Grade 6: Expressions & Equaons...Grade 6: Expressions & Equaons NCTM Interac3ve Ins3tute,...

Grade6:Expressions&Equa3onsNCTMInterac3veIns3tute,2016

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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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