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

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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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