Problem of the Month: First Rate - Inside Mathematics
Transcript of Problem of the Month: First Rate - Inside Mathematics
ProblemoftheMonth FirstRate©NoyceFoundation2015.ThisworkislicensedunderaCreativeCommonsAttribution-NonCommercial-NoDerivatives3.0UnportedLicense(http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
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ProblemoftheMonth:FirstRate
TheProblemsoftheMonth(POM)areusedinavarietyofwaystopromoteproblemsolvingandtofosterthefirststandardofmathematicalpracticefromtheCommonCoreStateStandards:“Makesenseofproblemsandpersevereinsolvingthem.”ThePOMmaybeusedbyateachertopromoteproblemsolvingandtoaddressthedifferentiatedneedsofherstudents.AdepartmentorgradelevelmayengagetheirstudentsinaPOMtoshowcaseproblemsolvingasakeyaspectofdoingmathematics.POMscanalsobeusedschoolwidetopromoteaproblem-solvingthemeataschool.Thegoalisforallstudentstohavetheexperienceofattackingandsolvingnon-routineproblemsanddevelopingtheirmathematicalreasoningskills.Althoughobtainingandjustifyingsolutionstotheproblemsistheobjective,theprocessoflearningtoproblemsolveisevenmoreimportant.TheProblemoftheMonthisstructuredtoprovidereasonabletasksforallstudentsinaschool.ThePOMisstructuredwithashallowfloorandahighceiling,sothatallstudentscanproductivelyengage,struggle,andpersevere.ThePrimaryVersionisdesignedtobeaccessibletoallstudentsandespeciallyasthekeychallengeforgradesK–1.LevelAwillbechallengingformostsecondandthirdgraders.LevelBmaybethelimitofwherefourthandfifth-gradestudentshavesuccessandunderstanding.LevelCmaystretchsixthandseventh-gradestudents.LevelDmaychallengemosteighthandninth-gradestudents,andLevelEshouldbechallengingformosthighschoolstudents.Thesegrade-levelexpectationsarejustestimatesandshouldnotbeusedasanabsoluteminimumexpectationormaximumlimitationforstudents.Problemsolvingisalearnedskill,andstudentsmayneedmanyexperiencestodeveloptheirreasoningskills,approaches,strategies,andtheperseverancetobesuccessful.TheProblemoftheMonthbuildsonsequentiallevelsofunderstanding.AllstudentsshouldexperienceLevelAandthenmovethroughthetasksinordertogoasdeeplyastheycanintotheproblem.TherewillbethosestudentswhowillnothaveaccessintoevenLevelA.Educatorsshouldfeelfreetomodifythetasktoallowaccessatsomelevel.OverviewIntheProblemoftheMonthFirstRate,studentsusemeasurement,ratesofchange,andalgebraicthinkingtosolveproblemsinvolvingproportionalrelationships,metrics,andmultiplicativerelationships.ThemathematicaltopicsthatunderliethisPOMarerepeatedaddition,multiplication,division,percents,linearmeasurement,proportionalreasoning,rates,distance-time-velocity,change,functions,algebraicreasoning,andrelatedrates.InthefirstlevelsofthePOM,studentsarepresentedwithameasurementproblem.Intheproblem,studentsareaskedtodeterminewhowinsaracebetweentwobrothersjumpingupthestairway,whereonejumpsfartherthantheother.The
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studentshavetoreasonbetweennumberofjumpsandlengthofjumps.InLevelB,studentsarechallengedwithaprobleminvolvingtworunnerswhorundifferentdistancesfordifferenttimes.Theywillneedtoreasonabouttherelationshipbetweenthetworatestodeterminewhoisfaster.InLevelC,thestudentsarepresentedwiththechallengeofdetermininghowtwostudentsworkingtogetheratdifferentratescanworktogethertocompleteatask.Thestudentsareaskedtodeterminehowfastthetaskcanbecompleted.InLevelD,studentsanalyzetrackraceswhererunnersarerunningatdifferentpaces.Thestudentsinvestigatewhenarunnerneedstomakeachangeinordertoovertakearunnerinfrontofhim.InthefinallevelofthePOM,studentsarepresentedwithasituationthatinvolvesrelatedratesinthecontextofafootballgame.Studentsareaskedtodeterminethetimeandspeedafootballmustbethrowntocompleteapasstothetightend.MathematicalConceptsThemajormathematicalideasofthisPOMaremeasurement,proportionalreasoningandrateofchange.Studentsusemeasuringtechniques,addition,multiplication,division,andrepresentationsofrationalnumberssuchasfractions,decimals,andpercents,aswellasratios,proportion,rates,equations,andlinearfunctions.Studentssolveproblemsinvolvingratesforcompletingajob,aswellasdistance,rateandtime
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ProblemoftheMonth
FirstRateLevelADylanandAustinarebrothers.Theyplayaracinggameonthestairs.Theyjumpandlandontwofeetastheyraceupthestaircase.WhenAustinjumps,helandsoneachstep.WhenDylanjumps,heskipsastepandlandsoneveryotherstep.
1. Whohastotakemorejumpstogettothetopofthestairs?2. WhenDylanjumpsupthestaircase,howmanyjumpsdoeshemake?3. WhenAustinjumpsupthestaircase,howmanyjumpsdoeshemake?4. IfAustinandDylaneachtook5jumps,whowouldbefurtherupthestairs?5. Attheendoftheracewhotooklessjumps?6. Whodoyouthinkwontherace?Explainyouranswer.
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LevelBTomandDianestartedtorace.Tomtook4secondstorun6yards.Dianeran5yardsin3seconds.
Iftheycontinuedtorunatthesamespeeds,whowouldgetto30yardsfirst?Showhowyoufiguredoutyouranswer.Whorunsfaster?Howcanyoucomparetheirspeeds?
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LevelCTheEnvironmentalClubatschoolattendsanannualcommunityclean-upevent.Theyhaverecyclinggames.Ateamisassignedanareaoflandthatisscatteredwithlitter.Thegoalisforapairofparticipantstocleanuptheareainthefastesttimepossible.
Tammy,workingalone, could cleanone-half thearea inonehour.Herpartner,Melissa -workingalone-couldcleanone-thirdoftheareainonehour.Duringthecontestwhentheyworktogether,howlongwill it takethemtocleanthearea?Explainhowyoufoundyoursolution.
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Level D YouareanOlympicrunner.Youhave justqualified tobe in the finalsof the1,500-meterrace. The track is 400meters in an oval shape. The race is three and three-fourths lapsaroundthetrack.
The favorite to win the race is a Kenyan who holds the current best time, which is 3minutes29.4seconds.TheKenyanrunsaverysteadyrace.EachoftheKenyan’slaptimes(400meters)iswithinasecondofeachother.Yourunacompletelydifferenttypeofrace.Youhaveaverystrongkick,whichmeansyouusuallylagbehindforthefirstthreelapstosaveenergy,andthenwhentheleaderhas300meterstogoyoupouritontowinatthetape.Youliketosaveenergyinthefirstthreelaps,butyoudon’twanttobemorethan50metersbehindwhenyoustartyourkicktothefinishline.Determineyourstrategytowinthisrace.Whatistheaveragespeedyouneedtorunduringthe firstpartof therace?What is theaveragespeedyouneedtorunduringyourkicktowintherace?HowmightyourracechangeiftheKenyanrunstwosecondsfaster?
Start ->
Finish
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Level E It’s the third quarter, with ten yards to go for a first down. The quarterback calls hisfavoriteplay,arollouttotherightandasquare-outpasstohistightend.Seethediagramoftheplaybelow:On the snap from center, the tight end runs straight ahead for ten yards,makes a sharpright turn and runs toward the side lines. The quarterback rolls to his right and stopsdirectlybehindwherethetightendbegan,butsixyardsbehindthelineofscrimmage.Thequarterbackdoesnotmakethepassuntilafterthetightendmakeshisbreaktowardthesidelines.The tight end is running toward the sideline at a speedof8 yardsper second.Thequarterbacktracksthereceiver,decidingwhentothrowthepassandtheflightpathofthe ball. If the tight end makes the catch 12 yards after the break, how far does thequarterbackthrowthepass(inastraightline)andatwhatrateisthedistancebetweenthereceiverandquarterbackchanging?Suppose thequarterback threwthepasssooner,and thereceiver ranat thesamespeed.Thedistancetheballtraveledwas17.3yards.Howmanyyardsafterthebreakwastheballcaughtandatwhatratewasthedistancebetweenthereceiverandquarterbackchanging?Giventheconstantspeedofthereceiver,considerseverallocationswherethesquare-outpasscouldbecompleted.Explaintherelationshipbetweenthespotofthecompletion,thedistanceofthepass,andatwhatratethedistancebetweenthereceiverandquarterbackischanging?
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ProblemoftheMonthFirstRate
PrimaryVersion
Materials:Apictureofthestaircasewithfootprints,tape,measuringtape,paperandpencil.Discussionontherug:Teacherusestapetomaketwomarksontheflooraboutfiveyardsapart.Theteacherchoosestwoclassmatestostartatonemarkandmakeonejumptoward theothermarkwithboth feet together.The students jumpand the teacherasks, “Who jumped farther?” One at a time, the students jump until they reach the secondmark as the class counts. The teacher asks, “Who took the most jumps?” The teacherholdsupthepicturesof thestaircaseandthewet footprintandasks, “Who can tell me what they see in this picture?” The students respond. “This is a picture of a jumping race up the stairs. Two boys, both with wet feet race up the stairs. They jump landing on two feet as they race up the staircase. When the first boy jumps, he lands on each step. When the second boy jumps, he skips a step and lands on every other step.” In small groups: Each student has access to the picture of the staircase, paper andpencil. Theteacherexplainsthattheyneedtothinkaboutthejumpingraceupthestairs.Theteacherasksthefollowingquestions: “Who jumps farther? Who takes more jumps? How many jumps does the first boy take? Who do you think won the race? At the end of the investigation have students either discuss or dictate a response to theprompt:“Tell me how you know who won the race.”
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First Boy
Second Boy
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ProblemoftheMonth
FirstRateTaskDescription–LevelA
Thistaskchallengesstudentstodeterminewhowinsaracebetweentwobrothersjumpingupthestairwaywhereonebrotherjumpsfartherthantheother.Thestudenthastoreasonbetweennumberofjumpsandlengthofjumps.
CommonCoreStateStandardsMath-ContentStandardsMeasurementandDataDescribeandcomparemeasurableattributes.K.MD.1Describemeasurableattributesofobjects,suchaslengthorweight.Describeseveralmeasurableattributesofasingleobject.K.MD.2Directlycomparetwoobjectswithameasurableattributeincommon,toseewhichobjecthas“moreof”/“lessof”theattribute,anddescribethedifference.Forexample,directlycomparetheheightsoftwochildrenanddescribeonechildastaller/shorter.Measurelengthsindirectlyandbyiteratinglengthunits.1.MD.2Expressthelengthofanobjectasawholenumberoflengthunits,bylayingmultiplecopiesofashorterobject(thelengthunit)endtoend;understandthatthelengthmeasurementofanobjectisthenumberofsame-sizelengthunitsthatspanitwithnogapsoroverlaps.Limittocontextswheretheobjectbeingmeasuredisspannedbyawholenumberoflengthunitswithnogapsoroverlaps.
CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.2Reasonabstractlyandquantitatively.Mathematicallyproficientstudentsmakesenseofquantitiesandtheirrelationshipsinproblemsituations.Theybringtwocomplementaryabilitiestobearonproblemsinvolvingquantitativerelationships:theabilitytodecontextualize—toabstractagivensituationandrepresentitsymbolicallyandmanipulatetherepresentingsymbolsasiftheyhavealifeoftheirown,withoutnecessarilyattendingtotheirreferents—andtheabilitytocontextualize,topauseasneededduringthemanipulationprocessinordertoprobeintothereferentsforthesymbolsinvolved.Quantitativereasoningentailshabitsofcreatingacoherentrepresentationoftheproblemathand;consideringtheunitsinvolved;attendingtothemeaningofquantities,notjusthowtocomputethem;andknowingandflexiblyusingdifferentpropertiesofoperationsandobjects.MP.5 Useappropriatetoolsstrategically.Mathematicallyproficientstudentsconsidertheavailabletoolswhensolvingamathematicalproblem.Thesetoolsmightincludepencilandpaper,concretemodels,aruler,aprotractor,acalculator,aspreadsheet,acomputeralgebrasystem,astatisticalpackage,ordynamicgeometrysoftware.Proficientstudentsaresufficientlyfamiliarwithtoolsappropriatefortheirgradeorcoursetomakesounddecisionsaboutwheneachofthesetoolsmightbehelpful,recognizingboththeinsighttobegainedandtheirlimitations.Forexample,mathematicallyproficienthighschoolstudentsanalyzegraphsoffunctionsandsolutionsgeneratedusingagraphingcalculator.Theydetectpossibleerrorsbystrategicallyusingestimationandothermathematicalknowledge.Whenmakingmathematicalmodels,theyknowthattechnologycanenablethemtovisualizetheresultsofvaryingassumptions,exploreconsequences,andcomparepredictionswithdata.Mathematicallyproficientstudentsatvariousgradelevelsareabletoidentifyrelevantexternalmathematicalresources,suchasdigitalcontentlocatedonawebsite,andusethemtoposeorsolveproblems.Theyareabletousetechnologicaltoolstoexploreanddeepentheirunderstandingofconcepts.
CCSSMAlignment:ProblemoftheMonth FirstRate©NoyceFoundation2015.ThisworkislicensedunderaCreativeCommonsAttribution-NonCommercial-NoDerivatives3.0UnportedLicense(http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
ProblemoftheMonth
FirstRateTaskDescription–LevelB
Thistaskchallengesstudentstoanalyzeaproblemwheretworunnersrundifferentdistancesindifferenttimes.Thestudentswillneedtoreasonabouttherelationshipbetweenthetworatestodeterminewhoisfaster.
CommonCoreStateStandardsMath-ContentStandardsMeasurementandDataMeasureandestimatelengthsinstandardunits.2.MD.4Measuretodeterminehowmuchlongeroneobjectisthananother,expressingthelengthdifferenceintermsofastandardlengthunit.RatiosandProportionalRelationshipsUnderstandratioconceptsanduseratioreasoningtosolveproblems.6.RP.1Understandtheconceptofaratioanduseratiolanguagetodescribearatiorelationshipbetweentwoquantities.Forexample,“Theratioofwingstobeaksinthebirdhouseatthezoowas2:1,becauseforevery2wingstherewas1beak.”“ForeveryvotecandidateAreceived,candidateCreceivednearlythreevotes.”6.RP.2Understandtheconceptofaunitratea/bassociatedwitharatioa:bwithb= ̸0,anduseratelanguageinthecontextofaratiorelationship.Forexample,“Thisrecipehasaratioof3cupsofflourto4cupsofsugar,sothereis3/4cupofflourforeachcupofsugar.”“Wepaid$75for15hamburgers,whichisarateof$5perhamburger.”16.RP.3Useratioandratereasoningtosolvereal-worldandmathematicalproblems,e.g.,byreasoningabouttablesofequivalentratios,tapediagrams,doublenumberlinediagrams,orequations.6.RP.3a.Maketablesofequivalentratiosrelatingquantitieswithwhole-numbermeasurements,findmissingvaluesinthetables,andplotthepairsofvaluesonthecoordinateplane.Usetablestocompareratios.6.RP.3b.Solveunitrateproblemsincludingthoseinvolvingunitpricingandconstantspeed.Forexample,ifittook7hourstomow4lawns,thenatthatrate,howmanylawnscouldbemowedin35hours?Atwhatratewerelawnsbeingmowed?Analyzeproportionalrelationshipsandusethemtosolvereal-worldandmathematicalproblems.7.RP.1Computeunitratesassociatedwithratiosoffractions,includingratiosoflengths,areasandotherquantitiesmeasuredinlikeordifferentunits.Forexample,ifapersonwalks1/2mileineach1/4hour,computetheunitrateasthecomplexfraction1/2/1/4milesperhour,equivalently2milesperhour.7.RP.2Recognizeandrepresentproportionalrelationshipsbetweenquantities.7.RP.2a.Decidewhethertwoquantitiesareinaproportionalrelationship,e.g.,bytestingforequivalentratiosinatableorgraphingonacoordinateplaneandobservingwhetherthegraphisastraightlinethroughtheorigin.7.RP.2b.Identifytheconstantofproportionality(unitrate)intables,graphs,equations,diagrams,andverbaldescriptionsofproportionalrelationships.7.RP.2c.Representproportionalrelationshipsbyequations.Forexample,iftotalcosttisproportionaltothenumbernofitemspurchasedataconstantpricep,therelationshipbetweenthetotalcostandthenumberofitemscanbeexpressedast=pn.
CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.2Reasonabstractlyandquantitatively.Mathematicallyproficientstudentsmakesenseofquantitiesandtheirrelationshipsinproblemsituations.Theybringtwocomplementaryabilitiestobearonproblemsinvolvingquantitativerelationships:theabilitytodecontextualize—toabstractagivensituationandrepresentitsymbolicallyandmanipulatetherepresentingsymbolsasiftheyhavealifeoftheirown,withoutnecessarilyattendingtotheirreferents—andtheabilitytocontextualize,topauseasneededduringthemanipulationprocessinordertoprobeintothereferentsforthesymbolsinvolved.Quantitativereasoningentailshabitsofcreatingacoherentrepresentationoftheproblemathand;consideringtheunitsinvolved;attendingtothemeaningofquantities,notjusthowtocomputethem;andknowingandflexiblyusingdifferentpropertiesofoperationsandobjects.MP.5 Useappropriatetoolsstrategically.Mathematicallyproficientstudentsconsidertheavailabletoolswhensolvingamathematicalproblem.Thesetoolsmightincludepencilandpaper,concretemodels,aruler,aprotractor,acalculator,aspreadsheet,acomputeralgebrasystem,astatisticalpackage,ordynamicgeometrysoftware.Proficientstudentsaresufficientlyfamiliarwithtoolsappropriatefortheirgradeorcoursetomakesounddecisionsaboutwheneachofthesetoolsmightbehelpful,recognizingboththeinsighttobegainedandtheirlimitations.Forexample,mathematicallyproficienthighschoolstudentsanalyzegraphsoffunctionsandsolutionsgeneratedusingagraphingcalculator.Theydetectpossibleerrorsbystrategicallyusingestimationandothermathematicalknowledge.Whenmakingmathematicalmodels,theyknowthattechnologycanenablethemtovisualizetheresultsofvaryingassumptions,exploreconsequences,andcomparepredictionswithdata.Mathematicallyproficientstudentsatvariousgradelevelsareabletoidentifyrelevantexternalmathematicalresources,suchasdigitalcontentlocatedonawebsite,andusethemtoposeorsolveproblems.Theyareabletousetechnologicaltoolstoexploreanddeepentheirunderstandingofconcepts.
CCSSMAlignment:ProblemoftheMonth FirstRate©NoyceFoundation2015.ThisworkislicensedunderaCreativeCommonsAttribution-NonCommercial-NoDerivatives3.0UnportedLicense(http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
ProblemoftheMonth
FirstRateTaskDescription–LevelC
Thistaskchallengesstudentstodeterminehowtwostudentsworkingatdifferentratescanworktogethertocompleteataskandhowfastthetaskcanbecompleted.
CommonCoreStateStandardsMath-ContentStandardsRatiosandProportionalRelationshipsUnderstandratioconceptsanduseratioreasoningtosolveproblems.6.RP.2Understandtheconceptofaunitratea/bassociatedwitharatioa:bwithb= ̸0,anduseratelanguageinthecontextofaratiorelationship.Forexample,“Thisrecipehasaratioof3cupsofflourto4cupsofsugar,sothereis3/4cupofflourforeachcupofsugar.”“Wepaid$75for15hamburgers,whichisarateof$5perhamburger.”6.RP.3Useratioandratereasoningtosolvereal-worldandmathematicalproblems,e.g.,byreasoningabouttablesofequivalentratios,tapediagrams,doublenumberlinediagrams,orequations.6.RP.3a.Maketablesofequivalentratiosrelatingquantitieswithwhole-numbermeasurements,findmissingvaluesinthetables,andplotthepairsofvaluesonthecoordinateplane.Usetablestocompareratios.6RP.3b.Solveunitrateproblemsincludingthoseinvolvingunitpricingandconstantspeed.Forexample,ifittook7hourstomow4lawns,thenatthatrate,howmanylawnscouldbemowedin35hours?Atwhatratewerelawnsbeingmowed?6.RP.3c.Findapercentofaquantityasarateper100(e.g.,30%ofaquantitymeans30/100timesthequantity);solveproblemsinvolvingfindingthewhole,givenapartandthepercent.6.RP.3d.Useratioreasoningtoconvertmeasurementunits;manipulateandtransformunitsappropriatelywhenmultiplyingordividingquantities.Analyzeproportionalrelationshipsandusethemtosolvereal-worldandmathematicalproblems.7.RP.1Computeunitratesassociatedwithratiosoffractions,includingratiosoflengths,areasandotherquantitiesmeasuredinlikeordifferentunits.Forexample,ifapersonwalks1/2mileineach1/4hour,computetheunitrateasthecomplexfraction1/2/1/4milesperhour,equivalently2milesperhour.
CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.2Reasonabstractlyandquantitatively.Mathematicallyproficientstudentsmakesenseofquantitiesandtheirrelationshipsinproblemsituations.Theybringtwocomplementaryabilitiestobearonproblemsinvolvingquantitativerelationships:theabilitytodecontextualize—toabstractagivensituationandrepresentitsymbolicallyandmanipulatetherepresentingsymbolsasiftheyhavealifeoftheirown,withoutnecessarilyattendingtotheirreferents—andtheabilitytocontextualize,topauseasneededduringthemanipulationprocessinordertoprobeintothereferentsforthesymbolsinvolved.Quantitativereasoningentailshabitsofcreatingacoherentrepresentationoftheproblemathand;consideringtheunitsinvolved;attendingtothemeaningofquantities,notjusthowtocomputethem;andknowingandflexiblyusingdifferentpropertiesofoperationsandobjects.MP.5 Useappropriatetoolsstrategically.Mathematicallyproficientstudentsconsidertheavailabletoolswhensolvingamathematicalproblem.Thesetoolsmightincludepencilandpaper,concretemodels,aruler,aprotractor,acalculator,aspreadsheet,acomputeralgebrasystem,astatisticalpackage,ordynamicgeometrysoftware.Proficientstudentsaresufficientlyfamiliarwithtoolsappropriatefortheirgradeorcoursetomakesounddecisionsaboutwheneachofthesetoolsmightbehelpful,recognizingboththeinsighttobegainedandtheirlimitations.Forexample,mathematicallyproficienthighschoolstudentsanalyzegraphsoffunctionsandsolutionsgeneratedusingagraphingcalculator.Theydetectpossibleerrorsbystrategicallyusingestimationandothermathematicalknowledge.Whenmakingmathematicalmodels,theyknowthattechnologycanenablethemtovisualizetheresultsofvaryingassumptions,exploreconsequences,andcomparepredictionswithdata.Mathematicallyproficientstudentsatvariousgradelevelsareabletoidentifyrelevantexternalmathematicalresources,suchasdigitalcontentlocatedonawebsite,andusethemtoposeorsolveproblems.Theyareabletousetechnologicaltoolstoexploreanddeepentheirunderstandingofconcepts.
CCSSMAlignment:ProblemoftheMonth FirstRate©NoyceFoundation2015.ThisworkislicensedunderaCreativeCommonsAttribution-NonCommercial-NoDerivatives3.0UnportedLicense(http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
ProblemoftheMonth:
FirstRateTaskDescription–LevelD
Thistaskchallengesstudentstoanalyzetrackraceswhererunnersarerunningatdifferentpaces.Studentsmustinvestigatewhenarunnerneedstomakeachangetoovertaketherunnerinthefront.
CommonCoreStateStandardsMath-ContentStandardsRatiosandProportionalRelationshipsUnderstandratioconceptsanduseratioreasoningtosolveproblems.6.RP.3Useratioandratereasoningtosolvereal-worldandmathematicalproblems,e.g.,byreasoningabouttablesofequivalentratios,tapediagrams,doublenumberlinediagrams,orequations.6.RP.3a.Maketablesofequivalentratiosrelatingquantitieswithwhole-numbermeasurements,findmissingvaluesinthetables,andplotthepairsofvaluesonthecoordinateplane.Usetablestocompareratios.6.RP.3b.Solveunitrateproblemsincludingthoseinvolvingunitpricingandconstantspeed.Forexample,ifittook7hourstomow4lawns,thenatthatrate,howmanylawnscouldbemowedin35hours?Atwhatratewerelawnsbeingmowed?6.RP.3c.Findapercentofaquantityasarateper100(e.g.,30%ofaquantitymeans30/100timesthequantity);solveproblemsinvolvingfindingthewhole,givenapartandthepercent.6.RP.3d.Useratioreasoningtoconvertmeasurementunits;manipulateandtransformunitsappropriatelywhenmultiplyingordividingquantities.Analyzeproportionalrelationshipsandusethemtosolvereal-worldandmathematicalproblems.7.RP.1Computeunitratesassociatedwithratiosoffractions,includingratiosoflengths,areasandotherquantitiesmeasuredinlikeordifferentunits.Forexample,ifapersonwalks1/2mileineach1/4hour,computetheunitrateasthecomplexfraction1/2/1/4milesperhour,equivalently2milesperhour.7.RP.3Useproportionalrelationshipstosolvemultistepratioandpercentproblems.
CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.2Reasonabstractlyandquantitatively.Mathematicallyproficientstudentsmakesenseofquantitiesandtheirrelationshipsinproblemsituations.Theybringtwocomplementaryabilitiestobearonproblemsinvolvingquantitativerelationships:theabilitytodecontextualize—toabstractagivensituationandrepresentitsymbolicallyandmanipulatetherepresentingsymbolsasiftheyhavealifeoftheirown,withoutnecessarilyattendingtotheirreferents—andtheabilitytocontextualize,topauseasneededduringthemanipulationprocessinordertoprobeintothereferentsforthesymbolsinvolved.Quantitativereasoningentailshabitsofcreatingacoherentrepresentationoftheproblemathand;consideringtheunitsinvolved;attendingtothemeaningofquantities,notjusthowtocomputethem;andknowingandflexiblyusingdifferentpropertiesofoperationsandobjects.MP.5 Useappropriatetoolsstrategically.Mathematicallyproficientstudentsconsidertheavailabletoolswhensolvingamathematicalproblem.Thesetoolsmightincludepencilandpaper,concretemodels,aruler,aprotractor,acalculator,aspreadsheet,acomputeralgebrasystem,astatisticalpackage,ordynamicgeometrysoftware.Proficientstudentsaresufficientlyfamiliarwithtoolsappropriatefortheirgradeorcoursetomakesounddecisionsaboutwheneachofthesetoolsmightbehelpful,recognizingboththeinsighttobegainedandtheirlimitations.Forexample,mathematicallyproficienthighschoolstudentsanalyzegraphsoffunctionsandsolutionsgeneratedusingagraphingcalculator.Theydetectpossibleerrorsbystrategicallyusingestimationandothermathematicalknowledge.Whenmakingmathematicalmodels,theyknowthattechnologycanenablethemtovisualizetheresultsofvaryingassumptions,exploreconsequences,andcomparepredictionswithdata.Mathematicallyproficientstudentsatvariousgradelevelsareabletoidentifyrelevantexternalmathematicalresources,suchasdigitalcontentlocatedonawebsite,andusethemtoposeorsolveproblems.Theyareabletousetechnologicaltoolstoexploreanddeepentheirunderstandingofconcepts.
CCSSMAlignment:ProblemoftheMonth FirstRate©NoyceFoundation2015.ThisworkislicensedunderaCreativeCommonsAttribution-NonCommercial-NoDerivatives3.0UnportedLicense(http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
ProblemoftheMonth
FirstRateTaskDescription–LevelE
Thistaskchallengesstudentstodeterminethetimeandspeedafootballmustbethrowntocompleteapasstoatightendwhenpresentedwithasituationthatinvolvesrelatedratesinthecontextofafootballgame.
CommonCoreStateStandardsMath-ContentStandardsGeometryUnderstandandapplythePythagoreanTheorem.8.G.7ApplythePythagoreanTheoremtodetermineunknownsidelengthsinrighttrianglesinreal-worldandmathematicalproblemsintwoandthreedimensions.HighSchool–Functions-BuildingFunctionsBuildafunctionthatmodelsarelationshipbetweentwoquantitiesF-BF.1Writeafunctionthatdescribesarelationshipbetweentwoquantities.★F-BF.1c.(+)Composefunctions.Forexample,ifT(y)isthetemperatureintheatmosphereasafunctionofheight,andh(t)istheheightofaweatherballoonasafunctionoftime,thenT(h(t))isthetemperatureatthelocationoftheweatherballoonasafunctionoftime.HighSchool–Functions-Linear,Quadratic,andExponentialModelsConstructandcomparelinear,quadratic,andexponentialmodelsandsolveproblems.Interpretexpressionsforfunctionsintermsofthesituationtheymodel.F-LE.1Distinguishbetweensituationsthatcanbemodeledwithlinearfunctionsandwithexponentialfunctions.
CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.2Reasonabstractlyandquantitatively.Mathematicallyproficientstudentsmakesenseofquantitiesandtheirrelationshipsinproblemsituations.Theybringtwocomplementaryabilitiestobearonproblemsinvolvingquantitativerelationships:theabilitytodecontextualize—toabstractagivensituationandrepresentitsymbolicallyandmanipulatetherepresentingsymbolsasiftheyhavealifeoftheirown,withoutnecessarilyattendingtotheirreferents—andtheabilitytocontextualize,topauseasneededduringthemanipulationprocessinordertoprobeintothereferentsforthesymbolsinvolved.Quantitativereasoningentailshabitsofcreatingacoherentrepresentationoftheproblemathand;consideringtheunitsinvolved;attendingtothemeaningofquantities,notjusthowtocomputethem;andknowingandflexiblyusingdifferentpropertiesofoperationsandobjects.MP.4 Modelwithmathematics.Mathematicallyproficientstudentscanapplythemathematicstheyknowtosolveproblemsarisingineverydaylife,society,andtheworkplace.Inearlygrades,thismightbeassimpleaswritinganadditionequationtodescribeasituation.Inmiddlegrades,astudentmightapplyproportionalreasoningtoplanaschooleventoranalyzeaprobleminthecommunity.Byhighschool,astudentmightusegeometrytosolveadesignproblemoruseafunctiontodescribehowonequantityofinterestdependsonanother.Mathematicallyproficientstudentswhocanapplywhattheyknowarecomfortablemakingassumptionsandapproximationstosimplifyacomplicatedsituation,realizingthatthesemayneedrevisionlater.Theyareabletoidentifyimportantquantitiesinapracticalsituationandmaptheirrelationshipsusingsuchtoolsasdiagrams,two-waytables,graphs,flowchartsandformulas.Theycananalyzethoserelationshipsmathematicallytodrawconclusions.Theyroutinelyinterprettheirmathematicalresultsinthecontextofthesituationandreflectonwhethertheresultsmakesense,possiblyimprovingthemodelifithasnotserveditspurpose.MP.5 Useappropriatetoolsstrategically.Mathematicallyproficientstudentsconsidertheavailabletoolswhensolvingamathematicalproblem.Thesetoolsmightincludepencilandpaper,concretemodels,aruler,aprotractor,acalculator,aspreadsheet,acomputeralgebrasystem,astatisticalpackage,ordynamicgeometrysoftware.Proficientstudentsaresufficientlyfamiliarwithtoolsappropriatefortheirgradeorcoursetomakesounddecisionsaboutwheneachofthesetoolsmightbehelpful,recognizingboththeinsighttobegainedandtheirlimitations.Forexample,mathematicallyproficienthighschoolstudentsanalyzegraphsoffunctionsandsolutionsgeneratedusingagraphingcalculator.Theydetectpossibleerrorsbystrategicallyusingestimationandothermathematicalknowledge.Whenmakingmathematicalmodels,theyknowthattechnologycanenablethemtovisualizetheresultsofvaryingassumptions,exploreconsequences,andcomparepredictionswithdata.Mathematicallyproficientstudentsatvariousgradelevelsareabletoidentifyrelevantexternalmathematicalresources,suchasdigitalcontentlocatedonawebsite,andusethemtoposeorsolveproblems.Theyareabletousetechnologicaltoolstoexploreanddeepentheirunderstandingofconcepts.
CCSSMAlignment:ProblemoftheMonth FirstRate©NoyceFoundation2015.ThisworkislicensedunderaCreativeCommonsAttribution-NonCommercial-NoDerivatives3.0UnportedLicense(http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
ProblemoftheMonth
FirstRateTaskDescription–PrimaryLevel
Thistaskchallengesstudentstoexploretheideaofmeasurementofdistance.Afterobservingtwostudentsjumpingagivendistance,theclassdiscussestheirobservationsanddetermineswhichstudentjumpedfartherthantheother.Thistaskchallengesstudentstodeterminewhowinsaracebetweentwobrothersjumpingupthestairwaywhereonebrotherjumpsfartherthantheother.Astudenthastoreasonbetweennumberofjumpsandlengthofjumps.
CommonCoreStateStandardsMath-ContentStandardsMeasurementandDataDescribeandcomparemeasurableattributes.K.MD.1Describemeasurableattributesofobjects,suchaslengthorweight.Describeseveralmeasurableattributesofasingleobject.K.MD.2Directlycomparetwoobjectswithameasurableattributeincommon,toseewhichobjecthas“moreof”/“lessof”theattribute,anddescribethedifference.Forexample,directlycomparetheheightsoftwochildrenanddescribeonechildastaller/shorter.Measurelengthsindirectlyandbyiteratinglengthunits.1.MD.2Expressthelengthofanobjectasawholenumberoflengthunits,bylayingmultiplecopiesofashorterobject(thelengthunit)endtoend;understandthatthelengthmeasurementofanobjectisthenumberofsame-sizelengthunitsthatspanitwithnogapsoroverlaps.Limittocontextswheretheobjectbeingmeasuredisspannedbyawholenumberoflengthunitswithnogapsoroverlaps.
CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.1Makesenseofproblemsandpersevereinsolvingthem.Mathematicallyproficientstudentsstartbyexplainingtothemselvesthemeaningofaproblemandlookingforentrypointstoitssolution.Theyanalyzegivens,constraints,relationships,andgoals.Theymakeconjecturesabouttheformandmeaningofthesolutionandplanasolutionpathwayratherthansimplyjumpingintoasolutionattempt.Theyconsideranalogousproblems,andtryspecialcasesandsimplerformsoftheoriginalprobleminordertogaininsightintoitssolution.Theymonitorandevaluatetheirprogressandchangecourseifnecessary.Olderstudentsmight,dependingonthecontextoftheproblem,transformalgebraicexpressionsorchangetheviewingwindowontheirgraphingcalculatortogettheinformationtheyneed.Mathematicallyproficientstudentscanexplaincorrespondencesbetweenequations,verbaldescriptions,tables,andgraphsordrawdiagramsofimportantfeaturesandrelationships,graphdata,andsearchforregularityortrends.Youngerstudentsmightrelyonusingconcreteobjectsorpicturestohelpconceptualizeandsolveaproblem.Mathematicallyproficientstudentschecktheiranswerstoproblemsusingadifferentmethod,andtheycontinuallyaskthemselves,“Doesthismakesense?”Theycanunderstandtheapproachesofotherstosolvingcomplexproblemsandidentifycorrespondencesbetweendifferentapproaches.MP.5 Useappropriatetoolsstrategically.Mathematicallyproficientstudentsconsidertheavailabletoolswhensolvingamathematicalproblem.Thesetoolsmightincludepencilandpaper,concretemodels,aruler,aprotractor,acalculator,aspreadsheet,acomputeralgebrasystem,astatisticalpackage,ordynamicgeometrysoftware.Proficientstudentsaresufficientlyfamiliarwithtoolsappropriatefortheirgradeorcoursetomakesounddecisionsaboutwheneachofthesetoolsmightbehelpful,recognizingboththeinsighttobegainedandtheirlimitations.Forexample,mathematicallyproficienthighschoolstudentsanalyzegraphsoffunctionsandsolutionsgeneratedusingagraphingcalculator.Theydetectpossibleerrorsbystrategicallyusingestimationandothermathematicalknowledge.Whenmakingmathematicalmodels,theyknowthattechnologycanenablethemtovisualizetheresultsofvaryingassumptions,exploreconsequences,andcomparepredictionswithdata.Mathematicallyproficientstudentsatvariousgradelevelsareabletoidentifyrelevantexternalmathematicalresources,suchasdigitalcontentlocatedonawebsite,andusethemtoposeorsolveproblems.Theyareabletousetechnologicaltoolstoexploreanddeepentheirunderstandingofconcepts.