ProblemoftheMonth:Courtney’sCollection
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.ThePrimaryVersion
LevelAisdesignedtobeaccessibletoallstudentsandespeciallythekeychallenge
forgradesK–1.LevelAwillbechallengingformostsecondandthirdgraders.
LevelBmaybethelimitofwherefourthandfifth‐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
IntheProblemoftheMonthCourtney’sCollection,studentsusenumbertheory,
numberoperations,organizedlists,andcountingmethodstosolveproblems.The
mathematicaltopicsthatunderliethisPOMareknowledgeofnumbersense,
numberproperties,comparisonsubtraction,division,factorsanddivisibility,
countingprinciples,systematiccharting,andgeneralizations.Themathematicsthat
includescountingprinciplesandsystematicchartingisoftenreferredtoasdiscrete
mathematics.
InthefirstlevelofthePOM,studentsarepresentedwithaproblemthatinvolves
summingthreevalues.Theirtaskinvolvesasituationwheretheyaretoselectthree
coinsfromalargesetofpennies,nickels,anddimes.Thestudentsareaskedto
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ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
selectthreecoinsanddeterminehowmuchmoneytheyhavealtogether.InLevelB,
studentsstarttoexaminehowmanydifferentwaystheycanselectthreecoinsand
thesumofeachsetofthreecoins.InLevelC,studentsconsideracollectionof5cent,
6cent,and7centstamps.Theyareaskedtodetermine,whichpostageamountsthey
couldmakeandwhichpostageamountstheycouldn’tmakeiftheyhadanunlimited
numberofeachstamp.InLevelD,thestudentsexaminethreedifferentsetsof
stamps.Theyareaskedtodeterminewhetherthesetsproduceafinitesetof
impossiblevaluesoraninfinitesetofimpossiblevalues.Forthestampswithafinite
amountofimpossiblevalues,thestudentsareaskedtofindthelargestimpossible
value.InLevelE,studentsareaskedtogeneralizetheirfindingsfromLevelD.
Studentsareaskedtopredictandjustifywhetherasetofstamps(cardinality3)has
afinitesetofimpossiblevaluesoraninfinitesetofimpossiblevalues.Forthe
stampswithafiniteamountofimpossiblevalues,thestudentsareaskedtodescribe
aprocessforfindingthelargestimpossiblevalue.
©NoyceFoundation2014.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
Problem of the Month
Courtney’s
Collection
Level A
Courtney has a bank of pennies, nickels and dimes. Courtney pulled out three coins from
the bank. Name the coins she picked. How much money does she have?
Show how you figured it out.
Show a different way that Courtney could pick three coins. How much does Courtney
have now? Show how you figured it out.
Problem of the Month
Courtney’s Collection
Page 1
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License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
Level B
How many different ways can Courtney pick three coins out of her bank? Show all the
ways you can find.
What are the different amounts of money Courtney would have by picking any three
coins?
How do you know you found all the possible ways?
Explain your answer.
Problem of the Month
Courtney’s Collection
Page 2
(c) 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 C
Courtney visited her grandmother. Her grandmother used to collect stamps. She had a
shoebox full of 5 cent, 6 cent and 7 cent stamps. Courtney thought, “I could mail a lot of
different-sized letters and postcards with these stamps.” She tried to figure out the
different amount of postage she could make with a combination of those stamps. What
totals could she make and what totals are impossible?
Explain how you found your solution. How do you know your solution is correct?
Problem of the Month
Courtney’s Collection
Page 3
(c) 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
Courtney’s grandmother said, “Your grandpa has different shoeboxes with other stamps.
His shoebox has just 4 cent, 6 cent and 8 cent stamps. Which totals can you make with
these stamps?”
Courtney said, “I wonder why there is a difference between Grandma’s and Grandpa’s
shoeboxes. I like finding the largest total I can’t make, but Grandpa’s box has an
unlimited amount of totals I can’t make. I wonder why it works for some and not for
others.” She continues, “I am going to investigate which three stamp amounts are like
Grandma’s shoebox and which three stamp amounts are like Grandpa’s shoebox. I am
going to compare three different sets. I will try 6 cents, 7 cents and 8 cents. Then I will
try 6 cents, 9 cents and 12 cents. Finally I will try 6 cents, 8 cents and 9 cents.”
Explain how the three different sets are like either Grandma’s shoebox or Grandpa’s
shoebox.
Problem of the Month
Courtney’s Collection
Page 4
(c) 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
Determine a method for predicting whether a given set of any three stamps would have a
largest impossible total, or infinite impossible totals. Justify your method using
mathematical reasoning.
For those sets of three stamps that have a largest impossible total, make a conjecture
about how to find that total.
Problem of the Month
Courtney’s Collection
Page 5
(c) 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
Courtney’s
Collection
Materials: A set of coins (pennies, nickels, dimes - several of each kind) in
a container for each pair of students, paper and pencil, crayons, or markers
Discussion on the rug: “What is a piggy bank?” Teacher asks the
students questions about where there might be a collection of coins. If a
bank is not familiar, then maybe introduce a jar of coins. Teacher holds a
collection of coins in a container. “Suppose I pick three coins from my
bank/jar. Name the coins I might pick.” The teacher encourages the
students to find answers for different selections of three coins.
In small groups: Each group has a set of coins in a container. Teacher asks
the following questions. Go on to the next question when students have
success.
1. “Name the three coins you picked out of your bank/jar. How
much money do you have in all? Show how you know.”
2. “Now name three more coins you can pick from your bank/jar.
How much money do you have in all?”
At the end of the investigation, have students either draw a picture or dictate
to you to represent their solutions.
Problem of the Month
Courtney’s Collection
Page 6
(c) 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
Courtney’sCollection
TaskDescription–LevelA
Thistaskchallengesstudentstoselectthreecoinsandthendeterminehowmuchmoneythethree
coinsarealltogether.Thesetofcoinsincludespennies,nickels,anddimes.Thestudentsareasked
toshowtwodifferentcombinationsofcoinsandtheirrespectivesums.
CommonCoreStateStandardsMath‐ContentStandards
CountingandCardinality
Knownumbernamesandthecountsequence.
K.CC.3Writenumbersfrom0to20.…
OperationsandAlgebraicThinking
Understandadditionasputtingtogetherandaddingto,andunderstandsubtractionastakingapartand
takingfrom.
K.OA.1Representadditionandsubtractionwithobjects,fingers,mentalimages,drawings,sounds(e.g.,claps),
actingoutsituations,verbalexplanations,expressions,orequations.
Representandsolveproblemsinvolvingadditionandsubtraction.
1.OA.1Useadditionandsubtractionwithin20tosolvewordproblemsinvolvingsituationsofaddingto,
takingfrom,puttingtogether,takingapart,andcomparing,withunknownsinallpositions,e.g.,byusingobjects,
drawings,andequationswithasymbolfortheunknownnumbertorepresenttheproblem.
1.OA.2Solvewordproblemsthatcallforadditionofthreewholenumberswhosesumislessthanorequalto20,
e.g.,byusingobjects,drawings,andequationswithasymbolfortheunknownnumbertorepresenttheproblem.
2.OA.1Useadditionandsubtractionwithin100tosolveone‐andtwo‐stepwordproblemsinvolvingsituations
ofaddingto,takingfrom,puttingtogether,takingapart,andcomparing,withunknownsinallpositions,e.g.,by
usingdrawingsandequationswithasymbolfortheunknownnumbertorepresenttheproblem.
MeasurementandData
Workwithtimeandmoney.
2.MD.8Solvewordproblemsinvolvingdollarbills,quarters,dimes,nickels,andpennies,using$and¢symbols
appropriately.Example:Ifyouhave2dimesand3pennies,howmanycentsdoyouhave?
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.3 Constructviableargumentsandcritiquethereasoningofothers.
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,eventhoughtheyarenotgeneralizedormadeformaluntillatergrades.Later,studentslearnto
determinedomainstowhichanargumentapplies.Studentsatallgradescanlistenorreadtheargumentsof
others,decidewhethertheymakesense,andaskusefulquestionstoclarifyorimprovethearguments.
©NoyceFoundation2014.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
ProblemoftheMonth
Courtney’sCollection
TaskDescription– LevelB
Thistaskchallengesstudentstoexaminehowmanydifferentwaysthreecoinscanbeselectedfroma
bankcontainingpennies,nickelsanddimes,andtodeterminethesumofeachsetofthreecoins.The
studentsareaskedtoexplainhowtheyknowtheyhavefoundallthepossiblewaysorcombinations.
CommonCoreStateStandardsMath‐ ContentStandards
CountingandCardinality
Knownumbernamesandthecountsequence.
K.CC.3Writenumbersfrom0to20.…
OperationsandAlgebraicThinking
Understandadditionasputtingtogetherandaddingto,andunderstandsubtractionastakingapartand
takingfrom.
K.OA.1Representadditionandsubtractionwithobjects,fingers,mentalimages,drawings,sounds(e.g.,claps),
actingoutsituations,verbalexplanations,expressions,orequations.
Representandsolveproblemsinvolvingadditionandsubtraction.
1.OA.1Useadditionandsubtractionwithin20tosolvewordproblemsinvolvingsituationsofaddingto,
takingfrom,puttingtogether,takingapart,andcomparing,withunknownsinallpositions,e.g.,byusingobjects,
drawings,andequationswithasymbolfortheunknownnumbertorepresenttheproblem.
1.OA.2Solvewordproblemsthatcallforadditionofthreewholenumberswhosesumislessthanorequalto20,
e.g.,byusingobjects,drawings,andequationswithasymbolfortheunknownnumbertorepresenttheproblem.
2.OA.1Useadditionandsubtractionwithin100tosolveone‐andtwo‐stepwordproblemsinvolvingsituations
ofaddingto,takingfrom,puttingtogether,takingapart,andcomparing,withunknownsinallpositions,e.g.,by
usingdrawingsandequationswithasymbolfortheunknownnumbertorepresenttheproblem.
MeasurementandData
Workwithtimeandmoney.
2.MD.8Solvewordproblemsinvolvingdollarbills,quarters,dimes,nickels,andpennies,using$and¢symbols
appropriately.Example:Ifyouhave2dimesand3pennies,howmanycentsdoyouhave?
Solveproblemsinvolvingmeasurementandconversionofmeasurementsfromalargerunittoasmaller
unit.
4.MD.2Usethefouroperationstosolvewordproblemsinvolvingdistances,intervalsoftime,liquidvolumes,
massesofobjects,andmoney,…
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.3 Constructviableargumentsandcritiquethereasoningofothers.
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,eventhoughtheyarenotgeneralizedormadeformaluntillatergrades.Later,studentslearnto
determinedomainstowhichanargumentapplies.Studentsatallgradescanlistenorreadtheargumentsofothers,decide
whethertheymakesense,andaskusefulquestionstoclarifyorimprovethearguments.
©NoyceFoundation2014.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
ProblemoftheMonth
Courtney’sCollection
TaskDescription– LevelC
Thistaskchallengesstudentstodetermine,givenanunlimitednumberof5¢,6¢,and7¢stamps,whichpostage
amountscanbemadeandwhichpostageamountscan’tbemade.Thestudentisaskedtoexplaintheirsolutions
andtojustifyhowtheyknowtheyarecorrect.
CommonCoreStateStandardsMath‐ ContentStandards
CountingandCardinality
Knownumbernamesandthecountsequence.
K.CC.3Writenumbersfrom0to20.…
OperationsandAlgebraicThinking
Understandadditionasputtingtogetherandaddingto,andunderstandsubtractionastakingapartand
takingfrom.
K.OA.1Representadditionandsubtractionwithobjects,fingers,mentalimages,drawings,sounds(e.g.,claps),
actingoutsituations,verbalexplanations,expressions,orequations.
Representandsolveproblemsinvolvingadditionandsubtraction.
1.OA.1Useadditionandsubtractionwithin20tosolvewordproblemsinvolvingsituationsofaddingto,
takingfrom,puttingtogether,takingapart,andcomparing,withunknownsinallpositions,e.g.,byusingobjects,
drawings,andequationswithasymbolfortheunknownnumbertorepresenttheproblem.
1.OA.2Solvewordproblemsthatcallforadditionofthreewholenumberswhosesumislessthanorequalto20,
e.g.,byusingobjects,drawings,andequationswithasymbolfortheunknownnumbertorepresenttheproblem.
2.OA.1Useadditionandsubtractionwithin100tosolveone‐andtwo‐stepwordproblemsinvolvingsituations
ofaddingto,takingfrom,puttingtogether,takingapart,andcomparing,withunknownsinallpositions,e.g.,by
usingdrawingsandequationswithasymbolfortheunknownnumbertorepresenttheproblem.
MeasurementandData
Workwithtimeandmoney.
2.MD.8Solvewordproblemsinvolvingdollarbills,quarters,dimes,nickels,andpennies,using$and¢symbols
appropriately.Example:Ifyouhave2dimesand3pennies,howmanycentsdoyouhave?
Solveproblemsinvolvingmeasurementandconversionofmeasurementsfromalargerunittoasmaller
unit.
4.MD.2Usethefouroperationstosolvewordproblemsinvolvingdistances,intervalsoftime,liquidvolumes,
massesofobjects,andmoney,…
ExpressionsandEquations
Solvereal‐lifeandmathematicalproblemsusingnumericalandalgebraicexpressionsandequations.
7.EE.4Usevariablestorepresentquantitiesinareal‐worldormathematicalproblem,andconstructsimple
equationsandinequalitiestosolveproblemsbyreasoningaboutthequantities.
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.3 Constructviableargumentsandcritiquethereasoningofothers.
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,eventhoughtheyarenotgeneralizedormadeformaluntillatergrades.Later,studentslearnto
determinedomainstowhichanargumentapplies.Studentsatallgradescanlistenorreadtheargumentsofothers,decide
whethertheymakesense,andaskusefulquestionstoclarifyorimprovethearguments.
©NoyceFoundation2014.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
ProblemoftheMonth
Courtney’sCollection
TaskDescription – LevelD
Thistaskchallengesstudentstoexaminethreedifferentsetsofstamps ‐ 4¢,6¢,and8¢stamps.Thestudents are
askedtodeterminewhetherthesetsproduceafinitesetofimpossiblevaluesoraninfinitesetofimpossible
values.Forthestampswithafiniteamountofimpossiblevalues,thestudentisaskedtofindthelargest
impossiblevalue.
CommonCoreStateStandardsMath‐ ContentStandards
CountingandCardinality
Knownumbernamesandthecountsequence.
K.CC.3Writenumbersfrom0to20.…
OperationsandAlgebraicThinking
Understandadditionasputtingtogetherandaddingto,andunderstandsubtractionastakingapartandtakingfrom.
K.OA.1Representadditionandsubtractionwithobjects,fingers,mentalimages,drawings,sounds(e.g.,claps),actingout
situations,verbalexplanations,expressions,orequations.
Representandsolveproblemsinvolvingadditionandsubtraction.
1.OA.1Useadditionandsubtractionwithin20tosolvewordproblemsinvolvingsituationsofaddingto,takingfrom,
puttingtogether,takingapart,andcomparing,withunknownsinallpositions,e.g.,byusingobjects,drawings,andequations
withasymbolfortheunknownnumbertorepresenttheproblem.
1.OA.2Solvewordproblemsthatcallforadditionofthreewholenumberswhosesumislessthanorequalto20,e.g.,byusing
objects,drawings,andequationswithasymbolfortheunknownnumbertorepresenttheproblem.
2.OA.1Useadditionandsubtractionwithin100tosolveone‐andtwo‐stepwordproblemsinvolvingsituationsofaddingto,
takingfrom,puttingtogether,takingapart,andcomparing,withunknownsinallpositions,e.g.,byusingdrawingsand
equationswithasymbolfortheunknownnumbertorepresenttheproblem.
MeasurementandData
Workwithtimeandmoney.
2.MD.8Solvewordproblemsinvolvingdollarbills,quarters,dimes,nickels,andpennies,using$and¢symbolsappropriately.
Example:Ifyouhave2dimesand3pennies,howmanycentsdoyouhave?
Solveproblemsinvolvingmeasurementandconversionofmeasurementsfromalargerunittoasmallerunit.
4.MD.2Usethefouroperationstosolvewordproblemsinvolvingdistances,intervalsoftime,liquidvolumes,massesof
objects,andmoney,…
ExpressionsandEquations
Solvereal‐lifeandmathematicalproblemsusingnumericalandalgebraicexpressionsandequations.
7.EE.4Usevariablestorepresentquantitiesinareal‐worldormathematicalproblem,andconstructsimpleequationsand
inequalitiestosolveproblemsbyreasoningaboutthequantities.
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.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,eventhoughtheyarenotgeneralizedormadeformaluntillatergrades.Later,studentslearnto
determinedomainstowhichanargumentapplies.Studentsatallgradescanlistenorreadtheargumentsofothers,decide
whethertheymakesense,andaskusefulquestionstoclarifyorimprovethearguments.
©NoyceFoundation2014.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
ProblemoftheMonth
Courtney’sCollection
TaskDescription– LevelE
Thistaskchallengesstudentstoselectthreecoinsandthendeterminehowmuchmoneythethreecoinsareall
together.Thesetofcoinsincludespennies,nickels,anddimes.Studentsareaskedtoshowtwodifferent
combinationsofcoinsandtheirrespectivesums.
CommonCoreStateStandardsMath‐ ContentStandards
CountingandCardinality
Knownumbernamesandthecountsequence.
K.CC.3Writenumbersfrom0to20.…
OperationsandAlgebraicThinking
Understandadditionasputtingtogetherandaddingto,andunderstandsubtractionastakingapartandtakingfrom.
K.OA.1Representadditionandsubtractionwithobjects,fingers,mentalimages,drawings,sounds(e.g.,claps),actingout
situations,verbalexplanations,expressions,orequations.
Representandsolveproblemsinvolvingadditionandsubtraction.
1.OA.1Useadditionandsubtractionwithin20tosolvewordproblemsinvolvingsituationsofaddingto,takingfrom,
puttingtogether,takingapart,andcomparing,withunknownsinallpositions,e.g.,byusingobjects,drawings,andequations
withasymbolfortheunknownnumbertorepresenttheproblem.
1.OA.2Solvewordproblemsthatcallforadditionofthreewholenumberswhosesumislessthanorequalto20,e.g.,byusing
objects,drawings,andequationswithasymbolfortheunknownnumbertorepresenttheproblem.
2.OA.1Useadditionandsubtractionwithin100tosolveone‐andtwo‐stepwordproblemsinvolvingsituationsofaddingto,
takingfrom,puttingtogether,takingapart,andcomparing,withunknownsinallpositions,e.g.,byusingdrawingsand
equationswithasymbolfortheunknownnumbertorepresenttheproblem.
Gainfamiliaritywithfactorsandmultiples.
4.OA.4Findallfactorpairsforawholenumberintherange1‐100.Recognizethatawholenumberisamultipleofeachofits
factors.Determinewhetheragivenwholenumberintherange1‐100isamultipleofagivenone‐digitnumber.
MeasurementandData
Workwithtimeandmoney.
2.MD.8Solvewordproblemsinvolvingdollarbills,quarters,dimes,nickels,andpennies,using$and¢symbolsappropriately.
Example:Ifyouhave2dimesand3pennies,howmanycentsdoyouhave?
Solveproblemsinvolvingmeasurementandconversionofmeasurementsfromalargerunittoasmallerunit.
4.MD.2Usethefouroperationstosolvewordproblemsinvolvingdistances,intervalsoftime,liquidvolumes,massesof
objects,andmoney,…
ExpressionsandEquations
Solvereal‐lifeandmathematicalproblemsusingnumericalandalgebraicexpressionsandequations.
7.EE.4Usevariablestorepresentquantitiesinareal‐worldormathematicalproblem,andconstructsimpleequationsand
inequalitiestosolveproblemsbyreasoningaboutthequantities.
HighSchool–Algebra‐CreatingEquations
A‐CED.2Createequationsintwoormorevariablestorepresentrelationshipsbetweenquantities…
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.3 Constructviableargumentsandcritiquethereasoningofothers.
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,eventhoughtheyarenotgeneralizedormadeformaluntillatergrades.Later,studentslearnto
determinedomainstowhichanargumentapplies.Studentsatallgradescanlistenorreadtheargumentsofothers,decide
whethertheymakesense,andaskusefulquestionstoclarifyorimprovethearguments.
©NoyceFoundation2014.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
ProblemoftheMonth
Courtney’sCollection
TaskDescription–PrimaryLevel
Thistaskchallengesstudentstoselectthreecoinsandthendeterminehowmuchmoneythethree
coinsarealltogether.Thesetofcoinsincludespennies,nickels,anddimes.Thestudentsarethen
askedtochooseasecondsetofthreecoinsanddeterminethetotalforbothsets.
CommonCoreStateStandardsMath‐ContentStandards
CountingandCardinality
Knownumbernamesandthecountsequence.
K.CC.3Writenumbersfrom0to20.…
OperationsandAlgebraicThinking
Understandadditionasputtingtogetherandaddingto,andunderstandsubtractionastaking
apartandtakingfrom.
K.OA.1Representadditionandsubtractionwithobjects,fingers,mentalimages,drawings,sounds
(e.g.,claps),actingoutsituations,verbalexplanations,expressions,orequations.
CommonCoreStateStandardsMath–StandardsofMathematicalPractice
MP.1Makesenseofproblemsand persevereinsolvingthem.
Mathematicallyproficientstudentsstartbyexplainingtothemselvesthemeaningofaproblemand
lookingforentrypointstoitssolution.Theyanalyzegivens,constraints,relationships,andgoals.
Theymakeconjecturesabouttheformandmeaningofthesolutionandplanasolutionpathway
ratherthansimplyjumpingintoasolutionattempt.Theyconsideranalogousproblems,andtry
specialcasesandsimplerformsoftheoriginalprobleminordertogaininsightintoitssolution.They
monitorandevaluatetheirprogressandchangecourseifnecessary.Olderstudentsmight,depending
onthecontextoftheproblem,transformalgebraicexpressionsorchangetheviewingwindowon
theirgraphingcalculatortogettheinformationtheyneed.Mathematicallyproficientstudentscan
explaincorrespondencesbetweenequations,verbaldescriptions,tables,andgraphsordraw
diagramsofimportantfeaturesandrelationships,graphdata,andsearchforregularityortrends.
Youngerstudentsmightrelyonusingconcreteobjectsorpicturestohelpconceptualizeandsolvea
problem.Mathematicallyproficientstudentschecktheiranswerstoproblemsusingadifferent
method,andtheycontinuallyaskthemselves,“Doesthismakesense?”Theycanunderstandthe
approachesofotherstosolvingcomplexproblemsandidentifycorrespondencesbetweendifferent
approaches.
MP.3Constructviableargumentsandcritiquethereasoningofothers.
Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,and
previouslyestablishedresultsinconstructingarguments.Theymakeconjecturesandbuildalogical
progressionofstatementstoexplorethetruthoftheirconjectures.Theyareabletoanalyze
situationsbybreakingthemintocases,andcanrecognizeandusecounterexamples.Theyjustify
theirconclusions,communicatethemtoothers,andrespondtotheargumentsofothers.Theyreason
inductivelyaboutdata,makingplausibleargumentsthattakeintoaccountthecontextfromwhich
thedataarose.Mathematicallyproficientstudentsarealsoabletocomparetheeffectivenessoftwo
plausiblearguments,distinguishcorrectlogicorreasoningfromthatwhichisflawed,and—ifthereis
aflawinanargument—explainwhatitis.Elementarystudentscanconstructargumentsusing
concretereferentssuchasobjects,drawings,diagrams,andactions.Suchargumentscanmakesense
andbecorrect,eventhoughtheyarenotgeneralizedormadeformaluntillatergrades.Later,
studentslearntodeterminedomainstowhichanargumentapplies.Studentsatallgradescanlisten
orreadtheargumentsofothers,decidewhethertheymakesense,andaskusefulquestionstoclarify
orimprovethearguments.
©NoyceFoundation2014.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).