Mathematics: Modeling Our World - COMAP

Background & Rationale 1

Modeling: The Applications-Based Curriculum 2

Core Curriculum 3

Student-Centered Content 3

Authentic Assessment 4

Technology/Multimedia 5

Field Testing 5

Teacher Training 6

Mathematical Concepts in Mathematics: Modeling Our World, Course 1 (Grade 9) 6



Mathematics: Modeling Our World correlation to the NCTM Standards 10

Components of Mathematics: Modeling Our World 12

Sample Pages 14

Mathematics: Modeling Our World

1B a c kg r o u n d & R a t i o n a l e

Since its inception in 1980, COMAP has beendedicated to presenting mathematics throughcontemporary applications. We have producedhigh school and college texts, hundreds ofsupplemental modules, and three televisioncourses—all with the purpose of showing studentshow mathematics is used in their daily lives.

After the publication of the NCTM Standards in1989,the National Science Foundation began tofund major curriculum projects at the elementary,middle, and secondary levels. The purpose of all ofthese programs is to turn the vision of theStandards into the curriculum of today’sclassrooms. Given the Standards’ emphasis onmodeling and applications and our commitmentto the reform movement,COMAP wanted todevelop a curriculum at the secondary level. Wesubmitted a proposal to the NSF to create aStandards-based secondary school mathematicscurriculum: Applications Reform in SecondaryEducation. In 1992,the ARISE project was one ofonly four such programs selected by the NSF forfunding.

Over the past five years, we have worked to developthis curriculum with a team of over 20 authors,almost all practicing high school teachers,including several Presidential Award recipients andWoodrow Wilson Fellows. We have field-testedthese materials with over 5,000 students across thecountry. Both our author team and our field-testers come from an amazingly diverse collectionof schools with a full range of student populations,from large urban schools in Philadelphia, PA andPortland, OR, to a small private school in Texas.Without the authors’ and teachers’ dedication andboundless energy, none of our work would havebeen possible.

The result of these labors is Mathematics: ModelingOur World. In the COMAP spirit, Mathematics:Modeling Our World demonstrates mathematicalconcepts in the contexts in which they are actually

used. The word “modeling” is the key. Realproblems do not come at the end of chapters. Realproblems don’t look like math problems. Realproblems are messy. Real problems ask questionssuch as: How do we create computer animation?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 ata problem,finding a mathematical core, workingwithin that core,and coming back to see whatmathematics tells us about the original problem.We do not know in advance what mathematics toapply. The mathematics we settle on may be a mixof geometry, algebra, trigonometry, data analysis,and probability. We may need to use computers orgraphing calculators, spreadsheets, or otherutilities. Because Mathematics: Modeling Our Worldbrings to bear so many different mathematicalideas and technologies, this approach is trulyintegrated.

At COMAP, we firmly believe in applying theNCTM Standards to both content and pedagogy.Mathematics: Modeling Our World features hands-on activities as well as collaborative learning.Simply put, many problems are solved moreefficiently by people working in groups. In today’sworld that is what work looks like. Moreover, theunits in this text are arranged by context andapplication rather than mathematical topic. Wehave done this to reemphasize our primary goal: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 willlearn. More importantly, we hope to demonstratethat using mathematics to solve interestingproblems about how our world works can be atruly enjoyable and rewarding experience.

M o d e l i n g :T h e A p p l i c a t i o n s - Ba s e dC u r r i c u l u m

Students learn best when they are actively involvedin the process. In Mathematics: Modeling OurWorld, each unit is based on engaging, real-lifesituations and the problems and conditionsassociated with them. Students

➤ analyze various voting methods used throughoutthe world.

➤ predict changes in the Florida manateepopulation relative to powerboat use.

➤ analyze the effectiveness of pooling samples inmedical testing.

Using technology and group work, students exploresituations that offer a wide variety of mathematicalconcepts. In the problem-solving process,they

➤ build a simple model,

➤ test it against various criteria,and

➤ modify the model in an effort to improve it.

The modeling cycle may be repeated several timesas new information is added. In the process,students learn and apply concepts from othersubject areas as well.

By integrating technology into the learning process,working with others to solve problems,andpresenting their findings in a variety of ways,students are better prepared than ever to enter thereal world of work.

FI E L D- TE S T TE AC H E R::

To me, modeling is like representing the problem in a

fundamental way, and showing various

representations is like making the problem accessible

to as many students as possible.The power of models

is one of the things about this program that I keep

getting goose bumps about!

2

3C o r e C u r r i c u l u m

A core curriculum is one that is suitable for allstudents. All students study a common “core” ofmaterial, but some students will have a moresophisticated conceptual understanding than others.Some field-test teachers were skeptical at first aboutusing one curriculum for all students. They feared thatsome students would be totally lost with this newapproach.

What we found in analyzing field-test results is thatmany students exceed expectations. Students withpreviously low math grades and test scores were oftenable to grasp the idea of multiple approaches toproblem-solving faster than students with traditionallyhigh math scores. Group problem-solving resulted indeeper understanding of concepts for all students. Theevaluations show that, through the interesting contentand its applications in the real world, students realizethat math is critical in solving real problems and is anintegral part of their own lives. In addition, studentslearn about a wide variety of careers, many of whichhave never occured to them before.

FI E L D- TE S T TE AC H E R 1 ::

I really had my doubts that any material could engage all

levels of kids simultaneously, but I have seen this

curriculum do it!


The depth of the sense of “why you do it” and the breadth

of “where you use it” becomes a tapestry of understanding

that isn’t ever going to fade.


There was a definite increase in student attendance, but

the big difference was in attitude and self-esteem.The

students enjoyed math,some for the first time.They were

proud of themselves:they bragged to other students and

teachers about their work. They were well aware that they

were dealing with concepts which were not covered in the

traditional classes in our school,and they were proud to be

doing so.

S t u d e nt - Ce n t e r e d C o n t e n t

The content of Mathematics: Modeling Our Worldappeals to students with topics that they findinteresting and challenging. Students are able tointegrate what they are learning in math into othercontent areas, such as English, science, social sciences,political science, physical education, and family andconsumer sciences.

In today’s workplace, people need to solve problemscooperatively as well as independently. And in real life,it is clear that there are many ways to solve a givenproblem. Mathematics: Modeling Our World is activity-based: students are given the opportunity to discoverproblems and devise a variety of strategies to solvethem. Students are also taught to use differentresources to solve problems, and they learn to chooseresources that meet the needs of a particular situation.They are frequently asked to present and defend theirsolutions in a number of ways, including oral reports,written summaries, projects,and demonstrations.

FI E L D- TE S T ST U D E N T 1 ::

The curriculum’s purpose is to actually make us think and

sort out the problem . . . . It’s all about thinking and

working with the questions. I’ve gathered and learned so

much information in this class. I think it’s the first time I

can say I’m really enjoying math (honestly).


This program is better because you don’t just add, subtract,

multiply and divide and get confused with a bunch of

numbers.They teach you things you will probably use in

your life. I think most people fail something because they

have no interest in it. But this program helps you to think

and understand. I want to continue this course until I

graduate . . . .

A u t h e n t i c A s s e s s m e n t

In Mathematics: Modeling Our World assessment isan integral part of the curriculum. Throughout thetext, both activities and individual workassignments offer multiple opportunities toevaluate student progress. Working together andalone, students analyze information, collect andcheck data, and predict outcomes based on whatthey have learned.Students share their preliminaryfindings by presenting oral reports, producingcharts and graphs, building models, and writingsummaries to explain how they reached theirconclusions.

The Teacher’s Resources provides assessmentproblems for use throughout each unit,as well asunit projects. Each unit concludes with a writtensummary of the major mathematical conceptsdeveloped and applied in the unit. A “WrappingUp” problem set provides more opportunities forstudents to extend what they have learned to newproblems that help them analyze and synthesize thetopics studied in the unit.

A separate Solutions Manual contains answers toall of the questions in the text,including theConsider questions. Because the majority of thequestions and problems are open-ended,theSolutions Manual offers a variety of responses.


I love this program. It’s very appealing to me. Before I

was getting C, D, F, now I’m getting As and Bs. I

recommend it.


I think this is a very good program.It’s helped a lot. I

was getting Ds and Cs in regular algebra,but now I’m

getting Bs and As.


I’m much more open about how to evaluate a

student’s progress. Journals, portfolios, authentic

assessment, computer programs, projects, speaking

opportunities—they all are fair game for any student

to demonstrate understanding, personal growth,and

accomplishments.

4

Te c h n o l o g y / Mu l t i m e d i a

Every unit of Mathematics: Modeling Our World beginswith a short, motivational video segment accompaniedby a brief written guide with questions for students toconsider as they view the video.

Both graphing calculators and computers are usedextensively throughout the curriculum. By using avariety of tools, students learn to select the mosteffective tool for the problems they are solving. Softwarewritten specifically for Mathematics: Modeling OurWorld is provided with the program; other software maybe downloaded from the Internet at no charge. While itis strongly recommended that computers be used withthis curriculum, material is provided to teach thelessons without computers as well.

It is also recommended that classroom sets of graphingcalculators and a calculator-based laboratory (CBL) bepurchased for students using Mathematics: ModelingOur World—it is a cost-effective way to serve a largenumber of students.


Only one student ever asked,“When will I ever have to use

this?”, a question which I heard often in prior years.This

time it was asked by a student who had just transferred

into my class in the middle of the year.We were in the

Animation unit and he resented having to learn how to use

the graphing calculator. So he and I had a little talk about

technology and the role it plays in his chosen career: auto

mechanics. He then reluctantly agreed to try to master the

calculator.Within two days he was begging to take home a

TI-82 (but he couldn’t because I have only one classroom

set.) He was soon the class expert, happily explaining the

work to other students. In the end, he was the first person

in the class to complete his fireworks animation. He ran

down the hall to show his English teacher what he had

done on the calculator. At the end of the year, he made sure

his advisor put him in my class for next year.

F i e l d T e s t i n g

The first draft of Mathematics: Modeling Our Worldwas completed in the summer of 1994, and fieldtesting began in 15 high schools at the beginning ofthe 1994–1995 school year. The high schools chosenfor this field test represent a wide range of studentpopulations. There are large,urban schools inPhiladelphia, PA and Portland,OR, medium-sizeschools in suburban and rural districts, and a smallprivate school in Texas. In the first three years of fieldtesting, more than 5,000 students have used or arecurrently using this curriculum.

Multimedia Research, an independent evaluator,conducted a formative evaluation of the text andprovided written reports for every unit of the text. AsCOMAP made changes to various units,field-testteachers incorporated the revisions into their workwith students. The teachers meet with COMAP on aregular basis and provide valuable feedback.

These teachers have now been using Mathematics:Modeling Our World for three years, and both teachersand students remain enthusiastic about thiscurriculum. The field-test teachers are also spreadingthe word about the program to other school dist ricts.At a recent meeting of field-test teachers, one teacherannounced that he had teachers from 15 other schooldistricts visit his classroom to learn about thecurriculum. Others reported that they have giventeacher workshops in neighboring school districts.COMAP will continue to work with these teachers torefine this dynamic curriculum.


This past school year was the most fun I ’ve had in 30 years

of teaching. Every day was like an adventure where we

never knew what we would discover.The students

continually surprised me.They love using the TI-82s and

amazed me with how rapidly they learned to per form

various operations. Sometimes they found an easier or

faster way to do something on the calculators, as they were

continuously exploring the possibilities.

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6Te ac h e r Tr a i n i n g

Mathematics: Modeling Our World may be differentfrom curriculum you have taught in the past, butthe mathematical concepts are here, plus a lotmore. We recognize that adopting a new program isexciting, but it also involves learning new material.COMAP realizes that helping you feel comfortablewith a new curriculum benefits everyone, so wehave addressed the need for training in a variety ofways:

➤ COMAP’s Leadership Institute to immersesupervisors in the program with two- to three-day meetings that provide hands-on experiencewith the curriculum.

➤ For adopters of Mathematics: Modeling OurWorld, COMAP provides on-site training led bysomeone well-versed in the program: authors,field-test teachers, or COMAP personnel.

➤ COMAP has developed teacher-training videosto provide background in all of the new NSF-sponsored curricula, including specific examplesfrom Mathematics: Modeling Our World.

➤ COMAP’s Website at www.comap.com providesupdated information about the program.

➤ Call COMAP toll free at (800) 772-6627 to speakdirectly to someone well-versed in the program.

For those considering adoption of Mathematics:Modeling Our World, funding may be availablefrom a number of sources. The NSF, theEisenhower Mathematics and Science RegionalConsortia, and other sources might pay for an on-site trainer; your school district may also havefunding available.


Career awareness, interdisciplinary connections,

higher-order thinking skills, communication

experiences, exposure to technology—plus “fresh”

math to boot! It almost seems indecent!


This is definitely a more involved, successful way of

teaching absolute-value equations (and in 9th or 10th

grade, rather than in 10th or 11th). Our students are

very good at this.

Unit 1

Pick a Winner: Decision Making in a Democracy

Number sense 1–4Percentages 1–4Preference diagram representation 1–4Graph theory 2–4Paradox 2–4Matrices 3

Unit 2

Secret Codes and the Power of Algebra

Mathematical modeling 1–7Functions and linear functions 2–7Representations of functions: tables, graphs,symbolic equations, arrow diagrams 2–6Algebraic expressions 2–7Matrix operations: addition, subtraction,scalar multiplication 2,5, 6, 7Modular arithmetic 3, 5Solving equations 3–6Inverse of a function 3, 5, 6, 7Frequency distributions 4–7Order of operations 5, 6Equivalent expressions, distributive property 5, 6

Unit 3

Landsat

Distance 1, 3, 4Scale 1–4Graphical interpretation 1–3Unit conversion 1, 4Scale factor 1–4Ratios 1–4Precision 1Significant figures 1Relative size 2, 3Pixel 2Digitization 2, 4Corresponding parts 3Shape 3Similarity 3Proportionality 3Solving proportions 3, 4Pythagorean theorem 3Coordinates 3Dilation 3Translation 3

Mathematical Concepts Lesson(s)

Mathematical Concepts in Course 1

Area 4Length-area relationship 4Approximation 4Monte Carlo methods 4

Unit 4

Prediction

Dot plots 2Scatter plots 1–4Mean 2Slope 1Variable 1Linear equations 1Graphing lines 1Collecting data 2Interpreting data 1–4Fitting a line to data 1–4Residuals 3, 4

Unit 5

Animation/Special Effects

Coordinate systems 1–5Continuous and discrete representations 1–5Rates of change 2–6Variables and constants 2–5Recursive and closed-form representations 2–6Linear functions 2–6Elementary programming 3–6Matrices 3–6Parametric equations 4–6Graphs of functions (time series, state-space, time lapse) 4, 5Systems of linear euqations 5Averages 6

Unit 6

Wildlife

The modeling cycle 1, 2, 3, 5Representing quantitative information:graphical, numerical, symbolic 1–5Linear functions 1, 2Function notation 1Parametric graphs 1Additive processes 1–3Recursive and closed-form descriptions 2–4Inequalities 2Simulation 2–5

Translations 2–4Scale changes 2–4Sensitivity 3–5Exponential functions 3–5Relative rate 3–5Growth factor 3–5Growth and decay 4Exponential graphs 4Properties of exponential functions 4Laws of Exponents 4Identifying exponential data 4, 5Probabilistic models 5

Unit 7

Imperfect Testing

Decimals, fractions, percentages 1–4Estimating a population percentage 1–3Variability due to sampling 1, 2Probability 1–4Mutually exclusive and complementary events 1Randomized response technique 2Conditional probability 2–4Two-way tables and tree diagrams 2–4Independence 2Linear models 3Domain of a model 3, 4Inverses of linear functions 3Rational functions 4Bayes theorem (informally only) 4

Unit 8

Testing 1,2, 3

Probability 1–3Expected value 1–3Curve-fitting and residuals 1–4Law of large numbers 2Quadratic function 2–5Area model 3, 5Solving quadratic equations 4, 5Transforming graphs 4, 5Vertex form of a quadratic function 4, 5Completing the square 5Quadratic formula 5

Mathematical Concepts Lesson(s) Mathematical Concepts Lesson(s)

7Mathematical Concepts in Course 1

Unit 9

Gridville

Distance (fire-truck geometry)Shortest path

(fire-truck geometry)Round-trip distanceTotal distanceAverage distanceAbsolute value as distanceGraphs on a number lineAbsolute-value function graphsInequalitiesSlopeLinear equationsSolving absolute-value equationsAbsolute-value inequalitiesPiecewise-defined functionsTransformationsMedianMidrangeMinimax location

Unit 10

Strategies

Zero-sum gamesNonzero-sum gamesPayoff matricesFair gamesExpected payoffAverage (mean)OptimizationProbabilitySimulationsLinear functions and modelsSystems of equationsProbability distributionsTree diagramsValue of a gameFunctions of two variables

Unit 11

Hidden Connections

Elementary graph theoryOptimizationAlgorithms and heuristicsVertex coloringProofMinimum spanning treesKruskal’s algorithmStable matchingsGale-shapley algorithmTraveling salesperson problem

Unit 12

The Right Stuff

AnglesMeasurementBisectorTangentsArea and perimeter30-60-90 triangle relationshipsRight trianglesPythagorean theoremParallel and perpendicular linesAlternate interior anglesCorresponding anglesVertical anglesSymmetry: reflection

and rotationSimilarityInductive reasoningDeductive reasoningCounterexampleIndirect proof

Unit 13

Proximity

DistanceArithmetic meanVoronoi diagramAreaVolumeWeighted averagesAlgorithmsGeometric constructionsPolygonsPerpendicular bisectorReflectionCoordinate geometry of points,

lines, and polygonsDeriving formulasSystems of equationsCongruenceIterationSupplementary angles

Unit 14

Growth in the News

Conversion of units of measureSequencesSubscript notationRecursive formulasFirst differencesRate of growthLinear graphsAdditive growthArithmetic sequences

Closed formulasMultiplicative growthGeometric sequencesExponential graphsPartial sumsPercentage rate of growthQuadratic sequencesQuadratic growthQuadratic graphs (parabolas)Mixed growthMixed sequencesUnbounded growthBounded growthConstant sequence

Unit 15

Motion

Rate of changeLinear equationsPiecewise equationsData analysis and residualsQuadratic equationsSystems of equationsParametric equations

8

Unit 16

The Geometry of Art

ProjectionAnglePerspectiveParallel linesPerpendicular lines and planesScale and proportionSimilar trianglesConstructionRight triangle trigonometryParallelograms

Unit 17

Fairness and Apportionment

Fair divisionMeasuring fairnessAlgorithmic thinkingModelingTabular reasoningApportionment methodsFractional partsRoundingRatioParadoxMinimizing unfairnessRelative unfairness

Unit 18

Sampling

Simulation methodsFrequency distributionReference distributionLikely-sample groupSampling methodsProportionData collectionInterpretation of data

Unit 19

Mind Your Own Business

OptimizationSubstitutionFunctions of more than one variableAverageRational functionsAsymptotic behaviorStep functions

Unit 20

Oscillation

Periodic functionsSineCosineUnit circleDegrees and radiansTrigonometric equations and graphsDamped harmonic motionExponential decayGeometric progressionsPiecewise functionsEquations of best fitAnalysis of residualsTrend analysis

Unit 21

Feedback

Data analysisRegressionResidualsCausal loop diagramsFeedbackPositive and negative feedbackInterpretation of graphsPredictionLine diagramsLevelsRatesTime series graphsFlow diagramsRate equationsAsymptotic behaviorLogistic curveStabilityEquilibriumDynamicsFixed pointCarrying capacityWeb diagramsPhase plotsOne-population modelsInteracting population modelsSimulation

Unit 22

Modeling

Mathematical modeling

Other concepts depend upon individual projectsstudents select to research, construct models, test,and prepare in report form.

9

10This chart addresses the first course of the curriculum Mathematics: Modeling Our World.

NCTM STANDARDS Unit 1 Unit 2 Unit 3

Pick a Winner: Secret Codes LandsatDecision Making and the Powerin a Democracy of Algebra

STANDARD 1MATHEMATICS AS

PROBLEM SOLVINGX X X


COMMUNICATIONX X X


REASONINGX X

STANDARD 4MATHEMATICAL

CONNECTIONSX X

STANDARD 5ALGEBRA

X X

STANDARD 6FUNCTIONS X X

STANDARD 7GEOMETRY FROM

A SYNTHETIC

PERSPECTIVE

STANDARD 8GEOMETRY FROM

AN ALGEBRAIC

PERSPECTIVE

X

STANDARD 9TRIGONOMETRY

STANDARD 10STATISTICS

STANDARD 11PROBABILITY

X

STANDARD 12DISCRETE

MATHEMATICSX X

STANDARD 13CONCEPTUAL

UNDERPINNINGS

OF CALCULUS

X

STANDARD 14MATHEMATICAL

STRUCTURE

Decide which voting systemis appropriate in differenttypes of elections.

Apply arrow diagrams,symbolicequations, tables,and graphs toevaluate the effectiveness of codingtechniques.

Understand variables, expressions,and equations in the context of thehistory and future of secret codes.

Examine satellite pictures usingproperties of regular polygonsto estimate the area of anirregularly shaped region.

Utilize properties of similartriangles to study the effects ofzooming on a satellite image.

Create a digraph to model a run-off election.

Unit 4 Unit 5 Unit 6 Unit 7 Unit 8

Prediction Animation/ Wildlife Imperfect Testing 1, 2, 3Special Effects Testing

X X X X

X X X X

X X X X X

X X X X X

X X X X X

X X X X

X X

X

X X X X

X X X

X

X X X

X

Use a variety ofmathematical tools toanalyze why a moosepopulation is dwindling.

Create and presentanimations using graphingcalculators,matrices,andparametric equations.

Use linear functions to explorerelationships using actualforensic data.

Experiment withtransformations ofparabolas to analyzethe cost-effectivenessof pooling samples formedical tests.

Analyze data from powerboatregistrations and manatee deathsto explore a possible relationship.

Use conditional probability tounderstand the elusive “false positives”in athletes tested for performanceenhancing drugs.

Explore price andrevenue variables todetermine theminimum andmaximum points in agraph.

Compare the recursive vs.closed-formrepresentation methodsfor animating an image.

11

Components of Mathematics: Modeling Our World12

ST U D E N T TE XT:

• Mathematical conceptsare presented in 8 units inthe context of real worldproblems.

• Open-ended questionsand problems encouragestudents to workindependently and ingroups to find better waysto improve upon theoriginal model.

VI D E O SU P P O RT:

• Video segmentsaccompany each unit andare used to motivatestudents as they begin aunit, or to provideadditional information fora specific problem.

C D - RO M:

• Mac and IBM formats areavailable.

• Contains softwareprograms writtenspecifically forMathematics: Modeling Our World.

Components of Mathematics: Modeling Our World 13

TE AC H E R’S RE S O U RC E S:

FOR TEACHERS

• Ideas for presenting videosegments

• Additional notes,Background Readingsand use of Assessmentproblems

• Transparencies

FOR STUDENTS

• Supplemental Activities

• Handouts

• Assessment Problems

AN N OTAT E D TE AC H E R ’S ED I T I O N:

• Background informationabout mathematicalconcepts and unit content

• Page-by-page suggestionsin the wrap-around tohelp teachers presentstudent material

• Notes with additionalbackground about a topic

• References to the Teacher’sResources

SO LU T I O N S MA N UA L:

• Answers to all of theConsider questions,Activities, IndividualWorks, and allsupplementary materials.

• A variety of answers areprovided for the manyopen-ended questions.

14

FI E L D- TE S T ST U D E N T::

We will use this in life. Neat to know ratios of shapes.

FEATURES

Using technology

Producing a table anda graph from data

Developingmeasurement andrecording skills


These students really enjoyed this computer work.They have

difficulty reading, but it is amazing how independent of me

they have become as they do their computer work.I also

overheard one student explain in terms of reflectance what it

meant when she sees “blue”. She did an excellent job.

182


Very helpful simulation. The students were wondering how we were

going to determine numbers of births and deaths.Once I had helped

some pairs, I think all groups caught on to the process.This had high

appeal . . . a very productive group activity.

FEATURES

Modeling a realproblem

Understanding visualrepresentations

Writing aboutmathematics

15

475

16FI E L D- TE S T ST U D E N T::

One problem in particular comes to mind. It is that of the marquees.This problem was about

the timing of lights, lights turning on and off.We had to figure out when the light was

supposed to come on and when it was going to go off. I have to be honest;at first, I wasn’t too

sure about trying something new. Now that I have done the math,I am amazed. I never

thought that I would be interested in math;now I am.

FEATURES

Using data to describereal-world phenomena

Interpreting data

321


A wonderful experiment with a lot of positive student responses.

FEATURES

Applying conditionalprobability

Working with treediagrams

17

558


My favorite lesson would have to be the first lesson;I think it was the voting lesson.To

me, it was fun interacting with people and asking them about what they like, but also

it was a chance to get acquainted with everyone else. After we were done with the

tallying of votes, we ranked them from greatest to least.Then we arranged them into a

preference schedule and into a matrix.But it wasn’t just about voting;I also learned

how to change a losing vote into a winning vote, and how to make a practical decision.

FEATURES

Using actual data

Creating andinterpreting diagrams

Writing aboutmathematics


We would be able to use our own ingenuit y, instincts, and new knowledge to work out a code.The interaction with

your peers during this section worked well; your partner would do his best to come up with a code that would be

difficult to crack,then your task is to break it. It was a competition;try to be faster than your partner.Then you’d

work together at cracking a code that the whole class is working on.I learned a lot from this section.I learned how

to graph equations, what is entailed in the equations, and how to decode.Then we learned about matrices, a way

to code that no one could break easily without the key. I think I enjoyed this section because it exemplifies what

[Mathematics: Modeling Our World] is all about: a new way of learning, where you explain your thinking and

then use it;instead of letting those skills sit on the shelf, you see where they can be put to use.

FEATURES

Reading about real-world situations

Connectingmathematics to otherdisciplines


Uneasiness has disappeared. Discussion and participation are up in class. I don’t have to call on

students, they volunteer information.Voting insincerely and the concept of paradox surprised

them. Having a local 3-way race helped students see the need for alternate voting methods.

FEATURES

Exploring amathematical ideausing the Internet

Learning howmathematics plays arole in society

Teaching problem-solving techniques

20

21

FEATURES

Teachingcommunication skills

Developing topics forclass discussion

Participating inhands-on activities


The class was telling ME what each step was (logically) as well as what

keys to press in order to perform them.I get the feeling that they are

getting really comfortable doing these regressions and they really

understand using different regressions to determine the best model.

FEATURES

Using graphingcalculators

Applying regressionmodels

Graphing functions

22

611

2,

23

FI E L D- TE S T ST U D E N T::

I learned many things, but the one thing I feel really helped me was that I

found other ways to approach and look at math . . . I learned a lot of ways

and methods to approach equations, graphs, charts, etc. So, I think my

highlight from last year was that my mathematical knowledge was

expanded to more than one area of math.

FEATURES

Creating a diagram toexplain the steps in anequation

Using algebra to solvereal-world problems

Understandingequivalentrepresentations of aconcept

267

24


I am finally encouraged to think about and

discover about math rather than being “spoon-fed”

about math.

FEATURES

Working on a groupproject

Using the Internet forresearch

Exploring factors toinclude in amathematical model


I think that this program not only helps students

prepare themselves mathematically for college, but

it also helps in regular daily lives . . . because it

gives self confidence to those who don’t have it.

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Mathematics: Modeling Our World - COMAP