When I teach students to solve linear equations, I’m aiming to help move them from guess-and-check to using the properties of equality. I’m in search of the ways of thinking that could help students move towards the more sophisticated techniques.

Balance problems (i.e. mobile problems) seem promising. They seem promising because they can be solved using guess-and-check, just as linear equations can. They can also be solved using the properties of equality, if you translate the situation into a linear equation.

In short, the thinking that students can do with balance problems is close enough to the thinking used to solve equations to be useful. If there happens to be a special way of thinking about balance problems, that way of thinking might help students develop facility with properties of equality. Much in the same way that visual pattern problems can help shepherd students from recursive to functional thinking, balance problems might help students move from guess-and-check to equality-property-thinking.

And, in fact, there is a special way of thinking that balance problems allow for. I don’t know what to call it. Maybe I’ll call it “balance reasoning.” That seems almost tautological, but whatever, balance reasoning! Balance reasoning is available for balance problems.

This is the reasoning that goes like, “If the left side has four circles and the right side has one circle and two triangles, then it’ll have to stay balanced if we take one circle off of each side. So it must be that three circles on the left side and two triangles on the right side are still balanced.”

In sum, students can use three types of reasoning for balance problems:

- Guess-and-check
- Balance Reasoning
- Properties of Equality

Noticing patterns in what we do with balance reasoning can help us develop properties of equality within balance problems. With enough experience with the sort of “remove the same things from each side” thinking we could formulate a technique: “If you can, take off the same number of things from each side.” This is a close analogue to the addition property of an equation.

With enough experience with other balance problems, we could get something like the division property of equality: “If you have all squares (or whatever) on one side and all triangles (or whatever) on the other, split up one side and give it to the other.”

Yay, kids can now solve equations! Right?

Well, not so fast. All of this is taking place in an unusual, foreign context. Balance problems are not equations. How do we make the move back into solving equations? I think we need to design activities that make explicit the connections between balance problems and equations. We translate back and forth between them. We formulate new rules for solving equations that explicitly mirror the rules that we know and love from balance problems.

So, now kids can solve equations! Right?

No, because balance problems aren’t natural settings for a lot of equations. Anything with negatives or fractions or subtraction is awkward to use with balance problems. Sure, we can extend balances to work with all sorts of signed variables or subtraction (like -2x – 3 = 0) but it’s awkward. It’s mathematically *fascinating *and valuable for its own sake, but unlikely to be much help when a kid is staring at an equation and feeling the urge to guess-and-check.

In short, balance problems can be solved using three modes of reasoning. One of those modes has promise for helping students think using properties of equality, but not *all *the properties of equality.

The goal of using balance problems in this context would be to develop properties-of-equality-thinking. Our feedback should be oriented towards that. Balance problems are useful, but only in a very particular way.

In the next post, I’ll think a bit about developing facility with other properties of equality.

Similar to your noted difficulties in moving students from balance problems to equations involving rational coefficients and constants, I observed my students having similar challenges in transferring their understanding of balance from Hands-On Equations (which are basically physical balance problems) to using equality in manipulating equations. Hands-On equations includes different pieces for negative integers and a negative variable, but some students never really saw the connection between removing pieces from both sides of the balance and the analogous act of adding/subtracting from both sides of an equation. I did my best to remind students of this parallel, but it does feel like a strained connection.

I’m wondering if the progression might be strengthened by including a creative step in what you described. After students have worked with balance problems, offer them the chance to create problems for each other. Without a lot of fanfare, I would move into solving equations using the concept of equality. Once a handful of basic examples are complete and students are questioning my sanity, I would cycle back to the basic equation and draw a balance diagram for the equation. Once we discussed the parallel, I would direct students to draw diagrams for the other examples we completed. Perhaps by adding this creative element would strengthen the formation of the equality concept in students?

At this point, the obstacle of solving more interesting equations using equality over guess-and-check remains to be resolved. Curiously, I found with one of my classes last year that they adopted equality more rapidly when I increased the complexity of equations. The use of equality remained bothersome for some students who wanted to use calculator kung-fu, but I found that more students adopted equality when I stated that it was a method of “proving” the solution. I’m wondering if this situation might have developed even earlier if I used balance problems, followed my suggestions mentioned above, and asked students devise a diagram for one of these more complex equations. Once students struggled with developing an accurate diagram for the equation, I would direct students to solve the equation algebraically. In this way, the weaknesses of balance problems in regards to their relationship to equations would be evident; however, the utility value of equality for solving equations would be put on full display for students.

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