I want to help students move from guess-and-check to using equality properties in the equation work. There are some problems that might be helpful in this work — like balance problems — but there are limitations to their usefulness. Balance problems allow for a sort of “balance thinking” that has close analogies in the world of solving equations — when you take things off of both sides things stay balanced (subtraction property) and when you divide each side equally each divided part is balanced (division property).
It’s hard, though, to represent negative numbers or fractions in balance problems, and there’s little reason to use anything resembling either the addition or multiplication property in that work. So there are limits to the usefulness of balance problems.
What other problems might be helpful? Another promising problem type is the “pick a number” problem, as seen at the top of the post. (There has got to be a better name for this type of problem. They’re not number tricks. “Closed number tricks”?)
I’d like to get precise about what makes pick a number problems promising for learning to solve equations. There are two things that I’m looking for to gauge potential usefulness of a new scenario or problem type: resemblance and novelty.
- Resemblance: My students will able to use guess-and-check to solve equations, but I want them to be able to use equality properties. Can guess-and-check and equality properties both be used for this problem type?
- Novelty: To be useful, some different way of thinking about this problem type, beyond guess-and-check and equality properties, needs to be available to students. Otherwise, the problem is just functionally equivalent to my equations. Is there a new strategy students can use to think about this problem type?
“Pick a number” problems allow for guess-and-check, and you could also translate the steps into an equation and solve it conventionally. So there is resemblance.
The novelty is “backtracking” thinking. This is the thinking that, step by step, reverses the steps of the recipe until the original “pick a number” is revealed.
So, kids are likely (I bet) to think about backtracking problems in one of three ways:
- Guess and check
- Equality properties
Suppose that kids got very, very good at these “pick a number” problems. What benefits could that have for their solving equation skills? Could backtracking help them develop equality property strategies?
I see two possible teaching strategies along these lines:
- Ask students to translate equations into steps, and then have students use backwards thinking in its natural setting.
- Develop generalizations in the “pick a number” problem that can be extended to solving equations.
The first option seems like a bad idea, since there are all sorts of equations that cannot be usefully translated into steps. Even basic equations like 4 – x = 10 don’t really work well with backtracking reasoning. (Pick a number. Start with 4. Subtract your number. You get 10.) You’d have to help students understand a specific class of equations that work well with backtracking, and that sounds like a headache.
I like the second option more. We could form generalizations from backtracking that are easily applicable to equations. “To find your number, you need to undo the operations that are done to it” is as true for “pick a number” problems as it is for equations. Of course, we would need to clearly show how that reasoning applies. But work in these problems could help students use the inverse of operations to reveal the starting number. Balance problems only gave you the subtraction and the division properties, but “pick a number” problems also give you multiplication, addition, squares and square roots properties of equality.
The big limitation is that variables have to be on one side for “pick a number” problems to work. These problems are unlikely to develop tools that are helpful for anything that has a variable on each side, because backtracking only works if we’re imagining the equation to have one, single picked number. How can you reverse the steps in “I multiply my number by 7, I add 18, I get 10 times my starting number”?
That’s the first big limitation of this problem type.
The second big limitation is the same one that balance problems had: this is a new scenario. Any skills that we want students to take away from “pick a number” or balance problems to equations will not come for free. We will need to spend time helping students see how the skills they use in one context are applicable in another. There’s an art to this, but maybe it’s not worth it.
I want to think about this more in the fourth post in this series.