You probably noticed in the Interactive Activity that in order to keep balance, you must do the same thing to both sides of the scale.
In an algebraic equation, the balance is represented by the equal sign, and only doing the same thing to both sides of the equation will preserve the balance. Note 6
Make up your own bag and block balance puzzle. If you are working with someone else, exchange puzzles with your partner and solve your partner's puzzle.
Draw a balance puzzle that represents 3b + 7 = 3b + 2. Now solve the equation. Explain what happens. Which equation below from Problem A1 is most like this one?
Note 8
Draw a balance puzzle that represents 4b + 3 = 4b + 3. How is this different from what happened in Problem C5? Which equation below from Problem A1 is most like this one?
Can you draw a balance puzzle to represent the equation 4b - 2 = 5b - 3? Why or why not?
Note 9
Problem C7 brings out some of the limitations of the balance model. The method of doing the same thing to both sides may still be used to solve problems that are difficult to represent with balance puzzles.
One method of teaching how to solve equations is that "if you don't like which side a number is on, move it to the other side and switch the sign." How is this related to the method of doing the same thing to both sides?
Video Segment In this video segment, Sue-Anne says that the balance puzzle helped her see why it was so important to do the same thing to both sides, and emphasized the importance of using this analogy with her students.
Think about the problems you've worked on in this session. Will the method of doing the same thing to both sides solve every problem in this session, or just some of them?
There are many strategies that people use to solve equations: guessing and checking, backtracking or inverting operations, and doing the same thing to both sides. For any particular problem, one method may be easier than another. The word "easier," however, has two different meanings. It might mean "more conceptually understandable," or it might mean "more efficient to compute." The guess-and-check method is rarely efficient, but students understand it. Backtracking or inverting operations doesn't always work. Doing the same thing to both sides always works, but sometimes the computation is messy.
Problems in Part C taken from IMPACT Mathematics Course 2, developed by Education Development Center, Inc. (New York: Glencoe/McGraw-Hill, 2000), p. 389. www.glencoe.com/sec/math
