SECONDARY MATH 1 // MODULE 9

MODELING WITH DATA – 9.7

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9.7 Getting Schooled

A Solidify Understanding Task

InGettingMore$,LeoandAracelinoticeda

differenceinmen’sandwomen’ssalaries.Araceli

thoughtthatitwasunfairthatwomenwerepaidless

thanmen.Leothoughtthattheremustbesome

goodreasonforthediscrepancy,sotheydecidedtodigdeeperintotheCensusBureau’sincome

datatoseeiftheycouldunderstandmoreaboutthesedifferences.

First,theydecidedtocomparetheincomeofmenandwomenthatgraduatedfromhighschool(or

equivalent),butdidnotpursuefurtherschooling.Theycreatedthescatterplotbelow,withthex

valueofapointrepresentingtheaveragewoman’ssalaryforsomeyearandtheyvalue

representingtheaverageman’ssalaryforthesameyear.Forinstance,theyear2011isrepresented

onthegraphbythepoint(17887,30616).Youcanfindthispointonthegraphinthebottomleft

corner.

1. Baseduponthegraph,estimatethecorrelationcoefficient.

Women’sincome($)

Men’sincome($)

CCBYSteven

Isaacson

https://flic.kr/p/2M3fF

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2. Estimatetheaverageincomeformeninthistimeperiod.Describehowyouusedthegraph

tofindit.

3. Whatistheaverageincomeforwomeninthistimeperiod?Describehowyouusedthe

graphtofindit.

4. LeoandAracelicalculatedthelinearregressionforthesedatatobe! = 2.189! − 6731.8.Whatdoestheslopeofthisregressionlinemeanabouttheincomeofmencomparedto

women?Usepreciseunitsandlanguage.

“Hmmmm,”saidAraceli,“It’sjustasIsuspected.Thewholesystemisunfairtowomen.”“No,wait,”

saidLeo,“Let’slookatincomesformenandwomenwithbachelor’sdegreesormore.Maybeithas

somethingtodowithlevelsofeducation.”

5. LeoandAracelistartedwiththedataformenwithbachelor’sdegreesormore.Theyfound

thecorrelationcoefficientfortheaveragesalaryvsyearfrom2000-2011wasr=-.894.

Predictwhatthegraphmightlooklikeanddrawithere.Besuretoscaleandlabeltheaxes

andput12pointsonyourgraph.

36

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MODELING WITH DATA – 9.7

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Theactualscatterplotforsalariesformenwithbachelor’sdegreesfrom2000-2011isbelow.Howdidyoudo?

6. BothLeoandAraceliweresurprisedatthisgraph.Theycalculatedtheregressionlineand

got ! = −588.46! + 69978.Whatdoesthisequationsayabouttheincomeofmenwithbachelor’sdegreesfrom2000-2011?Useboththeslopeandthey-interceptofthelineof

regressioninyouranswer.

Next,theyturnedtheirattentiontothedataforwomenwithbachelor’sdegreesormorefrom

2000-2011.Here’sthedata:

Year 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000IncomeforWomen($)

41338 42409 42746 42620 44161 44007 42690 42539 42954 42871 42992 43293

37

SECONDARY MATH 1 // MODULE 9

MODELING WITH DATA – 9.7

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7. Analyzethedataforwomenwithbachelor’sdegreesbycreatingascatterplot,interpreting

thecorrelationcoefficientandtheregressionline.Forconsistencywiththemen’sgraphabove,use

0fortheyear2000,1fortheyear2001,etc.Drawthegraphandreporttheresultsofyouranalysis

below:

8. Nowthatyouhaveanalyzedtheresultsforwomen,comparetheresultsformenand

womenwithbachelor’sdegreesandmoreovertheperiodfrom2000-2011.

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MODELING WITH DATA – 9.7

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9. Leobelievesthatthedifferenceinincomebetweenmenandwomenmaybeexplainedby

differencesineducation,butAracelibelievestheremustbeotherfactorssuchasdiscrimination.

BasedonthedatainthistaskandGettingMore$,makeaconvincingcasetosupporteitherLeoor

Araceli.

10. Whatotherdatawouldbeusefulinmakingyourcase?Explainwhattolookforandwhy.

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9.7 Getting Schooled – Teacher Notes

A Solidify Understanding Task

SpecialNotetoTeachers:Thistaskrequirestheuseoftechnologythatcancalculatethe

correlationcoefficient,r,andalinearregression.Mostgraphingcalculatorswillworkwell.

GeoGebraorDesmos,bothpowerful,freecomputerappswouldbeveryhelpfulandeasytouseon

thistask.

Purpose:Thepurposeofthistaskistosolidifystudentsunderstandingoflinearmodelsfordata

byinterpretingtheslopesandinterceptsofregressionlineswithvariousunits.Studentsareasked

touselinearmodelstocompareandanalyzedata.Inthetasktheydrawconclusionsandjustify

argumentsaboutdata.Inadditiontheyareaskedtoconsideradditionaldatathatcouldbe

collectedtoinformtheirconclusions.

CoreStandardsFocus:

S.ID.6Representdataontwoquantitativevariablesonascatterplot,anddescribehowthe

variablesarerelated.

a.Fitafunctiontothedata;usefunctionsfittedtodatatosolveproblemsinthecontextof

thedata.Usegivenfunctionsorchooseafunctionsuggestedbythecontext.Emphasize

linear,quadratic,andexponentialmodels.

c.Fitalinearfunctionforascatterplotthatsuggestsalinearassociation.

S.ID.7Interprettheslope(rateofchange)andtheintercept(constantterm)ofalinearmodelinthe

contextofthedata.

S.ID.8Compute(usingtechnology)andinterpretthecorrelationcoefficientofalinearfit.

StandardsforMathematicalPracticeofFocusintheTask

SMP3-Constructviableargumentsandcritiquethereasoningofothers.

SMP4–Modelwithmathematics.

SECONDARY MATH 1 // MODULE 9

MODELING WITH DATA – 9.7

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Launch(WholeClass):

Remindstudentsoftheirworkwithmen’sandwomen’smedianannualincomesfromtheprevious

task.Askthemtorecallsomeoftheconclusionsthatcouldbemadefromthedata.Introducethis

taskbytellingthemthattheywillbedrawingupontheirexperiencewithcorrelationcoefficients

andlinearregressionstoanalyzeandcomparedata.Bythetimetheyhavefinishedthetaskthey

shouldbepreparedtousethedatatomakeanargumentaboutthedifferencesinmen’sand

women’ssalary,baseduponeducationandotherpossiblefactors.

Explore(SmallGroup):

Monitorstudentsastheywork,ensuringthattheyareestimatingasrequestedinthetaskbefore

makingthecalculations.Thiswillhelptodrawthemintothedatasothattheycanmakesenseofit

anddeepentheirunderstanding.Keepstudentsfocusedonusingtheunitsofslopebasedonthe

graphs.Theymaybemorefamiliarwithgraphsthathavetimeacrossthex-axis,butstruggleto

interpretthefirstgraphthatcomparessalariesofmenandwomenwheretheyearthedatawas

obtainedisnotevident.

Discuss(WholeClass):

Actualcorrelationcoefficientfor#1isr=0.6421.

Beginthediscussionwiththemeaningoftheslopeofthelinearregressioninthefirstgraph.

Studentsshouldbeabletoarticulatetheideathattheslopeinthiscasemeansthatthemedian

salaryformenwas2.189timesthemediansalaryforwomenofthesameeducationlevel.Inthis

casetheslopeisaratioofmen’ssalariestowomen’ssalariesortheratethatmen’ssalarieschange

inrelationtowomen’ssalaries.

Thenextslopetointerpretisin#6.Studentsshouldbeabletoarticulatethatthemediansalaryfor

menwentdownbyabout$588.49eachyearduringthetimeperiod.Inthiscasetheslopeisthe

rateofchangeofmen’ssalarieseachyear.

Thebulkofthediscussionshouldbeanopportunityforstudentstodigdeeplyintheanalysisofthe

datatomakethecasethateducationexplainsthedifferencesinmedianincomesbetweenmenand

SECONDARY MATH 1 // MODULE 9

MODELING WITH DATA – 9.7

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womenorthatthereareotherfactorsthatexplainthedifferences.Organizetheclasssothat

studentsareassignedtoonesideoftheargumentortheotherandthentaketurnspresentingone

pieceofevidencefromtheiranalysis.Recordtheclaimsandallowtheothersidetorefuteanyclaim

thattheyfeelisinerror.

AlignedReady,Set,Go:ModelingwithData9.7

SECONDARY MATH I // MODULE 9

MODELING DATA – RSG 9.7

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9.7

READY Topic:FindingdistancesandaveragesThegraphbelowshowsseveralpointsandtheline! = !.Usethegraphtoanswereachquestion.

1.TheverticaldistancebetweenpointNandtheline! = !onthegraphis3.Findalloftheverticaldistancesbetweenthepointsandtheline! = !.

B:

D:

E:

G:

I:

L:

N:

X:

2.Calculatethesumofallthedistancesyoufoundinexerciseone.

3.Whatistheaverageverticaldistanceofthepointsfromtheline! = !?

4.Isthelineshownonthegraphthelineofbestfit?Explainwhyorwhynot.Ifitisnotthebestline,drawonethatisbetterfittothedata.

5.Estimatethecorrelationcoefficientforthissetofdatapoints.Ifyouhaveawaytocalculateitexactly,checkyourestimate.(Youcoulduseagraphingcalculatorordatasoftware.)

READY, SET, GO! Name PeriodDate

40

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MODELING DATA – RSG 9.7

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9.7

SET Topic:CreatingandanalyzingscatterplotsDeterminewhetheralinearoranexponentialmodelwouldbebestforthegivenscatterplot.Thensketchamodelonthegraphthatcouldbeusedtomakepredictions.6.

7.

8.a)Usethedatainthetablebelowtomakeascatterplot.

b)Isthecorrelationofthegraphpositiveornegative?Why?

c)Whatwouldyouestimatethecorrelationcoefficienttobe?Why?

d)Createaregressionlineandwritetheregressionequation.

e)Whatdoestheslopeoftheregressionequationmeanintermsofthevariables?

f)Mostschoolyearsare36weeks.Iftherateofspendingiskeptthesame,howmuchmoremoneyneedstobesavedduringthesummerinorderfortheretobemoneytolastall36weeks?

20

200

Money

Weeks

41

SECONDARY MATH I // MODULE 9

MODELING DATA – RSG 9.7

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9.7

GO Topic:Determiningwhentouseatwo-waytableandwhenuseascatterplot9.Inwhichsituationsdoesitmakethemostsensetouseatwo-waytableandlookattherelativefrequencies.

10.Inwhichsituationsdoesitmakethemostsensetouseascatterplotandalinearorexponentialmodeltoanalyzeandmakedecisionsordrawconclusions?

Labeleachrepresentationbelowasafunctionornotafunction.Ifitisafunction,labelitaslinear,exponential,orneither.Ifisdoesnotrepresentafunction,explainwhy.11.

! !0 121 122 12

3 12

4 12

12.! !

1 152 303 152 201 25

13.! !

-6 -2-5 -3-4 -4-3 -5-2 -6

14.! + 12! = 4

15.! = 3 ∙ 4 !!!

16.Theamountofmedicineinthebloodstreamofacatastimepasses.Theinitialdoseofmedicineis80mmandthemedicinebreaksdownat35%eachhour.

17.

Time 0 1 2 3 4

Moneyinbank $250 $337.50 $455.63 $615.09 $830.38

42