Problem of the Month Squirreling It Away

ProblemoftheMonth
SquirrelingItAway
TheProblemsoftheMonth(POM)areusedinavarietyofwaystopromoteproblem
solvingandtofosterthefirststandardofmathematicalpracticefromtheCommon
CoreStateStandards:“Makesenseofproblemsandpersevereinsolvingthem.”
ThePOMmaybeusedbyateachertopromoteproblemsolvingandtoaddressthe
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
IntheProblemoftheMonthSquirrelingItAway,studentsusenumberoperations,
organizedlists,andcountingmethodstosolveproblems.Themathematicaltopics
thatunderliethisPOMareknowledgeofnumbersense,comparisonsubtraction,
division,factorsanddivisibility,countingprinciples,systematiccharting,and
closed‐formequations.Themathematicsthatincludescountingprinciplesand
systemicchartingisoftenreferredtoasdiscretemathematics.
InthefirstlevelofthePOM,studentsarepresentedwithasituationthatinvolves
makingsenseoftotalsandcomparisondifferences.Thetaskinvolvesmakinga
ProblemoftheMonth SquirrelingitAway
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1
numberstoryaboutgivingacornstotwodifferent‐sizedgroupsofsquirrelsand
thendetermininghowmanywereleftoverfromtheoriginaltotalacorns.InLevel
B,studentsstarttoexaminehowtheacornscanbepartitionedintodifferentsets.
Thestudentsaretoldthatdifferentsquirrelscancarrydifferentamountsofacorns
ongiventrips.Studentsareaskedtofindthenumberoftripsittakesforeachtype
ofsquirreltocarrytheacorns.InLevelC,studentsareaskedhowmanywaysthree
differenttypesofsquirrelscancarryaway24acorns.InLevelD,thestudents
determinethenumberofways24acornscanbedividedbetweenthreespecific
squirrels.InLevelE,studentsareaskedtofindandjustifyaclosed‐formequation
thatwilldeterminethenumberofwaysthatnacornscanbedividedbetweenthree
squirrels.
ProblemoftheMonth SquirrelingitAway
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2
Problem of the Month
Squirreling It
Away
Level A:
Austin had a bag of 17 acorns. Eight squirrels came up to him. He gave each squirrel an
acorn. Then five more squirrels came up to him and he gave away one acorn to each of
them. How many more squirrels could he still feed?
Show how you figured it out?
How do you know you have the right answer?
Problem of the Month
Squirreling It Away
Page 1
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License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
Level B:
Austin likes to watch squirrels find and store acorns for the winter. Brown Squirrels can
carry two acorns at a time. Gray Squirrels can carry three acorns at a time and Black
Squirrels can carry five acorns at a time. There is a pile of 24 acorns.
How many trips would a Brown Squirrel need to make to store all of the acorns in the
pile?
How many trips would a Gray Squirrel need to make to store all of the acorns in the pile?
How many trips would a Black Squirrel need to make to store all of the acorns in the
pile?
If all three squirrels worked together to store the acorns how many trips would the
squirrels need to make to store all of the acorns?
Explain your solution.
Problem of the Month
Squirreling It Away
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:
Brown Squirrels can carry 2 acorns at a time. Gray Squirrels can
carry 3 acorns at a time. Black Squirrels can carry 5 acorns at a
time.
Suppose the three squirrels all wanted to store acorns for the winter. Depending on how
motivated each squirrel was, they would end up with different amounts. For instance
suppose the Brown Squirrel took 4 trips, the Gray Squirrel took 2 trips and the Black
Squirrel took 2 trips. The Brown Squirrel would end up with 8 acorns, the Gray Squirrel
would have 6 acorns and the Black Squirrel would have 10. Between them they took
every one of the 24 acorns.
How many different ways could the three Squirrels divide up the 24 acorns and not leave
any left over? Each Squirrel must carry his maximum load on each trip.
How do you know that you have found all of the ways?
Problem of the Month
Squirreling It Away
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:
The squirrels are rather smart. They realize that they can carry less than their maximum
loads. How many different ways could the squirrels divide up the 24 acorns?
Explain your solution.
Problem of the Month
Squirreling It Away
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:
Suppose there are a different number of acorns than 24. Determine a generalization for
finding how 3 squirrels can divide up any given number of acorns.
Explain your solutions.
Problem of the Month
Squirreling It Away
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
Squirreling It Away
Primary Version
Materials: A set of acorns or cubes (1-10) for each pair of students, paper
and pencil, crayons, or markers
Discussion on the rug: “Here are some acorns. What animal likes
to eat acorns?” Teacher continues to ask children to name animals who
like acorns. Teacher holds five acorns in her hand. “Suppose I had five
acorns and one squirrel came up to me and I gave it an acorn
How many would I have left?” The teacher encourages the students to
find answers for different amounts of acorns and to ask the students to
explain how they know.
In small groups: Each group has a set of acorns or cubes.
Teacher asks the following questions and goes on to the next question when
students have success.
1. “You have 10 acorns. Four squirrels come to you. You give
each squirrel one acorn. How many acorns did you give? How
many are left? Now two more squirrels come to you and you
give them each an acorn. How many did you give now? How
many are left? How many more squirrels can you feed?”
2. Select a set of numbers that is reasonable for your class. “You have __
acorns. __ squirrels come to you. You give the squirrel each
one acorn. Now __ more squirrels come to you and you give
them each an acorn. How many more squirrels can you feed?” At
the end of the investigation, have students either draw a picture or dictate to
you to represent their solutions.
Problem of the Month
Squirreling It Away
Page 6
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License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
ProblemoftheMonth
SquirrelingItAway
TaskDescription–LevelA
Thistaskchallengesastudenttoreasonaboutrepeatedsubtractioninawordproblemtofindthe
numberofacornsleftaftergivingsetamountstotwodifferentsetsofsquirrels.
CommonCoreStateStandardsMath‐ContentStandards
OperationsandAlgebraicThinking
Representandsolveproblemsinvolvingadditionandsubtraction.
2.OA.1Useadditionandsubtractionwithin100tosolveone‐andtwo‐stepwordproblemsinvolving
situationsofaddingto,takingfrom,puttingtogether,takingapart,andcomparing,withunknownsin
allpositions,e.g.byusingdrawingsandequationswithasymbolfortheunknownnumberto
representtheproblems.
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.2Reasonabstractlyandquantitatively.
Mathematicallyproficientstudentsmakesenseofquantitiesandtheirrelationshipsinproblem
situations.Theybringtwocomplementaryabilitiestobearonproblemsinvolvingquantitative
relationships:theabilitytodecontextualize–toabstractagivensituationandrepresentit
symbolicallyandmanipulatetherepresentingsymbolsasiftheyhavealifeoftheirown,without
necessarilyattendingtotheirreferents–andtheabilitytocontextualize,topauseasneededduring
themanipulationprocessinordertoprobeintothereferentsforthesymbolsinvolved.Quantitative
reasoningentailshabitsofcreatingacoherentrepresentationoftheproblemathand;considering
theunitsinvolved;attendingtothemeaningofquantities,notjusthowtocomputethem;and
knowingandflexiblyusingdifferentpropertiesofoperationsandobjects.
ProblemoftheMonth SquirrelingitAway
©NoyceFoundation2014.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
1
ProblemoftheMonth
SquirrelingItAway
TaskDescription–LevelB
Thistaskchallengesstudentstosolveproblemsinvolvingsubdividingawhole amountintosub‐sets.
Studentsneedtothinkaboutequalgroupsormaximum‐sizedgroupsandtointerpretremaindersin
context.
CommonCoreStateStandardsMath‐ContentStandards
OperationsandAlgebraicThinking
Representandsolveproblemsinvolvingmultiplicationanddivision.
3.OA.1.Interpretproductsofwholenumbers,e.g.interpret5x7asthetotalnumberofobjectsin5groupsof7
objectseach.
3.OA.2Interpretwholenumberquotientsofwholenumbers,e.g.,interpret568asthenumberofobjectsin
eachsharewhen56objectsarepartitionedequallyinto8shares,,orasanumberofobjectsineachsharewhen
56objectsarepartitionedintoequalsharesof8objectseach.
3.OA.3Usemultiplicationanddivisionwithin100tosolvewordproblemsinsituationsinvolvingequalgroups,
arrays,andmathematicalquantities,e.g.,byusingdrawingsandequationswithasymbolfortheunknown
numbertorepresenttheproblems.
Solveproblemsinvolvingthefouroperations,andidentifyandexplainpatternsinarithmetic.
3.OA.8Solvetwo‐stepwordproblemsusingthefouroperations.Representtheseproblemsusingequationswith
aletterstandingfortheunknownquantity.Assthereasonablenessofanswersusingmentalcomputationand
estimationstrategiesincludingrounding.
Usefouroperationswithwholenumberstosolveproblems.
4.OA.3Solvemultistepwordproblemsposedwithwholenumbersandhavingwhole‐numberanswersusingthe
fouroperations,includingproblemsinwhichremaindersmustbeinterpreted.Representtheseproblemsusing
equationswithaletterstandingfortheunknownquantity.Assessthereasonablenessofanswersusingmental
computationandestimationstrategiesincludingrounding.
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.2Reasonabstractlyandquantitatively.
Mathematicallyproficientstudentsmakesenseofquantitiesandtheirrelationshipsinproblemsituations.They
bringtwocomplementaryabilitiestobearonproblemsinvolvingquantitativerelationships:theabilityto
decontextualize–toabstractagivensituationandrepresentitsymbolicallyandmanipulatetherepresenting
symbolsasiftheyhavealifeoftheirown,withoutnecessarilyattendingtotheirreferents–andtheabilityto
contextualize,topauseasneededduringthemanipulationprocessinordertoprobeintothereferentsforthe
symbolsinvolved.Quantitativereasoningentailshabitsofcreatingacoherentrepresentationoftheproblemat
hand;consideringtheunitsinvolved;attendingtothemeaningofquantities,notjusthowtocomputethem;and
knowingandflexiblyusingdifferentpropertiesofoperationsandobjects.
ProblemoftheMonth SquirrelingitAway
©NoyceFoundation2014.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
2
ProblemoftheMonth
SquirrelingItAway
TaskDescription– LevelC
Thistaskchallengesastudenttousedivisionandequal‐sized groupstofindallthepossiblewaysthe3squirrels
cancarryawayexactly24acornsiftheytakethemaximumamounteverytrip.Studentsarechallengedto
organizetheirthinkingtodevelopaconvincingargumentabouthowtheyknowtheyhavefoundallthe
possibilities.
CommonCoreStateStandardsMath‐ ContentStandards
OperationsandAlgebraicThinking
Representandsolveproblemsinvolvingmultiplicationanddivision.
3.OA.1.Interpretproductsofwholenumbers,e.g.interpret5x7asthetotalnumberofobjectsin5groupsof7
objectseach.
3.OA.2Interpretwholenumberquotientsofwholenumbers,e.g.,interpret568asthenumberofobjectsin
eachsharewhen56objectsarepartitionedequallyinto8shares,,orasanumberofobjectsineachsharewhen
56objectsarepartitionedintoequalsharesof8objectseach.
3.OA.3Usemultiplicationanddivisionwithin100tosolvewordproblemsinsituationsinvolvingequalgroups,
arrays,andmathematicalquantities,e.g.,byusingdrawingsandequationswithasymbolfortheunknown
numbertorepresenttheproblems.
Solveproblemsinvolvingthefouroperations,andidentifyandexplainpatternsinarithmetic.
3.OA.8Solvetwo‐stepwordproblemsusingthefouroperations.Representtheseproblemsusingequationswith
aletterstandingfortheunknownquantity.Assthereasonablenessofanswersusingmentalcomputationand
estimationstrategiesincludingrounding.
Usefouroperationswithwholenumberstosolveproblems.
4.OA.3Solvemultistepwordproblemsposedwithwholenumbersandhavingwhole‐numberanswersusingthe
fouroperations,includingproblemsinwhichremaindersmustbeinterpreted.Representtheseproblemsusing
equationswithaletterstandingfortheunknownquantity.Assessthereasonablenessofanswersusingmental
computationandestimationstrategiesincludingrounding.
Gainfamiliaritywithfactorsandmultiples.
4.OA.4Findallfactorpairsforawholenumberintherange1‐100.Recognizethatawholenumberisamultiple
ofeachofthefactors.Determinewhetheragivenwholenumberintherange1‐100isamultipleofagivenone‐
digitnumber.Determinewhetheragivenwholenumberintherange1‐100isaprimeorcomposite.
StatisticsandProbability
Investigatechanceprocessesanddevelop,use,andevaluateprobabilitymodels.
7.SP.8findprobabilitiesofcompoundeventsusingorganizedlists,tables,treediagramsandsimulation.
CommonCoreStateStandardsMath– StandardsofMathematicalPractice
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.8Lookforandexpressregularityinrepeatedreasoning.
Mathematicallyproficientstudentsnoticeifcalculationsarerepeated,andlookbothforgeneralmethodsandfor
shortcuts.Upperelementarystudentsmightnoticewhendividing25by11thattheyarerepeatingthesame
calculationsoverandoveragain,andconcludetheyhavearepeatingdecimal.Bypayingattentiontothe
calculationofslopeastheyrepeatedlycheckwhetherpointsareonthelinethrough(1,2)withslope3,middle
schoolstudentsmightabstracttheequation(y‐2)/(x‐1)=3.Noticingtheregularityinthewaytermscancel
whenexpanding(x‐1)(x+1),(x‐1)(x2+x+1),and(x‐1)(x3+x2+x+1)mightleadthemtothegeneralformula
forthesumofageometricseries.Astheyworktosolveaproblem,mathematicallyproficientstudentsmaintain
oversightoftheprocess,whileattendingtothedetails.Theycontinuallyevaluatethereasonablenessoftheir
intermediateresults.
ProblemoftheMonth SquirrelingitAway
©NoyceFoundation2014.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
3
ProblemoftheMonth
SquirrelingItAway
TaskDescription–LevelD
Thistaskchallengesastudenttodeterminethenumberofways24acornscanbedividedbetween
threespecificsquirrels,iftheydon’thavetocarrythemaximumoneverytrip.Studentsmustdevelop
aconvincingargumenttoexplaintheirstrategyandhowtheyknowtheyhaveeverypossibility.
CommonCoreStateStandardsMath‐ContentStandards
OperationsandAlgebraicThinking
Representandsolveproblemsinvolvingmultiplicationanddivision.
3.OA.1Interpretproductsofwholenumbers,e.g.interpret5x7asthetotalnumberofobjectsin5groupsof7
objectseach.
3.OA.2Interpretwholenumberquotientsofwholenumbers,e.g.,interpret568asthenumberofobjectsin
eachsharewhen56objectsarepartitionedequallyinto8shares,,orasanumberofobjectsineachsharewhen
56objectsarepartitionedintoequalsharesof8objectseach.
3.OA.3Usemultiplicationanddivisionwithin100tosolvewordproblemsinsituationsinvolvingequalgroups,
arrays,andmathematicalquantities,e.g.,byusingdrawingsandequationswithasymbolfortheunknown
numbertorepresenttheproblems.
Solveproblemsinvolvingthefouroperations,andidentifyandexplainpatternsinarithmetic.
3.OA.8Solvetwo‐stepwordproblemsusingthefouroperations.Representtheseproblemsusingequationswith
aletterstandingfortheunknownquantity.Assthereasonablenessofanswersusingmentalcomputationand
estimationstrategiesincludingrounding.
Usefouroperationswithwholenumberstosolveproblems.
4.OA.3Solvemultistepwordproblemsposedwithwholenumbersandhavingwhole‐numberanswersusingthe
fouroperations,includingproblemsinwhichremaindersmustbeinterpreted.Representtheseproblemsusing
equationswithaletterstandingfortheunknownquantity.Assessthereasonablenessofanswersusingmental
computationandestimationstrategiesincludingrounding.
StatisticsandProbability
Investigatechanceprocessesanddevelop,use,andevaluateprobabilitymodels.
7.SP.8findprobabilitiesofcompoundeventsusingorganizedlists,tables,treediagramsandsimulation.
CommonCoreStateStandardsMath– StandardsofMathematicalPractice
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.8Lookforandexpressregularityinrepeatedreasoning.
Mathematicallyproficientstudentsnoticeifcalculationsarerepeated,andlookbothforgeneralmethodsandfor
shortcuts.Upperelementarystudentsmightnoticewhendividing25by11thattheyarerepeatingthesame
calculationsoverandoveragain,andconcludetheyhavearepeatingdecimal.Bypayingattentiontothe
calculationofslopeastheyrepeatedlycheckwhetherpointsareonthelinethrough(1,2)withslope3,middle
schoolstudentsmightabstracttheequation(y‐2)/(x‐1)=3.Noticingtheregularityinthewaytermscancel
whenexpanding(x‐1)(x+1),(x‐1)(x2+x+1),and(x‐1)(x3+x2+x+1)mightleadthemtothegeneralformula
forthesumofageometricseries.Astheyworktosolveaproblem,mathematicallyproficientstudentsmaintain
oversightoftheprocess,whileattendingtothedetails.Theycontinuallyevaluatethereasonablenessoftheir
intermediateresults.
ProblemoftheMonth SquirrelingitAway
©NoyceFoundation2014.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
4
ProblemoftheMonth
SquirrelingItAway
TaskDescription–LevelE
Thistaskchallengesastudenttofindandjustifyaclosedformequationthatwilldeterminethe
numberofwaysthatnacornscanbedividedbetweenthreesquirrels.
CommonCoreStateStandardsMath‐ContentStandards
OperationsandAlgebraicThinking
Analyzepatternsandrelationships.
5.OA.3Generatetwonumericalpatternsusingtwogivenrules.Identifyapparentrelationships
betweencorrespondingterms.Formorderedpairsconsistingofcorrespondingtermsfromthetwo
patternsandgraphtheorderedpairsonacoordinateplane.
ExpressionsandEquations
Solvereal‐lifeandmathematicalproblemsusingnumericalandalgebraicexpressionsand
equations.
7.EE.4Usevariablestorepresentquantitiesinareal‐worldormathematicalproblem,andconstruct
simpleequationsandinequalitiestosolveproblemsbyreasoningaboutthequantities.
StatisticsandProbability
Investigatechanceprocessesanddevelop,use,andevaluateprobabilitymodels.
7.SP.8findprobabilitiesofcompoundeventsusingorganizedlists,tables,treediagramsand
simulation.
HighSchool–Algebra–CreatingEquations
Createequationsthatdescribenumbersorrelationships.
A‐CED.1Createequationsandinequalitiesinonevariableandusethemtosolveproblems,include
equationsarisingfromlinearandquadraticfunctions,andsimplerationalandexponentialfunctions.
CommonCoreStateStandardsMath–StandardsofMathematicalPractice
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.
MP.8Lookforandexpressregularityinrepeatedreasoning.
Mathematicallyproficientstudentsnoticeifcalculationsarerepeated,andlookbothforgeneral
methodsandforshortcuts.Upperelementarystudentsmightnoticewhendividing25by11that
theyarerepeatingthesamecalculationsoverandoveragain,andconcludetheyhavearepeating
decimal.Bypayingattentiontothecalculationofslopeastheyrepeatedlycheckwhetherpointsare
onthelinethrough(1,2)withslope3,middleschoolstudentsmightabstracttheequation(y‐2)/(x‐
1)=3.Noticingtheregularityinthewaytermscancelwhenexpanding(x‐1)(x+1),
(x‐1)(x2+x+1),and(x‐1)(x3+x2+x+1)mightleadthemtothegeneralformulaforthesumofa
geometricseries.Astheyworktosolveaproblem,mathematicallyproficientstudentsmaintain
oversightoftheprocess,whileattendingtothedetails.Theycontinuallyevaluatethereasonableness
oftheirintermediateresults.
ProblemoftheMonth SquirrelingitAway
©NoyceFoundation2014.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
5
ProblemoftheMonth
SquirrelingItAway
TaskDescription–PrimaryLevel
Thistaskchallengesastudenttothinkaboutsubtractionandtakingaway.Studentsusecubestohelp
themactoutthesituationandrecordtheirideaswithpaperandmarkers.
CommonCoreStateStandardsMath‐ContentStandards
CountingandCardinality
Knownumbernamesandthecountsequence.
K.CC.1Countto100byonesandbytens.
Counttotellthenumberofobjects.
K.CC.4Understandtherelationshipbetweennumbersandquantities;connectcountingtocardinality.
K.CC.5Counttoanswer“howmany?”questionsaboutasmanyas20thingsarrangedinaline,arectangular
array,oracircle,orasmanyas10thingsinascatteredconfiguration;givenanumberfrom1‐20countoutthat
manyobjects.
OperationsandAlgebraicThinking
Understandadditionasputtingtogetherandaddingto,andunderstandsubtractionastakingapartand
takingfrom.
K.OA.1Representadditionandsubtractionwithobjects,fingers,mentalimages,drawings,sounds,actingout
situationsverbalexplanations,expressionsorequations.
Representandsolveproblemsinvolvingadditionandsubtraction.
1.OA.2Solvewordproblemsthatcallforadditionofthreewholenumbers,whosesumislessthanorequalto20,
e.g.,byusingobjects,drawings,andequationswithasymbolfortheunknownnumbertorepresenttheproblem.
2.OA.1Useadditionandsubtractionwithin100tosolveone‐andtwo‐stepproblemsinvolvingsituationsof
addingto,takingfrom,puttingtogether,takingapart,andcomparing,withunknownsinallpositions,e.g.by
usingdrawingsandequationswithasymbolfortheunknownnumbertorepresenttheproblem.
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.
ProblemoftheMonth SquirrelingitAway
©NoyceFoundation2014.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
6

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