fostering algebraic reasoning rather than algebraic skills · 2017. 5. 10. · what might algebraic...
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Whatdoesitlooklike?
Fosteringalgebraicreasoningratherthanalgebraicskills
MarianSmallMay2017
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Whatisit?
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Whatmightalgebraicreasoninglooklike?
•Itcouldbeanumericalgeneralization:•Tosquareanumberoftheform[]5,youmultiply[]by[]+1andsticka25attheend.•Forexample,352 =1225.•Howcome?
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Avisual(butstillalgebraic)approach
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Avisual(butstillalgebraic)approach
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Avisual(butstillalgebraic)approach
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Usingvariables
• (10a+5)(10a+5)=100a2 +50a+50a+25• =100a(a+1)+25
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Whatmightalgebraicreasoninglooklike?
•Itcouldbesomethingsimple,e.g.
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Whatmightalgebraicreasoninglooklike?
•Imightask:•YouwantAx– Btobeworthalotwhenxisonlyaround10.•WhatvalueswouldyouuseforAandB?Why?
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Whatmightalgebraicreasoninglooklike?
•ImightchoosealargepositivevalueforAandanegativevalueforB
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OrPlayGuessmyRule
• 0
–8 8 17
54
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Maybex2 – 8ORx3−8x2 +20x– 8
• 0
–8 8 17
54
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OrImightasksomethingalittlemorecomplex•Twolinesintersectat(20,1).•Couldbothslopesbesteep?
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Oralittlemorecomplex
•Iftheyare,whatelsedoyouknowaboutthoselines?•Ifnot,whynot?
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Orevenmorecomplex,butcool
•Let’slookatDesmos RacingDots.
• https://teacher.desmos.com/activitybuilder/custom/56d139907e51c4ed1014b51f
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So…
•Whatisthedifferencebetweenfocusingonalgebraicreasoningvs.algebraicskills?
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Youwillseereasoningin
•Exploringgeneralizations•Exploringequality
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Youwillseereasoningin
•Creatingandtestingconjectures•Justifyingandproving
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MAYBE
•A)Solve3x– 80=2x– 40•VS
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MAYBE
•B)WITHOUTSOLVING,predictwhichtwosolutionswillbecloserandwhy.•#1:3x– 80=2x– 40•#2:4x– 70=3x– 40•#3:10x– 80=2x– 40
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OR
•A)Graphy=3(x– 3)2 +8•VS
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OR
•B)Withoutdrawing,predicthowthesegraphsarealikeanddifferent?•y=2(x– 3)2 +8•y=0.5(x– 8)2 +3
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Sohowcanyou…
•SwitchtoincludeMOREalgebraicreasoninginwhichtoembedworkonalgebraicskills.
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Startwithanexpectation
•Grade7:•Makepredictionsaboutlineargrowingpatterns,throughinvestigationwithconcretematerials
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Askillquestion
•Imightaskstudentstopredictthe20th termofthepatternbelow.Theymightormightnotusereasoning.
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Whatkindofmorereasoningquestioncouldyouask?•Alineargrowingpatternismadeusingsquaretiles.•Thefirsttermusesmorethan2tiles.
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Whatkindofmorereasoningquestioncouldyouask?•Oneofitstermsuses25tiles.•Anotherofitstermsuses49tiles.•Whatothernumbersofsquaretilescouldbeusedinotherterms?Why?
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S/hethinks
•Ifonetermhas25tilesandonehas49tiles,thedifferenceis24tiles.
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S/hethinks
•Imustgoupbyafactorof24,like8.
•Mypatterncouldbe9,17,25,33,41,49,….
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Grade7
•Comparepatternrulesthatgenerateapatternbyaddingorsubtractingaconstant,….togetthenextterm… withpatternrulesthatusethetermnumbertodescribethegeneralterm
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SoImightask
•Whichwouldyoufindeasiertodeterminethe1000th termof?Why?•Pattern1rule:Startwith8andkeepadding3.•Pattern2rule:Multiplythetermnumberby4andthenadd3.
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Grade7
•Translatephrasesdescribingsimplemathematicalrelationshipsintoalgebraicexpressions
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Askillquestion
•WritethephraseOnemorethanthreetimesanumber algebraically.
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Amorereasoningquestion
•Describeaseriesoftwocomputationsyoumightusetogetfrom10to35.•Usevariablestodescribethatrelationshipsoitcouldbeappliedtoothernumbersaswell.
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Grade8
•Evaluatealgebraicexpressionswithuptothreeterms,bysubstitutingfractions,decimalsandintegersforthevariables
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Askillquestion
•Whichvalueisgreaterwhenyousubstitutex=–8:
•–3x– 4•–10x+36•–2x+6
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OrImightask
•Createanalgebraicexpressionwhichhasavaluecloseto–100whenxiscloseto19.
•Youthink:–100isaboutsubtracting5setsof20,soIwoulduse1– 5x.
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Grade8
•Solveandverifylinearequationsinvolvingaone-variabletermandhavingsolutionsthatareintegers,byusinginspection,guessandtestanda“balance”model
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Askillquestion
•Whatisthesolutionto3x– 18=–2x+32?•Explainyourstrategy.
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ButImightask
•HowdoyouknowthatthesolutionhastobenegativeWITHOUTSOLVINGorevenALMOSTSOLVING?
•5x+80=3x+10
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Grade8
•Representlinearpatternsgraphicallyusingavarietyoftools
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Askillquestion
• Continuethetableofvaluestoshowthispatternandthengraphit.• Pattern:3,7,11,15,19,…
• x y• 13• 27
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Areasoningquestion
• Idrewthegraphofapattern.• Itendedupbeingalinethatpassedthrough(8,12).•Whatmightthepatternhavebeen?
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Grade9
•Expresstheequationofalineintheformy=mx+bgiventheformAx+By+C=0.
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SoImightask
•ThelineAx+By+C=0hasanegativeslope.•HowmanyandwhichofA,B,andCcanbenegativetomakethishappen?
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Grade9
•Identifythegeometricsignificanceofmandbiny=mx+b
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SoImightask
•Youknowthatthepoint(4,–5)isonacertainline.•Youwanttofigureoutanotherpointontheline,butIwillonlytellyouONEofmorbiftheequationisy=mx+b.•Whichwouldyouaskmefor?Whythatone?
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Grade10
•solvesystemsoftwolinearequationsinvolvingtwovariables,usingthealgebraicmethodofsubstitutionorelimination
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SoImightask
•Whymightsomeonesayit’seasiertosolvethefirstpairofequationsthanthesecond?•Pair1: Pair2•6x– 20y=18 3x– 2y=10•3x+20y=244x+5y=20
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Grade10
•Factorpolynomialexpressionsinvolvingcommonfactors,trinomials,anddifferencesofsquares
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Imightask…
•Explorethisnumberpattern.•Relateittofactoringalgebraicexpressions.
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Imightshow…
•3x5=15•4x6=24•5x7=35•6x8=49•7x9=63
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Or….
•3x5isonelessthan4x4•4x6isonelessthan5x5•5x7isonelessthan6x6
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Visuals
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Visuals
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Visuals
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Visuals
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Otherreasoningsituations
•Variablesvs.unknowns
Howaretheseequationsalikeanddifferent?•2x+1=5•2x+1=y•2x+1=x+x+1
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Otherreasoningsituations
•Generalizations
Calculateandsimplify.Whatgeneralizationdoyousee?Whydoesitmakesense?
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Pattern
•Calculateandsimplify.2/3x3/44/6x6/93/8x8/152/10x10/12
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Equality
•Whatdoyouknowabouttherelationshipbetweenthemassoftheyellowandredboxes?
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Estimatingsolutionstoequations
Youcouldaskforareasonableestimateforthese:•42+2x =150• 3x – 40=2x +1• 8x =4– x•3/4x +40=x
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Thereisalwaysmorethanoneway.
•Onewaytodescribetheperimeterofarectanglewherethelengthisdoublethewidthis:•P=3L.•Whatisanotherway?
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Relatingalgebratonumber
•Createtwoalgebraicexpressionsthatfitoneoftheseruleswhenyousubstituteinwholenumbers.•Thevaluesarealwaysevennegativeintegers.•Thevaluesarealwayspositivemultiplesof3.
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Relatingalgebraicsituations
•Supposeyouknowthat3x+4=10.•Withoutsolvingtheequation,tellwhatelseyouknowaboutx.
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So…
• Ratherthanthinkingabouttheskillsastheendgame,• Inthisscenario,theskillsareameanstotheend.• Theendisreallyaboutalgebraicreasoning.
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Insummary
•Thereareskills,butitiseasytoembedtheminalgebraicactivitiesthatfocusonreasoning.
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Downloadat
www.onetwoinfinity.caRecentPresentationsOAME2017IS