Download - Module 5 - Mathematical Modeling · 2015-11-04 · 2015 Mathematics Teaching Institute Module 5 Topic: Authentic Mathematical Modeling The purpose of this session is to provide 9‐12th

2015MathematicsTeachingInstitute

Module5

Topic:AuthenticMathematicalModelingThepurposeofthissessionistoprovide9‐12thgradeteacherswithafewopportunitiestoengageintheprocessofmathematicalmodelingtosolveauthenticproblemsrootedintherealworld.Theprimaryfocusisforteacherstodevelopandrefinemathematicalmodelsasstudentlearnerstoprovidethemasenseofwhatitmightlooklikewhenimplementedintheirclassroom.

MathematicalElements Applyingmathematicstosolvemathematicalmodelingproblems Generatingmathematicalrepresentationsofareal‐worldsystem(mathematizingspace) Learningaboutassumption‐makingandcontrollingforvariablesinamodelingproblem

PedagogicalElements Developingskillsinfacilitatingmathematicalmodelingactivities Helpingstudentsinmakingassumptionstocontrolforthelargenumberofvariables

presentinauthenticmodelingproblems

ActivitiesTasksandGuidedQuestionsforDiscussion

Discussion:DefinitionofMathematicalModeling–Thisactivityprovidesanopportunityforteacherstodiscusstheirideasaboutwhatmathematicalmodelingis,andwhyitisimportant.Thepurposeofthissessionistoprovidetheteacherswithacommondefinitionformathematicalmodelingandexposuretoauthenticmathematicalmodelingtasks.Thisdiscussionbuildsupontheprevioussessionsasproblemsencounteredhavebecomemoreopentomultiplesolutionstrategiesandavarietyofanswers.ThisspacewillbeusedforteacherstoaskquestionsaboutmathematicalmodelingandframetheKrogerFuelPointProblemactivity.

MathematicalModelingContexts–Thepurposeofthistaskistoexposetheclassroomteacherstoavarietyofmodelingcontextstoexplore.Teacherswillbesplitintogroupsof3‐4,andprovidedaminimumoftwomodelingcontexts(listedbelow).Followingthecompletionofthesetwotasks,responseswillbesharedtotheentiregroups.Thepurposeofthisactivityistogenerateadiscussionabouttheskillsneededinsolvingtheseproblems,andhowthesemaybeenactedinaclassroomenvironment.Theteacherswillbeaskedtoprovideamathematicalandpedagogicalanalysisofeachproblemcontext.Supportstaffwillbeavailableinthesolvingoftheseproblemstohelppushteachersmathematicallyandassisttheminconsideringandaccountingforalternatevariables.

KrogerFuelPointsProblem–GivetheteachersacopyoftheKrogerFuelPointsHandoutandputthemintogroupsoffour.Allowtheteacherstoworkthroughthinkingabouthow


tomaximizetheirsavingsfortheyearusingtheconstraintsset.Oncegroupshaveconstructedamodeloftheproblemscenario,teachersshouldbeencouragedtoengageinadditionaliterationsofthemathematicalmodelingcycletorefinetheirmodel.OSUMathematician(Fowler)willbeavailabletohelpgroupsextendtheirmathematicalrepresentationsoftheproblemscenario.Twothingstohighlight:whatassumptionsweremadeinconstructingthemodelandwhatvariablesdidthemodelaccountfor.Theteachersshouldbeaskedattheend:

‐ Whatfactorsdidyouoriginallyconsiderwhencomingupwithasolution?Arethereanythatyoudismissed,andwhy?

‐ Wouldyouimplementyourstrategy?Whyorwhynot?‐ Isthereanythingthatcouldimproveyourstrategy?

RainfallProblem–GiveteachersacopyoftheRainfallProblemHandout.Inthiscontextteachersareaskedtoexplorewhetheritwouldbemorebeneficialtowalkorrunifcaughtinarainstorm.Whatfactorscandeterminethis?Aretherebettertimestowalkversusrun?Buildamodeloftheproblemsscenarioanduseittodetermineforwhatconditionswouldbemoreoptimaltowalkversusrun.

CostofMe(Dossey)–Buildamodeltodeterminethe“costofme.”Thisproblemcouldbeinterpretedfromavarietyofstandpoints(elementsthatmakeapersonup,costofyoureconomicfootprint,budgeting,determinationofhowmuchmoneyisspentbyotherstosustainyou,etc.).Constructamathematicalmodeltohelpyoudeterminethecost.Insharingouttothelargegroup,teachersshouldemphasizehowtheyinterpretedthequestion,andhowtheywentaboutsolvingit.Astheywork,teachersshouldbemindfulofthemathematicsrequiredtodetermineasolutiontothiscontextaswellashowthistypeofproblemmightbeimplementedinaclassroom.

Postman’sLetterProblem‐Supposethatastreethas10mailboxesoneachsideofthestreet.Apostmanindeliveringletterscaneitherwalkalongonesideofthestreetuntiltheyreachtheendandcrosswalkingbacktheotherway,orcandelivertoamailbox,crossthestreetbetweenmailboxes.Canyoudetermineanoptimalpathforthepostmantotaketodeliverhismail?Whatassumptionsdidyouaccountforinyoursolution?

ConcertProblem‐Arockbandisputtingonaconcertatyourlocaltheater,andyouhavebeenhiredtodeterminehowmuchtochargefortickets.Youhavebeenprovidedalayoutofthefloorplanofthevenue.Whatfactorswillyouaccountforinordertodetermineaticketprice?Whatadditionalinformationdoyouneedinsolvingthisproblem?Thelastvenuetheyplayedatofferedyouaphotoofthecrowdtoassistyouinyourplanning.Howcanthisbeusedtodeterminethepriceofeachticket?

PretzelJarProblem–TeacherswillremainintheirworkgroupsfrompreviousactivityandgivenacopyofthePretzelJarProblemHandout.Inthisactivityteacherswillbeaskedtodescribeatleastthreemethodstodeterminethenumberofpretzelsinajar.Teacherswillthenranktheirapproachesbylevelofmathematicalsophistication,andconsiderhowtheir9‐12studentsmightaccomplishthistask.Againinimplementationofthistaskteacherswillbeaskedtoconsiderthoseassumptionsmadeduringtheactivityandidentifywhichvariablestheydecidedtocontrolforineachiterationofthetask.

PondDesignProblem–[getinfofromMonelleorDr.M]


ModelingGalleryWalk:Afterteachershavehadanopportunitytoworkontwomodelingcontexts,teacherswillsharetheirsolutionsviagallerywalkbyhangingtheirworkonthewalls.Oneortwoteachersfromeachgroupwillstaywiththeirworkwhileothersvisittheworkofothergroupstogetasenseforhowdifferentgroupstackledthemodelingcontexts.After10minutesofvisitingothergroups,thepairsofteacherswillswitchsothatallteacherswillhaveanopportunitytoseeanddescribetheirwork.Followingthegallerywalk,abriefdiscussionwillbehadsurroundingsimilaritiesanddifferencesinthesolutionstotheproblem.Thiswillleadintotheguideddebriefofthemodelingscenario(questionslistedbelow).GuidedQuestionsforDebrief

Inwhatwaysdidyouidentifyvariablesthatwouldimpactyourmodel? Howdidyouidentifyandselect(asagroup)whichvariableswerethemostimportantto

accountforinyourmodel? Whattypesofknowledgeaboutthesystembeingmodeleddidyoudrawoninthisactivity? Whatmathematicalconceptsdidyouusewhenconstructingyourmodel? Howmightyourstudentsreacttotheseproblems?Whatvariablesdoyouanticipatewould

theyaccountforintheirmodel? Howwouldyoufacilitatethistypeoftaskinyourclassrooms? Whenininstructiondoyouseethisactivitybeingthemosteffective? Whatbarriersmightyouconfrontinimplementingthistaskinaclassroom? Whenteachingmathematicalmodeling,howcanwepreparetoassiststudentsinthis

process? Wasyourmodelsimilartoothergroupsordrasticallydifferent?Isthistypeofdiscrepancy

productive?Why? Howdidthismathematicalmodelingtaskdifferfromtraditionalapplicationproblemsthat

havebeenusedpreviously?

MaterialsTechnology

Laptop iPads DocumentProjector DocumentCamera

Supplies

ChartPaper Markers Activityhandouts


MathematicalModeling

KrogerFuelPointsWhenyoushopatKrogerwithaKrogercard,youcanearn1fuelpointforevery$1youspend.Foreach100pointsyousave10centspergallonoffuel,upto$1pergallon.Eachmonthisaseparateaccumulationperiodandpointsdonotcombine.Thepointsexpireattheendofthefollowingmonth(i.e.thefuelpointsfromwhatyouspentinMarchwillexpireattheendofApril).

WhenyougotoaKrogerFuelCenterwithaKrogercard,youautomaticallysave3centspergallonoffuel.Inaddition,ifyouhave100ormorefuelpointsaccumulated,youcanchoosehowmuchofyourdiscountyouwanttouseinincrementsof10cents,so10¢,20¢,30¢,etc.

Forexample,ifyouspentatotal$320inthemonthofMarchthenyouhave30centsoffasyourdiscount.Youcouldtakeall30centsoffofonetankofgasnow.Youcouldleaveitforanothertime(andsoonlygettheusual3centdiscountthistime).Youcouldtake10centsoffnow,10centsoffanothertime,and10centsoffanothertimetospreaditout.Youcouldeventake10centsoffnowand20centsofflater.Yougettochoose.

DevelopastrategytomaximizeyoursavingsongasfromKrogerFuelPointsoverthecourseofayearandexplainwhyitwillmaximizeyoursavings.


FollowupQuestions:

1) Whatfactorsdidyouoriginallyconsiderwhencomingupwithasolution?Arethereanythatyoudismissedandwhy?

2) Wouldyouimplementyourstrategy?Whyorwhynot?

3) Isthereanythingthatcouldimproveyourstrategy?



RainfallProblemSupposeyouareoutwalkingwithoutanumbrellaanditstartstorain.Ifyouwanttominimizehowwetyougetduringyourwalk,woulditbemorebeneficialtowalkorrunandunderwhatconditions?

Developastrategytominimizehowwetyougetduringyourtravel.



TheCostofMeYourtaskistodevelopamathematicalmodelthatcandeterminethe“thecostofme”.Thiscouldbeinterpretedinanumberofways.

1. Thinkofatleastthreedifferentwaysthatyoucouldinterpretwhatthisproblemisasking.Oftheseinterpretations,selectoneofthesethatyouwishtoexplore.

2. Basedonyourinterpretationoftheproblem,thinkaboutwhatvariablesyouneedtoconsiderindevelopingyourmodel?Whatassumptionsdoyouhavetomakeregardingthissituation?

DevelopamodelthatyoucouldusetodetermineTheCostofMebasedonyourinterpretationoftheproblem.



ThePostman’sProblemOnisdeliveryrouteapostmanhastodelivertomailboxesonbothsidesofthestreet.Thepostmancaneitherdeliveralltheletterstoonesideofthestreet,cross,thendelivertotheothersideofthestreet,ORdelivertoonemailboxcrossthestreetanddelivertoanotheroneortwomailboxes,thencrossthestreetagain.Underwhatconditionsareeitherofthesetheoptimalmethodsfordelivery?Whatassumptionsdidyoumakeinaccountingforthisproblem?

Developamodeltodeterminetheoptimalrouteforthepostmantotravelonhismaildeliveryroute.



TheConcertProblemArockbandisputtingonaconcertatyourlocaltheater,andyouhavebeenhiredtodeterminehowmuchtochargefortickets.Youhavebeenprovidedalayoutofthefloorplanofthevenue.Whatfactorswillyouaccountforinordertodetermineaticketprice?Whatadditionalinformationdoyouneedinsolvingthisproblem?Thelastvenuetheyplayedatofferedyouaphotoofthecrowdtoassistyouinyourplanning.Howcanthisbeusedtodeterminethepriceofeachticket?

Developastrategyfordetermininghowmuchtochargeforeachticketandjustifyyourdecision.

NationwideArena,ColumbusOhio



PondDesignProblemThere is a new residential development that is being constructed, and the designers want to include a

pond. Suppose that the residential development will only include 5 story apartment complexes, and

that the community will include approximately 300 families (at maximum capacity). How large should

the pond be and what will be the design? The total square footage of the complex is 360,000 square

feet.

Developastrategytodeterminehowlargethepondshouldbeintheapartmentcomplex.Whatfactorsshouldyouconsiderindeterminingasolutiontothistask?Justifyyoursolution.Whatotherquestionsariseinsolvingthisproblem?



PretzelJarTaskYoubroughtabigjarofpretzelstoshareataparty.Theyareahugehitandeveryoneissnackingonthem!Wheneveryouwalkby,younoticelessandlesspretzelsinthejar.Howwouldyouestimatethenumberofpretzelsleftinthejaratanytimeduringtheparty?(Don’tforgetaboutthebrokenpieces!)

Task:

Describethreemethodstoestimatethenumberofpretzelsinthejar.

Rankyourthreeapproachesbythelevelofmathematicalsophistication.

Describehowyouanticipatestudentsindifferentgradebandswouldestimatethenumberofpretzelsinthejar.



InvestmentTask

WhyshouldIinvest?“Oneofthemostcompellingreasonsforyoutoinvestistheprospectofnothavingtoworkyourentirelife!Bottomline,thereareonlytwowaystomakemoney:byworkingand/orbyhavingyourmoneyworkforyou”–Investopedia.

Moneycanearnyoumoneyintwoways.First,yougivesomeonemoneyforaperiodoftimeandthatpersonpaysyoubackwithinterest.Second,whenyougivesomeonemoney,inreturnyouareapartowneroftheirbusiness.Inotherwords,younowpossesssomefinancialassets.Thisisinvesting!

Task:

Youhave$1,000andtwofriendswithbusinessplans.Supposeyoulikethemequallyasfriends.Bothbusinessplanssoundexciting,butultimatelyyourgoalistomakemoneyoffofyourinvestment.Ifyouhavetochooseonlyone,howdoyoupickthebusinessplantoinvestin?

Writeaninvestmentplanthatincludesanoutlineofwhatyouconsidernecessarydataforyourchoiceofinvestment(i.e.factorsthatwouldaffectinvestmentchoices).

Selectthemostviablefactorsandexplainwhyeachpieceofinformationisimportanttothecomparisons.

Ranktheimportanceofeachpieceofinformationforyourinvestmentdecisionbecauseyoumaynotbeabletocollectallinformationthatyouneed.

Buildmathematicalconnectionsofeachfactorforinvestmentdecisionmaking. Usespreadsheettobuildaninvestmentcalculatortohelpyoudecideinwhichbusinessto

invest.



CollegeSavingProblemSuppose you want to start a college savings plan for your child(ren). Consider a variety of options and

determine an optimal method for savings. How much money will you be putting away per paycheck?

How much money will you save by the time your children are in school? How can you maximize the

amount of money to put away?

Developastrategyforsavings,anddeterminehowmuchmoneyyouwillhavesavedbythetimeyourchildrenfinishschool.