Or even more basically, it’s also a good idea to delve into the different ways of doing division with integers (e.g., long division, short division, chunking), and ask them which ones they prefer. I feel like this can be integral in the current landscape where procedural approaches are discarded or de-emphasized.

1. Squaring both sides of a trig equation and then checking an answer graphically to see that the solutions don’t match up.

2. Solving a quadratic inequality (like x^2 – 5x + 14 > 0) using the Zero Product Property only to find that it doesn’t match up with the graph.

3. Finding out that there could be two possible triangles using the Law of Sines.

In all cases I try to build the drama. For example, for #1 I sell it as an afterthought that we are going to check our answer graphically to give them a way to check their work for an upcoming test.

The first of these problems arose from a situation in which I was told the night before that parents would be touring through my class in period 4 on the next day and that I had to have the class using the graphing calculators. The parents touring wasn’t an issue, but the graphing calculators were — the students I had in period 4 had never used the graphing calculators (long story), and we were heading into the final unit of the course: rational expressions. On the fly, I decided that we would spend that period graphing rational expressions (despite rational functions being in the next grade), just to see what the students thought (so far they were familiar with linear and quadratic functions). They very quickly figured out how to enter functions into the graphing calculators and were on to the first rational function, y = 1/x. “Something’s wrong with my calculator”, “mine too”, “hey they are all busted”. I immediately said, well then, I’ll do it up on the one hooked up to the projector. Second later “yours is busted too” — the investigation was on — why does the graph look that way? Soon they were pros at identifying non-permissible values (exclusions from the domain or undefined values, if you prefer), but then I threw in a function that had a common factor on the top and bottom, resulting in another non-permissible value which was not undefined, but indeterminant. “What’s going on now”…. We had over 50 parents through that room (not bad for a community of 800 and school of 300), and neither the students nor I remember any of them coming through; however, I never had to remind any of those students to state and consider non-permissible values when working with rational expressions — beautiful!

The second problem was similar, but happened around trigonometric identities. Before talking about trig identities or proving them, I gave each student a paper with a list of trig functions on it. Their parter’s paper had a list of trig functions that could be matched to the other to create an identity. All of the matching graphs brought great curiousity: “how can those graphs be the same?” I ended by asking them how they would like to show that the graphs were the same on paper, and they created the identities that we went on to prove.

I think WTF is a good description of this type of problem, but not quite classroom appropriate. I usually think of them as WAM problems — wait a minute….

I then tell them to divide a given polynomial by 2x-1 using either method. I asked students to share their answers given the two methods and ask what happened and why. Comes to an interesting discussion.