8.3 the bungee jump simulator - utah education networksecondary math iii // module 8 modeling with...

SECONDARY MATH III // MODULE 8

MODELING WITH FUNCTIONS – 8.3

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

8.3 The Bungee Jump

Simulator

A Solidify Understanding Task

Asarewardforhelpingtheengineersatthelocalamusementparkselectadesignfortheir

nextride,youandyourfriendsgettovisittheamusementparkforfreewithoneoftheengineers

asatourguide.Thistimeyouremembertobringyourcalculatoralong,incasetheengineersstart

tospeakin“mathequations”again.

Sureenough,justasyouareabouttogetinlinefortheBungeeJumpSimulator,yourguide

pullsoutagraphandbeginstoexplainthemathematicsoftheride.Topreventinjury,theridehas

beendesignedsothatabungeejumperfollowsthepathgiveninthisgraph.Jumpersarelaunched

fromthetopofthetowerattheleft,anddismountinthecenterofthetowerattherightaftertheir

upanddownmotionhasstopped.Thecabletowhichtheirbungeecordisattachedmovesthe

ridersafelyawayfromthelefttowerandallowsforaneasyexitattheright.

Yourtourguidewon’tletyouandyourfriendsgetinlinefortherideuntilyouhave

reproducedthisgraphonyourcalculatorexactlyasitappearsinthis

diagram.

1. Workwithapartnertotryandrecreatethisgraphonyourcalculatorscreen.Makesureyoupayattentiontotheheightofthejumperateachoscillation,asgiveninthetable.

Recordyourequationofthisgraphhere:

CC

BY

Mar

k H

arki

n

http

s://f

lic.k

r/p/

dhg8

ZN

18




AfterathrillingrideontheBungeeJumpSimulator,youaremetbyyourhostwhohasa

newpuzzleforyou.“Asyouareaware,”saystheengineer,“temperaturesaroundherearevery

coldatnight,butverywarmduringtheday.Whendesigningrideswehavetotakeintoaccount

howthemetalframesandcablesmightheatupthroughouttheday.Ourcalculationsarebasedon

Newton’sLawofHeating.Newtonfoundthatwhilethetemperatureofacoldobjectincreases

whentheairiswarmerthantheobject,therateofchangeofthetemperatureslowsdownasthe

temperatureoftheobjectgetsclosertothetemperatureofitssurrounding.”

Ofcoursetheengineerhasagraphofthissituation,whichhesays“representsthedecayof

thedifferencebetweenthetemperatureofthecablesandthesurroundingair.”

Yourfriendsthinkthisgraphremindsthemofthepointsatthebottomofeachofthe

oscillationsofthebungeejumpgraph.

2. Usingthecluegivenbytheengineer,“Thisgraphrepresentsthedecayofthedifferencebetweenthetemperatureofthecablesandthesurroundingair,”trytorecreatethisgraphonyourcalculatorscreen.(Hint:Whattypesofgraphsdoyougenerallythinkofwhenyouaretryingtomodelagrowthordecaysituation?Whattransformationsmightmakesuchagraphlooklikethisone?)

Recordyourequationofthisgraphhere:

19




8.3 The Bungee Jump Simulator – Teacher Notes A Solidify Understanding Task

Purpose:Thepurposeofthistaskistosolidifythinkingaboutcombiningfunctiontypesusing

suchoperationsasadditionormultiplication.Intheprevioustaskstudentshavenoticedhowthe

characteristicsofbothtypesoffunctionsaremanifestedintheresultinggraphs.Inthistaskthey

becomemorepreciseabouthowthefunctionscombinebytryingtodeterminetheexactequation

thatwillproduceagivengraph.

CoreStandardsFocus:

F.BF.1bWriteafunctionthatdescribesarelationshipbetweentwoquantities.�

Combinestandardfunctiontypesusingarithmeticoperations.Forexample,buildafunctionthat

modelsthetemperatureofacoolingbodybyaddingaconstantfunctiontoadecayingexponential,

andrelatethesefunctionstothemodel.

StandardsforMathematicalPractice:

SMP4–Modelwithmathematics

SMP5–Useappropriatetoolsstrategically

SMP6–Attendtoprecision

Vocabulary:Ifyouhavenotpreviouslydoneso,usethewordmodelingtodescribetheprocessof

applyingmathematicsweknowtosolvereal-worldproblems,usuallyrequiringustomakeand

varyassumptionsbasedonhowwellourmathematicalmodelfitsdatafromthecontext.

TheTeachingCycle:

Launch(WholeClass):

Engagestudentsinthescenarioofthetask.Pointoutthetwographsinthetask,anddiscusswhat

theyaresupposedtomodelaccordingtotheengineerinthestory.Informstudentsthattheyare




notfinishedwiththetaskuntiltheycanproducetheseexactgraphsontheircalculatorsand

explainwhatfunctionstheyusedandhowtheytweakedeachfunctiontomakeitfitthedetailsof

thegivengraphs.

Explore(SmallGroup):

Forbothofthegraphs,encouragestudentstotrysomething.Theyhaveexperimentedwith

similargraphsontheprevioustask,andguessandcheckcanprovidealotofinsightsintothese

graphs.Studentsshouldtrysomethingandthenrefinetheirinitialguesses.Thefollowingspecific

promptsmayhelp.

Ifstudentsarestrugglingwiththefirstgraph,helpthemdeconstructitintoindividualfunctionsby

askingquestionssuchas,“Whatiftheoscillationsofthebungeejumperdidn’tleveloff,butkept

returningtothesameheightordistancefromthegroundasinthepreviousoscillation?What

functionwouldmodelthatbehavior?”Or,“Whatifwejustexaminethepointsatthetopofeach

oscillationofthegraph?Whatkindoffunctionwouldpassthroughthosepoints?”Thesepoints

arelistedinboldfacedtypeinthetable.Thecolumnonthetablelistedas‘distancefrommidline’

illustratesthedecayoftheamplitudeofthecosinegraph.

Forthesecondgraph,remindstudentsthattheyhavetwohints—theengineerusedtheword

“decay”indescribingthegraph,andoneoftheirfriendssaidthegraphlookedlikethepointsatthe

bottomofeachoftheoscillationsofthebungeejumpgraph.Askstudentshowthesepointsare

relatedtothepointsatthetopofeachoscillation,whichwehadtoconsiderinthepreviousgraph.

Perhapshavingstudentsdrawthemidlineofthebungeejumpgraphwillhelpthemseethe

reflectionacrossthismidlineofthegraphthatgoesthroughthemaximumpointsandthegraph

thatgoesthroughtheminimumpoints.Ifneeded,askstudentsiftheycanthinkofaseriesof

transformationsthatcouldtakeanexponentialdecayfunctionandmakeitlooklikethisgraphthat

isapproachingthemidlinefrombelow.




Discuss(WholeClass):

Havestudentswhohavesuccessfullyduplicatedthegraphsdescribetheirwork.Ifnoonehas

beensuccessful,focusonthefirstgraphandhavestudentscreateanequationforthe

undampened,sinusoidalgraph,! = 40 cos()*) + 80.Thenhavethemcreateanequationfortheexponentialdecaygraphthatwouldfitthepointsatthetopofeachcurve,! = 40 ∙ (0.8)0 + 80.Thisisanexponentialdecayfunctionthathasbeenshifteduptotheheightofthemidline.The

exponentialdecayfunction! = 40 ∙ (0.8)0canbefoundfromthedatagivingthedistancefromthemidline.Finally,havestudentsconsiderhowtheymightcombinethetwofunctions.The

importantissuehereistonotethatfunctionsoftheform! = 1 sin(4*) + 5wouldoscillatearoundthemidlineaty=cwithanamplitudegivenbya.Ifwereplacetheparameterawithan

exponentialdecayfunction,wecancausetheamplitudetodampendownto0,justasthe

exponentialdecayfunctionapproaches0.Notethatwedon’tneedtoincludetheverticalshift

twice.Itisasifwehadcreatedtheentiregraphtooscillateanddecayaroundthex-axis,andthen

shiftedthiswholebehavioruptothelocationofthemidline.Itmightbehelpfultoprovide

studentswithacopyofthegraphonthefollowingpageastheyconsiderhowtheexponential

decayfunctiondescribesthedecayoftheamplitudeawayfromthemidline.

Ifthereistimeyoucanalsoworkthroughthesecondgraphusingthehintsasdiscussedinthe

exploresection.Theimportantissuehereistofocusontheexponentialdecayfunctionusedinthe

previousgraph—theversionthatdecaystothex-axis.Ifthisfunctionisreflectedoverthex-axis

andthenshifteduptothemidline,itwouldpassthroughtheminimumpointsofthebungeejump

graph,whicharethesamepointsasintheheatingupgraph.

AlignedReady,Set,Go:ModelingwithFunctions8.3




8.3

Needhelp?Visitwww.rsgsupport.org

READY Topic:Evaluatingfunctions

Evaluateeachfunctionasindicated.Simplifyyouranswerswhenpossible.Stateundefinedwhenapplicable.

1.!(#) = #& − 8#

a)!(0) b)!(−10) c)!(5)d)!(8) e)!(# + 2)

2./(#) = 01231

a)/(−1) b)/(10) c)/ 4506d)/(0) e)/(2# + 4)

3.ℎ(#) = 9:;(#)

a)ℎ(<) b)ℎ 40=&6 c)ℎ 455=

>6d)ℎ 43=

?6 e)ℎ 4cos25 425

&6 , # < <6

4.E(#) = FG;(#)

a)E(<) b)E40=&6 c)ℎ 4H=

>6d)ℎ 40=

?6 e)ℎ 4cos25 425

&6 , # < <6

READY, SET, GO! Name PeriodDate

20




8.3


SET Topic:Dampingfunctions

Twofunctionsaregraphed.Graphathirdfunctionbymultiplyingthetwofunctionstogether.Usethetableofvaluestoassistyou.Itmayhelpyoutochangethefunctionvaluestodecimals.

5.# I5 = # I& = 9:;# I0 = (#)9:;#

-2π

−3<2

−<

−<2

0

<2

π

3<2

2π

6.AfteryouhavegraphedI0,graphthelineI? = −#.WhatdoyounoticeaboutthegraphofI0inrelationtothegraphsofI5G;LI??

I5

I&

21




8.3


GO Topic:Comparingmeasuresofcentraltendency(mean,median,andmode)

Duringsalarynegotiationsforteacherpayinaruralcommunity,thelocalnewspaper

headlinesannounced:Greedy Teachers Demand More Pay!Thearticlewentontoreportthatteacherswereaskingforapayhikeeventhoughdistrictemployees,includingteachers,werepaidanaverageof$70,000.00peryear,whiletheaverageannualincomeforthecommunitywascalculatedtobe$55,000perhousehold.The65schoolteachersinthedistrictrespondedbydeclaringthatthenewspaperwasspreadingfalseinformation.

Usethetablebelowtoexplorethevalidityofthenewspaperreport.

JobDescription Numberhavingjob AnnualSalarySuperintendent 1 $258,000BusinessAdministrator 1 $250,000FinancialOfficer 1 $205,000TransportationCoordinator 1 $185,000Districtsecretaries 5 $55,000SchoolPrincipals 5 $200,000AssistantPrincipals 5 $175,000GuidanceCounselors 10 $85,000SchoolNurse 5 $83,000SchoolSecretaries 10 $45,000Teachers 65 $48,000Custodians 10 $40,000

7.Whichmeasureofcentraltendency(mean,median,mode)doyouthinkthenewspaperusedtoreporttheteachers’salaries? Justifyyouranswer.

8.Whichmeasureofcentraltendencydoyouthinktheteacherswouldusetosupporttheirargument? Justifyyouranswer.

9.Whichmeasuregivestheclearestpictureofthesalarystructureinthedistrict?Justify.

10.Makeupaheadlineforthenewspaperthatwouldbemoreaccurate.

22

8.3 the bungee jump simulator - utah education networksecondary math iii // module 8 modeling with...

Documents