ProblemoftheMonth:OnceUponaTime
TheProblemsoftheMonth(POM)areusedinavarietyofwaystopromoteproblem
solvingandtofosterthefirststandardofmathematicalpracticefromtheCommon
CoreStateStandards:“Makesenseofproblemsandpersevereinsolvingthem.”The
POMmaybeusedbyateachertopromoteproblemsolvingandtoaddressthe
differentiatedneedsofherstudents.Adepartmentorgradelevelmayengagetheir
studentsinaPOMtoshowcaseproblemsolvingasakeyaspectofdoing
mathematics.Itcanalsobeusedschoolwidetopromoteaproblem‐solvingthemeat
aschool.Thegoalisforallstudentstohavetheexperienceofattackingandsolving
non‐routineproblemsanddevelopingtheirmathematicalreasoningskills.Although
obtainingandjustifyingsolutionstotheproblemsistheobjective,theprocessof
learningtoproblemsolveisevenmoreimportant.
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
IntheProblemoftheMonthOnceUponaTime,studentsusemeasurement,number
properties,andcirculargeometrytosolveproblemsinvolvingtimeandangles.The
mathematicaltopicsthatunderliethisPOMaretimemeasurement,including
conversionbetweenyears,months,weeks,days,hours,minutes,andseconds;
modulararithmeticthatinvolvesdivisibilityandremainders;patternrecognition,as
wellascirclesandangularmeasurement.
InLevelAofthePOM,studentsarepresentedwiththetaskofdetermininghow
manyminuteswillpassbeforethelargehandcatchesuptothesmallhandona
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clock.Theirtaskinvolvescountingupandunderstandinghowaclockmeasures
time.AtLevelB,thestudentsareaskedtoconverttheiragefromyearsintoseasons,
months,andweeks.Studentsarealsoaskedtodeterminewhatdaynumberthe
currentdateisintheyear.InLevelC,studentsaregivenaproblemthatrequiresan
understandingofdivisibilityandmaybedeterminedbyusingknowledgeof
relativelyprimefactors.InLevelD,thestudentispresentedwithaproblemthat
involvesthreedifferent‐sizedalarmclocksthatringatvariedintervals.Thetaskis
todetermineiforwhenthethreeclockschimesimultaneously.Somestudentsmay
useanunderstandingofmodulardivisiontosolvetheproblem.InLevelE,students
areaskedtodeterminethetimesinadaythattheclockhandsformanangleof48
degrees.
MathematicalConcepts
ThisPOMinvolvescyclicalpatternsandmeasurement.Themostcommoncyclical
patternistimemeasurement.Timemeasurementandcirclesareintertwined.The
factthatcircleshave360degreeshasitsrootsintheoriginalcalendarsthat
determinedayearwas360days.Cyclicalrelationshipsareoftenrepresentedby
circlesandtheircomponentparts,suchasanglesandsectors.Cyclicalrelationships
arecommonthroughoutmathematicswithlinksbetweennumber,geometry,
algebra,trigonometry,andmeasurement.
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Problem of the Month
Once Upon a Time
Level A:
When it is four o’clock, how many minutes must pass before the big hand
(minute hand) gets to where the little hand (hour hand) was at four o’clock?
How did you figure it out?
When it is six-thirty, how many minutes must pass before the big hand
(minute hand) gets to where the little hand (hour hand) was at six-thirty?
Explain the way you figured it out.
Problem of the Month
Once Upon a Time
Page 1
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Level B:
How old are you?
• State your answer in years.
• State your answer in seasons.
• State your answer in months.
• State your answer in weeks.
What date is it?
What number day of the year is it?
How many more days until January 1?
Problem of the Month
Once Upon a Time
Page 2
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License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
Level C:
I met a man who said, “If you can guess my age, I will pay you one dollar
for each year that I have lived. I will also give you two hints. If you take
my age and divide it by any odd number greater than 1 and less than 9, you
will get a remainder of 1. But if you take my age and divide it by any even
number greater than 1 and less than 9, you will not get a remainder of 1.”
How much money could you earn?
Explain your solution and how you know it is the only correct answer.
Problem of the Month
Once Upon a Time
Page 3
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License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
Level D:
An eccentric clockmaker built three different clocks.
The first clock was a five-minute clock designed with an alarm set to sound
each time the hand reached the number 2.
The second clock was a six-minute clock designed to sound each time the
hand reached the number 3.
The third clock was a seven-minute clock designed to sound each time the
hand reached the number 4.
The clockmaker started the clocks simultaneously one day, and each clock
began to sound at its appropriate time. Was there a time when all three
clocks sounded their alarms together? If so, tell when it occurred and
explain why. If not, explain why not.
4
7
6
5
1
1
6
1
5
2
5

3
2

2
4
3


4
3
Problem of the Month
Once Upon a Time
Page 4
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License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
Level E:
The minute hand and the hour hand on a clock form a 48˚angle. What time
is it?
At what other times during the day do the hands on the clock form a 48˚
angle?
How many times in a day (24 hour period) do the hands form a 48˚ angle?
Explain your reasoning.
Problem of the Month
Once Upon a Time
Page 5
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License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
Problem of the Month
Once Upon a Time
PrimaryVersionLevelA
Materials:Alargedemonstrationclock,asmallclockpergroup
Discussionontherug:Teacherholdsuptheclocksetat4:00.“What time does
this clock say?”Teachersolicitsanswersfromstudents.“Which is the minute
hand? Which is the hour hand?”Teachersolicitsanswersfromstudents;teacher
mayrefertohandsasbigandsmall.“How do the hands move around the
clock?”Studentsdemonstrate.“Which hand moves faster?”
Insmallgroups:Eachgrouphasanindividualclock.Theteacherstatesthe
following:“Set your clock to 4 o’clock. Which direction do the hands move?
Which is the minute hand (big hand)? How many minutes must pass before
the minute hand gets to where the hour hand (little hand) is now? Draw a
picture or use words to explain how you know.”
“Set your clock to 6:30. Which direction do the hands move? Which is the
minute hand (big hand)? How many minutes must pass before the minute
hand gets to where the hour hand (little hand) is now? Draw a picture or
use words to explain how you know.”
Attheendoftheinvestigationhavestudentseitherdiscussordictatearesponseto
thesesummaryquestions.
Problem of the Month
Once Upon a Time
Page 6
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ProblemoftheMonth
OnceUponaTime
TaskDescription–LevelA
Thistaskchallengesastudenttodeterminehowmanyminuteswillpassbeforethelargehandwill
catchupwiththesmallhandonananalogclock.Astudentwillunderstandhowananalogclock
measurestimeandwillusingcountingprinciplestoworkonthistask.
CommonCoreStateStandardsMath‐ContentStandards
CountingandCardinality
Knownumbernamesandthecountsequence.
K.CC.1Countto100byonesandbytens.
K.CC.5Counttoanswer“howmany?”questionsaboutasmanyas20thingsarrangedinaline,a
rectangulararray,oracircle,orasmanyas10thingsinascatteredconfiguration;givenanumber
from1–20,countoutthatmanyobjects.
OperationsandAlgebraicThinking
Understandadditionasputtingtogetherandaddingto,andunderstandsubtractionastaking
apartandtakingfrom.
K.OA.1Representadditionandsubtractionwithobjects,fingers,mentalimages,drawings,sounds
(e.g.,claps),actingoutsituations,verbalexplanations,expressions,orequations.
Addandsubtractwithin20.
1.OA.5Relatecountingtoadditionandsubtraction(e.g.,bycountingon2toadd2).
Representandsolveproblemsinvolvingadditionandsubtraction.
2.OA.1Useadditionandsubtractionwithin100tosolveone‐andtwo‐stepwordproblemsinvolving
situationsofaddingto,takingfrom,puttingtogether,takingapart,andcomparing,withunknownsin
allpositions,e.g.,byusingdrawingsandequationswithasymbolfortheunknownnumberto
representtheproblem.
MeasurementandData
Tellandwritetime.
1.MD.3Tellandwritetimeinhoursandhalf‐hoursusinganaloganddigitalclocks.
Workwithtimeandmoney.
2.MD.7Tellandwritetimefromanaloganddigitalclockstothenearestfiveminutes,usinga.m.and
p.m.
CommonCoreStateStandardsMath–StandardsofMathematicalPractice
MP.7Lookforandmakeuseofstructure.
Mathematicallyproficientstudentslookcloselytodiscernapatternorstructure.Youngstudents,forexample,
mightnoticethatthreeandsevenmoreisthesameamountassevenandthreemore,ortheymaysorta
collectionofshapesaccordingtohowmanysidestheshapeshave.Later,studentswillsee7x8equalsthewell‐
remembered7x5+7x3,inpreparationforlearningaboutthedistributiveproperty.Intheexpressionx2+9x+
14,olderstudentscanseethe14as2x7andthe9as2+7.Theyrecognizethesignificanceofanexistinglinein
ageometricfigureandcanusethestrategyofdrawinganauxiliarylineforsolvingproblems.Theyalsocanstep
backforanoverviewandshiftperspective.Theycanseecomplicatedthings,suchassomealgebraicexpressions,
assingleobjectsorasbeingcomposedofseveralobjects.Forexample,theycansee5–3(x–y)2as5minusa
positivenumbertimesasquareandusethattorealizethatitsvaluecannotbemorethan5foranyrealnumbers
xandy.
MP.8 Lookforandexpressregularityinrepeatedreasoning.
Mathematicallyproficientstudentsnoticeifcalculationsarerepeated,andlookbothforgeneralmethodsandfor
shortcuts.Upperelementarystudentsmightnoticewhendividing25by11thattheyarerepeatingthesame
calculationsoverandoveragain,andconcludetheyhavearepeatingdecimal.Bypayingattentiontothe
calculationofslopeastheyrepeatedlycheckwhetherpointsareonthelinethrough(1,2)withslope3,middle
schoolstudentsmightabstracttheequation
(y–2)/(x–1)=3.Noticingtheregularityinthewaytermscancelwhenexpanding(x–1)(x+1),(x–
1)(x2+x+1),and(x–1)(x3+x2+x+1)mightleadthemtothegeneralformulaforthesumofageometricseries.As
theyworktosolveaproblem,mathematicallyproficientstudentsmaintainoversightoftheprocess,while
attendingtothedetails.Theycontinuallyevaluatethereasonablenessoftheirintermediateresults.
©NoyceFoundation2013.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
ProblemoftheMonth
OnceUponaTime
TaskDescription–LevelB
Thistaskchallengesastudenttoconverttheiragefromyearsintoseasons,months,andweeks.A
studentisalsoaskedtodeterminewhatdaynumberthecurrentdateisintheyear.Astudentmay
useunitconversionsandvariables.
CommonCoreStateStandardsMath‐ContentStandards
OperationsandAlgebraicThinking
Representandsolveproblemsinvolvingadditionandsubtraction.
2.OA.1Useadditionandsubtractionwithin100tosolveone‐andtwo‐stepwordproblemsinvolvingsituations
ofaddingto,takingfrom,puttingtogether,takingapart,andcomparing,withunknownsinallpositions,e.g.,by
usingdrawingsandequationswithasymbolfortheunknownnumbertorepresenttheproblem.
NumberandOperationsinBaseTen
Useplacevalueunderstandingandpropertiesofoperationstoaddandsubtract.
2.NBT.5Fluentlyaddandsubtractwithin100usingstrategiesbasedonplacevalue,propertiesofoperations,
and/ortherelationshipbetweenadditionandsubtraction.
Useplacevalueunderstandingandpropertiesofoperationstoperformmulti‐digitarithmetic.
3.NBT.2Fluentlyaddandsubtractwithin1000usingstrategiesandalgorithmsbasedonplacevalue,properties
ofoperations,and/ortherelationshipbetweenadditionandsubtraction.
Useplacevalueunderstandingandpropertiesofoperationstoperformmulti‐digitarithmetic.
4.NBT.4Fluentlyaddandsubtractmulti‐digitwholenumbersusingthestandardalgorithm.
MeasurementandData
Solveproblemsinvolvingmeasurementandconversionofmeasurementsfromalargerunittoasmaller
unit.
4.MD.1Knowrelativesizesofmeasurementunitswithinonesystemofunitsincludingkm,m,cm;kg,g;lb,oz.;l,
ml;hr,min,sec.Withinasinglesystemofmeasurement,expressmeasurementsinalargerunitintermsofa
smallerunit.…
4.MD.2Usethefouroperationstosolvewordproblemsinvolvingdistances,intervalsoftime,liquid
volumes,massesofobjects,andmoney,includingproblemsinvolvingsimplefractionsordecimals,and
problemsthatrequireexpressingmeasurementsgiveninalargerunitintermsofasmallerunit
ExpressionsandEquations
Applyandextendpreviousunderstandingsofarithmetictoalgebraicexpressions.
6.EE.2Write,read,andevaluateexpressionsinwhichlettersstandfornumbers.
CommonCoreStateStandardsMath–StandardsofMathematicalPractice
MP.7Lookforandmakeuseofstructure.
Mathematicallyproficientstudentslookcloselytodiscernapatternorstructure.Youngstudents,forexample,
mightnoticethatthreeandsevenmoreisthesameamountassevenandthreemore,ortheymaysorta
collectionofshapesaccordingtohowmanysidestheshapeshave.Later,studentswillsee7x8equalsthewell‐
remembered7x5+7x3,inpreparationforlearningaboutthedistributiveproperty.Intheexpressionx2+9x+
14,olderstudentscanseethe14as2x7andthe9as2+7.Theyrecognizethesignificanceofanexistinglinein
ageometricfigureandcanusethestrategyofdrawinganauxiliarylineforsolvingproblems.Theyalsocanstep
backforanoverviewandshiftperspective.Theycanseecomplicatedthings,suchassomealgebraicexpressions,
assingleobjectsorasbeingcomposedofseveralobjects.Forexample,theycansee5–3(x–y)2as5minusa
positivenumbertimesasquareandusethattorealizethatitsvaluecannotbemorethan5foranyrealnumbers
xandy.
MP.8 Lookforandexpressregularityinrepeatedreasoning.
Mathematicallyproficientstudentsnoticeifcalculationsarerepeated,andlookbothforgeneralmethodsandfor
shortcuts.Upperelementarystudentsmightnoticewhendividing25by11thattheyarerepeatingthesame
calculationsoverandoveragain,andconcludetheyhavearepeatingdecimal.Bypayingattentiontothe
calculationofslopeastheyrepeatedlycheckwhetherpointsareonthelinethrough(1,2)withslope3,middle
schoolstudentsmightabstracttheequation
(y–2)/(x–1)=3.Noticingtheregularityinthewaytermscancelwhenexpanding(x–1)(x+1),(x–
1)(x2+x+1),and(x–1)(x3+x2+x+1)mightleadthemtothegeneralformulaforthesumofageometricseries.As
theyworktosolveaproblem,mathematicallyproficientstudentsmaintainoversightoftheprocess,while
attendingtothedetails.Theycontinuallyevaluatethereasonablenessoftheirintermediateresults.
©NoyceFoundation2013.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
ProblemoftheMonth
OnceUponaTime
TaskDescription–LevelC
Thistaskchallengesastudenttodetermineaman’sageandtoexplainthesolutionandhowitisthe
onlycorrectsolution.Twohintsaregiven.Astudentwillusebasicoperationsandunderstand
remaindersinareal‐lifecontext.Astudentwilluseanunderstandingofdivisibilityandthe
knowledgeofrelativelyprimefactorstoworkonthistask.
CommonCoreStateStandardsMath‐ContentStandards
OperationsandAlgebraicThinking
Gainfamiliaritywithfactorsandmultiples.
4.OA.4Findallfactorpairsforawholenumberintherange1–100.Recognizethatawholenumber
isamultipleofeachofitsfactors.Determinewhetheragivenwholenumberintherange1–100isa
multipleofagivenone‐digitnumber.Determinewhetheragivenwholenumberintherange1–100
isprimeorcomposite.
Analyzepatternsandrelationships.
5.OA.3Generatetwonumericalpatternsusingtwogivenrules.Identifyapparentrelationships
betweencorrespondingterms.Formorderedpairsconsistingofcorrespondingtermsfromthetwo
patterns,andgraphtheorderedpairsonacoordinateplane.Forexample,giventherule“Add3”and
thestartingnumber0,andgiventherule“Add6”andthestartingnumber0,generatetermsinthe
resultingsequences,andobservethatthetermsinonesequencearetwicethecorrespondingterms
intheothersequence.Explaininformallywhythisisso.
TheNumberSystem
Computefluentlywithmulti‐digitnumbersandfindcommonfactorsandmultiples.
6.NS.2Fluentlydividemulti‐digitnumbersusingthestandardalgorithm.
6.NS.4Findthegreatestcommonfactoroftwowholenumberslessthanorequalto100andtheleast
commonmultipleoftwowholenumberslessthanorequalto12.
CommonCoreStateStandardsMath–StandardsofMathematicalPractice
MP.7Lookforandmakeuseofstructure.
Mathematicallyproficientstudentslookcloselytodiscernapatternorstructure.Youngstudents,for
example,mightnoticethatthreeandsevenmoreisthesameamountassevenandthreemore,or
theymaysortacollectionofshapesaccordingtohowmanysidestheshapeshave.Later,students
willsee7x8equalsthewell‐remembered7x5+7x3,inpreparationforlearningaboutthe
distributiveproperty.Intheexpressionx2+9x+14,olderstudentscanseethe14as2x7andthe9
as2+7.Theyrecognizethesignificanceofanexistinglineinageometricfigureandcanusethe
strategyofdrawinganauxiliarylineforsolvingproblems.Theyalsocanstepbackforanoverview
andshiftperspective.Theycanseecomplicatedthings,suchassomealgebraicexpressions,assingle
objectsorasbeingcomposedofseveralobjects.Forexample,theycansee5–3(x–y)2as5minusa
positivenumbertimesasquareandusethattorealizethatitsvaluecannotbemorethan5forany
realnumbersxandy.
MP.8 Lookforandexpressregularityinrepeatedreasoning.
Mathematicallyproficientstudentsnoticeifcalculationsarerepeated,andlookbothforgeneral
methodsandforshortcuts.Upperelementarystudentsmightnoticewhendividing25by11thatthey
arerepeatingthesamecalculationsoverandoveragain,andconcludetheyhavearepeatingdecimal.
Bypayingattentiontothecalculationofslopeastheyrepeatedlycheckwhetherpointsareonthe
linethrough(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)mightleadthemtothegeneralformulaforthesumofageometric
series.Astheyworktosolveaproblem,mathematicallyproficientstudentsmaintainoversightofthe
process,whileattendingtothedetails.Theycontinuallyevaluatethereasonablenessoftheir
intermediateresults.
©NoyceFoundation2013.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
ProblemoftheMonth:
OnceUponaTime
TaskDescription–LevelD
Thistaskchallengesastudenttodeterminewhenthealarmsonthreedifferentmodularclocksthat
ringatvariedintervalswillallsoundatthesametime.Astudentwillneedadeepunderstandingof
placevalueandquantitiesforworkingonthistask.Somestudentsmayuseanunderstandingof
modulardivisiontosolvetheproblem.
CommonCoreStateStandardsMath‐ContentStandards
OperationsandAlgebraicThinking
Gainfamiliaritywithfactorsandmultiples.
4.OA.4Findallfactorpairsforawholenumberintherange1–100.Recognizethatawholenumberisa
multipleofeachofitsfactors.Determinewhetheragivenwholenumberintherange1–100isamultipleofa
givenone‐digitnumber.Determinewhetheragivenwholenumberintherange1–100isprimeorcomposite.
Analyzepatternsandrelationships.
5.OA.3Generatetwonumericalpatternsusingtwogivenrules.Identifyapparentrelationshipsbetween
correspondingterms.Formorderedpairsconsistingofcorrespondingtermsfromthetwopatterns,andgraph
theorderedpairsonacoordinateplane.Forexample,giventherule“Add3”andthestartingnumber0,and
giventherule“Add6”andthestartingnumber0,generatetermsintheresultingsequences,andobservethat
thetermsinonesequencearetwicethecorrespondingtermsintheothersequence.Explaininformallywhythis
isso.
TheNumberSystem
Computefluentlywithmulti‐digitnumbersandfindcommonfactorsandmultiples.
6.NS.2Fluentlydividemulti‐digitnumbersusingthestandardalgorithm.
6.NS.4Findthegreatestcommonfactoroftwowholenumberslessthanorequalto12.…
ExpressionsandEquations
Solvereal‐lifeandmathematicalproblemsusingnumericalandalgebraicexpressionsand
equations.
7.EE.4Usevariablestorepresentquantitiesinareal‐worldormathematicalproblem,andconstruct
simpleequationsandinequalitiestosolveproblemsbyreasoningaboutthequantities.
Analyzeandsolvelinearequationsandpairsofsimultaneouslinearequations.
8.EE.7Solvelinearequationsinonevariable.
CommonCoreStateStandardsMath–StandardsofMathematicalPractice
MP.7Lookforandmakeuseofstructure.
Mathematicallyproficientstudentslookcloselytodiscernapatternorstructure.Youngstudents,forexample,
mightnoticethatthreeandsevenmoreisthesameamountassevenandthreemore,ortheymaysorta
collectionofshapesaccordingtohowmanysidestheshapeshave.Later,studentswillsee7x8equalsthewell‐
remembered7x5+7x3,inpreparationforlearningaboutthedistributiveproperty.Intheexpressionx2+9x+
14,olderstudentscanseethe14as2x7andthe9as2+7.Theyrecognizethesignificanceofanexistinglinein
ageometricfigureandcanusethestrategyofdrawinganauxiliarylineforsolvingproblems.Theyalsocanstep
backforanoverviewandshiftperspective.Theycanseecomplicatedthings,suchassomealgebraicexpressions,
assingleobjectsorasbeingcomposedofseveralobjects.Forexample,theycansee5–3(x–y)2as5minusa
positivenumbertimesasquareandusethattorealizethatitsvaluecannotbemorethan5foranyrealnumbers
xandy.
MP.8 Lookforandexpressregularityinrepeatedreasoning.
Mathematicallyproficientstudentsnoticeifcalculationsarerepeated,andlookbothforgeneralmethodsandfor
shortcuts.Upperelementarystudentsmightnoticewhendividing25by11thattheyarerepeatingthesame
calculationsoverandoveragain,andconcludetheyhavearepeatingdecimal.Bypayingattentiontothe
calculationofslopeastheyrepeatedlycheckwhetherpointsareonthelinethrough(1,2)withslope3,middle
schoolstudentsmightabstracttheequation
(y–2)/(x–1)=3.Noticingtheregularityinthewaytermscancelwhenexpanding(x–1)(x+1),(x–
1)(x2+x+1),and(x–1)(x3+x2+x+1)mightleadthemtothegeneralformulaforthesumofageometricseries.As
theyworktosolveaproblem,mathematicallyproficientstudentsmaintainoversightoftheprocess,while
attendingtothedetails.Theycontinuallyevaluatethereasonablenessoftheirintermediateresults.
©NoyceFoundation2013.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
ProblemoftheMonth
OnceUponaTime
TaskDescription–LevelE
Thistaskchallengesastudenttodeterminethetimesinadaywhenthehandsofananalogclockwill
forma48degreeangle.Astudentwillusetheirunderstandingofangleandangularmeasuresto
workonthistask.
CommonCoreStateStandardsMath‐ContentStandards
Geometry
Drawandidentifylinesandangles,…
4.G.1Drawpoints,lines,linesegments,rays,angles(right,acute,obtuse),…
RatiosandProportionalRelationships6.RP
Understandratioconceptsanduseratioreasoningtosolveproblems.
6.RP.1Understandtheconceptofaratioanduseratiolanguagetodescribearatiorelationship
betweentwoquantities.Forexample,“Theratioofwingstobeaksinthebirdhouseatthezoowas
2:1,becauseforevery2wingstherewas1beak.”“ForeveryvotecandidateAreceived,candidateC
receivednearlythreevotes.”
6.RP.3Useratioandratereasoningtosolvereal‐worldandmathematicalproblems,e.g.,by
reasoningabouttablesofequivalentratios,tapediagrams,doublenumberlinediagrams,or
equations.
6.RP.3.dUseratioreasoningtoconvertmeasurementunits;manipulateandtransformunits
appropriatelywhenmultiplyingordividingquantities.
TheNumberSystem
Computefluentlywithmulti‐digitnumbersandfindcommonfactorsandmultiples.
6.NS.2Fluentlydividemulti‐digitnumbersusingthestandardalgorithm.
6.NS.4Findthegreatestcommonfactoroftwowholenumberslessthanorequalto12.…
ExpressionsandEquations
Solvereal‐lifeandmathematicalproblemsusingnumericalandalgebraicexpressionsand
equations.
7.EE.4Usevariablestorepresentquantitiesinareal‐worldormathematicalproblem,andconstruct
simpleequationsandinequalitiestosolveproblemsbyreasoningaboutthequantities.
Analyzeandsolvelinearequationsandpairsofsimultaneouslinearequations.
8.EE.7Solvelinearequationsinonevariable.
HighSchool–Algebra‐CreatingEquations
Createequationsthatdescribenumbersorrelationships.
A‐CED.1Createequationsandinequalitiesinonevariableandusethemtosolveproblems.
HighSchool–Geometry‐Congruence
Experimentwithtransformationsintheplane
G‐CO.1Knowprecisedefinitionsofangle,circle,perpendicularline,parallelline,andlinesegment,
basedontheundefinednotionsofpoint,line,distancealongaline,anddistancearoundacirculararc.
G‐CO.4Developdefinitionsofrotations,reflections,andtranslationsintermsofangles,circles,
perpendicularlines,parallellines,andlinesegments.
Makegeometricconstructions
G‐CO.12Makeformalgeometricconstructionswithavarietyoftoolsandmethods(compassand
straightedge,string,reflectivedevices,paperfolding,dynamicgeometricsoftware,etc.).
CommonCoreStateStandardsMath–StandardsofMathematicalPractice
MP.7Lookforandmakeuseofstructure.
Mathematicallyproficientstudentslookcloselytodiscernapatternorstructure.Youngstudents,for
example,mightnoticethatthreeandsevenmoreisthesameamountassevenandthreemore,or
theymaysortacollectionofshapesaccordingtohowmanysidestheshapeshave.Later,students
willsee7x8equalsthewell‐remembered7x5+7x3,inpreparationforlearningaboutthe
distributiveproperty.Intheexpressionx2+9x+14,olderstudentscanseethe14as2x7andthe9
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as2+7.Theyrecognizethesignificanceofanexistinglineinageometricfigureandcanusethe
strategyofdrawinganauxiliarylineforsolvingproblems.Theyalsocanstepbackforanoverview
andshiftperspective.Theycanseecomplicatedthings,suchassomealgebraicexpressions,assingle
objectsorasbeingcomposedofseveralobjects.Forexample,theycansee5–3(x–y)2as5minusa
positivenumbertimesasquareandusethattorealizethatitsvaluecannotbemorethan5forany
realnumbersxandy.
MP.8 Lookforandexpressregularityinrepeatedreasoning.
Mathematicallyproficientstudentsnoticeifcalculationsarerepeated,andlookbothforgeneral
methodsandforshortcuts.Upperelementarystudentsmightnoticewhendividing25by11thatthey
arerepeatingthesamecalculationsoverandoveragain,andconcludetheyhavearepeatingdecimal.
Bypayingattentiontothecalculationofslopeastheyrepeatedlycheckwhetherpointsareonthe
linethrough(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)mightleadthemtothegeneralformulaforthesumofageometric
series.Astheyworktosolveaproblem,mathematicallyproficientstudentsmaintainoversightofthe
process,whileattendingtothedetails.Theycontinuallyevaluatethereasonablenessoftheir
intermediateresults.
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ProblemoftheMonth
OnceUponaTime
TaskDescription–PrimaryLevel
Thistaskchallengesastudenttoworkwithanalogclockstolayafoundationforunderstandingtime.A
discussionisheldsothatastudentwouldunderstandthemeasurementoftimewithananalogclock.The
teacheraskstheclass,whenshownananalogclock,whichistheminutehandandwhichisthehourhand.A
teacheralsoneedstoaskaboutthemovementofthehandsandwhichhandmovesfasterthantheother.Each
groupofstudentshasanindividualclockwhichtheysetat4o’clock.Thentheyareaskedhowmanyminutes
mustpassbeforetheminutehandgetstowherethehourhandisnow.Eachstudentisaskedtodrawapicture
orexplainhowtheyknow.Thegroupsaretheninstructedtosettheclockat6:30andasked,“Howmany
minutesmustpassbeforetheminutehandgetstowherethehourhandisnow?”Eachstudentdrawsapicture
orexplainshowtheyknow.
CommonCoreStateStandardsMath‐ContentStandards
CountingandCardinality
Knownumbernamesandthecountsequence.
K.CC.1Countto100byonesandbytens.
K.CC.5Counttoanswer“howmany?”questionsaboutasmanyas20thingsarrangedinaline,arectangular
array,oracircle,orasmanyas10thingsinascatteredconfiguration;givenanumberfrom1–20,countoutthat
manyobjects.
OperationsandAlgebraicThinking
Understandadditionasputtingtogetherandaddingto,andunderstandsubtractionastakingapartand
takingfrom.
K.OA.1Representadditionandsubtractionwithobjects,fingers,mentalimages,drawings,sounds(e.g.,claps),
actingoutsituations,verbalexplanations,expressions,orequations.
Addandsubtractwithin20.
1.OA.5Relatecountingtoadditionandsubtraction(e.g.,bycountingon2toadd2).
Representandsolveproblemsinvolvingadditionandsubtraction.
2.OA.1Useadditionandsubtractionwithin100tosolveone‐andtwo‐stepwordproblemsinvolvingsituations
ofaddingto,takingfrom,puttingtogether,takingapart,andcomparing,withunknownsinallpositions,e.g.,by
usingdrawingsandequationswithasymbolfortheunknownnumbertorepresenttheproblem.
MeasurementandData
Tellandwritetime.
1.MD.3Tellandwritetimeinhoursandhalf‐hoursusinganaloganddigitalclocks.
Workwithtimeandmoney.
2.MD.7Tellandwritetimefromanaloganddigitalclockstothenearestfiveminutes,usinga.m.andp.m.
CommonCoreStateStandardsMath– StandardsofMathematicalPractice
MP.7Lookforandmakeuseofstructure.
Mathematicallyproficientstudentslookcloselytodiscernapatternorstructure.Youngstudents,forexample,
mightnoticethatthreeandsevenmoreisthesameamountassevenandthreemore,ortheymaysorta
collectionofshapesaccordingtohowmanysidestheshapeshave.Later,studentswillsee7x8equalsthewell‐
remembered7x5+7x3,inpreparationforlearningaboutthedistributiveproperty.Intheexpressionx2+9x+
14,olderstudentscanseethe14as2x7andthe9as2+7.Theyrecognizethesignificanceofanexistinglinein
ageometricfigureandcanusethestrategyofdrawinganauxiliarylineforsolvingproblems.Theyalsocanstep
backforanoverviewandshiftperspective.Theycanseecomplicatedthings,suchassomealgebraicexpressions,
assingleobjectsorasbeingcomposedofseveralobjects.Forexample,theycansee5–3(x–y)2as5minusa
positivenumbertimesasquareandusethattorealizethatitsvaluecannotbemorethan5foranyrealnumbers
xandy.
MP.8 Lookforandexpressregularityinrepeatedreasoning.
Mathematicallyproficientstudentsnoticeifcalculationsarerepeated,andlookbothforgeneralmethodsandfor
shortcuts.Upperelementarystudentsmightnoticewhendividing25by11thattheyarerepeatingthesame
calculationsoverandoveragain,andconcludetheyhavearepeatingdecimal.Bypayingattentiontothe
calculationofslopeastheyrepeatedlycheckwhetherpointsareonthelinethrough(1,2)withslope3,middle
schoolstudentsmightabstracttheequation
(y–2)/(x–1)=3.Noticingtheregularityinthewaytermscancelwhenexpanding(x–1)(x+1),(x–
1)(x2+x+1),and(x–1)(x3+x2+x+1)mightleadthemtothegeneralformulaforthesumofageometricseries.As
theyworktosolveaproblem,mathematicallyproficientstudentsmaintainoversightoftheprocess,while
attendingtothedetails.Theycontinuallyevaluatethereasonablenessoftheirintermediateresults.
©NoyceFoundation2013.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).