M1 Mod9 SE 62016f · 2. 3. If the box plots above represent the results of two different classes on...

The Mathematics Vision Project Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius

© 2016 Mathematics Vision Project Original work © 2013 in partnership with the Utah State Off ice of Education

This work is licensed under the Creative Commons Attribution CC BY 4.0

MODULE 9

Modeling Data

SECONDARY

MATH ONE

An Integrated Approach

SECONDARY MATH 1 // MODULE 9

MODELING DATA

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

MODULE 9 - TABLE OF CONTENTS

MODELING DATA

9.1 Texting by the Numbers – A Solidify Understanding Task

Use context to describe data distribution and compare statistical representations (S.ID.1, S.ID.3)

READY, SET, GO Homework: Modeling Data 9.1

9.2 Data Distribution – A Solidify/Practice Understanding Task

Describe data distributions and compare two or more data sets (S.ID.1, S.ID.3)


9.3 After School Activity – A Solidify Understanding Task

Interpret two way frequency tables (S.ID.5)


9.4 Relative Frequency – A Solidify/Practice Understanding Task

Use context to interpret and write conditional statements using relative frequency tables (S.ID.5)


9.5 Connect the Dots – A Develop Understanding Task

Develop an understanding of the value of the correlation co-efficient (S.ID.8)


9.6 Making More $ – A Solidify Understanding Task

Estimate correlation and lines of best fit. Compare to the calculated results of linear regressions and the

correlation co-efficient (S.ID.7, S.ID.8)



MODELING DATA


9.7 Getting Schooled – A Solidify Understanding Task

Use linear models of data and interpret the slope and intercept of regression lines with various units

(S.ID.6, S.ID.7, S.ID.8)


9.8 Rocking the Residuals – A Develop Understanding Task

Use residual plots to analyze the strength of a linear model for data (S.ID.6)


9.9 Lies and Statistics – A Practice Understanding Task

Use definitions and examples to explain understanding of correlation coefficients, residuals, and linear

regressions (S.ID.6, S.ID.7, S.ID.8)


SECONDARY MATH I // MODULE 9

MODELING DATA – 9.1


9.1 Texting by the Numbers A Solidify Understanding Task

Technologychangesquicklyandyethasalargeimpactonour

lives.Recently,Rachelwasbusychattingwithherfriendsviatext

messagewhenhermomwastryingtoalsohaveaconversationwithher.Afterward,theyhada

discussionaboutwhatisanappropriatenumberoftextstosendeachday.Sincetheycouldnot

agree,theydecidedtocollectdataonthenumberoftextspeoplesendonanygivenday.Theyeach

asked24oftheirfriendsthefollowingquestion:“WhatistheaveragenumberoftextsyouSEND

eachday?”Thedataandhistogramrepresentingall48responses:

{0,2,3,3,5,5,5,5,5,5.5,6,6,6,10,12,13,15,15,16,20,25,35,36,70,80,85,110,130,137,138,

138,140,142,143,145,150,150,150,150,150,150,150,155,162,164,165,175,275}

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PartI:

1. Whatinformationcanyouconcludebasedonthehistogramabove?

2. Representthesamedatabycreatingaboxplotabovethehistogram.

3. Whatstorydoestheboxplottell?Describetheprosandconsofeachrepresentation(histogramandboxplot).Inotherwords,whatinformationdoeseachrepresentationhighlight?Whatinformationdoeseachrepresentationhideorobscure?

PartII:Priortotalkingaboutthedatawithhermom,Rachelhadcreatedaboxplotusingherown

datashecollectedanditlookedquitedifferentthanwhentheycombinedtheirdata.

Averagenumberoftextssenteachday

4. DescribethedataRachelcollectedfromherfriends.Whatdoesthisinformationtellyou?

5. Comparethetwoboxplots(Rachel’sdatavsalldata).

6. Rachelwantstocontinuesendinghernormalnumberoftexts(averageof100perday)andhermomwouldlikehertodecreasethisbyhalf.Presentanargumentforeachside,usingmathematicstojustifyeachperson’srequest.

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9.1 Texting by the Numbers – Teacher Notes A Solidify Understanding Task

Purpose:Inthistask,studentswillusepriorknowledgetointerpretdatausingahistogram,and

thenrepresentthesamedatawithaboxplot.Studentswilldiscusstheshape(bimodal),center,and

spread(outliers)ofthedata,theinformationhighlightedorhiddenbyeachrepresentation,and

comparetwodatasetsusingdifferentrepresentations.Comparingdatasetsisthefocusofthetask.

CoreStandardsFocus:

S.ID.1Representdatawithplotsontherealnumberline(dotplots,histograms,andboxplots).S.ID.3Interpretdifferencesinshape,center,andspreadinthecontextofthedatasets,accountingforpossibleeffectsofextremedatapoints(outliers).RelatedStandards:S.ID.2

StandardsforMathematicalPracticeofFocusintheTask:

SMP1–Makesenseofproblemsandpersevereinsolvingthem

SMP3–Constructviableargumentsandcritiquethereasoningofothers

SMP5–Useappropriatetoolsstrategically

TheTeachingCycle:

Launch(WholeClass):

Accessbackgroundknowledge:Readthescenariofromthetask,thenhavestudentsquietlywrite

downtheirobservationsfromthehistogram,thensharewithapartner.Listenforcommentstouse

duringwholegroupdiscussionaboutshape,center,andspread.Havestudentsmoveontoanswer

theremainingquestions.

Note:Ifmoststudentsseemstuck,havethewholegroupcometogethertopopcorn(quicklyshare)

observationstogetideasaboutshapeandspreadout,thenhavestudentsmoveontoanswerthe

remainingquestionsrelatedtothedatainpartnersorsmallgroups.




Explore(SmallGroup),part1:

Asyoumonitor,listentotheinterpretationofthehistogramandpressstudentstodescribethe

distribution(shape,center,andspread).Lookforstudentswhotalkaboutthedatahavingtwo

‘modes’andtheirconjecturesforwhythatmaybe(havethemsharethetwomodeconversation

duringthewholegroupdiscussion).Whenstudentsarecreatingtheboxplot,remindthemtolabel

eachquartileandlistenforcommentsaboutthethedatapointof275(seeifsomeonelabelsthisas

anoutlier).Studentshavecreatedboxplotsbefore.Aftermoststudentshavecreatedtheboxplotto

gowiththesamedataandseveralhavealreadywrittenabouttheinformationeachrepresentation

highlights,bringtheclassbacktogetherforthefirstwholegroupdiscussion.

Discuss(WholeClass),partI:

Forthisdiscussion,besuretohavethehistogramdisplayedsoeveryonecanmakeavisual

connectiontothedescriptionofthedata.Asthefirststudentchosensharestheirinterpretationof

thehistogram,makesuretheypointtothehistogramastheycommunicatetheirinterpretationof

thedata.Studentsmaynotusetheacademicvocabularyofbimodal,butnowisagoodtimetobring

thisupandhavestudentswritethisintheirjournal.Nexthaveastudentsharetheirboxplotandgo

overtheirinterpretationofthesamedata(quartiles,median,variability,andpossiblyoutlier).

Ifnooneinyourclasshasaboxplotthatshows275asanoutlier,thentheirboxplotwilllooklike

theoneabove.Ifyouhavestudentswhomade275anoutlier,havetheoutlierdiscussionfirst,then

comparewhateachrepresentation(histogram,boxplot)highlightsandwhateachrepresentation

‘hides’orobscures.Ifnoonemade275anoutlier,thenstartwithcomparingrepresentationsthen

discusswhatdeterminesanoutlier.




Insummary,forthiswholegroupdiscussion,besuretogetoutthefollowing:

• twomodeswithadescriptionthatpeopleeitherseemtoonlydoafewtexts(lessthan20)oralot(between140-180)perday,

• thedatamostlyliesbetween0and180textsperday;onevalueappearstobeanoutlierof275textsperday.

• themeanforthewholesetofdatais81.05-doesthisseemrelevant?• Prosofhistogram:showsfrequency,canseebimodaldata.

Prosofboxplot:showsquartilerangesandthemedian• Inthissituation,theboxplotobscuresthebimodaldataandmayseemlikethereisamoreeven

distribution.Iftheoutlierdiscussionhasn’thappened,nowisthetimetotalkaboutdatapointsthatseemtobeoutliers.Acommonruleofthumbtodetermineifadatapointisanoutlieristousetheequation:1.5(valueofinterquartilerange)+valueofq3forupperextremesor1.5(valueofinterquartilerange)-valueofq1forlowerextremes.Askstudentsifthisdatahasanyoutliers?Becausetheinterquartilerangeissolarge,thevalueof275isnotanoutlier.Afterward,explainhowoutlierscanberepresentedinaboxplotandshowwhattheboxplotwouldlooklikeif275hadbeenanoutlier.

Explore(SmallGroup),partII:

ForpartII,studentsshouldconcludethatRachel’sfriendsaremostlyrepresentingthe‘uppermode’

ofthedatawhilehermom’sfriendsarethe‘lowermode’.Studentsmayhavealready‘guessed’this

duringthefirstpartofthetask,however,theyshouldnowusethedatafromtheboxplotinpartII

tojustifythisstatement.

IfyouwouldlikeyourstudentstohavethedataafteranalyzingRachel’sboxplot,itislistedhere:

Rachel’smom:{150,5.5,6,5,3,10,150,15,20,15,6,5,3,6,0,5,12,25,16,35,5,2,13,5}

Rachel:{130,145,155,150,162,80,140,150,165,138,175,275,85,137,110,143,138,142,164,

70,150,36,150,150}

Discuss(WholeClass),partII:

ThefocusofthisdiscussionisinitiallyontheinterpretationofRachel’sdata,thenonthe

comparisonofthetwosetsofdata(Rachel’sfriendsversusRachel’smomsfriends).

AlignedReady,Set,Go:Features9.1


MODELING DATA – RSG 9.1

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

9.1

READY Topic:MeasuresofcentraltendencySam’stestscoresforthetermwere60,89,83,99,95,and60.

1.SupposethatSam’steacherdecidedtobasethetermgradeonthemean.

a.WhatgradewouldSamreceive?

b.Doyouthinkthisisafairgrade?Explainyourreasoning.

2.SupposethatSam’steacherdecidedtobasethetermgradeonhismedianscore.



3.SupposethatSam’steacherdecidedtobasethetermgradeonthemodescore.



4.Aiden’stestscoresforthesametermwere30,70,90,90,91,and99.WhichmeasureofcentraltendencywouldAidenwanthisteachertobasehisgradeon?Justifyyourthinking.

5.Mostteachersbasegradesonthemean.Doyouthinkthisisafairwaytoassigngrades?Whyorwhynot?

READY, SET, GO! Name PeriodDate

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9.1

SET Topic:Examiningdatadistributionsinabox-and-whiskerplot.

6.Makeabox-and-whiskerplotforthefollowingtestscores.

60,64,68,68,72,76,76,80,80,80,84,84,84,84,88,88,88,92,92,96,96,96,96,96,96,96,100,100

7a.Howmuchofthedataisrepresentedbythebox?

b.Howmuchisrepresentedbyeachwhisker?

8.Whatdoesthegraphtellyouaboutstudentsuccessonthetest?

GO Topic:Creatinghistograms.

UsethedatafromtheSETsectiontoanswerthefollowingquestions.

9.Makeafrequencytablewithintervals.Useanintervalof5.10.Makeahistogramofthedatausingyourintervalsof5.

11.Whatinformationishighlightedinthehistogram?12.Whatinformationishighlightedinthebox-and-whiskerplot?

50 55 60 65 70 75 80 85 90 95 100

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SECONDARY MATH I // MODULE 3 MODELING DATA—9.2


9.2 Data Distribution A Practice Understanding Task Alotofinformationcanbeobtainedfromlookingatdataplotsandtheirdistributions.Itisimportantwhendescribingdatathatweusecontexttocommunicatetheshape,center,andspread.Shapeandspread:

• Modes:uniform(evenlyspread-noobviousmode),unimodal(onemainpeak),bimodal(twomainpeaks),ormultimodal(multiplelocationswherethedataisrelativelyhigherthanothers).

• Skeweddistribution:whenmostdataistoonesideleavingtheotherwitha‘tail’.Dataisskewedtosideoftail.(iftailisonleftsideofdata,thenitisskewedleft).

• Normaldistributionandstandarddeviation:curveisunimodalandsymmetric.Datathathasanormaldistributioncanalsodescribethedatabyhowfaritisfromthemeanusingstandarddeviation.

• Outliers:valuesthatstandawayfromthebodyofthedistribution.Forabox-and-whiskeroutliersdeterminediftheyaremorethan1.5timestheinterquartilerange(lengthofbox)beyondquartiles1and3.Alsoconsideredanoutlinerifdataismorethantwostandarddeviationsfromthecenterofanormaldistribution.

• Variability:valuesthatareclosetogetherhavelowvariability;valuesthatarespreadaparthavehighvariability.

Center:• Analyzethedataandseeifonevaluecanbeusedtodescribethedataset.Normal

distributionsmakethiseasy.Ifnotanormaldistribution,determineifthereisa‘center’valuethatbestdescribesthedata.Bimodalormultimodaldatamaynothaveacenterthatwouldprovideusefuldata.

Therearerepresentationsoftestscoresfromsixdifferentclassesfoundbelow,foreach:

1. Describethedatadistribution.2. ComparedatadistributionsbetweenAndersonandWilliams.3. ComparedatadistributionsbetweenWilliamsandLemon.4. ComparedatadistributionsbetweenCroftandHurlea.5. ComparedatadistributionsbetweenJones,Spencer,andAnderson.6. ComparedatadistributionsbetweenSpencerandtheotherhistograms.7. Whichdistributionsaremostsimilar?Different?Explainyouranswer.

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DatasetI:Williams’sclass DatasetII:Lemon’sclass

DatasetIII:Croft’sClass DatasetIV:Anderson’sClass

DatasetV:Hurlea’sclass DatasetVI:Jones’class

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DatasetVII:Spencer’sclass

DatasetVIII:OverallAchievementTestScores

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9.2 Data Distribution – Teacher Notes A Practice Understanding Task Purpose:Studentsarealreadyfamiliarwithdotplots,boxplots,andhistograms.Thistaskhasthemdescribedatadistributionsandcompareshape,center,andspreadoftwoormoresetsofdata.

CoreStandardsFocus:

S.ID.2Usestatisticsappropriatetotheshapeofthedatadistributiontocomparecenter(median,mean)andspread(interquartilerange,standarddeviation)oftwoormoredifferentdatasets.S.ID.3Interpretdifferencesinshape,center,andspreadinthecontextofthedatasets,accountingforpossibleeffectsofextremedatapoints(outliers).

RelatedStandards:S.ID.1

StandardsforMathematicalPractice:


SMP4–Modelwithmathematics

SMP8–Lookforandexpressregularityinrepeatedreasoning

TheTeachingCycle:

Note:Itwouldbegoodtohavethedatayouwanttocompareinaformatthatislargeandvisibleforthewholegroupdiscussion.Forexample,youcouldcopythetwodatasetsyouwishtocompareandplacethemnexttoeachotherinaformatthatcanbeprojectedsothatwhenstudentsaresharingduringwholegroup,thevisualrepresentationisavailableforeveryonetosee.

Note:Studentshavebeenaskedtoidentifyandinterpretunivariatedatausingdotplots,histograms,andboxplotssincesixthgrade.Inthiscourse,studentsareaskedtocomparedatasetsusingtheirknowledgeofshape,center,andspreadandhavebecomemorecomfortablewiththeseattributes.Outliers,skeweddata,andnormaldistributionmaybenewthisyearaswell.Launch(WholeClass):

Havestudentsreadthevocabularytodescribedatadistributionsandaskthemtounderlineinformationthatisnewtothem.Havethemworkindividuallyforawhileonquestion1thathasthemdescribeeachdatasetbeforehavingthemworktogetherwithapartnerorsmallgrouptoanswertheremainingquestions(wheretheycomparedatasets).



Explore(SmallGroup):

Givestudentstimetoanswerthequestionscomparingdatasets.Listenforstudentstousevocabularyindescribingagivendataset,andtocompareshape,center,andspreadoftwoormoredatasets.Listenforstudentstocomparedatasets,notjustlistattributesofeach.Pressstudentstomakecomparisonsshowingtheyunderstandwhentousedatatodescribeandcompareshape,center,andspreadbetweendatasets.Examplesincludenoticingoutliers,variabilityandspreadbetweendata(noticethatHurleaandSpencerhaveascalethatisdifferentthantheothers),andothertrends.Again,makesurestudentsdonotjustlistcharacteristicsofeachdistributionandthinktheyare‘comparing’.Discuss(WholeClass):

Beginthewholegroupdiscussionbyselectingproblemsfromquestionsthatcomparedatasets.Basedonsmallgroupconversations,choosewhichcomparisonstoshareoutinwholegroup.Thefocusofthewholegroupdiscussionistodothefollowing:

o Showstudentunderstandingofusingstatisticsappropriatetotheshapeofthedatadistributiontocomparecenterandspread.

o Showstudentunderstandingofwhatinformationisprovidedwhengivenahistogram,boxplot,dotplot.

AlignedReady,Set,Go:ModelingData9.2






9.2

READY Topic:Drawingconclusionsfromdata.Inproblems1–4youaretoselectthebestanswerbasedonthegivendata.Belowyourchosenanswerisaconfidencescale.Circlethestatementthatbestdescribesyourconfidenceinthecorrectnessoftheansweryouchose.Thegoalistogainawarenessofhowitseemseasiertodrawconclusionsinsomecasesthaninothers. 1.Data:1,2,4,8,16,32, Thenextnumberinthelistwillbe:________

a.largerthan32 b.positive c.exactly64 d.lessthan32

IamcertainIamcorrect. Iamalittleunsure. IhadnoideasoIguessed.

Whataboutthedatamadeyoufeelthewayyoudidabouttheansweryoumarked?

2.Data:47,-13,-8,9,-23,14, Thenextnumberinthelistwillbe:________

a.positive b.negative c.lessthan100 d.lessthan-100



3.Data:-10,¾,38,-10,½,-81,-10,¼,93,-10, Thenextnumberinthelistwillbe:______

a.morethan93 b.negative c.afraction d.awholenumber


4.Data:50,-43,36,-29,22,-15 Thenextnumberinthelistwillbe:______

a.odd b.lessthan9 c.two-digits d.greaterthan-15




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9.2

SET Topic:Creatinghistograms.

Mr.Austingaveaten-pointquiztohis9thgrademathclasses.Atotalof50studentstookthequiz.Mr.Austinscoredthequizzesandlistedthescoresalphabeticallyasfollows.

1stPeriodMath 2ndPeriodMath 3rdPeriodMath

6,4,5,7,5,

9,5,4,6,6,

8,5,7,5,8,

1,8,7,10,9

4,5,8,6,8,

9,5,8,5,1,

5,5,7,5,7

9,8,10,5,9,

7,8,9,8,5,

8,10,8,8,5

5.UseALLofthequizdatatomakeafrequencytablewithintervals.Useanintervalof2.

Score Frequency

0-1

2–3

4–5

6–7

8–9

10-11

6.Useyourfrequencytabletomakeahistogramforthedata

7.Describethedatadistributionofthehistogramyoucreated.Includewordssuchas:mode,skewed,outlier,normal,symmetric,center,andspread,iftheyapply.(Hint:Don’tforgetstandarddeviation.)

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9.2

8.Createagraphofyourchoice(histogram,boxplot,dotplot)for1stand3rdperiod.

9.Whichclassperformedbetter? Justifyyouranswerbycomparingtheshape,center,andspreadofthetwoclasses.(Hint:Don’tforgetstandarddeviation.)

GO

Topic:Figuringpercentages

10.Whatpercentof97is11? 11.Whatpercentof88is132?

12.Whatpercentof84is9? 13.Whatpercentof88.6is70?

14.Whatis270%of60? 15.Whatis84%of25?

18

16

14

12

10

8

6

4

2

1 2 3 4 5 6 7 8 9 10

18

16

14

12

10

8

6

4

2

1 2 3 4 5 6 7 8 9 10

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9.3 After School Activity A Develop Understanding Task PartI

Rashidisinchargeofdeterminingtheupcomingafterschoolactivity.Todeterminethetypeof

activity,Rashidaskedseveralstudentswhethertheyprefertohaveadanceorplayagameof

soccer.AsRashidcollectedpreferences,heorganizedthedatainthefollowingtwo-wayfrequency

table:

Girls Boys Total

Soccer 14 40 54

Dance 46 6 52

Total 60 46 106

Rashidisfeelingunsureoftheactivityheshouldchoosebasedonthedatahehascollectedandis

askingforhelp.Tobetterunderstandhowthedataisdisplayed,itisusefultoknowthattheouter

numbers,locatedinthemarginsofthetable,representthetotalfrequencyforeachroworcolumn

ofcorrespondingvaluesandarecalledmarginalfrequencies.Valuesthatarepartofthe‘inner’bodyofthetablearecreatedbytheintersectionofinformationfromthecolumnandtherowandthey

arecalledthejointfrequencies.

1. Usingthedatainthetable,constructaviableargumentandexplaintoRashidwhichafter

schooleventheshouldchoose.

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PartII:Twowayfrequencytablesallowustoorganizecategoricaldatainordertodraw

conclusions.Foreachsetofdatabelow,createafrequencytable.Wheneachfrequencytableis

complete,writethreesentencesaboutobservationsofthedata,includinganytrendsor

associationsinthedata.

2. Dataset:Thereare45totalstudentswholiketoreadbooks.Ofthosestudents,12ofthem

likenon-fictionandtherestlikefiction.Fourgirlslikenon-fiction.Twentyboyslikefiction.

Fiction Nonfiction Total

Boys

Girls

Total

Observation1:

Observation2:

Observation3:

3. Dataset:35seventhgradersand41eighthgraderscompletedasurveyabouttheamountoftimetheyspendonhomeworkeachnight.50studentssaidtheyspentmorethananhour.

12eighthgraderssaidtheyspendlessthananhoureachnight.

Total

Morethanonehour

Lessthanonehour

Total

Observation1:

Observation2:

Observation3:

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9.3 After School Activity – Teacher Notes A Develop Understanding Task Purpose:Thepurposeofthistaskisforstudentstomakesenseoftwowayfrequencytables,touse

thedatatomakeaninformeddecision,andthenconstructaviableargumentjustifyingtheirchoice.

Studentswillfocusondifferentareasofthetwowaytablesoitisimportantthattheyareprecise

withtheircommunication.

CoreStandardsFocus:

S.ID.5Summarizecategoricaldatafortwocategoriesintwo-wayfrequencytables.Interpret

relativefrequenciesinthecontextofthedata(includingjoint,marginal,andconditionalrelative

frequencies).Recognizepossibleassociationsandtrendsinthedata.


SMP1–Makesenseofproblemsandpersevereinsolvingthem


SMP6–Attendtoprecision

TheTeachingCycle:

Launch(WholeClass):

Readthescenarioandclarifyhowatwowayfrequencytableiscreated.Explaintostudentsthat

theirjobistointerpretthetable,choosetheafterschoolactivitythatmakesthemostsensetothem,

andthenprovidemathematicalreasoningthatwouldconvinceRashidtomakethesameselection.


Givestudentstimetointerpretthedata,movingfromgrouptogroupmakingsuretheyareusing

mathematicstomakesenseofthedata(forexample,showingthat14outof106girlschosesoccer

meansthatonly14%ofallgirlswouldlikesoccertobethechosenafterschoolactivity).Asyou

monitor,listenfordifferentgroupstoselectoppositeafterschoolactivities.Pressstudentstobe



veryclear,usingpreciselanguagetodescribetheirmathematics.Thiswillbeimportantduringthe

wholegroupdiscussionsincethepercentageforeachsituationvariesdependingonwhich‘total’

studentschoose.Thistaskismoreaboutbecomingfamiliarwithhowtofinddifferentpercentages

inatwowaytableandnotaboutconditionalprobabilities.AsstudentsmovetopartII,helpgroups

thatstrugglebyasking“Whatarethetwotypesofcategoricaldatabeingcompared?”orhavethem

readonesentenceonly,thenask“Whichcellofthetablecanbefilledinbasedonthisinformation?”

Discuss(WholeClass):

Asawholeclass,havetwodifferentgroupssharetheirrecommendationsfortheafterschool

activity.Havethefirstgroupsharethatselectedtheactivitythatwasleastchosenbytheclass.Ask

theclassiftheyhaveanyquestionsforthegroupwhopresented,thenasktheclassifanyonewho

hadchosentheotherafterschoolactivityhaschangedtheirmind,andifso,explainwhy.Next,have

agroupsharethatchosetheotheractivity.Thepurposeofthisdiscussionistohighlighthowto

summarizedatainatwowaytable,sobesurethatthepresenterscommunicatehowtheyfound

eachpercentagepresentedandthatallstudentscansummarizeatwowaytable.Movetopart2of

thetaskandhavesomeoneexplainhowtheysetupthetwowaytableforoneoftheproblemsin

part2.Asawholeclass,summarizetheprocessforfillinginatwowaytable.







9.3

READY

Topic:Interpretingdatafromascatterplot

1.Thescatterplotcomparesshoesizeandheightinadultmales.Basedonthegraph,doyouthinkthereisarelationshipbetweenaman’sshoesizeandhisheight?

Explainyouranswer.

2.Thescatterplotcomparesleft-handednesstobirthweight.Basedonthegraph,doyouthinkbeingleft-handedisrelatedtoaperson’sbirthweight?

Explainyouranswer.


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9.3

SET Topic:Two-wayfrequencytablesHereisthedatafromMr.Austin’s10-pointquiz.Studentsneededtoscore6orbettertopassthequiz.

1stPeriodMath 2ndPeriodMath 3rdPeriodMath

6,4,3,7,5,

9,5,4,6,6,

8,5,7,3,6,

2,8,7,10,9

3,3,8,6,6,

9,5,8,5,3,

5,5,7,5,7

9,8,10,5,9,

7,8,9,8,3,

8,10,8,7,5

3.Makeatwo-wayfrequencytableshowinghowmanystudentspassedthequizandhow

manystudentsfailedthequizineachclass. 1stperiod 2ndperiod 3rdperiod TotalPassed Failed Total

Useacoloredpenciltolightlyshadethecellscontainingthejointfrequencynumbersinthetable.Theun-shadednumbersarethemarginalfrequencies.(Usethesetermstoanswerthefollowingquestions.)4.IfMr.Austinwantedtoseehowmanystudentsinall3classescombinedpassedthequiz,

wherewouldhelook?

5.IfMr.Austinwantedtowritearatioofthenumberofpassingstudentscomparedtothenumberoffailingstudentsforeachclass,wherewouldhefindthenumbershewouldneedtodothis?

6.Makeatwo-wayfrequencytablethatgivestherelativefrequenciesofthequizscoresforeachclass.

1stPeriod 2ndPeriod 3rdPeriod Total

Passed

Failed

Total

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9.3

GO Topic:Organizingdata.7.Sophiesurveyedallofthe6thgradestudentsatReaganElementarySchooltofindoutwhichTVNetworkwastheirfavorite.Shethoughtthatitwouldbeimportanttoknowwhethertherespondentwasaboyoragirlsosherecordedherinformationthefollowingway.AnimalPlanet CartoonNetwork Disney Nickelodeon

GGBBBBBGBBBGBBBGGBBBBBBBB

BBBBBBBBBGGGBBBGBGBGGGBGG

GGGGGGBBBBBBGBGBGGBBBGGBGGGGGBBBGGGGGB

BBBBGGGGGGGGGGGGGGGBBGGGGBGGGGGGGGGBBBBBGGGGGGGG

Sophieplannedtouseherdatatoanswerthefollowingquestions:

I.Aretheremoregirlsorboysinthe6thgrade? II.Whichnetworkwastheboys’favorite? III.Wasthereanetworkthatwasfavoredbymorethan50%ofonegender?Butwhenshelookedatherchart,sherealizedthatthedatawasn’ttellingherwhatshewantedtoknow.Herteachersuggestedthatherdatawouldbeeasiertoanalyzeifshecouldorganizeitintoatwo-wayfrequencychart.HelpSophieoutbyputtingthefrequenciesintothecorrectcells.

FavoriteTVNetworks Girls Boys Totals

AnimalPlanet

CartoonNetwork

Disney

Nickelodeon

Totals

NowthatSophiehasherdataorganized,usethetwo-wayfrequencycharttoanswerher3questions.

a.Aretheremoregirlsorboysinthe6thgrade?

b.Whichnetworkwastheboys’favorite?

c.Wasthereanetworkthatwasfavoredbymorethan50%ofonegender?

15



9.4 Relative Frequency A Solidify/Practice Understanding Task Rachelisthinkingaboutthedatasheandhermomcollectedfortheaveragenumberoftextsapersonsendseachdayandstartedthinkingthatperhapsatwo-waytableofthedatatheycollectedwouldhelpconvincehermomthatshedoesnotsendanexcessiveamountoftextsforateenager.Thetableseparateseachdatapointbyage(teenagerandadult)andbytheaveragenumberoftextssent(morethan100perdayorlessthan100perday).

1. Writetwoobservationstatementsofthistwowaytable.

Tofurtherprovideevidence,Racheldecidedtodosomeresearch.Shefoundthatonly43%ofpeoplewithphonessendover100textsperday.Shewasdisappointedthatthedatadidnotsupporthercaseandconfusedbecauseitdidnotseemtomatchwhatshefoundinhersurvey.

2. Whatquestionsdothesestatisticsraiseforyou?WhatdatashouldRachellookfortosupporthercase?

Afterlookingmorecloselyatthedata,Rachelfoundotherpercentageswithinthesamedatathatseemedmoreaccuratewiththedatashecollectedfromherteenagefriends.

Averageismorethan100textssentperday

Averageislessthan100textssentperday

Total

Teenager 20 4 24

Adult 2 22 24

Total 22 26 48

cc b

y ht

tps:

//flic

.kr/

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fMA

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16



3. HowmightRachelusethedatainthetwowaytabletofindpercentagesthatwouldbeusefulforhercase?

PartII:OnceRachelrealizedtherearealotofwaystolookatasetofdatainatwowaytable,shewasmotivatedtolearnaboutrelativefrequencytablesandconditionalfrequencies.Whenthedataiswrittenasapercent,thisiscalledarelativefrequencytable.Inthissituation,the‘inner’valuesrepresentapercentandarecalledconditionalfrequencies.Theconditionalvaluesinarelativefrequencytablecanbecalculatedaspercentagesofoneofthefollowing:

• thewholetable(relativefrequencyoftable)• therows(relativefrequencyofrows)• thecolumns(relativefrequencyofcolumn)

SinceRachelwantstoemphasizethataperson’sagemakesadifferenceinthenumberoftextssent,thefirstthingshedecidedtodoisfocusontheROWofvaluessoshecouldwriteconditionalstatementsaboutthenumberoftextsapersonislikelytosendbasedontheirage.Thisiscalledarelativefrequencyofrowtable.

4. Fillinthepercentageofteenagersforeachoftheconditionalfrequenciesinthehighlightedrowbelow:

SincethePERCENTAGEScreatedfocusonROWvalues,allconditionalobservationsarespecifictotheinformationintherow.Completethefollowingsentencefortherelativefrequencyofrow:

5. Ofallteenagersinthesurvey,_______%averagemorethan100textsperday.

6. Writeanotherstatementbasedontherelativefrequencyofrow:

Average is more than 100 texts sent

per day

Average is less than 100 texts sent per day

Total

Teenager % of

teenagers

20

__ %

4

__%

24

100%

% of Adults

2 8%

22 92%

24 100%

% of People

22 46%

26 54%

48 100%

Row

17



Belowistherelativefrequencyofcolumnusingthesamedata.Thistime,allofthepercentagesarecalculatedusingthedatainthecolumn.

7. Writetwoconditionalstatementsusingtherelativefrequencyofcolumn.

Thisdatarepresentstherelativefrequencyofwholetable:

8. Createtwoconditionaldistributionstatementsfortherelativefrequencyofwholetable.

9. Whatinformationishighlightedwhendataisinterpretedfromrelativefrequencytables?

Averageismorethan100textssentperday

Averageislessthan100textssentperday

Total

%ofTeenagers 2042%

48%

2450%

%ofAdults 24%

2246%

2450%

%ofTotal 2246%

2654%

48100%

Average is more than 100 texts sent per day

Average is less than 100 texts sent per day

Total

Teenagers 20 91%

4 15%

24 50%

Adults 2 9%

22 85%

24 50%

Total 22 100%

26 100%

48 100%

18



9.4 Relative Frequency – Teacher Notes A Solidify Understanding Task Purpose:Inthistaskstudentswillexaminedifferentwaystointerpretrelativefrequencytables

andwillwriteconditionaldistributionstatementsbasedontherelativefrequencyofrow,column,

orwholetable.UsingdatafromTextingBytheNumbers,studentswillseehowtwowaytablescan

showinformationthatisoftenhiddeninboxplotsorhistograms.Theywillalsolearnhow

conditionalfrequenciescanprovidespecificinformationaboutasubgroupofthedata(callingfor

moreprecisionofdescribingthedata).

CoreStandardsFocus:

S.ID.5Summarizecategoricaldatafortwocategoriesintwo-wayfrequencytables.Interpret

relativefrequenciesinthecontextofthedata(includingjoint,marginal,andconditionalrelative

frequencies).Recognizepossibleassociationsandtrendsinthedata.

RelatedStandards:S.ID.1,S.ID.2.S.ID.3


SMP2–Reasonabstractlyandquantitatively

SMP6–Attendtoprecision

SMP7–Lookandmakeuseofstructure

TheTeachingCycle:

Launch(WholeClass):

Aspartofaccessingbackgroundknowledge,youmaywishtoaskstudentswhattheyremember

aboutthedatafromthetaskTextingBytheNumbers.Thepurposeisonlytohavestudentsmention

datafromRachelandhermom,withcommentsrelatedtothestorythedatatold(specificsnot

needed).Readthescenariofromthistaskandhavestudentsanswerthefirstquestionbywriting

andsharingafewobservationsaboutthetwowaytable(reviewfromthetaskAfterSchoolActivity).



Asstatementsareshared,writethemonstripsofpaper(canbesortedduringwholegroup

discussion).


Asstudentsworkthroughthetask,listenfortheirconjecturesaboutthedataRachelshouldfocus

ontomakehercase.Afterafewminutes,bringtheclassbacktogethertodiscussthetypesof

relativefrequencytables.Explainhoweachvalueisdeterminedintherelativefrequencyofrow.

Havestudentsworkinpairstocompletethesentenceframesandwriteconditionaldistribution

statementsforeachofthethreerelativefrequencytables.Givestudentstimetoconsiderallthree

tablesandcreatestatementsabouteachtable.Listenforunderstandingofeachrelativefrequency

table.Toassistinwritingsentences,remindstudentstopayattentiontothefocusofthetable

(whetherthefocusistherow,thecolumn,ortheentiretable).


Theintentionofthewholegroupdiscussionistohighlightthefollowing:

• differencesbetweenrow,column,andwholetablerelativefrequencystatements

• becomepreciseinourlanguageasweuseconditionalfrequencystatements

• tellastoryusingtwowaytables.

Onewaytoorchestratethisdiscussionistoselectastudenttoshareaspecificrelativefrequencyof

columnstatementtheyhavecreated(letthemknowthisduringtheexplorephaseofthetask)and

havetheclassdetermineifthestatementisfromtherelativefrequencyofrow,column,orwhole

table.Thenaskastudenttoshareanotherrelativefrequencyofcolumnstatement.Askthegroup,

whatdoesthedataspecifictocolumntellus?

Movetoshowingtherelativefrequencyofrowstatementsandaskwhatdoesthedataspecificto

rowtellus?Continuediscussionwithrelativefrequencyofwholetable.

Toconclude,discussthelastquestionfromthetask:Whatinformationishighlightedwhendatais

interpretedfromrelativefrequencytables?Ifthereistime,alsodiscusshowtwowaytables

comparetootherunivariatemodelswehaveused(dotplots,boxplots,histograms).







9.4

READY

Topic:WritingexplicitfunctionrulesforlinearrelationshipsWritetheexplicitlinearfunctionforthegiveninformationbelow.

1.(3,7)(5,13)

2.Mikeearns$11.50anhour

3.(-5,-2)(1,10)

4.(-2,12)(6,8)

5.

6.


19






9.4

SET Topic:RelativeFrequencytables

Foreachtwo-waytablebelow,createtheindicatedrelativefrequencytableandalsoprovidetwoobservationswithregardtothedata.7.Thistablerepresentssurveyresultsfromasampleofstudentsregardingmodeoftransportationtoandfromschool.

Createtherelativefrequencyofcolumntable.Thenprovidetwoobservationstatements.8.Thetwo-waytablecontainssurveydataregardingfamilysizeandpetownership.

Createtherelativefrequencyofrowtable.Thenprovidetwoobservationstatements.

Walk Bike CarPool Bus Total

Boys 37 47 27 122 233

Girls 38 22 53 79 192

Total 75 69 80 201 425

Walk Bike CarPool Bus Total

Boys

Girls

Total 100% 100% 100% 100% 100%

NoPets OwnonePetMorethanonepet

Total

Familiesof4orless 35 52 85 172Familiesof5ormore 15 18 10 43

Total 50 70 95 215

NoPets OwnonePetMorethanonepet

Total

Familiesof4orless 100%Familiesof5ormore 100%Total 100%

20






9.4

9.Thetwo-waytablebelowcontainssurveydataaboutboysandgirlsshoes.

Createtherelativefrequencyofwholetable.Thenprovidetwoobservationstatements.

GO Topic:Onevariablestatisticalmeasuresandcomparisons

Foreachsetofdatadeterminethemean,median,mode,range,andstandarddeviation.Thencreateeitherabox-and-whiskerplotorahistogram.

10.23,24,25,20,25,29,24,25,30 11.20,24,10,35,25,29,24,25,33

12.Howdothedatasetsinproblems10and11comparetooneanother?

13.2,3,4,5,3,4,7,4,4 14.1,1,3,5,5,10,5,1,14

15.Howdothedatasetsinproblems13and14comparetooneanother?

Athleticshoes Boots DressShoe Total

Girls 21 35 60 116

Boys 50 16 10 76

Total 71 51 70 192

Athleticshoes Boots DressShoe Total

Girls

Boys

Total 100%

21




9.5 Connect the Dots

A Develop Understanding Task

Foreachsetofdata:• Graphonascatterplot.• Usetechnology(graphingcalculatororcomputer)tocalculatethecorrelationcoefficient.

SetA2 2.3 3.3 3.7 4.2 4.6 4.5 5 5.5 5.7 6.1 6.41 1.5 2.5 1.9 2.8 3.2 4.5 3.7 1.7 4.8 2.7 2.3SetB2 2.3 3.3 3.7 4.2 4.6 4.5 5 5.5 5.7 6.1 6.41 1.5 2.5 1.9 2.8 3.2 4.5 3.7 4 4.8 5 4.6SetC2 2.3 3.3 3.7 4.2 4.6 4.5 5 5.5 5.7 6.1 6.44.7 4.9 4.2 3.9 3.5 3.2 3.1 2.6 3.2 2.1 1.3 0.8SetD2 2.3 3.3 3.7 4.2 4.6 4.5 5 5.5 5.7 6.1 6.44.7 4.9 3.6 3.9 2.1 4.5 3.1 1.7 3.7 2.1 1.3 1.8SetE2 2.3 3.3 3.7 4.2 4.6 4.5 5 5.5 5.7 6.1 6.44.7 4 4.2 3.9 2.8 3.2 4.5 3.7 3.2 4.8 5 4.4SetF2 2.3 3.3 3.7 4.2 4.6 4.5 51.8 2.22 3.62 4.18 4.88 5.44 5.3 6SetG2 2.3 3.3 3.7 4.2 4.6 4.5 54.4 4.01 2.71 2.19 1.54 1.02 1.15 0.5

1. Putthescatterplotsinorderbaseduponthecorrelationcoefficients.

2. Compareeachscatterplotwithitscorrelationcoefficient.Whatpatternsdoyousee?

CCBY

https://flic.kr/p/jAZ

BNr

22




3. UsethedatainSetAasastartingpoint.Keepingthesamex-values,modifythey-valuestoobtainacorrelationcoefficientascloseto0.75asyoucan.Recordyourdatahere:

2 2.3 3.3 3.7 4.2 4.6 4.5 5 5.5 5.7 6.1 6.4

Whatdidyouhavetodowiththedatatogetagreatercorrelationcoefficient?

4. Thistime,againstartwiththedatainSetA.Keepthesamex-values,butthistime,modifytheyvaluestoobtainacorrelationcoefficientascloseto0.25asyoucan.Recordyourdatahere:

2 2.3 3.3 3.7 4.2 4.6 4.5 5 5.5 5.7 6.1 6.4

Whatdidyouhavetodowiththedatatogetacorrelationcoefficientthatiscloserto0?

5. Onemoretime:startwiththedatainSetA.Keepthesamex-values,modifythey-valuestoobtainacorrelationcoefficientascloseto-0.5asyoucan.Recordyourdatahere:

2 2.3 3.3 3.7 4.2 4.6 4.5 5 5.5 5.7 6.1 6.4

Whatdidyouhavetodowiththedatatogetacorrelationcoefficientthatisnegative?

6. Whataspectsofthedatadoesthecorrelationcoefficientappeartodescribe?

7. Onthenightbeforethelastmathtest,Shaniquaheldastudygroupatherhouse.Itwasa

greatnight;theyatealotofpizza,didmath,andlaughedalot.Shaniquascoredbetteron

hertestthanusualandthoughtitmightberelatedtopizza.Shecollectedthefollowingdata

fromherfriendsinthestudygroup:

23




Shaniqua David Susana Ruby Deion Oscar

Numberof

Pizza

Slices

Eaten

2 6 1 4 3 5

Increase

inTest

Score

5 9 4 7 6 8

Createascatterplotofthisdataandcalculatethecorrelationcoefficient.Basedonthesedata,wouldyourecommendeatingpizzaonthenightbeforeatestto

increasescores?Whyorwhynot?

8. Describeasituationwithtwovariablesthatmayhaveahighcorrelation,butnotbe

causallyrelated.

9. Whataresomereasonsthattwovariablesmaybehighlycorrelatedbutnothaveacausal

relationship?

24




9.5 Connect the Dots – Teacher Notes


SpecialNotetoTeachers:Thistaskrequirestheuseoftechnologythatcancalculatethe

correlationcoefficient,r.Mostgraphingcalculatorswillworkwell.Freecomputerappswouldbe

veryhelpfulandeasytouseonthistaskaswell(GeoGebraandDesmos,etc.).

Purpose:Thepurposeofthistaskistodevelopanunderstandingofthecorrelationcoefficient.

Thetaskasksstudentstoplotvariousdatasetsandusetechnologytocalculatethecorrelation

coefficient.Theywillorderthegraphsandcreatenewdatasetstodeveloptheideathatthe

correlationcoefficientindicatesthestrengthanddirectionofalinearrelationshipinthedata.

Studentsalsoconsidersituationsinwhichtwovariablesarehighlycorrelated,buttherelationship

isnotnecessarilycausal.

CoreStandardsFocus:

S-ID.8Compute(usingtechnology)andinterpretthecorrelationcoefficientofalinearfit.

S-ID.9Distinguishbetweencorrelationandcausation.

S.IDNotes:Buildonstudents’workwithlinearrelationshipsineighthgradeandintroducethe

correlationcoefficient.Thefocushereisonthecomputationandinterpretationofthecorrelation

coefficientasameasureofhowwellthedatafittherelationship.Theimportantdistinctionbetween

astatisticalrelationshipandacause-and-effectrelationshiparisesinS.ID.9.

RelatedStandards:S-ID.6

StandardsForMathematicalPracticeofFocusintheTask:

SMP-1Makesenseofproblemsandpersevereinsolvingthem.

SMP-5Useappropriatetoolsstrategically.

TheTeachingCycle

Launch(WholeClass):Sincethisisthefirsttaskinthemodulethatusesscatterplotsfor

bivariatedata,beginbyremindingstudentsofthetermandhowtheyareconstructed.Tellthem




thatthepurposeofthistaskistocomeupwiththeirownhypothesisaboutwhatfeaturesofthe

datathecorrelationcoefficientdescribes.Showstudentshowtoenterdataandcalculatea

correlationcoefficientusingwhatevertechnologyyouhaveselectedforyourclass.Alsotell

studentsthatcorrelationcoefficientscanonlybecalculatedfortwoquantitativevariables.Forthe

purposeoftheclassroomdiscussion,eachstudentshouldplotthedataandrecordthecorrelation

coefficientonpaper(eitherbyhandorusingaprinter)foreachproblem.Thiswillfacilitate

comparingandorderingofthegraphs,aswellasusethedatafromproblems4-7toconfirmtheir

hypothesisaboutthecorrelationcoefficient.

Explore(SmallGroup):Monitorstudentsastheyworktoseethattheyareabletousethe

technologyproperlyandarerecordingthegraphsonpaper.Oncetheyhavegraphedandordered

eachofthefirst6datasets,encouragethemtospeculateandsharetheirideasinthegroups.Listen

forstudentsthatarenoticingthatthevaluesofrarebetween-1and1,thatnegativevalues

describedecreasingtrends,positivevaluesdescribeincreasingtrends,thatvaluesofrnear0

correspondtodatawithoutnoticeablepatterns,andrvaluesnear1or-1describedatathatappear

tofitalinearmodel.

Discuss(WholeClass):PrepareforthediscussionbyreproducingthescatterplotsforsetsA-G

(givenbelow)sothattheycanbedisplayedfortheentireclass.Posttheplotsinorderfrom-1to1

(G,C,D,E,A,B,F).Askstudentsfortheirideasabouttheaspectsofthedatadescribedbythe

correlationcoefficient.Recordalistthatshouldincludesomeorallofthefollowing:

• rvaluesrangebetween-1and1,

• negativevaluesofrdescribenegativeassociation,

• positivevaluesofrdescribepositiveassociation,

• valuesofrnear0correspondtodatawithaveryweaklinearrelationship

• rvaluesnear1or-1describedatathatfitalinearmodel

Theymayalsohavesomeideasthatmaybeabandonedlater,baseduponthediscussion.

Turnthediscussionto#4.Askthreestudentstodisplaythescatterplotstheymadethathavea

correlationcoefficientof.75.Asktheclasstoseewhatthethreegraphshaveincommon,

emphasizingobservationsaboutthedirectionoftheassociationandtheappearanceoflinearity.

AskstudentshowtheyadjustedthedatainsetA,forwhichr=0.49,toincreaser.Asktheclasshow




theexperiencein#4eitherconfirmsordeniestheirhypothesesaboutthecorrelationcoefficient.

Movethroughquestions5and6insimilarfashion.

Tellstudentsthatacorrelationof1or-1isaperfectcorrelation,andaskwhatthatmeansforthe

relationshipbetweenthetwovariables.Discusstheconclusionsthattheydrewinquestion7.

Endthediscussionbyeliminatinganyremainingincorrectstatementsinthelistofstudentideas

aboutthecorrelationcoefficientandbywritinganddiscussingthemeaningofthefollowing

statement:Thecorrelationcoefficientmeasuresthestrengthanddirectionofalinear

relationshipbetweentwoquantitativevariables.







9.5

READY

Topic:EstimatingthelineofbestfitExaminethescatterplotbelow.Imaginethatyoudrewastraightlinethroughthegeneralpatternofthepoints,keepingascloseaspossibletoallpointswithasmanypointsabovethelineasbelow.

1.Predictapossibley-interceptandslopeforthelineyouimagined.

a.y-intercept:____________________b.slope:___________________________

2.Sketchthelinethatyouimaginedforquestion#1andwriteanequationforthatline.

SET Topic:Estimatingthecorrelationcoefficient Matchthefollowingscatterplotswiththecorrectcorrelationcoefficient.Possiblecorrelationcoefficients:a.0.05 b.0.97 c.-0.94 d.-0.49 e.0.68 f.-0.25


© 2012 http://en.wikipedia.org/wiki/File:Scatter_diagram_for_quality_characteristic_XXX.svg

25






9.5

3.

4.

5.

6.

7.

8.

26






9.5

GO Topic:Visuallycomparingslopesoflines.

Followtheprompttosketchthegraphofalineonthesamegridwiththegivencharacteristics.

8.Agreaterslope

9.Alesserslope

10.Alargery–interceptandalesserslope

11.Slopeistheoppositereciprocal.

27


MODELING WITH DATA – 9.6


9.6 Making More $

A Solidify Understanding Task

EachyeartheU.S.CensusBureauprovidesincomestatisticsfortheUnited

States.Intheyearsfrom1990to2005,theyprovidedthedatainthe

tablesbelow.(Alldollaramountshavebeenadjustedfortherateof

inflationsothattheyarecomparablefromyear-to-year.)

1. Createascatterplotofthedataformen,setting1991asyear1.

Whatisyourestimateofthecorrelationcoefficientforthesedata?

Year

Median Income for All Women

2005 23970 2004 23989 2003 24065 2002 23710 2001 23564 2000 23551 1999 22977 1998 22403 1997 21759 1996 20957 1995 20253 1994 19158 1993 18751 1992 18725 1991 18649

Year

Median Income for All Men

2005 41196 2004 41464 2003 40987 2002 40595 2001 41280 2000 41996 1999 42580 1998 42240 1997 40406 1996 38894 1995 38607 1994 38215 1993 37712 1992 37528 1991 38145

CCBY40

1KCa

lculator.org

https://flic.kr/p/aFD

grH

28




2. Onaseparategraph,createascatterplotofthedataforwomen,setting1991asyear1.Whatisyourestimateofthecorrelationcoefficientforthesedata?

3. Estimateanddrawlinesthatmodeleachsetofdata.

4. Describehowyouestimatedthelineformen.Ifyouchosetorunthelinedirectlythroughanyparticularpoints,describewhyyouselectedthem.

5. Describehowyouestimatedthelineforwomen.Ifyouchosetorunthelinedirectly

throughanyparticularpoints,describewhyyouselectedthem.

6. Writetheequationforeachofthetwolinesinslopeinterceptform.

a. Equationformen:

b. Equationforwomen:

7.Usetechnologytofindtheactualcorrelationcoefficientformen.

Whatdoesittellyouabouttherelationshipbetweenincomeandyearsformen?

8. Whatistheactualcorrelationcoefficientforwomen?

a. Whatdoesittellyouabouttherelationshipbetweenincomeandyearsforwomen?

b. Whatdothecorrelationcoefficientsformenandwomentellusabouthowthedatacompares?

29




9.Usetechnologytocalculatealinearregressionforeachsetofdata.Addtheregressionlinestoyourscatterplots.

c. Linearregressionequationformen:

d. Linearregressionequationforwomen:

10. Compareyourmodeltotheregressionlineformen.Whatdoestheslopemeanineachcase?(Includeunitsinyouranswer.)

11.Compareyourmodeltotheregressionlineforwomen.Whatdoesthey-interceptmeanineachcase?(Includeunitsinyouranswer.)

12.Comparetheregressionlinesformenandwomen.Whatdothelinestellusabouttheincomeofmenvswomenintheyearsfrom1991-2005?

13.Whatdoyouestimatewillbethemedianincomeformenandwomenin2015?

30




14.TheCensusBureauprovidedthefollowingstatisticsfortheyearsfrom2006-2011.

Withtheadditionofthesedata,whatwouldyounowestimatethemedianincomeofmenin2015tobe?Why?

15.Howappropriateisalinearmodelformen’sandwomen’sincomefrom1991-2011?Justifyyouranswer.

Year

Median Income for All Men

2011 37653 2010 38014 2009 38588 2008 39134 2007 41033 2006 41103

Year

Median Income for All Women

2011 23395 2010 23657 2009 24284 2008 23967 2007 25005 2006 24429

31




9.6 Making More $ – Teacher Notes



correlationcoefficient,r.Mostgraphingcalculatorswillworkwell.Freecomputerappswouldbe

veryhelpfulandeasytouseonthistaskaswell(GeoGebraandDesmos,etc.).

Purpose:Thepurposeofthistaskistosolidifyunderstandingofcorrelationcoefficientandto

developlinearmodelsfordata.Studentsareaskedtoestimateandcalculatecorrelation

coefficients.Inthetasktheyestimatelinesofbestfitandthencomparethemtothecalculated

linearregression.Thetaskdemonstratesthedangersofusingalinearmodeltoextrapolatewell

beyondtheactualdata.Thetaskendswithanopportunitytousethecorrelationcoefficientand

scatterplottodeterminetheappropriatenessofalinearmodel.

CoreStandardsFocus:

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

contextofthedata.

S-ID.8Compute(usingtechnology)andinterpretthecorrelationcoefficientofalinearfit.

RelatedStandards:S.ID.6

StandardsForMathematicalPracticeofFocusintheTask:

SMP2–Reasonabstractlyandquantitatively.

SMP4–Modelwithmathematics.

TheTeachingCycle

Launch(WholeClass):Introducethetasktellingstudentsthatthistaskextendswhattheyhave

doneinpreviousmodulestomodelsituationswithlines.Inthiscase,theywillbemodelingreal




data,whichisnotusuallyperfectlylinear(correlationcoefficientof1or-1).Beforeactually

beginningthetaskofmakingscatterplots,askstudentstomakesomeobservationsaboutthedata

inthetwotablesthatshowthemedianincomeformenandwomen.Theymaynoticethatwomen’s

salariesarelowerthanmen’sorthattheybothappeartobeincreasingovertime.Whatquestions

areraisedbytheseobservations?Askstudentstoworkonquestions1-4.

Explore(SmallGroup):Monitorstudentsastheyareworking,observingtheirthinkingaboutthe

plots.Encouragethemtodiscussthecorrelationcoefficientwiththeirgroup,noticingboththe

directionandthestrengthofthelinearrelationship.Manystudentsmaynotfeelthatalinearmodel

isappropriateforthemen’sdatabecauseoftheshapeofthedistribution.(Bothscatterplotsare

shownbelow).Listenasstudentstalkabouttheirstrategiesforplacingthelineofbestfitonthe

twoscatterplotsandbepreparedtocallonstudentswithdifferentstrategiesforthediscussion.

Somestrategiesthatcanbeanticipatedare:

• Tryingtogetthegreatestnumberofpointsontheline

• Selectingapointatthebeginningandendofthedistributionandconnectingthem.

• Tryingtogetasmanypointsabovethelineasbelowtheline.

DiscussPartOne(WholeClass):

Beginthediscussionbydisplayingthetwoscatterplotsandbrieflydiscussionthecorrelation

coefficientandwhatitisdescribingaboutthedata.Thegraphsareshownbelow.

MedianAnnualIncomeforMenfrom

1991-2005: r=0.814




MedianAnnualIncomeforWomen

from1991-2005r=0.964

Focusingonthescatterplotformen’sincome,askstudentsthathaveuseddifferentstrategiesfor

placingthelineofbestfittosharetheirstrategiesanddrawtheirlinesonthegraphs.Asstudents

sharedifferentstrategiesasktheclasstocomparestrengthsandweaknessesoftheapproachin

modelingthetrendsinthedata.Askstudentstocompareandinterprettheslopeofthelineofbest

fitthattheyselected.

Re-launch:Demonstratehowtousetechnologytocalculatealinearregressionandgraphtheline

alongwiththedataonascatterplot.Thengivestudentstimetoworkontherestofthetask,

monitoringtheirdiscussionsastheywork.

Discuss(Part2):

Beginthesecondpartofthediscussionbydisplayingthegraphsandregressionlines,shownbelow.

Askstudentswhatobservationstheymakeabouttheregressionlines.Theymaybesurprisedthat

thelineforthemen’sdatadoesn’tactuallyintersectanyofthedatapoints.Askstudentshowthey

thinkthattheregressionlinewascalculated.Listenforstudentsthatarenoticingthatthelinesof

regressionseemtocutthedata“inhalf”,leavingasmuch“spacebetweenthepointsandtheline”




aboveandbelow.Thiswillleadnaturallytotheideaofresidualsandthewaythattheleastsquares

regressionlineiscalculated.

Regressionlineformen’sincome:! = !"#.!"# + !",!!"

Regressionlineforwomen’sincome:! = !"!.!"# + !",!"#

Askstudentstoconsidertheslopesofeachoftheregressionlines.Whatdoeseachslopemean?

Sincethewomen’sslopeisgreaterthanthemen’s,doesthismeanthatwomenmakemoremoney?

Alsodiscussthey-interceptonthemen’sincomegraph.Theyinterceptinthiscaseisthevalue

givenbythelinearmodelfortheyear0(1990).




Studentswereaskedtousethemodeltopredictthemen’sincomeforyear2015.Askstudentsfor

thepredictedvaluebasedontheirregressionline.

Afterdisplayingthegraphsthatshowtheadditionaldata,askhowtheywouldmodifythat

predictionnowthattheyhavemoredata.Theywillnoticethatthetrendformen’sincomefrom

2005to2011isdownward.Thishighlightsthedangerofextrapolating,whichmeanstoextendthe

modelwellbeyondtheactualdata.

Finally,discussthelastquestion.Students’justificationoftheiranswershouldincludetheuseof

thecorrelationcoefficient,andcomparetrendsinthedatathatcanbeobservedinthescatterplot

versustheslopeoftheregressionline.

Graphsandlinearregressionsareshownbelow.

MedianIncomeforMen1991-2011

r=0.17

y=47.16x+39,398

MedianAnnualIncomeforMen($)




MedianAnnualIncomeforWomen1991-2011

r=0.88

y=300.76x+19131

AlignedReady,Set,Go:ModelingwithData9.6

MedianAnnualIncomeforWomen($)






9.6

READY

Topic:FindingdistanceandaveragesUsethenumberlinebelowtoanswerthequestions.

1.FindthedistancebetweenpointAandeachofthepointsonthenumberline.

AF=______ AC=______ AG=______ AB=_______ AD=______ AE=______

2.WhatisthetotalofallthedistancesfrompointAthatyoufoundinexercisenumberone?3.Findtheaverageofthedistancesthatyoufoundinexercise1.4.WhichpointorpointsonthenumberlineislocatedtheaveragedistanceawayfrompointA?5.CirclethelocationorlocationsonthenumberlinethatistheaveragedistanceawayfromA.6.FindthedistancebetweenpointDandeachofthepointsonthenumberline.

DF=______ DC=______ DG=______ DA=_______ DB=______ DE=______

7.WhatisthetotalofallthedistancesfrompointDthatyoufoundinexercisenumbersix?8.Findtheaverageofthedistancesthatyoufoundinexercise6.9.IsthereapointonthenumberlinelocatedtheaveragedistanceawayfrompointD?10.LabelalocationonthenumberlinethatistheaveragedistanceawayfrompointD,labelitY.


32






9.6

SET Topic:Scatterplotsandlinesofbestfitortrendlines 11.Createascatterplotforthedatainthetable.

12.DotheEnglishandhistoryscoreshave

apositiveornegativecorrelation?13.DotheEnglishandhistoryscoreshaveastrongorweakcorrelation?14.Whichgraphbelowshowsthebestmodelforthedataandwillcreatethebestprediction?

Explainwhyyourchoiceisthebestmodelforthedata.

a.

b.

c.

English Score History Score

60 65

53 59

44 57

61 61

70 67

33






9.6

15.Whichgraphbelowshowsthebestmodelforthedataandwillcreatethebestprediction?Explainwhyyourchoiceisthebestmodelforthedata.

a.

b. c.

16.Whichgraphbelowshowsthebestmodelforthedataandwillcreatethebestprediction?

Explainwhyyourchoiceisthebestmodelforthedata.

a.

b. c.

GO Topic:Creatingexplicitfunctionrulesforarithmeticandgeometricsequences.Usethegiveninformationbelowtocreateanexplicitfunctionruleforeachsequence.17.! 2 = 7;commondifference=3 18.! 1 = 8;commonratio=2

19.ℎ 6 = 3;commonratio=-3 20.! 5 = −3;commondifference=7

21.! 7 = 1;commondifference=-9 22.! 1 = 5;commonratio=!!

34




9.7 Getting Schooled


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

35




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




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




7. Analyzethedataforwomenwithbachelor’sdegreesbycreatingascatterplot,interpreting

thecorrelationcoefficientandtheregressionline.Forconsistencywiththemen’sgraphabove,use

0fortheyear2000,1fortheyear2001,etc.Drawthegraphandreporttheresultsofyouranalysis

below:

8. Nowthatyouhaveanalyzedtheresultsforwomen,comparetheresultsformenand

womenwithbachelor’sdegreesandmoreovertheperiodfrom2000-2011.

38




9. Leobelievesthatthedifferenceinincomebetweenmenandwomenmaybeexplainedby

differencesineducation,butAracelibelievestheremustbeotherfactorssuchasdiscrimination.

BasedonthedatainthistaskandGettingMore$,makeaconvincingcasetosupporteitherLeoor

Araceli.

10. Whatotherdatawouldbeusefulinmakingyourcase?Explainwhattolookforandwhy.

39




9.7 Getting Schooled – Teacher Notes



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.




Launch(WholeClass):

Remindstudentsoftheirworkwithmen’sandwomen’smedianannualincomesfromtheprevious

task.Askthemtorecallsomeoftheconclusionsthatcouldbemadefromthedata.Introducethis

taskbytellingthemthattheywillbedrawingupontheirexperiencewithcorrelationcoefficients

andlinearregressionstoanalyzeandcomparedata.Bythetimetheyhavefinishedthetaskthey

shouldbepreparedtousethedatatomakeanargumentaboutthedifferencesinmen’sand

women’ssalary,baseduponeducationandotherpossiblefactors.


Monitorstudentsastheywork,ensuringthattheyareestimatingasrequestedinthetaskbefore

makingthecalculations.Thiswillhelptodrawthemintothedatasothattheycanmakesenseofit

anddeepentheirunderstanding.Keepstudentsfocusedonusingtheunitsofslopebasedonthe

graphs.Theymaybemorefamiliarwithgraphsthathavetimeacrossthex-axis,butstruggleto

interpretthefirstgraphthatcomparessalariesofmenandwomenwheretheyearthedatawas

obtainedisnotevident.


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




womenorthatthereareotherfactorsthatexplainthedifferences.Organizetheclasssothat

studentsareassignedtoonesideoftheargumentortheotherandthentaketurnspresentingone

pieceofevidencefromtheiranalysis.Recordtheclaimsandallowtheothersidetorefuteanyclaim

thattheyfeelisinerror.







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.)


40






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






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




9.8 Rockin’ the Residuals


Thecorrelationcoefficientisnottheonlytoolthat

statisticiansusetoanalyzewhetherornotalineisagood

modelforthedata.Theyalsoconsidertheresiduals,

whichistolookatthedifferencebetweentheobserved

value(thedata)andthepredictedvalue(they-valueon

theregressionline).Thissoundsalittlecomplicated,but

it’snotreally.Theresidualsarejustawayofthinking

abouthowfarawaytheactualdataisfromtheregressionline.

Startwithsomedata:

x 1 2 3 4 5 6y 10 13 7 22 28 19Createascatterplotandgraphtheregressionline.In,thiscasethelineis! = 3! + 6.

CCBYJamieAdkins

https://flic.kr/p/aRzLKP

43




Drawalinefromeachpointtotheregressionline,likethesegmentsdrawnfromeachpointbelow.

1. Theresidualsarethelengthsofthesegments.Howcanyoucalculatethelengthofeach

segmenttogettheresiduals?

2. Generally,ifthedatapointisabovetheregressionlinetheresidualispositive,ifthedata

pointisbelowtheline,theresidualisnegative.Knowingthis,useyourplanfrom#1to

createatableofresidualvaluesusingeachdatapoint.

44




3. Statisticiansliketolookatgraphsoftheresidualstojudgetheirregressionlines.So,you

getyourchancetodoit.Graphtheresidualshere.

Now,thatyouhaveconstructedaresidualplot,thinkaboutwhattheresidualsmeanandanswer

thefollowingquestions.

4. Ifaresidualislargeandnegative,whatdoesitmean?

5. Whatdoesitmeanifaresidualisequalto0?

45




6. Ifsomeonetoldyouthattheyestimatedalineofbestfitforasetofdatapointsandallofthe

residualswerepositive,whatwouldyousay?

7. Ifthecorrelationcoefficientforadatasetisequalto1,whatwilltheresidualplotlooklike?

Statisticiansuseresidualplotstoseeiftherearepatternsinthedatathatarenotpredictedbytheir

model.Whatpatternscanyouidentifyinthefollowingresidualplotsthatmightindicatethatthe

regressionlineisnotagoodmodelforthedata?Basedontheresidualplotarethereanypoints

thatmaybeconsideredoutliers?

8.

46




9.

10.

11.

47




9.8 Rockin’ the Residuals – Teacher Notes


Purpose:Thepurposeofthistaskistodevelopanunderstandingofresidualsandhowtouse

residualplotstoanalyzethestrengthofalinearmodelfordata.

CoreStandardsFocus:

S.ID.6:Representdataontwoquantitativevariablesonascatterplot,anddescribehowthe

variablesarerelated.

S.ID.6b:Informallyassessthefitofafunctionbyplottingandanalyzingresiduals.

RelatedStandards:S.ID.6a,S.ID.6c


SMP7-Lookforandmakeuseofstructure.

TheTeachingCycle:

Launch(WholeClass):

Beginthetaskbywalkingthroughthefirstpartofthetaskwithstudents,explainingwhataresidual

isusingthegraphicalrepresentation.Givestudentstimetocompletequestions1-3andthen

discusstheresidualplotthattheyhavecreated.Howdoestheresidualplotcomparewiththe

scatterplotofthedatawiththeregressionlinedrawn?Whatinformationcouldbedrawnfromjust

lookingattheresidualplotiftheyhadnotseenthescatterplotandlinearregression?


Allowstudentstimetodiscusstheremainingquestionsandfinishthetask.Listenforthewaysthat

theyaremakingsenseoftheideathattheresidualisthedifferencebetweentheactualpointand

correspondingpointonthelinearmodelofthedata.





Discusseachoftheremainingquestions.Finally,asktheclass:Whatinformationisobtainedfrom

lookingataresidualplotthatisnotgivenbythecorrelationcoefficient?Howdotheywork

togethertoinformtheanalysisofbivariatequantitativedata?







9.8

READYTopic:DescribingspreadDescribethespreadofthedatasetshownineachboxplotshownbelow.Includethemedian,therange,andtheinterquartilerange.1.

2.

3.Iftheboxplotsaboverepresenttheresultsoftwodifferentclassesonthesameassessment,whichclassdidbetter?Justifyyouranswer.

4.ThetwoboxplotsbelowshowthelowtemperaturesfortwocitiesintheUnitedStates.CityDistheboxplotontopandCityEonthebottom.

a.Whichcitywouldbeconsideredthecoldest,CityDorCityE? Why?

b.Dothesecitieseverexperiencethesametemperature?Howdoyouknow?

c.Isthereawaytoknowtheexacttemperatureforanygivendayfromtheboxplots?

d.Whatadvantage,ifany,couldahistogramoftemperaturedatahaveoveraboxplot?


C

i

C

i

48






9.8

SETTopic:Residuals,residualplotsandcorrelationcoefficientsThedatasheetsinexercise5andexercise6arescatterplotsthathavetheregressionlineandtheresidualsindicated.Foreachexercise,a)Markonthegraphwhere !,! wouldbelocated.b)Usethegivendatasheettocreatearesidualplot.c)Predictthecorrelationcoefficient.5.Datasheet1a)mark !,!

b)residualplot1

C)Correlationcoefficient?

6.Datasheet2a)mark !,!

B)residualplot2

C)Correlationcoefficient?

30

25

20

15

10

5

5 10 15 20

15

10

5

–5

–10

–15

5 10 15 200

54

52

50

48

46

44

42

40

38

36

34

32

302 4 6 8 10

12

10

8

6

4

2

–2

–4

–6

–8

–10

–12

5 10

49






9.8

Thefollowinggraphsareresidualplots.Analyzetheresidualplotstodeterminehowwellthepredictionline(lineofbestfit)describesthedata.7.Plot1 analysis

8.Plot2 analysis

50






9.8

GOTopic:Geometricconstructions9.Constructanisoscelestrianglewithacompassandastraightedge.10.Constructasquareusingacompassandastraightedge.11.Useacompassandastraightedgetoconstructahexagoninscribedinacircle.

51




9.9 Lies and Statistics

A Practice Understanding Task

Decidewhethereachstatementis:

• Sometimestrue• Alwaystrue• Nevertrue

Giveareasonforyouranswer.

1. Theslopeofthelinearregressionlinecanbecalculatedusinganytwopointsinthedata.

_________________________________________________________________________________________________________

2. Ifthecorrelationcoefficientforasetofdatais0,thenthelineofbestfitishorizontal.

_________________________________________________________________________________________________________

3. Thesumoftheresidualsforthelineofbestfitis0.

_________________________________________________________________________________________________________

4. Ifthecorrelationcoefficientisverylarge,thentheremustbeanoutlierinthedata.

_________________________________________________________________________________________________________

5. Anegativecorrelationcoefficientmeansthatthedatapointsareveryrandomanddon’treallyfitalinearmodel._________________________________________________________________________________________________________

6. Anegativeresidualmeansthattheregressionlineisveryfarfromtheactualdatapoint.

_________________________________________________________________________________________________________

7. Ifthecorrelationcoefficientispositive,thentheslopeofthelineofbestfitwillprobablybepositive._________________________________________________________________________________________________________

CCBYU.S.D

ept.ofAgriculture

https://flic.kr/p/jPo

bb4

52




8. Ifthereisaperfectcorrelationbetweenvariablesinthedata,thenthecorrelationcoefficientis1.

_________________________________________________________________________________________________________

9. Tofindthevalueofaresidualforapoint(a,b)givenalineofbestfit,f(x):a. Find!(!)b. Find! − !(!)c. Iftheanswerispositive,thenthepointisabovetheline.d. Iftheanswerisnegative,thenthepointisbelowtheline.

_________________________________________________________________________________________________________

10. Thelargertheresidualforagivenpoint,thefurtherawaythepointisfromthelineofbestfit.

_________________________________________________________________________________________________________

11. Ifthereisaperfectcorrelationbetweentwovariablesaandb,theneitheracausedborbcauseda._________________________________________________________________________________________________________

53




9.9 Lies and Statistics – Teacher Notes

A Practice Understanding Task

Purpose:Thepurposeofthistaskistorefinestudents’understandingofcorrelationcoefficients,residuals,andlinearregression.Asstudentsreasonthroughthestatementsthathavebeengiven,theywillhavetoconsidervariouscases,alongwithconsideringthedefinitionofthestatisticaltermsused.Theywillmakeargumentstojustifytheiranswers,citingexamplesanddefinitions.CoreStandardsFocus:

S-ID.6Representdataontwoquantitativevariablesonascatterplot,anddescribehowthevariablesarerelated.

a.Fitafunctiontothedata;usefunctionsfittedtodatatosolveproblemsinthecontextofthedata.Usegivenfunctionsorchooseafunctionsuggestedbythecontext.Emphasizelinear,quadratic,andexponentialmodels.b.Informallyassessthefitofafunctionbyplottingandanalyzingresiduals.c.Fitalinearfunctionforascatterplotthatsuggestsalinearassociation.


SMP6-Attendtoprecision.

SMP3-Constructviableargumentsandcritiquethereasoningofothers.

TeachingCycle

Launch(WholeClass):Beginbytellingstudentsthatthistaskisanopportunitytothinklikestatisticiansandrefinethewaytheyusethetermsofstatistics.Theirjobistotesteachofthestatementsgiventodetermineiftheyarealways,sometime,ornevertrue.Ineverycase,theyneedtojustifytheirchoicewithexamplesandreasoning.Givestudentssometimetoworkontheirownbeforesharingsothattheyhaveachancetodeveloptheirownargumentsforeachproblem.

Explore(SmallGroup):Asstudentsaresharing,listenformisconceptionsthatmayarisesothattheycanbesharedintheclassdiscussion.Alsowatchforstatementsthataregeneratingdisagreement,becausetheseareopportunitiesforproductivereasoningandengagingdiscussion.




Discuss(WholeClass):Begintheclassdiscussionwithanyofthestatementsthatwerecontroversial.Ineverycase,havestudentspresenttheirargumentsbeforeguidingtheclasstothecorrectanswer.Problems2,4,and5oftenbringoutmisunderstandingsandshouldbediscussed.Intheremainingtime,workthroughasmanyotherproblemsaspossible.







9.9

READY

Topic:IdentifyingtypesoffunctionsandwritingtheexplicitequationsForeachrepresentationofafunction,decideifthefunctionislinear,exponential,orneither.

Justifyyouranswer.

1.! ! !

1 1176492 168073 24014 3435 49

2.Thefeeforataxirideis$7forgettingintothetaxiplus$2permile.

3.

4.

6.

! ! !

1 1

4 2

9 3

16 4

25 5

7.! 1 = 7; ! ! = 5 ∙ ! ! − 1

8.

ℎ ! = 3 ! − 1 + 2

9.! ! = 3!! − ! − 3!! + 1


54






9.9

SET Topic:ReviewingkeytopicsinstatisticsDecidewhethereachstatementissometimestrue,alwaystrue,ornevertrue.Ifthestatementissometimestruegiveoneexampleofwhenitistrueandanexampleofwhenitisnot.10.Thelinearregressionlinepassesthroughtheaverageofthexvaluesandtheaverageofthey

values. 11.Apositivecorrelationcoefficientmeansthatthepointsinthescatterplotareveryclosetogether.12.Anegativeresidualmeansyourpredictedvalueistoolow.13.Acorrelationcoefficientcloseto1meansthatalinearmodelismostappropriateforthedata. GO Topic:SolvingliteralequationsSolveeachequationforx.

14. !" = ! 15.! + !" = ! 16.!" + !" = !

Solveeachequationfory.

17.4! + ! = 3 18.2! = 6! + 9 19.5! − 2! = 10

Solveeachequationfortheindicatedvariable.

20.! = !!!; Solve for !. 21.! = !"!! ; Solve for ℎ. 22.! = !"! !

!" ;SolveforV.

55

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