developing calculaon and problem solving skills · 2015-11-25 · background • some of you...

Developingcalcula.onandproblemsolvingskills

MarianSmallNovember,2015

Background

•  SomeofyouexpressedaninterestintalkingaboutdevelopingbeAercalcula.onskills.

•  SomeofyoutalkedaboutdevelopingbeAerproblemsolvingskills.

•  Todayisamix.•  Wewillalsoblendintheac.vi.esyoutriedandarerepor.ngon.

Let’sstartwith…

•  Someexamplesofopenques.onsyouusedthatwentwell.

•  Talktothepeopleatyourtable.•  Thetablewillchoosetwotopresenttoeveryone.

Calcula7onskills

•  Myfocuswillbeondevelopingtheimportantideasthatunderliecalcula.ons,ratherthanthemechanics.Iwillalwaysbereferringtomeaning.

•  Butwewilllookatvarie.esofstrategiesforperformingcalcula.on.

Facts(SmallNumbers)

•  Yourcurriculumspeaksclearlytotheimportanceofdevelopingfluencywithsmallfacts.

•  TheseareESSENTIALfores.ma.on,workwithpropor.onalthinking,workwithfrac.ons,etc.etc.

BUT

•  Weneedflexibilitywhichcomeswithunderstandingandnotjustinstantaneousresponses.

•  InyourMinistrydocuments,thereismen.onmadeofstrategieslikeMaking10,Usingdoubles,AddingtoSubtract,etc.

Addi7on/subtrac7onissues

•  Everysubtrac.oncanbesolvedusingaddi.on.

•  Howwouldyousolve13–8thisway?

Addi7on/subtrac7onissues

•  Orderdoesn’tmaAer.

Addi7on/subtrac7onissues

•  Youcanaddorsubtractinparts.•  Youcan“juggle”whenyouadd.

Addi7on/subtrac7onissues

•  Youcanuseaconstantdifferencewhenyousubtract.

•  Addingorsubtrac.ng0isespeciallyeasy.•  Addingorsubtrac.ng1ispreAyeasy.

Sohowwouldyouencourage

•  8+3•  6+7•  9–4•  12–9

ButIwouldalsoask

•  Showme6sothat:•  Therearemorethan2parts.•  Thereare2parts-oneliAleandonebig.•  Thereare2parts-similarinsize.

•  Repeatabovewith25.

ButIwouldalsoask

•  Showme10andshowme12sothat:•  Eachisshownintwoparts.•  Onepartfor10isthesamesizeasonepartfor12.

ButIwouldalsoask

•  Whatnumberscouldyoudecomposeintoanumberandtwiceasmuch?

•  Whatnumberscouldyoudecomposeintothreenext-to-each-othernumbers?

ButIwouldalsoask

•  Predicthowmuchmore8+3isthan6+2withoutgeengtheanswers.

Addi7onandsubtrac7onprocedures

•  Howwouldyouusebasetenblocksfor:•  35+48•  76+99•  81–17•  241–167

Mul7plica7onanddivisionconcepts

•  Everydivisioncanbesolvedbymul.plying

•  Howwouldyouusemul.plyingtosolve36÷4?

•  Tosolve546÷6?

Mul7plica7onanddivisionconcepts

•  Orderdoesn’tmaAer

Mul7plica7onanddivisionconcepts

•  Youcanmul.plyinparts(distribu.veandassocia.ve).

Mul7plica7onanddivisionconcepts

•  Youcan“juggle”factors(e.g.half/double)•  Youcandouble/triplebothdivisoranddividend.

Mul7plica7onanddivisionconcepts

•  Youcandivideinparts•  It’seasytodivide0.•  It’seasytodivideby1.

ButIwouldalsoask

•  Buildarectangularprismthatis2x2x3cubes.

•  Howdoesthishelpyouseethat4x3=2x6?

ButIwouldalsoask

•  Makeabigarrayofcounters.•  CanyoualwaysbreakitupintoliAlearrays?

Models

•  Howwouldyouusebasetenblockstomodel•  8x49?•  21x21?•  512÷4•  312÷4

Whataboutintegers?

•  Howdoyoumodel5–(–3)?•  5x(–3)?•  (–15)÷(–3)?

Whataboutfrac7ons?

•  Howdoyoumodel•  2/3+1/5•  2/3–1/5•  2/3x1/5•  2/3÷1/5

2/3+1/5

2/3+1/5

2/3+1/5

2/3–1/5

2/3–1/5

2/3–1/5

2/3x1/5

2/3x1/5

2/3÷1/5

Now

•  Let’stalkaboutexamplesofques.onsyouusedtopushthinking.

•  Shareatyourtable.•  Yourtablewillchoosetwotopresenttoeveryone.

Nowforproblemsolving

Characteris7csofgoodproblems

•  Thereissomeelementofdecisionmaking.•  Thereissomereasoning.•  Communica.ngaboutprocesswillworkformoststudents.

•  It’sniceifyoucandrawavisual.

Characteris7csofgoodproblems

•  It’sniceifthereismorethanoneapproachorevenmorethanoneanswer.

•  It’sniceifyoucaneasilyextend.

Forexample…

•  KaylahadjustafewmorepencilsthanLevi.•  Togethertheyhad20pencils.•  Howmanydoyouthinkeachonehad?

So…

•  Decisionmaking?•  Reasoning?•  Communica.ng?•  Whatsortofvisual?•  Morethanoneapproach?•  Extendable?

Forexample

•  Wemightask:Amyis4.Sindyis6yearsolder.HowoldisSindy?

OR•  WemightaskwhetherSindycanbeboth6yearsolderthanAmyandalsotwiceasold.

Possiblevisual

So…

•  Decisionmaking?•  Reasoning?•  Communica.ng?•  Whatsortofvisual?•  Morethanoneapproach?•  Extendable?

Or…

•  Whatnumberscanyoumakebyusingonly2s,5s,+sand–s?

•  e.g.3=5–2•  10=5+5•  9=5+2+2•  Arethereanycoun.ngnumbersyoucannotmake?

So…

•  Decisionmaking?•  Reasoning?•  Communica.ng?•  Whatsortofvisual?•  Morethanoneapproach?•  Extendable?

Maybe

•  Es.matethenumberofsquarecen.metresofpizzathatallofthekidsinWinnipegeatinoneweek.

So…

•  Decisionmaking?•  Reasoning?•  Communica.ng?•  Whatsortofvisual?•  Morethanoneapproach?•  Extendable?

Maybe

•  Onerectanglehashalftheareaofanother.•  Wouldtheperimeteralsobehalf?•  Isitever?Usually?Omen?

Maybe

•  Createrectangleswherethenumberofcen.metresintheperimeterisveryclosetothenumberofcm2inthearea.

Maybe

•  Yourepresentanumberwithbasetenblocks.

•  Youusethreemorerodsthanflats.•  Youusetwofeweronesthanrods.•  Whatcouldyournumberbe?

Maybe

•  UsepaAernblocks.•  Createadesignthatis1/5blueand1/2green.

Maybe

Maybe

•  AngelaandBenbothhavecoins.•  Benhas2½.mesasmanycoins,butAngela’samountisworth8.mesasmuch.

•  Whatcoinsdoeachhave?

Maybe

•  Elizabethcanrun200min2[].[]seconds.Youcanchoosethenumbersintheblanks.

•  Howlongwouldittakeher,atthatrate,torun1000m?

Maybe

•  AgrowingnumberpaAernincludesthenumbers25and40somewhereinthepaAern.

•  HowlongwillittakeyourpaAerntoreach1000?

Meanandmedian

•  Themeanofasetof10numbersisdoublethemedian.

•  Whatmightthenumbersbe?

Visual

__________44____________Ifthemeanif8,thenthesumofthemissingnumbers=72

Represen7ngintegers

•  Youusesome+1.lesandfour.mesasmany–1.les.

•  Whatintegerscanyourepresent?Whichcan’tyou?

Visual

Horseproblem

•  Amanbuysahorsefor$50.•  Hedecideshewantstosellhishorselaterandgets$60.

•  Hethendecidestobuyitbackagainandpaid$70.

•  However,hecouldnolongerkeepitandhesolditfor$80.

•  Isheahead,behind,oreven?

Times

•  Howmany.mesina12hourperioddoesthesumofthedigitsonadigitalclockequal6?Trytothinkofawaytosolvethiswithoutgoingthrougheverysingle.me.Describewhatyoudid.

Legs

•  Alexbuildsstoolsandtables.•  Heused72legs.•  Hebuilt3morestoolsthantables.•  Howmanyofeach?

Somevideoproblems

•  hAp://threeacts.mrmeyer.com/sugarpackets/

•  hAp://mrmeyer.com/threeacts/cokevsprite/

Mathalicious

•  Anothersource•  Whatdoyouthink?

Wealwaysstartwithcurriculum

•  Let’slookatsomeoutcomesandseeifwecandevelopsomegoodproblems.

Gr1outcome

•..compareandordersetscontainingupto20elementstosolveproblems….

Problem

•  Youpar..onanumberintothreeparts.•  Onepartissmall.•  Onepartisabouttwiceasmuchasanotherpart.

•  Whatnumbercoulditbeandhowisitpar..oned?

Possibili7es

•  15as1,2and12•  13as1,4and8•  18as3,5,and10

Let’sconsolidate

•  Whatdowewanttobringout?•  Thatsmallisrela.ve•  Thattwiceasmuchisanumberaddedtoitself

•  Thatthesmallonecouldbethetwiceornot•  Thatthe3rdnumbercouldbetwicethesmalloneortwicethe2ndnumber.

•  HowtoaAacktheproblem

Grade2outcome

•  Demonstrateanunderstandingofaddi.onandthecorrespondingsubtrac.onby…

•  Crea.ngandsolvingproblemsthatinvolveaddi.onandsubtrac.on

Firsttrythis

•  Chooseanumber.•  Add20.•  Subtract4.•  Subtractthenumberyoustartedwith.•  Subtract8.•  Subtract3.

Whatshouldtheyno7ce?

•  Totalamountofsubtrac.onvsaddi.on•  Howsubtrac.ngtheoriginalnumbermadeitirrelevant

Anotheridea

•  Makeupatrickinvolvingatleast5stepssothatamerfollowingyoursteps,theresultisalways10.

Agrade3outcome

•  Demonstrateanunderstandingofmul.plica.onto5x5by…

•  Crea.ngandsolvingproblemsincontextthatinvolvemul.plica.on

Maybe

•  Inoneclassroom,therewas1morebookinthelibrarythaninanother.

•  Eachshelfineachlibraryheldthesamenumberofbooks.

•  Howmanyshelvesandhowmanybooksmighthavebeenineachclassroomlibrary?

Agrade4outcome

•  Demonstrateanunderstandingofareaofregularandirregular2-Dshapesby..

construc.ngdifferentrectanglesforagivenareainordertodemonstratethatmanyrectanglesmayhavethesamearea

Maybe

•  Tworectangleshadanareaof20cm2butonewastwiceaslongastheother.

•  Howisthatpossible?

Agrade5outcome

•  Demonstrateanunderstandingoffrac.onsbyusingconcreteandpictorialrepresenta.onsto

•  Comparefrac.onswithlikeandunlikedenominators.

Maybe

•  5/[]isalotmorethan12/[].•  Whatnumberscouldgointheblanks?

Agrade6outcome

•  Demonstrateanunderstandingofpercentconcretely,pictoriallyandsymbolically.

Maybe

•  Chooseanumberthanisabout25%ofonenumberbutabout75%ofadifferentnumber.

•  Whatisyournumber?•  Whatarethenumbersitis25%and75%of?•  Showthatyou’reright.

Agrade7outcome

•  Demonstrateanunderstandingofaddingandsubtrac.ngposi.vefrac.onsandmixednumbers,withlikeandunlikedenominators,concretely,pictoriallyandsymbolically

Maybe

•  Youaddtwofrac.onswithdifferentdenominators.

•  Thedenominatorofthesumwas5morethanthenumerator.

•  Whatfrac.onsmightyouhaveadded?

Agrade8outcome

•  solveproblemsthatinvolverates,ra.os,andpropor.onalreasoning.

Maybe

•  Aboxofcookiescosts1½.mesasmuchasapackageofmuffins.

•  Youboughtsomecookiesandsomemuffinsandpaid$43.

•  Howmanyofeachmightyouhaveboughtandatwhatprice?

Doyouwanttotry?

•  Createaproblemthatmeetsanoutcomeofyourchoice.

Now

•  Shareatyourtablethelessonyoureorganizedtofocusmoreonbigideasthanskills.

•  Yourtablewillchoose1or2lessonstosharewitheveryone.

Fornext7me

•  Let’sgoatitagain,butwithaslightlydifferentslant.

•  1)Useanopen-endedproblemthatyouthinkisvaluableforyourstudents.

•  2)Teachonecomputa.onlessonbasedonideas,butwhichprac.sesskills.

Download

•  www.onetwoinfinity.caLouisRiel2