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Problem of the Month Got Your Number TheProblemsoftheMonth(POM)areusedinavarietyofwaystopromoteproblem solvingandtofosterthefirststandardofmathematicalpracticefromtheCommon CoreStateStandards:“Makesenseofproblemsandpersevereinsolvingthem.” POMsmaybeusedbyateachertopromoteproblemsolvingandtoaddressthe differentiatedneedsofherstudents.Adepartmentorgradelevelmayengagetheir studentsinaPOMtoshowcaseproblemsolvingasakeyaspectofdoing mathematics.POMscanalsobeusedschoolwidetopromoteaproblem‐solving themeataschool.Thegoalisforallstudentstohavetheexperienceofattacking andsolvingnon‐routineproblemsanddevelopingtheirmathematicalreasoning skills.Althoughobtainingandjustifyingsolutionstotheproblemsistheobjective, theprocessoflearningtosolveproblemsisevenmoreimportant. TheProblemoftheMonthisstructuredtoprovidereasonabletasksforallstudents inaschool.ThestructureofaPOMisashallowfloorandahighceiling,sothatall studentscanproductivelyengage,struggle,andpersevere.ThePrimaryVersionis designedtobeaccessibletoallstudentsandespeciallyasthekeychallengefor gradesK–1.LevelAwillbechallengingformostsecondandthirdgraders.LevelB maybethelimitofwherefourthandfifth‐gradestudentshavesuccessand understanding.LevelCmaystretchsixthandseventh‐gradestudents.LevelDmay challengemosteighthandninth‐gradestudents,andLevelEshouldbechallenging formosthighschoolstudents.Thesegrade‐levelexpectationsarejustestimatesand shouldnotbeusedasanabsoluteminimumexpectationormaximumlimitationfor students.Problemsolvingisalearnedskill,andstudentsmayneedmany experiencestodeveloptheirreasoningskills,approaches,strategies,andthe perseverancetobesuccessful.TheProblemoftheMonthbuildsonsequentiallevels ofunderstanding.AllstudentsshouldexperienceLevelAandthenmovethroughthe tasksinordertogoasdeeplyastheycanintotheproblem.Therewillbethose studentswhowillnothaveaccessintoevenLevelA.Educatorsshouldfeelfreeto modifythetasktoallowaccessatsomelevel. Overview IntheProblemoftheMonth,GotYourNumber,studentsusenumberproperties, numberoperations,organizedlists,countingmethods,andprobabilitytosolve problems.ThemathematicaltopicsthatunderliethePOMarenumbersense, numberproperties,addition,subtraction,multiplication,division,fractions, representationsofrationalnumbers,countingprinciples,systematiccharting, probability,andgeneralizations. InLevelAofthePOM,studentsarepresentedwithaproblemthatinvolves summingthreevalues.Theirtaskinvolvespickingthreecardsfromfivethatwere dealtinordertomakeasumcloseto20.Thetaskisdesignedasagamefortwo ©NoyceFoundation2014. ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0 UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US). studentswiththegoalofkeepingthedifferencefrom20assmallaspossible. Studentsconcludethegamebydescribingtheirstrategy.InLevelB,theirtask involvespickingfourcardsfromsixthatweredealtinordertomaketwotwo‐digit numberswhosesumiscloseto100.Thetaskisdesignedasagameforstudents withthegoalofkeepingthetotaldifferencefrom100assmallaspossible.InLevel C,studentspickfourone‐digitnumbersfromastackofcards.Theyalsospina spinnerthatidentifiesoneofthefourbasicoperations.Theyareaskedtodetermine theequationthatproducesthesmallestresultusingthosefourdigitsandapplying theidentifiedoperationtoformtwofractions.LevelDinvolvesagamebetweentwo students.Eachstudenthastocreatethelargestpossiblethree‐digitnumber.When acardispickedthestudentmustplaceitineithertheones,tens,orhundredsplace. Onceacardisplaced,itcan’tbemoved.Studentsplayindependentlyandcan’tsee oneanother’scardsuntiltheendwhentheycomparetheirnumberstodetermine whichnumberisgreater.InLevelE,studentsareaskedtogeneralizetheirfindings fromLevelD.Studentsareaskedtopredictandjustifythebeststrategyforwinning thegame. ©NoyceFoundation2014. ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0 UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US). Problem of the Month Got Your Number LevelA CarolandMelissaareplayingagame.Theyhaveadeckof36cardswith justthenumbers1through9.Aftertheymixupthecards,theyputthem intoapile.Belowaretherules: Dealfivenumbercardstoeachplayer. Useanythreeofyourcards. Pickthreenumbersthataddtoanumbernear20. Writeanumbersentencewithyourthreecardsandthetotalthatis near20. Findyourscore.Yourscoreisthedifferencebetweenyourtotaland 20. Forexample,ifyoupickedthecards6,9,7,6+9+7=22.Soyour totalis22.Tofindyourscore,subtract20from22.22–20=2. Shufflethecardsandplayanotherround. Playthegameseventimes.Attheendofthegame,sumallsevenscoresfor eachplayer.Theplayerwiththelowesttotalisthewinner. Problem of the Month Got Your Number Page 1 © Noyce Foundation 2014. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 Unported License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US). LevelB SandyandSallyareplayingagame.Theyhaveadeckof36cardswithjust thenumbers1through9.Aftertheymixupthecards,theyputthemintoa pile.Belowaretherules: Dealsixcardstoeachplayer. Selectanyfourofyourcardstomaketwonumbers.Eachnumber wouldbeatwo‐digitnumber. Arrangethenumbersandthenaddthemtogetasumasnear100as possible. Onceyouhaveselectedthetwonumbersandfoundthesum,write outtheequations. Determineyourscorebyfindingthedifference(distance)between yournumberand100. Shufflethecardsandplayanotherround. Playthegameseventimes.Attheendofthegame,sumallseven scoresforeachplayer.Theplayerwiththelowesttotalisthewinner. Explainthestrategyyouusedtotrytowinthegames.Explainwhyyou chosethatstrategy. Problem of the Month Got Your Number Page 2 © Noyce Foundation 2014. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 Unported License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US). LevelC JakeandLindaareplayingagame.Theyhaveadeckof36cardswithjust thenumbers1through9.Aftertheymixupthecards,theyputthemintoa pile.Belowaretherules: Dealfourcardstoeachplayer. Spinthespinnertoselectanoperation. Arrangethedigitsintotwofractions,suchthattheresultofthat operationuponthetwofractionswillproducethesmallestpossible outcome. Onceyouhaveselectedthetwofractionsandfoundtheoutcome, writeouttheequations. Thecalculatedoutcomebecomesyourscoreforthatround. Shufflethecardsandplayanotherround. Playthegameseventimes.Attheendofthegame,sumallseven scoresforeachplayer.Theplayerwiththelowesttotalisthewinner. Explainthestrategyyouusedtotrytowinthegames.Explainwhyyou chosethatstrategy. Problem of the Month Got Your Number Page 3 © Noyce Foundation 2014. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 Unported License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US). LevelD JeanandFordareplayingagame.Theyeachhaveadeckofsixcardswith justthenumbers1through6.Aftertheyeachmixupthecardstheyput themintoonepile.Thegoalofthegameistomakethelargestthree‐digit number.Thefirstplayerpicksthetopcardandplacesthenumberineither theones,tensorhundredsplace.Thesecondplayerthenpicksacardand hasthesameoptions.Onceacardisputinalocation,itcannotbemoved. Continueplayinguntilalltheplacesarefilled.Wheneachhascreateda three‐digitnumber,determinewhichplayerhasthelargestnumber.That playeristhewinner!Note:Theplayersarenotabletoseetheother player’scardsorplacementuntilallthecardsaredrawnandplayed. Playthegameseveraltimes.Keeptrackoftheresults. Writeadetailedstrategyforthisgame.Givenspecificcards,explainwhere youwouldputthatnumbertoinsurethebestprobabilityofwinning.Be thoroughindevelopingastrategy.Forexample,ifyoudrewa4asyour secondcard,wherewouldyouputit?Ofcourseyouwouldneedtoexplain optionsdependingonwhereyourfirstcardwasplaced. Problem of the Month Got Your Number Page 4 © Noyce Foundation 2014. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 Unported License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US). LevelE InthegamethatJeanandFordareplaying,supposeyoudrewa4onyour firstturn.Explainwhere(hundreds,tensorones)youwouldplacethat cardinordertogiveyouthebestchanceofwinningthegame.Justifyyour answerwithamathematicalargument. Supposeyoudrewa3onyourfirstturn.Explainwhere(hundreds,tensor ones)youwouldplacethatcardandjustifyyouranswer. Justifyyourstrategyprovingwhyyouwouldplaceanygivencardinany givenlocation.Yourjustificationshouldbecompleteandprovideavalid argumentforwhereeachcard should be placed given any situation. Problem of the Month Got Your Number Page 5 © Noyce Foundation 2014. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 Unported License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US). Problem of the Month Got Your Number PrimaryVersionLevelA Materials:Thedeckofcards(1‐9)foreachpair. Discussion on the rug: Theteacherstartsadiscussionaboutthenumber 10. “Why is the number 10 an important number?” Theteacherinvites ideasfromtheclass. “We are going to play a fun game today. It is called Make Ten.” Theteacherdemonstrateshowtoplaythegamewithtwoplayers.“We play in pairs with a deck of number cards. Each player picks seven cards from the deck. You look at your cards and find two cards that add-up (count up) to 10. For example: 8 and 2 make 10. If you can make ten, then put those two cards together in a 10-pairs pile and then pick two more cards. If you can’t make ten, then say ‘pass’ and pick a new card. Switch turns with each other until all the cards on the deck are picked and all the pairs that make ten are found. We will examine our 10-pairs pile all together.” Insmallgroups:Eachgroupplaysthegameuntilthedeckisusedandall pairsoftenaremade.Havestudentslookoverandcountuphowmany10‐ pairstheymade.Afterthegame,theteacheraskstheclasstolistthe10‐ pairsthatthestudentsmade. Theteacherasksthefollowingquestions: “Suppose you are playing this game with a new friend. Explain to your friend how you play the game and which cards you need to put together to make 10-pairs.” Attheendoftheinvestigationhavestudentseitherdiscussordictatea responsetothesummaryquestion. Problem of the Month Got Your Number Page 6 © Noyce Foundation 2014. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 Unported License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US). Operation Spinner Problem of the Month + - X ÷ Got Your Number Page 7 © Noyce Foundation 2014. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 Unported License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US). 1 1 1 1 2 2 2 2 3 3 3 3 Problem of the Month Got Your Number Page 8 © Noyce Foundation 2014. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 Unported License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US). 4 4 4 4 5 5 5 5 6 6 6 6 Problem of the Month Got Your Number Page 9 © Noyce Foundation 2014. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 Unported License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US). 7 7 7 7 8 8 8 8 9 9 9 9 Problem of the Month Got Your Number Page 10 © Noyce Foundation 2014. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 Unported License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US). ProblemoftheMonth GotYourNumber TaskDescription–LevelA Thistaskchallengesstudentstoestimateusingadditioninordertochoose3cardsoutof5that make asumcloseto20.Studentsneedtocompareoptionsto20beforemakingthefinalchoice.Students needtoaddaccuratelyandthencomparethesumsto20usingsubtractiontogetascore. CommonCoreStateStandardsMath‐ContentStandards OperationsandAlgebraicThinking Representandsolveproblemsinvolvingadditionandsubtraction. 2.OA.1Useadditionandsubtractionwith100tosolveone‐andtwo‐stepproblemsinvolving situationsofaddingto,takingfrom,puttingtogether,takingapartandcomparing,withunknownsin allpositions,e.g.byusingdrawingsandequationswithasymbolfortheunknownnumberto representtheproblem. Addandsubtractwithin20. 2.OA.2Fluentlyaddandsubtractwithin20usingmentalstrategies.ByendofGrade2,knowfrom memoryallsumsoftwoone‐digitnumbers. CommonCoreStateStandardsMath–StandardsofMathematicalPractice MP.1Makesenseofproblemsandpersevereinsolvingthem. Mathematicallyproficientstudentsstartbyexplainingtothemselvesthemeaningofaproblemand lookingforentrypointstoitssolution.Theyanalyzegivens,constraints,relationships,andgoals. Theymakeconjecturesabouttheformandmeaningofthesolutionandplanasolutionpathway ratherthansimplyjumpingintoasolutionattempt.Theyconsideranalogousproblems,andtry specialcasesandsimplerformsoftheoriginalprobleminordertogaininsightintoitssolution.They monitorandevaluatetheirprogressandchangecourseifnecessary.Olderstudentsmight, dependingonthecontextoftheproblem,transformalgebraicexpressionsorchangetheviewing windowontheirgraphingcalculatortogettheinformationtheyneed.Mathematicallyproficient studentscanexplaincorrespondencesbetweenequations,verbaldescriptions,tables,andgraphsor drawdiagramsofimportantfeaturesandrelationships,graphdata,andsearchforregularityor trends.Youngerstudentsmightrelyonusingconcreteobjectsorpicturestohelpconceptualizeand solveaproblem.Mathematicallyproficientstudentschecktheiranswerstoproblemsusinga differentmethod,andtheycontinuallyaskthemselves,“Doesthismakesense?”Theycanunderstand theapproachesofotherstosolvingcomplexproblemsandidentifycorrespondencesbetween differentapproaches. MP.7Lookforandmakeuseofstructure. Mathematicallyproficientstudentstrytolookcloselytodiscernapatternorstructure.Young students,forexample,mightnoticethatthreeandsevenmoreisthesameamountassevenandthree more,ortheymaysortacollectionofshapesaccordingtohowmanysidestheshapeshave.Later, studentswillsee7x8equalsthewell‐remembered7x5+7x3,inpreparationforlearningabout thedistributiveproperty.Intheexpressionx2+9x+14,olderstudentscanseethe14as2x7and the9as2+7.Theyrecognizethesignificanceofanexistinglineinageometricfigureandcanusethe strategyofdrawinganauxiliarylineforsolvingproblems.Theyalsocanstepbackforanoverview andshiftperspective.Theycanseecomplicatedthings,suchassomealgebraicexpressions,assingle objectsorbeingcomposedofseveralobjects.Forexample,theycansee5–3(x‐y)2as5minusa positivenumbertimesasquareandusethattorealizethatitsvaluecannotbemorethan5forany realnumbersxandy. ©NoyceFoundation2014. ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0 UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US). ProblemoftheMonth GotYourNumber TaskDescription–LevelB Thistaskchallengesstudentstochoose4cardsfrom6tomake2 two‐digit numbersthatwilladd closeto100.Studentsmustuseplace‐valueknowledgetoestimateandmaketheirchoices.Students mustthenbeabletoaccuratelyusecomparisonsubtractiontofindthedistancefrom100.Finally studentsuseplace‐valueunderstandingtogeneralizethesituationbydescribingastrategyfor choosingandarrangingthecardstoformthetwo‐digitnumbers. CommonCoreStateStandardsMath‐ContentStandards NumberandOperationsinBaseTen Useplacevalueunderstandingandpropertiesofoperationstoaddandsubtract. 2.NBT.5fluentlyaddandsubtractwithin100usingstrategiesbasedonplacevalue,propertiesof operations,and/ortherelationshipbetweenadditionandsubtraction. 2.NBT.6Adduptofourtwo‐digitnumbersusingstrategiesbasedonplacevalueandpropertiesof operations. CommonCoreStateStandardsMath–StandardsofMathematicalPractice MP.3Constructviableargumentsandcritiquethereasoningofothers. Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,and previouslyestablishedresultsinconstructingarguments.Theymakeconjecturesandbuildalogical progressionofstatementstoexplorethetruthoftheirconjectures.Theyareabletoanalyze situationsbybreakingthemintocases,andcanrecognizeandusecounterexamples.Theyjustify theirconclusions,communicatethemtoothers,andrespondtotheargumentsofothers.They reasoninductivelyaboutdata,makingplausibleargumentsthattakeintoaccountthecontextfrom whichthedataarose.Mathematicallyproficientstudentsarealsoabletocomparetheeffectiveness oftwoplausiblearguments,distinguishcorrectlogicorreasoningfromthatwhichisflawed,and–if thereisaflawinanargument–explainwhatitis.Elementarystudentscanconstructarguments usingconcretereferentssuchasobjects,drawings,diagrams,andactions.Suchargumentscanmake senseandbecorrect,eventhroughtheyarenotgeneralizedormadeformaluntillatergrades.Later, studentslearntodeterminedomainstowhichanargumentapplies.Studentsatallgradescanlisten orreadtheargumentsofothers,decidewhethertheymakesense,andaskusefulquestionstoclarify orimprovethearguments. MP.7Lookforandmakeuseofstructure. Mathematicallyproficientstudentstrytolookcloselytodiscernapatternorstructure.Young students,forexample,mightnoticethatthreeandsevenmoreisthesameamountassevenandthree more,ortheymaysortacollectionofshapesaccordingtohowmanysidestheshapeshave.Later, studentswillsee7x8equalsthewell‐remembered7x5+7x3,inpreparationforlearningabout thedistributiveproperty.Intheexpressionx2+9x+14,olderstudentscanseethe14as2x7and the9as2+7.Theyrecognizethesignificanceofanexistinglineinageometricfigureandcanusethe strategyofdrawinganauxiliarylineforsolvingproblems.Theyalsocanstepbackforanoverview andshiftperspective.Theycanseecomplicatedthings,suchassomealgebraicexpressions,assingle objectsorbeingcomposedofseveralobjects.Forexample,theycansee5–3(x‐y)2as5minusa positivenumbertimesasquareandusethattorealizethatitsvaluecannotbemorethan5forany realnumbersxandy. ©NoyceFoundation2014. ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0 UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US). ProblemoftheMonth GotYourNumber TaskDescription–LevelC Thistaskchallengesstudentstotakefourdigitsandmakethemintotwofractionsthat yieldthelowestresult whensubjectedtoagivenoperation.Studentsmusthaveanunderstandingofequivalencetocomparetheir resultstothoseoftheirpartners.Studentsmightchoosetousecommondenominatorsandequivalentfractions orstudentsmightchoosetochangeresultstodecimalstomakethecomparisons.Finally,studentsmustreflect onhowtheoperationsaffecttheanswersandexplainastrategyforcomposingthefractions.Theymustalso decideifordermattersinhowthefractionsareplacedintheproblem. CommonCoreStateStandardsMath‐ContentStandards NumberandOperations‐Fractions Extendunderstandingoffractionequivalenceandordering. 4.NF.2Comparetwofractionswithdifferentnumeratorsanddifferentdenominators,e.g.bycreatingcommon denominatorsornumerators,orbycomparingtoabenchmarkfractionsuchas½.Recognizetheresultsofthe comparisonswithsymbols,=,or,andjustifytheconclusions,e.g.byusingavisualfractionmodel. Useequivalentfractionsasastrategytoaddandsubtractfractions. 5.NF.1Addandsubtractfractionswithunlikedenominators(includingmixednumbers)byreplacinggiven fractionswithequivalentfractionsinsuchawayastoproduceanequivalentsumordifferenceoffractionswith likedenominators. Applyandextendpreviousunderstandingsofmultiplicationanddivisiontomultiplyanddivide fractions. 5.NF.4Applyandextendpreviousunderstandingsofmultiplicationtomultiplyafractionorwholenumberbya fraction. 5.NF.7Applyandextendpreviousunderstandingsofdivisiontodivideunitfractionsbywholenumbersand wholenumbersbyunitfractions. NumberandOperationsinBaseTen Performoperationswithmulti‐digitwholenumbersandwithdecimalstohundredths. 5.NBT.7Add,subtract,multiplyanddividedecimalstohundredths,usingconcretemodelsordrawingsand strategiesbasedonplacevalue,propertiesofoperations,and/ortherelationshipbetweenadditionan subtraction;relatethestrategytoawrittenmethodandexplainthereasoningused. TheNumberSystem Applyandextendpreviousunderstandingsofmultiplicationanddivisiontodividefractionsby fractions. 6.NS.1Interpretandcomputequotientsoffractions,andsolvewordproblemsinvolvingdivisionoffractionsby fractions,e.g.byusingvisualfractionmodelsandequationstorepresenttheproblem. CommonCoreStateStandardsMath–StandardsofMathematicalPractice MP.3Constructviableargumentsandcritiquethereasoningofothers. Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,andpreviouslyestablishedresultsin constructingarguments.Theymakeconjecturesandbuildalogicalprogressionofstatementstoexplorethetruthoftheir conjectures.Theyareabletoanalyzesituationsbybreakingthemintocases,andcanrecognizeandusecounterexamples. Theyjustifytheirconclusions,communicatethemtoothers,andrespondtotheargumentsofothers.Theyreasoninductively aboutdata,makingplausibleargumentsthattakeintoaccountthecontextfromwhichthedataarose.Mathematically proficientstudentsarealsoabletocomparetheeffectivenessoftwoplausiblearguments,distinguishcorrectlogicor reasoningfromthatwhichisflawed,and–ifthereisaflawinanargument–explainwhatitis.Elementarystudentscan constructargumentsusingconcretereferentssuchasobjects,drawings,diagrams,andactions.Suchargumentscanmake senseandbecorrect,eventhroughtheyarenotgeneralizedormadeformaluntillatergrades.Later,studentslearnto determinedomainstowhichanargumentapplies.Studentsatallgradescanlistenorreadtheargumentsofothers,decide whethertheymakesense,andaskusefulquestionstoclarifyorimprovethearguments. MP.7Lookforandmakeuseofstructure. Mathematicallyproficientstudentstrytolookcloselytodiscernapatternorstructure.Youngstudents,forexample,might noticethatthreeandsevenmoreisthesameamountassevenandthreemore,ortheymaysortacollectionsofshapes accordingtohowmanysidestheshapeshave.Later,studentswillsee7x8equalsthewell‐remembered7x5+7x3,in preparationforlearningaboutthedistributiveproperty.Intheexpressionx2+9x+14,olderstudentscanseethe14as2x7 andthe9as2+7.Theyrecognizethesignificanceofanexistinglineinageometricfigureandcanusethestrategyofdrawing anauxiliarylineforsolvingproblems.Theyalsocanstepbackforanoverviewandshiftperspective.Theycansee complicatedthings,suchassomealgebraicexpressions,assingleobjectsorbeingcomposedofseveralobjects.Forexample, theycansee5–3(x‐y)2as5minusapositivenumbertimesasquareandusethattorealizethatitsvaluecannotbemorethan 5foranyrealnumbersxandy. ©NoyceFoundation2014. ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0 UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US). ProblemoftheMonth GotYourNumber TaskDescription–LevelD Thistaskchallengesstudentstoplayagameofchancetoforma3‐digitnumberbydrawingcards fromadeckwithdigits1to6.Aftereachdrawthestudentmustdecidewhethertoplacethecardin theones,tens,orhundredsplace.Onceanumberisplaced,itcan’tbemoved.Studentsmustthen comparenumberstodecidewhichislarger.Afterplayingthegameafewtimes,studentsmustthink aboutprobabilitiestodetermineadetailedstrategyforplayingthegame. CommonCoreStateStandardsMath‐ContentStandards NumberandOperationsinBaseTen Understandplacevalue. 2.NBT.4Comparetwothree‐digitnumbersbasedonmeaningsofhundreds,tens,andonesdigitsusing,=, symbolstorecordtheresultsofcomparisons. StatisticsandProbability Investigatechanceprocessesanddevelop,use,andevaluateprobabilitymodels. 7.SP.5Understandthattheprobabilityofachanceeventisanumberbetween0and1thatexpressesthe likelihoodoftheeventoccurring.Largernumbersindicatesgreaterlikelihood.Aprobabilitynear0indicatesan unlikelyevent,aprobabilityaround½indicatesaneventthatisneitherunlikelynorlikely,andaprobability near1indicatesalikelyevent. 7.SP.6Approximatetheprobabilityofachanceeventbycollectingdataonthechanceprocessthatproducesit andobservingitslong‐runrelativefrequencyandpredicttheapproximaterelativefrequencygiventhe probability. 7.SP.7Developaprobabilitymodelanduseittofindprobabilitiesofevents.Compareprobabilitiesfromamodel toobservedfrequencies;iftheagreementisnotgood,explainpossiblesourcesofdiscrepancy. 7..SP.7aDevelopauniformprobabilitymodelbyassigningequalprobabilitytoalloutcomes,andusethemodel todetermineprobabilitiesofevents. 7.SP.8Findprobabilitiesofcompoundeventsusingorganizedlists,tables,treediagramsandsimulation. 7.SP.8.aUnderstandthat,justaswithsimpleevents,theprobabilityofacompoundeventisthefractionof outcomesinthesamplespaceforwhichthecompoundeventoccurs. 7SP.8.bRepresentsamplespacesforcompoundeventsusingmethodssuchasorganizedlists,tables,andtree diagrams.Foraneventdescribedineverydaylanguage(e.g.,rollingdoublesixes),identifytheoutcomesinthe samplespace,whichcomposetheevent. HighSchool–StatisticsandProbability–UsingProbabilitytoMakeDecisions Useprobabilitytoevaluateoutcomesofdecisions. S‐MD.7Analyzedecisionsandstrategiesusingprobabilityconcepts(e.g.producttesting,medicaltesting,pulling ahockeygoalieattheendofagame). HighSchool–StatisticsandProbability–ConditionalProbabilityandtheRulesofProbability Understandindependenceandconditionalprobabilityandusethemtointerpretdata. S‐CP.1Describeeventsassubsetsofasamplespace(thesetofoutcomes)usingcharacteristics(orcategories)of theoutcomes,orasunions,intersections,orcomplementsofotherevents(“or”,“and”,“not”). S‐CP.2UnderstandthattwoeventsAandBareindependentiftheprobabilityofAandBoccurringtogetheris theproductoftheirprobabilities,andusethischaracterizationtodetermineiftheyareindependent. Usetherulesofprobabilitytocomputeprobabilitiesofcompoundeventsinauniformprobability model. S‐CP.6FindtheconditionalprobabilityofAgivenBasthefractionB’soutcomesthatalsobelongtoA,and interprettheanswerintermsofthemodel. S‐CP.8ApplythegeneralMultiplicationRuleinauniformprobabilitymodel,P9AandB)= P(A)P(BIA)=P(B)P(AIB),andinterprettheanswerintermsofthemodel. CommonCoreStateStandardsMath– StandardsofMathematicalPractice MP.3Constructviableargumentsandcritiquethereasoningofothers. Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,andpreviously establishedresultsinconstructingarguments.Theymakeconjecturesandbuildalogicalprogressionof statementstoexplorethetruthoftheirconjectures.Theyareabletoanalyzesituationsbybreakingtheminto cases,andcanrecognizeandusecounterexamples.Theyjustifytheirconclusions,communicatethemtoothers, andrespondtotheargumentsofothers.Theyreasoninductivelyaboutdata,makingplausibleargumentsthat takeintoaccountthecontextfromwhichthedataarose.Mathematicallyproficientstudentsarealsoableto ©NoyceFoundation2014. ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0 UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US). comparetheeffectivenessoftwoplausiblearguments,distinguishcorrectlogicorreasoningfromthatwhichis flawed,and–ifthereisaflawinanargument–explainwhatitis.Elementarystudentscanconstructarguments usingconcretereferentssuchasobjects,drawings,diagrams,andactions.Suchargumentscanmakesenseand becorrect,eventhroughtheyarenotgeneralizedormadeformaluntillatergrades.Later,studentslearnto determinedomainstowhichanargumentapplies.Studentsatallgradescanlistenorreadtheargumentsof others,decidewhethertheymakesense,andaskusefulquestionstoclarifyorimprovethearguments. MP.4Modelwithmathematics. Mathematicallyproficientstudentscanapplythemathematicstheyknowtosolveproblemsarisingineveryday life,society,andtheworkplace.Inearlygradesthismightbeassimpleaswritinganadditionequationto describeasituation.Inmiddlegrades,astudentmightapplyproportionalreasoningtoplanaschooleventor analyzeaprobleminthecommunity.Byhighschool,astudentmightusegeometrytosolveadesignproblemor useafunctiontodescribehowonequantityofinterestdependsonanother.Mathematicallyproficientstudents whocanapplywhattheyknowarecomfortablemakingassumptionsandapproximationstosimplifya complicatedsituation,realizingthatthesemayneedrevisionlater.Theyareabletoidentifyimportant quantitiesinapracticalsituationandmaptheirrelationshipsusingsuchtoolsasdiagrams,two‐waytables, graphs,flowcharts,andformulas.Theycananalyzethoserelationshipsmathematicallytodrawconclusions. Theyroutinelyinterprettheirmathematicalresultsinthecontextofthesituationandreflectonwhetherthe resultsmakesense,possiblyimprovingthemodelifithasnotserveditspurpose. MP.7Lookforandmakeuseofstructure. Mathematicallyproficientstudentstrytolookcloselytodiscernapatternorstructure.Youngstudents,for example,mightnoticethatthreeandsevenmoreisthesameamountassevenandthreemore,ortheymaysort acollectionsofshapesaccordingtohowmanysidestheshapeshave.Later,studentswillsee7x8equalsthe well‐remembered7x5+7x3,inpreparationforlearningaboutthedistributiveproperty.Intheexpressionx2 +9x+14,olderstudentscanseethe14as2x7andthe9as2+7.Theyrecognizethesignificanceofanexisting lineinageometricfigureandcanusethestrategyofdrawinganauxiliarylineforsolvingproblems.Theyalso canstepbackforanoverviewandshiftperspective.Theycanseecomplicatedthings,suchassomealgebraic expressions,assingleobjectsorbeingcomposedofseveralobjects.Forexample,theycansee5–3(x‐y)2as5 minusapositivenumbertimesasquareandusethattorealizethatitsvaluecannotbemorethan5foranyreal numbersxandy. ©NoyceFoundation2014. ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0 UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US). ProblemoftheMonth GotYourNumber TaskDescription–LevelE Thistaskchallengesstudentstothinkaboutprobabilityinthecontextofaplace‐valuegameanduse mathematicstojustifyastrategyforplayingthegame.Thestudentsmustfirstanalyzespecific choicepointswithinthegametofindtheprobabilitiesfortheevents.Thenstudentsmustusetheir analysisoftheprobabilitiestodevelopafullstrategyforthegame. CommonCoreStateStandardsMath‐ContentStandards StatisticsandProbability Investigatechanceprocessesanddevelop,use,andevaluateprobabilitymodels. 7.SP.5Understandthattheprobabilityofachanceeventisanumberbetween0and1thatexpressesthe likelihoodoftheeventoccurring.Largernumbersindicatesgreaterlikelihood.Aprobabilitynear0indicatesan unlikelyevent,aprobabilityaround½indicatesaneventthatisneitherunlikelynorlikely,andaprobability near1indicatesalikelyevent. 7.SP.6Approximatetheprobabilityofachanceeventbycollectingdataonthechanceprocessthatproducesit andobservingitslong‐runrelativefrequencyandpredicttheapproximaterelativefrequencygiventhe probability. 7.SP.7Developaprobabilitymodelanduseittofindprobabilitiesofevents.Compareprobabilitiesfromamodel toobservedfrequencies;iftheagreementisnotgood,explainpossiblesourcesofdiscrepancy. 7.SP.7aDevelopauniformprobabilitymodelbyassigningequalprobabilitytoalloutcomes,andusethemodel todetermineprobabilitiesofevents. 7.SP.8Findprobabilitiesofcompoundeventsusingorganizedlists,tables,treediagramsandsimulation. 7.SP.8aUnderstandthat,justaswithsimpleevents,theprobabilityofacompoundeventisthefractionof outcomesinthesamplespaceforwhichthecompoundeventoccurs. 7SP.8bRepresentsamplespacesforcompoundeventsusingmethodssuchasorganizedlists,tables,andtree diagrams.Foraneventdescribedineverydaylanguage(e.g.,rollingdoublesixes),identifytheoutcomesinthe samplespace,whichcomposetheevent. HighSchool–StatisticsandProbability–UsingProbabilitytoMakeDecisions Useprobabilitytoevaluateoutcomesofdecisions. S‐MD.7Analyzedecisionsandstrategiesusingprobabilityconcepts(e.g.producttesting,medicaltesting,pulling ahockeygoalieattheendofagame.) HighSchool–StatisticsandProbability–ConditionalProbabilityandtheRulesofProbability Usetherulesofprobabilitytocomputeprobabilitiesofcompoundeventsinauniformprobability model. S‐CP.6FindtheconditionalprobabilityofAgivenBasthefractionB’soutcomesthatalsobelongtoA,and interprettheanswerintermsofthemodel. S‐CP.8ApplythegeneralMultiplicationRuleinauniformprobabilitymodel,P9AandB)= P(A)P(BIA)=P(B)P(AIB),andinterprettheanswerintermsofthemodel. CommonCoreStateStandardsMath– StandardsofMathematicalPractice MP.3Constructviableargumentsandcritiquethereasoningofothers. Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,andpreviously establishedresultsinconstructingarguments.Theymakeconjecturesandbuildalogicalprogressionof statementstoexplorethetruthoftheirconjectures.Theyareabletoanalyzesituationsbybreakingtheminto cases,andcanrecognizeandusecounterexamples.Theyjustifytheirconclusions,communicatethemtoothers, andrespondtotheargumentsofothers.Theyreasoninductivelyaboutdata,makingplausibleargumentsthat takeintoaccountthecontextfromwhichthedataarose.Mathematicallyproficientstudentsarealsoableto comparetheeffectivenessoftwoplausiblearguments,distinguishcorrectlogicorreasoningfromthatwhichis flawed,and–ifthereisaflawinanargument–explainwhatitis.Elementarystudentscanconstructarguments usingconcretereferentssuchasobjects,drawings,diagrams,andactions.Suchargumentscanmakesenseand becorrect,eventhroughtheyarenotgeneralizedormadeformaluntillatergrades.Later,studentslearnto determinedomainstowhichanargumentapplies.Studentsatallgradescanlistenorreadtheargumentsof others,decidewhethertheymakesense,andaskusefulquestionstoclarifyorimprovethearguments. MP.4Modelwithmathematics. Mathematicallyproficientstudentscanapplythemathematicstheyknowtosolveproblemsarisingineveryday life,society,andtheworkplace.Inearlygradesthismightbeassimpleaswritinganadditionequationto describeasituation.Inmiddlegrades,astudentmightapplyproportionalreasoningtoplanaschooleventor ©NoyceFoundation2014. ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0 UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US). analyzeaprobleminthecommunity.Byhighschool,astudentmightusegeometrytosolveadesignproblemor useafunctiontodescribehowonequantityofinterestdependsonanother.Mathematicallyproficientstudents whocanapplywhattheyknowarecomfortablemakingassumptionsandapproximationstosimplifya complicatedsituation,realizingthatthesemayneedrevisionlater.Theyareabletoidentifyimportant quantitiesinapracticalsituationandmaptheirrelationshipsusingsuchtoolsasdiagrams,two‐waytables, graphs,flowcharts,andformulas.Theycananalyzethoserelationshipsmathematicallytodrawconclusions. Theyroutinelyinterprettheirmathematicalresultsinthecontextofthesituationandreflectonwhetherthe resultsmakesense,possiblyimprovingthemodelifithasnotserveditspurpose. MP.7Lookforandmakeuseofstructure. Mathematicallyproficientstudentstrytolookcloselytodiscernapatternorstructure.Youngstudents,for example,mightnoticethatthreeandsevenmoreisthesameamountassevenandthreemore,ortheymaysort acollectionsofshapesaccordingtohowmanysidestheshapeshave.Later,studentswillsee7x8equalsthe well‐remembered7x5+7x3,inpreparationforlearningaboutthedistributiveproperty.Intheexpressionx2 +9x+14,olderstudentscanseethe14as2x7andthe9as2+7.Theyrecognizethesignificanceofanexisting lineinageometricfigureandcanusethestrategyofdrawinganauxiliarylineforsolvingproblems.Theyalso canstepbackforanoverviewandshiftperspective.Theycanseecomplicatedthings,suchassomealgebraic expressions,assingleobjectsorbeingcomposedofseveralobjects.Forexample,theycansee5–3(x‐y)2as5 minusapositivenumbertimesasquareandusethattorealizethatitsvaluecannotbemorethan5foranyreal numbersxandy. ©NoyceFoundation2014. ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0 UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US). ProblemoftheMonth GotYourNumber TaskDescription–PrimaryLevel Thistaskchallengesstudentstothinkabouttheimportanceof thenumber10.Studentsusedrawing, counting,andadditiontofindcombinationsoftwonumbercardsthatmakeexactly10. CommonCoreStateStandardsMath‐ContentStandards OperationsandAlgebraicThinking Understandadditionasputtingtogetherandaddingtoandunderstandsubtractionastaking apartandtakingfrom. K.OA.1Representadditionandsubtractionwithobjects,fingers,mentalimages,drawings,sounds, actingoutsituations,verbalexplanations,expressions,orequations. K.OA.2Solveadditionandsubtractionwordproblemsandaddandsubtractwith10,e.g.byusing objectsordrawingstorepresenttheproblem. K.OA.4Foranynumberfrom1to9,findthenumberthatmakes10whenaddedtothegivennumber, e.g.byusingobjectsordrawings,andrecordtheanswerwithadrawingorequation. Representandsolveproblemsinvolvingadditionandsubtraction. 1.OA.1Useadditionandsubtractionwithin20tosolvewordproblemsinvolvingsituationsofadding to,takingfrom,puttingtogether,takingapartandcomparing,withunknownnumbertorepresent theproblem. Addandsubtractwithin20. 1.OA.6Addandsubtractwithin20,demonstratingfluencyforadditionandsubtractionwith10.Use strategiessuchascountingon;;makingten,decomposinganumberleadingtoaten;usingthe relationshipbetweenadditionandsubtraction;andcreatingequivalentbuteasierorknownsumsby creatingtheknownequivalent. CommonCoreStateStandardsMath–StandardsofMathematicalPractice MP.1Makesenseofproblemsandpersevereinsolvingthem. Mathematicallyproficientstudentsstartbyexplainingtothemselvesthemeaningofaproblemandlookingfor entrypointstoitssolution.Theyanalyzegivens,constraints,relationships,andgoals.Theymakeconjectures abouttheformandmeaningofthesolutionandplanasolutionpathwayratherthansimplyjumpingintoa solutionattempt.Theyconsideranalogousproblems,andtryspecialcasesandsimplerformsoftheoriginal probleminordertogaininsightintoitssolution.Theymonitorandevaluatetheirprogressandchangecourseif necessary.Olderstudentsmight,dependingonthecontextoftheproblem,transformalgebraicexpressionsor changetheviewingwindowontheirgraphingcalculatortogettheinformationtheyneed.Mathematically proficientstudentscanexplaincorrespondencesbetweenequations,verbaldescriptions,tables,andgraphsor drawdiagramsofimportantfeaturesandrelationships,graphdata,andsearchforregularityortrends.Younger studentsmightrelyonusingconcreteobjectsorpicturestohelpconceptualizeandsolveaproblem. Mathematicallyproficientstudentschecktheiranswerstoproblemsusingadifferentmethod,andthey continuallyaskthemselves,“Doesthismakesense?”Theycanunderstandtheapproachesofotherstosolving complexproblemsandidentifycorrespondencesbetweendifferentapproaches. MP.7Lookforandmakeuseofstructure. Mathematicallyproficientstudentstrytolookcloselytodiscernapatternorstructure.Youngstudents,for example,mightnoticethatthreeandsevenmoreisthesameamountassevenandthreemore,ortheymaysort acollectionsofshapesaccordingtohowmanysidestheshapeshave.Later,studentswillsee7x8equalsthe well‐remembered7x5+7x3,inpreparationforlearningaboutthedistributiveproperty.Intheexpressionx2 +9x+14,olderstudentscanseethe14as2x7andthe9as2+7.Theyrecognizethesignificanceofanexisting lineinageometricfigureandcanusethestrategyofdrawinganauxiliarylineforsolvingproblems.Theyalso canstepbackforanoverviewandshiftperspective.Theycanseecomplicatedthings,suchassomealgebraic expressions,assingleobjectsorbeingcomposedofseveralobjects.Forexample,theycansee5–3(x‐y)2as5 minusapositivenumbertimesasquareandusethattorealizethatitsvaluecannotbemorethan5foranyreal numbersxandy. ©NoyceFoundation2014. ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0 UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).