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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