Simplifying with Complexity

Sometimes the quickest and easiest way to solve an easier problem isn’t the best way to approach a hard problem. Today (first day of course content in my Pre-Calculus class) we were reviewing sets of numbers, categories of numbers, and notation. I gave my class the following set:

{-2, -1, 0, 1, 2, 3}

and asked them to turn it into set builder notation. Every student was able to write the set as some variant of the following:

Set Builder AV

I asked if there were other ways of writing it, and some people changed the set of integers to whole numbers or natural numbers, but that was about it.

Then I asked the class to do the same thing with the following set:

{3, 6, 9, 12 … }

And they struggled, and struggled. I saw a lot of students just going nowhere. As they worked, I went back to that original set, and wrote up a more complicated solution:

Set Builder EW

One student looked up, then another. A couple of students started working out how my solution worked. A couple of others asked for clarification. And then, one by one, I started to hear those magical “Ohhh!” sounds that we all love – you know that light bulb moment. Students started to remember that they could match each term to a number, and most of them came up with something like this:

Set Builder RBFrom a desk off to the side, I heard a student exclaim, “Whoah! When you showed us a more complicated way to do the easy problem, it made it easy to do the hard problems!”