WherearetheCookies?

Grades4-­‐6

MathematicsFormativeAssessmentLesson

DesignedandrevisedbyKentuckyDepartmentofEducationMathematicsSpecialistsField-­‐testedbyKentuckyMathematicsLeadershipNetworkTeachers

Ifyouencountererrorsorotherissues,pleasecontacttheKDEteamat:

CreatedforthesolepurposeofassistingteachersastheydevelopstudentunderstandingofKentucky’sCoreAcademicStandardthroughtheuseofhighlyeffectiveteachingandlearning.

Notintendedforsale.

WherearetheCookies?–Grades4-6

Mathematicalgoals

Thislessonisintendedtohelpyouassesshowwellstudentsareabletousefractionsinaproblemsolvingcontext.Inparticularthislessonaimstoidentifyandhelpstudentswithdifficulties:

  • Conceptualizingfractionalpartsofdifferentwholes
  • Choosinganappropriate,systematicwaytocollectandorganizetodisplaymultiplesteptasks
  • Conceptualizingfractionalcomparisonsandfractionalpartsofotherfractions

CommonCoreStateStandards

ThislessoninvolvesarangeofStandardsforMathematicalPractice,withemphasison:

1.Makesenseofproblemsandpersevereinsolvingthem.

4.Modelwithmathematics.

7.Lookforandmakeuseofstructure.

ThislessoninvolvesStandardsforMathematicalContentinthestandardsfromacrossthegrades,withemphasison:

4.NFBuildfractionsfromunitfractionsbyapplyingandextendingpreviousunderstandingsofoperationsonwholenumbers.

5.NFApplyandextendpreviousunderstandingsofmultiplicationanddivisiontomultiplyanddividefractions.6.NSApplyandextendpreviousunderstandingsofmultiplicationanddivision.

Thislessonunitisstructuredinthefollowingway:

•Beforethelesson,studentsattemptthetaskindividually.Youthenreviewtheirworkandformulatequestionsforstudentstoanswerinorderforthemtoimprovetheirwork.

•Studentsusewhiteboardstoreviewfractiontasksimilartocookietask.

•Inthelessontheyworkcollaboratively,insmallgroups,tocritiqueexamplesofotherstudents’work.

•Inawhole-classdiscussion,studentsexplainandcomparethealternativeapproachestheyhaveseenandused.

•Finally,studentsworkaloneagaintoimprovetheirindividualsolutions.

Materialsrequired

•EachindividualstudentwillneedtwocopiesofthehandoutWherearetheCookies?

•Eachstudentneedsawhiteboarddryerasemarkerfortheshortintrolesson.

•EachsmallgroupofstudentswillneeddacopyofSampleResponsestoDiscussandwhicheversamplesofstudentworkchosen.

Timeneeded

Approximatelyfifteenminutesbeforethelesson,aone-hourlesson,andtenminutesinafollow-uplesson.Alltimingsareapproximate.Exacttimingswilldependontheneedsoftheclass.

Beforethelesson

Assessmenttask:

Havethestudentsdothistaskinclassadayormorebeforetheformativeassessmentlesson.Thiswillgiveyouanopportunitytoassesstheworkandtofindoutthekindsofdifficultiesstudentshavewithit.Thenyouwillbeabletotargetyourhelpmoreeffectivelyinthefollow-uplesson.

GiveeachstudentacopyofWherearethecookies?Introducethetaskbrieflyandhelptheclasstounderstandtheproblemanditscontext. Pg.1

Spendfifteenminutesonyourown,answeringthequestion.Showyourwork.Don’tworryifyoucan’tfigureitout.Therewillbealessononthismaterial[tomorrow]thatwillhelpyouimproveyourwork.

Yourgoalistobeabletoanswerthisquestionwithconfidencebytheendofthatlesson.

Itisimportantthatstudentsanswerthequestionwithoutassistance,asfaraspossible.Studentswhosittogetheroftenproducesimilaranswers,andthen,whentheycometocomparetheirwork,theyhavelittletodiscuss.Forthisreason,wesuggestthatwhenstudentsdothetaskindividually.

Assessingstudents’responses

Collectstudents’responsestothetask.Makesomenotesonwhattheirworkrevealsabouttheircurrentlevelsofunderstanding,andtheirdifferentproblemsolvingapproaches. Wesuggestthatyoudonot scorestudents’work.Theresearchshowsthatthiswillbecounterproductive,asitwillencouragestudentstocomparetheirscores,andwilldistracttheirattentionfromwhattheycandotoimprovetheirmathematics. Instead,helpstudentstomakefurtherprogressbysummarizingtheirdifficultiesasaseriesofquestions.

Somesuggestionsforfeedbackquestionsbasedoncommonmisconceptionsaregiveninthechartbelow.Wesuggestthatyouwritealistofyourownquestions,basedonyourstudents’work,usingtheideasthatfollow.Youmaychoosetowritequestionsoneachstudent’swork.Ifyoudonothavetimetodothis,selectafewquestionsthatwillbeofhelpthemajorityofstudents.Thesecanbewrittenontheboardattheendofthelesson.

CommonIssues / Suggestedquestionsandprompts
Studentwhohastroublegettingstarted. / •Howmightyouworkbackwardstobeginthistask?
Studentdoesnotadjustforanewwholewhendeterminingthenewamounteaten. / •Howcanyoumakesureyouareonlygettingafractionofwhatisleftandnottheoriginalamountofcookies?
Studentconfusesamounteatenwithamountleft. / •If1/3areeatenwhatfractionareleft?
Studentworkisunsystematic. / •Whatisthesameandwhatisdifferentaftereachpersoneatssomecookies?
•Howcanyouorganizeyourwork?
Studentusesthewrongfractionoperations. / •Inthisproblemarewemultiplyingoradding,dividingorsubtracting?Howdoyouknow?
Studentwritesanswerwithoutexplanation. / •Howcouldyouexplain/showhowyoureachedyourconclusionssothatsomeoneinanotherclassunderstands?
•Howcanyouusenumbers,words,ordiagramstodescribehowthecookieswereeaten?
Studentcorrectlyidentifieswhenthenumberofcookiesthatstartedonthetray. / •Thinkofanotherwayofsolvingtheproblem.Isthismethodbetterorworsethanyouroriginalone? Explainyouranswer.
•Canyoumakeanewproblemwithadifferentnumberofcookiesleft?

Suggestedlessonoutline

Pg.2

IntroductoryLesson(10minutes)

Usingwhiteboards,havestudentsrespondtothefollowing:

Sallyhas3applesandeats½ofthem. Jimeats1/3ofwhatSallyhadleft.Howmuchoftheapplesarenowleft?

Havestudentsshareresponsestothistaskandexplaintheirpicturesornumbersintheirsolutions.Sampleresponse: Sallyate1½oftheapples,sothere1½leftwhenshewasfinished.Jimate1/3ofthe1½thatwereleft sotherewere2 1/2sor1wholeappleleftwhenhewasfinished.

Collaborativeactivity:(20minutes)

Returnthestudents’workontheWherearetheCookies?problem.Askstudentstore-readboththeWherearetheCookies?problemandtheirsolutions.Havestudentssharetheirworkwithapartner,andhaveeachpersonaskanyclarifyingquestionstounderstandtheirpartner’sapproachtothetask.

MakeanoteofstudentapproachestothetaskGiveeachsmallgroupofstudentsacopyoftheSampleResponsestoDiscusshandout.Choosethesamplesofstudentworkthatmatchyourstudents’levelofunderstanding. Beginwithacoupleofincorrectorincompleteworksamples(Robin,Britney,CharlieDawn)andacoupleofcorrectsamples(Katrina,Jim,EddieAndrew)dependingonthemethodsandmisconceptionsheldbyeachgroup. ThestudentworksamplebyBrandonusesequationsandshouldbeusedasanextensiononlyforgroupswhounderstandtheothersamplesfirst.

Displaythefollowingquestionsusingtheprovidedsheet:SampleResponsestoDiscuss.

Describetheproblemsolvingapproachthestudentused.Forexample,youmight:

•Describethewaythestudenthasorganizedthesolution.

•Describewhatthestudentdidtocalculatethenumberofcookiesstartingonthetray.Explainwhatthestudentneedstodotocompleteorcorrecthisorhersolution.

Thisanalysistaskwillgivestudentsanopportunitytoevaluateavarietyofalternativeapproachestothetask.Duringsmall-groupwork,supportstudentthinkingasbefore.Also,checktoseewhichoftheexplanationsstudentsfindmoredifficulttounderstand.Identifyoneortwooftheseapproachestodiscussintheplenarydiscussion.Notesimilaritiesanddifferencesbetweenthesampleapproaches.

Plenarywhole-­‐classdiscussioncomparingdifferentapproaches(20minutes)

Organizeawhole-classdiscussiontoconsiderdifferentapproachestothetask.Theintentionisforyoutofocusongettingstudentstounderstandthemethodsofworkingouttheanswers,ratherthaneithernumericalorpictorialsolutions.Focusyourdiscussiononthetasksstudentsfounddifficult.

Let’sstopandtalkaboutdifferentapproaches.

Askthestudentstocomparethedifferentsolutionmethods.Beginwiththemostpictorialandworkthroughtothemoreabstractapproaches–makeconnectionsbetweenapproachesthroughoutthediscussion.Questionsyoumightposetostudentsduringthediscussion:

Whichapproachdidyoulikebestofthestudentworkyouanalyzed?Why?Whichapproachdidyoufinditmostdifficulttounderstand?

Readthroughyouroriginalresponsesandthinkaboutwhatyouhavelearnedthislesson.Didanyoneuseadifferentmethodthanthesamplesofstudentworkyouanalyzed?

Improvingindividualsolutionstotheassessmenttask(10minutes)

Ifyouarerunningoutoftime,youcouldschedulethisactivitytostartthenextday.

Pg.3

MakesurestudentshavetheiroriginalindividualworkontheWherearethecookies?taskonhand.Givethemafresh,blankcopyoftheWherearethecookies? tasksheet.

Ifastudentissatisfiedwithhisorhersolution,askthestudenttotryadifferentapproachtotheproblemandtocomparetheapproachalreadyused.

SolutionDiscussionofStudentworksamples

Thesolutiontothetaskis24cookiesstartedonthetray. Studentscouldworkbackwardstosolvethetaskusingnumbersand/orpicturesbuttheyhavetokeeptrackoftheamounteatenandtheamountleftaftereachpersoneatssomeofthecookies. Itispossibleastudentmightjustguesschecktokeeptryingnumberstobeginonthetrayandworkthroughthefractionsuntiltheyfindananswerthatresultsin6remainingcookies. Usingmultiplicationoffractionsorequationsaremoresophisticatedmethods,butarenotrequiredtosolvethistask. Theyarehowever,efficientapproachesthatsomestudentswillbereadytoinvestigate.

ThemethodsusedbyKatrina,Dawn,Jim,Eddie,AndrewBrandonallleadtothecorrectsolution.However,Dawn’sworkisincompletebecauseitdoesnotgivethefinalanswerof24.Dawn’spictorialmodelusesthetrayastheonewholeandsubtractsamountswhileatthesametimeshowingthefractionalpartsofthewhole.Intheendtherewouldbe12equalsizesectionsand2cookiescouldfitineachsotheapproachcouldleadtocorrectsolution,itisjustincomplete. KatrinaJimusesimilarpictorialmethodsbyactuallydrawingoutcookiesbyworkingbackwards. Eddieusesabarmodeltoworkbackwards. AndrewusesfractionmultiplicationandBrandonsolveswithequations.

Robin,BritneyCharliealluseincorrectmethods. Robinattemptstousepictures,Britneyusesdivision

Charlieconfusestheamounteatenwiththeamountleft.

Mrs.Jamesleftatrayofcookiesonthecounterearlyonemorning.Larrywalkedbybeforelunchanddecidedtotake1/3ofthecookiesonthetray.LaterthatafternoonBarrycameinandate1/4oftheremainingcookies.

AftersupperTerrysawthetrayofcookiesandate1/2ofthecookiesremainingatthattime. ThenextmorningMrs.Jamesfoundthetraywithonly6cookiesleft. HowmanycookieswereonthetraywhenMrs.Jamesfirstleftitonthecounter?

SampleResponsestoDiscuss

HereissomeworkonWherearetheCookies?fromstudentsinanotherclass.Foreachpieceofwork:

1.Writethenameofthestudentwhosesolutionyouareanalyzing.

2.Describetheproblemsolvingapproachthestudentused.Forexample,youmight:

•Describethewaythestudenthasorganizedthesolution.

•Describewhatthestudentdidtocalculatethenumberofcookiesthatstartedonthetray.

3.Explainwhatthestudentneedstodotocompleteorcorrecthisorhersolution.

’s_Solution

’s_Solution

’s_Solution

Katrina’sSolution

Dawn’sSolution

Robin’sSolution

Britney’sSolution

Charlie’sSolution

Jim’sSolution

Eddie’sSolution

Andrew’sSolution

Brandon’sSolution