professor of Mathematics Gary D.Watts and Natural Philosophy Art McArdle Educational Consultant Department of Mathematics Northern California Surveyors Harvard University Apprentice Committee We want to thank Dr.Norman Webb of the Wisconsin Center for Education Research for his leadership in our evaluation program,and our Evaluation Advisory Board,whose expertise was so valuable in that aspect of our work. • David Clarke,University of Melbourne • George Hein,Lesley College • Robert Davis,Rutgers University • Mark St.John,Inverness Research Associates Finally,we want to thank Steve Rasmussen,President of Key Curriculum Press, Casey FitzSimons,Key’s Project Editor for the IMP curriculum,and the many others at Key whose work turned our ideas and words into published form. iv Interactive Mathematics Program FOREWORD Students must be prepared for the world that they will inherit. Whether or not they choose to enter college immediately after high school,we must equip them to handle new problems with confidence and perseverance.Our ever-changing world requires that students grow into critically thinking adults who are prepared to absorb new ideas and who will become lifelong learners.The Interactive Mathematics Program (IMP) aids in this development. IMP enhances students’understanding of mathematics by obliging them to present reasoned arguments.The group activities in IMP foster teamwork and the development of oral and written communication skills.These skills are honed by requiring students to write intelligible explanations about the processes that they followed to reach their conclusions. As a parent of an IMP student,I have found that IMP enables students to experience mathematics in action and to recognize that mathematics is not simply an esoteric subject.On the other hand,IMP also offers students the opportunity to experience how beautiful and open-ended mathematics is. As a professional mathematician,I believe that IMP teaches mathematics in the way that it should be taught.Mathematics does not arise naturally in nicely defined semester-long modules labeled Algebra I,Geometry,Algebra II,and Trigonometry/Precalculus.IMP effectively breaks down the artificial barriers created by such divisions. I have found the Problems of the Week exceedingly interesting and intellectually stimulating—sufficiently so that I have shared several of them with members of my faculty.It is so refreshing to interact with my son around mathematics that is quite challenging to me also.He can appreciate my excitement and that mathematics can be fun. As a parent and educator,I know the concerns that students,parents, school officials,and others have about colleges’expectations of entering students.What I value most,as do many of my colleagues at other top institutions,is that students have experienced good teaching Interactive Mathematics Program v Foreword Year 3 in well-constructed courses that emphasize communication and creative thinking,and in which the learning that takes place is genuine and meaningful. At Colorado School of Mines,a school of engineering and applied science,we require that our students develop strong communication skills and learn to work effectively as team members.To help our students enhance these skills further,we have established a writing center staffed by qualified professionals.In the beginning courses in calculus in our Department of Mathematical and Computer Sciences, we emphasize the working of real problems provided by the science and engineering disciplines.Students learn to think creatively and not be tied to one notation system.We also require our seniors to take turns at presenting reports on a research topic at weekly seminars. The other students submit reviews of their classmates’presentations and learn from the preparation of their assessments,in addition to providing valuable feedback to the presenter. We expect that our students will not simply reflect their professors’ thinking.Students have a responsibility to engage in independent thinking and to understand the power of thought as distinct from the power of authority.Students have a head start when they enter college courses with prior knowledge in solving complex problems that go beyond calculation and in coping with ambiguity. The Interactive Mathematics Program helps prepare students for life, not just for college calculus.Because the Program emphasizes creative thinking,communication skills,and teamwork,it should serve our students well. Graeme Fairweather Professor and Head Department of Mathematical and Computer Sciences Colorado School of Mines Golden,Colorado vi Interactive Mathematics Program CONTENTS Note to Students . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xix Fireworks DAYS 1–3: The World of Quadratics . . . . . . . . . . . . .3 Fireworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 The Standard POW Write-up . . . . . . . . . . . . . . . . . . . . . . . .6 Homework 1:A Corral Variation . . . . . . . . . . . . . . . . . . . . .8 POW 1:Growth of Rat Populations . . . . . . . . . . . . . . . . . . .9 The Ups and Downs of Quadratics . . . . . . . . . . . . . . . . . .11 Quadratics and Other Polynomials . . . . . . . . . . . . . . . . . .13 Homework 2:Rats in June . . . . . . . . . . . . . . . . . . . . . . . . .16 Homework 3:Product Equations . . . . . . . . . . . . . . . . . . . .17 DAYS 4–6: Factoring and Solving . . . . . . . . . . . . . .18 Factored Intercepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19 Homework 4:Make Your Own Intercepts . . . . . . . . . . . . . .21 Revisiting a Mystery . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22 Homework 5:Factoring Begun . . . . . . . . . . . . . . . . . . . . .24 Who’s Perfect? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25 Homework 6:More About Perfection . . . . . . . . . . . . . . . . .26 DAYS 7–11: The Algebra of the Vertex . . . . . . . . . .27 The Same but Different . . . . . . . . . . . . . . . . . . . . . . . . . . .28 Homework 7:Make Your Own Vertices . . . . . . . . . . . . . . . .29 Vertex Form Begun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30 Homework 8:How Much Can They Drink? . . . . . . . . . . . .31 Homework 9:Corrals and Pens Again . . . . . . . . . . . . . . . .33 Fireworks Height Revisited . . . . . . . . . . . . . . . . . . . . . . . .34 Homework 10:Quadratic Query . . . . . . . . . . . . . . . . . . . .35 Homework 11:“Fireworks”Portfolio . . . . . . . . . . . . . . . . . .36 Interactive Mathematics Program vii Contents Year 3 Appendix: Supplemental Problems . . . . . . . . . . . . .37 Check It Out! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38 Imagine a Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40 More and More Mysterious . . . . . . . . . . . . . . . . . . . . . . . .42 Factors of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43 Twin Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44 Number Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46 Vertex Forms Everywhere . . . . . . . . . . . . . . . . . . . . . . . . . .47 Vertex Form and Intercepts Together . . . . . . . . . . . . . . . . .48 Quadratic Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50 Equilateral Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52 What About One? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55 Orchard Hideout DAYS 1–4: Orchards and Mini-Orchards . . . . . . .59 Orchard Hideout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .60 A Geometric Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .62 Homework 1:Geometry and a Mini-Orchard . . . . . . . . . .68 POW 2:Equally Wet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69 Homework 2:Only Two Flowers . . . . . . . . . . . . . . . . . . . . .71 A Perpendicularity Proof . . . . . . . . . . . . . . . . . . . . . . . . . .72 Homework 3:From Two Flowers to Three . . . . . . . . . . . . .73 More Mini-Orchards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74 Homework 4:In,On,or Out? . . . . . . . . . . . . . . . . . . . . . . .75 DAYS 5–7: Coordinates and Distance . . . . . . . . . .76 Homework 5:Other Trees . . . . . . . . . . . . . . . . . . . . . . . . . .77 Sprinkler in the Orchard . . . . . . . . . . . . . . . . . . . . . . . . . .78 Homework 6:The Distance Formula . . . . . . . . . . . . . . . . .79 How Does Your Orchard Grow? . . . . . . . . . . . . . . . . . . . . .80 Homework 7:A Snack in the Middle . . . . . . . . . . . . . . . . .81 viii Interactive Mathematics Program