Last semester I gave a nifty test question. Students had to reason through whether something I presented them is valid using a combination of things we have talked about. Some of them followed the lines of thinking I intended and a few went beyond my expectations.

A similar thing happened recently. Last year in my Algebra II revamp I wanted to put more emphasis on the purpose of graphs. My current Pre Cal group didn't necessarily get that connection during their Algebra II experience. Early early on I brought it up as we began with quadratics. Now that it was time for trig equations I brought it up again. Why would trig equations have a series of repeatable solutions?

I had them hack it out via Desmos first:

I gave them some equations and showed them how to plot it. They jotted down a ton of the solutions and I asked if they noticed anything. Hoping they caught on to the presence of a pattern, even if they didn't necessarily know what was causing it.

Anyway, we spent some time solving trig equations algebraically, representing the solutions as a multiple of the period. Some of that was on the test. But then I threw this at them.

The expected process would involve taking the equation given, solving it algebraically, and seeing if the intersection points in the picture appear. Simple.

A few demonstrated some great understanding of trig functions. In this particular example, the student solved the equation, noted the presence of an intersection point and agreed. But, BUT. Do you see the part where they verified that the GRAPH itself makes sense, that the presence of an intersection point might not tell the whole story? I mean, I had to stop for a second. This was so great.