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

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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]

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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

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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.

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FollowupQuestions:

1) Whatfactorsdidyouoriginallyconsiderwhencomingupwithasolution?Arethereanythatyoudismissedandwhy?

2) Wouldyouimplementyourstrategy?Whyorwhynot?

3) Isthereanythingthatcouldimproveyourstrategy? 

2015MathematicsTeachingInstitute  

MathematicalModeling

RainfallProblemSupposeyouareoutwalkingwithoutanumbrellaanditstartstorain.Ifyouwanttominimizehowwetyougetduringyourwalk,woulditbemorebeneficialtowalkorrunandunderwhatconditions?

Developastrategytominimizehowwetyougetduringyourtravel.

 

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MathematicalModeling

TheCostofMeYourtaskistodevelopamathematicalmodelthatcandeterminethe“thecostofme”.Thiscouldbeinterpretedinanumberofways.

1. Thinkofatleastthreedifferentwaysthatyoucouldinterpretwhatthisproblemisasking.Oftheseinterpretations,selectoneofthesethatyouwishtoexplore.

2. Basedonyourinterpretationoftheproblem,thinkaboutwhatvariablesyouneedtoconsiderindevelopingyourmodel?Whatassumptionsdoyouhavetomakeregardingthissituation?

DevelopamodelthatyoucouldusetodetermineTheCostofMebasedonyourinterpretationoftheproblem.

 

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MathematicalModeling

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

Developamodeltodeterminetheoptimalrouteforthepostmantotravelonhismaildeliveryroute.

 

2015MathematicsTeachingInstitute  

MathematicalModeling

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

Developastrategyfordetermininghowmuchtochargeforeachticketandjustifyyourdecision.

NationwideArena,ColumbusOhio

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2015MathematicsTeachingInstitute  

MathematicalModeling

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?

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MathematicalModeling

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

Task:

Describethreemethodstoestimatethenumberofpretzelsinthejar.

Rankyourthreeapproachesbythelevelofmathematicalsophistication.

Describehowyouanticipatestudentsindifferentgradebandswouldestimatethenumberofpretzelsinthejar.

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MathematicalModeling

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.

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MathematicalModeling

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.