Partners for Mathematics Learning

1. 1 PARTNERS forMathematicsLearning Grade1 Module6 Partners forMathematicsLearning

2. 2 OverviewofSession Fluencywithfactsand operations Equipartitioning Problemsolving Professionalreading Problemtypes Partners forMathematicsLearning

3. 3 Getto25 Startat0;firstperson enters1,2,3,4,or5 Secondpersonmayadd1,2,3,4,or5 ONLYifthenumberis(a)notshowingin thedisplayor(b)isnotthesumofthe digitsinthedisplaywindow Taketurnsaddingnumbers Winneristhepersonwhoreaches25 Partners forMathematicsLearning

4. 4 Getto25 Doesitmatterwhogoes firstinthegame? Isthereanumbertoavoid? Isthereanumbertotrytoreach? Whatstrategiesdidyouuseintryingto reach25? Partners forMathematicsLearning

5. 5 BigIdeainNumber:Fluency Fluency(accuracy,efficiency,flexibility)is reasoningaboutandusingrational numberoperationswithunderstanding Fluencyinvolvesknowingstrategiesfor retrievingbasicfactsandbeingabletoapply theminothercomputations Fluencyisbuiltuponnumberrelationships, placevalue,properties,andoperation understandings forMathematicsLearning

6. 6 FluencywithNumberRelationships Magnitudeofnumbers:64isgreaterthan63 Onemorethananynumberisthenextcounting number:49+1issimilartocounting49,50 Onelessthananynumberistheprevious countingnumber:49-1isknowingthat48 comesbefore49 Countingbyconsistentgroups(ex.3’sor5’s) tonamethemultiplesofthatnumber Partners forMathematicsLearning

7. 7 Fluency:KeyBehaviors Studentswhoarefluentusenumberfacts withoutbeingprompted Flexibility:3+4hasthesamesumas4+3 Efficiently:4+5is4+4+1(not4+1+1+1+1=1) Accurately:8-5=4can’tberightbecause4+4=8) Knowledgeofsumsto10 iscriticalforsuccesswith additionandsubtractionof multi-digitnumbers Partners forMathematicsLearning

8. 8 DevelopingFluencyinClassroom ConstanceKamii’sresearchonlearning numbercombinationsfoundthat1stgrade studentsdemonstrated 55%AccuracyintheMemorizationClass 76%AccuracyintheRelational ThinkingClass Partners forMathematicsLearning

9. 9 Part-Part-WholeRelationships Toconceptualizeanumberasbeingmade upoftwoormorepartsisthemostimportant relationshipthatcanbedevelopedabout numbers Sixandthreearethe sameamountasnine Partners forMathematicsLearning

10. 10 HowManyObjectsDoYouSee? Partners forMathematicsLearning

11. 11 HowCouldYouSolveThis? Isolveditthisway…..Iknewthat4+4=8 4+3+4+2= 8 ThenIaddedthe2tothe8toget10 8+3+2= 10 So10+3is13andthat’stheanswer! 10+3=13 Partners forMathematicsLearning

12. 12 DropandDecide Viewhandoutworksheets Oncestudentsunderstandformat,use sheetsmultipletimesfordifferenttarget numbers GrabBag Concretetosymbolicinaworksheet Multipleaddendspresentaunique challengeforsomestudents Partners forMathematicsLearning

13. 13 SpinningMoreorLess Partners forMathematicsLearning

14. 14 TenTurnsRolling Reviewhandout Twostudentsshareapairofdice Eachstudentmayhaveaworksheetor studentsmayshare Studentscanverifysums Roll,record,andadd Howcouldtheactivitybeusedinacenter? Partners forMathematicsLearning

15. 15 6 KnowIt!ShowIt! 3 Ingroupsofthree,read throughthehandoutandmakecertain everyoneunderstandsthedirections Playthreeorfourrounds,exchangingroles When,duringtheyear,mightyouhave studentsreadytoplaythisgame? Partners forMathematicsLearning

16. 16 RelationalThinking 12-7= 7+__=12 9+5=10+4 Partners forMathematicsLearning

17. 17 NumberTalks… Areclassconversationsaboutarithmetic problemsanddiscussionsthatcritique solutionstrategies Areworkdonementallywithminimum writingtorecordstrategies Helpstudentslearnbasicfactsthrough reasoninganddiscussion(notisolateddrill) Havestudentsusereasoningtodetermine ifastrategyiseffective Partners forMathematicsLearning

18. 18 78+95 …NumberTalks Provideopportunitiesforchildrentoshare howtheythinkaboutnumbers Increasefluencyinoperationswithsmall numbersinordertoincreasefluencywith largenumbers Canbeadaptedforanygroup 801-347 26x52 Partners forMathematicsLearning

19. 19 NumberTalks:StudentDirections Solvetheprobleminyourhead Putyourthumbupinfrontofyourchest whenyouhaveasolution Trytosolveinadifferentway Foreachdifferentsolution,putupanother finger Shareyoursolutionswithyourpartner Partners forMathematicsLearning

20. 20 What’sMyNumber? Guessthesecretnumberonanumberline Ifguessistoobig,thelargetriangleisplaced abovetheguess Iftheguessistoosmall,thesmallertriangle isplacedabovethenumber Whatdoyouknowaboutnumbersbetween? 012345678910 Partners forMathematicsLearning

21. 21 TryThisOne… 4243444546474849 Partners forMathematicsLearning

22. 22 NumberoftheDay 6+6 5+5+1+1 dozen 12 dimeand 2pennies 13-1 7+5 Partners forMathematicsLearning

23. 23 MeaningfulFactStrategies One-more-thanandtwo-more-than Countinguporcountingback Factswithzero Doublesandneardoubles Maketenfacts Factfamilies part-part Commutativepropertywhole Compensation Partners forMathematicsLearning        

25. 25 DevelopingFactFluency Expectations: Byendofgrade1,fluencywithaddition andrelatedsubtractionfactsto10 Explorefactsto18 Exploresumsbeyond10bybuildingon10s Byendofgrade2,fluency withadditionandrelated subtractionfactsto20 Partners forMathematicsLearning

26. 26 AssessmentIsKey Determiningeachstudent’sunderstanding ofnumbercombinations Knowingwhichfactseachchildhasnotyet learnedandreassessingweeklytoencourage studentstomoveon Helpingstudentstake responsibilityfor learningfacts Informingparents withspecifics Partners forMathematicsLearning

28. 28 HowDoYouRecognizeFluency? Talkwithapartneraboutnumberfluency in1stgrade Whatdostudentsdothatletsyouknowthey havenumberfluency? Whatdostudentsdothatlets youknowtheydonothave numberfluency? Howdoyoukeeptrack? Partners forMathematicsLearning

29. 29 SupportingYourWork Considerusingthepowerpoint,discussion notes,andotherarticlesforCrossingfrom Grade1intoGrade2ingrade-levelmeetings Eachpersoncouldbring afavoritenumberfact gametoshare Discussionsofcommontasks andstudentworkalso supportsteacherplanning Partners forMathematicsLearning

30. 30 FairSharesforTwo FairSharesforJulieandJennifer ReferbacktoliteraturebookJustLikeMe Discusshowsisterssharedthings Whathappenedtothe“leftovers”? WhichnumberscouldJulieand Jennifersharefairly? Partners forMathematicsLearning

31. 31 BumpyorNotBumpy Cutoutthetwo-columncards Whatcanyoutellaboutthepieces? Howcanyouorganizethesepieces? Orderpieces Sortthepiecesintotwogroups;givetherule Determineifyournumbersareorarenot bumpy Howdoyouknow? Partners forMathematicsLearning

32. 32 EvenorOdd Evennumber:anamountthatcanbe madeoftwoequalpartswithnoleftovers Oddnumber:anamountthatcannotbe madeupoftwoequalparts Observableattribute:NOTadefinition Numbersendingin0,2,4,6,8 Numbersendingin1,3,5,7,9 Partners forMathematicsLearning

33. 33 “SenseofBalance” Eachshapehasauniquewhole numberweightmorethan0 Differentshapeshavedifferent weights;identicalshapeshavesameweights Thesizeofshapesisnotrelatedtoweight Ashapehangingdirectlybelowthefulcrumdoes notaffectthebalanceofarmstoleftorrightofthe fulcrum Distancefromfulcrumdoesnotaffectweight Partners forMathematicsLearning

34. 34 “SenseofBalance” Totalweightis32units Eachshapeweighslessthan10units Partners forMathematicsLearning

35. 35 “SenseofBalance” Whatamounts(24,25,26)couldnot bethetotalweightforthispuzzle? Howdoyouknow? Partners forMathematicsLearning

36. 36 Equipartitioning Equipartitioningreferstodividingawhole intoequalpartsormakingfairshares Childrenoftenbegintothinkofpartitioning intermsofsharingwithfriendsandeach gettingthesameamount Partners forMathematicsLearning

37. 37 PaperFolding–Part1 Howmanytimesdoyouthinkyoucanfold thepattypaperinhalf? Howmanyequalpartswouldbecreated withthenumberoffoldsyoupredicted? Explainyourreasoning Beginfolding! Recordthetotalnumber offoldsandthenumberofpartscreated Partners forMathematicsLearning

38. 38 Equipartitioning-Part2 Whatwouldhappenifyoufoldedinthirds eachtimeinsteadofhalves? Whatwouldhappenifyoufoldedthepaper inhalfandtheninthirds? Howmanydifferentwayscouldyoufolda pieceofpapertocreate24equalparts? Whichwayusestheleastnumberoffolds? Thegreatest? Partners forMathematicsLearning

39. 39 Fair“Sharing” Wereeveryone’sfoldsthesame? Whatsimilaritiesanddifferencesarethere inthehalffoldsandthethirdfolds? Ifyouhavelargerpaper,could youfolditinhalfmoretimes thanyoucanfoldsmallpaper? Partners forMathematicsLearning

40. 40 Equipartitioning LookattheEssentialStandardthatrelates toequipartitioning Whatexpectationsdotheclarifyingobjectives identify? Whatcanyoulearnfromtheassessment prototypes? Whataspectsofthisstandardarenewfor yourclassroom? Partners forMathematicsLearning

41. 41 Equipartitioning Noticehowspatialreasoning,equality, measurement,andnumbercometogether inthisstandard Languageassociatedwithdivisionand fractionsareneededindiscussingthis standard,butfraction(symbolic)notationis NOTthepurpose Partners forMathematicsLearning

42. 42 ProblemBasedLessons Researchhasindicatedthat beginningwithproblem situationsyieldsgreater problem-solvingcompetence andequalorbetter computationalcompetence forMathematicsLearning

43. 43 BasicStructureofProblems AccordingtoCGI,therearefourbasic structuresforproblems,regardlessofthe magnitudeofnumbers JoinProblems SeparateProblems Part-Part-WholeProblems CompareProblems Recognizingtheproblemstructuresand identifyingonesthatseemtobemore difficultforstudentsishelpfulinplanning forMathematicsLearning

44. 44 JoinProblems Join:ResultUnknown Sandrahad8pennies.Georgegaveher4more. HowmanypenniesdoesSandrahavealtogether? Join:ChangeUnknown Sandrahad8pennies.Georgegavehersome more.NowSandrahas12pennies.Howmany didGeorgegiveher? Join:StartUnknown Sandrahadsomepennies.Georgegaveher4 more.NowSandrahas12pennies.Howmany penniesdidSandrahavetobeginwith? forMathematicsLearning

45. 45 SeparateProblems Separate:ResultUnknown Sandrahad12marbles.Shegave4marblesto George.HowmanymarblesdoesSandrahavenow? Separate:ChangeUnknown Sandrahad12marbles.ShegavesometoGeorge. Nowshehas8marbles.Howmanydidshegiveto George? Separate:StartUnknown Sandrahadsomemarbles.Shegave4toGeorge. NowSandrahas8marblesleft.Howmanymarbles didSandrahavetobeginwith? Partners forMathematicsLearning

46. 46 Part-Part-WholeProblems Part-Part-Whole:WholeUnknown Georgehas4penniesand8nickels.How manycoinsdoeshehave? Part-Part-Whole:PartUnknown Georgehas12coins.Eightofhiscoinsare pennies,andtherestarenickels.Howmany nickelsdoesGeorgehave? Partners forMathematicsLearning

47. 47 CompareProblems Compare:DifferenceUnknown Georgehas12penniesandSandrahas8 pennies.Howmanymorepenniesdoes GeorgehavethanSandra? Compare:LargerorSmallerUnknown Georgehas4morepenniesthanSandra. Sandrahas8pennies.Howmanypennies doesGeorgehave? Georgehas4morepenniesthanSandra. Georgehas12pennies.Howmanypennies doesSandrahave? Partners forMathematicsLearning

48. 48 Teacher’sRoleinProblemSolving     Choosesappropriateproblems Listenstostudents’strategies Asksquestionsandsupportsthinking Ensuresthatdifferentstrategiesareshared Howstrategiesarealike Howstrategiesaredifferent Encouragesstudentstosolve problemsinmorethanoneway Givesfeedback Partners forMathematicsLearning

49. 49 ProfessionalReading MakingtheMostofStoryProblems TeachingChildrenMathematics December2008/January2009 Pages260-266 NationalCouncilofTeachersofMathematics Preparetoshareyourassignedpart Firstportionofarticle SupportingandExtendingMathematicalThinking Partners forMathematicsLearning

50. 50 DebriefingTeacher’sRole Sharewithothersatyourtablethemain pointsfromyoursectionofthearticle Whatisdifferentaboutteachers’comments andquestionsbeforecorrectanswersare givenandaftercorrectanswersaregiven? Whatthreethingsimpressed youthemost? Whatideaswillyoutrythisyear? Partners forMathematicsLearning