workshop 2: solving equaons · 2016. 9. 9. · workshop 2: solving equaons nctm interac5ve...
TRANSCRIPT
Workshop2:SolvingEqua5ons
NCTMInterac5veIns5tute,2016
NameTitle/Posi5onAffilia5on
EmailAddress
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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