8.5. Solving Equations II. Goal Solve equations by balancing.

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1 8.5 Solvig Equatios II Goal Solve equatios by balacig. STUDENT BOOK PAGES Direct Istructio Prerequisite Skills/Cocepts Solve a equatio by ispectio or systematic trial. Perform operatios usig itegers, e.g., add, subtract, multiply, ad divide. Evaluate simple algebraic expressios usig substitutio. Assessmet for Feedback Studets will solve equatios usig a balace model Preparatio ad Plaig Pacig (allow 5 mi for previous homework) Materials Vocabulary/ Symbols 5 mi Itroductio 25 mi Teachig ad Learig 10 mi Cosolidatio paper bags red ad blue coloured couters calculators Optioal: Scaffoldig for Lesso 8.5 (Master) p. 64 balacig a equatio Workbook pp Recommeded 8, 9* (Kowledge ad Practice Uderstadig), (Applicatio of Learig) Additioal Practice Extedig: 19, 20 Learig Skills Key Assessmet of Learig Questio Mathematical Processes Use of Iformatio, Problem Solvig Questio 9, Kowledge ad Uderstadig Selectig Tools ad Computatioal Strategies, Reflectig * Key Assessmet of Learig Questio (See chart o p. 39.) Expectatios describe differet ways i which algebra ca be used i real-life situatios traslate statemets describig mathematical relatioships ito algebraic expressios ad equatios solve ad verify liear equatios ivolvig a oe-variable term ad havig solutios that are itegers, by usig ispectio, guess ad check, ad a balace model What You Will See Studets Doig Whe studets uderstad Studets will uderstad that i a equatio the equal sig meas that the quatity o the left side is the same as the quatity o the right side. Whe somethig is doe to oe side of the equatio, i order to keep it balaced as with a pa balace, the same thig must be doe to the other side of the equatio. Meetig Idividual Needs If studets misuderstad Studets may ot uderstad that the quatity o the left side of a equatio is the same as that o the right side. Use a pa balace ad a ukow object to represet the variable. Whe studets kow that both sides must remai balaced, it will be easier for studets to determie what value the variable represets. Whe oe side of a equatio has a term such as 4t, some studets may subtract 4 from both sides. Explai that 4t meas that there are 4 groups of t objects. To determie what t is, both sides have to be divided by 4. Extra Challege Studets ca explore how to use a pa balace to model a equatio whe two variables are ivolved. For example, ask studets how they would model Questio 10 i Lesso 8.1 o Studet Book page 255, where the umber of each colour of tile is represeted by a uique variable. Alteratively, studets could determie how to model the cois ad bills problem i Lesso 8.2 o Studet Book page 256, or the hardcover ad paperback books problem i the Example i Lesso 8.3 o Studet Book page 259. Extra Support Have studets use a pa balace, couters, ad empty film caisters to model the equatios i this lesso. The film caister ca represet the ukow amout, because the umber of couters iside it is ukow. This will be particularly helpful whe solvig equatios where it is ecessary to isolate the variable by divisio, such as 4 = 12. Math Backgroud I this lesso, pa balaces are used to help studets see that the left side of a equatio must equal the right side. As a result, the value of the expressio with the variable o oe side must equal the kow amout o the other side. The key to idetifyig the value of the variable is to isolate the variable o oe side of the equatio. This may ivolve simply addig or subtractig a amout to each side, or it may ivolve a umber of steps. Studets eed to be careful whe the variable is multiplied by a costat, such as 4. Copyright 2006 by Thomso Nelso Lesso 5: Solvig Equatios II 35

2 Dealig with Homework (Whole Class/Pairs) about 5 mi If the Follow-Up ad Preparatio was assiged i Lesso 8.4, have studets share their examples of real-life situatios ivolvig balacig. Some examples might iclude usig a balace i a sciece experimet, acrobats balacig o a tight rope, the famous Chiese bicycle balacig act, balacig a cheque book, ad so o. Talk about why maitaiig the balace is importat i these situatios, ad what happes whe somethig does ot balace. 1. Itroductio (Whole Class/Pairs) about 5 mi Review with the class what they already kow from previous grades about balacig equatios. Do a few examples together, ad have studets explai their thikig processes i fidig the solutios: a) 2 + = 20 b) 3c = 12 c) 2s 4 = 12 Sample Discourse How did you solve the equatio 3c = 12? I kow that 3c meas 3 groups of c items. So if I divide the left side of the equatio by 3, the I have to divide the right side by divided by 3 is 4, so c = 4. Tell studets that i this lesso they will use balacig to solve equatios. 2. Teachig ad Learig (Whole Class/Pairs) about 25 mi Lear about the Math Read the problem ad cetral questio o Studet Book page 268 with studets. Have studets work i pairs to read the examples ad solutios i the Studet Book. Both equatios ca be solved by balacig, but the first equatio ivolves additio ad the secod equatio ivolves subtractio. Have studets thik about what is the same betwee these two examples ad what is differet. Explai that what is the same is that o matter what the operatio sigs are withi a equatio, you balace the equatio by doig the same thig to both sides. What is differet is what you decide to do, whether you add, subtract, multiply, or divide. You decide that based o each idividual equatio. Sample Discourse Why did Stefa decide to take away two couters from each side? Because i the equatio o the side with the variable, there was a + 2. So to get the variable by itself, Stefa subtracted 2. The he had to subtract 2 from the other side to keep the equatio balaced. How were solvig these two examples the same? Differet? I order to keep the equatios balaced, you always did the same thig to both sides. I the first example, Stefa had to subtract couters, but i the secod example Kayley had to add couters. Reflectig Discuss the questios with the class. You may wat to have studets talk about the questios first i pairs ad the as a class. Aswers to Reflectig 1. For example, a equatio is like a pa balace because if both sides are equal, you ca say it is balaced. If oe side is heavier (pa balace) or greater (equatio), it will ot be balaced. The variable ca be o either the left or right side. 36 Chapter 8: Equatios ad Relatioships Copyright 2006 by Thomso Nelso

3 2. For example, addig or subtractig the same umber of couters o both sides of a pa balace does ot chage the equatio. It s just rewritig it aother way, but the equatio has the same meaig. The solutio to the equatio does ot chage because you did the same thig to both sides. It maitais the balace. 3. For example, the umber of bags represets the umber of groups. All of the couters i all of the bags altogether equals the total umber of couters. So if you group the couters to match the umber of bags, it is like dividig the total umber of couters by the umber of bags. Whe you have a equatio like 3 = 15, the 3 tells you how may bags (or groups) you have ad the tells you how may couters are i each bag. So if you divide 15 ito 3 groups, you ca tell that there will be 5 couters i each bag. 4. For example, if the side of the equatio with the variable has more couters tha the other, you eed to subtract. If the side of the equatio with the variable has fewer couters, you eed to add. If the side of the equatio with the variable has groups of couters that equal a total amout o the other side, you eed to separate couters ito groups. 3. Cosolidatio about 10 mi Work with the Math Solved Example (Whole Class) Example 3 shows a equatio that ivolves two steps to solve. Go through this example carefully with the class. Ask why it is ot possible for the first step to be to separate the left side ito three groups. Use the pa balace model to show why this would ot work. I this example, studets add first ad the divide. However, poit out that other equatios that ivolve two steps to solve may ivolve other operatios i a differet order. This example also ivolves decimals. Talk about how workig with decimals affects solvig the equatio. A Checkig (Whole Class/Pairs) Have studets work i pairs. Questios 5, 6, ad 7 c) ivolve two-step solutios. Make sure that studets do the additio or subtractio of couters first before they divide. You may wat to do 7 d) together as a class, sice it is the first time i this lesso that the divisio symbol is used i a equatio. Ask a studet to explai how that would be modelled o the pa balace. Aswers to Checkig 5. a) For example, if you double the umber of couters o the left side, you must also double the umber of couters o the right side to keep the balace. b) For example, if you subtract 4 couters from the left side, you must also subtract 4 couters from the right side to keep the balace. 6. Let represet the umber of couters i each bag. Sice there are 2 bags ad 3 extra couters o the left side, a expressio to represet this situatio is Sice there are 17 couters o the right side, a expressio to represet this situatio is 17. So the equatio is = 17. To solve, first subtract 3 couters from each side = = 14 Sice there are two bags, divide the couters o each side by 2 to determie how may couters are i each bag. 2 2 = 14 2 = 7 (Aswers to Lesso 5, Checkig, cotiued o p. 87) Copyright 2006 by Thomso Nelso Lesso 5: Solvig Equatios II 37

4 B Practisig (Idividual) These questios give studets opportuities to create, model, ad solve equatios by balacig. If extra support is required, provide copies of Scaffoldig for Lesso 8.5, Questio 8, p. 64. Aswers to Key Assessmet of Learig Questio 9. (Kowledge ad Uderstadig) a) 2m 3 = 31 2m = m = 34 2m 2 = 34 2 m = 17 2m 3 31 = 2(17) 3 = 34 3 = 31 The solutio is m = 17. b) 5a + 15 = 75 5a = a = 60 5a 5 = 60 5 a = 12 5a = 5(12) + 15 = = 75 The solutio is a = 12. c) = 37 ( 8) = ( 8) = 160 ( 8) 8 = = = = = 37 The solutio is = d) 214 = = = = = = = 6(32.5) + 19 = = 214 The solutio is = Key Assessmet of Learig Questio (See chart o p. 39.) e) 10w 145 = w = w = w 10 = w = w = 10(110) 145 = = 955 The solutio is w = 110. f) = 13 ( 4) = 13 7 ( 4) = 6 ( 4) 4 = 6 4 = = (24 4) + 7 = = 13 The solutio is = 24. C Extedig (Idividual) Questios 19 ad 20 challege studets to model ad solve equatios ivolvig itegers. 38 Chapter 8: Equatios ad Relatioships Copyright 2006 by Thomso Nelso

5 Closig (Whole Class) Have studets share their aswers ad methods of solutio to Questio 14. Although the itetio of the questio is to have studets write ad solve the equatio 3b + $1.15 = $10.00, be ope to other ways studets may have writte or solved the equatio (for example, = 3b). Some studets may say that sice $10.00 $1.15 = $8.85 ad Deborah bought three biders, each bider must cost $ = $2.95. Still others may say that three biders at $3 each would be $9.00. But that would oly be $1 chage ad Deborah received $1.15. So you eed to take aother $0.15 divided by three off the price of each bider. That would be $3 $0.05 = $2.95. Chapter Project Lik I this lesso, studets lear about solvig equatios usig balacig. They ca apply ad exted their uderstadig of this cocept i the Chapter Project pp Follow-Up ad Preparatio for Next Class At the ed of the class, have studets share with a parter how usig a pa balace model helps solve equatios. Assessmet of Learig What to Look for i Studet Work Assessmet Strategy: short aswer Kowledge ad Uderstadig Key Assessmet Questio 9 Solve each equatio. Show your steps. a) 2m 3 = 31 c) = 37 e) 10w 145 = 955 b) 5a + 15 = 75 d) 214 = f) = Carry Out the Pla demostrates limited or iaccurate kowledge of solvig equatios by balacig demostrates some kowledge of solvig equatios by balacig demostrates cosiderable kowledge of solvig equatios by balacig demostrates sophisticated kowledge of solvig equatios by balacig Copyright 2006 by Thomso Nelso Lesso 5: Solvig Equatios II 39

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