So today in an uncharacteristic head-fake toward efficiency over laborious, painstaking discovery, I threw them this:
If you didn't notice, take a look at those measurements and that diagram and you will see the problem. I did not think much about it, because diagrams are not to scale and perfect reasoning on imperfect pictures and blah blah blah. A few years ago my students wouldn't have batted an eye at this, they would have just plugged and chugged, but these guys were NOT. HAVING. IT.
"What is going on with your diagram?"
"That is pitiful, Nowak."
"This is not physically possible."
This made me ridiculously happy. Just that they view the quantities we work with as physical things they can measure and verify and use their spatial intuition on. And they don't see them as arbitrary rules devoid of meaning that they have to memorize because the grumpy lady makes them. Maybe this is obvious and it just means that I'm now a mediocre Geometry teacher instead of the worst Geometry teacher ever. But, you know, baby steps. I may or may not have also shown them Rico Suave on youtube. It is the end of May. We are punchy.