Chapter 1 Mathematical Modeling - Denton ISD

Chapter 1Mathematical ModelingMathematical OverviewThis chapter opens the door for students to better understand the use ofmathematical modeling when presented with a situation or problem to solve. Theybegin by examining a presentation of different forms of mathematical models andproceed to using ratios and proportions to create a model for estimating animalpopulations. Then students explore how proportions can be used to model a varietyof other real-world situations. In the last lesson of this chapter, students examinethe use of words, graphs, and tables as models to relate one real-world quantityto another. Understanding which model best describes a situation, looking closelyat that model, discovering patterns in the model, and describing the patternsmathematically are steps students will use throughout this book to solve problems.Lesson SummariesLesson 1.1 Activity: Animal PopulationsIn this Activity, students use beans to represent a wildhorse population. Each bean represents one wild horse ina population to be estimated. During the Activity, studentssimulate a technique known as capture-recapture to estimatehow many horses there are without actually countingeach horse. Students write ratios, set up proportions usinga variable to represent the total number in the horsepopulation, and then solve for the variable.Lesson 1.2 Proportions as ModelsIn this lesson, students extend their knowledge ofproportions by representing and solving a variety of realworld situations. The first real-world situation they examineis the cost of driving a given distance when the cost permile is constant. Then students examine using a scale froma scale drawing to find the actual width of a room. A thirdsituation draws on students’ recall of geometry. Studentsuse corresponding sides of similar polygons to calculatethe scale factor, set up a proportion, and then solve for anunknown length for one of the polygons.Lesson 1.3 R.A.P.In this lesson, students Review And Practice solvingproblems that require the use of skills and concepts taughtin previous math levels. The skills reviewed in this lessonare skills that are needed as a basis for solving problemsthroughout this course.Lesson 1.4 Investigation: Patterns and ExplanationsIn this Investigation, students are given a situation andasked to choose a graph or a table that best models therelationship between the variables in the situation. Theydiscuss the features of several qualitative graphs andidentify the two variables in each situation. They alsoexamine patterns in graphs and tables to better understandhow to use mathematics to describe the relationshipbetween the variables in a given situation.1aCOMAP2e ATE ch01.indd 1a25/02/12 12:05 PM

Lesson GuideLesson/ObjectivesMaterialsChapter 1 Opener: What Is a Mathematical Model? recognize that many different representations can beused to model real-world situations.Per group: white beans or other small objects that canbe marked with a marker (about 150) small paper bag or other container forholding the beans permanent marker1.1 Activity: Animal Populations use ratios and proportions to create mathematicalmodels. use mathematical models to estimate the sizes ofpopulations. solve proportions.1.2 Proportions as Models use proportions to model real-world situations. solve problems that involve scale drawings. solve problems that involve similar polygons.Optional: TRM table shell forQuestion 5.Optional: TRM shell for theVocabulary Organizer an item that shows ascale (map, blueprint,model car, etc.)1.3 R.A.P. (Review and Practice) solve problems that require previously learnedconcepts and skills.1.4 Investigation: Patterns and Explanations use multiple representations to model real-worldsituations.Pacing GuideDay 1Day 2Day 3Day 4Day 5Day 6Basicp. 2, 1.11.21.31.4projectreviewStandardp. 2, 1.11.21.31.4projectreviewBlockp. 2, 1.1, 1.21.3, 1.4project,reviewSupplement SupportSee the Book Companion Website at www.highschool.bfwpub.com/ModelingwithMathematics and the Teacher’s Resource Materials (TRM)for additional resources.1bCOMAP2e ATE ch01.indd 1b24/02/12 11:44 AM

CHAPTER 1CHAPTERCHAPTERChapter 2 MathematicalDirect delingCONTENTSChapter Opener:CONTENTSWhatIs a MathematicalHowIs MathematicsRelatedModel?toLesson1.1Bungee Jumping?ACTIVITY:Animal PopulationsLesson2.1Lesson1.2ACTIVITY: Bungee Jumping235337Proportionsas RelationshipsR.A.P. (ReviewAnd Practice)63912LessonLesson2.31.4DirectVariation FunctionsINVESTIGATION:PatternsLesson2.4 and ExplanationsRAPModelingProject:Lesson2.5 Is Worth a Thousand WordsA Picture431447Slope185319It’s Only Water Weight57Chapter ReviewModeling Project:Chapter Review60Extension:Inverse VariationComap2e Modeling Ch01.indd 1672/3/12 7:04 PM1COMAP2e ATE ch01.indd 124/02/12 11:44 AM

How isIsMathematicsRelatedtoWhata MathematicalModel?Bungee Jumping?CHAPTER 1OPENERThe process of starting with a situation or problem and gainingAbungee cordaboutis an elasticcord thatcan betomathematicssecure objectsunderstandingthe situationthroughtheuseduse ofisEngagewithoutastyingknots. Specializedbungeecords are usedin theknownmathematicalmodeling.The mathematicaldescriptionssport of bungeejumping.of the cord isattachedto aobtainedin the processare Onecalledendmathematicalmodels.TheseLesson Objective recognize that many differentrepresentations can be used tomodel real-world situations.bridgeother endattachedto themodelsoraretower,often andbuilttheto explainwhyisthingshappenin ajumper.certain ll. At theThey are also created to make predictions about the future.bottom of the jump, the cord recoils and the jumper bounces upMathematical models can take many different forms. Among them are:and down at the end of the cord.VocabularyThe strength of the cord used for a bungee jump must beTimeTemperature“The U.S. Postal Service acceptsverbalaccurately known. The cord must be adjustedthe height(minutes) for ( F)a package for mailing only if thedescriptions040Otherwise, thesum of its length and girthofis thenot jump and for the weight of the jumper.548more than 108 inches.” consequences can be disastrous. In one 10well-publicizedcase, a54tables of1559woman diedinformationpracticing a bungee jump exhibitionforthe19972062 mathematical modeling mathematical modelsDescriptionThis chapter sets the stage forthe entire course. The chapter ispurposely short, yet it shows mostof the different types of lessons:Activities, Investigations, R.A.P.lessons, Modeling Projects, andChapter Reviews. The mathematicalskills reviewed in this chapter arenecessary for student success in futurechapters.This reading introduces studentsto the process of mathematicalmodeling and the different forms thatmathematical models can take.Super Bowl half-time show. The bungee cord was supposed toequationsstop her 100-footfall just above the floor of the Superdome inyh 2 1.5nNew Orleans. At the time, officials were quoted in The BostonGlobe as sayingc40'– 1"aApparently, she graphsmade an earlier jump and didn’t come as close as theywanted. They made some adjustments, and somebody made a miscalcuBedroomMasterBedroomlation. I think it was human error. 9'8" 11'6" Living room 13'3" 17'2"bx15'6" 14'0"ToiletBungee safety is a product of simpleBathmathematicsDiningthat factors height andHallweight in itsformulascalculations. It’s so predictable.Bath IA r2KitchenBedroom9'8" 11'6"Walk–in5e54'– 0"Ratios can be used to model bungee jumping. Knowing howmuch the cord stretches for different jumper weightscan helpGarageensure thatbungeeandjumps are safe.drawings19'8" 22'2"diagramsphysicalmodelsTEACHING TIPAfter students have read the ChapterOpener and examined the examples ofthe different types of representationsdiscussed in the reading, lead a wholeclass discussion asking students to giveother examples of each type of model.A good mathematical model is one that helps you betterunderstand the situation under investigation.2Chapter 1Comap2e Modeling Ch01.indd 2M AT H E M AT I C A L M O D E L I N G03/02/12 11:20 AM2COMAP2e ATE ch01.indd 225/02/12 8:19 AM

Lesson 1.12.1ACTIVITY:Total: ming(4,381)Colorado(767)New Mexico(115)LESSON 1.1Animal Populations5eRecall that a ratio is a comparison of two numbers bydivision. A ratio can be written in the form of a fraction.In this lesson, you will use ratios to create a model forestimating animal populations.Lesson ObjectivesIn 1900, there were about 2 million mustangs (wild horses)in the western United States. By 1950, there were fewerthan 100,000, and at the start of the 21st century, onlyabout 37,000 remain.It would be almost impossible to actually count thenumber of horses in a given region. Instead, a techniquecalled capture-recapture (or mark-recapture) is used.In this Investigation, you will use beans or other small objects in acontainer to represent a wild horse population.1. Each bean in your container represents one horse in a populationto be estimated. Scoop out some of the beans and count them.How many beans did you scoop out of your container?2. Let the beans that you scooped out represent the horses that willbe marked. To simulate marking horses, mark each bean that wasremoved from the container with a permanent marker. Then putthe marked beans back in the container with the “uncaptured”beans. Mix the beans in the container well. This is equivalent toletting marked horses mix in with the population of unmarkedhorses. How many marked beans did you put back in yourcontainer?Lesson 1.1 Activity Answers1. Sample answer: 372. Sample answer: 37, the samenumber that were taken out3. Sample answer: 424. Sample answer: 5Materials ListPer group: white beans or other small objectsthat can be marked with a marker(about 150) small paper bag or other containerfor holding the beans permanent markerPreparation:This lesson is designed as a wholeclass/small group activity (2–4students). Prior to class, place anunknown number of white beans ineach bag (one bag of about 120–150beans per group).4. How many of these beans are marked beans that have been“recaptured?”Comap2e Modeling Ch01.indd 3 proportion ratio variableDescription3. After the beans have been thoroughly mixed, scoop out a secondgroup of beans and count the number in this group. How manydid you scoop out?Lesso n 1 .1 use ratios and proportions to createmathematical models. use mathematical models toestimate the sizes of populations. solve proportions.VocabularyIn this process, a number of animals are captured andmarked in some way. Horses are often branded on aneasily-seen part of the body. Then the marked animals are released.After they have had time to mix in with the rest of the animals in aregion, a second group is captured. Finding the number of markedhorses in this group makes it possible to make an estimate of theentire population.A N I M A L P O P U L AT I O N SEngage303/02/12 11:20 AMCONNECTIONMethods similar to the capture-recapturemethod used in this lesson are used toestimate the population of homelesspeople in large cities. A known quantityof people acting as decoys is planted inthe street population. Then the numberof decoys later spotted during a searchfor homeless people is recorded.During the Activity:Give each group one bag of beans andpoint out that the number of beans isunknown. Explain that the object is todetermine the number of beans in thebag without counting them.As students remove some of theirbeans, have them count and markthem with a permanent marker. Oncedone, they should place all of themarked beans back into the bag.Closing the Activity:Remind students that this procedureis used when you are physicallyunable to count an unknownpopulation. Reinforce the modelingaspect of this Activity by askingstudents to explain why they thinkthis procedure provides them with areasonable estimate of the number ofbeans in their bag.3COMAP2e ATE ch01.indd 324/02/12 11:44 AM