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Partners for Mathematics Learning

1. PARTNERS for Mathematics Learning Grade Four Module 5. Partners for Mathematics Learning. 2. Data Essential Standards  Look at statistics and probability across the grades  What do you notice?  What are the similarities and differences between grades 3, 4 and 5?. Partners

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Partners for Mathematics Learning

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  1. 1 PARTNERS forMathematicsLearning GradeFour Module5 Partners forMathematicsLearning

  2. 2 DataEssentialStandards Lookatstatisticsandprobabilityacross thegrades Whatdoyounotice? Whatarethesimilaritiesanddifferences betweengrades3,4and5? Partners forMathematicsLearning

  3. 3 BigIdeas Datacanbeeithercategoricalornumerical Pose,Collect,Analyze,Interpret(PCAI)is amodelfortheprocessofstatistical investigations Differentrepresentationsandgraphs classifyandcommunicatedata Understandingbasicconceptsof probabilityallowsustomakemore accuratepredictions Partners forMathematicsLearning

  4. 4

  5. 5 NumericalData Valuesthatarenumberssuchascounts, measurements,andratings Representobjectsorindividualsby numbersassignedtocertainmeasurable propertiesExamples: Numberofchildreninfamilies Pulseratesoftopathletes Timeinminutesthatstudentsspend watchingtelevisioneachday Numberofpets Partners forMathematicsLearning

  6. 6 PCAIModel Partners forMathematicsLearning

  7. 7 PCAIStep1:PosetheQuestion Identifyaspecificquestionto exploreanddecidewhat datatocollecttoaddress thequestion Partners forMathematicsLearning

  8. 8 PCAIStep2:CollecttheData Studentsshould Collecttheirowndata Haveaplanfordatacollection Understandwheredatacomefrom Partners forMathematicsLearning

  9. 9 FlawsinDataCollection Whatkindsofthings needtobeconsidered whencollectingdata? Partners forMathematicsLearning

  10. 10 What’stheProblem? Billyconductsasurveytodetermine computerownershipoutsideacomputer repairshop Whymightthedatacollectedinthis samplebeinaccurate? Partners forMathematicsLearning

  11. 11 What’stheProblem? Keishasurveyseveryhouseina5block radiusofcityhallaskingifsubsidiesshould becontinuedfordairyfarmers Whymightthedatacollectedinthis samplebeinaccurate? Partners forMathematicsLearning

  12. 12 What’stheProblem? Alocalradiostation askslistenerstocallin andexpresstheirviews onexpandingthelocalairport Whymightthedatacollected inthissamplebeinaccurate? Partners forMathematicsLearning

  13. 13 What’stheProblem? Fromthepagesofhistory… Inthe1936presidentialcampaignbetween FDRandAlfredLandon,awell-respected publicationconductedasurveysendingten millionballotstoasamplepopulation selectedfromclubmemberships,telephone directories,andmagazinesubscriptions ItincorrectlydeterminedthatLandonwould win;FDRwonbyalandslide Partners forMathematicsLearning

  14. 14 ImplementingthePCAIModel “GettingtoKnowUs”projectwillinclude… Part1 Posingaquestionandcollectingdata Part2 Analyzingandinterpretthedata Part3 Presentation Partners forMathematicsLearning

  15. 15 FormingTeams Formteamsof3-4people Theteamscanbeallgrade4andallgrade 5orteamswhoarefromthesameLEA Partners forMathematicsLearning

  16. 16 ImplementingthePCAIModel “GettingtoKnowUs”-First… Generateatleastonenumericalquestion Formulateanhypothesis Preparetosampletwogroupsofpeople Partners forMathematicsLearning

  17. 17 ImplementingthePCAIModel “GettingtoKnowUs” Second… Collectthedata Posethequestiontotwodifferentpopulations Partners forMathematicsLearning

  18. 18 ImplementingthePCAIModel “GettingtoKnowUs” Maintainnotes abouttheprocess Bepreparedtoshare yourexperience Thinkaboutimplementing thiswithstudents

  19. 19 ImplementingthePCAIModel “GettingtoKnowUs” Atanothersessionwewill… Analyzethedata Interpretthedata Presentthedatatotheclass Partners forMathematicsLearning

  20. 20 ImplementingthePCAIModel “GettingtoKnowUs” Thingstokeepinmind… Havedatacollectedby… Anyquestions? Partners forMathematicsLearning

  21. 21 PARTNERS forMathematicsLearning Movinginto fractions Partners forMathematicsLearning

  22. 22 WhatAboutThoseFractions? BigFractionIdeas Fractionalpartsareequalsharesorequal-sized partsofawhole Fractionalpartshavespecialnamesthattellhow manyequalpartsareinthewhole Fractionsymbols(numeratoranddenominator) tellhowmanyandwhat Themorefractionalpartstomakeupthewhole, thesmallertheparts EquivalentFractions–differentfractionsexpress thesameamount Partners forMathematicsLearning

  23. 23 Fractions:WhatCanTheyMean? Values Fractionsarerationalnumbersandcanbe ordered Operators Findingafractionalpartofavalue,afraction asadivisionproblem Ratios Acomparisonoftwoquantities Partners forMathematicsLearning

  24. 24 FractionActions Compareandorderfractions FractionComputation Addition,Subtraction, Multiplication(fractionalpartsofwhole numbers)     Fractionasdivision Mixednumbersandequivalentfractions Fractionsinproblemsolvingsituations Fractionsrelatedtodecimals Partners forMathematicsLearning

  25. 25 Fractions:TeachingandLearning Whatmakesfractionssohardfor students?  Taught tooabstractlywithlimitedmodels Taughtwithrotememorizationof procedures Nottaughtinmeaningfulcontexts Moreattentiontoalgorithmsratherthanto developingnumbersenseandreasoning Partners forMathematicsLearning

  26. 26 FractionNumberSense:Representations Area/RegionModels LinearorMeasurementModels SetModels Symbols(withmeaning) 3 4 7 8 1 2 Partners forMathematicsLearning

  27. 27 2 3 howmany what FractionSymbols Thebottomnumbertellsuswhatisbeing counted–howmanypartswearetalkingabout Itcountsthepartsorsharesofawhole DenominatorisfromtheLatinwordfor“namer” Thetopnumberisthecountingnumber Ittellshowmanysharesorpartswearetalking about Itcountsthepartsorsharesbeingconsidered NumeratorisfromtheLatinwordfor“number” Partners forMathematicsLearning

  28. 28 Closeto… Nameafractioncloseto1butnotmorethan1 Nameafractionthatisevencloserto1thanthat Whydoyoubelieveitiscloser? Nameafractionthatisevencloserthanthe previousfraction Again… Partners forMathematicsLearning

  29. 29 FractionsonaNumberLine Numberlinesshowrelativemagnitudeof fractions Wherewouldyouplace1/3? 1 0abcde Whatfractionwouldbeatpointa? Howfarapartareaandb? Partners forMathematicsLearning

  30. 30 HowFull? Estimatehowfulleachcontaineris: Whatfractionofeachcontainerholdsliquid? Partners forMathematicsLearning

  31. 31 ComparingFractions Whichisgreater?ExplainandTest Partners forMathematicsLearning

  32. 32 FourthsonaGeoboard Partners forMathematicsLearning

  33. 33 FractionalParts–NotAlwaysCongruent Partners forMathematicsLearning

  34. 34 WhatPartIsRed? Whatpartisred? green?blue?yellow? Howdoyouknow? Whymightastudentsay 2/9ofthefigureisred? Ifyoudoublethisfigure, whatfractionisred? Partners forMathematicsLearning

  35. 35 FractionalPartsofaWhole Take24chips Dividethemintothirdsifyoucan Howmanyinonethird? …twothirds? Dividethemintofourthsifyoucan Howmanyinonefourth? …threefourths? Partners forMathematicsLearning

  36. 36 FractionalPartsofaWhole Take24chips Dividethemintofifthsifyoucan Whycan’tyoudividethemintofifths? Intowhatotherfractionalpartscan24 bedivided? Partners forMathematicsLearning

  37. 37 SetModel Caution:Whenpartitioningsets,children frequentlyconfusethenumberofcountersina sharewiththenumberofshares Example: Say:Divide12countersintofourths: (Thechildcorrectlymakesfourequalgroups.) Say:Showmethreefourths (Somechildrenwhocorrectlydividedthesetintofourequalgroups abovewillnowregroupthe12chipsintothreegroupsoffour.) Partners forMathematicsLearning

  38. 38 SetModel Say:"Showmethree-fourthsof12 Correctexample: Incorrectexample: Partners forMathematicsLearning

  39. 39 Fractions:ParttoWhole Makethewholelineifthisisonethird Makethewholeshapeifthisisthree fourths Partners forMathematicsLearning

  40. 40 Fractions:ParttoWhole Thewholeline... Thewholeshapemightlooklike... Partners forMathematicsLearning

  41. 41 Fractions:ParttoWhole Setmodel Ifthisistwofifthsofaset,makethewholeset Think:Howmanymoonsinonefifthofthe set? Partners forMathematicsLearning

  42. 42 Fractions:ParttoWhole Setmodel Thewholeset:fivefifths(fivegroupsof three) Partners forMathematicsLearning

  43. 43 FractionsGreaterthanOne Howmuchisshaded? Howcouldyounametheamountasa fraction? Asawholenumberandafraction? Partners forMathematicsLearning

  44. 44 FractionsGreaterthanOne Howmuchisshaded? Howcouldyouprovethis: 3 4 15 4 3 = Partners forMathematicsLearning

  45. 45 EquivalentFractions TheConcept: Twofractionsareequivalentiftheyare representationsforthesameamountor quantity–iftheyarethesamenumber TheAlgorithm: Multiplyordividethetopandbottom numbersbythesamenonzeronumber Intuitivemethodsarealwaysbestatfirst VandeWalleandLovin,TeachingStudent-CenteredMathematics,Grades3-5 Partners forMathematicsLearning

  46. 46 EquivalentFractions Allstudentsshouldeventuallybeabletowritean equivalentfractionforagivenfraction Atthesametime,therulesshouldneverbe taughtoruseduntilthestudentsunderstand whattheresultmeans Inaproblem-basedclassroom,studentscan developanunderstandingofequivalentfractions andalsodevelopfromthatunderstandinga conceptuallybasedalgorithm Partners forMathematicsLearning

  47. 47 EquivalentFractions DevelopingtheConcept Therateseries:aproblembasedapproach Partners forMathematicsLearning

  48. 48 EquivalentFractions DevelopingtheConcept TheRateSeries:AProblemBasedApproach 3x2=6 5x2=10 15x2=30 25x2=50 3x10=30 5x10=50 Partners forMathematicsLearning

  49. 49 ConstructingUnderstanding Developmentandgrowthoccurthroughan enormouslycomplicatedandcontinuous processofinteractionwiththeenvironment Throughactivitythepersondiscovers understandingbyre-inventingwhatheor shewantstounderstand Thisprocesstakestime -JeanPiaget Partners forMathematicsLearning

  50. 50 FractionComputation Aproblem-basednumbersenseapproach Beginwithsimpletasksincontexts Connectthemeaningoffractioncomputation withwhole-numbercomputation Developstrategiesusingestimationand informalmethods Usemodelstoexploreeachoftheoperations AdaptedfromVandeWalleandLovin,TeachingStudent-CenteredMathematics,2006 Partners forMathematicsLearning

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