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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?
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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.
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MathematicalModeling
TheConcertProblemArockbandisputtingonaconcertatyourlocaltheater,andyouhavebeenhiredtodeterminehowmuchtochargefortickets.Youhavebeenprovidedalayoutofthefloorplanofthevenue.Whatfactorswillyouaccountforinordertodetermineaticketprice?Whatadditionalinformationdoyouneedinsolvingthisproblem?Thelastvenuetheyplayedatofferedyouaphotoofthecrowdtoassistyouinyourplanning.Howcanthisbeusedtodeterminethepriceofeachticket?
Developastrategyfordetermininghowmuchtochargeforeachticketandjustifyyourdecision.
NationwideArena,ColumbusOhio
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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.