Mathematics: Modeling Our World · Landy Godbold, Bruce Grip EVALUATION Barbara Flagg MULTIMEDIA...

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

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Solomon GarfunkelCOMAP, INC., BEDFORD, MA

Landy GodboldTHE WESTMINSTER SCHOOLS, ATLANTA, GA

Henry PollakTEACHERS COLLEGE, COLUMBIA UNIVERSITY, NY, NY

Mathematics: Modeling Our World

© Copyright 1998

by COMAP, Inc.

The Consortium for Mathematics and Its Applications (COMAP)

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Bedford, MA 01730

Published and distributed by

The Consortium for Mathematics and Its Applications (COMAP)

Bedford, MA 01730

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

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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 0-538-68219-1

Printed in the United States of America.

1 2 3 4 5 6 7 8 VH 02 01 00 99 98

ii Mathematics: Modeling Our World Annotated Teacher’s Edition

®

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


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


ASSESSMENT

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

REVISION TEAM

Marsha Davis, Gary Froelich, Landy Godbold, Bruce Grip

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,Amy Novit, Daiva Kiliulis, David Barber, Gail Wessell, Gary Feldman, Clarice Callahan,Brenda McDonald, George Jones, Rafael Aragón, Peter Bousquet, Linda Vahey

INDEX EDITOR

Seth MaislinFOCUS PUBLISHING SERVICES, WATERTOWN, MA

iv Mathematics: Modeling Our World Annotated Teacher’s Edition

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

vAnnotated Teacher’s Edition Mathematics: Modeling Our World

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 PollakCO-PRINCIPAL INVEST

IGATOR

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 2 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 Gridville T23

Overview for Unit 2 Strategies T26

Overview for Unit 3 Hidden Connections T31

Overview for Unit 4 The Right Stuff T35

Overview for Unit 5 Proximity T39

Overview for Unit 6 Growth T43

Overview for Unit 7 Motion T46

vi Mathematics: Modeling Our World Annotated Teacher’s Edition

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 four such programsselected 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


190

Often in your studyof mathematicsyou have seenthat the way a problem is

represented can be

important to the solution

of the problem. For

example, when you

analyze a relationship that

you have described with a

mathematical function, a

graph of the function

helps you visualize the

relationship and answer

questions about it.

However, it isn’t always

possible to describe a

relationship with a

function. Mathematicians

sometimes use another

type of graph to visualize

relationships among

objects. In this unit, you

will apply this new type of

graph to a variety of real-

world problems.

HiddenConnections

LESSON ONEConnections

LESSON TWOProcedures

LESSON THREEMinimum Spanning TreeAlgorithms

LESSON FOURColoring to AvoidConflicts

LESSON FIVETraveling SalespersonProblems

LESSON SIXMatching

Unit Summary

3U N I T

The Image Bank

PREPARATION READING

Testing Strategies

The optimal strategies in a game can be found by testingvarious strategies repeatedly. In Lesson 3, you built alarge table of results by trying the strategies 0, 0.2, 0.4,0.6, 0.8, and 1 for both players. It is a fairly rough table: youdid not test common strategies such as 0.5 and 0.25. Even withthe aid of technology, building this rough table can take quite abit of time, particularly if you use calculators rather thancomputers. Simulating a thousand games with a particularstrategy may seem like a lot, but the expected payoffs youcalculated from your table would be more reliable if you haddone many more than a thousand trials of each game.

How can you find optimal strategies more easily and moreaccurately? You can learn much by observing the effects on thegame when you hold one strategy constant and allow theopponent’s strategy to vary. Scanning your tables of dataprovides some good information, but how can you “see” thepatterns even better?

148 Mathematics: Modeling Our World UNIT TWO

LESSON FOUR

OptimalStrategies

KEY CONCEPTS

Strictly-determined game

Dominant strategy

Linear equations

Strategy lines

Solving systems of equations

The Image Bank

191

MAKING CONNECTIONS

Many real-world problems involveobjects that are related or connected insome way. For example, cities areconnected by airline flights andhighways; homes and businesses are connected by phone

lines. People are connected because they are relatives, but

also because they work for the same employer or live in the

same neighborhood. Many problems that are suitable for

mathematical analysis arise in these situations: a person who

is traveling to several cities wants to use the flights that have

the lowest total cost; the phone company wants to connect

homes and businesses in the most efficient way; a company

wants to schedule its employees in a way that makes the best

use of employee skills and company facilities.

In this unit you will learn how objects and the connections

among them can be represented in a simple but helpful way,

and you will develop problem-solving procedures that can be

applied to these representations.

UnitOpener• Sets tone for

unit

• Piques studentinterest

LessonOpener• Lists key

concepts onwhich theunit is based

PreparationReading• Provides

background aboutthe main focus ofthe lesson

Student Edition Features

573

In Item 2 of Individual Work 8, you found that akicked soccer ball went higher with air resistancethan without it. That can’t be correct in reality!

CONSIDER:

1.Do you think that air resistance is more of a factor with alight plastic ball or a heavier basketball? How does air resistance affect the acceleration of a ball as it travels upward?How does it affect the ball as it falls? Explain.

2.Does your answer to the previous item suggest a refinementto your models of Activity 9 and Individual Work 7, particu-larly if air resistance is not negligible? How might you goabout using your data to find a better model?

When modeling, it is important to assess which factors are mostinfluential. What can you ignore and what must you include inyour model? For example, you know that real stunts take place inair, so there must be some air resistance. But does it matter? Howmuch does air resistance affect a ball’s motion when it is tossedupward and allowed to fall?

Depending on the kinds of balls that were used in Activity 9, youmay already have the data that you need to determine the effectof air resistance. If a variety of balls were used in that experi-ment, move directly to Item 2. Otherwise you will need to gathermore data.

1.Obtain a ball that is similar in size but significantly differentin weight than the one used in Activity 9. Use the sameequipment set-up and procedures as you used in Activity 9 toobtain data from the toss of this new ball.

11

ACTIVITY

MOTION Mathematics: Modeling Our World LESSON FOUR

ASSESSING THE MODELConsider• Raises issues

related toideaspresented inthe lessons

• Encouragesstudents toask questionsthroughoutthe unit


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.

44 LESSON TWO Mathematics: Modeling Our World UNIT ONE

17.The context is a linear village and the graphs below (see Figure 1.51)represent information about distance between locations x in linearvillage and two Automated Teller Machines (ATMs).

a) Explain the meaning of Graph A.

b) Explain the meaning of Graph B.

c) Explain the meaning of the intersection of Graphs A and B.

d) A person who services or monitors the two ATM machines mightinvestigate total distance. Sketch a new graph showing the sum ofthe distances, along with Graph A and Graph B.

18.The graph of y = |x + 2| + |x – 1| + |x – 3| is shown in Figure 1.52.

INDIVIDUAL WORK 3

–4 –2 0 2 4 6 8 10 12 14 16

2

4

6

8

Dis

tanc

e

Location of ATM machine

Graph A Graph B

Graph A

Graph B

Figure 1.51. Distance to ATM machines.

5

10

15

20

–4 –3 –2 –1 0 1 2 3 4 5 6 7 8Figure 1.52. Graph for total distance to three houses.

381

PROXIMITYMathematics: Modeling Our World

GLOSSARYGlossaryACUTE TRIANGLE: A triangle in which all of the angles mea-

sure less than 90˚, but more than 0˚.CENTER OF INFLUENCE: A point used to establish boundaries ofregions of influence. All points in a regionare closer to that region’s center than toany other region’s center.COLLINEAR POINTS: Points that lie on the same line.CONCAVE POLYGON: A polygon in which some of its sides, when

extended, intersect other sides. CONVEX POLYGON: A polygon in which none of its sides, whenextended, intersect other sides. For everypair of points in the interior of a convexpolygon, the segment connecting the pointsis completely in the interior.DOMAIN: A region in which centers of influence are

located. The domain is the area that isbeing divided into regions of influence.HERON’S FORMULA: The area of a triangle is

where a, b, and care the lengths of the triangle’s sides and sis half the triangle’s perimeter.ITERATION (ITERATIVE PROCEDURE):a procedure that repeats the same sequenceof steps over and over. Each cycle is con-sidered one iteration.

MIDPOINT: A point that is halfway along a segment(equidistant from the segment’s two end-points). In coordinate geometry, the coordi-nates of a midpoint are found by averagingthe coordinates of the two endpoints.OBTUSE TRIANGLE: A triangle with one angle that measures

more than 90˚, but less than 180˚.PERPENDICULAR BISECTOR: A line that passes through the midpoint ofa given line segment and forms rightangles with it.PICK’S FORMULA: If the vertices of a polygon are points of a

grid, then the area of the polygon is 0.5b + i – 1, where b is the number of gridpoints on the polygon’s border, and i is thenumber of points in its interior.REGION OF INFLUENCE: A region in which each point is closer to

the region’s center of influence than to anyother center of influence.VORONOI BOUNDARY: A boundary between two centers of

influence.

VORONOI CENTER: A center of influence.VORONOI DIAGRAM: A diagram composed of several centers of

influence and their regions of influence.VORONOI REGION: A region of influence.VORONOI VERTEX: A point at which Voronoi boundaries

intersect.

WEIGHTED AVERAGE:The average found by multiplying eachcategory by the decimal weight attached tothat category and finding a total.

s(s − a)(s − b)(s − c) ,

296

Much of the geometry used in package designinvolves right angles. At first glance, some noveldesigns such as the triangular package in

Figure 4.50 appear to involve no right angles.However, it is not obvious how to find the pack-age’s area because the only known length in thefigure is the radius of each can. When you draw aradius and another segment to form a triangle,the triangle appears to be a right triangle. To findthe area of this small triangle you must be certainthat it is a right triangle. Once you know that it isa right triangle, you must find the length of onemore side before you can calculate its area.

Because right angles are so important in packagedesign, much of the geometry used in packagedesign could be called “the right stuff.” In thisactivity you will consider an important conclu-sion about right angles and an important conclu-sion about the sides of right triangles.

To analyze the design in Figure 4.50, you must beable to conclude that the small triangle is a righttriangle. This conclusion involves the relationshipbetween a tangent to a circle and a radius of thecircle (Figure 4.51).

Answer the following questions to prove that theradius and tangent form a right angle.

1. In Figure 4.52, the radius in Figure 4.51 hasbeen extended across the circle. The tangenthas been tilted so that it intersects the circlein a second point A and a radius has beendrawn to A. (Now the tangent is no longer atangent.) What kind of triangle is formed?

THE RIGHT GEOMETRIC STUFF

7

LESSON FOUR Mathematics: Modeling Our World UNIT FOUR

ACTIVITY

Figure 4.50.

Figure 4.51. A circle, a tangent, and a radius.

A

1

2

Figure 4.52.

297

2.How is ∠2 related to ∠1? (Hint: review Item 9 of IndividualWork 6.)

3.Suppose point A is closer to the original point of tangency(Figure 4.53). Do either of the answers you gave to Items 1or 2 change?

4.What happens to ∠1 as point A moves closer to the point oftangency? What can you conclude about ∠2?

Now that you have established that the angle formed by a radiusand tangent is a right angle, you know that the small triangle inFigure 4.50 is indeed a right triangle. Based on your work inActivity 6 and Individual Work 6, you can conclude that the sumof the measures of the other two angles of the triangle is 90°. Youalso know the length of one of the triangle’s sides, but all thesethings are not enough to determine the triangle’s area. You needto understand how the sides of a right triangle are related, whichyou will do in this activity. You also need to understand how therelationship applies in this particular situation; Individual Work 7explains that.

Among all the facts in geometry, perhaps in all of mathematics,none are as important as the Pythagorean formula (also called thePythagorean Theorem), which says that in a right triangle, thesquare of the hypotenuse is equal to the sum of the squares of theother two sides. (Note: the hypotenuse is the longest side, whichis also across from the right, or 90°, angle.)

It is easier to state the formula whenreferring to a figure. The triangle inFigure 4.54 is a right triangle. (Note thatin a right triangle a small square is some-times used to mark the right angle.) Thelower-case letters a, b, and c represent themeasures of its sides. The Pythagoreanformula says that as long as the triangleis a right triangle, c2 = a2 + b2.

7ACTIVITY

THE RIGHT STUFF Mathematics: Modeling Our World LESSON FOUR

THE RIGHT GEOMETRIC STUFF

A

1

2

Figure 4.53.

The Pythagoreanformula is namedfor Pythagoras, aGreek mathematicianborn on the island ofSamos around 540 B.C.Pythagoras settled inCrotona, an Italianseaport town, where heestablished thePythagorean school.Members of the schoolstudied mathematicsand applied it toscience, philosophy,music, and other topics.

b

a

c

Figure 4.54. A right triangle.

Individual Work• Vary both in difficulty and

purpose

• Items review, reinforce,extend, practice, andforeshadow conceptsdeveloped in the lesson.

• Key terms appear inboldface type.

• Provides opportunities forindividual students toprocess the results ofactivities at their own pace

379

PROXIMITYMathematics: Modeli

ng Our WorldUNIT SUMMARY

Mathematical Summary

The modeling problem

from which the mathematics in this un

it

arises is that of estimating the rainfall for

the entire state of

Colorado from readings taken at eight r

ain gauges scattered

around the state.

The solution is geometric: divide the stat

e into eight regions so that the

points in a region are closer to its gauge

than to any other gauge. Weight

the rainfall measured at each region’s ga

uge according to the portion of

the state’s area in that region. The weigh

ted average of the rainfall at the

eight gauges estimates the rainfall for th

e state.

Proximity problems like the Colorado ra

in gauge problem involve

Voronoi diagrams, which are named afte

r the mathematician Georgii

Voronoi. To use Voronoi diagrams, you m

ust determine the boundaries

of regions from their centers of influenc

e.

The boundaries can be drawn roughly b

y hand, but answers obtained

from rough drawings lack precision. The

refore, the boundaries should be

constructed. There are several means of

constructing the boundaries.

Every Voronoi boundary is the perpend

icular bisector of a segment

joining two centers. Perpendicular bisec

tors can be constructed by

several methods: 1) by folding a piece o

f paper and creasing it so that

two centers coincide; 2) by placing a Ple

xiglas® mirror so that the

reflection of one center coincides with th

e other center; 3) by striking

intersecting compass arcs from the cente

rs and joining the two points of

intersection; 4) or by using the segment,

midpoint, and perpendicular

construction features of a drawing utilit

y.

Perpendicular bisectors are lines, but Vo

ronoi boundaries are either rays

or line segments. Moreover, the perpend

icular bisector for some pairs of

centers is not a boundary in the Voronoi

diagram. Therefore, when

perpendicular bisectors are constructed

, they must be analyzed carefully

to determine which portions to keep. Al

gorithms for establishing

Voronoi boundaries often divide the pro

blem into several smaller

problems of, say, three or four vertices, t

hen combine the diagrams that

result.

Voronoi regions are usually polygons. (A

n exception occurs when the

boundary of the domain is curved.) Many

modeling problems, including

the Colorado rain gauge problem, requir

e determination of each regions’

area. One way to find a region’s area is t

o divide it into triangles and

apply Heron’s formula, which finds the

area of a triangle from the

lengths of its sides. Another method is t

o apply Pick’s formula, which

373

PROXIMITY

Mathematics: Modeling Our World UNIT SUMMARY

UNITSUMMARY

Wrapping Up Unit Five1.A person studying Voronoi diagrams has developed the following

algorithm for situations with four centers.

Step 1.Arbitrarily select one of the four centers and label it A. Movearound the centers in either clockwise or counterclockwise fash-ion. As you go, label the remaining points B, C, and D.

Step 2. Construct the perpendicular bisector of each adjacent pair of centers.

Step 3. Determine the portions of each perpendicular bisector to keep.

A sample implementation of the first two steps of the algorithm isshown in Figure 5.71.

Is this a good algorithm for four-center situations? Explain.

A

BC

D

Figure 5.71.

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.


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

G ridville, a city laid out in a rectangular grid ofstreets, needs a fire station. Students arechallenged to find the optimum placement for this firestation. Throughout the unit, the role of mathematicsand the role of community values are consideredtogether in the search for the best location. Lessonsfollow the modeling process of simplifying theproblem to study the essential conditions andmathematics in detail. Finally, students return to theoriginal problem and use their new mathematicalunderstanding to provide optimal solutions.

Mathematical Development

I n order to attack the two-dimensional problem,students first consider a simplified model, a one-dimensional village with houses spread along onestreet. Despite its simplicity, the linear model gives risein a natural way to important mathematical ideas ofabsolute value, average, median, and midrange.

Students explore the problem numerically, graphically,symbolically, and logically, first seeking locations forwhich total distance to all houses is minimized.Absolute-value notation is introduced as a symbolicway to represent distances between locations in onedimension. Students determine that a median locationmust minimize the total distance and averagedistance.

Since the use of the absolute-value function torepresent distance is essential throughout the unit,the absolute-value function is studied in detail.Students learn to recognize the graph of the basicfunction y = |x| and of simple transformations of thatfunction, including y = a|x – h| + k.

Because the graphs of the absolute-value functionand of sums of absolute-value functions resemblecombinations of linear functions, students areintroduced to piecewise descriptions for functions.Students develop equations to match each portion ofa graph and identify, using inequalities, the intervalfor which each equation is valid.

Still within the context of a linear village, students findlocations minimizing the maximum distance a firetruckmust travel to reach all the houses, noting that themidrange determines minimax locations in onedimension.

With two models and two examples of optimization,students consider the issue of fairness. Which locationis “best” becomes “best for whom?” or “best underwhat circumstances?”

Students return to the two-dimensional context ofGridville and apply what they learned in the linearvillage. “Firetruck” geometry, introduced in the firstlesson, is further developed as students exploredistance in two dimensions. Students discover thatthe method to minimizing the total distance in twodimensions is to find the solutions to two one-dimensional problems. The search for minimax in twodimensions leads to methods for finding the centersof the smallest circles that contain all the houses inGridville.

Students are now prepared better to address the keyquestion, “What is the best location for the fire stationin Gridville?” Given a map of Gridville, students canfind locations that minimize total distance andlocations that minimize maximum distance. Findingthe best location involves establishing criteria whichdepend on community values. Once the criteria havebeen determined, students may use mathematics tohelp find and evaluate locations based on the criteria.

Related Disciplines

T he real-world contexts of this unit provide manyopportunities for students to apply what they arelearning to other subject areas and to applyknowledge from other areas to the development ofnew mathematical ideas.

Business and Finance

Optimizing routes for delivery services.Lesson 1

Chemistry

Using absolute value to represent fluid andtemperature levels.Lesson 3

English

Preparing, writing, and presenting reports.Lessons 1, 2, 3, 4, 5

Environmental Science

Determining the best location for a recycling center.Lesson 1

Urban Planning

Determining the best location for a fire station.Lessons 1 and 2

Locating a fire station or regional mall.Lesson 5

T24 Mathematics: Modeling Our World UNIT ONE: GRIDVILLE Annotated Teacher’s Edition T25

LESSON ONE

In Case of Fire2–3 DAYS

This unit explores the problem of optimizing theposition of an object on a grid. The first lessonestablishes the context for the unit, poses the keyquestion, and sets the stage for the modeling process.Gridville, a town whose streets are laid out in a grid-like manner, needs to determine where to build a firestation. A group activity challenges students todetermine the best location for the fire station.

LESSON TWO

Linear Village4–6 DAYS

The purposes of this lesson is to begin the modelingprocess and to introduce multiple representations tosolving the location problem. Students begin themodeling process by simplifying the two-dimensionalGridville model to the one-dimensional Linear Village.The first activity challenges students to use tables andgraphs to observe patterns and develop a procedurefor finding locations which minimize the total distancea firetruck would travel to reach each house in thecommunity. The second activity introduces students toabsolute-value notation as a symbolic method forexploring the location problem in one dimension.

LESSON THREE

Absolute Value2–3 DAYS

The purpose of this lesson is to study the absolute-value function in detail. In the context of a game,students transform absolute-value function graphs bychanging the control numbers a, h, k in the generalequation y = a|x – h| + k. Piecewise descriptions areused to specify the linear components of absolute-value graphs.

LESSON FOUR

Minimax Village2–3 DAYS

While still in the context of Linear Village, students areintroduced to another criterion for optimization. Thepurpose of this lesson is to develop procedures forfinding locations that minimize the maximum distancea firetruck must travel to reach any house in thecommunity. The issue of fairness is raised as thecompeting criteria lead to different choices for bestlocation.

LESSON FIVE

Return to Gridville3–5 DAYS

In search of the best location to build the fire stationin Gridville, students investigate finding both themedian and the minimax location in two dimensions.The purposes of this lesson are to apply the conceptslearned in the one-dimensional Linear Village to thetwo-dimensional Gridville and to develop proceduresfor finding locations easily that optimize distance.

Unit Summary1–2 DAYS

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

Annotated Teacher’s Edition UNIT ONE: GRIDVILLE Mathematics: Modeling Our World

Annotated Teacher’s Edition UNIT ONE: GRIDVILLE Mathematics: Modeling Our World T23

MAJOR CONTEXTUAL THEMES

GovernmentPublic PolicyMAJOR MATHEMATICAL THEMES

OptimizationDistance

RELATED DISCIPLINES

Business and FinanceChemistryEnglishEnvironmental ScienceUrban Planning

1U N I T

Gridville

Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5

Distance in firetruck geometry ● ●

Circles in firetruck geometry ● ●

Total and average distance in two dimensions ● ●

Total and average distance in one dimension ●

Addition of functions ●

Slope and rate of change ●

Absolute-value equations and inequalities ● ●

Absolute-value graphs ● ● ●

Piecewise-defined functions ● ● ●

Median ● ●

Transformation of functions ●

Maximum distance in one dimension ●

Midrange ● ●

Maximum distance in two dimensions ●

Scope and Sequence Chart


254

Packages aregeometric: softdrinks, vegetables,and soups are packaged

in metal cylinders; boxes

that contain everything

from electronic equipment

to shoes are rectangular.

Sometimes packages that

are geometric shapes

contain packages that are

other geometric shapes.

For example, the cylinders

that contain soft drinks

are often sold in

rectangular boxes that

contain six or more of the

cylinders.

The design of efficient

packages requires a

knowledge of geometry.

Since packages are three-

dimensional objects with

two-dimensional sides,

volume and area play

important roles in the

geometry of packaging. In

this unit you will consider

how geometry can be

used to create a definition

of efficient packaging and

how a knowledge of

geometry can be used to

improve package design.

The Right Stuff

LESSON ONEPackaging Models

LESSON TWODesigning a Package

LESSON THREETechnological Solutions

LESSON FOURGetting the Facts

LESSON FIVEPackaging Spheres

UNIT SUMMARY

4U N I T

LESSON 4CD-ROM:SIMDEMO.GSP (similarity demo)

RDTNDEMO.GSP (radius & tangent demo)PYTHDEMO.GSP (Pythagoras demonstrator)TRIAN6PK.GSP (triangular 6-pack)MODTR6PK.GSP (mod triangular 6-pack)HEX7PK.GSP (hexagonal 7-pack)

TRIANGCS.GSP (triangular case)MODTRCS.GSP (mod triangular case)

Transparencies T4.11–T4.16

Supplemental Activities S4.1 and S4.2

Assessment Problems A4.8–A4.13

TEACHER PROVIDED MATERIALS

Geometric drawing utility (e.g.,Geometer’s Sketchpad, Cabri)

Card stock or heavy paper

Coins: pennies, nickels, dimes,and quarters

Graph paper or dot paper

Plexiglas® mirrors

Rulers

Scissors

Six soda cans

Small blocks such as patternblocks (or copies of HandoutH4.2 printed on card stock)

Tape

MATERIALS PROVIDED

LESSON 1Video, Handout H4.1, andVideo Support

Handout H4.2

Assessment Problem A4.1

LESSON 2Handouts H4.3 and H4.4


Assessment ProblemsA4.2–A4.7

LESSON 3CD-ROM:RECT6PK.GSP (rectangular

6-pack)SQUAR9PK.GSP (square 9-pack)TRIAN3PK.GSP (triangular 3-pack)CYL7PK.GSP (cylindrical 7-pack)HEX7PK.GSP (hexagonal 7-pack)

Handout H4.5


254 Mathematics: Modeling Our World UNIT FOUR: THE RIGHT STUFF Annotated Teacher’s Edition

VIDEO SUPPORT

See Video Support in the Teacher’sResources along with

Handout H4.1.

ROMCD

ROMCD

ROMCD

255

IT’S A PACKAGE DEAL

It is difficult to imagine life without packaging. Youencounter packaging every day of your life. Indeed,packaging seems more a necessity than a convenience.The packages that contain the food you eat keep thefood from spoiling and protect it from insects, thereby

reducing the risk of certain kinds of diseases. Packaging,

however, can create problems. It accounts for about 30% of

the material in U. S. landfills, many of which are

overburdened. By contributing to the volume of the goods it

contains, packaging puts additional demands on space in

warehouses, in delivery vehicles, and on store shelves. To

minimize the problems created by packaging, packages must

be well designed.

The Image Bank

LESSON 5CD-ROM:MLN4PKR.GSP (four melon packer)

MLN5PKR.GSP (five melon packer)MLN6PKR.GSP (six melon packer)MLN8PKR.GSP (eight melon packer)LSQDEMO.GSP (least squares demo)LSQDEMSQ.GSP (least squares demo with squares)

Supplemental Activities S4.3 and S4.4

Assessment Problem A4.14

255Annotated Teacher’s Edition UNIT FOUR: THE RIGHT STUFF Mathematics: Modeling Our World

ROMCD

PREPARATION READING

When One Player’s Loss is Not the Other Player’s Gain

Games of strategy are very common in everyday life.Some of these games are strictly determined becausethere is really only one option that makes sense. Inothers, however, the best way to play is to use a strategy thatmixes plays in a random way. Sometimes a 50-50 mix is best,but in other cases it is best to play one option more often thananother. If you know the optimal mix, you can prevent youropponent from capitalizing on your strategy.

You have spent a lot of time in this unit learning to find theoptimal strategies in a wide variety of situations. But all thesesituations have one thing in common—one player’s gain is theother’s loss. In other words, these situations are zero sum orconstant sum.

Many real-world situations, however, are non-zero sum; thesum of the corresponding payoffs is not constant. Often theplayers have something to gain by cooperating with eachother; the interests of the players are not completely opposed.How do you tell if the interests of players are not completelyopposed? What strategies are best in games in which interestsare not completely opposed?

Mathematics: Modeling Our World LESSON SIX

LESSON SIX

Games That AreNot Zero

Sum

KEY CONCEPTS

Games that are neither zero sum nor constant sum

Dilemmas

Cooperation and defection

Tit-for-tat strategy

177

The Image Bank

LESSON SIX

Games That AreNot Zero Sum3–4 days

LESSON PURPOSE

To consider games that are neither zerosum nor constant sum.

To see that non-zero-sum games have avariety of structures.

To study games that pose dilemmas.

LESSON STRUCTURE

Preparation ReadingWhen One Player’s Loss is Notthe Other Player’s Gainreviews facts about constant-sumgames, then considers strategic situa-tions in which players’ interests are notcompletely opposed.Activity 11The Haircutshows that non-constant-sum games can exhibit different structures,including dilemmas.Individual Work 12Non-zero-sum Gamesdifferentiates between situations thatpose dilemmas and situations that don’t,and emphasizes the roles of repeatedplay and the specific game matrix.Activity 12Strategies in Dilemmasprovides an opportunity to test students’strategies in a dilemma situation againstother students and against a calculatorprogram.Individual Work 13Four Kinds of Gamesreviews zero-sum and non-zero-sumgames, and optimal strategies.

MATERIALS PROVIDED

CD-ROM: PD1.83P (or .82P)and PD2.83P (or .82P)

Handout H2.14

TEACHING SUGGESTIONS

Preparation Reading

When One Player’sLoss is Not theOther Player’s Gain

T his reading reviews whatstudents have learnedabout games in whichthe players’ interests are com-pletely opposed. Then it asksstudents to think about othertypes of strategic situations.

Assign the reading prior tobeginning Activity 11.

177Annotated Teacher’s Edition UNIT TWO: STRATEGIES Mathematics: Modeling Our World

ROMCD

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


Annotated Teacher’s Edition Features

22 LESSON TWO Mathematics: Modeling Our World UNIT ONE

INDIVIDUAL WORK 2

Odd or Even1.Suppose Linear Village has houses located at 3, 6, 7, 10, and 12 (see

Figure 1.24).

a) Find the total distance and average distance when you place thefire station at location 8.

b) Find the total distance and average distance when you place thefire station at location 10.

c) Which location, 8 or 10, is the better location for building the firestation?

d) Is there a location that yields a smaller total distance? Explainyour answer.

2.The Linear Village in Item 1 is growing! Add a new house at location15 (see Figure 1.25).

a) Calculate the change in the total distance for the fire station atlocation 8 when the new house is added at location 15.

b) Calculate the change in the total distance for the fire station atlocation 10 when the new house is added at location 15.

c) Which location is better for the fire station now that a new houseis added at location 15?

d) What do you predict will happen to the total distance for loca-tions 8 and 10 when you add another house at location 1?

A B C

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

D EFigure 1.24. Linear Village for Item 1.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

A B C D E G

Figure 1.25. Linear Village for Item 2.

Individual Work 2

Odd or Even

T his individual work intro-duces graphical and tabular presentationsthat will be developed through-out the unit, and providesopportunities for review andchallenge.

Items 6, 7, 8, 9, 13 are criticalto student understanding or themathematical development ofthe unit and should be assignedto all students. Items 14 and 15review elementary inequalitynotation, first introduced inCourse 1, Unit 6, Wildlife.Assign these depending on student comfort with interpret-ing inequalities using numberlines. Additional items may beassigned as needed for review.Some items provide additionalchallenges for developing higher-level thinking skills.

Items 1 and 2 review the skillsof calculating average and totaldistance for different locationsof the fire station and differentnumbers of houses.

22 Mathematics: Modeling Our World UNIT ONE: GRIDVILLE Annotated Teacher’s Edition

23GRIDVILLE Mathematics: Modeling Our World LESSON TWO

Note: The next three items involve an in-depth study of distances for lin-ear villages with 2 or 3 houses. Look for patterns that you can extend toa linear village of any size.

3.The houses in Linear Village are located at 3 and 7 (see Figure 1.26).

For any particular Linear Village, your search for a fire station locationassumes that the houses do not move, but trial locations for the station dochange. Thus, station location acts as an explanatory variable and totaldistance is a response variable.

a) Predict the location that minimizes the total distance.

b) Prepare a table that you can use to look for patterns while you searchfor the location that minimizes the total distance. An example table lay-out is shown in Figure 1.27.

A B

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Figure 1.26. Linear Village withtwo houses.

Figure 1.27. Table for Item 3(b)

0 3 7 10 5 20

1 2 6 8 4 16

2 1 5 6 3 12

3 0 4 4 2 8

4 1 3 4 2 8

5 2 2 4 2 8

6 3 1 4 2 8

7 4 0 4 2 8

8 5 1 6 3 12

9 6 2 8 4 16

10 7 3 10 5 20

11 8 4 12 6 24

12 9 5 14 7 28

13 10 6 16 8 32

14 11 7 18 9 36

Fire Station Distance Distance Total Average Location From F to A From F to B Distance Distance

Items 3, 4, and 5 direct stu-dents to use tables and graphsto investigate locations that min-imize total distance in a linearvillage with two houses or threehouses. These three items aredesigned especially for studentswho were absent during theactivity or need some extrawork in review or assistanceduring Activity 2. Blank tablesand graphs are designed toaccompany these items and are available as Handouts H1.5–H1.7 in theTeacher’s Resources and asTransparencies T1.4–T1.6.

23Annotated Teacher’s Edition UNIT ONE: GRIDVILLE Mathematics: Modeling Our World

A B

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 T 4

Mathematics: Modeling Our World Unit 1: GRIDVILLE TRANSPARENCY

T1.4

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Fire Station Distance Distance Total Average Location From F to A From F to B Distance Distance

1 2 3 4 10 20

1234

10

0 5 6 7 8 9 11 12 1314 15 16 17 1819

56789

11121314151617181920

Fire station location

Dis

tanc

e


T1.5

A B

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 T 6

C


T1.6

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Fire Station Distance Distance Distance Total AverageLocation F From F to A From F to B From F to C Distance Distance

Individual Work• Provides the same teacher

support elements as thosefor Activities

350

In this activity, you will determine the Voronoiregions for each of the eight Colorado raingauges and find the area of each region.

In Figure 5.43, Colorado is represented on a grid in which one unit = 25 miles. The gauges are labeled with letters instead ofnumerals.

The coordinates of the gauges (in grid units) are

A: (5.5, 5.8) B: (7.4, 6.3)

C: (3.7, 4.0) D: (3.1, 8.5)

E: (5.8, 9.1) F: (9.9, 8.6)

G: (10.5, 6.3) H: (6.0, 1.3)

1.Use any of the methods you have developed to find theVoronoi regions. (Handout H5.3 is an enlarged copy of Figure5.43 that you may find convenient.) Write a description ofyour method.

2.When you have finished constructing the regions, determineapproximate values for the area of each and the portion of thestate’s area in each. Several methods that you might use aredescribed below.

a. Area formulas from Unit 4, The Right Stuff.

b. A drawing utility.

c. Heron’s formula. Heron’s formula says that the area of atriangle is where s is half the trian-gle’s perimeter and a, b, c are the lengths of its three sides.For example, if the sides of a triangle are 3, 4, and 5 units,s is 0.5( 3 + 4 + 5) = 6, and the area is

6(6 − 3)(6 − 4)(6 − 5) = 6 square units.

s(s − a)(s − b)(s − c) ,

GETTING THE RAIN RIGHT

5

LESSON THREE Mathematics: Modeling Our World UNIT FIVE

ACTIVITY

DE F

GBA

C

H

Figure 5.43. Eight rain gauges in Colorado.

TEACHING SUGGESTIONS

Activity 5

Getting the RainRight

CD-ROM: COLORADO.83p (or .82p)

A one-pound bag of rice Handouts H5.3 and H5.4 Transparency T5.17

T he purposes of this activity are to establishthe Voronoi boundariesfor the eight Colorado raingauges and estimate the area ofeach of the associated Voronoiregions.

The time required for this activity will vary with themethod chosen by your students and the number of methods you choose to consider. You might allow oneday for construction of theboundaries and a second fordetermining area. If necessary, athird day could be used to dis-cuss and compare results.Consider dividing the class into groups and encouraging avariety of methods.

In Item 1, methods includepaper folding, using Plexiglas®

mirrors, compass and straight-edge, and using a drawing utility.

In Item 2, the choice ofmethod is likely to be affectedby the choice of method in Item1. For example, students whouse a drawing utility to con-struct the boundaries are likelyto use the utility to measure theareas of the regions. Other pair-ings that are appropriateinclude Heron’s formula withmeasurements taken from acompass construction and Pick’sformula with paper folding orPlexiglas mirror constructions.See the Teacher’s Resources forextended discussions of severalsample approaches.

350 Mathematics: Modeling Our World UNIT FIVE: PROXIMITY Annotated Teacher’s Edition

ROMCD

Some historians considerHeron’s result a significantintellectual accomplishmentfor its time. For example,William Dunham devotesone of the 12 chapters ofhis Journey Through Genius:The Great Theorems ofMathematics (New York:John Wiley & Sons, 1990)to the original proof.Shorter proofs can befound in many geometrytexts. See, for example,Harold Jacobs’ Geometry, 2ded. (New York: W.H.Freeman, 1987), pp.336–337.

351

The formula is named after Heron, a Greek mathematicianwho lived in the first century A.D. An advantage to thisformula is that it requires knowing only the lengths of thetriangle’s sides and none of its altitudes.

d. Pick’s formula. Pick’s formula says that the area of anypolygon whose vertices are points of a rectangular grid is0.5b + i – 1, where b is the number of grid points on thepolygon’s boundary (including the vertices) and i is thenumber of grid points in the polygon’s interior. For exam-ple, the polygon in Figure 5.44 has 12 grid points on itsboundary and 10 grid points in its interior. Therefore, itsarea is 0.5 x 12 + 10 – 1 = 15 square units. One advantageof Pick’s formula is that it works for any polygon. A disad-vantage is that the vertices must be grid points. Pick’s for-mula was discovered and proved by Georg Pick in 1899.

e. A simulation. There are a variety of ways a simulation canbe done. One method is to scatter small objects such asgrains of rice over a completed diagram and determine thepercentage that fall in each region. Keep in mind that asimulation becomes more reliable as the number of trialsincreases. You might, for example, use 100 grains of riceand repeat several times.

3.When you have found the eight areas and percentages, writea description of the method you used. Include in yourdescription a discussion of reasons why your answers maylack precision.

CONSIDER:

1.The estimates of the eight areas in Colorado probably varysomewhat in your class. Which estimate do you think is best?Why?

2.For which region(s) do the area estimates in your class varymost? Why?

5ACTIVITY

PROXIMITY Mathematics: Modeling Our World LESSON THREE

GETTING THE RAIN RIGHT

Figure 5.44. A polygon whose vertices are pointsof a grid.

The TI graphing calculator pro-gram COLORADO on theCourse 2 CD–ROM can be usedfor the simulation. See theTeacher’s Resources for a discussion. Since some studentsmay want to redo the programfor use in other problems, anannotated copy of the code isprovided on Handout H5.4.Note that simulations are usedto find areas of irregular figuresin Course 1, Unit 3, Landsat.

The area of a polygon in aplane can be found quicklyfrom a matrix of its coordinates.See the Teacher’s Resources for adiscussion of the method.

Transparency T5.17 can beused to discuss the results ofthis Activity.

The Consider questions can beused to discuss differences inestimates obtained by the class.In Individual Work 5, studentsare asked to estimate theColorado rainfall by applyingthe area estimates in which theyhave greatest confidence.Therefore, the discussion shouldproduce closure on the matterof which estimate is best.

351Annotated Teacher’s Edition UNIT FIVE: PROXIMITY Mathematics: Modeling Our World

Although few of the Voronoi boundaries in thissituation have endpoints at grid points, Pick’s for-mula can be used if the boundaries are redrawnusing the grid points nearest the endpoints. Seethe Teacher’s Resources for a discussion. Cautionmust be used in applying Pick’s formula since itcan be difficult to determine visually whether apoint is on a boundary. Attention to slopes canhelp resolve such cases. Note that some of yourstudents may have seen Pick’s formula in ageoboard geometry unit in elementary or middleschool.

Mathematics: Modeling Our World Unit 5: PROXIMITY TRANSPARENCY

T5.17

E

F

GBA

D

C

H

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


373

PROXIMITY

Mathematics: Modeling Our World UNIT SUMMARY

UNITSUMMARY

Wrapping Up Unit Five1.A person studying Voronoi diagrams has developed the following

algorithm for situations with four centers.

Step 1.Arbitrarily select one of the four centers and label it A. Movearound the centers in either clockwise or counterclockwise fash-ion. As you go, label the remaining points B, C, and D.

Step 2. Construct the perpendicular bisector of each adjacent pair of centers.

Step 3. Determine the portions of each perpendicular bisector to keep.

A sample implementation of the first two steps of the algorithm isshown in Figure 5.71.

Is this a good algorithm for four-center situations? Explain.

A

BC

D

Figure 5.71.

UNIT SUMMARY

Wrapping UpUnit Five1–2 days

MATERIALS PROVIDED

Handout H5.10Supplemental Activity S5.1

H andout H5.10 is a listof projects for thisunit. For students whoare interested in examiningVoronoi diagrams in Gridville(Item 2 on the project list), youmay want to have them beginwith Supplemental ActivityS5.1. This activity has severalbasic questions about thenature of Gridville Voronoi dia-grams in situations with twocenters.

373Annotated Teacher’s Edition UNIT FIVE: PROXIMITY Mathematics: Modeling Our World379PROXIMITY Mathematics: Modeling Our World UNIT SUMMARY

Mathematical Summary

T he modeling problem from which the mathematics in this unitarises is that of estimating the rainfall for the entire state ofColorado from readings taken at eight rain gauges scatteredaround the state.

The solution is geometric: divide the state into eight regions so that thepoints in a region are closer to its gauge than to any other gauge. Weightthe rainfall measured at each region’s gauge according to the portion ofthe state’s area in that region. The weighted average of the rainfall at theeight gauges estimates the rainfall for the state.

Proximity problems like the Colorado rain gauge problem involveVoronoi diagrams, which are named after the mathematician GeorgiiVoronoi. To use Voronoi diagrams, you must determine the boundariesof regions from their centers of influence.

The boundaries can be drawn roughly by hand, but answers obtainedfrom rough drawings lack precision. Therefore, the boundaries should beconstructed. There are several means of constructing the boundaries.Every Voronoi boundary is the perpendicular bisector of a segmentjoining two centers. Perpendicular bisectors can be constructed byseveral methods: 1) by folding a piece of paper and creasing it so thattwo centers coincide; 2) by placing a Plexiglas® mirror so that thereflection of one center coincides with the other center; 3) by strikingintersecting compass arcs from the centers and joining the two points ofintersection; 4) or by using the segment, midpoint, and perpendicularconstruction features of a drawing utility.

Perpendicular bisectors are lines, but Voronoi boundaries are either raysor line segments. Moreover, the perpendicular bisector for some pairs ofcenters is not a boundary in the Voronoi diagram. Therefore, whenperpendicular bisectors are constructed, they must be analyzed carefullyto determine which portions to keep. Algorithms for establishingVoronoi boundaries often divide the problem into several smallerproblems of, say, three or four vertices, then combine the diagrams thatresult.

Voronoi regions are usually polygons. (An exception occurs when theboundary of the domain is curved.) Many modeling problems, includingthe Colorado rain gauge problem, require determination of each regions’area. One way to find a region’s area is to divide it into triangles andapply Heron’s formula, which finds the area of a triangle from thelengths of its sides. Another method is to apply Pick’s formula, which


Unit SummaryWrapping Up the Unit

• Notes suggest ways to reviewskills and concepts

Mathematical Summary• Appropriate notes as needed to

guide this review

Glossary

381PROXIMITY Mathematics: Modeling Our World GLOSSARY

GlossaryACUTE TRIANGLE: A triangle in which all of the angles mea-sure less than 90˚, but more than 0˚.

CENTER OF INFLUENCE: A point used to establish boundaries ofregions of influence. All points in a regionare closer to that region’s center than toany other region’s center.

COLLINEAR POINTS: Points that lie on the same line.

CONCAVE POLYGON: A polygon in which some of its sides, whenextended, intersect other sides.

CONVEX POLYGON: A polygon in which none of its sides, whenextended, intersect other sides. For everypair of points in the interior of a convexpolygon, the segment connecting the pointsis completely in the interior.

DOMAIN: A region in which centers of influence arelocated. The domain is the area that isbeing divided into regions of influence.

HERON’S FORMULA: The area of a triangle is

where a, b, and care the lengths of the triangle’s sides and sis half the triangle’s perimeter.

ITERATION (ITERATIVE PROCEDURE):a procedure that repeats the same sequenceof steps over and over. Each cycle is con-sidered one iteration.

MIDPOINT: A point that is halfway along a segment(equidistant from the segment’s two end-points). In coordinate geometry, the coordi-nates of a midpoint are found by averagingthe coordinates of the two endpoints.

OBTUSE TRIANGLE: A triangle with one angle that measuresmore than 90˚, but less than 180˚.

PERPENDICULAR BISECTOR: A line that passes through the midpoint ofa given line segment and forms rightangles with it.

PICK’S FORMULA: If the vertices of a polygon are points of agrid, then the area of the polygon is 0.5b + i – 1, where b is the number of gridpoints on the polygon’s border, and i is thenumber of points in its interior.

REGION OF INFLUENCE: A region in which each point is closer tothe region’s center of influence than to anyother center of influence.

VORONOI BOUNDARY: A boundary between two centers of influence.

VORONOI CENTER: A center of influence.

VORONOI DIAGRAM: A diagram composed of several centers ofinfluence and their regions of influence.

VORONOI REGION: A region of influence.

VORONOI VERTEX: A point at which Voronoi boundaries intersect.

WEIGHTED AVERAGE:The average found by multiplying eachcategory by the decimal weight attached tothat category and finding a total.

s(s − a)(s − b)(s − c) ,



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

74 VIDEO SUPPORT

UNIT FIVE: PROXIMITY

Teacher’s Guide


Before viewing the video for thi

s unit, give each

student a copy of Handout H5.1, Video V

iewing

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 are some similarities in the st

udy of birds

and the study of robots that are portraye

d in the

video?

Bird nests and obstacles in a robot’s pa

th are both

centers of influence. They establish re

gions that

birds inhabit in one case and that rob

ots should

avoid in the other case.

2. Name some other situations that yo

u think are

similar to these two. What kinds of thin

gs are the

centers of influence in each of your exam

ples?

For example, schools in a community

are centers

of influence for the regions from whic

h they draw

students.

3. Situations in which proximity is imp

ortant

involve distance. What kind of distance

state-

ments can you make about centers of in

fluence

and the regions about them?

The ideal answer is that when you are

in a particular

center’s (bird nest, obstacle, school) re

gion, you are

probably closer to it than to any other

center (bird

nest, obstacle, school).

Video Support VIDEO SUPPORT

In Figure 6, the points have been pasted into XYZGeoBench. When pasted, the points are automaticallyselected.

Figure 6.

Figure 7 shows the results of a Voronoi diagram rou-tine chosen from XYZ GeoBench’s Operations menu.The program offers several Voronoi diagram algo-rithms, including options that allow the user toobserve the construction in steps and others thatdynamically change the diagram as the user drags apoint. The algorithm chosen for this construction isPlane Sweep Fortune.

Figure 7.

If desired, the results can be transported back to thedrawing utility in which the problem originated.Select and copy the diagram by choosing Select All,then Copy from the program’s Edit menu. Paste thediagram into the drawing utility. Drag the pasted pic-ture so that the points coincide with the points in theoriginal drawing utility sketch. The result is shown inFigure 8. Since the utility does not interpret theGeoBench boundaries as lines, additional work mightinclude drawing points at intersections, constructingthe appropriate segments, then deleting the GeoBenchpicture.

Figure 8.

ASSESSMENT PROBLEM A5.1Dominance

T he problem presents another way to constructVoronoi diagrams: by finding intersections ofdominance regions.ASSESSMENT PROBLEMS A5.2–A5.4 Lookout 1, Lookout 2, Lookout 3L ookout 1 and Lookout 2 are identical except thatthe latter is presented on a grid. Lookout 3 ismore challenging than either of the other two. Allthree ask students to construct Voronoi diagrams withtwo centers and to estimate areas.

One option for using these three problems is to havestudents do them at three different times in this les-son. The first, for example, could be used afterActivity 3, the second after students have been intro-duced to a drawing utility in Individual Work 3, andthe third at the end of the lesson. Another option is touse the second and third later in the unit. The second,for example, can be used in Lesson 4 after a coordi-nate approach has been developed.

78 LESSON TWO

UNIT FIVE: PROXIMITY

Teacher’s Guide


82

Teacher’s Guide Mathematics: Modeling Our World

UNIT FIVE: PROXIMITY LESSON FOUR

ACTIVITY 6

Getting a Line on Voronoi Diagrams

See Annotated Teacher’s Edition.

INDIVIDUAL WORK 6

Connect the Dots

T hree sample programs are supplied on theCourse 2 CD–ROM. PERP.83P (or .82P) finds themidpoint of the segment connecting two points andthe slope and y-intercept of the perpendicular bisec-tor. HERON.83P (or .82P) finds the area of a trianglefrom either the lengths of the sides or the coordinatesof the vertices. POLYAREA.83P (or .82P) finds thearea of a polygon from the coordinates of its vertices.Handouts H5.5, H5.6, and H5.7 are documented listings of these programs.

PERP prompts theuser for x- and y-coordinates of twopoints. Figure 16shows the input ofVoronoi centers (2,4)and (3,6).

Figure 16.

When the user presses ENTER aftertyping the second y-coordinate, theprogram displays themidpoint, in thiscase (5/2, 5) andpauses (Figure 17).

Figure 17.

Pressing ENTER displays the slopeand y-intercept ofthe perpendicular bisector (Figure 18).In this example, theequation isy = –1/2x + 25/4.

Figure 18.

HERON offers theuser a choice ofinput: the lengths ofthe sides of the trian-gle or the coordi-nates of its vertices.In Figure 19, theuser is choosingcoordinates.

Figure 19.

The programprompts the user foreach coordinate(Figure 20). In thiscase, the triangle’svertices are at (1, 2),(5, 6), and (8, 3).

Figure 20.

When the user presses ENTER afterthe third y-coordi-nate, the programdisplays the area(Figure 21).

Figure 21.

LESSON FOUR

A Method of a Different Color

Teacher’s GuideVideo Support• Provides discussion questions for use

after viewing the video

Lesson-by-lesson information• Provides additional information to

supplement teaching suggestionscovered in the Annotated Teacher’sEdition

• Provides background readings forteachers to present content and softwareinformation not found in othercomponents of the program

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

• Suggests alternate approaches to studentActivities

CALCULATOR REVIEW: PARAMETRIC EQUATIONS

T he following calculator displays show possible settings for usingparametric equations and your calculator to graph the motion of cars sailing off ramps.

Mathematics: Modeling Our World Unit 7: MOTION HANDOUT

H7.7

Select paramtric MODE.

Either graph andTRACE . . .

Now you can be preciseto the 0.01 second.

Or you could changethe table set-up to bemore precise.

. . . or use tables toanswer the question.

You might need tochange the window tobe more precise withyour graph.

Enter the equations ofthe x and y motion.

Select a window thatwill show the x and ymotion for a reasonabletime.

Handouts• As with all supplemental materials, these

are numbered consecutively within eachunit

• First handout is always a Video ViewingGuide with questions for students

• Additional masters contain directions forusing software and hardware,background reading for students, andrecord sheets with particular explorations


PROJECT IDEAS AND REFERENCES

1.The Federal Communications Commission (FCC) has used game theoryto design the auction of communication licenses. Bidders have, in turn,used game theory to decide how to bid. Report on how game theory hasbeen used and the effect it has had on the licensing process.

2.Trade wars between countries are common. In 1995, for example, theUnited States raised tariffs on Japanese automobiles in an effort to gaintrade concessions from Japan. Report on the way game theory is used, ormight be used, to determine strategies in trade disputes.

3. In this unit, optimal strategies are conservative ones because theyminimize the risk of loss. For many people, minimizing the risk of loss isthe most important concern, but not for others. For example, some peopletake big risks in hopes of maximizing their winnings. Others seem mostinterested in beating their opponents, and others tend to cooperate.Report on the effect the personality of the players has on outcomes ofgames, particularly non-zero-sum games.

4. In 1994, the Nobel prize in economics was awarded to John Nash, JohnHarsanyi, and Reinhard Selten for work in game theory. Research andreport on the contributions of one of these individuals.

5.Game theory is a relatively new branch of mathematics. It originated inthe United States around the 1940s. Research and report on the originsand history of game theory.

6. In this unit, many of the games you studied are games in which theplayers make their choices independently. In some games, choices are notindependent. For example, in some games one player chooses first andthe other knows the first player’s choice. In other situations, the playersmay be allowed to negotiate their choices of strategy. Report on the effectsthat dependent choices or negotiations have on the outcomes of games.

7.Many applications of game theory occur in political science. Politics, ofcourse, is an area where many people have opinions. Sometimes somepeople feel that a game is zero sum, but others feel that it is not. Forexample, the North America Free Trade Agreement (NAFTA) wascontroversial because some Americans felt the U.S. would lose dollars andjobs to Mexico, but others felt the result would be a gain for bothcountries. Select NAFTA or some other topic and report on evidencesupporting both beliefs.

8.Most of the games in this unit are games in which each player has onlytwo options. Report on methods of analyzing games in which there aremore than two options.

Mathematics: Modeling Our World Unit 2: STRATEGIES HANDOUT

H2.15page 1 of 2

PROBLEM A3.2

Archaeology

Archaeologists study

old civilizations. In t

he past there were tra

nsitions as one

civilization evolved in

to another. Some civ

ilizations had more th

an one

successor.

In Figure 1 you can

see, for example, tha

t civilization Matri-H

awaiian

evolved into the civi

lizations: Bi-Eskimo,

Patri-Eskimo, and No

rmal

Hawaiian. From thes

e three, only Patri-Es

kimo had a successo

r that is shown:

Normal Eskimo.

The figure can be rea

d like a matrix: a bla

ck box means that th

ere was a direct

succession; a white b

ox means that direct

succession was imp

ossible.

This information can a

lso be represented by

a graph.

The graph in Figure

2 represents just a pa

rt of the information

in Figure 1.

Copy and complete

the graph by using a

ll the information in

Figure 1.

Figure 2. The beg

inning of a civilizatio

n graph.

page 1 of 2

Mathematics: Modelin

g Our World

Unit 3: HIDDEN C

ONNECTIONS

ASSESSMENT

A3.2

From

[NoE] Normal Esk

imo

[BiE] Bi-Eskimo

[MaE] Matri-Eskim

o

[PaE] Patri-Eskim

o

[NoH] Normal Haw

aiian

[MaH] Matri- Hawa

iian

[NeH] Neo-Hawaii

an

[PaH] Patri-Hawa

iian

[NoY] Normal Yum

an

[BiY] Bi-Yuman

[MaY] Matri-Yuma

n

[NeY] Neo-Yuman

To [NoE

]N

orm

al E

skim

o

[BiE

]B

i-E

skim

o

[MaE

] M

atri

-Esk

imo

[PaE

]

Patr

i-E

skim

o

[NoH

] N

orm

al H

awai

ian

[MaH

] M

atri

- Haw

aiia

n

[NeH

]

Neo

-Haw

aiia

n

[PaH

]

Patr

i-H

awai

ian

[NoY

] N

orm

al Y

uman

[BiY

]

Bi-

Yum

an

[MaY

] M

atri

-Yum

an

[NeY

] N

eo-Y

uman

3 A 3

MaH

NoH PaE BiE

NoE

3 A 4

Figure 1. Several c

ivilizations and their

successors.

Unit Projects• Usually a culminating

exploration drawingon all the conceptsdeveloped throughoutthe unit

PROBLEM A4.8

Shorties 2For the next series of short problems, use one or more of the following facts.

Fact 1: The sum of the angles of a triangle is 180˚.

Fact 2: If two parallel lines are cut by a transversal, then

(a) the alternate interior angles are equal (in Figure 1, ∠3 = ∠5).

(b) angles that form a straight line total 180˚ (in Figure 1, ∠3 + ∠4 = 180˚).

(c) corresponding angles are equal (in Figure 1, ∠1 = ∠5).

Fact 3: If all angles in a triangle are equal, then the triangle isequilateral.

Fact 4: In an isosceles triangle, the angles at the base are equal.

Fact 5: The Pythagorean formula states that in a right triangle: a2 + b2 = c2. See Figure 2.

1. In Figure 3, side BC is parallel to line l, ∠B = 64˚ and ∠1 = 53˚.Calculate the measure of ∠3.

2. In triangle ABC (Figure 4) ∠A = 40˚, ∠B = 50˚, and AD = CD. Find the measure of ∠CDB.

page 1 of 2

Mathematics: Modeling Our World Unit 4: THE RIGHT STUFF ASSESSMENT

A4.8

12

3 4

56

7 8

Figure 1.

c

a

b

90°

Figure 2.

A

C

B3

2

1

l

Figure 3.

A D B

C

40˚ 50˚

Figure 4.

AssessmentProblems• Wide variety of

problems to usethroughout or atend of the unit

• Suggestions forplacement arefound in theAnnotated Teacher’sEdition

• Many open-endedproblems


Teacher’s Resources Features

Deductiv

e reasoni

ng can b

e used to

establish

area form

ulas.

1.Many

area pro

ofs are ba

sed on a

fact with

which m

ost

people ag

ree name

ly, that th

e area of

a rectang

le is the

length of

its base t

imes the

length of

its heigh

t (or just

written a

s bh). Us

e the 3 x

6 rectang

le in Figu

re 1to

explain w

hy the are

a can be f

ound by m

ultiplying

the

base tim

es the he

ight.

2.The sol

id lines in

Figure 2

form a pa

rallelogr

am.

a) Identif

y the par

allelogra

m’s base

and heig

ht.

b) Use d

eductive

reasonin

g to expl

ain why th

e

parallelo

gram use

s the sam

e area for

mula as

a rectang

le.

3.Figure

3 is triang

le.

a) Identif

y a base a

nd the c

orrespon

ding heig

ht.

b) Does a

triangle

have on

ly one ba

se? Expla

in. Is you

r

answer th

e same for

a paralle

logram?

AREA P

ROOFS

S4.1

page 1

of 2

Mathemat

ics: Modeli

ng Our W

orld

Unit 4: T

HE RIG

HT STU

FF

SUPPLEM

ENTAL A

CTIVITY

Figure 1

. A 3x 6 r

ectangle

.

Figure 2

. A parall

elogram

.

Figure 3

. A triang

le.

Many graphing calculators

use what are called Boolea

n

functions in addition to the

more usual kinds of functi

ons

you have studied. For the p

urposes of this activity, a Bo

olean

function is one that has onl

y two possible values, 0 an

d 1, having the value 1 whe

n

a particular condition is tru

e, and having the value 0 w

hen the condition is false.

Thus the Boolean function

amounts to a “yes” or “no

” rule.

To illustrate the use of this

idea in graphing piecewise

-defined functions, conside

r

the absolute value function

y = |x|. Recall that one cha

racterization of this

function is y = x when x ≥ 0 a

nd y = –x when x < 0. Defi

ne Y1 = X*(X ≥ 0) and

define Y2 = -X*(X


Unit 1Pick a Winner: DecisionMaking in a DemocracyNumber sensePercentagesPreference diagram representationGraph theoryParadoxMatrices

Unit 2Secret Codes and the Powerof AlgebraMathematical modelingFunctions and linear functionsRepresentations of functions: tables,graphs, symbolic equations, arrowdiagramsAlgebraic expressionsMatrix operations: addition,subtraction, scalar multiplicationModular arithmeticSolving equationsInverse of a functionFrequency distributionsOrder of operationsEquivalent expressions, distributiveproperty

Unit 3LandsatDistanceScaleGraphical interpretationUnit conversionScale factorRatiosPrecisionSignificant figuresRelative sizePixelDigitizationCorresponding partsShapeSimilarityProportionalitySolving proportionsPythagorean theoremCoordinatesDilationTranslationAreaLength-area relationshipApproximationMonte Carlo methods

Unit 4PredictionDot plotsScatter plotsMeanSlopeVariableLinear equationsGraphing linesCollecting dataInterpreting dataFitting a line to dataResiduals

Unit 5Animation/Special EffectsCoordinate systemsContinuous and discreterepresentationsRates of changeVariables and constantsRecursive and closed-formrepresentationsLinear functionsEl

Mathematics: Modeling Our World · Landy Godbold, Bruce Grip EVALUATION Barbara Flagg MULTIMEDIA...

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Transcript of Mathematics: Modeling Our World · Landy Godbold, Bruce Grip EVALUATION Barbara Flagg MULTIMEDIA...