ProblemoftheMonth
PartyTime
TheProblemsoftheMonth(POM)areusedinavarietyofwaystopromoteproblem
solvingandtofosterthefirststandardofmathematicalpracticefromtheCommon
CoreStateStandards:“Makesenseofproblemsandpersevereinsolvingthem.”The
POMmaybeusedbyateachertopromoteproblemsolvingandtoaddressthe
differentiatedneedsofherstudents.Adepartmentorgradelevelmayengagetheir
studentsinaPOMtoshowcaseproblemsolvingasakeyaspectofdoing
mathematics.POMscanalsobeusedschoolwidetopromoteaproblem‐solving
themeataschool.Thegoalisforallstudentstohavetheexperienceofattacking
andsolvingnon‐routineproblemsanddevelopingtheirmathematicalreasoning
skills.Althoughobtainingandjustifyingsolutionstotheproblemsistheobjective,
theprocessoflearningtoproblemsolveisevenmoreimportant.
TheProblemoftheMonthisstructuredtoprovidereasonabletasksforallstudents
inaschool.ThestructureofaPOMisashallowfloorandahighceiling,sothatall
studentscanproductivelyengage,struggle,andpersevere.ThePrimaryVersionis
designedtobeaccessibletoallstudentsandespeciallythekeychallengeforgrades
K–1.LevelAwillbechallengingformostsecondandthirdgraders.LevelBmay
bethelimitofwherefourthandfifth‐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
IntheProblemoftheMonthPartyTime,studentsusemathematicalconceptsof
logic,reasoning,andcountingmethods.Themathematicaltopicsthatunderliethis
POMareknowledgeoflogic,deductivereasoning,countingprinciples/strategies,
andavarietyofmathematicalrepresentationssuchastreediagrams,Venn
diagrams,tables,charts,andmatrices.
InthefirstlevelofthePOM,studentsdeterminethenumberofguestsinvitedtoa
partythroughexaminationofsetinvites‐guestsinvitingothersetsofguests.In
LevelB,studentsareaskedtodeterminethenumberofgirlsatthepartywithshort
redhair,givenanumberoflogicclues.Thestudentsneedtopartitionawholeusing
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License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
simplefractionalamounts(1/2and1/4).Thestudentsapplylogicalreasoningto
determinethenumberofred‐headedgirlsattheparty.InLevelC,studentsare
presentedwithasetofcluesregardingtheguests’names,costumes,andtimeof
arrivalattheparty.Thestudentsareaskedtomatcheachofthepartygoerstotheir
names,costumesandarrivaltimes.InLevelD,thestudentsareaskedtodetermine
whenagameisfairforbothplayers.Studentsjustifytheirfindingsandexplain
whenthegameisfair.Inthefinallevel,studentsareaskedtosolveacomplexlogic
puzzle.Studentsmustdefendtheirsolutionandexplainhowtheysolvedthepuzzle.
© 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
Party Time
Level A
Cindy had a party. She invited two guests. Her guests each invited four guests, and
then those guests each invited three guests.
How many people were at Cindy’s party?
Explain how you determined your solution.
Problem of the Month
Party Time
P1
© 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).
Level B
At Leslie’s party ¼ of the people had long hair. One half of the people at the party
were boys, and ¼ of the girls had short blond hair. None of the boys had long hair.
If there were 32 guests, what is the maximum number of girls who could have had
short red hair?
Show how you determined you answer and why you know you have a correct
solution.
Problem of the Month
Party Time
P2
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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).
Level C
Mia, Jake, Carol, Barbara, Ford and Jeff are all going to a costume party. Figure
out what costume each person is wearing and when they arrived at the party.









The person that arrived fourth was wearing a bathing suit.
Barbara was the last to arrive.
Jake and Mia arrived and stayed together.
The first person was dressed as a French maid.
Superman arrived right before Barbara.
The potato heads were always together at the party.
Ford was a surfer dude.
The French maid was not Carol.
The vampire arrived after Superman.
Problem of the Month
Party Time
P3
© 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).
Level D
Your aunt is having a baby. You have created a party game for a baby shower. It
is called Pick the Gender. You put pink and blue tiles into a bag. You ask two
guests to pick one tile out of the bag without looking. You tell your guests that if
they are the same color, Player A wins and if they are two different colors then
Player B wins.
How many tiles of which colors did you put into the bag to make sure that both
players have an equal chance of winning?
Explain your solution and why it is fair.
Problem of the Month
Party Time
P4
© 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).
Level E
A man and his wife invite 5 other couples to a dinner party. As the guests arrive to
visit before dinner, they shake hands. Not everybody shakes everybody's hands,
and of course, no one shakes hands with his own spouse. Later, as they sit down to
dinner, the host asks each other person, including his wife, “How many hands did
you shake?” He notices, to his surprise, that each respondent shook a different
number of hands.
How many did his wife shake?
Explain your solution and justify your reasoning.
Problem of the Month
Party Time
P5
© 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
Party Time
PrimaryVersionLevelA
Materials:Setsofcounters.
Discussionontherug:Teacheraskstheclass,“Who likes to have parties in our
class? We are going to solve a problem that is about inviting friends to a party.
Who would like to be our party host?”Teacherinvitesastudenttocomeforward.The
teachersaystothehost,“Let’s start by inviting three friends to the party.”The
studentpicksthreefriendstocomeforward.“How many people are at the party?”the
teacheraskstheclass.Studentssharetheiranswersandexplainhowtheyknow.The
teachersays,“Suppose each of the friends phones two people to come to the party.
How many will be at the party altogether?”Studentssharetheirideasanddiscuss
solutions.Thentheyactuallyactitoutandcountthetotalnumberofpeopleattheparty.
Insmallgroups:Studentshavecountersavailable.Teachersays,“Cindy had a party.
She invited two guests. Her guests each invited four guests, and then those guests
each invited three guests. How many people were at Cindy’s party?”Studentswork
togethertofindasolution.Afterthestudentsaredone,theteacherasksstudentstoshare
theiranswersandmethod.
Attheendoftheinvestigation:Studentseitherdiscussordictatearesponsetothis
summaryquestion:“Explain and show how you know how many people are at the
party.”
Problem of the Month
Party Time
P6
© 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
PartyTime
TaskDescription–LevelA
Thistaskchallengesstudentstouselogicandcountingprinciplestofindthenumberofguestsata
partyifguestsinviteguestsatagivenratio.
CommonCoreStateStandardsMath‐ContentStandards
OperationsandAlgebraicThinking
Representandsolveproblemsinvolvingadditionandsubtraction.
2.OA.1Useadditionandsubtractionwithin100tosolveone‐andtwo‐stepproblemsinvolving
situationsofaddingto,takingfrom,puttingtogether,takingapart,andcomparing,withunknownsin
allpositions.
3.OA.1Interpretproductsofwholenumbers,e.g.interpret5x7asthetotalnumberofobjectsin5
groupsof7objectseach.
3.OA.3Usemultiplicationanddivisionwithin100tosolvewordproblemsinsituationsinvolving
equalgroups,arrays,andmeasurementquantities,e.g.,byusingdrawingsandequationswitha
symbolfortheunknownnumbertorepresenttheproblem.
StatisticsandProbability
Investigatechanceprocessesanddevelop,use,andevaluateprobabilitymodels.
7.SP.8Findprobabilitiesofcompoundeventsusingorganizedlists,tables,treediagrams,and
simulation.
7.SP.8bRepresentsamplespacesforcompoundeventsusingmethodssuchasorganizedlists,tables
andtreediagrams.Foraneventdescribedineverydaylanguage,identifytheoutcomesinthesample
space,whichcomposetheevent.
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.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.
© 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
PartyTime
TaskDescription–LevelB
Thistaskchallengesstudentstouselogicandfractionstofindapartofasetthatmeetsseveral
constraints.Studentsmustusefractionalpartsofthesettohelpcalculatetheiranswers.Students
needtofindasystematicwaytoorganizetheirresultsastheyconsidereachconstraint.
CommonCoreStateStandardsMath‐ContentStandards
NumberandOperations‐Fractions
Buildfractionsfromunitfractionsbyapplyingandextendingpreviousunderstandingsof
operationsonwholenumbers.
4.NF.4Applyandextendpreviousunderstandingsofmultiplicationtomultiplyafractionbyawhole
number.
Applyandextendpreviousunderstandingsofmultiplicationanddivisiontomultiplyand
dividefractions.
5.NF.4Applyandextendpreviousunderstandingsofmultiplicationandtomultiplyafractionor
wholenumberbyafraction.
5.NF.6Solverealworldproblemsinvolvingmultiplicationoffractionsandmixednumbers,e.g.,by
usingvisualfractionmodelsorequationstorepresenttheproblem.
StatisticsandProbability
Investigatechanceprocessesanddevelop,use,andevaluateprobabilitymodels.
7.SP.8Findprobabilitiesofcompoundeventsusingorganizedlists,tables,treediagrams,and
simulation.
7.SP.8bRepresentsamplespacesforcompoundeventsusingmethodssuchasorganizedlists,tables
andtreediagrams.Foraneventdescribedineverydaylanguage,identifytheoutcomesinthesample
space,whichcomposetheevent.
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.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.
© 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
PartyTime
TaskDescription–LevelC
Thistaskchallengesstudentstouselogicandlogicstrategiessuchastables,charts,andmatricesto
solveaproblemwithaseriesofcluesaboutguests,theircostumes,andtheirarrivaltimes.Students
needtomakesenseofmultipleconstraintstofindasolutionandcheckthesolutionagainstallthe
constraints.
CommonCoreStateStandardsMath‐ContentStandards
StatisticsandProbability
Investigatechanceprocessesanddevelop,use,andevaluateprobabilitymodels.
7.SP.8Findprobabilitiesofcompoundeventsusingorganizedlists,tables,treediagrams,and
simulation.
7.SP.8bRepresentsamplespacesforcompoundeventsusingmethodssuchasorganizedlists,tables
andtreediagrams.Foraneventdescribedineverydaylanguage,identifytheoutcomesinthesample
space,whichcomposetheevent.
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.4Modelwithmathematics.
Mathematicallyproficientstudentscanapplythemathematicstheyknowtosolveproblemsarising
ineverydaylife,society,andtheworkplace.Inearlygradesthismightbeassimpleaswritingan
additionequationtodescribeasituation.Inmiddlegrades,astudentmightapplyproportional
reasoningtoplanaschooleventoranalyzeaprobleminthecommunity.Byhighschool,astudent
mightusegeometrytosolveadesignproblemoruseafunctiontodescribehowonequantityof
interestdependsonanother.Mathematicallyproficientstudentswhocanapplywhattheyknoware
comfortablemakingassumptionsandapproximationstosimplifyacomplicatedsituation,realizing
thatthesemayneedrevisionlater.Theyareabletoidentifyimportantquantitiesinapractical
situationandmaptheirrelationshipsusingsuchtoolsasdiagrams,two‐waytables,graphs,
flowcharts,andformulas.Theycananalyzethoserelationshipsmathematicallytodrawconclusions.
Theyroutinelyinterprettheirmathematicalresultsinthecontextofthesituationandreflecton
whethertheresultsmakesense,possiblyimprovingthemodelifithasnotserveditspurpose.
© 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
PartyTime
TaskDescription–LevelD
Thistaskchallengesstudentstoreasonaboutafairgameinthecontextofpickingtilesofdifferent
colorsfromabag.Studentsmustdefineasamplespacethatwillgiveanequalprobabilityofgetting
matchingornonmatchingcolorswhen2tilesaredrawnfromthebagwithoutreplacement.
CommonCoreStateStandardsMath‐ContentStandards
StatisticsandProbability
Investigatechanceprocessesanddevelop,useandevaluateprobabilitymodels.
7.SP.5Understandthattheprobabilityofachanceeventisanumberbetween0and1thatexpressesthe
likelihoodoftheeventoccurring.Largernumbersindicategreaterlikelihood.Aprobabilitynear0indicatesan
unlikelyevent,aprobabilityaround½indicatesaneventhatisneitherunlikelynorlikely,andaprobabilitynear
1indicatesalikelyevent.
7.SP.7Developaprobabilitymodelanduseittofindprobabilitiesofevents.Compareprobabilitiesfromamodel
ofobservedfrequencies;iftheagreementisnotgood,explainpossiblesourcesofthediscrepancy.
7.SP.8Findprobabilitiesofcompoundeventsusingorganizedlists,tables,treediagrams,andsimulation.
7.SP.8bRepresentsamplespacesforcompoundeventsusingmethodssuchasorganizedlists,tablesandtree
diagrams.Foraneventdescribedineverydaylanguage,identifytheoutcomesinthesamplespace,which
composetheevent.
HighSchool–StatisticsandProbability–ConditionalProbabilityandtheRulesofProbability
Understandindependenceandconditionalprobabilityandusethemtointerpretdata.
S‐CP.1Describeeventsassubsetsofasamplespace(thesetofoutcomes)usingcharacteristicsoftheoutcomes,
orasunions,intersections,orcomplementsofotherevents(“or”,“and”,“not”).
S‐CP.3UnderstandtheconditionalprobabilityofAgivenBasP(AandB)/P(B),andinterpretindependenceofA
andBassayingthattheconditionalprobabilityofAgivenBisthesameastheprobabilityofA,andthe
conditionalprobabilityofBgivenAisthesameastheprobabilityofB.
S‐CP.4Constructandinterprettwo‐wayfrequencytablesofdatawhentwocategoriesareassociatedwitheach
objectbeingclassified.Usethetwo‐waytableasasamplespacetodecideifeventsareindependentandto
approximateconditionalprobabilities.
Usetherulesofprobabilitytocomputeprobabilitiesofcompoundeventsinauniformprobability
model.
S‐CP.8applythegeneralMultiplicationRuleinauniformprobabilitymodel,P(AandB)=
P(A)P(B/A)=P(B)P(A/B),andinterprettheanswerintermsofthemodel.
CommonCoreStateStandardsMath– StandardsofMathematicalPractice
MP.1Makesenseofproblemsandpersevereinsolvingthem.
Mathematicallyproficientstudentsstartbyexplainingtothemselvesthemeaningofaproblemandlookingforentrypointsto
itssolution.Theyanalyzegivens,constraints,relationships,andgoals.Theymakeconjecturesabouttheformandmeaningof
thesolutionandplanasolutionpathwayratherthansimplyjumpingintoasolutionattempt.Theyconsideranalogous
problems,andtryspecialcasesandsimplerformsoftheoriginalprobleminordertogaininsightintoitssolution.They
monitorandevaluatetheirprogressandchangecourseifnecessary.Olderstudentsmight,dependingonthecontextofthe
problem,transformalgebraicexpressionsorchangetheviewingwindowontheirgraphingcalculatortogettheinformation
theyneed.Mathematicallyproficientstudentscanexplaincorrespondencesbetweenequations,verbaldescriptions,tables,
andgraphsordrawdiagramsofimportantfeaturesandrelationships,graphdata,andsearchforregularityortrends.Younger
studentsmightrelyonusingconcreteobjectsorpicturestohelpconceptualizeandsolveaproblem.Mathematicallyproficient
studentschecktheiranswerstoproblemsusingadifferentmethod,andtheycontinuallyaskthemselves,“Doesthismake
sense?”Theycanunderstandtheapproachesofotherstosolvingcomplexproblemsandidentifycorrespondencesbetween
differentapproaches.
MP.4Modelwithmathematics.
Mathematicallyproficientstudentscanapplythemathematicstheyknowtosolveproblemsarisingineverydaylife,society,
andtheworkplace.Inearlygradesthismightbeassimpleaswritinganadditionequationtodescribeasituation.Inmiddle
grades,astudentmightapplyproportionalreasoningtoplanaschooleventoranalyzeaprobleminthecommunity.Byhigh
school,astudentmightusegeometrytosolveadesignproblemoruseafunctiontodescribehowonequantityofinterest
dependsonanother.Mathematicallyproficientstudentswhocanapplywhattheyknowarecomfortablemakingassumptions
andapproximationstosimplifyacomplicatedsituation,realizingthatthesemayneedrevisionlater.Theyareabletoidentify
importantquantitiesinapracticalsituationandmaptheirrelationshipsusingsuchtoolsasdiagrams,two‐waytables,graphs,
flowcharts,andformulas.Theycananalyzethoserelationshipsmathematicallytodrawconclusions.Theyroutinelyinterpret
theirmathematicalresultsinthecontextofthesituationandreflectonwhethertheresultsmakesense,possiblyimproving
themodelifithasnotserveditspurpose.
© 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
PartyTime
TaskDescription–LevelE
Thistaskchallengesstudentstosolve alogicproblemaboutshakinghandswitheachnewsetof
arrivingguests.Therearemultipleconstraints,suchastherequirementthateachpersonmusthavea
differentnumberoftotalhandshakesattheend.Studentsmustdefendtheirsolutionandexplain
howtheysolvedthepuzzle.
CommonCoreStateStandardsMath‐ContentStandards
StatisticsandProbability
Investigatechanceprocessesanddevelop,use,andevaluateprobabilitymodels.
7.SP.8Findprobabilitiesofcompoundeventsusingorganizedlists,tables,treediagrams,and
simulation.
7.SP.8bRepresentsamplespacesforcompoundeventsusingmethodssuchasorganizedlists,tables
andtreediagrams.Foraneventdescribedineverydaylanguage,identifytheoutcomesinthesample
space,whichcomposetheevent.
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.4Modelwithmathematics.
Mathematicallyproficientstudentscanapplythemathematicstheyknowtosolveproblemsarising
ineverydaylife,society,andtheworkplace.Inearlygradesthismightbeassimpleaswritingan
additionequationtodescribeasituation.Inmiddlegrades,astudentmightapplyproportional
reasoningtoplanaschooleventoranalyzeaprobleminthecommunity.Byhighschool,astudent
mightusegeometrytosolveadesignproblemoruseafunctiontodescribehowonequantityof
interestdependsonanother.Mathematicallyproficientstudentswhocanapplywhattheyknoware
comfortablemakingassumptionsandapproximationstosimplifyacomplicatedsituation,realizing
thatthesemayneedrevisionlater.Theyareabletoidentifyimportantquantitiesinapractical
situationandmaptheirrelationshipsusingsuchtoolsasdiagrams,two‐waytables,graphs,
flowcharts,andformulas.Theycananalyzethoserelationshipsmathematicallytodrawconclusions.
Theyroutinelyinterprettheirmathematicalresultsinthecontextofthesituationandreflecton
whethertheresultsmakesense,possiblyimprovingthemodelifithasnotserveditspurpose.
© 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
PartyTime
TaskDescription–PrimaryLevel
Thistaskchallengesstudentstouselogicandcountingprinciplestofindthenumberofguestsata
partyifguestsinviteguestsatagivenratio.Studentsusepeopleandobjectstohelpmakethe
mathematicsaccessible.
CommonCoreStateStandardsMath‐ContentStandards
CountingandCardinality
Knownumbernamesandthecountsequence.
K.CC.1Countto100byonesandbytens.
Counttotellthenumberofobjects.
K.CC.4Understandtherelationshipbetweennumbersandquantities;connectcountingtocardinality.
K.CC.5Counttoanswer“howmany?”questionsaboutasmanyas20thingsarrangedinaline,a
rectangulararray,oracircle,orasmanyas10thingsinascatteredconfiguration;givenanumber
from1‐20countoutthatmanyobjects.
OperationsandAlgebraicThinking
Understandadditionasputtingtogetherandaddingto,andunderstandsubtractionastaking
apartandtakingfrom.
K.OA.1Representadditionandsubtractionwithobjects,fingers,mentalimages,drawings,sounds,
actingoutsituationsverbalexplanations,expressionsorequations.
Representandsolveproblemsinvolvingadditionandsubtraction.
1.OA.2Solvewordproblemsthatcallforadditionofthreewholenumbers,whosesumislessthanor
equalto20,e.g.,byusingobjects,drawings,andequationswithasymbolfortheunknownnumberto
representtheproblem.
2.OA.1Useadditionandsubtractionwithin100tosolveone‐andtwo‐stepproblemsinvolving
situationsofaddingto,takingfrom,puttingtogether,takingapart,andcomparing,withunknownsin
allpositions,e.g.byusingdrawingsandequationswithasymbolfortheunknownnumberto
representtheproblem.
CommonCoreStateStandardsMath–StandardsofMathematicalPractice
MP.1Makesenseofproblemsandpersevereinsolvingthem.
Mathematicallyproficientstudentsstartbyexplainingtothemselvesthemeaningofaproblemandlookingforentrypointsto
itssolution.Theyanalyzegivens,constraints,relationships,andgoals.Theymakeconjecturesabouttheformandmeaningof
thesolutionandplanasolutionpathwayratherthansimplyjumpingintoasolutionattempt.Theyconsideranalogous
problems,andtryspecialcasesandsimplerformsoftheoriginalprobleminordertogaininsightintoitssolution.They
monitorandevaluatetheirprogressandchangecourseifnecessary.Olderstudentsmight,dependingonthecontextofthe
problem,transformalgebraicexpressionsorchangetheviewingwindowontheirgraphingcalculatortogettheinformation
theyneed.Mathematicallyproficientstudentscanexplaincorrespondencesbetweenequations,verbaldescriptions,tables,
andgraphsordrawdiagramsofimportantfeaturesandrelationships,graphdata,andsearchforregularityortrends.Younger
studentsmightrelyonusingconcreteobjectsorpicturestohelpconceptualizeandsolveaproblem.Mathematicallyproficient
studentschecktheiranswerstoproblemsusingadifferentmethod,andtheycontinuallyaskthemselves,“Doesthismake
sense?”Theycanunderstandtheapproachesofotherstosolvingcomplexproblemsandidentifycorrespondencesbetween
differentapproaches.
MP.4Modelwithmathematics.
Mathematicallyproficientstudentscanapplythemathematicstheyknowtosolveproblemsarisingineverydaylife,society,
andtheworkplace.Inearlygradesthismightbeassimpleaswritinganadditionequationtodescribeasituation.Inmiddle
grades,astudentmightapplyproportionalreasoningtoplanaschooleventoranalyzeaprobleminthecommunity.Byhigh
school,astudentmightusegeometrytosolveadesignproblemoruseafunctiontodescribehowonequantityofinterest
dependsonanother.Mathematicallyproficientstudentswhocanapplywhattheyknowarecomfortablemakingassumptions
andapproximationstosimplifyacomplicatedsituation,realizingthatthesemayneedrevisionlater.Theyareabletoidentify
importantquantitiesinapracticalsituationandmaptheirrelationshipsusingsuchtoolsasdiagrams,two‐waytables,graphs,
flowcharts,andformulas.Theycananalyzethoserelationshipsmathematicallytodrawconclusions.Theyroutinelyinterpret
theirmathematicalresultsinthecontextofthesituationandreflectonwhethertheresultsmakesense,possiblyimproving
themodelifithasnotserveditspurpose.
© Noyce Foundation 2014.
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 Unported
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