TeachingtoBigIdeasinIntermediateMath

MarianSmallOakville

June,2017

•Ourmissiontoday:•WhatareSOME oftheBigIdeasinIntermediateMath?•Howdoweorganizeteachingaroundthem?

Tostart

•Onapapernobodyelsesees,youoryouandapartnermakealistoftwoideasthatyouthinkareimportantin:•- number•- algebra•Putitaside.

Let’sconsidernumeration

•Everycanberepresentedinmanyways.Eachwayhighlightssomethingdifferentaboutthatwholenumber.

So..

•Show5/3inalotofways.•Whichofyourwaysmadeiteasytoseethat:

• 5/3isfive1/3s• 5/3ismorethan1½• 5/3is5÷ 3• 5/3islessthan2

01/35/3

Oritmightbe

•Show40%differentways.•Whatdoeseachwayshow?

Whichshows

•That40%islessthanhalf•That40%is2/5•That40%is4setsof10%•That40%is10%lessthan50%•That40%canbealotoralittle

20%20%20%

20%20%

Itmeansthat..

•Wheneveryouhavekidsdoarepresentationinanyunit,youaskwhattheysee.

Forexample…

•Theymodel3/5+1/3likethis:

Forexample…

•Theymodel3/5+1/3likethis:

Or

•Theymodel(-12)÷ (–3)as

Let’sconsidernumeration

•Everynumbercanbedecomposedandrecomposedindifferentways;thisoftenmakesiteasiertoestimatethesizeofthenumberorcalculatewithit.

Itcomesupwhen…

•StudentsaretryingtofigureouttheLCMof12and20.•Kidshavetoseethatthenumber60canbethoughtofbothas3x20oras5x12.

Itcomesupwhen…

•StudentsaretryingtofigureouttheGCFof12and20.•Kidshavetoseethatthenumber12canbewrittenas4x3and20as4x5.

Soyoumightask…

•Whatrepresentationsof36and42wouldhelpyoufigureouttheirGCF?

Or

•YouhaveLOTSofbagsof6counters.•YouhaveLOTSofbagsof9counters.•YouhaveLOTSofbagsof20counters.•Whichamountscanyoushow?WhichcanyouNOTshow?

Cans

•6,12,18,24,30,…•9,18,27,36,45,…•20,40,60,80,…•15,33,51,26,46,etc.

can’ts 6,9,20

•1,2,3,4,5,7,8,9,10,11,13,14,16,17,19,….

Youmightenjoythis

•https://www.youtube.com/watch?v=vNTSugyS038

Whenever…

•youaddorsubtractormultiplyordivide,youaredoingcompositionordecomposition.•youuseplacevalueideas,youaredoingcompositionordecomposition.

Whenever…

•youworkwithmoney,youaredoingcompositionordecomposition.

Let’sconsidernumeration

•Whenyouusefractionstocomparetwoamounts,youneedtobeclearabouttheattributeyoucareabout.

Imightask

•Usepatternblockstomakeadesignthatis2/3yellow.•Whymightsomeoneuseadifferentfractiontodescribetheyellow?

Perhaps

•Youarecarefulabouttalkingaboutwhatattributetoconsiderwhenusingfractions.

Let’sconsidernumeration

•Thesizeofafractioncanbethoughtaboutintermsoftheratioofthenumeratortothedenominator.

Youmightask

•Whyareyousurewithoutdoingmuchofanythingthat1/10<5/7?•Whyis2/3=20/30?•Supposethenumeratorofafractionis2/5ofthedenominator.Whatcouldthefraciton be?

Let’sconsidernumeration

•Additionandsubtractionarerelated.Theyalwaysinvolvepartsandwholes.

SoImightask…

•Describeasituationwhereyoumightsubtract12/3from31/2.•Whatadditionalsodescribesthatsituation?Why?

SoImightask…

•Fillinoneoftheboxeswiththenumber5/3.Thenfillintheotherboxessothetotallengthsarethesame.

Somaybe

•Youteachadditionandsubtractiontogether.•Yourepeatedlyaskforwhatadditionisrelatedtoasubtractionsituationkidsaredealingwithorthereverse.

Let’sconsidernumeration

•Multiplicationanddivisionarerelatedtoeachotherandhavethesamemeaningsnomatterwhatsortsofnumbersareinvolved.

SoImightask…

•Drawapictureorbuildamodelthatshowswhat2/3÷ 1/2means.•Whatmultiplicationdoyouseeinyourpictureormodel?

Itmightbe:

Itmightbe:

Perhaps…

•Frequentlywhenstudentsaredoingacomputationinvolvingoneoperation,youcanaskhowadifferentoperationcouldhavebeenused.

Let’sconsidernumeration

•Themultiplicativecomparisonoftwonumberscanalwaysbethoughtofintwoways– howmanyb’smakeanaORhowmuchofanaisab?

Forexample

•Fillintheblanksusingmixednumberstomakeeachtrueinlotsofways.•____is3timesasmuchas____.•____is1/3asmuchas____.

Later

•Fillintheblankswithimproperfractionstomakeeachtrueinlotsofways.•____is5/2timesasmuchas____.•____is2/5asmuchas____.

Let’sconsidernumeration

•Knowingarelationshipbetweenbetweentwovaluesautomaticallytellsyoutherelationshipbetweenothervalues.

Forexample

•SupposeAis20%ofB.•WhatpercentofBis2A?•Whatpercentof2BisA?•Whatpercentof2Bis2A?•WhatpercentofAisB?

Forexample

•Whatotherpercentrelationshipsdoyouknowbasedonthefirstone?

OR

•Yousaved20%whenyouboughtshoes.•Yousaved40%whenyouboughtajacket.•Youpaidthesameforeach.•Howweretheoriginalpricesrelated?

Perhaps

•Wheneverkidsareworkingwithratios,youbringupfractionstheyalsosee,andviceversa.

Let’sconsidernumeration

•Percents canbeusedtostandardizetherelationshipbetweentwonumbers.

Soyoumightask

•WhichsituationwouldmakeiteasiertofigureoutwhetherLeviorAaronisclosertohisgoal:

Situation1

•Levihassaved4/7ofthemoneyheneedstobuyahoverboard.•Aaronhassaved5/8ofthemoneyheneedstobuyahoverboard.OR

Situation2

•Levihassaved53%ofthemoneyheneedstobuyahoverboard.•Aaronhassaved61%ofthemoneyheneedstobuyahoverboard.

Considermeasurement

•Whenyoudescribethesamemeasurementwithdifferentunits,youneedmoreofalittleunitandfewerofabigunittodescribeit.

Forexample

•Isthisabigshoppingmall?•Area:2000000cm2

Or

•Fillintheblankswithunitsinlotsofways:6.4____=6400_____.

Considermeasurement

•Somemeasurementsofanobjectareindependentofothermeasurements,butsomearenot.

SoImightask

•Youknowthistriangleisisosceles.•Whatothermeasurementsofitdoyouknownow?Whichnot?

10cm70°

OR

•Whatothermeasurementsareyousureof?

108°

OR

•Onemeasurementofacircleis20cm.Whatareothermeasurementsassociatedwiththatcircle?

Perhaps

•Youfrequentlybringupsituationswhereknowingonemeasurementofanobjecttellsyouanotheroneandtalkabouthowtheyknow.Youalsobringupsituationswheremeasurementsareindependentanddiscussthose.

Considermeasurement

•Formulasareusefulsinceyoucanfigureoutmeasurementsthatarehardtotakefromonesthatareeasy.

Youmightask

•Sarahdoesn’tknowtheformulaforthesurfaceareaofacylinder.•Rickdoes.•Whowillhaveaneasiertimefiguringoutthissurfacearea?Why?

Considerpattern

•Everypatterninvolvesrepetition.

Youmightask

•Whymightyouthinkthisisapattern?•4,7,11,15,…?

Perhaps

• Frequentlywhenpatternsarebeingaddressed,yougobacktowhytheyarepatterns--- focusonpredictabilitybecauseofrepetition.

Considerpattern

•Thereisnowaytobecertainhowapatterncontinueswithoutapatternrule.

SoImightask

•Continuethistoformapatterninthreedifferentways:•5,10,…

Perhaps

•Youarecarefultonotsay:Whatisthetenthshapeinthispattern?•Instead,youaskwhatMIGHTbethetenthshape.

Consideralgebra

•Equalityisanexpressionofbalance.

Youmightask

•Putinnumberstomakethistrue.Whatdoyounoticeaboutthenumbers?•[]+48={}+50•3x[]=9x{}

Perhaps

•Whenyouread4x=12,yousay“Foursetsofanumbermake12.Whatcouldthenumberbe?”

Consideralgebra

•Anequationdescribesmanysituations.

Youmightask

•Describedifferentsituationsthataredescribedbythisequation:• 2x– 8=32

Consideralgebra

•Therearemanythingsthataretrueabout“all”numbersorallofaparticularsetofnumbers.

Youmightask

•Iaddtwonumbers.•Isubtractthesametwonumbers.•The“add”answeris10lessthanthe“subtract”answer.•Whatnumberscouldtheybe?

Youmightask

•Chooseanumber.•Multiplyitbythenumberthatis2greater.•Nowmultiplythenumberbetweenthembyitself.•Whatdoyounotice?

Forexample

•Whenyoutalkaboutideaslikemultiplyingnumbersinanyorder,youaskkidsforothermathematicalthingsthatare“alwaystrue”.

Considerdata

•Oftenanorganizedvisualdisplaymakesiteasierto“see”data.

Youmight..

• provideatablewithsomedataandthesamedatainasimplebargraphandaskaquestionaboutthedata;discusswhichorganizationtheyfindmosthelpfulandwhy.

Considerprobability

•Probabilitiesarealwayspredictionsthatarerarelycertain.

Youmightask

•IflippedacoinandgotH,T,T.•Whatwillhappennext?

Or

•Ichose10cubesandgot2red,8blue.•WhatwillhappennexttimeIchoose10cubes?

Whatdoesplanninglooklike?

•Iprobablyusetheexpectationsinconjunctionwiththesebigideastocreatelearninggoals.

Whatdoesplanninglooklike?

•Ibuildlessonswiththecommitmenttoaddresssomeessentialunderstandings(aswellasexpectations)eachtime.

Forexample

•Ichooseanexpectation

Idecide

•Toconnectthistotheideathatittakesfewercopiesofabigunitthanalittleunittodescribeameasurement

Icreatesomeactivitieslikethese:

• Youareusingfractionstrips.ThestripforfractionAfitsintothestripforfractionB2½times.•Whatcouldthetwofractionsbe?

Icreatesomeactivitieslikethese:

•SupposefractionCis¾offractionA.HowmanytimeswillitfitintoB?Why?

•AND/OR

•Ilookatthebigideas.•Ifigureoutwhichexpectationstheymightmarrywith.•Ibuildlessonsaroundthatbigideathatcoverthoseexpectations.

So

•Let’stryit.•Youeitherstartwithanexpectation,marryittoabigideaandbuildalesson.OR•Startwiththebigidea,findrelatedexpectations,andbuildalesson.

Now

•Takeoutyouroldpaperaboutyourideasofbigideas.•Havetheychanged?•Doyoufeelthereareimportantoneswemissed?

•Questions?

Download•www.onetwoinfinity.ca•Recentpresentations•HCINT