Download Problem of the Month: Fair Games

ProblemoftheMonth:FairGames
TheProblemsoftheMonth(POM)areusedinavarietyofwaystopromote
problem‐solvingandtofosterthefirststandardofmathematicalpracticefromthe
CommonCoreStateStandards:“Makesenseofproblemsandpersevereinsolving
them.”ThePOMmaybeusedbyateachertopromoteproblem‐solvingandto
addressthedifferentiatedneedsofherstudents.Adepartmentorgradelevelmay
engagetheirstudentsinaPOMtoshowcaseproblem‐solvingasakeyaspectof
doingmathematics.Itcanalsobeusedschoolwidetopromoteaproblem‐solving
themeataschool.Thegoalisforallstudentstohavetheexperienceofattacking
andsolvingnon‐routineproblemsanddevelopingtheirmathematicalreasoning
skills.Althoughobtainingandjustifyingsolutionstotheproblemsistheobjective,
theprocessoflearningtoproblem‐solveisevenmoreimportant.
TheProblemoftheMonthisstructuredtoprovidereasonabletasksforallstudents
inaschool.ThestructureofaPOMisashallowfloorandahighceiling,sothatall
studentscanproductivelyengage,struggle,andpersevere.ThePrimaryVersion
LevelAisdesignedtobeaccessibletoallstudentsandespeciallythekeychallenge
forgradesK–1.LevelAwillbechallengingformostsecondandthirdgraders.
LevelBmaybethelimitofwherefourthandfifthgradestudentshavesuccessand
understanding.LevelCmaystretchsixthandseventhgradestudents.LevelDmay
challengemosteighthandninthgradestudents,andLevelEshouldbechallenging
formosthighschoolstudents.Thesegrade‐levelexpectationsarejustestimates
andshouldnotbeusedasanabsoluteminimumexpectationormaximumlimitation
forstudents.Problem‐solvingisalearnedskill,andstudentsmayneedmany
experiencestodeveloptheirreasoningskills,approaches,strategies,andthe
perseverancetobesuccessful.TheProblemoftheMonthbuildsonsequentiallevels
ofunderstanding.AllstudentsshouldexperienceLevelAandthenmovethroughthe
tasksinordertogoasdeeplyastheycanintotheproblem.Therewillbethose
studentswhowillnothaveaccessintoevenLevelA.Educatorsshouldfeelfreeto
modifythetasktoallowaccessatsomelevel.
Overview:
IntheProblemoftheMonthFairGames,studentsusemathematicalconceptsof
probabilityandexpectedvalue.ThemathematicaltopicsthatunderliethisPOMare
knowledgeofsamplespaces,fairness,probabilityratios,experimentaland
theoreticalprobability,countingprinciples/strategies,andexpectedvalue.
InthefirstlevelofthePOM,studentsplayaracegameusinganunfairspinner.The
studentsthroughexperimentingdeterminewhetherthegameisfairforallplayers
andwhoisfavored.InpartB,thestudentskeeptrackofthewinsandlossesofeach
player,thengraphthedataanddeterminewhoismorelikelytowinbasedonthe
ProblemoftheMonth
FairGames
©NoyceFoundation2013.
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UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
experimentalresults.Thestudentsalsoexamineareasofthespinnertoanswer
questionsregardinglikelihoodofwinningtherace.InlevelC,studentsare
presentedwithacasewheretwoprobabilitiesareindependentandareaskedto
determinewhethertheraceisfairandtoexplaintheirconclusions.InlevelD,the
studentsareaskedtoredesignthespinnerinordertomakethegamefair.Inthe
finallevelstudentsareaskedtoredesigntheraceboardtomakethegamefairby
usingexpectedvaluewithunequalprobabilitiesbalancedwithlongerdistances.
MathematicalConcepts:
Probabilityisthemathematicsisusedtoevaluatesituationsinvolvingchance.A
probabilityratioisanumericvalueusedtojudgethelikelihoodofanevent
occurring.Theprobabilityratioisanumberbetweenandincluding0and1.An
eventthatwillalwayshappenhasaprobabilityof1or100%,andtheprobabilityof
animpossibleeventoccurringis0.Theprobabilityofaneventoccurringasoftenas
itdoesnotoccuris1/2or50%.Tocalculatetheprobabilitythataneventoccurs,
findtheratioofthenumberofwaysthataneventcanoccurtothetotalnumberof
possibleoutcomes.
Example:Whatistheprobabilitythatifyoufliptwofaircoins,theybothcomeup
heads?Theprobabilityratioiscalculatedbyexaminingtheoutcomes.Thereare
fourwaysthatthecoinscanland–headsandheads,headsandtails,tailsandheads
tailsandtails.However,onlyoneofthefouroccurrencesproducesasuccessful
outcome.Therefore,theratioisoneoutoffour,1/4or25%chance.
ProblemoftheMonth
FairGames
©NoyceFoundation2013.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
Problem of the Month
Fair Games
The Race
Rules:
There are three players: Brick, Stripe, and Diamond. Each player puts a token on
the Start square of their pattern path. The players take turns by spinning the
spinner. The player whose pattern comes up on the spinner moves their token one
space on the game board. The other players do not move their tokens. The game
continues as each player takes turns with the spinner and a move is made, until one
player reaches the Finish Line. That player wins.
Start
Finish
Problem of the Month
Fair Games
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).
Problem of the Month
Fair Games
Level A:
What is a fair game? Explain.
In groups of three, play The Race game five times. Keep track of who won
each game, who came in second and who came in third.
Which player won the most? How many times?
Which player came in second the most?
Is this a fair game? Explain why this game is fair or not fair.
How could you change this game to make it better?
Problem of the Month
Fair Games
Page 2
© Noyce Foundation 2013.
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:
Play The Race game five times. This time keep track of each spin, listing
who moved each time.
Draw a bar graph showing the outcome of the spins. Label the horizontal
axis by pattern and the vertical axis by the number of spins that occurred.
How many more times did Stripe move than Brick?
Between Brick and Diamond, which player moved more times in all 10
games?
How much of the spinner is a stripe pattern?
How much of the spinner is a diamond pattern?
How much of the spinner is a brick pattern?
How could you change the spinner to make the game more fair?
Problem of the Month
Fair Games
Page 3
© Noyce Foundation 2013.
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 Unported
License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
Level C:
Your friend Alex says that he can make the game more fair. He makes a
second spinner with numbers on it. He says the number stand for the
number of spaces a token is moved. He modified the rules as follows: First
the pattern spinner is spun to find out who moves. Then the number spinner
is spun to find out how many spaces the token is moved. Below is the
spinner he made.
2
1
2
What is the probability that Diamond comes up on the first spinner? Explain.
What is the probability that 1 comes up on the second spinner? Explain.
What is the probability that Stripe moves 1 space on any turn? Explain.
What is the probability that Brick moves 2 spaces on any turn? Explain.
Alex says that the new spinner makes the game fair because Stripe will
mostly move 1 space and the Brick and Diamond players will mostly move
2 spaces. Explain why you either agree or disagree with Alex.
Problem of the Month
Fair Games
Page 4
© Noyce Foundation 2013.
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:
Another friend, Dolores, says that she made the game fair by drawing just
one new spinner, but using both patterns and numbers on the same spinner.
When the spinner is spun, the player with that pattern moves the number of
spaces indicated in that sector of the spinner. She said she made the spinner
by first drawing the diameter and then making the central angle of the brick
sector 120 degrees.
What is the probability of Brick moving on any spin?
How many times would Diamond have to move to win a game?
How much bigger in area is the Stripe sector than the Diamond sector?
Dolores’ spinner makes the game unfair. Use mathematics to explain why
her game is unfair.
How could you change the spinner to make her game fair?
Problem of the Month
Fair Games
Page 5
© Noyce Foundation 2013.
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 Unported
License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
Level E:
Dolores has already made several copies of her spinner. You don’t want to
hurt her feelings by not using her spinner, but you need to make sure the
game is fair. You decide to make a new track in the shape of an oval where
racers near the inside of the track have fewer spaces to travel than racers
near the outside of the track. Design an oval track that can be used with
Dolores’ spinner. Design the game board track so that the game will be fair
to all players, but Dolores’ spinner isn’t changed.
Use mathematics to justify why your game board makes the game fair to all
players.
Problem of the Month
Fair Games
Page 6
© Noyce Foundation 2013.
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 Unported
License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
Problem of the Month
Fair Games
Primary Version Level A
Materials: The Race game board and spinner; paper clip and
brad to make the spinner; and Brick, Stripe, and Diamond
tokens for each group.
Discussion on the rug: (Teacher starts a discussion about fair
games.) “What makes a game fair? Tell us about a game that
is unfair.” (Teacher demonstrates how to play the game.)
“Each player takes a turn spinning the spinner. The pattern
the spinner points to gets to move. They move their racer
one square. Then it is time for the next player to spin and
that pattern gets to move next. Keep playing until someone
gets to the finish line.”
In small groups: (Each group plays the game several times,
switching patterns after each game. Have them indicate the
pattern that was first, second and then third for each game.)
(After the games, the teacher polls the class and records the
outcomes for the entire class. Teacher asks the following
questions.)
“Which pattern won the most? How many times?
Which pattern came in second most?
Is this a fair game?
Why this game is fair or not fair?
How could you change this game to make it better?”
(At the end of the investigation, have students either discuss
or dictate a response to the summary questions above.)
Problem of the Month
Fair Games
Page 7
© Noyce Foundation 2013.
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).
ProblemoftheMonth
FairGames
TaskDescription–LevelA
Thistaskchallengesastudenttodiscusswhatisafairgame,toplayagameseveraltimes,tocollect
data,todeterminethefairnessofthegameandtodiscusshowtochangethegametomakeitmore
fair.
CommonCoreStateStandardsMath‐ContentStandards
MeasurementandData
Representandinterpretdata.
1.MD.4Organize,represent,andinterpretdatawithuptothreecategories;askandanswerquestions
aboutthetotalnumberofdatapoints,howmanyineachcategory,andhowmanymoreorlessarein
onecategorythaninanother.
CommonCoreStateStandardsMath–StandardsofMathematicalPractice
MP.1Makesenseofproblemsandpersevereinsolvingthem.
Mathematicallyproficientstudentsstartbyexplainingtothemselvesthemeaningofaproblemand
lookingforentrypointstoitssolution.Theyanalyzegivens,constraints,relationships,andgoals.
Theymakeconjecturesabouttheformandmeaningofthesolutionandplanasolutionpathway
ratherthansimplyjumpingintoasolutionattempt.Theyconsideranalogousproblems,andtry
specialcasesandsimplerformsoftheoriginalprobleminordertogaininsightintoitssolution.They
monitorandevaluatetheirprogressandchangecourseifnecessary.Olderstudentsmight,depending
onthecontextoftheproblem,transformalgebraicexpressionsorchangetheviewingwindowon
theirgraphingcalculatortogettheinformationtheyneed.Mathematicallyproficientstudentscan
explaincorrespondencesbetweenequations,verbaldescriptions,tables,andgraphsordraw
diagramsofimportantfeaturesandrelationships,graphdata,andsearchforregularityortrends.
Youngerstudentsmightrelyonusingconcreteobjectsorpicturestohelpconceptualizeandsolvea
problem.Mathematicallyproficientstudentschecktheiranswerstoproblemsusingadifferent
method,andtheycontinuallyaskthemselves,“Doesthismakesense?”Theycanunderstandthe
approachesofotherstosolvingcomplexproblemsandidentifycorrespondencesbetweendifferent
approaches.
MP.3 Constructviableargumentsandcritiquethereasoningofothers.
Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,and
previouslyestablishedresultsinconstructingarguments.Theymakeconjecturesandbuildalogical
progressionofstatementstoexplorethetruthoftheirconjectures.Theyareabletoanalyze
situationsbybreakingthemintocases,andcanrecognizeandusecounterexamples.Theyjustify
theirconclusions,communicatethemtoothers,andrespondtotheargumentsofothers.Theyreason
inductivelyaboutdata,makingplausibleargumentsthattakeintoaccountthecontextfromwhich
thedataarose.Mathematicallyproficientstudentsarealsoabletocomparetheeffectivenessoftwo
plausiblearguments,distinguishcorrectlogicorreasoningfromthatwhichisflawed,and—ifthereis
aflawinanargument—explainwhatitis.Elementarystudentscanconstructargumentsusing
concretereferentssuchasobjects,drawings,diagrams,andactions.Suchargumentscanmakesense
andbecorrect,eventhoughtheyarenotgeneralizedormadeformaluntillatergrades.Later,
studentslearntodeterminedomainstowhichanargumentapplies.Studentsatallgradescanlisten
orreadtheargumentsofothers,decidewhethertheymakesense,andaskusefulquestionstoclarify
orimprovethearguments.
CCSSMAlignment:ProblemoftheMonth
FairGames
©NoyceFoundation2013.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
ProblemoftheMonth
FairGames
TaskDescription–LevelB
ThistaskchallengesastudenttoplaythesamegameasinLevelA,butkeepingtrackofwhomoved
eachtimeforeachspin.Astudentwillmakeabargraphandanswerquestionsbasedontheirdata.
CommonCoreStateStandardsMath‐ContentStandards
MeasurementandData
Representandinterpretdata.
1.MD.4Organize,represent,andinterpretdatawithuptothreecategories;askandanswerquestions
aboutthetotalnumberofdatapoints,howmanyineachcategory,andhowmanymoreorlessarein
onecategorythaninanother.
2.MD.10Drawapicturegraphandabargraph(withsingle‐unitscale)torepresentadatasetwithup
tofourcategories.Solvesimpleput‐together,take‐apart,andcompareproblemsusinginformation
presentedinabargraph.
3.MD.3Drawascaledpicturegraphandascaledbargraphtorepresentadatasetwithseveral
categories.Solveone‐andtwo‐step“howmanymore”and“howmanyless”problemsusing
informationpresentedinscaledbargraphs.Forexample,drawabargraphinwhicheachsquarein
thebargraphmightrepresent5pets.
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.3 Constructviableargumentsandcritiquethereasoningofothers.
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.
CCSSMAlignment:ProblemoftheMonth
FairGames
©NoyceFoundation2013.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
ProblemoftheMonth
FairGames
TaskDescription–LevelC
Thistaskchallengesastudenttodeterminewhetherthegameisreallymorefairbaseduponchanges
madebyAlex.Probabilitiesaredetermined,andstudentsthenneedtojustifywhetherornotAlex’s
changesreallydomakethegamemorefair.Thetwoprobabilitiesareindependent.
CommonCoreStateStandardsMath‐ContentStandards
StatisticsandProbability
Investigatechanceprocessesanddevelop,use,andevaluateprobabilitymodels.
7.SP.5Understandthattheprobabilityofachanceeventisanumberbetween0and1thatexpressesthe
likelihoodoftheeventoccurring.Largernumbersindicategreaterlikelihood.Aprobabilitynear0
indicatesanunlikelyevent,aprobabilityaround1/2indicatesaneventthatisneitherunlikelynorlikely,
andaprobabilitynear1indicatesalikelyevent.
7.SP.6Approximatetheprobabilityofachanceeventbycollectingdataonthechanceprocessthat
producesitandobservingitslong‐runrelativefrequency,andpredicttheapproximaterelativefrequency
giventheprobability.Forexample,whenrollinganumbercube600times,predictthata3or6wouldbe
rolledroughly200times,butprobablynotexactly200times.
7.SP.7Developaprobabilitymodelanduseittofindprobabilitiesofevents.Compareprobabilitiesfroma
modeltoobservedfrequencies;iftheagreementisnotgood,explainpossiblesourcesofthediscrepancy.
7.SP.8Findprobabilitiesofcompoundeventsusingorganizedlists,tables,treediagrams,andsimulation.
c.Designanduseasimulationtogeneratefrequenciesforcompoundevents
CommonCoreStateStandardsMath–StandardsofMathematicalPractice
MP.1Makesenseofproblemsandpersevereinsolvingthem.
Mathematicallyproficientstudentsstartbyexplainingtothemselvesthemeaningofaproblemandlookingforentry
pointstoitssolution.Theyanalyzegivens,constraints,relationships,andgoals.Theymakeconjecturesabouttheform
andmeaningofthesolutionandplanasolutionpathwayratherthansimplyjumpingintoasolutionattempt.They
consideranalogousproblems,andtryspecialcasesandsimplerformsoftheoriginalprobleminordertogaininsight
intoitssolution.Theymonitorandevaluatetheirprogressandchangecourseifnecessary.Olderstudentsmight,
dependingonthecontextoftheproblem,transformalgebraicexpressionsorchangetheviewingwindowontheir
graphingcalculatortogettheinformationtheyneed.Mathematicallyproficientstudentscanexplaincorrespondences
betweenequations,verbaldescriptions,tables,andgraphsordrawdiagramsofimportantfeaturesandrelationships,
graphdata,andsearchforregularityortrends.Youngerstudentsmightrelyonusingconcreteobjectsorpicturesto
helpconceptualizeandsolveaproblem.Mathematicallyproficientstudentschecktheiranswerstoproblemsusinga
differentmethod,andtheycontinuallyaskthemselves,“Doesthismakesense?”Theycanunderstandtheapproaches
ofotherstosolvingcomplexproblemsandidentifycorrespondencesbetweendifferentapproaches.
MP.3 Constructviableargumentsandcritiquethereasoningofothers.
Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,andpreviouslyestablished
resultsinconstructingarguments.Theymakeconjecturesandbuildalogicalprogressionofstatementstoexplorethe
truthoftheirconjectures.Theyareabletoanalyzesituationsbybreakingthemintocases,andcanrecognizeanduse
counterexamples.Theyjustifytheirconclusions,communicatethemtoothers,andrespondtotheargumentsof
others.Theyreasoninductivelyaboutdata,makingplausibleargumentsthattakeintoaccountthecontextfromwhich
thedataarose.Mathematicallyproficientstudentsarealsoabletocomparetheeffectivenessoftwoplausible
arguments,distinguishcorrectlogicorreasoningfromthatwhichisflawed,and—ifthereisaflawinanargument—
explainwhatitis.Elementarystudentscanconstructargumentsusingconcretereferentssuchasobjects,drawings,
diagrams,andactions.Suchargumentscanmakesenseandbecorrect,eventhoughtheyarenotgeneralizedormade
formaluntillatergrades.Later,studentslearntodeterminedomainstowhichanargumentapplies.Studentsatall
gradescanlistenorreadtheargumentsofothers,decidewhethertheymakesense,andaskusefulquestionstoclarify
orimprovethearguments.
CCSSMAlignment:ProblemoftheMonth
FairGames
©NoyceFoundation2013.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
ProblemoftheMonth:
FairGames
TaskDescription–LevelD
Thistaskchallengesastudenttodeterminewhetherthegameisreallymorefairbaseduponchanges
madebyDolores.ProbabilitiesaredeterminedandstudentsthenneedtojustifywhyDolores’changes
makethegamemoreunfair.
CommonCoreStateStandardsMath‐ContentStandards
StatisticsandProbability
Investigatechanceprocessesanddevelop,use,andevaluateprobabilitymodels.
7.SP.5Understandthattheprobabilityofachanceeventisanumberbetween0and1thatexpressesthe
likelihoodoftheeventoccurring.Largernumbersindicategreaterlikelihood.Aprobabilitynear0
indicatesanunlikelyevent,aprobabilityaround1/2indicatesaneventthatisneitherunlikelynorlikely,
andaprobabilitynear1indicatesalikelyevent.
7.SP.6Approximatetheprobabilityofachanceeventbycollectingdataonthechanceprocessthat
producesitandobservingitslong‐runrelativefrequency,andpredicttheapproximaterelativefrequency
giventheprobability.Forexample,whenrollinganumbercube600times,predictthata3or6wouldbe
rolledroughly200times,butprobablynotexactly200times.
7.SP.7Developaprobabilitymodelanduseittofindprobabilitiesofevents.Compareprobabilitiesfroma
modeltoobservedfrequencies;iftheagreementisnotgood,explainpossiblesourcesofthediscrepancy.
7.SP.8Findprobabilitiesofcompoundeventsusingorganizedlists,tables,treediagrams,andsimulation.
c.Designanduseasimulationtogeneratefrequenciesforcompoundevents
CommonCoreStateStandardsMath–StandardsofMathematicalPractice
MP.1Makesenseofproblemsandpersevereinsolvingthem.
Mathematicallyproficientstudentsstartbyexplainingtothemselvesthemeaningofaproblemandlookingforentry
pointstoitssolution.Theyanalyzegivens,constraints,relationships,andgoals.Theymakeconjecturesabouttheform
andmeaningofthesolutionandplanasolutionpathwayratherthansimplyjumpingintoasolutionattempt.They
consideranalogousproblems,andtryspecialcasesandsimplerformsoftheoriginalprobleminordertogaininsight
intoitssolution.Theymonitorandevaluatetheirprogressandchangecourseifnecessary.Olderstudentsmight,
dependingonthecontextoftheproblem,transformalgebraicexpressionsorchangetheviewingwindowontheir
graphingcalculatortogettheinformationtheyneed.Mathematicallyproficientstudentscanexplaincorrespondences
betweenequations,verbaldescriptions,tables,andgraphsordrawdiagramsofimportantfeaturesandrelationships,
graphdata,andsearchforregularityortrends.Youngerstudentsmightrelyonusingconcreteobjectsorpicturesto
helpconceptualizeandsolveaproblem.Mathematicallyproficientstudentschecktheiranswerstoproblemsusinga
differentmethod,andtheycontinuallyaskthemselves,“Doesthismakesense?”Theycanunderstandtheapproaches
ofotherstosolvingcomplexproblemsandidentifycorrespondencesbetweendifferentapproaches.
MP.3 Constructviableargumentsandcritiquethereasoningofothers.
Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,andpreviouslyestablished
resultsinconstructingarguments.Theymakeconjecturesandbuildalogicalprogressionofstatementstoexplorethe
truthoftheirconjectures.Theyareabletoanalyzesituationsbybreakingthemintocases,andcanrecognizeanduse
counterexamples.Theyjustifytheirconclusions,communicatethemtoothers,andrespondtotheargumentsof
others.Theyreasoninductivelyaboutdata,makingplausibleargumentsthattakeintoaccountthecontextfromwhich
thedataarose.Mathematicallyproficientstudentsarealsoabletocomparetheeffectivenessoftwoplausible
arguments,distinguishcorrectlogicorreasoningfromthatwhichisflawed,and—ifthereisaflawinanargument—
explainwhatitis.Elementarystudentscanconstructargumentsusingconcretereferentssuchasobjects,drawings,
diagrams,andactions.Suchargumentscanmakesenseandbecorrect,eventhoughtheyarenotgeneralizedormade
formaluntillatergrades.Later,studentslearntodeterminedomainstowhichanargumentapplies.Studentsatall
gradescanlistenorreadtheargumentsofothers,decidewhethertheymakesense,andaskusefulquestionstoclarify
orimprovethearguments.
CCSSMAlignment:ProblemoftheMonth
FairGames
©NoyceFoundation2013.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
ProblemoftheMonth
FairGames
TaskDescription–LevelE
ThistaskchallengesastudenttodesignanewtracktousewithDolores’spinnertomakethegame
fairbyusingexpectedvaluewithunequalprobabilitiesbalancedwithlongerdistances.
CommonCoreStateStandardsMath‐ContentStandards
StatisticsandProbability
Investigatechanceprocessesanddevelop,use,andevaluateprobabilitymodels.
7.SP.5Understandthattheprobabilityofachanceeventisanumberbetween0and1thatexpresses
thelikelihoodoftheeventoccurring.Largernumbersindicategreaterlikelihood.Aprobabilitynear
0indicatesanunlikelyevent,aprobabilityaround1/2indicatesaneventthatisneitherunlikelynor
likely,andaprobabilitynear1indicatesalikelyevent.
7.SP.6Approximatetheprobabilityofachanceeventbycollectingdataonthechanceprocessthat
producesitandobservingitslong‐runrelativefrequency,andpredicttheapproximaterelative
frequencygiventheprobability.Forexample,whenrollinganumbercube600times,predictthata3
or6wouldberolledroughly200times,butprobablynotexactly200times.
7.SP.7Developaprobabilitymodelanduseittofindprobabilitiesofevents.Compareprobabilities
fromamodeltoobservedfrequencies;iftheagreementisnotgood,explainpossiblesourcesofthe
discrepancy.
7.SP.8Findprobabilitiesofcompoundeventsusingorganizedlists,tables,treediagrams,and
simulation.
c.Designanduseasimulationtogeneratefrequenciesforcompoundevents
HighSchool–ProbabilityandStatistics‐UsingProbabilitytoMakeDecisions
Calculateexpectedvaluesandusethemtosolveproblems
S‐MD.1(+)Definearandomvariableforaquantityofinterestbyassigninganumericalvaluetoeach
eventinasamplespace;graphthecorrespondingprobabilitydistributionusingthesamegraphical
displaysasfordatadistributions.
S‐MD.2(+)Calculatetheexpectedvalueofarandomvariable;interpretitasthemeanofthe
probabilitydistribution.
S‐MD.3(+)Developaprobabilitydistributionforarandomvariabledefinedforasamplespacein
whichtheoreticalprobabilitiescanbecalculated;findtheexpectedvalue.Forexample,findthe
theoreticalprobabilitydistributionforthenumberofcorrectanswersobtainedbyguessingonall
fivequestionsofamultiple‐choicetestwhereeachquestionhasfourchoices,andfindtheexpected
gradeundervariousgradingschemes.
S‐MD.4(+)Developaprobabilitydistributionforarandomvariabledefinedforasamplespacein
whichprobabilitiesareassignedempirically;findtheexpectedvalue.Forexample,findacurrent
datadistributiononthenumberofTVsetsperhouseholdintheUnitedStates,andcalculatethe
expectednumberofsetsperhousehold.HowmanyTVsetswouldyouexpecttofindin100randomly
selectedhouseholds?
CommonCoreStateStandardsMath–StandardsofMathematicalPractice
MP.1Makesenseofproblemsandpersevereinsolvingthem.
Mathematicallyproficientstudentsstartbyexplainingtothemselvesthemeaningofaproblemandlookingfor
entrypointstoitssolution.Theyanalyzegivens,constraints,relationships,andgoals.Theymakeconjectures
abouttheformandmeaningofthesolutionandplanasolutionpathwayratherthansimplyjumpingintoa
solutionattempt.Theyconsideranalogousproblems,andtryspecialcasesandsimplerformsoftheoriginal
CCSSMAlignment:ProblemoftheMonth
FairGames
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probleminordertogaininsightintoitssolution.Theymonitorandevaluatetheirprogressandchangecourseif
necessary.Olderstudentsmight,dependingonthecontextoftheproblem,transformalgebraicexpressionsor
changetheviewingwindowontheirgraphingcalculatortogettheinformationtheyneed.Mathematically
proficientstudentscanexplaincorrespondencesbetweenequations,verbaldescriptions,tables,andgraphsor
drawdiagramsofimportantfeaturesandrelationships,graphdata,andsearchforregularityortrends.Younger
studentsmightrelyonusingconcreteobjectsorpicturestohelpconceptualizeandsolveaproblem.
Mathematicallyproficientstudentschecktheiranswerstoproblemsusingadifferentmethod,andthey
continuallyaskthemselves,“Doesthismakesense?”Theycanunderstandtheapproachesofotherstosolving
complexproblemsandidentifycorrespondencesbetweendifferentapproaches.
MP.3 Constructviableargumentsandcritiquethereasoningofothers.
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.
MP.4 Modelwithmathematics.
Mathematicallyproficientstudentscanapplythemathematicstheyknowtosolveproblemsarisingineveryday
life,society,andtheworkplace.Inearlygrades,thismightbeassimpleaswritinganadditionequationto
describeasituation.Inmiddlegrades,astudentmightapplyproportionalreasoningtoplanaschooleventor
analyzeaprobleminthecommunity.Byhighschool,astudentmightusegeometrytosolveadesignproblemor
useafunctiontodescribehowonequantityofinterestdependsonanother.Mathematicallyproficientstudents
whocanapplywhattheyknowarecomfortablemakingassumptionsandapproximationstosimplifya
complicatedsituation,realizingthatthesemayneedrevisionlater.Theyareabletoidentifyimportantquantities
inapracticalsituationandmaptheirrelationshipsusingsuchtoolsasdiagrams,two‐waytables,graphs,
flowchartsandformulas.Theycananalyzethoserelationshipsmathematicallytodrawconclusions.They
routinelyinterprettheirmathematicalresultsinthecontextofthesituationandreflectonwhethertheresults
makesense,possiblyimprovingthemodelifithasnotserveditspurpose.
CCSSMAlignment:ProblemoftheMonth
FairGames
©NoyceFoundation2013.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).
ProblemoftheMonth
FairGames
TaskDescription–PrimaryLevel
Thistaskchallengesastudenttoexploretheideaofthefairnessofagame.Thelessonbeginswitha
discussionwiththeclassontheirideasofafairorunfairgame.Astudentischallengedtoplaya
gamegivenspecificrules.Aftertheclassplaysthegame,theteacherrecordstheoutcomesordata
fortheentireclass.Discussionquestionsareaskedofthestudentssuchas,“Isthisafairgame?”and
“Whyisthisgamefairornotfair?”
CommonCoreStateStandardsMath‐ContentStandards
MeasurementandData
Representandinterpretdata.
1.MD.4Organize,represent,andinterpretdatawithuptothreecategories;askandanswer
questionsaboutthetotalnumberofdatapoints,howmanyineachcategory,andhowmanymoreor
lessareinonecategorythaninanother.
CommonCoreStateStandardsMath–StandardsofMathematicalPractice
MP.1Makesenseofproblemsandpersevereinsolvingthem.
Mathematicallyproficientstudentsstartbyexplainingtothemselvesthemeaningofaproblemand
lookingforentrypointstoitssolution.Theyanalyzegivens,constraints,relationships,andgoals.
Theymakeconjecturesabouttheformandmeaningofthesolutionandplanasolutionpathway
ratherthansimplyjumpingintoasolutionattempt.Theyconsideranalogousproblems,andtry
specialcasesandsimplerformsoftheoriginalprobleminordertogaininsightintoitssolution.They
monitorandevaluatetheirprogressandchangecourseifnecessary.Olderstudentsmight,depending
onthecontextoftheproblem,transformalgebraicexpressionsorchangetheviewingwindowon
theirgraphingcalculatortogettheinformationtheyneed.Mathematicallyproficientstudentscan
explaincorrespondencesbetweenequations,verbaldescriptions,tables,andgraphsordraw
diagramsofimportantfeaturesandrelationships,graphdata,andsearchforregularityortrends.
Youngerstudentsmightrelyonusingconcreteobjectsorpicturestohelpconceptualizeandsolvea
problem.Mathematicallyproficientstudentschecktheiranswerstoproblemsusingadifferent
method,andtheycontinuallyaskthemselves,“Doesthismakesense?”Theycanunderstandthe
approachesofotherstosolvingcomplexproblemsandidentifycorrespondencesbetweendifferent
approaches.
MP.3 Constructviableargumentsandcritiquethereasoningofothers.
Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,and
previouslyestablishedresultsinconstructingarguments.Theymakeconjecturesandbuildalogical
progressionofstatementstoexplorethetruthoftheirconjectures.Theyareabletoanalyze
situationsbybreakingthemintocases,andcanrecognizeandusecounterexamples.Theyjustify
theirconclusions,communicatethemtoothers,andrespondtotheargumentsofothers.Theyreason
inductivelyaboutdata,makingplausibleargumentsthattakeintoaccountthecontextfromwhich
thedataarose.Mathematicallyproficientstudentsarealsoabletocomparetheeffectivenessoftwo
plausiblearguments,distinguishcorrectlogicorreasoningfromthatwhichisflawed,and—ifthereis
aflawinanargument—explainwhatitis.Elementarystudentscanconstructargumentsusing
concretereferentssuchasobjects,drawings,diagrams,andactions.Suchargumentscanmakesense
andbecorrect,eventhoughtheyarenotgeneralizedormadeformaluntillatergrades.Later,
studentslearntodeterminedomainstowhichanargumentapplies.Studentsatallgradescanlisten
orreadtheargumentsofothers,decidewhethertheymakesense,andaskusefulquestionstoclarify
orimprovethearguments.
CCSSMAlignment:ProblemoftheMonth
FairGames
©NoyceFoundation2013.
ThisworkislicensedunderaCreativeCommonsAttribution‐NonCommercial‐NoDerivatives3.0
UnportedLicense(http://creativecommons.org/licenses/by‐nc‐nd/3.0/deed.en_US).