lesson 18: one-step equations with integers …ntnmath.kemsmath.com/level g teacher notes/grade 7-...

Mathematics Success – Grade 7 T413

LESSON 18: One-Step Equations with Integers

[OBJECTIVE]The student will solve one-step equations with integers in mathematical and real-

world situations.

[PREREQUISITE SKILLS]integer operations, writing and evaluating expressions

[MATERIALS]Student pages S212–S230Overhead unit tiles (red and yellow)

Red and yellow unit tiles for students

Cups

[ESSENTIAL QUESTIONS]1. Why do we use variables in solving equations?

2. What are the goals when solving equations?

3. In an addition equation, how do we isolate the variable?

4. In a multiplication equation, how do we isolate the variable?

[WORDS FOR WORD WALL] variable, equation, balance, zero pair, additive identity property

[GROUPING]Cooperative Pairs (CP), Whole Group (WG), Individual (I)

[LEVELS OF TEACHER SUPPORT]Modeling (M), Guided Practice (GP), Independent Practice (IP)

[MULTIPLE REPRESENTATIONS]SOLVE, Algebraic Formula, Concrete Representation, Pictorial Representation, Verbal

Description, Graphic Organizer

[WARM-UP] (I, IP, WG) S212 (Answers on T431.)• HavestudentsturntoS212intheirbookstobegintheWarm-Up.Studentswill

complete the problems with computation of integers. Monitor students to see if

anyofthemneedhelpduringtheWarm-Up.Havestudentscompletetheproblemsand then review the answers as a class. {Verbal Description}

[HOMEWORK] Taketimetogooverthehomeworkfromthepreviousnight.

This lesson has 2 parts.

[LESSON – PART 1] [2 – 3 Days (1 day = 80 minutes) – M, GP, WG, CP, IP]

Mathematics Success – Grade 7T414

SOLVE Problem (WG, GP) S214 (Answers on T433.)

HavestudentsturntoS214intheirbooks.ThefirstproblemisaSOLVEproblem.You are only going to complete the S step with students at this point. Tell students

that during the lesson they will learn how to solve one-step equations with integers.

Theywillusethisknowledgetocompletereal-worldSOLVEproblemsthroughoutthelesson. {SOLVE, Verbal Description, Graphic Organizer}

Addition Equations with Concrete and Pictorial Models(M, GP, WG, CP) S213, S214, S215 (Answers on T432, T433, T434.)

WG, M, GP, CP Students will use unit tiles to explore one- step addition

equations with integers. Pass out red and yellow unit

tiles and cups to student pairs. Students will be using

the cups, tiles and a copy of the balance scale to build

a foundational understanding of solving equations.

Assign the roles of Partner A and Partner B to students.

{Concrete Representation, Verbal Description, Graphic Organizer, Algebraic Formula}


MODELING

Addition Equations – Concrete Models

Step 1:HavestudentsturntoS213tousetheirbalancescale. • Havestudentsplaceoneyellowunittileontheleftsideofthescale. • PartnerA,identifythevalueofoneyellowtile.(1) • PartnerB,isthescalebalanced?Explainyourthinking.(No,because

the value on one side is positive 1 and the value on the other side is

zero.)

• Have student pairs discusswhatwe can do to balance the scale.

(Place a yellow unit tile on the right side.)

• Whatdoesitmeanforthescaletobebalanced?(Bothsidesneedtobe equal in value.) Remove the yellow tiles from the scale.

• PartnerA,whatisthevalueofoneredtile?(negative1)Placeoneredtile on the left side of the scale.

• PartnerB,isthescalebalanced?(No)Whatcanwedotobalancethescale? (Place one red tile on the right side of the scale.)

• Practicebalancingthescalebyplacingdifferentvaluesofyelloworred tiles on the left side of the scale and have the students identify

what color and number of tiles will balance the scale.

Step 2: Review the concept of zero pairs by placing one yellow tile and one red

tile on the overhead balance scale.

• PartnerA,whatisthevalueoftheyellow?(positive1) • PartnerB,whatisthevalueofthered?(negative1)


• Havestudentsdiscussthevalueoftheredandyellowtileswhentheyare combined. (The value is zero.)

• PartnerA,whatdowecalltheyellowandredtilestogether?(azeropair)

• Place2yellowtilesontheleftsideand2yellowtilesontherightsideof the scale. Then place a zero pair on the left.

• Partner B, does adding the zero pair to the left change the valueon the left?Explainyour thinking. (No, thevaluedoesnot changebecause the additive identity property for addition tells us that we

can add zero to any number and it will not change the sum or the total

value.)

Step 3: Write the equation c + 3 = 5 at the top of the balance scale.

Explain to students that when modeling equations with integers there

are two things that they must focus on.

*Teacher Note: Students have worked previously with equations inGrade 6.

• PartnerA,whatisthefirst?(Isolatethevariable.)Askstudentswhatit means to be isolated. (to be alone or by yourself)

• PartnerB,whatisthevariable? (c) • PartnerA,explainthemeaningofthewordvariable.(Avariableisa

symbolthatrepresentsanunknownvalueornumber.Variablesareusually written as letters.)

Step 4: Partner B, what is our second focus? (Keep the equation balanced.)

• PartnerA,whatdoestheequalsignmean?(Iftwothingsareequal,they have the same value.)

• Partner B, explain what this means when solving an equation.(Whatever is on one side of the equation must be equal to what is on

the other side.)

• Whateveryoudo toone sideof theequation, youmustdo to theothersideinordertokeeptheequationbalanced.

Step 5: Model the equation in Problem 1 on S214 using a cup for the c. • Havestudentsplaceacupontheirscaleontheleft. • Havestudentsdiscusshowwecanmodeladding3,orpositive3. (Place 3 yellow tiles on the left side.)

Step 6:Havestudentsdiscusswhatneedstobeplacedontherightsideofthescale to represent the 5. (5 yellow tiles)

• Modelhow toplace5yellow tileson the rightsideof thescaleasstudents place their 5 yellow tiles on the right side of their balance scale.



Step 7:What are the two goals when working with equations? (Isolate thevariableandkeeptheequationbalanced.)

• Askstudentsforideasabouthowtoisolatethevariable.Howcouldthey use the idea of opposite operations to isolate the variable?

• Explain that to isolate the variable, students perform the oppositeoperation. If the equation is an addition equation, students will

subtract. If the equation is a subtraction equation, students will add.

Step 8:Modelhowtosubtract,ortakeaway,the3yellowtilesbyremovingthemfrom the scale.

• PartnerA,isthevariablenowisolated,oralone?(Yes) • PartnerB,istheequationbalanced?Explainyourthinking.(No,because

the same operation must be performed on both sides.)

• PartnerA,howcanwebalance theequation? (Takeaway3yellowtiles from the right side.)

• PartnerB,istheequationbalancednow?Justifyyouranswer.(Yes,because we have subtracted 3 from both sides of the equation.)

Step 9: Askstudentswhatthevalueofc is. (c = 2)

• Modelforstudentshowtochecktheproblemusingthetiles. • Gobacktotheoriginalequation(c + 3 = 5) and model that on the scale.

• PartnerA,whatisthevalueofc? (2)

• Modelhowtosubstitute2yellowtilesforthec. • PartnerB,whatisthevalueontheleft?(5yellowtiles) • PartnerA,whatisthevalueisontheright?(5yellowtiles) • Istheequationbalanced?Isouranswercorrect?(Yes,because5

yellow=5yellow,theequationisbalancedandweknowtheansweris correct.)

Step 10: Repeat the modeling process for Problem 2 on S215 (c + -3 = -5). The

processwillbethesameasthefirstequationexceptthatthetilesareall red because the numbers are negative.

MODELING

Addition Equations – Pictorial Models

Step 1: Partner A, what letter do we use to represent the cup? (c) • PartnerB,whatletterdoweusetorepresentthepositivethree?(Y) • Modelforstudentshowtorepresenttheequationc + 3 = 5 pictorially

as students complete their pictorial example.

• PartnerA,whatisourfirststepinsolvingtheequation?(Isolatethevariable.)

• Havestudentpairsdiscussideasforisolatingthevariable.(Wecansubtract the 3 Ys by crossing them out.)



• Havestudentscrossoutthe3Ysontheleftsideoftheequation. • PartnerB,haveweisolatedourvariable?(Yes) • PartnerA,istheequationbalanced?Explainyouranswer.(No,because

we only subtracted the positive 3 from one side.)

• PartnerB,whatdoweneedtodotobalancetheequation?(Subtract3 Ys from the right side of the equation because whatever operation

you do to one side, you must do to the other.)

• Modelsubtractionofthe3Ys(yellows)bycrossingthemout. • PartnerB,whatisthevalueofc? [c = 2 yellows (YY)]

c + YYY = YYYYY

c + YYY = YYYYY

c = YY

Step 2:Modelforstudentshowtochecktheequationbysubstitutingthevalueof c (YY) into the original equation.

• PartnerA,istheequationbalanced?(Yes,because5Ys=5Ys.)

YY + YYY = YYYYY

YYYYY = YYYYY

Step 3:HavestudentsturntoS215intheirbooks. • Modelforstudentshowtorepresenttheequationc + -3 = -5 pictorially

for Problem 2 as they model the equation on S215.

• PartnerA,howcanweisolatethevariable?(CrossoutthethreeRsonthe left side of the equation.)

• PartnerB,istheequationnowbalanced?(No) • PartnerA,whatdoweneedtodotobalancetheequation?(Subtract

3 Rs from the right side of the equation, because whatever operation


• Modelsubtractionofthe3Rs(reds)bycrossingthemout. • PartnerB,whatisthevalueofc? [2 reds (RR)]

c + RRR = RRRRR

c + RRR = RRRRR

c = RR

Step 4:Modelforstudentshowtochecktheequationbysubstitutingthevalueof c (RR) into the original equation.

• PartnerB,istheequationbalanced?(Yes,because5Rs=5Rs.)

RR + RRR = RRRRR

RRRRR = RRRRR



Addition Equations – Concrete and Pictorial Models Using Zero Pairs(M, GP, CP, WG) S213, S215, S216 (Answers on T434, T435.)

WG, M, GP, CP Students will use unit tiles to explore one-step addition

equations with integers and zero pairs. Be sure students

knowtheirdesignationasPartnerAorPartnerB.{Concrete Representation, Verbal Description, Graphic Organizer, Algebraic Formula}

MODELING

Addition Equations – Concrete Models Using Zero Pairs

Step 1:HavestudentsturntoS213tousetheirbalancescale. • Writetheequationc + 3 = -5 at the top of the balance scale.

• Havestudentsidentifythevariable.(c) • PartnerA,howcanwemodelthevariable?(Placeonecupontheleft

side of the balance scale.)

• PartnerB,howcanwemodeladding3,orpositive3?(3yellowtiles) • Place3yellowtilesnexttothecuponthelefthandsideofthescale

as students place their 3 yellow tiles on the left side of their balance

scale.

Step 2: Askstudentswhatneedstobeplacedontherightsideofthescaletorepresent the -5. (5 red tiles)

• Place5redtilesontherighthandsideofthescaleasstudentsplacetheir 5 red tiles on the right side of their balance scale.

Step 3: PartnerA,howcanweisolatethevariable?(Bytakingawayorsubtractingthe 3 yellow tiles from the left side of the scale.)

• Havestudentsremovethe3tiles. • PartnerB,isthevariableisolatedoralone?(Yes) • PartnerA,istheequationbalanced?(No,becausethesameoperation

must be performed on both sides.)

Step 4: PartnerB,whatdoweneedtodotokeeptheequationbalanced?(Subtract3 yellow tiles from the right side.)

• PartnerA,isitpossibletotakeaway,orsubtract,3yellowtilesfromtherightside?(No,becauseallthetilesarered.)

• PartnerB,whatcanwedoto“createthepossibility”oftakingaway3yellow tiles? (We can use zero pairs.)

Step 5: Add one zero pair (1 red and 1 yellow) to the right side.

• Whatisthevalueoftherightside?(Itisstillnegative5becauseweadded zero which did not change the value.)

• PartnerA,canwenowtakeaway3yellows?(No) • Addanotherzeropairtotherightside.



• PartnerB,whatisthevalueoftherightside?(Itisstillnegative5,because we added zero which did not change the value.)

• PartnerA,canwenowtakeaway3yellows?(No) • Addanotherzeropairtotherightside. • PartnerB,whatisthevalueoftherightside?(Itisstillnegative5,

because we added zero which did not change the value.)

• PartnerA,canwenowtakeaway3yellows?(Yes)

Step 6:Modelhowtotakeawaythe3yellowsasstudentsworkontheirbalancescaletakingawaythe3yellowsfromtheright.

Step 7: Partner A, is the variable now isolated? (Yes)

• PartnerB,istheequationnowbalanced?Explainyouranswer.(Yes,because the same operation has been performed on both sides of the

equation.)

• PartnerA,whatisthevalueofcintheequation?(Negative8,becauseall the tiles are red.)

Step 8:Modelforstudentshowtochecktheproblemusingthetiles. • Gobacktotheoriginalequation(c + 3 = -5) and model that on the

scale.

• Substitute8redtilesforthec. • Havestudentpairsdiscusswhattodowiththetilesthataredifferent

colorsonthesamesideoftheequalsign.(Wecanmakezeropairsand remove those from the equation without changing the value.)

• Afterremovingthe3setsofzeropairs,askstudentstoidentifythevalue on the left side (negative 5) and the value on the right side

(negative5)sotheequationisbalancedandweknowtheansweriscorrect.

Step 9:HavestudentsturntoS216intheirbooks.Repeatthemodelingprocessfor Problem 4 (c + -3 = 5). The process will be the same as in Problem

3 except that the tiles on the left are red and on the right are yellow.

MODELING

Addition Equations – Pictorial Models Using Zero Pairs

Step 1:HavestudentsturnbacktoS215. • Modelforstudentshowtorepresenttheequationc + 3 = -5 pictorially

asstudentsworkwithyouonS215. • PartnerA,canwetake3Ys(3yellows)from5Rs(RRRRR)?(No) • Havestudentsdiscusshow theycouldpossibly takeaway the3Ys.

(Create the possibility with zero pairs.)



• Addthezeropairsoneatatime,askingstudentseachtimeif it ispossibletotakeaway3Ys(3yellows).

Step 2: Model subtraction of the 3 Ys (3 yellows) on the left side by crossing

them out.

• PartnerA,isthevariablenowbyitself?(Yes) • PartnerB,istheequationnowbalanced?(No) • Havestudentsdiscusswhattheyneedtodotobalancetheequation.

(Subtract 3 Ys from the right side of the equation because whatever

operation you do to one side, you must do to the other.)

• Modelsubtractionofthe3Ys(yellows)ontherightbycrossingthemout.

• Whatisthevalueofc? (c=8reds;RRRRRRRR).

c + YYY = RRRRR RRR

c + YYY = YYY

c = RRRRRRRR

Step 3:Modelforstudentshowtochecktheequationbysubstitutingthevalueof c(RRRRRRRR)intotheoriginalequation.Askstudentsiftheequationis balanced. (Yes, because 5 Rs = 5 Rs.)

RRRRRRRR + YYY = RRRRR

RRRRR = RRRRR

Step 4:HavestudentsturntoS216intheirbooks. • Modelforstudentshowtorepresenttheequationc + -3 = 5 pictorially

astheyworkonS216. • Askstudentsiftheycantake3Rs(3reds)from5Ys(YYYYY).(No) • Havestudentsdiscusshowtheycouldpossiblytakeawaythe3Rs.


• Addthezeropairsoneatatime,askingstudentseachtimeif it ispossibletotakeaway3Rs(3reds).

Step 5:Havestudentsdiscusshowtheycanmodelsubtractionofthe3Rs(3reds) on the left side. (Cross them out.)

• PartnerA,isthevariablenowbyitself?(Yes) • PartnerB,istheequationnowbalanced?(No) • Havestudentsdiscusswhattheyneedtodotobalancetheequation.

(Subtract 3 Rs from the right side of the equation because whatever


• Modelsubtractionofthe3Rs(reds)ontherightbycrossingthemout. • Whatisthevalueofc? (c=8yellows;YYYYYYYY).



c + RRR = YYYYY YYY

c + RRR = RRR

c = YYYYYYYYY

Step 6:Modelforstudentshowtochecktheequationbysubstitutingthevalueof c (YYYYYYYY) into the original equation.

• Istheequationnowbalanced?(Yes,because5Ys=5Ys.)

YYYYYYYY + RRR = YYYYY

YYYYY = YYYYY

Subtraction Equations with Pictorial Models(M, GP, WG, CP, IP) S216, S217 (Answers on T435, T436.)

WG, M, GP, CP Students will explore one-step subtraction equations

withintegersandzeropairs.Besurestudentsknowtheir designation as Partner A or Partner B. {Pictorial Representation, Verbal Description, Graphic Organizer, Algebraic Formula}

MODELING

Subtraction Equations with Pictorial Models

Step 1: Tell students that we do not model subtraction with tiles, but that we

can model it pictorially by changing the subtraction equations to addition

equations.

• Reviewtherulesforsubtractingintegers.(3–5isthesameas3+-5.)

• Modelforstudentshowtorepresenttheequationc –3=5pictoriallyby adding the opposite. (c–3=5isthesameasc + -3 = 5.)

• PartnerA,whatisthefirstgoalforsolvingourequations?(isolatethevariable)

• Modelsubtractionofthe3Rs(reds)ontheleftsideoftheequation. • PartnerA,isthevariablenowbyitself?(Yes) • PartnerB,canwetake3Rs(3reds)from5Ys(YYYYY)?(No) • Havestudentsdiscusshowtheycouldpossiblytakeawaythe3Rs.


• Addthezeropairsoneatatime,askingstudentseachtimeif it ispossibletotakeaway3Rs(3reds).

• PartnerA,istheequationnowbalanced?(No) • Havestudentsdiscusswhattheyneedtodotobalancetheequation.



c + RRR = YYYYY + YYY

c + RRR = RRR

c = YYYYYYYY



Step 2:Modelforstudentshowtochecktheequationbysubstitutingthevalueof c (YYYYYYYY) into the original equation.

• PartnerB,istheequationbalanced?(Yes,because5Ys=5Ys.)



YYYYY = YYYYY

Step 3:HavestudentsturntoS217intheirbooks. • Model for students how to represent the equation c – 3 = -5 by

changing it to an addition equation (c + -3 = -5).

• PartnerA,whatisthefirststepinsolvingtheequation?(isolatethevariable)

• Modelsubtractionofthe3Rs(reds)bycrossingthemout. • PartnerB,isthevariablenowbyitself?(Yes) • PartnerA,istheequationnowbalanced?(No) • Havestudentsdiscusswhattheyneedtodotobalancetheequation.



• Modelsubtractionofthe3Rs(reds)bycrossingthemout. • PartnerB,whatisthevalueofc? (c=2reds;RR).

c + RRR = RRRRR

c + RRR = RRRRR

c = RR

Step 4:Modelforstudentshowtochecktheequationbysubstitutingthevalueof c (RR) into the original equation.

• Askstudentsiftheequationisbalanced.(Yes,because5Rs=5Rs.)

RR + RRR = RRRRR

CP, IP, WG: HavestudentsworkinpartnerstocompleteProblems3–4onS217.Remindstudentsthatbeforecompletingthe pictorial they will need to change the equation to an

additionequation.Havestudentscomebacktogetheras a class and share their results. They should be able

to justify results pictorially. {Verbal Description, Pictorial Representation, Graphic Organizer}



Solve Addition Equations – Algebraic Model (M, WG, GP, CP, IP) S214, S215, S216 (Answers on T433, T434, T435.)

WG, M, GP, CP Students will solve one-step subtraction equations

with addition of integers by building on foundational

understanding using concrete and pictorial

representations.Besurestudentsknowtheirdesignation as Partner A or Partner B. {Pictorial Representation, Verbal Description, Graphic Organizer, Algebraic Formula}

MODELING

Solve Addition Equations – Algebraic Model

Step 1: Model for students how to represent the equation c + 3 = 5.

• PartnerA,whatisthefirststepinsolvinganequation?(Isolatethevariable.)

• PartnerB,explainhowtoisolatethevariable.(Performtheoppositeoperation.)

• PartnerA,whatoperationwillweuseinanadditionequationtoisolatethe variable? (subtraction)

Step 2: Model how to subtract 3 from the left side of the equation.

• PartnerB,istheequationnowbalanced?(No) • PartnerA,whatdoweneedtodotobalancetheequation?(Subtract

3 from the right side of the equation because whatever operation you

do to one side, you must do to the other.)

• PartnerB,whatisthevalueofc? (c = 2) As in the pictorial model,

substitute the value of cbackintotheoriginalproblemtocheck.

CP, IP, WG: HavestudentsworkinpartnerstocompleteProblems2–4onS215andS216.Remindstudentsthattheycan use the pictorial representation as a model for their

equation.Havestudentscomebacktogetherasaclassand share their results. {Verbal Description, Graphic Organizer, Pictorial Representation, Algebraic Formula}

Solve Subtraction Equations - Algebraic Model (M, WG, GP, CP, IP) S216, S217 (Answers on T435, T436.)

WG, M, GP, CP Students will solve one-step subtraction equations

with subtraction of integers by building on

foundational understanding using concrete and

pictorialrepresentations.Besurestudentsknowtheirdesignation as Partner A or Partner B to students.

{Pictorial Representation, Verbal Description, Graphic Organizer, Algebraic Formula}



MODELING

Solve Subtraction Equations - Algebraic Model

Step 1: Model for students how to represent the equation c–3=5bychangingit to an addition equation (c + -3 = 5).

• PartnerA,explainhowtoisolatethevariable.(Performtheoppositeoperation.)

• PartnerB,becausethisisanadditionequation,howcanweisolatethe variable? (Subtract a negative 3.)

• Subtractthenegative3fromtheleft.

Step 2: Askstudentsiftheequationisnowbalanced.(No) • PartnerA,whatdoyouneedtodotobalancetheequation?(Subtract

a negative 3 from the right side of the equation because whatever


• Subtractanegative3fromtherightside. • PartnerB,whatisthevalueofc? (c=8). • Asinthepictorialmodel,substitutethevalueofcbackintotheoriginal

problemtocheck.

CP, IP, WG: HavestudentsworkinpartnerstocompleteProblems2–4onS217.Remindstudentsthattheyneedtochange the equations to addition equations and that

they can use the pictorial representation as a model

fortheirequations.Havestudentscomebacktogetheras a class and share their results. {Verbal Description, Pictorial Representation, Graphic Organizer, Algebraic Formula}

Practice with Solving Equations (CP, IP, WG) S218, S219 (Answers on T437, T438.)

CP, IP, WG: HavestudentsworkinpartnerstocompleteProblems1–8onS218andS219.RemindstudentsthatinProblems4–8theyneedtochangetheequationstoadditionequations.Havestudentscomebacktogetheras a class and share their results. {Verbal Description, Pictorial Representation, Graphic Organizer, Algebraic Formula}




Remind students that the SOLVE problem is the same one from the beginning of

thelesson.CompletetheSOLVEproblemwithyourstudents.Askthemforpossibleconnections from the SOLVE problem to the lesson. (They will solve an addition equation

with integers.) {SOLVE, Verbal Description, Graphic Organizer, Algebraic Formula}

If time permits… (CP, IP) S221 (Answers on T440.)HavestudentscompleteProblems1–8onS221.

[HOMEWORK]AssignS222forhomework.(AnswersonT441.)

[LESSON – PART 2] [2 – 3 Days (1 day = 80 minutes) – M, GP, WG, CP, IP]

[HOMEWORK]Taketimetogooverthehomeworkfromthepreviousnight.


HavestudentsturntoS223intheirbooks.ThefirstproblemisaSOLVEproblem.Youare only going to complete the S step with students at this point. Tell students that

during the lesson they will learn how to solve one-step equations with multiplication

anddivision.TheywillusethisknowledgetocompletethisSOLVEproblemattheendof the lesson. {SOLVE, Verbal Description, Graphic Organizer}

Multiplication Equations with Concrete and Pictorial Models (M, GP, CP, WG) S223, S224 (Answers on T442, T443.)

WG, M, GP, CP Students will use unit tiles to explore one- step

multiplication equations with integers. Pass out red

and yellow algebra tiles and cups to student pairs.

Students will be using the cups, tiles, and a copy of the

balance scale to build a foundational understanding of

solving equations. Assign the roles of Partner A and

Partner B to students. {Concrete Representation, Verbal Description, Graphic Organizer, Algebraic Formula, Pictorial Representation}



MODELING

Multiplication Equations – Concrete Models

Step 1:HavestudentsturntoS223intheirbooks.Studentswillalsoneedtheirbalance scale from S213.

Step 2: Write the equation 2c = 6 at the top of the balance scale.

• Reviewthestepsofsolvingequations.(Isolatethevariableandkeepthe equation balanced.)

• PartnerA,whenweweresolvingadditionequations,whatoperationdid we use? (the opposite operation, subtraction)

• Havepartnersdiscusswhatoperationtheythinktheywillusewhensolving a multiplication equation. (the opposite operation, division)

Step 3: Model the equation using cups to represent c. • Havestudentsdiscussthefollowingquestion.Ifc = 1 cup, how can

we represent 2c? (2 cups)

• Modelforstudentshowtoplace2cupsontheleftsideofthescale.

Step 4: Partner B, what do we need to place on the right side of the scale to

represent the 6? (6 yellow tiles because the 6 is positive.)

• Place6yellowtilesontheright-handsideof thescaleasstudentsplace 6 yellow tiles on the right side of their balance scale.

Step 5:Havestudentsdiscussstrategiestoisolatethevariable.(Divisionbecauseit is a multiplication equation.)

• Modelhowtodividethe2cupsbyseparatingthemintotwoseparategroups. Partner A, have we isolated the variable? (Yes) Partner B, how

can we balance the equation? (Divide the 6 yellow tiles on the right

side of the equation.)

• Modeldividingthe6yellowtilesinto2equalgroups. • Askstudentsiftheequationisbalancednow.(Yes,becausetheyhave

performed the same operation on both sides of the equation.)

Step 7: What is the value of c? (c = 3).

• Modelforstudentshowtochecktheproblemusingthetiles.Gobackto the original equation (2c = 6) and model that on the scale.

• Substitute3yellowtilesforthec. • PartnerA,whatisthevalueontheleft?(6yellowtiles) • PartnerB,whatisthevalueontheright?(6yellowtiles) • Istheequationbalanced?Explainyouranswer.(Yes,because6yellow

= 6 yellow, and the answer is correct.)

Step 8: Repeat the modeling process for Problem 2 on S224 (2c = -6). The

processwillbethesameaswiththefirstequationexceptthatthetilesare all red because they are negative.



MODELING

Multiplication Equations – Pictorial

Step 1:HavestudentsturnbacktoS223intheirbooks. • Modelforstudentshowtorepresenttheequation2c = 6 pictorially as

students use S223 to complete their pictorial example.

• Havestudentpairsdiscusshowtomodelthedivision. • Modeldivisionofthe2cups(2c) by writing two separate c’s. Then,

model dividing the 6 Ys on the right side of the equation into 2 groups,

explaining that whatever operation is done on one side must also be

done on the other.

• Thereare3Ys,oryellows,ineachgroupsothevalueofc is 3.

2c = YYYYYY c = YYY

c = YYY

Step 2:Modelforstudentshowtochecktheequationbysubstitutingthevalueof c (YYY) into the original equation.

• PartnerA,istheequationbalanced?(Yes,because6Ys=6Ys.)

Check:YYY+YYY=YYYYYY YYYYYY = YYYYYY

Step 3:HavestudentsturntoS224intheirbooks. • Modelforstudentshowtorepresenttheequation2c = -6 pictorially

as students use S224 to complete their pictorial example.

• Havestudentsdiscusshowtomodeldivisionofthe2cups(bywritingtwo separate c’s)

• PartnerB,whatisthenextstep?(Dividethe6Rsontherightsideofthe equation into 2 groups, because whatever operation is done on

one side must also be done on the other.)

• Thereare3Rs,orreds,ineachgroupsothevalueofc is -3.

2c = RRRRRR c = RRR

c = RRR

Step 4:Modelforstudentshowtochecktheequationbysubstitutingthevalueof c (RRR) into the original equation.

• PartnerA,istheequationbalanced?(Yes,because6Rs=6Rs.)

RRR + RRR = RRRRRR

RRRRRR = RRRRRR



Solve Multiplication Equations (M, GP, CP, WG, IP) S223, S224 (Answers on T442, T443.)

WG, M, GP, CP Students will solve one-step equations with

multiplication of integers by building on foundational

understanding using concrete and pictorial

representations.Besurestudentsknowtheirdesignation as Partner A or Partner B. {Pictorial Representation, Verbal Description, Graphic Organizer, Algebraic Formula}

MODELING

Solve Multiplication Equations

Step 1:HavestudentsturnbacktoS223intheirbooks. • Havestudentslookatthesectionfor“SolvetheEquation.” • PartnerA,whatisthefirststepinsolvingourequation?(Isolatethe

variable.)

• PartnerB,whatoperationwillweuse?Explainyour thinking.(Thisis a multiplication equation, so we will use the opposite operation of

division.)

• Modelforstudentshowtodividetheleftsideoftheequationby2. • PartnerA,isourequationbalanced?(No) • PartnerB,whatdoweneedtodotobalancetheequation?(Dividethe

right side of the equation by 2.)

• PartnerA,whatisthevalueofc? (c = 3).

• Asinthepictorialmodel,substitutethevalueofcbackintotheoriginalproblemtocheck.

Step 2:HavestudentsturntoS224intheirbooks. • HavestudentslookatProblem3.(-2c = 6)

• Ask students if they think it is possible tomodel Problem 3. (No,because you cannot represent negative 2 with cups or pictorially.)

Step 3: Explain that because the equation in Problem 3 is a multiplication

equation, students can isolate the variable by dividing by -2.

• Havestudentsdividetheleftsideoftheequationby-2.

• Istheequationbalanced?(No) • Havestudentsdeterminewhattheyneedtodotobalancetheequation.

(Divide the right side of the equation by -2, because whatever operation


• Whatisthevalueofc? (c = -3).

• Modelhowtosubstitutethevalueofcbackintotheoriginalproblemtocheck.



MODELING

Solve Division Equations

Step 1:HavestudentpairsdiscussProblem1onS225. • Identifytheformatofthedivisionequation.(Theproblemiswrittenasa

fraction bar with the fraction bar representing the division symbol.)

• Explaintostudentsthattheywillnotmodelwithtilesoruseapictorialmodelfordivisionequations.However,studentswillfollowthesamesteps as with multiplication equations to solve the division equations.

Step 2: PartnerA,whatisthefirstgoalwhensolvingourequation?(Isolatethevariable.)

• Partner B, explain how we can isolate the variable. (Perform theopposite operation.)

• PartnerA,thisisadivisionequation.Whatoperationwillweusetoisolate the variable? (The opposite operation which is multiplication)

• PartnerB,whatnumberdoweneedtomultiplyby?(2) • Explaintostudentsthatwhentheymultiplyc

2 by 2, the product is

2c2

.

• PartnerA,what is the value of 2 divided by 2? (1, because any number

divided by itself is 1.)

• Have students complete the multiplication on the left side of theequation.

Step 3: Askstudentsiftheequationisnowbalanced.(No) • PartnerB,whatdoweneedtodotobalancetheequation?(Multiply

the right side of the equation by 2, because whatever operation you

do to one side, you must do to the other.)

• Havestudentsmultiplytherightsideoftheequationby2. • Whatisthevalueofc? (c = 6).

Step 4: Substitute the value of cbackintotheoriginalproblemtocheck.

c2 = 3,

6

2 = 3, 3 = 3

CP, IP, WG: HavestudentsworkwiththeirpartnerstocompleteProblems2and4onS224.Havestudentscomebacktogetherasaclassandsharetheirresults.{Verbal Description, Concrete Representation, Pictorial Representation, Algebraic Formula}

Solve Division Equations (M, WG, GP, CP, IP) S225 (Answers on T444.)

WG, M, GP, CP Students will solve one-step equations with division

ofintegers.Besurestudentsknowtheirdesignationas Partner A or Partner B. {Verbal Description, Graphic Organizer, Algebraic Formula}


lesson 18: one-step equations with integers …ntnmath.kemsmath.com/level g teacher notes/grade 7-...

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