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