I think what is really becoming clear to me is that the real work I have to do is in clarifying how kids can make sense of equations that go beyond the simplest cases.
The simplest cases are ax + b = c and ax + b = cx + d, where all the numbers are integers. They’re relatively simple for students because they can use balance reasoning or backtracking/unwinding to tackle them.
Things get more complex with equations of the type c x (ax + b) = d where the numbers are integers. I would expect these to be not-that-much harder for kids, since they can use unwinding and covering up to crack these. Still, these equations afford a new strategy, that of using the distributive property to transform this into an equation of ax + b = c type. I would bet that this is the best way to get manipulating expressions on the table, as a strategy.
b – ax = c is a lot harder, even with a, b, c as integers. It’s nearly impossible to directly use either unwinding or balancing on this equation. (Though if b > c then students can at least use the “cover up” method.) To use either balancing or unwinding, some arithmetic manipulation likely needs to be involved, treating b – ax = c as the same as b + (-ax) = c. This is useful for kids, but it’s a slippery move that many will struggle with. Another algebraic move could be transforming b – ax = c into b = c + ax. This involves having a strong grasp of the addition property of equations, something that students might develop from balance reasoning. Still, this is tough.
I don’t know whether it’s worth it to identify b- ax = c +/- dx as a distinctive type of problem. The same problems with balancing and unwinding apply, and covering up seems unlikely to help much. It seems likely to me that kids will be tempted to use balance reasoning, because of the surface resemblance of this type of equation to ax + b = cx + d. I would expect that to be the major source of errors. If we’re creative with the balance model, we can try to find a way to represent this and to use balance reasoning on this. That seems tricky to me, and I’m feeling like that might not be worth it. The other move, as before, is to use some algebraic manipulation to turn b – ax into b + (-ax). But, again, this is hard.
So far, so good. But where things get really complicated, I think, is when negative numbers and fractions get involved in any of these problem types. (As I write this, my thought is “well duh.”)
How kids think about equations with fractions, decimals and negatives in them is something I’m entirely unclear about. I think this is a place where I need to focus.
I think there’s a case to be made that all this work with integer equations should go before we spend much time working on equations with fractions/decimals/negatives involved. These trickier numbers might distract from the reasoning that we’re working on developing. This might be a situation where it makes sense to be as fluent as possible with something like balance reasoning in its natural setting before stretching the metaphor to include things like 1.4x + 9 = 0.7x + 8.
(Though, come to think about it, that’s actually pretty natural in a balance setting because weight can take on any positive value. A better example would have been -0.2x + 5 = 2x + 3. To me, this mistake in my writing underscores how little I know about non-integer equations and how kids can think about them.)
Next steps: non-integer equations, and thinking about feedback that could help students progress when working with different integer equations.