Missed an opportunity here. An unfortunate side effect this time of year. Some days I just have to coast. And with something like Pre-Cal that I've been doing for a while, it can be easy to overlook an obvious improvement.
It's polar equation time around here and it's one of my favorite things to teach. The graphs are just so bizarre to kids and yet the mechanics behind playing with polar systems are simple enough that we can explore lots of things. For example, this rose curve:
Among the last things I do with rose curves is have them write out some explanations of the properties: equation, petal count, petal separation angle, and x-axis offset (for sine functions).
Where I mess up is just launching right into an explanation of how to determine petal separation (360/no. of petals) and the offset (90/b where b is second parameter of r = a sin b theta). I do discuss that random 90 by showing them r = cos theta and r = sin theta and that 90 degrees is their "natural" offset. But the method could include some discovery. Busting out protractors, measuring this stuff, and collecting theories would've been significantly more interesting. Primarily, because while describing that set of 6, several kids offered alternate methods to the 90/b thing. Including several thoughts like "oh, it looks like it's always half the petal separation" or "can't I count how many gaps there are between 0 and 180, multiply it by the separation, subtract that number from 180 and the remainder has to be the offset?"
Great ideas! How many others kid noticed the same thing? How many didn't bother because I straight up told them?
Lessons learned for next year.