Graphs of trig functions always hit at the end of the first semester, and I have a pretty good progression in place right now. However, my push for modernism bit me a little bit recently. Early on the semester I gave the kids the gift of Desmos. Many of them proactively would pull out their phone to verify their algebra with a quadratic system or what have you. Some of them were so clever in their choice of Desmos for the given problem that I didn't want to discourage them.

Flash forward a few weeks and I had questions like this on the last test of the semester:

And within a few minutes of reaching this section, I start getting THE question: "can I use Desmos?" to which I wasn't sure how to reply. Here they are, instantly realizing they know the best way to graph these things and would like to demonstrate that to me. If you write your own tests (I've written 92 Pre-Cal tests with multiple versions up until now), there's always the nagging question, what am I trying to test here?

On the surface, I'd say the important things I was looking for in these questions is do they understand what the parameters affect and how thing are modified by vertical translations. The statement part takes care of that. Other than that, in an era with such nice graphing tools, I'm not sure what else I was going for here.

I can tell you that it was obvious who really understood how to annotate the graphs and who didn't. The kids who just slapped the thing in Desmos and copied the picture were pretty obvious in their inability to show what the frequency of a function really meant. Many just randomly plotting points on the x-axis (one student graphed a function with a period of pi/2 and the only point labeled on their x-axis was a tick at 10pi, because...reasons). A different set did base their graph off Desmos but were able to note what should line up where on the x-axis and how to properly illustrate a vertical shift. And then a third set did the whole thing by hand, like the 90s would've wanted.

The lesson, I think, is that in the future I need to change the emphasis of this assessment section. The time for assessing graphing questions like this may have ended. We've reached the point where I can assume they'll know *how* to generate a graph given any sort of tool and it'll look the part. Now I need a higher concept wrapper around the subject. Things like "what equation would cancel the functions?" or "what equation would amplify the functions?"

I've always wanted to restructure Pre-Cal around bigger, higher concept ideas (ie instead of Trig Graphs we study signals, instead of vectors we study static mechanics, etc), and I think this moment is going to push me to finally do it.