Intent

This activity further develops students’ abilities to simplify and solve equations. At the conclusion of this activity, students will have two related ways to think about solving equations. The pan-balance model provides a more concrete metaphor for manipulating symbols while the set of rules developed to maintain equivalent equations is useful in more varied situations.

Mathematics

The algebraic thinking developed during this unit emphasizes reasoning, multiple representations, and the connections among these representations. By this point in the unit, students will have solved a variety of equations, using both the pan-balance model and some general principles for symbol manipulation that maintain equivalent equations. These symbol-manipulation methods should be thought of as complementing other techniques students have for solving equations, including using a graph, a table, estimation, and other reasoning processes.

Progression

After students work on their own to use the rules of equivalent equations and the pan-balance model, they come together to share results and ask questions of one another. The teacher emphasizes the “doing” and “undoing” nature of these problems, some of which may remain unsolved by some students at this time.

Approximate Time

5 minutes for introduction

30 minutes for activity (at home or in class)

20 minutes for discussion

Classroom Organization

Individuals, followed by whole-class discussion

Doing the Activity

Tell students that they will be unscrambling a few more equations in this activity, following the rules for keeping equations equivalent. They will also try to solve a couple of mystery-bag problems, including one that may be especially challenging. Students should record, in some way, what they do at each step to unscramble the equation or solve the mystery-bag puzzle.

You may want to mention that Questions 2 through 4 were not necessarily scrambled in exactly three steps, as was student work in Scrambling Equations. Don’t expect all or even most students to be able to complete Question 3, 4, or 7 on their own at this point. The techniques they have been developing, especially as applied to more difficult problems, will be strengthened and refined in future work.

Discussing and Debriefing the Activity

The follow-up to this activity could begin with groups sharing results and asking questions of one another.

Then ask a few volunteers to describe what they did to unscramble one of Questions 2 through 4. It is not necessary that all students are confident with the procedure for Question 4, but they may be curious to see it solved if they haven’t figured it out yet.

During the discussion, reemphasize that each step students took is essentially the reverse of one of the steps that was used to create the equation and that they did these “backward” steps in the reverse order from the order of the original steps.