7.4*Composing*and*Decomposing* A"Develop"Understanding"Task"

Transcription

1 23 7.4ComposingandDecomposing A"Develop"Understanding"Task" Asthedaygetwarmer,youandyourfriendsdecidetocooloffby takingarideontheturbulent"waters"dive.asyouarewaitingin lineyourtourguideexplainsthemathematicsbehinddesigning thewaitingareaforaride. Asyoucansee, saystheengineer, thewaitingareacanbe enlargedorreducedbymovingafewchainsaround.theareaweneedforwaitingguestsdepends onthetimeofday.wecollectdataforeachridesowecanusefunctionstomodelthetypicalwait timeandhowmuchwaitingareaweneedtoprovideforourguests. Andofcourse,yourguidehasthefunctionsthatrepresentthisparticularride. AveragenumberofpeopleintheTWD$lineasafunctionoftime: pt) = 3000cos 1 5 t 3) ) tisthenumberofhoursbeforeorafternoon,sot"="2represents2:00p.m.and t=n2represents10:00a.m. p"representsthenumberofpeopleinline Waitingarearequiredasafunctionofthenumberofpeopleinline: a p) = 4 p +100 a,thewaitingarea,ismeasuredinsquarefeet p 1500 Waittimeforaguestasafunctionofthenumberofpeopleinline:W p) = W,thewaittime,ismeasuredinminutes 1. HowmuchwaitingareaisrequiredfortheguestsinlinefortheTurbulent"Waters"Dive"at eachofthetimeslistedinthefollowingtable? Time$of$Day$ 10:00a.m. 12:00noon 2:00p.m. 4:00p.m. 8:00p.m. Waiting$Area$Required$sq.ft.)$ 2014www.flickr.com/photos/Easterbilby MathematicsVisionProject MVP LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense. Modeling with Functions 23

2 24 a. Foreachinstantintimeyouhadtocompleteaseriesofcalculations.Describehow youfoundthewaitingareaatdifferenttimes. b. Canyoucreateasinglerulethatwilldeterminethewaitingareaasafunctionofthe timeofday? 2. WhatisthewaittimeforaguestthatarrivesattheendofthelinefortheTurbulent"Waters" Diveateachofthetimeslistedinthefollowingtable? Time$of$Day$ 10:00a.m. 12:00noon 2:00p.m. 4:00p.m. 8:00p.m. Wait$Time$minutes)$ a. Foreachinstantintimeyouhadtocompleteaseriesofcalculations.Describehow youfoundthewaittimeatdifferenttimesoftheday. b. Canyoucreateasinglerulethatwilldeterminethewaittimeasafunctionofthe timeofday? Tomaintaincrowdcontrolwhenthelinesgetlong,castmembersdressedaspiratestheTurbulent" Waters"Divehasapiratetheme)minglewiththewaitingguests.Theiranticsdistracttheguests wholistenattentivelytotheirpiratejokes.thenumberofcastmembersneededdependsonthe numberofpeoplewaitingintheline. Numberofushersneededasafunctionofthenumberofpeopleinline:ct) = p"representsthenumberofpeopleinline c"representsthenumberofcastmembersneeded p 150 MathematicsVisionProject MVP LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense. Modeling with Functions 24

3 25 3. Howmanycastmembersareneededtoentertainanddistractthewaitingguestsateachof thefollowingtimesoftheday? Time$of$Day$ Cast$Members$Needed$ 10:00a.m. 12:00noon 2:00p.m. 4:00p.m. 8:00p.m. t"hoursbeforeorafternoon t"<0beforenoon,t>0afternoon) Onwarm,sunnydaysmistersareusedtocooldownthewaitingguests.Thenumberofmistersthat needtobeturnedondependsonthesizeofthewaitingareathathasbeenopeneduptocontainthe numberofpeopleinline. Numberofmistersneededasafunctionofthewaitingarea: mt) = a,thewaitingarea,ismeasuredinsquarefeet m"representsthenumberofmisterstobeturnedon a Howmanymistersneedtobeturnedontocoolthewaitingguestsateachofthefollowing timesofday? Time$of$Day$ 10:00a.m. 12:00noon 2:00p.m. 4:00p.m. 8:00p.m. t"hoursbeforeorafternoon t"<0beforenoon,t>0afternoon) Misters$Needed$ MathematicsVisionProject MVP LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense. Modeling with Functions 25

4 26 5. Explainhowthefollowingdiagrammighthelpyouthinkabouttheworkyouhavebeen doingonthepreviousproblems.howdoesthenotationusedinthediagramsupportthe wayyouhavebeencombiningfunctionsinthistask?thiswayofcombiningfunctionsis calledfunction$composition. Interpreting$the$Functions$ 6. Atwhattimeofdayisthenumberofpeopleinlinethelargest? Whatisthemaximumnumberofpeopleinline,basedonthefunctionfortheaverage numberofpeopleinline? Whendoyouthinktheamusementparkopensandcloses,basedonthisfunction? 7. Intermsofthestorycontext,whatdoyouthinkthe4andthe100representinfunctionrule forwaitingarea? 8. Intermsofthestorycontext,whatmightbethemeaningofthe1500inthefunctionrulefor waittime? 9. Intermsofthestorycontext,whatmightbethemeaningofthe150inthefunctionrulefor castmembersneeded? 10. Intermsofthestorycontext,whatmightbethemeaningofthe1000inthefunctionrulefor thenumberofmistersneeded? MathematicsVisionProject MVP LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense. Modeling with Functions 26

5 7.4ComposingandDecomposing TeacherNotes A"Develop"Understanding"Task" Purpose:Thepurposeofthistaskistoconsideranotherwaythatfunctionscanbecombined by functioncomposition.whenfunctionsareaddedormultiplied,itistheoutputsoftheindividual functionsforaparticularinputthatareaddedormultipliedtogether.infunctioncomposition,the outputofonefunctionbecomestheinputoftheother.thisisrepresentedalgebraicallybywriting onefunctioninsidetheother: y = f gx)).notethattofindtheoutputvalueforaparticularinput valuex,wewouldfirstevaluategx),andthenusethisresultingvalueastheinputforfunctionf." Thefollowingdiagramillustratestheideaofanewfunctionrelationshipbeingcreatedbythe compositionoftwootherfunctions: CoreStandardsFocus: F.BF.1cWriteafunctionthatdescribesarelationshipbetweentwoquantities. +)Composefunctions.Forexample,ifTy)isthetemperatureintheatmosphereasa functionofheight,andht)istheheightofaweatherballoonasafunctionoftime,then Tht))isthetemperatureatthelocationoftheweatherballoonasafunctionoftime. RelatedStandards: LaunchWholeClass): Introducethescenarioforthistaskbyaskingstudentsiftheyhaveeverwaitedinlineforan amusementparkrideoranothertypeofevent,andifthevendormadeanyefforttomanagethe waittime.studentshaveprobablyobservedlinesthatweavebackandforthinparallelrowsuntil yougettothefrontoftheline.perhapsstudentshaveobservedhowlinescanbeadjustedby ropingoffcertainareas,oropeningupnewones.perhapstheyhavebeenonrideswherethe waitingareaprovidedaformofentertainmentitself. Longlines,crowds,andthehotsuncontributeinnegativewaystotheexperienceofaguestona ride.therefore,amusementparkscollectdatainordertodeterminewaystominimizethe discomfortoftheguests.thistaskdealswithsomeofthemathematicsofwaitinginlines. MathematicsVisionProject MVP LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.

6 Discussthepurposeofthefirstthreefunctionslistedinthetask,andthequantitiesrepresentedby theinputandoutputvariables. Note:Thesectionattheendofthetask,Interpreting"the"Functions,couldbeusedaspartofthe launchifyouwantstudentstounderstandmoreofthedetailsofwhattheconstantsandcoefficients ineachfunctionrepresent.thissectionisleftforlasttoallowstudentstogetintotheworkof composingfunctionsmorequickly,andtogivefastfinisherssignificantworktodoafterthey completethemainportionofthetask.feelfreetouseitaspartofthelaunchortorelaunchthe task)ifyoufeelitisneededtohelpstudentsworkwitheachfunction. ExploreSmallGroup): Therearetwoapproachestoworkingontheseproblems:wecandecomposethescenariointoits componentfunctionsandworkwiththemindividuallyandinsequence,orwecancomposeasingle functionthattakescareofthesequenceofcomputations.itisassumedthatstudentswillbeginby workingthroughatwonstepprocessastheyfilloutthetableonquestion1:first,calculatethe numberofpeopleinlineasafunctionofthegiventime;second,calculatetheareaofthewaiting regiondesignatedfortheguestswaitinginline.watchforstudentswhobegintoshortcutthis processbythinkingofasinglefunctionthattakescareofthissequenceofcomputationalsteps,as promptedbypartsaandb. Listenforstudentswhomaycreatethecompositionfunctionfirst,beforefillingoutthetablesfor questions2and3. Listenforhowstudentstacklequestion4.Dotheygobacktodecomposingthescenariointoa threenstepprocess:first,calculatethenumberofpeopleinlineasafunctionoftime;second, calculatethewaitingareaasafunctionofthenumberofpeople;third,calculatethenumberof mistersneededasafunctionofthearea?or,dotheyusetheoutputsfromtheirworkonquestion1 asinputstocompletethetableinquestion4?or,dotheycreateasinglefunctionthattakescareof theentiresequenceofcomputations? Thesection,Interpreting"the"Functions,isforstudentswhomaymovequicklythroughthetaskand needsomeadditionalsignificantworktodo.identifystudentswhohavediscussedthisportionof thetaskandcanassignameaningtoeachoftheparametersineachfunctionrule. DiscussWholeClass): Beginbydiscussingquestions6N8,togivestudentsadeeperunderstandingofthefunctionsthey areworkingwithandthestoriestheytell.usestudentexplanationsoftheparametersinthe functionsasmuchaspossible.forexample,thefunctionthatrepresentsthenumberofpeoplein linesuggeststhatat3:00p.m.theridehasthemaximumnumberofpeoplewaitinginline.this maximumnumberis3000,theamplitudeofthecosinegraph.sincethisisawaterride,withthe potentialofgettingwet,theridewouldbemorepopularduringthehotafternoonthanearlierinthe MathematicsVisionProject MVP LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.

7 morningoraftersunset.thisaccountsforthet" 3)inthefunctionrulesincethecosinehasa maximumvaluewhentheinputis0,andthisexpressionmakestheinput0whent"=3.discussing thisideaexplicitlygivesstudentsinsightintowhytheexpressiont 3insidethecosinefunction shiftsthecosinegraphtotheright.examiningthegraphofthisfunctionsuggeststhatthepark opensat7:00a.m.andclosesat11:00p.m.,sincethosearethehoursbetweenwhichthefunction givesapositivevalueforthenumberofpeoplewaitingintheline. Forquestion7studentsshouldidentifythatthe4represents4squarefeetperpersonwaitingin line,andthatthe100representsastartingareaof100squarefeet,whichmightbeaccountedforas theloadingareaoftheride. Itmaybemoredifficultforstudentstodeterminethemeaningofthe1500inthefunctionforwait time,asrequestedinquestion8.the1500givesthehourlycapacityoftheride.thereareonly peoplewaitinginlineinsteadofmovingthroughtheline)when p>1500.forexample,ifp"=2000,thereare500morepeopleinlinethancanfreelymovethrough 500 people theridewithoutwaiting people hour = 1 3 hour,whichweconverttominutesby multiplyingby60. Afterdiscussingthedetailsofthemeaningofeachofthesethreefunctions,askstudentstousea diagramsimilartothatgiveninquestion5todescribetheirworkonquestions1n4.itmaybe interesting,ifyouhavetime,tohavestudentsdetermineareasonabledomainforeachofthe compositefunctionsbasedontheactualstorycontext.forexample,woulditmakesensetoopen upmorewaitingareaifthenumberofpeopleliningupduringaparticularhourislessthat1500, sincethatnumberofpeoplecanmovefreelythroughthelinewithoutwaiting? Discussthenotationusedforfunctioncomposition y = f gx))fromthediagram.youmaychoose tointroduceanalternativenotationforfunctioncomposition: f g)x) = f gx)).notethatin thisnotationgisthe inside function,thefunctionthatisevaluatedfirst,andthentheresultofthat evaluationisusedastheinputintofunctionf.)"""makesurethatstudentscanmatchthediagramto thenotationusedforfunctioncompostion. AlignedReady,Set,Go:ModelingwithFunctions7.4 MathematicsVisionProject MVP LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.

8 NameModelingWithFunctions 7.4$ 27 Ready,$Set,$Go$ Ready$ Topic:Recognizingoperationsonavariable Each$expression$contains$2$operations.$One$of$the$ operations$will$be$ inside $the$second$operation.$$ Identify$the$ inside $operation$as$u$by$writing$$$$$$$$$$$$$$$$$$$$$$ u$=$.$$then$substitute$u$into$the$expression$so$ that$the$ outside $operation$is$being$performed$on$u.$$$ 2014www.flickr.com/photos/Easterbilby $Example:$$"#$:.$$$$ I$can$see$two$operations$on$x.$$$First$the$x$is$being$cubed$and$then$ is$multiplied$by$5.$$$$$$$$$ So$if$u$=$,$then$ = ".$ 1.Wouldtheanswerintheexamplehavebeendifferentifyouweregiven 5?Explain tan u"=" u"=" " " " " u"=" sin u"=" " " " " u"=" " " " " u"= $ MathematicsVisionProject MV P LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense Modeling with Functions 27

9 NameModelingWithFunctions 7.4$ Set$ Topic:Creatingformulasforcompositefunctions Recallthat = Let = 2 4"# = 5.Find: a) 1 b) 1 c) 2 d) 1 9.Let = " "# =. Find: a) b) c) d) 10.Usethefunctionsin#9. Find: a) 2 b) 5 11.Describetheproblemthatyouencounteredwhencalculatingthevaluesin# Describethedomainsfora) b) c) d) 13.Whatmakesthedomainforeachcompositiondifferent? $ MathematicsVisionProject MV P LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense Modeling with Functions 28

10 NameModelingWithFunctions 7.4$ Go$ Topic:Writingequationsofpolynomialsgiventhedegreeandtheroots$ Write$the$equation$of$the$polynomial$with$the$given$features.$ 29 Degreeof polynomial Givenrootsyou mayhaveto determineothers): W2,1,andW1 Leading coefficient 3 Equationinstandardform: , 4, multiplicity2, W1multiplicity2, and3 W , 2 W2 MathematicsVisionProject MV P LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense Modeling with Functions 29