Download Problem of the Month Got Your Number

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
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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.
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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
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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
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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
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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
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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
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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
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Operation Spinner
Problem of the Month
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Got Your Number
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License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
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Problem of the Month
Got Your Number
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Problem of the Month
Got Your Number
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Problem of the Month
Got Your Number
Page 10
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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
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ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
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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).