Table Of ContentIInntteerraaccttiivvee
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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
