Download Problem of the Month Rod Trains

ProblemoftheMonth
RodTrains
TheProblemsoftheMonth(POM)areusedinavarietyofwaystopromoteproblem
solvingandtofosterthefirststandardofmathematicalpracticefromtheCommon
CoreStateStandards:“Makesenseofproblemsandpersevereinsolvingthem.”The
POMmaybeusedbyateachertopromoteproblemsolvingandtoaddressthe
differentiatedneedsofherstudents.Adepartmentorgradelevelmayengagetheir
studentsinaPOMtoshowcaseproblemsolvingasakeyaspectofdoing
mathematics.Itcanalsobeusedschoolwidetopromoteaproblemsolvingthemeat
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
challengemosteighthandninthgradestudents,andLevelEshouldbechallenging
formosthighschoolstudents.Thesegrade‐levelexpectationsarejustestimatesand
shouldnotbeusedasanabsoluteminimumexpectationormaximumlimitationfor
students.Problemsolvingisalearnedskill,andstudentsmayneedmany
experiencestodeveloptheirreasoningskills,approaches,strategies,andthe
perseverancetobesuccessful.TheProblemoftheMonthbuildsonsequentiallevels
ofunderstanding.AllstudentsshouldexperienceLevelAandthenmovethroughthe
tasksinordertogoasdeeplyastheycanintotheproblem.Therewillbethose
studentswhowillnothaveaccessintoevenLevelA.Educatorsshouldfeelfreeto
modifythetasktoallowaccessatsomelevel.
Overview
IntheProblemoftheMonthRodTrains,studentsusemathematicalconceptsof
combinatorics,numbertheory,anddiscretemathematics.Themathematicaltopics
thatunderliethisPOMareknowledgeofnumbersense,numberpatterns,counting
principles,systematiccharting,andclosedformequations.Themathematicsthat
includescountingprinciplesandsystematicchartingisoftenreferredtoasdiscrete
mathematics.
InthefirstlevelsofthePOM,studentscomparethelengthofrodstodeterminea
numericalmeasurementofeachoftherods.Asonecontinuesthroughthelevels,
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studentsanalyzeproblemstodeterminethenumberoftrainsthatarecreatedby
arrangingtherodsindifferentordersanddifferentlengths.Inthefinallevelsofthe
POM,studentsarepresentedwithsituationsthatrequirethemtousecounting
principlesandorganizedliststodeterminethenumberofwaystrainscanbe
assembled.Inthefinallevel,studentsareaskedtogeneralizetheirfindingsinan
equationgivenatrainlengthofn.
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UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
Problem of the Month
Rod Trains
Level A:
You have 10 different rods - each a different color and a different length.
If you use just the red rods and put them together in a train (one next to each other), what
other length rods could you make? List the other rods by color. Explain why only some
rods work.
If the light green rod is 3 units long, determine the length of each of the other rods.
Explain the method you used to figure out the lengths.
Organize the rods in order from smallest to largest and draw each of them. Write the
length next to each of your drawings.
Problem of the Month
Rod Trains
Page 1
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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 B:
A rod train can be made with different sizes of rods. A rod train with a red rod first and a
purple rod second is different than a purple rod first and a red rod second. Which color
rod is the same length as a red rod next to a purple rod? What is the length of that rod?
How long is the brown rod?
Suppose you put two smaller rods together to make a rod train the same length as the
brown rod. How many different ways (order matters) can you put two rods together and
make it the same length as the brown?
Explain how you figured it out.
Write an addition number sentence for each of the combinations that you found.
What do you notice from the number sentences? Explain.
Problem of the Month
Rod Trains
Page 2
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License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
Level C:
Rod Trains can be just one rod, several rods of equal size, or several rods of differing
sizes. The order of the rods matters – different order makes Rod Trains unique from one
another. For example a Rod Train made up of a red on the left side and a purple on the
right is a different Rod Train from one that has a purple on the left side and a red on the
right.
Consider the yellow rod. Determine all the different combinations of rods that can be
arranged so that you have a rod train that is equal to the length of the yellow rod.
How many possible Rod Trains are equal in length to a yellow rod?
Explain your solution and the method you used to figure it out.
How do you know you have all the combinations?
Problem of the Month
Rod Trains
Page 3
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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 longest rod you have is the orange rod, which is 10 centimeters in length.
Determine the number of rod trains that make up each of the ten rods.
Illustrate or list the ten rods and all the rod trains that have equal length to that rod.
Explain your method for finding each set of rod trains.
How do you know you have them all?
What patterns or relationships do you see in the list of the sets of rod trains?
Problem of the Month
Rod Trains
Page 4
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License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
Level E:
Suppose you have rods whose length equals every natural number.
Determine the number of rod train combinations needed for a train of length n. Justify
mathematically how you got your answer.
Problem of the Month
Rod Trains
Page 5
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License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
Problem
of the Month
Primary Version Level A
Rod Trains
PrimaryVersionLevelA
Materials:Asetofrods(1‐10)foreachpair,paperandpencilstowriteordraw,
colorcrayons,markersorpencils
Discussionontherug:Teachergivesstudentsseveralrods.“Here are some
rods. What do you notice about them? What else do you notice about
them?”Teachercontinuestoaskchildrentonoticethattheyaredifferentcolorsand
differentlengths.Theteacherencouragesthestudentstoplaywiththemandmake
differentthings.
Insmallgroups:Eachgrouphasasetofrods.Teacherasksthefollowing
questions,goingontothenextquestiononlywhenstudentshavesuccess.
1.“How many rods do you have? How can you check to know for sure?”
Continueuntilyouthinkstudentsunderstandthatyouhaveten.If10istoomany,
youmightuseless‐suchas6.
2.“Put the rods in order of smallest to biggest?”
Attheendoftheinvestigationhavestudentsdrawapicturetorepresenttheir
solution.
3.“If this is a size 1 rod, how big are the others? Can you give each a
name?”Havestudentswriteanumbernexttoeachroddrawn.
4.“If you put red rods together, what other color rods can you make?
Explain.”
Problem of the Month
Rod Trains
Page 6
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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).
Cuisenaire Rods
Problem of the Month
Rod Trains
Page 7
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ProblemoftheMonth
RodTrains
TaskDescription–LevelA
Thistaskchallengesastudenttouserodsofdifferentlengths anduseoneasa“unit”ofmeasureto
determinethesizeoftheotherrods.Studentsalsocomparelengthsofrods.
CommonCoreStateStandardsMath‐ContentStandards
MeasurementandData
Measurelengthsindirectlyandbyiteratinglengthunits.
1.MD.1Orderthreeobjectsbylength;comparethelengthsoftwoobjectsindirectlybyusingathird
object.
1.MD.2Expressthelengthofanobjectasawholenumberoflengthunits,bylayingmultiplecopiesof
ashorterobject(thelengthunit)endtoend;understandthatthelengthmeasurementofanobjectis
thenumberofthesamesizelengthunitsthatspanitwithnogapsoroverlaps.
CommonCoreStateStandardsMath–StandardsofMathematicalPractice
MP.5Useappropriatetoolsstrategically.
Mathematicallyproficientstudentsconsidertheavailabletoolswhensolvingamathematical
problem.Thesetoolsmightincludepencilandpaper,concretemodels,aruler,aprotractor,a
calculator,aspreadsheet,acomputeralgebrasystem,astatisticalpackage,ordynamicgeometry
software.Proficientstudentsaresufficientlyfamiliarwithtoolsappropriatefortheirgradeorcourse
tomakesounddecisionsaboutwheneachofthesetoolsmightbehelpful,recognizingboththe
insighttobegainedandtheirlimitations.Forexample,mathematicallyproficienthighschool
studentsanalyzegraphsoffunctionsandsolutionsgeneratedusingagraphingcalculator.They
detectpossibleerrorsbystrategicallyusingestimationandothermathematicalknowledge.When
makingmathematicalmodels,theyknowthattechnologycanenablethemtovisualizetheresultsof
varyingassumptions,exploreconsequences,andcomparepredictionswithdata.Mathematically
proficientstudentsatvariousgradelevelsareabletoindentifyrelevantexternalmathematical
resources,suchasdigitalcontentlocatedonawebsite,andusethemtoposeorsolveproblems.They
areabletousetechnologicaltoolstoexploreanddeepentheirunderstandingofconcepts.
MP.6Attendtoprecision.
Mathematicallyproficientstudentstrytocommunicatepreciselytoothers.Theytrytouseclear
definitionsindiscussionwithothersandintheirownreasoning.Theystatethemeaningofsymbols
theychoose,includingusingtheequalsignconsistentlyandappropriately.Theyarecarefulabout
specifyingunitsofmeasure,andlabelingaxestoclarifythecorrespondencewithquantitiesina
problem.Theycalculateaccuratelyandefficiently,expressnumericalanswerswithadegreeof
precisionappropriatefortheproblemcontext.Intheelementarygrades,studentsgivecarefully
formulatedexplanationstoeachother.Bythetimetheyreachhighschooltheyhavelearnedto
examineclaimsandmakeexplicituseofdefinitions.
©NoyceFoundation2013.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
ProblemoftheMonth
RodTrains
TaskDescription–LevelB
Thistaskchallengesstudentstofindthetotalcombinationsoftworodsthatcanequalabrownrod
(length8)andrecordthecombinationsusingnumbersentences.Studentsarechallengedtolookfor
patternsinthenumbersentences.
CommonCoreStateStandardsMath‐ContentStandards
OperationsandAlgebraicThinking
Representandsolveproblemsinvolvingadditionandsubtraction.
1.OA.1Useadditionandsubtractionwithin20tosolvewordproblemsinvolvingsituationsofadding
to,takingfrom,puttingtogether,takingapart,andcomparingwithunknownsinallthepositions,e.g.
byusingobjects,drawings,andequationswithasymbolfortheunknownnumbertorepresentthe
problem.
Addandsubtractwithin20.
1.OA.5Relatecountingtoadditionandsubtraction.
MeasurementandData
Measurelengthsindirectlyandbyiteratinglengthunits.
1.MD.1Orderthreeobjectsbylength;comparethelengthsoftwoobjectsindirectlybyusingathird
object.
1.MD.2Expressthelengthofanobjectasawholenumberoflengthunits,bylayingmultiplecopiesof
ashorterobject(thelengthunit)endtoend;understandthatthelengthmeasurementofanobjectis
thenumberofthesamesizelengthunitsthatspanitwithnogapsoroverlaps.
StatisticsandProbability
Investigatechanceprocessanddevelop,use,andevaluateprobabilitymodels.
7.SP.8Representsamplespacesforcompoundeventsusingmethodssuchasorganizedlists,tables,
andtreediagrams.
CommonCoreStateStandardsMath–StandardsofMathematicalPractice
MP.5Useappropriatetoolsstrategically.
Mathematicallyproficientstudentsconsidertheavailabletoolswhensolvingamathematical
problem.Thesetoolsmightincludepencilandpaper,concretemodels,aruler,aprotractor,a
calculator,aspreadsheet,acomputeralgebrasystem,astatisticalpackage,ordynamicgeometry
software.Proficientstudentsaresufficientlyfamiliarwithtoolsappropriatefortheirgradeorcourse
tomakesounddecisionsaboutwheneachofthesetoolsmightbehelpful,recognizingboththe
insighttobegainedandtheirlimitations.Forexample,mathematicallyproficienthighschool
studentsanalyzegraphsoffunctionsandsolutionsgeneratedusingagraphingcalculator.They
detectpossibleerrorsbystrategicallyusingestimationandothermathematicalknowledge.When
makingmathematicalmodels,theyknowthattechnologycanenablethemtovisualizetheresultsof
varyingassumptions,exploreconsequences,andcomparepredictionswithdata.Mathematically
proficientstudentsatvariousgradelevelsareabletoindentifyrelevantexternalmathematical
resources,suchasdigitalcontentlocatedonawebsite,andusethemtoposeorsolveproblems.They
areabletousetechnologicaltoolstoexploreanddeepentheirunderstandingofconcepts.
MP.6Attendtoprecision.
Mathematicallyproficientstudentstrytocommunicatepreciselytoothers.Theytrytouseclear
definitionsindiscussionwithothersandintheirownreasoning.Theystatethemeaningofsymbols
theychoose,includingusingtheequalsignconsistentlyandappropriately.Theyarecarefulabout
specifyingunitsofmeasure,andlabelingaxestoclarifythecorrespondencewithquantitiesina
problem.Theycalculateaccuratelyandefficiently,expressnumericalanswerswithadegreeof
precisionappropriatefortheproblemcontext.Intheelementarygrades,studentsgivecarefully
formulatedexplanationstoeachother.Bythetimetheyreachhighschooltheyhavelearnedto
examineclaimsandmakeexplicituseofdefinitions.
©NoyceFoundation2013.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
ProblemoftheMonth
RodTrains
TaskDescription–LevelC
Thistaskchallengesastudenttofindthetotalcombinationsofrodsthatcanequalabrownrod
(length8)andrecordthecombinationsusingnumbersentences.Atthislevelthechallengeis
expandedtothepossibilityofusingmorethanonerodofagivencolorandmorethan2rods.The
studentmustalsomakeaconvincingargumentabouthavingfoundthetotalnumberofpossibilities.
CommonCoreStateStandardsMath‐ContentStandards
OperationsandAlgebraicThinking
Representandsolveproblemsinvolvingadditionandsubtraction.
1.OA.1Useadditionandsubtractionwithin20tosolvewordproblemsinvolvingsituationsofaddingto,taking
from,puttingtogether,takingapart,andcomparingwithunknownsinallthepositions,e.g.byusingobjects,
drawings,andequationswithasymbolfortheunknownnumbertorepresenttheproblem.
Addandsubtractwithin20.
1.OA.5Relatecountingtoadditionandsubtraction.
MeasurementandData
Measurelengthsindirectlyandbyiteratinglengthunits.
1.MD.1Orderthreeobjectsbylength;comparethelengthsoftwoobjectsindirectlybyusingathirdobject.
1.MD.2Expressthelengthofanobjectasawholenumberoflengthunits,bylayingmultiplecopiesofashorter
object(thelengthunit)endtoend;understandthatthelengthmeasurementofanobjectisthenumberofthe
samesizelengthunitsthatspanitwithnogapsoroverlaps.
StatisticsandProbability
Investigatechanceprocessanddevelop,use,andevaluateprobabilitymodels.
7.SP.8Representsamplespacesforcompoundeventsusingmethodssuchasorganizedlists,tables,andtree
diagrams.
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.3Constructviableargumentsandcritiquethereasoningofothers.
Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,andpreviously
establishedresultsinconstructingarguments.Theymakeconjecturesandbuildalogicalprogressionof
statementstoexplorethetruthoftheirconjectures.Theyareabletoanalyzesituationsbybreakingtheminto
cases,andcanrecognizeandusecounterexamples.Theyjustifytheirconclusions,communicatethemtoothers,
andrespondtotheargumentsofothers.Theyreasoninductivelyaboutdata,makingplausibleargumentsthat
takeintoaccountthecontextfromwhichthedataarose.Mathematicallyproficientstudentsarealsoableto
comparetheeffectivenessoftwoplausiblearguments,distinguishcorrectlogicorreasoningfromthatwhichis
flawed,and–ifthereisaflawinanargument–explainwhatitis.Elementarystudentscanconstructarguments
usingconcretereferentssuchasobjects,drawings,diagrams,andactions.Suchargumentscanmakesenseand
becorrect,eventhoughtheyarenotgeneralizedormadeformaluntillatergrades.Later,studentslearnto
determinedomainstowhichanargumentapplies.Studentsatallgradescanlistenorreadtheargumentsof
others,decidewhethertheymakesense,andaskusefulquestionstoclarifyorimprovethearguments.
©NoyceFoundation2013.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
ProblemoftheMonth
RodTrains
TaskDescription–LevelD
Thistaskchallengesastudenttofindallthecombinationsofrodtrainsthatcanmakeasize10rod.
Studentsneedtobeabletobreaktheproblemdownintosmallercasesandorganizethedatafor
eachcase.Studentsmustconstructaconvincingargumenttoshowhowtheyknowtheyhavefound
allthepossibilitiesandlookforpatternsorrelationshipsinthesetsofrodtrains.
CommonCoreStateStandardsMath‐ContentStandards
StatisticsandProbability
Investigatechanceprocessanddevelop,use,andevaluateprobabilitymodels.
7.SP.8Representsamplespacesforcompoundeventsusingmethodssuchasorganizedlists,tables,
andtreediagrams.
HighSchool–StatisticsandProbability–ConditionalProbabilityandtheRulesofProbability
Usetherulesofprobabilitytocomputeprobabilitiesofcompoundeventsinauniform
probabilitymodel.
S‐CP.9.Usepermutationsandcombinationstocomputeprobabilitiesofcompoundeventsandsolve
problems.
CommonCoreStateStandardsMath–StandardsofMathematicalPractice
MP.3Constructviableargumentsandcritiquethereasoningofothers.
Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,and
previouslyestablishedresultsinconstructingarguments.Theymakeconjecturesandbuildalogical
progressionofstatementstoexplorethetruthoftheirconjectures.Theyareabletoanalyze
situationsbybreakingthemintocases,andcanrecognizeandusecounterexamples.Theyjustify
theirconclusions,communicatethemtoothers,andrespondtotheargumentsofothers.They
reasoninductivelyaboutdata,makingplausibleargumentsthattakeintoaccountthecontextfrom
whichthedataarose.Mathematicallyproficientstudentsarealsoabletocomparetheeffectiveness
oftwoplausiblearguments,distinguishcorrectlogicorreasoningfromthatwhichisflawed,and–if
thereisaflawinanargument–explainwhatitis.Elementarystudentscanconstructarguments
usingconcretereferentssuchasobjects,drawings,diagrams,andactions.Suchargumentscanmake
senseandbecorrect,eventhroughtheyarenotgeneralizedormadeformaluntillatergrades.Later,
studentslearntodeterminedomainstowhichanargumentapplies.Studentsatallgradescanlisten
orreadtheargumentsofothers,decidewhethertheymakesense,andaskusefulquestionstoclarify
orimprovethearguments.
MP.5Useappropriatetoolsstrategically.
Mathematicallyproficientstudentsconsidertheavailabletoolswhensolvingamathematical
problem.Thesetoolsmightincludepencilandpaper,concretemodels,aruler,aprotractor,a
calculator,aspreadsheet,acomputeralgebrasystem,astatisticalpackage,ordynamicgeometry
software.Proficientstudentsaresufficientlyfamiliarwithtoolsappropriatefortheirgradeorcourse
tomakesounddecisionsaboutwheneachofthesetoolsmightbehelpful,recognizingboththe
insighttobegainedandtheirlimitations.Forexample,mathematicallyproficienthighschool
studentsanalyzegraphsoffunctionsandsolutionsgeneratedusingagraphingcalculator.They
detectpossibleerrorsbystrategicallyusingestimationandothermathematicalknowledge.When
makingmathematicalmodels,theyknowthattechnologycanenablethemtovisualizetheresultsof
varyingassumptions,exploreconsequences,andcomparepredictionswithdata.Mathematically
proficientstudentsatvariousgradelevelsareabletoindentifyrelevantexternalmathematical
resources,suchasdigitalcontentlocatedonawebsite,andusethemtoposeorsolveproblems.They
areabletousetechnologicaltoolstoexploreanddeepentheirunderstandingofconcepts.
©NoyceFoundation2013.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
ProblemoftheMonth
RodTrains
TaskDescription–LevelE
Thistaskchallengesastudenttofindaclosedequationforallthepossiblerodtrainsthatcouldequal
arodtrainoflengthn.
CommonCoreStateStandardsMath‐ContentStandards
ExpressionsandEquations
Applyandextendpreviousunderstandingsofarithmetictoalgebraicexpressions.
6.EE.1Writeandevaluatenumericalexpressionsinvolvingwhole‐numberexponents.
StatisticsandProbability
Investigatechanceprocessanddevelop,use,andevaluateprobabilitymodels.
7.SP.8Representsamplespacesforcompoundeventsusingmethodssuchasorganizedlists,tables,andtree
diagrams.
HighSchool–Algebra–CreatingEquations
Createequationsthatdescribenumbersorrelationships.
A‐CED.1Createequationsandinequalitiesinonevariableandusethemtosolveproblems,includeequations
arisingfromlinearandquadraticfunctions,andsimplerationandexponentialfunctions.
HighSchool–StatisticsandProbability–ConditionalProbabilityandtheRulesofProbability
Understandindependenceandconditionalprobabilityandusethemtointerpretdata.
S‐CP.1Describeeventsassubsetsofasamplespaceusingcharacteristicsoftheoutcomes,orasunions,
intersections,orcomplementsofotherevents(“or”,“and”,“not”).
S‐CP.4Constructandinterprettwo‐waytablesofdatawhentwocategoriesareassociatedwitheachobject
beingclassified.Usethetwo‐waytableasasamplespacetodecideifeventsareindependentandtoapproximate
conditionalprobabilities.
Usetherulesofprobabilitytocomputeprobabilitiesofcompoundeventsinauniformprobability
model.
S‐CP.9Usepermutationsandcombinationstocomputeprobabilitiesofcompoundeventsandsolveproblems.
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.
©NoyceFoundation2013.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
ProblemoftheMonth
RodTrains
TaskDescription–PrimaryLevel
Thistaskchallengesastudenttocountandcomparerodsofdifferentsizesandreasonabouthow
largerrodscanbemadewithsmallerrods.
CommonCoreStateStandardsMath‐ContentStandards
MeasurementandData
Measurelengthsindirectlyandbyiteratinglengthunits.
1.MD.1Orderthreeobjectsbylength;comparethelengthsoftwoobjectsindirectlybyusingathird
object.
1.MD.2Expressthelengthofanobjectasawholenumberoflengthunits,bylayingmultiplecopiesof
ashorterobject(thelengthunit)endtoend;understandthatthelengthmeasurementofanobjectis
thenumberofthesamesizelengthunitsthatspanitwithnogapsoroverlaps.
CommonCoreStateStandardsMath–StandardsofMathematicalPractice
MP.5Useappropriatetoolsstrategically.
Mathematicallyproficientstudentsconsidertheavailabletoolswhensolvingamathematical
problem.Thesetoolsmightincludepencilandpaper,concretemodels,aruler,aprotractor,a
calculator,aspreadsheet,acomputeralgebrasystem,astatisticalpackage,ordynamicgeometry
software.Proficientstudentsaresufficientlyfamiliarwithtoolsappropriatefortheirgradeorcourse
tomakesounddecisionsaboutwheneachofthesetoolsmightbehelpful,recognizingboththe
insighttobegainedandtheirlimitations.Forexample,mathematicallyproficienthighschool
studentsanalyzegraphsoffunctionsandsolutionsgeneratedusingagraphingcalculator.They
detectpossibleerrorsbystrategicallyusingestimationandothermathematicalknowledge.When
makingmathematicalmodels,theyknowthattechnologycanenablethemtovisualizetheresultsof
varyingassumptions,exploreconsequences,andcomparepredictionswithdata.Mathematically
proficientstudentsatvariousgradelevelsareabletoindentifyrelevantexternalmathematical
resources,suchasdigitalcontentlocatedonawebsite,andusethemtoposeorsolveproblems.They
areabletousetechnologicaltoolstoexploreanddeepentheirunderstandingofconcepts.
MP.6Attendtoprecision.
Mathematicallyproficientstudentstrytocommunicatepreciselytoothers.Theytrytouseclear
definitionsindiscussionwithothersandintheirownreasoning.Theystatethemeaningofsymbols
theychoose,includingusingtheequalsignconsistentlyandappropriately.Theyarecarefulabout
specifyingunitsofmeasure,andlabelingaxestoclarifythecorrespondencewithquantitiesina
problem.Theycalculateaccuratelyandefficiently,expressnumericalanswerswithadegreeof
precisionappropriatefortheproblemcontext.Intheelementarygrades,studentsgivecarefully
formulatedexplanationstoeachother.Bythetimetheyreachhighschooltheyhavelearnedto
examineclaimsandmakeexplicituseofdefinitions.
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