problem of the month: first rate - inside mathematics

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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

ProblemoftheMonth FirstRate©NoyceFoundation2015.ThisworkislicensedunderaCreativeCommonsAttribution-NonCommercial-NoDerivatives3.0UnportedLicense(http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).

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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.


LevelBTomandDianestartedtorace.Tomtook4secondstorun6yards.Dianeran5yardsin3seconds.

Iftheycontinuedtorunatthesamespeeds,whowouldgetto30yardsfirst?Showhowyoufiguredoutyouranswer.Whorunsfaster?Howcanyoucomparetheirspeeds?


LevelCTheEnvironmentalClubatschoolattendsanannualcommunityclean-upevent.Theyhaverecyclinggames.Ateamisassignedanareaoflandthatisscatteredwithlitter.Thegoalisforapairofparticipantstocleanuptheareainthefastesttimepossible.

Tammy,workingalone, could cleanone-half thearea inonehour.Herpartner,Melissa -workingalone-couldcleanone-thirdoftheareainonehour.Duringthecontestwhentheyworktogether,howlongwill it takethemtocleanthearea?Explainhowyoufoundyoursolution.


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


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?


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.”


First Boy

Second Boy

CCSSMAlignment:ProblemoftheMonth FirstRate©NoyceFoundation2015.ThisworkislicensedunderaCreativeCommonsAttribution-NonCommercial-NoDerivatives3.0UnportedLicense(http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).

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.


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.



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.



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.



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.


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.

problem of the month: first rate - inside mathematics

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