Try it and see if it works

I was tutoring a student who is using the AoPS Prealgebra textbook, and we were learning how to solve linear equations with one variable (p. 215).  This is the sort of problem that looks like this:
21n + 28 = 10n - 40.
My student understood that in order to solve the equation we first need to combine like terms and isolate the variable, which he successfully did:
11n = -68.
My student also understood that we need our variable n to have a coefficient of 1 so that we get something that looks like
n = some number.
But how to turn 11n = into n = ?
It turned out he had a lot of interesting ideas.  “How about we subtract 10n from both sides?” he suggested.
Those of you well-versed in algebraic manipulations can foretell that this will lead you farther away from a solution.  But rather than cut him off and tell him this is wrong, I went along with it.
“Okay, let’s give it a try and see what happens.  Subtracting 10n from both sides…”
11n - 10n = -68 - 10n
So: n = -68 - 10n .
Well, that didn’t work.  It turned out my student had many, many creative approaches to isolating n and solving this problem, none of which got us closer to actually solving the problem.  But rather than cutting him off and prematurely telling him he was wrong, I went along with the playful exploration.  This approach of “try it and see if it works” experimentation is something we want to cultivate in our students, not just in math, but in many academic fields.   Eventually he remembered that we could divide both sides by 11 to solve the problem.   While this sort of discovery approach can be inefficient and tedious, it’s fun to pop into a rabbit hole occasionally.
My hope is that in the future, if he sees another problem like this, he won’t panic, but will instead try to reason it out, just as he did when he learned it for the first time.

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