Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

Session 6, Part C:
Bags, Blocks, and Balance

In This Part: Balance Scale Activity | Balance Puzzles and Equations

 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

 Problem C1 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.

Problem C2

Look at the balance puzzle below.

 a. Assign a variable to the number of blocks in each bag. Write an equation that fits the puzzle. b. Using the drawing, find the number of blocks in each bag. c. Solve the equation by doing the same thing to both sides. How is the solution to the equation related to what you did in part b of this problem?

 If you are having trouble here, watch the video segment following the Interactive Activity.   Close Tip If you are having trouble here, watch the video segment following the Interactive Activity.

 Problem C3 Create a balance puzzle where the solution is not a whole number of blocks. How could you modify your model to fit this solution? Note 7

 Problem C4 Draw a balance puzzle that represents 2h + 3 = h + 8. Now solve the balance puzzle. In the puzzle, what is represented by the "h" in the equation?

Problem C5

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

 a. 5 + 3 = 8 b. 2 + 14 = 12 c. 5 + 3 = y d. x + 3 = y e. 3x = 2x + x f. 3x = 3x + 1

Problem C6

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?

 a. 5 + 3 = 8 b. 2 + 14 = 12 c. 5 + 3 = y d. x + 3 = y e. 3x = 2x + x f. 3x = 3x + 1

 Problem C7 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.

 Problem C8 Solve the equation in Problem C7 by doing the same thing to both sides.

 Problem C9 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

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 Session 6: Index | Notes | Solutions | Video