Name__________________________Date__________________________ Activity3 . HowDoYouMeasureUp?
ArmSpan Height Doesincreasingtheamountoftimepracticingasportincreaseperformancelevelsinthatsport?
Doesdecreasingthespeedatwhichacarisdrivenincreasethegasolinemileageforthatcar?
Asapersongetsolder,doestheperson’shatsizeincrease?
Insituationslikethese,questionsarebeingaskedabouthowonequantityisrelatedtoanother.Answeringsuchquestionsusuallyrequiresonetocollectandanalyzeseveralsetsofdatapairingthetwoquantities. TheProblem Inthisactivity,youwillinvestigatetherelationshipbetweenaperson’sheightandarmspan(thedistancefromfingertiptofingertipofaperson’soutstretchedarms). Tobegin,youwillneedtocollectdataonbothquantitiesfromseveralmembersofyourclass. Usingatapemeasureormeterstickmarkedincentimeters,determinetheheightsandthearmspans,tothenearestwholecentimeter,oftwoorthreemembersofyourclass. ✐RecordthemeasurementsinTable3.1intheQuestionssectionandpletethetablewithmeasurementscollectedbyothermembersofyourclass. ✐Examinethemeasurementsyouhaverecordedinthetablesandanswer#1through#4intheQuestionssectionofthisactivity. Inthisactivity,studentsareintroducedtotheuseofscatterplotsforinvestigatingpossiblerelationshipsbetweentwoquantitativevariables.Itmaytaketwotothreeclassperiodspletealloftheproblems. Thevariablesunderexaminationaretwobodymeasurements:heightandarmspan.Studentscollectdata,produceascatterplotofthedata,andinterpretanypatternsexhibitedbythescatterplot. ©1997TEXASINSTRUMENTSINCORPORATED 22 GraphingCalculatorActivitiesforEnrichingMiddleSchoolMathematics Thelanguageofcoordinatesystemsisusedinthisactivity.Youmaywanttoacquaintstudentswithsomeoftheassociatedterminologyeitherbeforeorduringthisactivity. Inadditiontoexamininglistsofnumbers,itisoftenhelpfultodisplaydatainagraphorplotandlookforvisualcluesthatmightsuggestpossiblerelationshipsbetweentwovariables.Onewaytographpaireddata,liketheheightandarmspanmeasurements,istoconstructascatterplot.Ascatterplotissimplyagraphofalloftheorderedpairsofdataonasinglecoordinatesystem.Beforeyouplotthedata,itisimportanttoexaminethedataformaximumandminimumvaluessothatappropriatemeasurementscalescanbedeterminedfortheaxesonthecoordinategrid. ✐Withthehelpofothersinyourgroup,determineanappropriatescaleforeachofthehorizontalandverticalaxesontheScatterplotofHeightvs.ArmSpanin#5oftheQuestionssection.Plottheinformationyouhaverecordedusingheightalongthehorizontalaxisandlengthofarmspanalongtheverticalaxis.Thenanswer#6through#10. Sincestudentshavehadtheopportunitytoproduceascatterplotonpaper,theywillbetterunderstandsomeofthedecisionsthatneedtobemadewheninstructingacalculatortoproduceascatterplotofthesamedata.Oncetheycanquicklyplotdatapointsonthecalculator,theycanusetheirtimetofocusoninterpretationofthegraphicaldisplays. Whenstudentsenterthedata,itwilllooksimilartothis: UsingtheCalculator Calculatortechnologyalsocanbeusefulinsearchingforrelationshipsbetweentwoquantitieslikeheightandarmspan.Infact,yourgraphingcalculatorcandrawscatterplotsliketheoneyoucreated.Todoso,youmustfirstplaceyourdataintothecalculator’sLISTstorage. CalculatingtheResults
1.Press…4:ClrList2[L1]¢y[L2]¢y[L3]Ítocleartheneededlists.(ThethirdlistL3willbeusedtostoreadditionalputedlaterinthisactivity.)
2.Press…1:Edittogainesstothecalculator’sliststorage.
3.Typethenumbersrepresentingstudentheightsintothefirstlist(L1)onyourcalculator.Typeanumber,pressÍ,andrepeatuntilallnumbershavebeenentered.
4.Press~tomovetothesecondlist.Typethenumbersrepresentingstudentarmspansintothesecondlist(L2).Besuretoenterthearmspanforeachstudentonthesamerowasthestudent’sheight. ©1997TEXASINSTRUMENTSINCORPORATED Activity3:HowDoYouMeasureUp?
23 Oncethepaireddatahavebeenentered,youneedtoenterinformationontwowindowsbeforethecalculatorcanproduceascatterplotgraph.First,youmusttellthecalculatoryouwantittodrawascatterplot,andthenyoumustdefinetheintervalsandthescalesyouwantittouseforthecoordinatesystem,justlikeyoudidwhenyoudrewascatterplotonpaper.
1.Pressy[STATPLOT]1:Plot1toselectPlot1.
2.Editthewindowsothatyourslooksliketheoneattheright.Tohighlightaselection,usethebluearrowkeystomovetheblinkingcursortothedesiredlocationandpressÍ. ByselectingthefirstoptionavailableinType,youareselectingascatterplot.
3.WhenyouhavefinishedmakingthechangesinPlot1,presspandeditthenumberstomatchthoseyouusedwhenyouconstructedthescatterplotonpaper. NotethatXminandXmaxrefertotheminimumandmaximumvaluestobeusedalongthehorizontalaxisheight.Xscldefinesthedistancebetweenreferenceortickmarksusedalongthataxis.Ymin,Ymax,andYscldefinethesameparametersfortheverticalaxis. ✐Recordyourselectedvaluesinthewindowshownin#11intheQuestionssection.
4.Pressstoviewthescatterplot.
5.Youmayprefertoviewtheplotwithgridpointsvisible.OnaTI-82,youcanaddgridpointstotheplotbypressingp,selectingFORMAT,pressing†and~toselectGridOn,andpressingÍ.
6.Presssagain,andgridpointsshouldappearonthescatterplot.ThedistancebetweengridpointsisdeterminedbythesettingsyouhavedefinedforXsclandYscl. ItisimportantthatallotherplotsandfunctionsbeturnedofforclearedbeforeattemptingtoproduceagraphofPlot1.Toturnoffallstatisticalplots,studentscanpressy[STATPLOT]4Í.Toturnoffallfunctions,theycanpressy[Y-VARS]5:On/Offthen2:FnOff. Thewindowscreensettingsforthesampledataareshownbelow: OntheTI-80calculator,gridpointscanbeaddedtoagraphbypressingy[DRAW]9Í.OntheTI-83,pressy[FORMAT],thenpress†and~tomovetoGridOn,andpressÍ. ©1997TEXASINSTRUMENTSINCORPORATED 24 GraphingCalculatorActivitiesforEnrichingMiddleSchoolMathematics Ascatterplotofthesampledata,alongwithagraphofy=x,asshownonaTI-82isgivenbelow.Studentswilladdthelineinstep8. Anexaminationofratiosisreasonableanytimeonenaturallyexpectsthatameasureofzeroononevariablewouldbeassociatedwithameasureofzeroontheother,thatis,wheneveronevariableisexpectedtovarydirectly(y=k*x)withtheother. ComputingtheratioofL2toL1usingthesampledataresultsinthefollowingontheTI-82: Althoughitisexpectedthatmostofthevalueswilllienear1,itispossiblethatsomewillnot.Thisisfine.Ratherthaneptingthetypicalor“average”ratioas1,youcouldinsteadusethearithmeticaverageofallputedratios.Thisaverage,ormean,canbefoundputingthemeanofL3.Fromthehomescreen,pressy[LIST],highlightMathandthenpress3y[L3]Íputeanddisplaythismeanvalue.Forthesampledatathismeanisapproximately1.005.
7.Youcanviewthecoordinatesofpointsonthisplotbypressingrandusing~and|tohighlightdifferentpoints.Checkthatthetracedvaluesagreewiththoseyouhavecollected. Whenyouexaminedyourpaperversionofthisscatterplot,youwereaskedtodrawalinethatconnectedpointswherethefirstandsecondcoordinateswereequal.Todothisonyourcalculator,youwillneedtoaddthegraphofthelinedefinedbyy=x.
8.Toaddthislinetoyourscatterplot,simplypress#„s. Thefactthattheheightandarmspandataseemtolieveryclosetotheliney=x(wherexrepresentsheightandyrepresentsarmspan)isaninterestingfinding.Infact,itisjustthistypeofalgebraicrelationshipthatstatisticiansareoftenlookingforwhentheyexaminesetsofpaireddata.Sometimesinthesearchforsuchrelationships,itishelpfultoexaminetheratiosoftwoquantitiesbeingstudied. Inthisstudy,youparingheightandarmspansolet’susethecalculatorputetheratioofarmspantoheightforeachclassmemberforwhomyouhavecollecteddata.
9.Toobtainthislistofratios: a.Press…1togainesstothecalculator’sliststorage. b.Press~twicetomovetoL3. 10.Press}tomovetosothatL3ishighlighted. 11.Pressy[L2]¥y[L1]ÍtodefineL3asalistofratiosofheighttoarmspan. ✐GototheQuestionssectionandanswer#12through#15. ©1997TEXASINSTRUMENTSINCORPORATED Activity3:HowDoYouMeasureUp?
25 Questions Table3.1.HeightandArmSpan Name Height(cm) ArmSpan(cm) Althoughstudentnamesarenotusedintheanalysisofanycollecteddata,itisimportantthatthebodymeasurementsbelinkedtothesameperson.IntheProblemsforAdditionalExploration,studentsareaskedtoinvestigatepossiblerelationshipsbetweenoverallheightandshoelength.Identifyingthedatawiththepersonfromwhomitwascollectedwillsavemakingheightmeasurementsagainifthatproblemisassigned.Samplemeasurementsfromsixrealstudentsareprovidedinthetablebelow. Name JillBrett JoJulieAshaCherise Height(cm) 158174164157157147 ArmSpan(cm)156178164156160148
1.Howoftenisaperson’sheightgreaterthantheirarmspan?
______________________________________________________________________________________________
2.Howoftenisheightlessthanarmspan?
______________________________________________________________________________________________ Inthesampledataprovided,heightwasgreaterthanarmspanone-thirdofthetime,heightwaslessthanarmspanone-halfthetime,andthetwomeasurementswereequalone-sixthofthetime. ©1997TEXASINSTRUMENTSINCORPORATED 26 GraphingCalculatorActivitiesforEnrichingMiddleSchoolMathematics
3.Howoftenareheightandarmspanequal?
______________________________________________________________________________________________ Ingeneral,armspanmeasurementsareapproximatelyequaltoheightmeasurementsonthesameindividual.Variationsfromthisgeneralrelationshiparenaturalandtobeexpectedwithinanysetofcollecteddata. Inthesampledata,heightsrangefrom147cmto174cmandarmspansfrom148cmto178cm.Eachdimensionofthegridisborderedby20gridsquaressoadistancewhichisamultipleof20wouldbeappropriateoneachscale.Using140cmto180cmoneachaxiswouldbeanappropriaterangeforthesetofsampledata.
4.Doyouseeanyotherrelationshipsbetweenheightandarmspan?
Describethesebelow. _______________________________________________ _______________________________________________ _______________________________________________ _______________________________________________ )
Returntopage22.
5. ScatterplotofHeightvs.ArmSpan ArmSpan(cm) Ascatterplotofthesampledataisgivenbelowalongwiththegraphoftheliney=xonthesamecoordinategrid. Height(cm) ©1997TEXASINSTRUMENTSINCORPORATED Activity3:HowDoYouMeasureUp?
27
6.Ingeneral,asheightincreases,whathappenstoarmspan?
Howisthisillustratedinthescatterplot?
____________________________________________________________________________________________________________________________________________________________________________________________ Ingeneral,asheightincreases(aswemovefromlefttorightacrossthegraph)theassociatedarmspanmeasurementsalsoincrease(gethigheronthegraph).
7.Inwhatwaysdoesthescatterplot,paredtothetableofdata,makeiteasiertoseerelationshipsbetweenheightandarmspan?
_______________________________________________ _______________________________________________ _______________________________________________ _______________________________________________ Since
thescatterplotordersthedataonthebasisofincreasingheights,andtablesofdatadonotnecessarilydoso,patternssuchasthisareoftenmucheasiertoseefromagraphicalpresentationthanfromanunorderednumericalpresentationofthedata.
8.Findthreeorfourpointsonyourcoordinatesystem(theydonothavetobepointsofthescatterplot)wherethehorizontalandverticalcoordinatesareequal,forexample(150,150).Drawastraightlinethroughthesepoints;thislinewillserveasareferencelineparingheightstoarmspans.Someofthepointsofthescatterplotlieabovethisline,someliebelowthisline,andothersmaylieonthisline.Findseveralpointsthatlieabovethisline.Whatistrueabouttherelationshipbetweenheightandarmspanforeachofthesepoints?
_______________________________________________ _______________________________________________ Be
certainthatstudentsunderstandthatthepointstheyaretolocatearenotnecessarilypointsthatarepartofthescatterplot.Theycanbeanypointslocatedonthecoordinategrid. Pointsthatlieabovethelinerepresentstudentswhosearmspansarelargerthantheirheights.
9.Howwouldyoudescribetherelationshipbetweenheightandarmspanforpointsthatliebelowtheline?
_______________________________________________ _______________________________________________ Points
thatliebelowthelinerepresentstudentswhosearmspansaresmallerthantheirheights. ©1997TEXASINSTRUMENTSINCORPORATED 28 GraphingCalculatorActivitiesforEnrichingMiddleSchoolMathematics Ifapointinthescatterplotliesontheline,thenthearmspanandheightofthestudentareequal. 10.Ifapointliesonthisline,whatrelationshipexistsbetweenheightandarmspan?
_______________________________________________ _______________________________________________
)Returntopage22,UsingtheCalculator.11.RecordyoursettingsforthePlotWindow: Sinceheightisrepresentedonthex-axisandarmspanonthey-axis,y=xindicatesthatarmspanandheightareequal. Iftheratioofarmspantoheightislessthanone,thenthemeasureofthearmspanislessthanthemeasureofthecorrespondingheight. IfL2¥L1≈1,thenaperson’sarmspanandheightareapproximatelyequalinmeasure. )Returntopage22,step4.12.Explainwhytheliney=xcontainspointswhere thetwocoordinatesareequal._____________________________________________________________________________________________________________________________________________ 13.FindavalueinL3thatislessthan1.Whatistherelationshipbetweentheheightandarmspanusedtocalculatethisvalue?
_____________________________________________________________________________________________________________________________________________ 14.IfweassumethatL2¥L1≈1,whatrelationshipexistsbetweenarmspanandheight?
Note:Thesymbol≈meansapproximatelyequalto. _____________________________________________________________________________________________________________________________________________ ©1997TEXASINSTRUMENTSINCORPORATED Activity3:HowDoYouMeasureUp?
29 15.Baseduponthisrelationship,predictthearmspanmeasurementforapersonwhoisknowntobe170cmtall?
Howoftendoyouthinkyourestimatewouldbetoohigh?
Toolow?
Explainyourresponses. _______________________________________________ _______________________________________________ _______________________________________________ _______________________________________________ _______________________________________________ _______________________________________________ A
personfoundtobe170cmtallwouldbeexpectedtohaveanarmspanofapproximately170cm.Thebestestimateofhowoftenthiswouldbetoohigh(ortoolow)couldbebaseduponthepercentofthepointsinthescatterplotthatlieabove(orbelow)theliney=x. ©1997TEXASINSTRUMENTSINCORPORATED 30 GraphingCalculatorActivitiesforEnrichingMiddleSchoolMathematics Theinstructionsforthisfirstproblemassumethatstudentsstillhavethedataonheightsandarmspansinthefirstthreelistsontheircalculators.Itisalsoassumedthatdataforshoelengthwillbeenteredforindividualsinthesameorderusedforheightsandarmspans. Somevariationintheratiosofshoelengthtooverallheightaretobeexpectedbut,aswiththearmspanandparisons,mostratiosshouldcenteraroundsomevalue.Theaverageoftheratiosshouldserveasagoodapproximationtouseindescribingarelationshipbetweenthetwoquantities.Thisratio(callitR)wouldserveasthebestguessfortheslopeofalinethat“fits”thepointsonthescatterplot.StudentscouldentertheexpressionY1=RX(whereRisreplacedwithputedmeanoftheshoelengthstooverallheightratios)intotheomenu.Theycouldthensboththescatterplotandthelineonthesamewindowtoobservethegoodnessofthe“fit.” ProblemsforAdditionalExploration
1.Doyouthinkthereisarelationshipbetweenaperson’sheightandthelengthoftheirshoe(shoelength,notshoesize)?
Investigatethisquestionbydoingthefollowing: a.Collectdataontheshoelengthsofclassmembers.EnterthisdatainTable2below.Makecertaintorecordthenewinformationwiththecorrectindividual. Table3.2.HeightandShoeLength Name Height(cm) ShoeLength(cm) ©1997TEXASINSTRUMENTSINCORPORATED Activity3:HowDoYouMeasureUp?
31 b.ClearlistsL4andL5onyourcalculator. c.EntertheshoelengthdataintoL4. d.Produceascatterplotofheight(L1)versusshoelength(L4)whereheightisrepresentedalongthehorizontalaxisandshoelengthisrepresentedalongtheverticalaxis. Dotheplottedpointsappeartoformapattern?
Describeanypatternsyou

see. ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ e.

CalculateL5astheratioofL4toL1;thatis,calculatetheratioofshoelengthtoheight. Findthemeanoftheratios.Fromthehomescreen,pressy[LIST],press~tohighlightMATH,press3y[L5]Í.Stateanapproximaterelationshipthatwouldpermityoutoestimateaperson’sshoelengthifyouknewtheperson’s

height. ____________________________________________ ____________________________________________
2.

Describeaprocessyoucouldusetodetermineanddescribearelationshipbetweenhandspan(thedistancebeweenthetipofthethumbandthetipofthelittlefingerwhenthefingersofthehandarestretchedopen)andlengthoftheforearm(thedistancefromtheinsideoftheelbowtothebaseofthehandatthewristjoint). _______________________________________________ _______________________________________________ _______________________________________________ _______________________________________________ We
wouldbeinterestedinseeingtheresultsofstudentinvestigationsintoproblems2and3.Encouragestudentstowritetousprovidingtheirdata,graphicaldisplays,andanalyses.Sendyourclassresultsto: Dr.ChristineBrowningDepartmentofMathematicsandStatisticsWesternMichiganUniv.Kalamazoo,MI49008 Wewillreturncorrespondence. Thevariablesheight,armspan,andshoelengthareexamplesofquantitativevariables.ThevariablesfavoritecolorofM&M,makeofautomobile,andgenderareexamplesofcategoricalvariables. Anotherinterestingratio thatstudentsmightenjoy investigatingisthatof heighttobellybutton height.Interestingly enough,thisratiowillbe veryclosetothegolden ratio 1+5≈1.618
2 . Thegoldenratio,alsocalledthedivineproportion,findsitswayintomanyothersettingsincludingart,architecture,andplantgrowth. ©1997TEXASINSTRUMENTSINCORPORATED 32 GraphingCalculatorActivitiesforEnrichingMiddleSchoolMathematics
3.Makeupaquestionthatconcernstwoquantitativevariables(variableswhosevaluesarenumbers)ofinteresttoyou.Collectdataonthesetwovariablesanduselistsandscatterplotstolookforanddescribepossiblerelationships. _______________________________________________ _______________________________________________ _______________________________________________ _______________________________________________ ©
1997TEXASINSTRUMENTSINCORPORATED