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Explore fluency with facts and operations, equipartitioning, problem solving, and professional reading in this module. Learn different problem types and strategies for reaching 25.

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## Partners for Mathematics Learning

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**1**PARTNERS forMathematicsLearning Grade1 Module6 Partners forMathematicsLearning**2**OverviewofSession Fluencywithfactsand operations Equipartitioning Problemsolving Professionalreading Problemtypes Partners forMathematicsLearning**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**Getto25 Doesitmatterwhogoes firstinthegame? Isthereanumbertoavoid? Isthereanumbertotrytoreach? Whatstrategiesdidyouuseintryingto reach25? Partners forMathematicsLearning**5**BigIdeainNumber:Fluency Fluency(accuracy,efficiency,flexibility)is reasoningaboutandusingrational numberoperationswithunderstanding Fluencyinvolvesknowingstrategiesfor retrievingbasicfactsandbeingabletoapply theminothercomputations Fluencyisbuiltuponnumberrelationships, placevalue,properties,andoperation understandings forMathematicsLearning**6**FluencywithNumberRelationships Magnitudeofnumbers:64isgreaterthan63 Onemorethananynumberisthenextcounting number:49+1issimilartocounting49,50 Onelessthananynumberistheprevious countingnumber:49-1isknowingthat48 comesbefore49 Countingbyconsistentgroups(ex.3’sor5’s) tonamethemultiplesofthatnumber Partners forMathematicsLearning**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**DevelopingFluencyinClassroom ConstanceKamii’sresearchonlearning numbercombinationsfoundthat1stgrade studentsdemonstrated 55%AccuracyintheMemorizationClass 76%AccuracyintheRelational ThinkingClass Partners forMathematicsLearning**9**Part-Part-WholeRelationships Toconceptualizeanumberasbeingmade upoftwoormorepartsisthemostimportant relationshipthatcanbedevelopedabout numbers Sixandthreearethe sameamountasnine Partners forMathematicsLearning**10**HowManyObjectsDoYouSee? Partners forMathematicsLearning**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**DropandDecide Viewhandoutworksheets Oncestudentsunderstandformat,use sheetsmultipletimesfordifferenttarget numbers GrabBag Concretetosymbolicinaworksheet Multipleaddendspresentaunique challengeforsomestudents Partners forMathematicsLearning**13**SpinningMoreorLess Partners forMathematicsLearning**14**TenTurnsRolling Reviewhandout Twostudentsshareapairofdice Eachstudentmayhaveaworksheetor studentsmayshare Studentscanverifysums Roll,record,andadd Howcouldtheactivitybeusedinacenter? Partners forMathematicsLearning**15**6 KnowIt!ShowIt! 3 Ingroupsofthree,read throughthehandoutandmakecertain everyoneunderstandsthedirections Playthreeorfourrounds,exchangingroles When,duringtheyear,mightyouhave studentsreadytoplaythisgame? Partners forMathematicsLearning**16**RelationalThinking 12-7= 7+__=12 9+5=10+4 Partners forMathematicsLearning**17**NumberTalks… Areclassconversationsaboutarithmetic problemsanddiscussionsthatcritique solutionstrategies Areworkdonementallywithminimum writingtorecordstrategies Helpstudentslearnbasicfactsthrough reasoninganddiscussion(notisolateddrill) Havestudentsusereasoningtodetermine ifastrategyiseffective Partners forMathematicsLearning**18**78+95 …NumberTalks Provideopportunitiesforchildrentoshare howtheythinkaboutnumbers Increasefluencyinoperationswithsmall numbersinordertoincreasefluencywith largenumbers Canbeadaptedforanygroup 801-347 26x52 Partners forMathematicsLearning**19**NumberTalks:StudentDirections Solvetheprobleminyourhead Putyourthumbupinfrontofyourchest whenyouhaveasolution Trytosolveinadifferentway Foreachdifferentsolution,putupanother finger Shareyoursolutionswithyourpartner Partners forMathematicsLearning**20**What’sMyNumber? Guessthesecretnumberonanumberline Ifguessistoobig,thelargetriangleisplaced abovetheguess Iftheguessistoosmall,thesmallertriangle isplacedabovethenumber Whatdoyouknowaboutnumbersbetween? 012345678910 Partners forMathematicsLearning**21**TryThisOne… 4243444546474849 Partners forMathematicsLearning**22**NumberoftheDay 6+6 5+5+1+1 dozen 12 dimeand 2pennies 13-1 7+5 Partners forMathematicsLearning**23**MeaningfulFactStrategies One-more-thanandtwo-more-than Countinguporcountingback Factswithzero Doublesandneardoubles Maketenfacts Factfamilies part-part Commutativepropertywhole Compensation Partners forMathematicsLearning **25**DevelopingFactFluency Expectations: Byendofgrade1,fluencywithaddition andrelatedsubtractionfactsto10 Explorefactsto18 Exploresumsbeyond10bybuildingon10s Byendofgrade2,fluency withadditionandrelated subtractionfactsto20 Partners forMathematicsLearning**26**AssessmentIsKey Determiningeachstudent’sunderstanding ofnumbercombinations Knowingwhichfactseachchildhasnotyet learnedandreassessingweeklytoencourage studentstomoveon Helpingstudentstake responsibilityfor learningfacts Informingparents withspecifics Partners forMathematicsLearning**28**HowDoYouRecognizeFluency? Talkwithapartneraboutnumberfluency in1stgrade Whatdostudentsdothatletsyouknowthey havenumberfluency? Whatdostudentsdothatlets youknowtheydonothave numberfluency? Howdoyoukeeptrack? Partners forMathematicsLearning**29**SupportingYourWork Considerusingthepowerpoint,discussion notes,andotherarticlesforCrossingfrom Grade1intoGrade2ingrade-levelmeetings Eachpersoncouldbring afavoritenumberfact gametoshare Discussionsofcommontasks andstudentworkalso supportsteacherplanning Partners forMathematicsLearning**30**FairSharesforTwo FairSharesforJulieandJennifer ReferbacktoliteraturebookJustLikeMe Discusshowsisterssharedthings Whathappenedtothe“leftovers”? WhichnumberscouldJulieand Jennifersharefairly? Partners forMathematicsLearning**31**BumpyorNotBumpy Cutoutthetwo-columncards Whatcanyoutellaboutthepieces? Howcanyouorganizethesepieces? Orderpieces Sortthepiecesintotwogroups;givetherule Determineifyournumbersareorarenot bumpy Howdoyouknow? Partners forMathematicsLearning**32**EvenorOdd Evennumber:anamountthatcanbe madeoftwoequalpartswithnoleftovers Oddnumber:anamountthatcannotbe madeupoftwoequalparts Observableattribute:NOTadefinition Numbersendingin0,2,4,6,8 Numbersendingin1,3,5,7,9 Partners forMathematicsLearning**33**“SenseofBalance” Eachshapehasauniquewhole numberweightmorethan0 Differentshapeshavedifferent weights;identicalshapeshavesameweights Thesizeofshapesisnotrelatedtoweight Ashapehangingdirectlybelowthefulcrumdoes notaffectthebalanceofarmstoleftorrightofthe fulcrum Distancefromfulcrumdoesnotaffectweight Partners forMathematicsLearning**34**“SenseofBalance” Totalweightis32units Eachshapeweighslessthan10units Partners forMathematicsLearning**35**“SenseofBalance” Whatamounts(24,25,26)couldnot bethetotalweightforthispuzzle? Howdoyouknow? Partners forMathematicsLearning**36**Equipartitioning Equipartitioningreferstodividingawhole intoequalpartsormakingfairshares Childrenoftenbegintothinkofpartitioning intermsofsharingwithfriendsandeach gettingthesameamount Partners forMathematicsLearning**37**PaperFolding–Part1 Howmanytimesdoyouthinkyoucanfold thepattypaperinhalf? Howmanyequalpartswouldbecreated withthenumberoffoldsyoupredicted? Explainyourreasoning Beginfolding! Recordthetotalnumber offoldsandthenumberofpartscreated Partners forMathematicsLearning**38**Equipartitioning-Part2 Whatwouldhappenifyoufoldedinthirds eachtimeinsteadofhalves? Whatwouldhappenifyoufoldedthepaper inhalfandtheninthirds? Howmanydifferentwayscouldyoufolda pieceofpapertocreate24equalparts? Whichwayusestheleastnumberoffolds? Thegreatest? Partners forMathematicsLearning**39**Fair“Sharing” Wereeveryone’sfoldsthesame? Whatsimilaritiesanddifferencesarethere inthehalffoldsandthethirdfolds? Ifyouhavelargerpaper,could youfolditinhalfmoretimes thanyoucanfoldsmallpaper? Partners forMathematicsLearning**40**Equipartitioning LookattheEssentialStandardthatrelates toequipartitioning Whatexpectationsdotheclarifyingobjectives identify? Whatcanyoulearnfromtheassessment prototypes? Whataspectsofthisstandardarenewfor yourclassroom? Partners forMathematicsLearning**41**Equipartitioning Noticehowspatialreasoning,equality, measurement,andnumbercometogether inthisstandard Languageassociatedwithdivisionand fractionsareneededindiscussingthis standard,butfraction(symbolic)notationis NOTthepurpose Partners forMathematicsLearning**42**ProblemBasedLessons Researchhasindicatedthat beginningwithproblem situationsyieldsgreater problem-solvingcompetence andequalorbetter computationalcompetence forMathematicsLearning**43**BasicStructureofProblems AccordingtoCGI,therearefourbasic structuresforproblems,regardlessofthe magnitudeofnumbers JoinProblems SeparateProblems Part-Part-WholeProblems CompareProblems Recognizingtheproblemstructuresand identifyingonesthatseemtobemore difficultforstudentsishelpfulinplanning forMathematicsLearning**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**SeparateProblems Separate:ResultUnknown Sandrahad12marbles.Shegave4marblesto George.HowmanymarblesdoesSandrahavenow? Separate:ChangeUnknown Sandrahad12marbles.ShegavesometoGeorge. Nowshehas8marbles.Howmanydidshegiveto George? Separate:StartUnknown Sandrahadsomemarbles.Shegave4toGeorge. NowSandrahas8marblesleft.Howmanymarbles didSandrahavetobeginwith? Partners forMathematicsLearning**46**Part-Part-WholeProblems Part-Part-Whole:WholeUnknown Georgehas4penniesand8nickels.How manycoinsdoeshehave? Part-Part-Whole:PartUnknown Georgehas12coins.Eightofhiscoinsare pennies,andtherestarenickels.Howmany nickelsdoesGeorgehave? Partners forMathematicsLearning**47**CompareProblems Compare:DifferenceUnknown Georgehas12penniesandSandrahas8 pennies.Howmanymorepenniesdoes GeorgehavethanSandra? Compare:LargerorSmallerUnknown Georgehas4morepenniesthanSandra. Sandrahas8pennies.Howmanypennies doesGeorgehave? Georgehas4morepenniesthanSandra. Georgehas12pennies.Howmanypennies doesSandrahave? Partners forMathematicsLearning**48**Teacher’sRoleinProblemSolving Choosesappropriateproblems Listenstostudents’strategies Asksquestionsandsupportsthinking Ensuresthatdifferentstrategiesareshared Howstrategiesarealike Howstrategiesaredifferent Encouragesstudentstosolve problemsinmorethanoneway Givesfeedback Partners forMathematicsLearning**49**ProfessionalReading MakingtheMostofStoryProblems TeachingChildrenMathematics December2008/January2009 Pages260-266 NationalCouncilofTeachersofMathematics Preparetoshareyourassignedpart Firstportionofarticle SupportingandExtendingMathematicalThinking Partners forMathematicsLearning**50**DebriefingTeacher’sRole Sharewithothersatyourtablethemain pointsfromyoursectionofthearticle Whatisdifferentaboutteachers’comments andquestionsbeforecorrectanswersare givenandaftercorrectanswersaregiven? Whatthreethingsimpressed youthemost? Whatideaswillyoutrythisyear? Partners forMathematicsLearning

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