C O M A P ’ S

Pre-CalculusT E A C H E R ’ S R E S O U R C E S

DEVELOPED BY

COMAP, Inc.175 Middlesex Turnpike, Suite 3B

Bedford, Massachusetts

01730

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 2000

by COMAP, Inc.

The Consortium for Mathematics and Its Applications (COMAP)

175 Middlesex Turnpike, Suite 3B

Bedford, Massachusetts

01730

Published and distributed by

The Consortium for Mathematics and Its Applications (COMAP)175 Middlesex Turnpike, Suite 3B

Bedford, Massachusetts 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 0-7167-4114-8

Printed in the United States of America.

EDITOR: Landy Godbold

AUTHORS: Nancy Crisler, PATTONVILLE SCHOOL DISTRICT, ST. ANN, MO; Marsha Davis, EASTERN CONNECTICUT STATE UNIVERSITY, WILLIMANTIC, CT

Gary Froelich, COMAP, INC., LEXINGTON, MA; Frank Giordano, U. S. MILITARY ACADEMY (RETIRED);

Jerry Lege, DR. JAMES HOGAN SENIOR HIGH SCHOOL, VALLEJO, CA

iiiMathematics: Modeling Our World

To the Teacher

Since its inception in 1980

, COMAP has been dedicated to presen

ting mathematics

through contemporary applications. We

have produced high school and college

texts,

hundreds of supplemental modules, an

d three television courses—all with the p

urpose

of showing students how mathematics i

s used in their daily lives.

After the publication of the NCTM Stan

dards in 1989, the National Science Foun

dation began to fund

major curriculum projects at the elemen

tary, middle, and secondary levels. The

purpose of all of these

programs is to turn the vision of the Sta

ndards into the curriculum of today’s cl

assrooms. In 1992, the

ARISE project was one of only five such

high school programs selected by the NS

F for funding.

Over the past eight years, we have work

ed to develop this curriculum with a tea

m of over 20 authors,

almost all practicing high school teacher

s, including several Presidential Awarde

es and Woodrow Wilson

Fellows. We have field-tested these mate

rials with over 10,000 students across th

e country. Both our author

team and our field-testers come from an

amazingly diverse collection of schools

with a full range of

student populations.

The result of these labors is Mathematics

: Modeling Our World. In the COMAP sp

irit, Mathematics: Modeling

Our World develops mathematical conce

pts in the contexts in which they are actu

ally used. The word

“modeling” is the key. Real problems d

o not come at the end of chapters. Real p

roblems don’t look like

mathematics problems. Real problems a

re messy. Real problems ask questions s

uch as: How do we create

computer animation? How do we effecti

vely control an animal population? Wha

t is the best location for a

fire station? What do we mean by “best

”?

Mathematical modeling is the process o

f looking at a situation, formulating a pr

oblem, finding a

mathematical core, working within that c

ore, and coming back to see what mathe

matics tells us about the

original problem. We do not know in ad

vance what mathematics to apply. The m

athematics we settle on

may be a mix of geometry, algebra, trigo

nometry, data analysis, and probability.

We may need to use

computers or graphing calculators, spre

adsheets, or other utilities.

At heart, we want to demonstrate to stud

ents that mathematics is the most usefu

l subject they will learn.

More importantly, we hope to demonstra

te that using mathematics to solve interes

ting problems about

how our world works can be a truly enj

oyable and rewarding experience. Ultima

tely, learning to model is

learning to learn.

For those of you who have taught the ea

rlier courses in this series, you will note

one major difference. In

Mathematics: Modeling Our World Cours

es 1, 2, and 3 all of the chapters were or

ganized around a major

context. We did this to emphasize furthe

r the broad applicability of the subject. I

n this text we have used

mathematical concepts as chapter titles.

We have done this to emphasize that m

athematics as a discipline

has a structure of its own, and that as st

udents go on into the study of mathema

tics they will learn more

and more of that structure and the pow

er it provides to solve an amazingly wid

e array of problems.

This course is the gateway to collegiate m

athematics. As such, students will see a

number of essential new

concepts and be asked to learn a numbe

r of important new skills.

While we have subtitled the course, pre

-calculus, we believe that the material in

this text can provide

students with a firm background for any

entry-level undergraduate mathematics

course—continuous or

discrete. For example, we have provide

d substantial material on matrices and v

ectors as well as a full

chapter on discrete dynamical systems.

We believe that the treatment of these t

opics will prepare students

for a deeper understanding of the conce

pts underlying the calculus as well as th

ose underlying discrete

mathematical structures.

What you will find here is a challenging

pre-calculus course for serious students

. And, in the COMAP

tradition, you will find exciting, contem

porary, applications and models presen

ted in novel ways to help

teach and motivate the further study of

our discipline.

Solomon Garfunkel

CO-PRINCIPAL INVESTIGATOR

Landy Godbold

CO-PRINCIPAL INVESTIGATOR

Henry PollakCO-PRINCIPAL INVEST

IGATOR

iv ACKNOWLEDGEMENTS Mathematics: Modeling Our World

AcknowledgementsPROJECT LEADERSHIP

Solomon Garfunkel COMAP, INC., LEXINGTON, MA

Landy Godbold, THE WESTMINSTER SCHOOLS, ATLANTA, GAHenry PollakTEACHERS COLLEGE, COLUMBIA UNIVERSITY, NY

EDITOR

Landy Godbold

AUTHORS

Nancy CrislerPATTONVILLE SCHOOL DISTRICT, ST. ANN, MO

Gary FroelichSECONDARY SCHOOL PROJECTS MANAGER, COMAP, INC., LEXINGTON, MA

Frank GiordanoU. S. MILITARY ACADEMY (RETIRED)

Jerry LegeDR. JAMES HOGAN SENIOR HIGH SCHOOL, VALLEJO, CA

REVIEWERS

Marsha Davis EASTERN CONNECTICUT STATE UNIVERSITY, WILLIMANTIC, CT

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

Henry PollakTEACHERS COLLEGE, COLUMBIA UNIVERSITY, NY

ASSESSMENT

Dédé de Haan, Jan de Lange, Anton Roodhardt, Henk van der KooijTHE FREUDENTHAL INSTITUTE, THE NETHERLANDS

EVALUATION

Barbara FlaggMULTIMEDIA RESEARCH, BELLPORT, NY

FIELD TEST SCHOOLS AND TEACHERS

Gresham Union High School, Gresham, ORKAY FRANCIS

Mills E. Godwin High School, Richmond, VAANN W. SEBRELL

Riverdale High School, Marylhurst, ORDAVID THOMPSON

Sam Barlow High School, Gresham, ORBRAD GARRETT

COMAP STAFF

Solomon Garfunkel, Laurie Aragón, Sheila Sconiers, Sue Rasala,Gary Froelich, Roland Cheyney, Sue Martin, Lynn Aro, Sue Judge,George Ward, Daiva Kiliulis, Gail Wessell, Pauline Wright, Gary Feldman, Clarice Callahan, Jan Beebe, Rafael Aragón, Kevin Darcy, Peter Bousquet

ART

Lianne DunnSANDWICH, MA

Mary ReillySOMMERVILLE, MA

COVER ART

CorbisBELLEVUE, WA

vMathematics: Modeling Our World CONTENTS

CHAPTER 1

FUNCTIONS IN MODELING

Teacher’s Guide 1.1Chapter Objectives 1.2Lesson 1.1 1.3Lesson 1.2 1.4Lesson 1.3 1.7Lesson 1.4 1.9Lesson 1.5 1.12Chapter 1 Review 1.14

Handouts H1.1–H1.4 1.15

Assessment Problems A1.1–A1.13 1.25

Transparency T1.1 1.59

CHAPTER 2

THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS

Teacher’s Guide 2.1Chapter Objectives 2.2Lesson 2.1 2.3Lesson 2.2 2.5Lesson 2.3 2.7Lesson 2.4 2.9Lesson 2.5 2.11Lesson 2.6 2.14Chapter 2 Review 2.16

Handouts H2.1–H2.5 2.17

Supplemental Activities S2.1–S2.2 2.27

Assessment Problems A2.1–A2.12 2.33

Transparencies T2.1–T2.9 2.67

fx( ) Table of Contents

vi CONTENTS Mathematics: Modeling Our World

CHAPTER 3

POLYNOMIAL MODELS

Teacher’s Guide 3.1Chapter Objectives 3.2Lesson 3.1 3.3Lesson 3.2 3.5Lesson 3.3 3.7Lesson 3.4 3.9Lesson 3.5 3.11Lesson 3.6 3.13

Handouts H3.1–H3.3 3.15

Assessment Problems A3.1–A3.9 3.21

CHAPTER 4

COORDINATE SYSTEMS AND VECTORS

Teacher’s Guide 4.1Chapter Objectives 4.2Lesson 4.1 4.3Lesson 4.2 4.5Lesson 4.3 4.6Lesson 4.4 4.8Lesson 4.5 4.9Lesson 4.6 4.11Chapter 4 Review 4.13

Handouts H4.1–H4.7 4.15

Supplemental Activities S4.1–S4.3 4.29

Assessment Problems A4.1–A4.6 4.39

Transparencies T4.1–T4.19 4.55

viiMathematics: Modeling Our World CONTENTS

CHAPTER 5

MATRICES

Teacher’s Guide 5.1Chapter Objectives 5.2Lesson 5.1 5.3Lesson 5.2 5.4Lesson 5.3 5.6

Handout H5.1 5.9

Assessment Problems A5.1–A5.10 5.11

Transparency T5.1 5.43

CHAPTER 6

ANALYTIC GEOMETRY

Teacher’s Guide 6.1Chapter Objectives 6.2Lesson 6.1 6.3Lesson 6.2 6.5Lesson 6.3 6.7Lesson 6.4 6.9Lesson 6.5 6.12Chapter 6 Review 6.15

Handouts H6.1–H6.5 6.17

Assessment Problems A6.1–A6.17 6.29

Transparency T6.1 6.69

viii CONTENTS Mathematics: Modeling Our World

CHAPTER 7

COUNTING AND THE BINOMIAL THEOREM

Teacher’s Guide 7.1Chapter Objectives 7.2Lesson 7.1 7.3Lesson 7.2 7.4Lesson 7.3 7.5

Assessment Problems A7.1–A7.12 7.7

CHAPTER 8

MODELING CHANGE

Teacher’s Guide 8.1Chapter Objectives 8.2Lesson 8.1 8.3Lesson 8.2 8.4Lesson 8.3 8.5Lesson 8.4 8.6

Assessment Problems A8.1–A8.11 8.7

ixMathematics: Modeling Our World

CHAPTER 1

Forestry, Forensic Science, Ecology, Audio Engineering

CHAPTER 2

Seismology, Personal Finance, PopulationBiology, Physical Science, Chemistry,Astronomy, Information Science, Medicine

CHAPTER 3

Physical Science, Geology, Cryptography,Medicine, History, Business, InformationScience

CHAPTER 4

Meteorology, Navigation, Astronomy, History,Orienteering, Physical Science, Surveying,Spatial Planning, Sports

CHAPTER 5

Information Science, Cryptography,Animation, Social Science, History,Management

CHAPTER 6

Carpentry, Robotics, Archeology, History,Navigation, Astronomy, Architecture,Personal Finance

CHAPTER 7

Recreation, Information Science,Management, Cryptography, History

CHAPTER 8

History, Personal Finance, Physical Science,Law Enforcement, Medicine, Management,Wildlife Biology

MMOW 4 Related Disciplines

x Mathematics: Modeling Our World

Chapter Objectives

CHAPTER 1

FUNCTIONS IN MODELING

Introduction:

Modeling Behavior: Explanations and Patterns

To reiterate the importance of mathematicalmodeling.

To explain the two types of mathematicalmodels, “theory driven” and “data driven.”

To review the modeling process.

To introduce the course’s first modelingsituation.

Lesson 1.1:

A Theory-Driven Model

To create a theory-driven model when given aspecific situation.

To test a model against actual data.

Lesson 1.2:

Building a Tool Kit of Functions

To create a tool kit of functions by identifyingkey features of five previously studiedfunctions: constant, direct variation, linear,absolute value, and sine.

To examine the equations, shapes of graphs,patterns in tables of values, and verbaldescriptions of these functions.

To review these functions both in purelymathematical exercises and in contextualsettings.

Lesson 1.3:

Expanding the Tool Kit of Functions

To expand the tool kit of functions that wasintroduced in Lesson 1.2.

To explore power functions, the Ladder ofPowers, and the effect of multiplying a powerfunction by a constant.

Lesson 1.4:

Transformations of Functions

To examine the effects of transformations onthe graphs of the tool kit functions.

Lesson 1.5:

Operations on Functions

To explore addition, subtraction,multiplication, and division of functions.

To refine models by adding or multiplyingtwo functions.

CHAPTER 2

THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS

Lesson 2.1:

Exponential Functions

To expand the tool kit of functions byreviewing exponential functions.

To review the order of operations, thestandard form for exponential functions, the characteristic properties of theirgraphs, and patterns in exponential data.

To introduce base e.

Lesson 2.2:

Logarithmic Scale

To introduce students to the problem ofworking with data that span several orders ofmagnitude.

To use earthquake measurements to illustratea need for relative comparisons.

To introduce the log10 scale and investigate itsproperties.

Lesson 2.3:

Changing Bases

To expand the concept of a logarithmic scaleto consider other bases, including the naturalbase e.

To formalize properties of logarithms anddevelop a procedure for changing bases.

xiMathematics: Modeling Our World

To expand methods for evaluatingexpressions and solving equations to usinglogarithms.

Lesson 2.4:

Logarithmic Functions

To expand the tool kit of functions to includelogarithmic functions.

To identify various properties, including theeffect of base on the graph and therelationship between logarithmic andexponential functions of the same base.

Lesson 2.5:

Modeling With Exponential and Logarithm Functions

To extend data analysis tools by usinglogarithms to develop log-log and semi-logtests through re-expression of original data.

Lesson 2.6:

Composition and Inverses of Functions

To introduce composition as an operation ontwo functions, and to formalize therelationship between a function and itsinverse with respect to composition.

To explore conditions for the existence of aninverse and methods for finding a function’sinverse.

CHAPTER 3

POLYNOMIAL MODELS

LESSON 3.1:

Modeling Falling Objects

To introduce polynomials.

To create and evaluate polynomial models forreal-world phenomena.

Lesson 3.2:

The Merits of Polynomial Models

To create and evaluate polynomial models forreal-world phenomena.

To find a polynomial function that capturesevery point in a set of data.

Lesson 3.3:

The Power of Polynomials

To create and evaluate polynomial models forreal-world phenomena.

To examine the relationship between thedegree of a polynomial function and theshape of its graph.

To find approximations of a polynomialfunction’s x-intercepts and exact x-interceptsin some cases.

To examine the relationship between themultiplicity of a zero and the shape of thegraph near that zero.

Lesson 3.4:

Zeroing in on Polynomials

To introduce complex numbers and theFundamental Theorem of Algebra in ahistorical context.

To find complex zeros of some polynomials.

To perform basic operations on complexnumbers.

Lesson 3.5:

Polynomial Divisions

To examine asymptotic behavior in rationalfunctions.

Lesson 3.6:

Polynomial Approximations

To introduce power series approximations ofexponential and trigonometric functions.

To explain the difference between convergentand divergent series.

CHAPTER 4

COORDINATE SYSTEMS AND VECTORS

Lesson 4.1:

Polar Coordinates

To introduce polar coordinates and methodsfor expressing directional angles.

To develop and apply formulas for convertingbetween rectangular and polar coordinates.

xii Mathematics: Modeling Our World

Lesson 4.2:

Polar Form of Complex Numbers

To represent complex numbers usingpolar coordinates.

To evaluate powers and roots of complexnumbers by use of De Moivre’s Theorem.

Lesson 4.3:

The Geometry of Vectors

To examine vectors from a geometricperspective.

To define magnitude, direction angle,vector addition and subtraction, andmultiplication by scalars.

To introduce basis vectors and resolutionof vectors into components.

Lesson 4.4:

The Algebra of Vectors

To re-examine the concepts, properties,and operations of vectors from analgebraic perspective.

To express vectors as linear combinationsof basis vectors.

To introduce the scalar (dot) product ofvectors

To introduce the Laws of Sines andCosines.

Lesson 4.5:

Vector Equations in Two Dimensions

To model motion (linear and nonlinear)using vector equations.

To relate vector equations for lines toparametric and two-variable forms ofequations for lines.

Lesson 4.6:

Vector Equations in Three Dimensions

To extend the Cartesian coordinate systemto three dimensions.

To generalize vectors to three-dimensionalspace.

To extend vector equation concepts,properties, and operations to model three-dimensional motion.

To introduce vector (cross) products.

CHAPTER 5

MATRICES

Lesson 5.1:

Matrix Basics

To review basic matrix concepts.

To provide practice with matrixoperations, including the use ofcalculators/computers to perform theseoperations.

Lesson 5.2:

The Multiplicative Inverse

To solve matrix equations.

To solve a system of equations by writingand solving a related matrix equation.

Lesson 5.3:

Systems of Equations in Three Variables

To represent systems of three equations inthree unknowns geometrically.

To determine whether a system of threeequations in three unknowns has nosolutions, one solution, or many solutions,and to describe existing solutions.

To solve systems of three equations inthree unknowns by writing and solving arelated matrix equation or by reducing anaugmented matrix.

CHAPTER 6

ANALYTIC GEOMETRY

Lesson 6.1:

Analytic Geometry and Loci

To investigate the usefulness of analytic(coordinate) geometry in solving problemsthat can be modeled geometrically.

xiiiMathematics: Modeling Our World

To introduce the concept of locus.

To introduce the idea of coordinate proof.

To apply coordinate geometry in threedimensions.

To review the distance formula, themidpoint formula, the slopes of parallellines, the slopes of perpendicular lines, thepoint-slope form of the equation of a line,and the definitions of some geometricfigures.

Lesson 6.2:

Modeling with Circles

To construct a circle and discover anequation for it using its locus definition.

To find the center and radius of a circlewhen given its equation.

To graph a circle when given its equation.

To find the equation of a circle when givencertain information.

To apply the knowledge of circles tospheres.

To investigate and apply the properties ofcircles to real-world situations.

Lesson 6.3:

Modeling with Parabolas

To use the locus definition for a parabolato construct and develop an equation forit.

To find the vertex, focus and directrix of a parabola when the equation is given.

To graph a parabola given its equation.

To find the equation of a parabola whengiven certain information.

To recognize and apply the reflectionproperty of the parabola.

To investigate and apply the properties of parabolas to real-world situations.

Lesson 6.4:

Modeling with Ellipses

To construct an ellipse and develop anequation for it using its locus definition.

To find the center, foci, and endpoints ofthe major and minor axes when given theequation of an ellipse.

To graph an ellipse when given itsequation.

To find the equation of an ellipse whengiven certain information.

To recognize and apply the reflectionproperty of the ellipse.

To investigate and apply the properties ofellipses to real-world situations.

Lesson 6.5:

Modeling with Hyperbolas

To construct a hyperbola and develop anequation for it using its locus definition.

To find the center, foci, vertices andasymptotes of a hyperbola when given itsequation.

To graph a hyperbola when given itsequation.

To find the equation of a hyperbola whengiven certain information.

To recognize and apply the reflectionproperty of the hyperbola.

To investigate and apply the properties ofhyperbolas to real-world situations.

To identify, compare, and contrast the fourconics.

xiv Mathematics: Modeling Our World

CHAPTER 7

COUNTING AND THE BINOMIAL THEOREM

Lesson 7.1:

Counting Basics

To solve counting problems by severalmethods: making a list, the basicmultiplication principle, permutations,and combinations.

To apply counting methods to probabilityproblems.

Lesson 7.2:

Compound Events

To use addition and multiplicationprinciples to determine the number ofways that compound events can occur.

To modify and apply the additionprinciple to counting problems in whichevents overlap.

Lesson 7.3:

The Binomial Theorem

To use the binomial theorem to raisealgebraic binomials to powers.

To establish connections amongcombinations, binomial coefficients, andPascal’s triangle.

CHAPTER 8

MODELING CHANGE WITHDISCRETE DYNAMICAL SYSTEMS

Lesson 8.1:

Modeling Change with Difference Equations

To define difference equations.

To use difference equations to build andsolve models that exactly model abehavior.

Lesson 8.2:

Approximating Change with Difference Equations

To define the first difference of a sequence.

To use the concept of proportionality tobuild difference equations that approximatea behavior.

Lesson 8.3:

Numerical Solutions

To find equilibrium values and investigatetheir stability numerically.

To determine long-term behavior byexamining and classifying equilibria interms of the forms of the definingdifference equations.

Lesson 8.4:

Systems of Difference Equations

To model economic, ecological, andpolitical systems with more than onedependent variable.

To investigate numerically the long-termbehaviors of the systems modeled.