Partners for Mathematics Learning

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

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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