C O M A P ’ S

A N N O T A T E D T E A C H E R ’ S E D I T I O N

DEVELOPED BY

COMAP, Inc.www.comap.com

PROJECT LEADERSHIP

Solomon GarfunkelCOMAP, INC., BEDFORD, MA

Landy GodboldTHE WESTMINSTER SCHOOLS, ATLANTA, GA

Henry PollakTEACHERS COLLEGE, COLUMBIA UNIVERSITY, NY, NY

Mathematics: Modeling Our World

© Copyright 1999

by COMAP, Inc.

The Consortium for Mathematics and Its Applications (COMAP)

175 Middlesex Turnpike, Suite 3B

Bedford, MA 01730

Published and distributed by

The Consortium for Mathematics and Its Applications (COMAP)

175 Middlesex Turnpike, Suite 3B

Bedford, MA 01730

ALL RIGHTS RESERVED

The text of this publication, or any part thereof, may not be reproduced

or transmitted in any form or by any means, electronic or

mechanical, including photocopying, recording, storage in an information retrieval system, or otherwise,

without prior written permission of the publisher.

This book was prepared with the support of NSF Grant ESI-9255252. However, any opinions,

findings, conclusions, and/or recommendations herein are those of the authors

and do not necessarily reflect the views of the NSF.

ISBN 0-538-68225-6

Printed in the United States of America.

iii

I t is almost impossible to know where to begin to acknowl-edge all of the people who have made this work possible.It is clear that any recognition must begin with the staff ofthe Division of Elementary, Secondary, and InformalScience Education at the National Science Foundation. Theirvision and continued support are the bedrock upon which all ofour efforts have rested. Our heartfelt thanks go to Margaret(Midge) Cozzens, whose championing of mathematics reform atall educational levels has made her the central figure in pre-col-lege mathematics and science education in the United States.

Members of the project leadership team are candidates for saint-hood. Their tireless efforts, creativity, and tenacity are the onlyreason we have been able to complete our work. LandyGodbold is truly the guiding light of our program. Not only hashe served as principal investigator, author, main editor, and direc-tor of our writing institutes, but he managed to keep us togetherand on task. His vision and gentle nature sustained us all.

Any mention of vision brings us to Henry Pollak, co-principalinvestigator of Mathematics: Modeling Our World. Henry hasbrought a perspective and wisdom to our work that is unpar-alled in the mathematics education community. He simply knowsmore about mathematical modeling and how to present it to stu-dents and teachers than any person alive. His willingness to rollup his sleeves and work with us as equals instilled a confidenceand joy of learning that saw us through these past five years.

Gary Froelich has also served in a myriad of capacities. He is notonly a writer on the program, but a key member of our revisionteam. And perhaps most importantly, he has been the COMAPhome office person, writing weekly updates and answering hun-dreds of telephone and e-mail queries from authors, field-testteachers, and users.

Our author, revision, and field-test teams are listed elsewhere inthese pages, but I cannot say enough about their efforts. Forfour summers our authors worked around the clock, first defin-ing the elements of our program and later writing all of the textthat makes up the content of Mathematics: Modeling Our World.Our revision team (Marsha Davis, Gary Froelich, Landy Godbold,and Bruce Grip) had to meet impossible deadlines in order toinsure our publication schedules.

Perhaps the bravest group of all is our field-test teachers. Theysigned on for three years of experimentation, based solely onour very preliminary outlines and our promise that we wouldproduce a curriculum they would be proud to use. Their faith inus was the single most important factor in sustaining our efforts.And they made it work. Despite the rough early drafts and theshort period for staff development, they made sense of what wedid and truly helped their students learn some real mathematics.

While our staff development sessions were not as long as theyshould have been, the success we achieved was due to theleadership of Beatriz D’Ambrosio. We gave Beatriz the impossibletask of presenting 30 teachers with a year’s worth of material ina little over a week’s time. Somehow she was able to organizethese sessions and keep up our spirits.

Our assessment team from the Freudenthal Institute in theNetherlands was wonderful. The “Cloggies,” as we affectionatelycall them, brought valuable experience in the design of open-ended assessment instruments under the direction of Jan deLange. Their ability to find new and exciting contexts for the

mathematical development was invaluable to our work. Jan deLange, along with Henk van der Kooij, Anton Roodhardt, Dedede Haan, and Kees Lagerwaard showed us how small the worldreally is as we worked together to solve common problems.

We acknowledge and thank Kay Merseth of Harvard Universityfor her early efforts to organize our summer writing and teacher-training institutes. We are grateful to Barbara Flagg for her in-depth evaluations of our field-test drafts, which aided us in ourrevision process. We especially wish to thank David Moore for hiscareful reading of our work as well as his practical and alwayson-target suggestions for revision. And last, but not least, weapplaud the efforts of the COMAP staff.

While I must thank people individually, I want to say for therecord that I am unbelievably fortunate. I work with the beststaff possible. Mathematics: Modeling Our World has been a pro-ject unlike any that COMAP has ever attempted. Everyone onthe COMAP staff has played an important role in making thistext a reality. I cannot say enough about their dedication andtheir efforts.

In a very real sense, the text you hold in your hands is the prod-uct of COMAP’s Creative Director, Roger Slade. Roger has over-seen the work of production, done all of the creative design,and the arduous task of composition. His efforts have been ablyaided by COMAP’s production manager, George Ward, who hassomehow managed to keep on schedule despite having to pro-duce all of COMAP’s other publications, marketing materials,newsletters, and journals as well. Thanks go to Susan Judge, our developmental editor, for pulling together all the pieces ofthe puzzle; to Pauline Wright for copyediting manuscripts; toDaiva Kiliulis and Dave Barber who so ably directed the layoutand art program for the book; and to Linda Vahey for her marketing input.

And of course, I need to thank the people who make COMAPwork. Roland Cheyney has the title of Director of Marketing, buteveryone at COMAP knows that he could have several differenttitles, not the least of which is making sure that all of our com-puter and telecommunications technology actually continue tofunction. And speaking of someone who could have severaltitles, no acknowledgment page would be complete withoutrecognizing the efforts of COMAP’s Business DevelopmentDirector, Laurie Aragón. Simply put, she makes sure COMAP andall of its projects work and work well.

If I have left anyone out, I apologize. If I have not been effusiveenough in my praise, the fault lies with me. It has been mygreat fortune to work with such people. No words can expressmy heartfelt thanks.

Sol Garfunkel

EXECUTIVE DIRECTOR, COMAP, INC.

Special Recognition

PROJECT LEADERSHIP

Solomon Garfunkel COMAP, INC., LEXINGTON, MA

Landy Godbold THE WESTMINSTER SCHOOLS, ATLANTA, GA

Henry PollakTEACHERS COLLEGE, COLUMBIA UNIVERSITY, NY

EDITOR

Landy Godbold

AUTHORS

Allan BellmanWATKINS MILL HIGH SCHOOL, GAITHERSBURG, MD

John BurnetteKINKAID SCHOOL, HOUSTON, TX

Horace ButlerGREENVILLE HIGH SCHOOL, GREENVILLE, SC

Claudia Carter MISSISSIPPI SCHOOL FOR MATH AND SCIENCE, COLUMBUS, MS

Nancy CrislerPATTONVILLE SCHOOL DISTRICT, ST. ANN, MO

Marsha Davis EASTERN CONNECTICUT STATE UNIVERSITY, WILLIMANTIC, CT

Gary FroelichCOMAP, INC., LEXINGTON, MA

Landy GodboldTHE WESTMINSTER SCHOOLS, ATLANTA, GA

Bruce GripETIWANDA HIGH SCHOOL, ETIWANDA, CA

Rick JenningsEISENHOWER HIGH SCHOOL, YAKIMA, WA

Paul KehleINDIANA UNIVERSITY, BLOOMINGTON, IN

Darien LautenOYSTER RIVER HIGH SCHOOL, DURHAM, NH

Sheila McGrailCHARLOTTE COUNTRY DAY SCHOOL, CHARLOTTE, NC

Geraldine OlivetoTHOMAS JEFFERSON HIGH SCHOOL FORSCIENCE AND TECHNOLOGY, ALEXANDRIA, VA

Henry PollakTEACHERS COLLEGE, COLUMBIA UNIVERSITY, NY

J.J. Price PURDUE UNIVERSITY, WEST LAFAYETTE, IN

Joan ReinthalerSIDWELL FRIENDS SCHOOL, WASHINGTON, D.C.

James SwiftALBERNI SCHOOL DISTRICT, BRITISH COLUMBIA, CANADA

Brandon ThackerBOUNTIFUL HIGH SCHOOL, BOUNTIFUL, UT

Paul ThomasMINDQ, FORMERLY OF THOMAS JEFFERSON HIGH SCHOOL FORSCIENCE AND TECHNOLOGY, ALEXANDRIA, VA

REVIEWERS

Dédé de Haan, Jan de Lange, Henk van der KooijFREUDENTHAL INSTITUTE, THE NETHERLANDS

David MoorePURDUE UNIVERSITY, WEST LAFAYETTE, IN

Henry PollakTEACHERS COLLEGE, COLUMBIA UNIVERSITY, NY

ASSESSMENT

Dédé de Haan, Jan de Lange, Kees Lagerwaard, Anton Roodhardt, Henkvan der KooijTHE FREUDENTHAL INSTITUTE, THE NETHERLANDS

REVISION TEAM

Nancy Chrisler, Marsha Davis, Gary Froelich, Landy Godbold, Bruce Grip, Rick Jennings

EVALUATION

Barbara FlaggMULTIMEDIA RESEARCH, BELLPORT, NY

TEACHER TRAINING

Allan Bellman, Claudia Carter, Nancy Crisler, Beatriz D’Ambrosio, Rick Jennings, Paul Kehle, Geraldine Oliveto, Paul Thomas

FIELD TEST SCHOOLS AND TEACHERS

Clear Brook High School, Friendswood, TX JEAN FRANKIE, TOM HYLE, LEE YEAGER

Clear Creek Middle School, Gresham, OR DAVID DROM, JOHN MCPARTIN, NICOLE RIGELMAN

Damascus Middle School, Boring, ORMARIAH MCCARTY, CLAUDIA MURRAY

Dexter McCarty Middle School, Gresham, OR CONNIE RICE

Dr. James Hogan Senior High School, Vallejo, CA GEORGIA APPLEGATE, PAM HUTCHISON, JERRY LEGE, TOM LEWIS

Foxborough High School, Foxborough, MABERT ANDERSON, SUE CARLE, MAUREEN DOLAN, JOHN MARINO, MARY PARKER, DAVE WALKINS, LEN YUTKINS

Frontier Regional High School, South Deerfield, MA LINDA DODGE, DON GORDON, PATRICIA TAYLOR

Gordon Russell Middle School, Gresham, OR MARGARET HEYDEN, TIFFANI JEFFERIS, KEITH KEARSLEY

Gresham Union High School, Gresham, ORDAVE DUBOIS, KAY FRANCIS, ERIN HALL, THERESA HUBBARD, RICK JIMISON, GAYLE MEIER, CRAIG OLSEN

Jefferson High School, Portland, ORSTEVE BECK, DAVE DAMCKE, LYNN INGRAHAM, MARTHA LANSDOWNE, JOHN OPPEDISANO, LISA WILSON

Lincoln School, Providence, RIJOAN COUNTRYMAN

Mills E. Godwin High School, Richmond, VAKEVIN O’BRYANT, ANN W. SEBRELL

New School of Northern Virginia, Fairfax, VAJOHN BUZZARD, VICKIE HAVELAND, BARBARA HERR, LISA TEDORA

Northside High School, Fort Wayne, INROBERT LOVELL, EUGENE MERKLE

Ossining High School, Ossining, NYJOSEPH DICARLUCCI

Pattonville High School, Maryland Heights, MOSUZANNE GITTEMEIER, ANN PERRY

Price Laboratory School, Cedar Falls, IADENNIS KETTNER, JIM MALTAS

Rex Putnam High School, Milwaukie, ORJEREMY SHIBLEY, KATHY WALSH

Sam Barlow High School, Gresham, OR BRAD GARRETT, KATHY GRAVES, COY ZIMMERMAN

Simon Gratz High School, Philadelphia, PALINDA ANDERSON, ANNE BOURGEOIS, WILLIAM ELLERBEE

Ursuline Academy, Dallas, TXSUSAN BAUER, FRANCINE FLAUTT, DEBBIE JOHNSTON, MARGARET KIDD, ELAINE MEYER, MARGARET NOULLET, MARY PAWLOWICZ, SHARON PIGHETTI, PATTY WALLACE, KATHY WARD

West Orient Middle School, Gresham, ORDAN MCELHANEY

COMAP STAFF

Solomon Garfunkel, Laurie Aragón, Sheila Sconiers, Gary Froelich, Roland Cheyney, Roger Slade, George Ward,Frank Giordano, Susan Judge, Emily Sacca,Pauline Wright, Daiva Kiliulis, David Barber, Gail Wessell, Gary Feldman, Clarice Callahan, Rafael Aragón, Peter Bousquet, Linda Vahey

INDEX EDITOR

Seth MaislinFOCUS PUBLISHING SERVICES, WATERTOWN, MA

iv

Project Directors, Authors, Reviewers, Field Test Teachers, and COMAP Staff

v

Dear Teacher,

COMAP has been dedic

ated to presenting mathematics through

contemporary applications since 1980. We ha

ve produced

high school and college texts, supplemental m

odules, and

television courses—all with the intention of s

howing students how

mathematics is used in their daily lives.

For the past five years, we have worked with

a team of over 20 authors, almost all

practicing high school teachers, to develop th

is curriculum. The authors include several

Presidential Awardees and Woodrow Wilson

Fellows. We have field-tested these

materials with over 5,000 students across the

country. Without the dedication and energy

of these authors and teachers, this work wou

ld not have been possible.

The result of these labors is Mathematics: Mo

deling Our World. In the COMAP spirit,

Mathematics: Modeling Our World develops m

athematical concepts in the context of how

they are actually used.

We are very much aware that Mathematics: M

odeling Our World is a very different kind of

book for a very different kind of course. We h

ave changed some of the standard content

and added material on both applications and

modeling. We are calling for more hands-

on activities and cooperative learning. Graph

ing calculators and computer software are

used where needed. Our assessments are mo

re open-ended. Teaching this course for the

first time will certainly take added preparatio

n.

The goals of the Mathematics: Modeling Our W

orld curriculum are not merely to provide

familiarity and facility with “mathematical o

perations.” A major goal of the curriculum is

the development of higher-order thinking sk

ills. And “thinking” is not the same as

“getting answers.” The ability to transfer idea

s from one context to another—to make

connections—is ultimately the skill that make

s mathematics valuable.

In order for students to develop these higher

-order thinking skills, other skills and

attitudes must be cultivated. Successful mod

eling requires the ability to generate

multiple possibilities from a single setting—t

o raise alternative assumptions for

consideration. It also involves intellectual ris

k-taking. Students must be willing to

become familiar with a situation; to explore it

s possibilities without first knowing how

things will turn out. They must be willing to

propose ideas, explain why they are

reasonable in terms of the assumptions that

led to them, and to revise assumptions and

conclusions after evaluating them using agree

d-upon criteria.

We deeply believe that the payoff in student

understanding and achievement will

make all of our efforts worthwhile. We know

the importance of a solid mathematics

education in today’s increasingly quantitative

world. We know conversely that a lack

of mathematical facility can be an enormous

handicap to our students when they face

the real world. We sincerely hope that you w

ill travel this brave new world with us.

We are dedicated to providing as much supp

ort as our energies allow. And as we

have said to the students, we hope that you fi

nd this work both an enjoyable and

rewarding experience.

Solomon Garfunkel

CO-PRINCIPAL INVESTIGATOR

Landy Godbold

CO-PRINCIPAL INVESTIGATOR

Henry Pollak

CO-PRINCIPAL INVESTIGATOR

Background of COMAP and Rationale for Mathematics: Modeling Our World T1

Components of the program T2

Student Edition features T4

Annotated Teacher’s Edition features T6

Teacher’s Resources features T10

Course 1 (Grade 9) Mathematical Concepts T13

Course 2 (Grades 10) Mathematical Concepts T14

Course 3 (Grades 11) Mathematical Concepts T15

NCTM Standards Correlation T16

Mathematics: Modeling Our World, Course 3 Pacing Chart T18

Frequently Asked Questions from Administrators, Counselors, and Parents T19

Interdisciplinary Curriculum T20

The Modeling-Based Curriculum T21

Core Curriculum T21

Student-Centered Content T21

Authentic Assessment T22

Technology/Multimedia T22

Overview for Unit 1 The Geometry of Art T23

Overview for Unit 2 Fairness & Apportionment T27

Overview for Unit 3 Sampling T31

Overview for Unit 4 Mind Your Own Business T37

Overview for Unit 5 Oscillation T40

Overview for Unit 6 Feedback T44

Overview for Unit 7 Modeling Your World T49

vi

CONTENTSMathematics: Modeling Our WorldA N N O T A T E D T E A C H E R ’ S E D I T I O N

T1

Since its inception in 1980, COMAP hasbeen dedicated to presenting mathematicsthrough contemporary applications. Wehave produced high school and collegetexts, hundreds of supplemental modules,

and three television courses—all with the purpose ofshowing students how mathematics is used in theirdaily lives.

After the publication of the NCTM Standards in 1989,the National Science Foundation began to fund majorcurriculum projects at the elementary, middle, andsecondary levels. The purpose of all of these programsis to turn the vision of the Standards into thecurriculum of today’s classrooms.Given the Standards’ emphasis onmodeling and applications and ourcommitment to the these ideas,COMAP wanted to developcurriculum at the secondary level.We submitted a proposal to the NSFto create a Standards-basedsecondary school mathematicscurriculum: Applications Reform inSecondary Education. In 1992, theARISE project was one of only foursuch programs selected by the NSF for funding.

Over the past five years, we have worked to developthis curriculum with a team of over 20 authors,almost all practicing high school teachers, includingseveral Presidential Awardees and Woodrow WilsonFellows. We have field-tested these materials withover 5,000 students across the country. Both ourauthor team and our field-testers come from anamazingly diverse collection of schools with a fullrange of student populations, from large urbanschools in Philadelphia, PA and Portland, OR, to asmall private school in Texas. Without the authors’and teachers’ dedication and boundless energy, noneof our work would have been possible.

The result of these labors is Mathematics: Modeling OurWorld. In the COMAP spirit, Mathematics: Modeling OurWorld develops mathematical concepts in the contextsin which they are actually used. The word “modeling”

is the key. Real problems do not come at the end ofchapters. Real problems don’t look like mathematicsproblems. Real problems are messy. Real problems askquestions such as: How do we create computeranimation? How do we effectively control an animalpopulation? What is the best location for a firestation? What do we mean by “best”?

Mathematical modeling is the process of looking at asituation, formulating a problem, finding amathematical core, working within that core, andcoming back to see what mathematics tells us aboutthe original problem. We do not know in advancewhat mathematics to apply. The mathematics we

settle on may be a mix of geometry,algebra, trigonometry, data analysis,and probability. We may need to usecomputers or graphing calculators,spreadsheets, or other utilities.Because Mathematics: Modeling OurWorld brings to bear so manydifferent mathematical ideas andtechnologies, this approach is trulyintegrated.

At COMAP, we firmly believe in applying the NCTMStandards to both content and pedagogy. Mathematics:Modeling Our World features hands-on activities as wellas collaborative learning. Simply put, many problemsare solved more efficiently by people working ingroups. In today’s world, that is what work looks like.Moreover, the units in this text are arranged bycontext and application rather than mathematicaltopic. We have done this to re-emphasize our primarygoal: presenting students with mathematical ideas theway they will see them as they go on in school andout into the work force.

At heart, we want to demonstrate to students thatmathematics is the most useful subject they will learn.More importantly, we hope to demonstrate that usingmathematics to solve interesting problems about howour world works can be a truly enjoyable andrewarding experience. Ultimately, learning to model islearning to learn.

Annotated Teacher’s Edition Mathematics: Modeling Our World

Background of COMAP and Rationale for Mathematics: Modeling Our World

“ Ultimately, learning to model islearning to learn.”

T2 Mathematics: Modeling Our World Annotated Teacher’s Edition

Components of Mathematics: Modeling Our World

STUDENT TEXT:

•Mathematical concepts aredeveloped in unitscentered in real-worldcontexts.

•Open-ended questions andproblems encouragestudents to workindependently and ingroups to improve uponoriginal models.

VIDEO SUPPORT:

•Video segmentsaccompany each unit tomotivate students as theybegin a unit, or to provideadditional information fora specific problem.

CD-ROM:

•Calculator and computersoftware writtenspecifically forMathematics: Modeling Our World

•Mac and IBM formats areavailable.

•TI-82 and TI-83 versions

•“Read me” files to explainthe software

•Software instruction andprogram codes appear inTeacher’s Resourcesmaterials.

•Software includes:graphing calculatorprograms, specialtycomputers, spreadsheettemplates, data sets, andgeometric drawing utilitysketches.

T3Annotated Teacher’s Edition Mathematics: Modeling Our World

TEACHER’S RESOURCES:

FOR TEACHERS

•Ideas for presenting videosegments

•A Teacher’s Guide withBackground Readings andadditional teaching suggestions

•Transparencies

FOR STUDENTS

•Supplemental Activities

•Handouts

•Assessment Problems

ANNOTATED TEACHER’S EDITION:

•Background information aboutmathematical concepts and unitcontent

•Page-by-page teachingsuggestions in the wrap-around

•Stated purposes for eachLesson, Activity, andIndividual Work

•References to the Teacher’sResources materials

SOLUTIONS MANUAL:

•Answers to all of the Considerquestions, Activities, IndividualWorks, Assessment Problems,and all supplementarymaterials

•Sample answers for the manyopen-ended questions

T4 Mathematics: Modeling Our World Annotated Teacher’s Edition

316

Running a business,even a small business,is a challenge becausethere are many variables that

contribute to its success or

failure. For example, if a

business sets the price of its

product too low, it will not

recover its costs of

production. If the price of the

product is set too high,

customers will buy from a

competitor. The price a

business sets for its product,

therefore, is one of the

variables that affects the

profits (or losses) that the

business realizes.

Most of the mathematical

models you have developed

in the Mathematics: ModelingOur World program haveinvolved functions of a single

variable. A business’s profits

depend on many variables. In

this unit, you will create

models that involve functions

of more than one variable

and develop techniques for

analyzing such models.

Mind Your Own Business

LESSON ONESo, You Want to Be in Business

LESSON TWOWho’s Minding the Store(room)?

LESSON THREEChanging Assumptions

LESSON FOURSlow Growth

Unit Summary

4U N I T

PREPARATION READING

Getting Your Fair Share

T his unit is about fairness. Even at an early age, peopleare concerned about being treated fairly. Children oftencomplain, “It’s not fair! Johnny got a bigger piece ofpizza than I did!” or “She got to stay up later than I did!” or“How come she gets a better bike than I do?” But the questionof dividing things fairly goes far beyond sharing a pizza. It is acontinuous concern to people of all ages and can be veryserious. For example, several roommates may need to sharetheir telephone bill fairly, or the heirs named in a will mayneed to divide an estate fairly. Areas of our world, such as theMiddle East, continually struggle with decisions about fairland division. Even the United States House of Representativeshas had difficulties in apportioning a fair number of seats tothe states.

126 Mathematics: Modeling Our World UNIT TWO

LESSON ONE

Heir Today, Gone

Tomorrow

KEY CONCEPTS

Fair division

Algorithms

Tabular reasoning

Measuring fairness

Paradox

The Image Bank

317

SMALL BUSINESS: NO SMALL MATTER

Small business is a vital part of the Americaneconomy. For example, according to the UnitedStates Small Business Administration, in 1996 therewere about 22.1 million small businesses in thecountry and about 75% of the 2.5 million new jobs created in

1995 were in small-business-dominated industries. Small

businesses provide most Americans with their first jobs and,

therefore, with much of the on-the-job training that develops

marketable skills. No doubt someone you know is employed

by a small business; perhaps someday you will operate one of

your own.

Because small business affects the lives of so many people, you

are likely to find knowledge of related matters useful at some

time in your life. In this unit, you will see how mathematical

models can help small businesses develop strategies to ensure

their growth and prosperity.

he Image Bank

UnitOpener• Sets tone for

unit

• Piques studentinterest

LessonOpener• Lists key

conceptscovered in the unit

PreparationReading• Provides

background tofocus the lesson

Student Edition Features

328 LESSON ONE Mathematics: Modeling Our World UNIT FOUR

CONSIDER:

1.Does it make the most sense to measure the success of a business interms of the revenues the business receives, in terms of the costs itincurs, or in terms of the profits it realizes? Explain.

2.Does the price a manufacturing business charges for its product havean important effect on the success of the business? Explain.

M A T H O N T H E J O B

Marilyn Hamilton is an inspiringexample of someone who hasmade a success of her ownsmall manufacturing business, andmuch more. A year after a hang glidingaccident left her a paraplegic, MarilynHamilton came up with the concept ofmanufacturing a more mobile,maneuverable, lightweight wheelchairusing materials like those used inmaking hang gliders. In 1986, six yearsafter she and two associates beganmanufacturing their high-performancechairs in a backyard garage, Marilyn’scompany, Motion Designs, had salesfigures of about $21 million. At thatpoint, her company was purchased bySunrise Medical Inc. where she retains aposition as Vice President of ConsumerDevelopment.

Among her many achievements outside the business world, Marilyn has won threetitles in the National Wheelchair Tennis Championships and is a six-time NationalDisabled Ski Champion. She is also the founder of Winners on Wheels, whichprovides young people in wheelchairs with challenging experiences and leadershiptraining.

Consider• Raises issues

about ideaspresented inthe lessons

• Encouragesstudents toask questionsthroughoutthe unit

T5Annotated Teacher’s Edition Mathematics: Modeling Our World

Activities• Numbered sequentially

throughout the unit

• Most designed to be completed inone class period

• Created as hands-on opportunitiesto introduce and develop newconcepts, to explore multipleaspects of a problem, workthrough the difficulties, and shareresults

• Designed for groups or pairs ofstudents to work together to solveproblems

• Key terms appear in boldface type.

205SAMPLING Mathematics: Modeling Our World LESSON TWO

INDIVIDUAL WORK 3

Simulated Sample Examples

In Activity 3, your group simulated the results from two experiments:tossing a coin 10 times and tossing a coin 20 times. Each of theseexperiments was repeated 100 times and the simulated data wereorganized into reference distributions. When you compared yourreference distribution for tossing a coin 10 times with another group’s,you probably found there were differences in the two referencedistributions. Your sample outcomes from 100 experiments differed from the other group’s.

1.Three groups of students gathered data on 100 throws of 10 coins.Figures 3.12–3.14 present their results. Because each group only col-lected 100 samples, and results differ from sample to sample, thetables differed from group to group. Variations in data sets due todifferent samples is called variability due to sampling. In this ques-tion, you will examine this source of variability.

a) Scan Figures 3.12–3.14. What features do each of these referencedistributions have in common? What are some differences?

0 1 2 3 4 5 6 7 8 9 10 Total

0 1 2 9 21 28 16 15 7 1 0 100

Number of headsin 10 tosses

FrequencyFigure 3.12. Group 1’s referencedistribution.

Figure 3.13. Group 2’s referencedistribution.

Figure 3.14. Group 3’s referencedistribution.

0 1 2 3 4 5 6 7 8 9 10 Total

0 1 4 12 13 31 16 16 6 1 0 100

Number of headsin 10 tosses

Frequency

0 1 2 3 4 5 6 7 8 9 10 Total

0 1 9 10 18 23 24 10 6 0 0 100

Number of headsin 10 tosses

Frequency

314 GLOSSARYMathematics: Modeling Our World

UNIT THREE

GlossaryBAILEY’S MODIFICATION:A modification to Peterson’s estimate that

is used when the number of recaptures isless than 10.

BIAS:A slanted point of view. Thus a biasedquestion is one worded in such a way thatit slants the responses toward a particularviewpoint.

BIASED SAMPLE:A sample that over-represents (or under-represents) one or more groups from thepopulation.

CLUSTER SAMPLE:A sampling design in which a simple ran-dom sample of groups is selected and thenall individuals within the selected groupsare surveyed.CONFIDENCE INTERVAL:A confidence interval consists of the per-

centage-Yes populations that contain agiven sample percentage in their likely-sample groups.CONVENIENCE SAMPLE:A sampling design in which you select any

sample that is easy to obtain.CROSS-TABULATION TABLES:Tables that break down responses by twoor more questions (or two or more variables).

DISTRIBUTION:The pattern of values of a numerical variable.

FRAME:A complete listing of the members of thetarget population.FREQUENCY:The number of times an outcome occurs.INFERENCE:Analysis that reveals information about a

population based on information obtainedfrom a sample.

INTERVAL ESTIMATE:An estimate for a population value, report-ed as an interval of reasonable values.LIKELY SAMPLE:The samples (of a particular size) likely to

be drawn from a specified Yes-No popula-tion. For example, the specified popula-tion’s 90% likely-sample group consists ofthe samples (of a particular size) from thatpopulation that are observed at least 90%of the time.

LIKELY-SAMPLE BAR:A bar that represents the likely-samplegroup. For example, a 90% likely-samplebar represents the samples in a Yes-Nopopulation’s 90% likely-sample group.LIKELY-SAMPLE TABLE:A collection of likely-sample bars from a

set of Yes-populations. A 90% likely-sampletable is based on 90% likely-sample bars.MARGIN OF ERROR: One-half the width of a confidence interval

(centered about the sample percentage).MARK-RECAPTURE METHOD:A method for estimating the size of an ani-mal population based on at least two sam-pling episodes. An initial sample of animalsis captured, marked, and then released.Later, a second sample is caught, and thefraction of marked animals recorded.MULTIMODAL HISTOGRAM:A histogram that appears to have more

than one peak.POINT ESTIMATE:A single-number estimate for a population

value.

POPULATION: The entire collection of individuals, ani-mals, or objects about which information isdesired.

34

In a transparent cube, with one face parallel tothe plane of vision, the edges perpendicular tothe plane of vision appear to converge. Forexample, look back at edges a, b, c, and d inFigure 1.45.

A similar effect is visible when you look down the parallel railsof a railroad track. They, too, appear to converge in the distance(see Figure 1.47).

Although the rails of the rail-road track, the sides of a hall-way, and the edges of a cubeare parallel in the real world,they appear to converge in a perspective view. In thisactivity, you will answer thequestion, Why do parallel linesappear to converge? InIndividual Work 4, “AdjustingYour Sights,” you will applythe principle of convergence torepresent depth in perspectivedrawings.

To understand the principle ofconvergence better, it can beuseful to think of the bottomface of a cube as being made of

a series of separate, equally-spaced, identical lines. Railroadcross-ties make a good model. Each tie represents the distancebetween the parallel rails. Similarly, the side faces of a cube canbe thought of as being made up of a series of poles. Each polerepresents the distance between the parallel top and bottomedges of the cube.

MORE THAN MEETS THE EYE

4

LESSON THREE Mathematics: Modeling Our World UNIT ONE

ACTIVITY

Figure 1.47. Railroad tracks illustrate convergence.

Vin

Cat

ania

35

1.Figure 1.48 is a top view of a person observing a railroadtrack and a series of equally-spaced railroad ties. The lines ofsight are drawn from the viewer to the ends of the first fourties.

a) Describe the relationshipbetween the two rails inreal life.

b) The projection of the firsttie in the picture plane isalmost as wide as the pic-ture (see Figure 1.48).Compare the length of theprojection of the fourth tieto the length of the projec-tion of the first tie. Howdo they differ?

c) Describe the projection ofthe railroad tie that is asfar as the eye can see fromthe viewer. (What does itlook like? Where is it?)Why must this be true?

d) Each tie connects the par-allel rails. What does youranswer to (c) tell youabout the images of therails in the picture?

e) What does your investigation of railroad ties tell youabout the appearance of the bottom edges, c and d, of thecube in Figure 1.45?

4ACTIVITY

THE GEOMETRY OF ART Mathematics: Modeling Our World LESSON THREE

MORE THAN MEETS THE EYE

Figure 1.48. Top view of person viewing aseries of equally-spaced rail-road ties.

Individual Work• Varies both in difficulty

and purpose

• Reviews, reinforces,extends, practices, orforeshadows conceptsdeveloped in the lesson.

• Key terms appear inboldface type.

• Provides opportunities forindividuals to workthrough activities at theirown pace

310 UNIT SUMMARY

Mathematics: Modeling Our World

UNIT THREE

Mathematical Summary

Two major uses of sam

pling are explored in this unit: sample

surveys and mark-recapture studies. On

e major goal of this unit is

to give you experience conducting your

own opinion poll. This

means writing the questionnaire, selecti

ng a sampling design,

administering the survey, analyzing the

results, and writing a report.

QUESTIONNAIRES

Questionnaires are intended to gather re

liable information from the

target population. To reach this goal, qu

estions must be worded and

ordered carefully to avoid bias in the re

sults. When critiquing a sur-

vey question, it can be helpful to ask the

following:

• Are the words in the question difficul

t to understand or does the

question contain a negative?

• Is the question ambiguous or vague?

• Does the question suggest how the res

pondent should answer or

try to sway the respondent with emotio

nally-loaded terms?

• Does the question contain a single ide

a?

• Is the question objectionable? too pers

onal? too sensitive?

• Does the question assume too much k

nowledge?

If the answer to any of these questions i

s Yes, then the survey ques-

tion should be rewritten.

SAMPLING DESIGNS

Reputable pollsters take great care in se

lecting the sample to be sur-

veyed. There are many sampling design

s to choose from, but all sam-

pling designs are not created equal. Som

e designs avoid bias by pro-

ducing samples that are likely to be repr

esentative of the population

as a whole. For example, simple random

samples and systematic

samples generally accomplish this goal.

Other designs produce sam-

ples that are likely to over- or under-rep

resent some group in the

population. Self-selecting samples and c

onvenience samples fre-

quently fall into this category.

Wrapping Up Unit Three1.The following article appeared in the Worcester (Massachusetts)

Telegram and Gazette.

a) Comment on the reliability of the results in this article. In particu-lar, what information is contained in this article that would makeyou believe the survey information is reliable? What might leadyou to believe the survey results are not reliable?

b) Only one-sixth of those surveyed would pay more taxes to fund asearch for extraterrestrial life, but 80% would be willing to payhigher taxes to search for cures for AIDS and cancer. Are these dif-ferences in sample percentages statistically significant? Why orwhy not?

SAMPLING

306 Mathematics: Modeling Our World UNIT THREE

UNITSUMMARY

Survey says taxpayers support science-related costs.

CAMBRIDGE—Americans’ enthusiasm for inventions andinventiveness appears alive and well and headed into

outer space.

One-sixth of adults surveyed would pay more taxes to fund a searchfor extraterrestrial life, according to a survey by an MIT-affiliatedgroup that promotes innovation.

Closer to home, more than 80 percent were willing to pay highertaxes to search for cures for AIDS and cancer.

“Americans are still adventurers at heart and understand that scienceis the endless frontier where there are always new explorations to bemade,“ said Lester Thurow, a professor of management at theMassachusetts Institute of Technology and chairman of theLemelson-MIT Awards, which conducted the survey.

UnitSummary•“Wrapping Up the Unit”

reviews concepts andmathematical skillspresented in the unit.

• The “MathematicalSummary” discussesimportant concepts inprose form.

• The “Glossary”contains key terms critical tounderstanding the unit.

T6 Mathematics: Modeling Our World Annotated Teacher’s Edition

Annotated Teacher’s Edition Features

ANNOTATED TEACHER’S EDITION

Unit Overview• Lists major contextual

theme, major mathematicaltheme, and disciplinesrelated to the unit

• Shows where skills andconcepts are taught

• Includes context overview,mathematical development,and ties related disciplinesto lesson content

• Provides brief descriptionsof every lesson withsuggested pacing

Context Overview

T his unit explores the representation of three-dimensional objects on a two-dimensional plane.In order of complexity, and following a modelingtheme, four basic elements of perspective drawing areexamined: overlapping, diminution, convergence, andforeshortening.

The modeling theme begins with the simplest view ofa simple three-dimensional object, the cube. Studentsview the cube in such a way that only one squareface is seen. Depth is achieved through overlappingand varying the size of the square to representvarying distance from the viewer. Convergence to onevanishing point is studied when the cube is madetransparent. Convergence to more than one vanishingpoint is studied when the cube is rotated or theposition of the viewer is changed so that two or threesides of the cube are visible to the viewer.Foreshortening is introduced as the student-artistrepresents objects tilted at an angle away from theplane of vision.

Photographs are used to identify various principles ofperspective. Works of art are examined to illustratehow artists from different cultures have representedperspective throughout the ages.

Mathematical Development

M athematics is used to create precision inperspective drawing and to verify the accuracyof techniques used by artists, draftspersons, andillustrators. This unit continues the study of similarityand proportions studied first in Course 1, Unit 3,Landsat. Ratios and proportions in similar righttriangles are used to introduce the sine, cosine, andtangent trigonometric ratios.

The development of mathematical topics follows themodeling development throughout the unit. Similarityand proportion are reviewed as students learntechniques of scaling and how to representdiminution. Size and shape are altered to representdistance from the viewer or artist.

Properties of parallelograms, rectangles, andtrapezoids, their angles and diagonals, lead toconstructions used by the artist to maintain theproper two-dimensional representation of objects thatare equally spaced in the real world and appear toconverge to a vanishing point in the artist’s view.Students use properties of rectangles and theirdiagonals to replicate a rectangle and assure equal

spacing of identical objects. The construction istranslated to perspective drawing to assure equalspacing of objects in a perspective world subject toprinciples of convergence.

Trigonometry, including sine, cosine, and tangentratios, is introduced when objects are tilted at anangle to the plane of vision. Students usetrigonometric ratios to represent accurately thelengths of objects tilted toward or away from theplane of vision. The concepts of trigonometry are builtupon the ratios of similar right triangles studied in theearlier lessons of the unit.

Related DisciplinesArchitecture

Helping an architect or builder visualize the design ofa project.Lessons 4, 5

Designing a viewing window and seating area for anaquarium.Lesson 6

Animation

Representing a perspective view of a road and signsin a video game.Lessons 2, 3

Illustration and Art

Producing or critiquing perspective representations ofa variety of scenes.Lessons 3-6.

T24 Mathematics: Modeling Our World UNIT ONE: THE GEOMETRY OF ART Annotated Teacher’s Edition T25

LESSON ONE

Keep It In Perspective

1-2 DAYS

The purpose of the first lesson is to introduce thecontext of perspective drawing by challengingstudents to identify the errors in a painting thatviolates principles of perspective drawing. Theprinciple of overlapping is introduced.

LESSON TWO

Drawn to Scale

3-4 DAYS

The purpose of this lesson is to determine the propersize of objects in a perspective drawing. Similarity andproportion are reviewed and applied to the scaling ofobjects to represent distance from the viewer. Theprinciple of diminution is introduced.

LESSON THREE

Vanishing Point

4-5 DAYS

The purpose of this lesson is to introduce the principleof convergence. The behavior of parallel lines in aperspective drawing is studied. Students concludethat parallel lines that increase in distance from theviewer must appear to converge to a point.

LESSON FOUR

The Right Space

4-5 DAYS

The purpose of this lesson is to produce a method fordetermining the correct perspective spacing of objectsthat are equally spaced in the real world. Trapezoidsare seen as a perspective view of rectangles.Properties of the diagonals of a rectangle aretranslated into perspective view as students usegeometric constructions and analytic techniques toassure proper spacing of objects.

LESSON FIVE

The View From the Edge

2-3 DAYS

The purpose of this lesson is to apply the principle ofconvergence to views of rectangular objects thatproduce two or more vanishing points.

LESSON SIX

Foreshortening

4-6 DAYS

The purpose of this lesson is to produce a method foraccurately representing the length of an object tiltedat an angle to the plane of vision. Students areintroduced to sine, cosine, and tangent ratios intrigonometry. Students use analytic methods to verifyresults from a simulation, and sketchpad technologyto confirm a trigonometric formula for determiningthe proper length.

Unit Summary

1-2 DAYS

This summary provides exercises to review conceptstaught in the unit, a written summary of themathematical concepts, and a unit glossary.

Annotated Teacher’s Edition UNIT ONE: THE GEOMETRY OF ART Mathematics: Modeling Our World

Annotated Teacher’s Edition UNIT ONE: THE GEOMETRY OF ART Mathematics: Modeling Our World T23

MAJOR CONTEXTUAL THEME

Perspective Drawing

MAJOR MATHEMATICAL THEME

Size and Shape

RELATED DISCIPLINES

ArchitectureAnimationIllustration and Art

1U N I T

The Geometry of Art

Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5 Lesson 6

Overlapping ●Point of view ●Projection ● ● ● ● ● ●Lines of sight ●Picture plane ●Diminution ● ● ● ●Scale ● ● ● ●Similarity ● ● ● ● ●Proportions ● ● ● ● ●Convergence ● ● ●Parallel lines ● ● ●Vanishing point ● ● ●Horizon ● ● ● ●Constructions ●Depth ● ●Foreshortening ●45˚-45˚-90˚ Right Triangles ●30˚-60˚-90˚ Right Triangles ●Sine, Cosine, Tangent ●

Scope and Sequence Chart

T7Annotated Teacher’s Edition Mathematics: Modeling Our World

554

Throughout your work in theMathematics: Modeling OurWorld program, and perhapseven in other courses, you have

worked with mathematical models

in a variety of contexts. You

probably have developed your

own models using the concepts

and modeling principles that you

learned. In this unit, you will

reexamine the modeling process

with a more critical eye. This time

you will need to focus on the

qualities that distinguish good

models from weak models. In

order to identify components and

characteristics of good models,

you will have a chance to examine

existing models created by other

students with an eye toward

evaluating those models. With

that background, you will be

asked to select a topic of interest

to you, define a question within

that topic, and develop a

mathematical model to answer

that question.

Modeling Your World

LESSON ONEThe Modeling Process

LESSON TWOAnalyzing MathematicalModels

LESSON THREEModeling Our World

LESSON FOURCreating Your Model

Unit Summary

7U N I T

TEACHER PROVIDED MATERIALS

Blank transparencies

Geometric drawing utility software

Graph paper

Internet access (if available)

Overhead pens

Spreadsheet software

Word processing software

MATERIALS PROVIDED

LESSON 1CD-ROM: CARBUY.XLS

Video, Handout H7.1, andVideo Support

Handout H7.2

Transparencies T7.1 and T7.2

Teacher Background Readings7.1-7.2

LESSON 2CD-ROM: VOTING76.XLS,VOTING80.XLS, VOTING84.XLS

Transparency T7.3

Assessment A7.1

LESSON 3CD-ROM: FENCING.XLS andFLU.XLS

Handouts H7.3-H7.6

LESSON 4Handout H7.7

Teacher Background Readings7.3-7.8

554 Mathematics: Modeling Our World UNIT SEVEN: MODELING YOUR WORLD Annotated Teacher’s Edition

VIDEO SUPPORT

See Video Support in theTeacher’s Resources along with

Handout H7.1.

555

CREATING YOUR OWN MODELS

As you know, mathematical modeling is a process used

in dealing with problems in the real world. In many

ways, the modeling process is a formal version of what

we all call thinking. It is a way of looking at situations

that helps answer questions. The idea is to simplify a problem

enough so you can understand it, but not so much that you lose the

essence. In this way a solution may be found, and such solutions will

be reasonable. The process of simplification involves sorting the

factors defining the problem. You may decide not to use some of the

factors, arguing that considering them will not significantly change

the problem. You might also explore the effect of one factor by

observing what happens as it changes and other factors remain

unchanged. Other kinds of explorations are possible. Much as

physical models use smaller or fewer parts or different materials

(wood, plastic, clay, etc.) to imitate characteristics of an original

object, a mathematical model uses mathematical structures (tables,

graphs, equations, algorithms, etc.) to capture the features of a

situation.

Mathematical models are used in all fields of human endeavor. From

planning traffic patterns, to anticipating population changes, to

approximating stock market or weather changes, mathematical

modeling is used to help us predict what might happen or to explain

what has happened. Throughout your modeling experience in the

Mathematics: Modeling Our World program, you have had quite a bit

of guidance. It is now your turn to create mathematical models on

your own. Using the tools you have developed over the years, you

will be asked to model a situation in which you are interested. Along

the way, you will need to refine your understanding of what makes a

good mathematical model.

NASA

555Annotated Teacher’s Edition UNIT SEVEN: MODELING YOUR WORLD Mathematics: Modeling Our World

PREPARATION READING

Diminution

R epresenting three-dimensional objects accurately on atwo-dimensional plane is complicated. The tricky partis how you represent the third dimension, depth. If youfail to represent depth accurately, your drawings may appearchildish.

In the previous lesson, you applied the principle ofoverlapping to show depth in a painting or drawing. Althoughoverlapping is a useful principle, it is not adequate forshowing the relative distances between the viewer and variousobjects in all situations. You need more principles ofperspective to produce accurate drawings.

Apply your experience with modeling and simplify the task ofrepresenting depth. Think about what goes into an image.What kinds of shapes are included? How many objects are

10 Mathematics: Modeling Our World UNIT ONE

LESSON TWO

Drawn to Scale

KEY CONCEPTS

Dimimution

Similarity

Lines of sight

Picture plane

Scale

The Image Bank

In the process of modeling, the cube is first viewedin the simplest way possible—with only one face vis-ible. From this view, the cube appears to be atwo-dimensional square. The artist varies the size ofthe two-dimensional square to show its distancefrom the viewer.

Keep the discussion of modeling decisions, begun inLesson 1, alive throughout the unit so students areconscious of how these decisions affect the ability tomake progress in difficult problems.

LESSON TWO

Drawn to Scale3–4 days

LESSON PURPOSE

To introduce the principle ofdiminution.

To review similarity, ratios, andproportion.

To determine the proper imagesizes for objects in a drawing orpainting.

LESSON STRUCTURE

Preparation ReadingDiminutiondefines and illustrates diminution, line ofsight, and picture plane, and uses asimple object, the cube, to guide themodeling process.Activity 2Beyond Imaginationchallenges students to determine theproper sizes and placement for objectsthat are to be added to a picture.Individual Work 2From a Distanceprovides practice identifying diminutionas applied in pictures.Activity 3Shrink to Fitapplies similar triangles and proportionsto determine the correct image sizes indrawings.Individual Work 3Similar Trianglesreviews the properties of similar trian-gles and uses proportions to determinethe proper sizes for images of objects.

MATERIALS PROVIDED

Handouts H1.2–H1.5Assessment Problem A1.2

TEACHING SUGGESTIONS

Preparation Reading

Diminution

T he preparation readingdefines and illustratesseveral terms usedthroughout the unit: diminu-tion, lines of sight, and plane ofvision (or picture plane).

The mathematics used inapplying the principle ofdiminution is similar to that ofdilation studied first in Course 1,Unit 3, Landsat.

10 Mathematics: Modeling Our World UNIT ONE: THE GEOMETRY OF ART Annotated Teacher’s Edition

Unit Opener Video Support• First lesson in every unit references

optional use of video segment

• Video provides motivational segment tointerest students in the unit content

Teacher Provided Materials• Lists additional materials needed to teach

the unit

Materials Provided• Lists materials included with

Mathematics: Modeling Our World, lessonby lesson

Teaching Suggestions• Provides ideas for presenting the content

of the text, additional backgroundinformation, and suggestions for usingmaterials from the Teacher’s Resources

PreparationReading• Provides suggestions for

introducing the lesson

• May refer to the Teacher’sGuide in the Teacher’sResources for alternateapproaches or additionalideas

Lesson Overview• Lists suggested pacing for

the lesson

• First lesson in every unitexplains Key Concepts andNew Terms

Lesson Purpose• Describes the concepts and

skills developed

Lesson Structure• States purposes of all major

elements of each lesson

Materials List• Informs about all materials

needed for each lesson

T8 Mathematics: Modeling Our World Annotated Teacher’s Edition

Annotated Teacher’s Edition Features

508 LESSON ONE Mathematics: Modeling Our World UNIT SIX

INDIVIDUAL WORK 2

The Cast of Characters

F irst differences are particularly useful in identifying patterns inequally-spaced data. For example, if you suspect quadraticbehavior, then graph Q(n + 1) – Q(n) versus n; that graph should be linear. For exponential and mixed growth, the graph of Q(n + 1) – Q(n) versus Q(n) is linear. If the line goes through (0, 0), then the data are exponential, otherwise it’s mixed.

Use this information about types of growth, and the data in Figure 6.5 of Activity 2 to respond to Items 1–3 below. Notice, too, that the timesconsidered in these items overlap. Your work in Activity 2 may alreadycontain answers to one or more of these items.

1. a) Plot the time-series graph for cumulative computer sales for quar-ters 12 through 18. If they appear linear, find an appropriate equa-tion. If they appear non-linear, check for mixed or quadraticgrowth using the appropriate graph described above.

b) Write an appropriate recursive equation for cumulative computersales during that interval, based on your linear graphs.

c) Put yourself in the position of an employee at the computer firm,and roll the clock back to the 19th quarter. Your boss hands youthe data for quarters 12–18 and asks you to predict cumulativesales for the end of the third year. Discuss how you would makeyour prediction and how accurate you likely would be.(Remember, the later data have not occurred yet.)

2. a) Plot the time-series graph for cumulative computer sales for quarters 6 through 15. It should not appear linear. Verify that aquadratic model is reasonably appropriate by graphing the firstdifferences versus n.

b) Write an appropriate recursive equation for computer sales during this interval, based on your graph in (a). Remember, Q(n + 1) – Q(n) represents the growth in Q, whether Q(n) represents computer sales or something else. Likewise, Q(n + 1) = Q(n) + growth, no matter what the model.

c) Again, put yourself in the place of an employee, this time duringquarter 16. Discuss how you would make your long-range predic-tion, and its likely accuracy.

Individual Work 2

The Cast ofCharactersComputers with spreadsheetsoftware

Supplemental Activity S6.2

Assessment Problems A6.2and A6.3

T he purposes of thisassignment are to provide practice usingthe linearity of particular kindsof graphs to help identify qua-dratic and mixed/exponentialpatterns, and to point out theweaknesses of piecewise-defined prediction models.

Items 1–3 of this assignmentduplicate work students mayhave done in Activity 2. If youchose to use the SupplementalActivity S6.1 instead of Activity 2,there will be an even greateroverlap. Assign these items asstudent needs dictate. Item 6provides a good check of stu-dent understanding of Activity2’s data analysis review inanother practical setting. Items4 and 5 are optional, but pro-vide useful practice.

Items 1–3 focus attention onthe recursive representations oflinear, quadratic, and exponen-tial (or mixed) growth patternsand on the connection betweenrecursive equations and growthgraphs. Use these items to becertain that all students under-stand the concepts and cantranslate those ideas intospreadsheet or calculator mod-els easily.

508 Mathematics: Modeling Our World UNIT SIX: FEEDBACK Annotated Teacher’s Edition

509FEEDBACK Mathematics: Modeling Our World LESSON ONE

3. a) Plot the time-series graph for cumulative computer sales for quar-ters 3 through 13. Again, it should not appear linear. Verify thatan exponential model is reasonably appropriate by graphing thefirst differences versus Q(n).

b) Write an appropriate recursive equation for computer sales during this interval.

c) Put yourself in the position of an employee during quarter 14.Again, discuss your boss’s request for a prediction.

4.Suppose human growth were modeled using a linear or exponentialfunction. A typical height of a newborn baby is 20 inches. In oneyear, the newborn grows to a height of 26 inches.

a) If the growth follows a linear pattern, write a recursive model ofthe person’s height.

b) Use the model to predict the height of the person at age 16. At age50. Are your predictions realistic?

c) If the growth follows an exponential pattern, write a recursivemodel of the person’s height.

d) Predict the height of the person at age 16. At age 50. Are yourpredictions realistic?

e) Use your recursive models from (a) and (c) to draw web diagrams for both types of growth. Carry each diagram farenough to show how the nature of the growth can be seen.

5.A closed-form rough approximation based on Olympic recordsfor the discus throw is d = 1.75t + 175, where d is distance in feetand t represents years since 1948 (so, t = 0 means 1948).

a) Predict what the winning distance should have been in the1996 Summer Olympics.

b) Write a corresponding recursive form, using n as years since1948.

c) Interpret your recursive equation into ordinary words.

6.The data in Figure 6.6 appeared in the form of a graph in an article in Scientific American, December 1982, “PersonalComputers” by Hoo-min D. Toong and Amar Gupta. The articleclaimed that the “sales of personal computers would continue itsexponential growth.” Apply your mathematical skills to deter-mine whether the claim that this represents exponential growth istrue or not.

1977 0.1

1978 0.3

1979 0.65

1980 1.1

1981 2.1

1982 3.0

1983 3.8

1984 4.8

1985 6.7

Year Total sales($ in billions)

Figure 6.6. Sales of Personal Computers.

Items 4 and 5 provide addi-tional practice making linearand exponential predictionmodels. The spreadsheets fromIndividual Work 1 may be usedeffectively here.

Item 4(e) asks for web dia-grams. If students are not famil-iar with these diagrams, assignSupplemental Activity S6.2for a thorough development.Such diagrams will be usedagain in later lessons. For stu-dents who have experiencewith web diagrams, but whoneed practice, use the equa-tions in Item 2 of S6.2 insteadof the entire activity. (Ask forweb diagrams instead of feed-back diagrams).

Item 5: The 1996 winner, LarsRiedel of Germany, had a tossof 227 feet, 8 inches.

Be sure students try recursivegraphs (Q(n + 1) versus Q(n))and graphs of growth (Q(n + 1) – Q(n) versus Q(n) and versus (n) in their decision-making in Item 6.

Assessment Problem A6.2focuses on interpreting graphsand thinking about extrap-olation of models. It may beused before or after IndividualWork 2.

Assessment Problem A6.3involves sketching graphs fromverbal descriptions of a context.A CLD and causal connectionscould also be requested.

509Annotated Teacher’s Edition UNIT SIX: FEEDBACK Mathematics: Modeling Our World

Individual Work• Provides the same teacher

support elements as thosefor Activities

200

In Lesson 1, you began drafting the questionsfor your seven-question survey. This lessonfocuses on interpreting sample results from such surveys. The difficulty in interpreting sample results lies in thefact that samples are seldom perfect miniatures of the populationfrom which they were taken.

CONSIDER:

Imagine that as part of a wolf restoration plan for your state, asurvey was sent to a sample of state residents. Suppose exactly80% of the respondents indicated they supported restoring thewolf population in your state.

1.Do the survey results indicate that exactly 80% of the peoplein your state support wolf restoration? Why or why not?

2.Based on the survey results, could you safely conclude that amajority of state residents support wolf restoration? Why orwhy not?

3.What other information about this survey would be useful inanswering questions 1 and 2?

In the situation described above, the population—the entire collection of individuals, animals, or objects about which infor-mation is desired—consists of the state residents (or more likely,the adults residing in the state). Questionnaires were mailed to aportion of a population, or sample, of the residents. Considerquestion 1 provided information about a sample (80% of thosesurveyed supported wolf restoration) and then asked you tomake an inference about the percentage of supporters in theentire population of residents. An inference is an analysis thatreveals information about a population based on informationobtained from a sample. Most likely, the percentage of supportersin the population is not exactly the same as in the sample.

A 50-50 PROPOSITION

3

LESSON TWO Mathematics: Modeling Our World UNIT THREE

ACTIVITYActivity 3

A 50-50PropositionVariety of materials that can beused as random devices:

CoinsRandom digit tablesCalculators with randomnumber generatorsContainers with two colors ofbeads (50%-50% color mix-ture)Dice (10-sided and/or 6-sided)

Handouts H3.8, H3.9Transparency T3.5

I n order to internalize theconcept of variability due tosampling, students need toobserve such variability. Thepurpose of this activity is to pro-vide that experience as studentsconduct experiments to approxi-mate reference distributions forsize-10 and size-20 samplesdrawn from a 50%-Yes popula-tion. Since simulations requirecareful planning and take timeto execute, this activity willprobably extend past one dayof class.

Students should have access toa variety of random devices tosimulate drawing many samplesfrom a 50%-Yes population. Forexample, students can viewtossing a coin as sampling froma 50%-Yes population by identi-fying the outcome Heads withthe response Yes. HandoutH3.9 is a random-digit table;students might interpret evendigits as Yes.

Begin the activity with a discus-sion of the Consider questions.Focus attention on the lack ofexperience with samples fromknown situations, and suggestthat more experience is what isneeded. Then, assign studentsto small groups (three to fivestudents).

200 Mathematics: Modeling Our World UNIT THREE: SAMPLING Annotated Teacher’s Edition

201

Furthermore, if you surveyed a second sample of residents, thenew sample percentage of wolf-restoration supporters might besomewhat different than 80%. So, what can you say about a population percentage based on a sample percentage?

You encountered a similar situation in the preparation readingwith your hypothetical survey results. What can you say aboutthe percentage of people in the population who would say Yeswhen you know that 16 in a sample of 20 did say Yes? Could theoverall percentage of Yes be only 20%? This last question is actu-ally easier to answer than some of the others posed in the prepa-ration reading. In fact, if you had experience with a real 20%-Yespopulation, you could say quite quickly whether you thoughtgetting a 16-of-20 sample might be likely from such a population.

More generally, what does an 80% sample percentage tell youabout a population percentage? Reference distributions createdby examining samples from various known populations can helpyou answer this question.

1. John tossed a coin 10 times and got 80% heads. A fair cointoss should result in 50% heads in the long run. John’s resultsare not 50% heads, but, of course, 10 tosses is not much of along run, either. This departure from perfect results could beexplained in one of two ways. First, John’s coin tosses are notfair. Alternatively, John’s results may be due to random luck.Do you think John was tossing the coin fairly? Explain.

2.Do you have much experience in watching someone toss acoin fairly 10 times? Do you know how often such a personwill have 8 of 10 tosses turn up heads? Design at least twoexperiments that could provide the experience on which youcould base decisions on how many heads are likely when youtoss a coin fairly 10 times. Include at least one experimentthat does not involve any coins. Be sure to indicate not onlywhat to do, but also what to record, and what everythingmeans. How do you know the coin in your experiment reallyhas a 50% chance of landing heads?

3ACTIVITY

SAMPLING Mathematics: Modeling Our World LESSON TWO

A 50-50 PROPOSITIONYou may decide to give studentstime to think about Items 1and 2 on their own and thendiscuss them briefly as a wholeclass. Answers to Item 1 shouldbe based on intuition, with Item2 aimed at gathering usefulexperience. Groups can thencomplete the remaining itemsindependently. Help studentsfocus on the interpretations oftheir simulation results, not juston doing the simulations.

Distribute Handout H3.8before students begin Items 3and 4. These items demonstratethe effect of sample size. Aftercollecting, organizing, andstudying large numbers of size-10 and size-20 samples from a50%-Yes population, studentsshould be aware that the likeli-hood of getting a particularsample percentage of Yesresponses depends upon thesize of the sample. For example,it is more likely to draw a size-10, 80%-Yes sample from a50%-Yes population than it is todraw a size-20, 80%-Yes sample.

Answers for Item 2 are thebasis for work in Item 3, socheck that students know howto use a variety of the randomdevices available to simulatesamples from 50%-Yes popula-tions. See Simulating Samples inthe Teacher’s Resources for fur-ther discussion. TransparencyT3.5 describes three experi-ments that could be used togenerate the samples.

201Annotated Teacher’s Edition UNIT THREE: SAMPLING Mathematics: Modeling Our World

Mathematics: Modeling Our World Unit 3: SAMPLING TRANSPARENCY

T3.5

Three Experiments

Experiment 1: Each member of a group of 5 students tosses 2 coins and records the number of heads. They repeat thisexperiment 100 times.

Experiment 2: A group uses a random number table. They letodd digits represent heads and even digits, tails. The groupexamines 10 digits and counts the number of even digits. Theyrepeat this experiment 100 times.

Experiment 3: A group writes a calculator or computer pro-gram. The program simulates the outcomes of throwing 10fair coins based on 10 outputs from a random number genera-tor. The random number generator returns numbers between0 and 1. Numbers less than or equal to 0.5 represent heads,and numbers greater than 0.5 tails. They run their program100 times.

Activities• Lists materials needed for the activity

• Provides notes to help guide studentsthrough the activity

• Suggests where to use additionalTeacher’s Resources materials

• Provides background or importantpoints for teachers to consider

• Provides reduced transparencymasters indicating most appropriatelocations for their use

T9Annotated Teacher’s Edition Mathematics: Modeling Our World

169Mathematics: Modeling Our World UNIT SUMMARY

Wrapping Up Unit Two1.When Alan, Beth, and Cathy were told it was their responsibility

to divide up the remaining items and $2,500 in cash from theirgrandparents’ estate, they decided to use the Estate-DivisionAlgorithm from Lesson 1. Their sealed bids are given in Figure 2.24.

Determine a fair distribution of the estate.

2.How does the distribution of items and cash change in Question 1if Cathy decides she wants all of the items and tries to outbideveryone else with bids of $700, $1500, $500, and $1000 for thequilt, dresser, lawn mower, and canoe respectively?

3.When asked about dividing items in an estate settlement, a familylawyer gave this advice: “Each party should prepare a list of itemshe or she wants, compare lists, and determine the items both want.Then a coin should be tossed with the winner taking first pick ofthe items wanted by both. The coin toss loser gets second pick.This process is continued until all items are given away.”

Keeping in mind criteria for judging fairness, is this a fair method?Why or why not?

4.As in many families, the Wallace children receive a weeklyallowance. In 1991, when the three children were ages 5, 7, and 10,their mother apportioned $22 among them according to their ages.

a) If the mother based her apportionment of the allowance onages, and she only had one-dollar bills to give the children,how much would each child receive? Explain your reasoning.

b) In 1993, two years later, the total amount of the allowance hadnot increased. Making the same assumptions as in Item 4(a),use the Hamilton method to reapportion the allowances.

FAIRNESS &APPORTIONMENT

UNIT SUMMARY

Handmade quilt 300 500 200

Antique dresser 800 1200 1000

Riding lawn mower 50 250 490

Canoe 700 500 750

Alan Beth Cathy

Figure 2.24. Estate bids.

Wrapping Up Unit Two1–2 days

MATERIALS PROVIDED

Handout H2.5

Handout H2.5 provides a pro-ject that can be used at the endof the unit.

T his project provides anopportunity for studentsto extend their thinkingbeyond apportioning indivisibleobjects, such as, objects in anestate or seats in a governmentbody. In this activity, studentsexplore dividing objects that aredivisible. Sharing a cake orcandy bar is a famous example.

Once again, ideas of fairness,and criteria for judging it play afundamental part in this explo-ration. Students shouldrecognize that a division amongn people is fair if each individualinvolved feels he or she hasreceived at least 1/nth of theobject.

This activity may be used as anindividual, partner, or small-group project.

169Annotated Teacher’s Edition UNIT TWO: FAIRNESS & APPORTIONMENT Mathematics: Modeling Our World175FAIRNESS & APPORTIONMENT Mathematics: Modeling Our World UNIT SUMMARY

Mathematical Summary

T his unit focuses on fairness. Fairness is an issue whenever anyonewishes to divide objects or groups of objects among severalpeople.Dividing estates among heirs is one such situation. If the estate is beingdivided equally among n heirs, all heirs want at least 1/nth of the estate.In reality, however, individual heirs are often dissatisfied with theirshare of the estate. Heirs often feel that they don’t receive their fairshare. Individuals sometimes value things differently. Items often are notdivisible and can’t be sold. Some people simply do not want particularitems. And the list goes on.

A second fair division situation is the problem of dividing identicalthings that cannot be split up, like computers or seats in Congress.Solving that apportionment problem involves adjusting non-integerquotas (ideal shares) to form integer shares in some fair manner.

One apportionment scheme, known as the Hamilton method, rounds allquotas (ideal shares) down. When the adjusted shares add up to fewerobjects than what is to be allocated, the remaining objects are assigned topeople, schools, or states whose quotas have the largest fractional parts.This method is simple and easy to use, but as you have seen, it is flawed.

As you searched for fairer methods, you discovered a group of methodsknown as divisor methods. All divisor schemes are similar in their initialapproach to the problem. If the goal is to apportion representatives in agovernmental body, begin by dividing the total constituent populationby the total number of seats. This gives the ideal ratio (divisor). Quotasare then calculated by dividing state populations by the ideal ratio. Thequotas are then rounded up or down according to the rounding methodspecified by the particular apportionment method being used. Figure2.31 summarizes standard rounding methods and their associatedapportionment type.

175Annotated Teacher’s Edition UNIT TWO: FAIRNESS & APPORTIONMENT Mathematics: Modeling Our World

Unit SummaryWrapping Up the Unit• Notes suggest ways to review

skills and concepts

Mathematical Summary• Appropriate notes as needed to

guide this review

Glossary

177FAIRNESS & APPORTIONMENT Mathematics: Modeling Our World GLOSSARY

GlossaryACTUAL SHARE:The amount or number of items one gets asa result of an apportionment.

ADAMS METHOD:A divisor method that rounds all fractionsup to the next integer.

ALABAMA PARADOX:An unexpected situation that occurs whenthe size of a legislative body increases andan individual state loses a seat even thoughthere are no population changes in any ofthe states.

APPORTIONMENT:The parceling out of a number of objectsamong several entities.

DISTRICT SIZE (U.S. HOUSE OF REPRESENTATIVES):The average number of people in a statewho are represented by one representativein the House. It is equal to the state popu-lation divided by the number of represen-tatives for that state.

DIVISOR METHOD:A method of apportionment that deter-mines quotas by dividing the populationsby an ideal ratio or an adjusted ratio, thenapplying a specified rounding rule.

ESTATE:Everything of value that a person possessesat the time of his or her death.

ESTATE-DIVISION ALGORITHM:A step-by-step procedure for distributingan estate among several heirs.

GEOMETRIC MEAN:The geometric mean of two positive num-bers is the square root of their product.

HAMILTON METHOD:A method of apportioning identical objectswhen the quotas are not whole numbers.Each person is first given the integer partof his or her quota. The remaining objectsare given (in order) to the persons whosequotas have the largest fractional parts.

HEIR:One who inherits all or part of the estate ofa person who has died.

HILL METHOD: A divisor method of apportionment thatrounds according to the geometric mean. If a quota (q) is between two integers s ands + 1, it is rounded up when q is greaterthan or equal to the geometric mean of sand s + 1 and down when q is less than thegeometric mean.

IDEAL SHARE:The exact amount or number of items aperson should receive as a result of anapportionment.

INDEX:A quantity, computed for each state, thatdetermines a state’s eligibility for the nextseat to be allocated. An index for a statedepends only on that state’s data and themeasure of unfairness being used.

177Annotated Teacher’s Edition UNIT TWO: FAIRNESS & APPORTIONMENT Mathematics: Modeling Our World

T10 Mathematics: Modeling Our World Annotated Teacher’s Edition

T he Teacher’s Resources package isdivided into sections with a contentslist at the beginning of each section.The Teacher’s Resources provide valuable

additional materials to enhance the core

curriculum presented in the Student Edition

and Annotated Teacher’s Edition.

Teacher’s Resources Features

48 VIDEO SUPPORT

UNIT THREE: SAMPLING

Teacher’s Guide

Mathematics: Modeling Our World

Before viewing the video for thi

s unit, give each

student a copy of Handout H3.1, Video

Viewing

Guide, which has questions for students

to answer.

The questions below may be used for dis

cussion after

students have seen the video.

1. What could happen to the bear pop

ulation if it

were mismanaged? What animals do w

ildlife

managers monitor in your state?

You could have an overabundance of b

ear, or worse,

you could wipe out the entire bear po

pulation.

Wildlife managers monitor endangere

d species

(wolf, desert tortoise, manatees) as wel

l as prevalent

species. They also monitor insect popul

ations such as

the mosquito.

2. Why do you think McDonald’s cond

ucts taste

tests before changing the recipe of one o

f its

products or before introducing a new fo

od prod-

uct?

Before McDonald’s changes a recipe, t

he company

wants to make sure a majority of its cu

stomers will

prefer the change. Otherwise, it might l

ose cus-

tomers. Before spending the money to

market a new

product, McDonald’s wants to make su

re a sizeable

portion of its customers will like it.

3. Why do you think McDonald’s feels

it is impor-

tant to rotate both the product being test

ed and

its three-digit code on the survey form?

People may have a preference for a pr

oduct based on

the order in which they taste it. For ex

ample, there

may be a natural tendency to like the

food you tasted

last the best. If McDonald’s didn’t rotat

e the order of

the foods tested, they might conclude

that people

liked the food tasted the best when, in

reality, the

selection was based on the order in wh

ich the foods

were tested and not on actual food pre

ference.

Video Support VIDEO SUPPORT

Funding, and a response of Yes to Q13 belongs in thecategory Funding Cuts. This was necessary in orderto compare the responses to these two survey ques-tions.

Furthermore, students should notice that the two piecharts indicate a conflict in the responses. Theresponses to Q12 indicate a majority of the respon-dents support federal funding of public television.However, the responses to Q13 indicate the opposite,a majority supports cutting federal funding of publictelevision. The reason for the apparent change ofheart of the respondents is most likely due to thenature of Q13. Although respondents support federalfunding for public television, they also support effortsto reduce federal spending. Part of respondents’change in heart is confusion over which item they aresupporting: public television funding or reduction infederal spending. Clearly, the phrase, as part of anoverall effort to reduce federal spending, is added tomanipulate the responses.

INDIVIDUAL WORK 1

Ask So You’ll Get What You WantSee Annotated Teacher’s Edition.

ACTIVITY 2

What Do You Want to Know?A fter groups have shared their seven-questionquestionnaires with the class comes the difficulttask of deciding which seven questions will be on thefinal questionnaire. If two schools are involved, then,the task of arriving at seven questions will be evenmore difficult. In this case, you may decide to expandthe questionnaire to eight questions, four from eachschool. Students from the two schools can communi-cate via e-mail. It may be best to encourage studentsto avoid linked sequences of questions so individualitems may be selected easily.To begin, look for questions that, in essence, appearon questionnaires from two or more groups. Select theremaining questions based on class interest. Once thepreliminary versions of the seven questions have beenselected, students should refine these questions toremove bias, provide information necessary to ensurethe survey participants make informed choices, estab-lish proper ordering, and remove any questions toosensitive to get honest answers. The seven questions

below are preliminary questions posed by students atPattonville High School in Missouri. Each question isfollowed by a critique and, when possible, a suggest-ed revision.1. Should students be required to do 50 hours ofcommunity service for graduation? (Right now,students at Pattonville are required to do 50 community service hours to graduate.)

Critique: In this situation, presenting the back-ground information after the question (and inparentheses) tends to make the respondent feel anegative response is desired. Instead, let thebackground information precede the questionand use neutral language. Suggested revision: Students at Pattonville arerequired to do 50 community service hours. Doyou agree or disagree that this is a reasonablerequirement? Agree or Disagree.

2. Pattonville High School sells candy during lunchperiods, yet, they put a ban on soda because it isunhealthy. Should they turn on the sodamachines during lunch periods?Critique: This is a leading question. The respon-dents know exactly what response is desired.Provide the necessary background in a neutralmanner.

Suggested revision: During lunch periods atPattonville High School, students can buy hotand cold lunches, milk, juice, desserts, and candy.At present, students cannot buy soda during thistime. Do you think that soda should be sold dur-ing lunch? Yes or No.3. Should parents’ opinions influence the decisionof the principals?

Critique: This question is vague. The respondenthas no idea what kinds of decisions are intended.These decisions could be teacher assignments,disciplinary issues, or school policies. Studentsneed to define more narrowly and clearly what ismeant by the word, decisions.Suggested revision: Should parents’ opinionsinfluence the decisions of the principals in disci-plinary actions involving students?

4. Do you believe equity exists between men andwomen at Pattonville High School?Critique: Although students could leave thisquestion as is, it is very broad. Students mayobtain more useful information if they make the

50 LESSON ONE

UNIT THREE: SAMPLING

Teacher’s Guide

Mathematics: Modeling Our World

49UNIT THREE: SAMPLING LESSON ONE

Mathematics: Modeling Our World Teacher’s GuideTeacher’s Guide

PREPARATION READING

Crying Wolf

See Annotated Teacher’s Edition.

ACTIVITY 1

Questionable QuestioningHarris Poll

Q 1–Q3 of the survey appeared as questions 11,12, and 13 on Harris Survey 951302, an actualsurvey conducted by Harris in February, 1995.Information about this poll was retrieved from theInstitute for Research in Social Science (IRSS) PublicOpinion Poll Question Database(http://www.irss.unc.edu:80/data_archive/) at theUniversity of North Carolina at Chapel Hill. In fact,this is a good place for students to search for surveyquestions on various topics.

The original survey was administered to a nationalsample of persons 18 years or older. The results pre-sented in Figure 1 are reproduced as TransparencyT3.1.

Notice that 35.7% of the participants in this surveyresponded that the government should be doing moreto protect and restore endangered species such assalmon and wolves. Only 12% said they were verywilling to pay higher taxes to support such programs.It is interesting to note that a higher percentage of therespondents would be willing to pay for endangered-species-protection efforts with higher electricity billsor food bills than with higher tax bills.

It would have been very interesting to see howrespondents would have answered these questionshad Q1 and Q3 been interchanged.

Comparative Pie Charts

When constructing pie charts, keep in mind that theangles of the pie wedges must be proportional to thepercentages the wedges represent. You can find theangle of a wedge by multiplying 360° by the decimalequivalent of the associated percentage. The anglesfor the pie chart wedges corresponding to Q12 areshown in Figure 2.

Figure 2. Pie chart construction.

Notice that for the comparative pie charts onTransparency T3.2 the original Yes and No responsesto Q12 and Q13 have been transformed into the cate-gories Funding and Funding Cuts. This means that aresponse of Yes to Q12 belongs in the category

LESSON ONE

It’s All in the Question

Responses Doing more Very willing Very willing(35.7%) (12%) (17.2%)

Doing less Somewhat Somewhatwilling willing

(49.7%) (28.1%) (33.6%)

Doing about Not very Not verythe same willing willing

Not sure Not willing Not willingat all at all

(2.4%) (38.10%) (30.9%)

Not sure Not sure(1.20%) (0.5%)

Number of751 751 751valid cases

Questions Q1 Q2 Q3

Figure 1. Harris survey results.

Responses Pie chart Percentage Angle ofcategories wedges

Yes Support 54% (0.54)(360°)≈194°funding

No Support 40% (0.40)(360°)≈100°fundingcuts

No opinion No opinion 6% (0.06)(360°)≈22°Teacher’s GuideVideo Support• Contains discussion questions for use

after viewing the video

Lesson-by-lesson information• Provides additional information to

supplement teaching suggestionscovered in the Annotated Teacher’sEdition

• Includes background readings forteachers containing content and softwareinformation not found in othercomponents of the program

• Presents suggestions for using handouts,supplemental activities, assessmentproblems, and transparencies

• Suggests alternate approaches to studentActivities

SURVEY: PEOPLE NEED NEWS BUT MANY DON’T TRUST THE MEDIAWASHINGTON—Americans have a growing need for news and most find ituseful in making everyday decisions, but many fewer trust the media thatdeliver it, a survey suggests.

In a poll sponsored by The Freedom Forum, 70% said they found news helpfulwhen making practical decisions. About 88% said their need for news stayedconstant or grew in the past year.

However, only about half of those surveyed said they trust what the news mediatell them.. About 53% said they believe their local television anchors and 45%said they trusted network anchors. Newspaper reporters had the trust of 31%and radio talk-show hosts 14%.

Few surveyed felt that freedom of the press should be unconditionallyguaranteed to reporters. About two-thirds, or 65%, said there are times when thenews media should be restricted from publishing or broadcasting stories.

“The survey shows that there is a gulf between the media and the public,” saidPeter S. Prichard, executive director of the Newseum, a news museum fundedby The Freedom Forum. “If journalists redouble their efforts to be fair andaccurate, it is likely that they will have more credibility with the public.

Other survey findings:

• Few respondents could name all five rights guaranteed by the FirstAmendment to the Constitution. Nearly 30% fai