In recent years I have become dissatisfied with the kind of assessment questions I ask on particular topics. This year I'm making an effort to increase the cognitive demand of my questions and focus less on mechanics.

As we wander through triangles, prior to inverse trig on the calculator, we cover approximating angles using a sin/cos/tan table. I want students to get a feel for the trends present as you move from 0-90º, something that's non-obvious if you just show them the calculator straight away.

I asked a question recently that sparked some really amazing answers. I offered a triangle and a suggested value for an angle. Students had to demonstrate whether or not they agreed with that value. The trig table students created is in steps of 5º, some judgement calls are necessary when getting intermediate values.

Two students started on the notion that the value was fact and used it as a check on the other angle:

Given the same triangle with the suggested value of 23º, another student took up the disagree position, likely using a calculator to play around with sin outputs (22º is not on their table):

Yet another student decided to convince me beyond all doubt it was wrong because look bro, it doesn't matter which one of these trig functions I use, they're ALL telling me your suggestion is junk:

Finally, a separate question about triangles that generate circles and what happens when your interior angle trips on multiples of 90º:

We generated the unit circle on the notion that a series of triangles with the same hypotenuse creates a circle with a radius equal to the hypotenuse. Here's a handy Desmos illustration. The magic collapsing triangle is something I save for after students are given something like sin(90º) and give me the "wait what, how is that a triangle?"

I was super pumped reading through all these as I graded.