Workshop2:SolvingEqua5ons

NCTMInterac5veIns5tute,2016

NameTitle/Posi5onAffilia5on

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WarmUp

Listtheseexpressionsfromleasttogreatest:2n2n+12(n+1)2n–12(n–1)

Reflec5on

Whatwouldstudentsneedtounderstandinordertosolvethewarmup?

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CommonCoreStandards

Thissessionwilladdressthefollowing:

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7.EE.1 ApplyproperGesofoperaGonsasstrategiestoadd,subtract,factor,andexpandlinearexpressionswithraGonalcoefficients.

7.EE.4 UsevariablestorepresentquanGGesinareal-worldormathemaGcalproblem,andconstructsimpleequaGonsandinequaliGesbyreasoningaboutthequanGGes.

SolvingEqua5ons

ThinkabouttheinstrucGonalsequenceyouuseinteachinghowtosolveanequaGon. Whatdostudentsdointhefirstlessons?

WhatarecriGcalbenchmarksorideasthatstudentsprogressthroughintheinstrucGonalsequence?

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AlgebraMagic•  Thinkofanumber.•  MulGplythenumberby3.•  Add8morethantheoriginalnumber.

•  Divideby4.•  Subtracttheoriginalnumber.

Compareyouranswertoothersatyourtable.Whydidthishappen?Find2differentwaystoexplainit.6

AlgebraMagic

Whatcouldbedonetothestepsinordertogetthenumberyoustartedwith?

•  Thinkofanumber.•  MulGplythenumberby3.•  Add8morethantheoriginalnumber.•  Divideby4.•  Subtracttheoriginalnumber.

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Wri5ngExpressions•  Enterthefirstthreedigitsofyourphonenumber.•  MulGplyby80.•  Add1.•  MulGplyby250.•  Addthelastfourdigitsofyourphonenumber.•  Repeattheabovestep.•  Subtract250.•  Divideby2.

Describethenumberyouhave.Howdidtheproblemwork?

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AlgebraMagic

Whichofthefollowingstepscanyoureversewithoutchangingtheresult?Why?

1)  Thinkofanumber.2)  Subtract7.3)  Add3morethantheoriginalnumber.4)  Add4.5)  MulGplyby3.6)  Divideby6.9

AlgebraMagic

Thefollowingtrickismissingthelaststep.•  Thinkofanumber.•  Takeitsopposite.•  MulGplyby2.•  Subtract2.•  Divideby2.•  ??????????

DecidewhatthelaststepshouldbeforthegivencondiGonsofinalresultis:a)  Onemorethan

originalnumber.b)  Oppositeoforiginal

number.c)  Always0.d)  Always-1.

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MatchingExpressions,Words,Tables,&Areas

WorkcollaboraGvelywithyourtablemates.•  Matchcardstomakeasetwithanexpression,words,table,andareacard.

•  Ifthereisnotacompleteset,makeacardforthemissingtype(s)withoneoftheblankcards.

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MatchingExpressions,Words,Tables,&AreasLargegroupdiscussion:

•  Which,ifany,ofthegroupsofexpressionsareequivalenttoeachother?Howdoyouknow?

•  WhatwillstudentslearnasaresultofthisacGvity?

•  WhatchallengesmightstudentencounterwiththisacGvity?

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ExpressionstoEqua5ons

8+4=+7

Whatresponsesdostudentsgiveforbox?

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Amajormisunderstanding

•  Manystudentsdonotunderstandtheequalssign.

•  Theybelieveitsignifiesthattheanswercomesnext.

2x–8=4x+6

EqualSign–TwoLevelsofUnderstanding

Opera5onal:Studentsseetheequalsignassignalingsomethingtheymust“do”withthenumberssuchas“givemetheanswer.”

Rela5onal:StudentsseetheequalsignasindicaGngtwoquanGGesareequivalent,theyrepresentthesameamount.MoreadvancedrelaGonalthinkingwillleadtostudentsgeneralizingratherthanactuallycompuGngtheindividualamounts.TheyseetheequalsignasrelaGngto“greaterthan,”“lessthan,”and“notequalto.”

Knuth, E. et. al (2008). The importance of equal sign understanding in the middle grades. Mathematics Teaching in the Middle School, 13, 514–519.

Whyisunderstandingtheequalsignimportant?

Knuth, E. et. al (2008). The importance of equal sign understanding in the middle grades. Mathematics Teaching in the Middle School, 13, 514–519.

Transi5oningtoRela5onalThinking

TrueorFalse:471–382=474–385674–389=664–379583–529=83–2937x54=38x535x84=10x4264÷14=32÷2842÷16=84÷32

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•  No calculators – No computations •  Use relational thinking to justify answer.

Transi5oningtoRela5onalThinking

Whatisthevalueofvariable?73+56=71+d67–49=c–46234+578=234+576+d94+87–38=94+85–39+f92–57=94–56+g68+58=57+69–b56–23=59–25–s

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•  No calculators – No computations •  Use relational thinking to justify answer.

SolvingEqua5ons

AnequaGonstatesthattwoexpressionsareequivalentforcertainvaluesofavariable.

EquaGonsbecomeusefulininvesGgaGngrelaGonshipsbetweentwoexpressions.

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SolvingEqua5ons

•  ManycurriculummaterialsbeginwithequaGonslikethis:

14–w=9

Foegen, A. & Dougherty, B. J. (2013). Algebra screening and progress monitoring study.

SolvingEqua5ons

14–w=9

48%ofstudents(1615)gotitcorrect.

(2ndgradeCCSSMstandard)

Foegen, A. & Dougherty, B. J. (2013). Algebra screening and progress monitoring study.

SolvingEqua5ons

RatherthanstarGngwith‘easy’equaGonsandapplyingalgebraicmanipulaGons,let’sconsideradevelopmentalapproach.

5+x=125–5+x=12–5

x=7

SolvingEqua5ons

5+x=12

Whatnumberaddedto5equals12?Whatbasicfactdoyouknowthatcouldtellyouthemissingaddend?

SolvingEqua5ons

WhenyouseeanequaGonlikethis,whatare3otherrelatedequaGonsyoucouldwrite?

5+x=12

SolvingEqua5ons

WhenyouseeanequaGonlikethis,whatare3otherrelatedequaGonsyoucouldwrite?

5+x=12

5 + x = 12 x + 5 = 12 12 – 5 = x 12 – x = 5

SolvingEqua5on

DiagramswithmanipulaGvesareanotherwaythatcansupportstudents’understandingofsolvingequaGons.

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SolvingEqua5ons

Workwithapartneratyourtabletocompletethelab.Bepreparedtoshareyourideas.

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SolvingEqua5ons

3x+2=4x–3

SolvingEqua5ons

Graph3x+2=4x–3Useyourgraphingcalculatortographthetwoexpressions.HowwouldyouidenGfythesoluGon?

SolvingEqua5ons

Graphing3x+2=4x–3

SolvingEqua5ons

1.  Logicalreasoning/inspecGon2.  Factfamilies/inverseoperaGons3.  Physicalmaterials/diagrams4.  Tables5.  Graphing

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SolvingEqua5ons

Howwouldyousolve3x+2=4x–3usingalgebraicsteps?

SolvingEqua5ons

3x+2=4x–33x+2+3=4x–3+3

3x+5=4x3x–3x+5=4x–3x

5=x

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SolvingEqua5ons

3x+2=4x–33x+5=4xA3

5=xS3x

SolvingEqua5ons

A:AddS:SubtractM:MulGplyD:DivideCLT:CombineLikeTermsDPMA:DistribuGvePropertyofMulGplicaGonoverAddiGon

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Whyisitimportanttounderstandsolvingequa5ons

DanchallengedAmytowriteanequaGonthathasasoluGonof3.WhichequaGoncouldAmyhavewriken?

a.4–x=10–3xb.3+x=–(x+3)c.–2x=6d.x+2=3

Sampleofstudentwork

DanchallengedAmytowriteanequaGonthathasasoluGonof3.WhichequaGoncouldAmyhavewriken?

a.4–x=10–3xb.3+x=–(x+3)c.–2x=6d.x+2=3

Reflec5on

•  Whatnewidea(s)doyouwanttoimplementintoyourclassroom?

•  Whatchallengesdidyouencounterduringthissession?

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Reflec5on

(Principles to Actions: Ensuring Mathematical Success for All [NCTM 2014], p. 47)

Reflec5on

(Principles to Actions: Ensuring Mathematical Success for All [NCTM 2014], p. 48)

Disclaimer The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all students. NCTM’s Institutes, an official professional development offering of the National Council of Teachers of Mathematics, supports the improvement of pre-K-6 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. It is a forum for the exchange of mathematics ideas, activities, and pedagogical strategies, and for sharing and interpreting research. The Institutes presented by the Council present a variety of viewpoints. The views expressed or implied in the Institutes, unless otherwise noted, should not be interpreted as official positions of the Council.

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