Grade6:Expressions&Equa3onsNCTMInterac3veIns3tute,2016
NameTitle/Posi3onAffilia3on
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Introduc3ons…
Withyourtable,decidethesimilari3esanddifferencesaboutthefourphrasesbelow:
• Numericalexpression• Numericalequa3on• Algebraicexpression• Algebraicequa3on
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same
different
CommonCoreStandards
Thissessionwilladdressthefollowing:
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6.EE.2 Write,read,andevaluateexpressioninwhichleCersstandfornumbers.
6.EE.4 Iden3fywhentwoexpressionsareequivalent.6.EE.9 Writeanequa3ontoexpressonequan3ty
(dependentvariable)intermsoftheotherquan3ty(independentvariable).
6.EE.7 Solvereal-worldandmathema3calproblemsbywri3ngandsolvingequa3onsoftheformx+p=qandpx=q.
AlgebraMagic
• Thinkofanumber.• Mul3plythenumberby3.• Add8morethantheoriginalnumber.• Divideby4.• Subtracttheoriginalnumber.
Compareyouranswertoothersatyourtable.Whydidthishappen?Showin2differentways.
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AlgebraMagic
Whatcouldbedonetothestepsinordertogetthenumberyoustartedwith?
• Thinkofanumber.• Mul3plythenumberby3.• Add8morethantheoriginalnumber.• Divideby4.• Subtracttheoriginalnumber.
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Wri3ngExpressions• Enterthefirstthreedigitsofyourphonenumber.• Mul3plyby80.• Add1.• Mul3plyby250.• Addthelastfourdigitsofyourphonenumber.• Repeattheabovestep.• Subtract250.• Divideby2.
Describethenumberyouhave.Howdidtheproblemwork?
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AlgebraMagic
Whichofthefollowingstepscanyoureversewithoutchangingtheresult?Why?
1) Thinkofanumber.2) Subtract7.3) Add3morethantheoriginalnumber.4) Add4.5) Mul3plyby3.6) Divideby6.7
AlgebraMagic
Thefollowingtrickismissingthelaststep.• Thinkofanumber.• Takeitsopposite.• Mul3plyby2.• Subtract2.• Divideby2.• ??????????
Decidewhatthelaststepshouldbeforthegivencondi3onsofinalresultis:a) Onemorethan
originalnumber.b) Oppositeoforiginal
number.c) Always0.d) Always-1.
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AlgebraMagic
Makeupaseparatealgebramagictrickwithatleastfivesteps
thatwillmeetoneofthebulletslistedbelow:• Finalresultisonemorethantheoriginalnumber.
• Finalresultis0.• Usesallfouropera3ons.• Resultissame,whetherstepsaredonebackwardsorforward.
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Interpre3ngAlgebraicExpressionsWhaterrorsmightoccurasstudentstranslatethefollowingsentencesintoalgebraicexpressions?
• Mul3plynby5thenadd4.• Add4tonthenmul3plyyouranswerby5.• Add4tonthendivideyouranswerby5.• Mul3plynbynthenmul3plyyouranswerby3.• Mul3plynby3thensquareyouranswer.
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MatchingExpressions,Words,Tables,&Areas
Workcollabora3velywithyourtablemates.• Matchcardstomakeacompletesetwithanequivalentexpression,descrip3on,table,andareacards.
• Ifthereisnotacompleteset,makeacardforthemissingtype(s)withoneoftheblankcards.
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MatchingExpressions,Words,Tables,&AreasLargegroupdiscussion:
• Which,ifany,ofthegroupsofexpressionsareequivalenttoeachother?Howdoyouknow?
• Whatwillstudentslearnasaresultofthisac3vity?
• Whatchallengesmightstudentencounterwiththisac3vity?
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ExpressionstoEqua3ons
8+4=+7
Whatresponsesdostudentsgiveforbox?
Opera3onalvsRela3onal“answer”vs“equivalence”
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Equality
Theno3onofequalityissurprisinglycomplex,
iso_endifficultforstudentstocomprehend,and
shouldbedevelopedthroughoutthecurriculum.
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Equality
• Manystudentsatallgradelevelshavenotdevelopedadequateunderstandingofthemeaningoftheequalsign.
“Limitedconcep:onofwhattheequalsignmeansisoneofthemajorstumblingblocksinlearningalgebra.Virtuallyallmanipula:onsonequa:onsrequireunderstandingthattheequal
signrepresentsarela:on.”
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Carpenter, Thomas, Megan Franke, and Linda Levi. Thinking Mathematically: Integrating Arithmetic and Algebra in the Elementary School. 2003
Equality
Isthenumberthatgoesintheboxthesamenumberinthefollowingtwoequa3ons?2X+15=312X+15–9=31–9Intheequa3on+18=35,thenumberthatgoesintheboxis17.Canyouusethisfacttofigureoutwhatnumbergoesinthisbox:+18+27=35+27
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Transi3oningtoRela3onalThinking
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.
Transi3oningtoRela3onalThinking
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.
Rela3onalThinking
Whatproper3esareimportanttodevelopingrela3onalthinkingwithstudents?a+0=aa–0=aax1=aa÷1=aa+b=b+aaxb=bxaa+b=(a+n)+(b–n)a+b=(a–n)+(b+n)a–b=(a+n)–(b+n)a–b=(a–n)–(b–n)ab=(na)( 1/𝑛 𝑏)𝑎/𝑏 = 𝑛𝑎/𝑛𝑏 19
EqualitySignCau3on
3+5=8+2=10+5=15
EqualitystringswriCenbystudents(andteachers!)provideopportunitytodiscuss
meaningofequalsignanditsproperuse.
3+5=88+2=1010+5=15
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Interpre3ngEqua3ons
Whichisgreater,xory?Explainyourreasoning.
y=4x
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x is greater because it’s multiplied by 4.
y is greater because it is four times the size of x.
Interpre3ngEqua3ons
Leterepresentthenumberofeggs.Letbrepresentthenumberofeggboxes.
Thereare6eggsineachbox.Findanequa3onlinkingeandb.
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b = 6e e = 6b
Interpre3ngEqua3ons
Leterepresentthecostofanegg.Letbrepresentthecostofaboxofeggs.
Thepricepereggisthesamewhetheryoubuythemseparatelyorinabox.
Findanequa3onlinkingeandb.
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b = 6e e = 6b
Interpre3ngEqua3ons
Workingtogetheratyourtables:• Matchanequa3oncardwithastatementcard.• Explain/challengereasoning.• Useblankcardstowriteequa3onorstatementcardssothateachcardisgroupedwithatleastoneothercard.
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SolvingEqua3onsStripDiagramMethod
• Helpsstudentsconceptualizethecharacteris3csoftheproblemtosolveMakesenseofvariabletorepresentunknownquan3ty
• Helpsstudentsformulateanalgebraicequa3ontosolvetheproblemAnalyzerela3onship(s)betweencomponentsofproblem
• HelpsempowerstudentsDevelopcompetenceandconfidenceinusingthealgebraicmethod.
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SolvingEqua3onsStripDiagramMethod
Thereare50childreninadancegroup.Ifthereare10moreboysthangirls,howmanygirlsarethere?
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50 Boys
Girls
SolvingEqua3onsStripDiagramMethod
Thereare50childreninadancegroup.Ifthereare10moreboysthangirls,howmanygirlsarethere?
Letxbethenumberofgirls.Whatcouldbepossiblealgebraicequa3on(s)?
2710
50 Boys
Girls
SolvingEqua3onsStripDiagramMethod
Thereare50childreninadancegroup.Ifthereare10moreboysthangirls,howmanygirlsarethere?
Letxbethenumberofgirls.Whatcouldbepossiblealgebraicequa3on(s)?
x+(x+10)=50
2810
50 Boys
Girls x
x + 10
SolvingEqua3onsStripDiagramMethod
Thereare50childreninadancegroup.Ifthereare10moreboysthangirls,howmanygirlsarethere?
Letxbethenumberofgirls.Whatcouldbepossiblealgebraicequa3on(s)?
(50–x)–x=10
2910
50 Boys
Girls x
50 - x
SolvingEqua3onsStripDiagramMethod
Thereare50childreninadancegroup.Ifthereare10moreboysthangirls,howmanygirlsarethere?
Letxbethenumberofboys.Whatcouldbepossiblealgebraicequa3on(s)?
3010
50 Boys
Girls
SolvingEqua3onsStripDiagramMethod
Thereare50childreninadancegroup.Ifthereare10moreboysthangirls,howmanygirlsarethere?
Letxbethenumberofboys.Whatcouldbepossiblealgebraicequa3on(s)?
x+(x–10)=50
3110
50 Boys
Girls
x
x - 10
SolvingEqua3onsStripDiagramMethod
Thereare50childreninadancegroup.Ifthereare10moreboysthangirls,howmanygirlsarethere?
Letxbethenumberofboys.Whatcouldbepossiblealgebraicequa3on(s)?
𝒙 −(𝟓𝟎 −𝒙)=𝟏𝟎
3210
50 Boys
Girls
x
50 - x
SolvingEqua3onsStripDiagramMethod
UsetheStripDiagramMethodtosolvetheproblemsonthehandout.Setupthediagram/algebraicequa3onsinasmanywaysaspossible.
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SolvingEqua3ons
Coverupmethod:
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SolvingEqua3onsCoverUpMethod
Coverupmethod:
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5+=7 =2
SolvingEqua3onsCoverUpMethod
Coverupmethod:
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5+=7 =2
=8
SolvingEqua3onsCoverUpMethod
Coverupmethod:
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5+=7 =2
=8
-1=8 =9
SolvingEqua3onsCoverUpMethod
Coverupmethod:
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5+=7 =2
=8
-1=8 =9 3=9 =3
SolvingEqua3onsCoverUpMethod
• Prac3cesolvingequa3onsusingtheCoverUpMethodwithyourtablemates.
• Whatwillstudentslearnasaresultofthisac3vity?
• Whatchallengesmightstudentencounterwiththisac3vity?
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Reflec3on
• Whatnewidea(s)doyouwanttoimplementintoyourclassroomasaresultofthissession?
• Whatchallengesdidyouencounterduringthissession?
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Reflec3on
(Principles to Actions: Ensuring Mathematical Success for All [NCTM 2014], p. 47)
Reflec3on
(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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