I made a minor tweak to my Pre-Cal curriculum. A discussion at TMC prompted a change where polar equations are taught right after vectors. All the math mechanics are the same, so it made sense. It also bumps polar to a time of year where the seniors haven't checked out yet and may remember something.
I did a detailed explanation of the unit last year. The part in the middle is a tad different now and man, it was the best way I've seen to dissect polar equation behavior.
The idea came from something I tried in Algebra II. That is, make equation discovery more of a puzzle. Instead of fiddling with sliders and recording observations, I would provide equation frameworks and leave the rest up to them.
Handed this out (screenshots generated from the answer key):
Put this up on the screen:
a+bsin(theta), a-bsin(theta), a+bcos(theta), a-bcos(theta), acos(theta), asin(theta), acos(b*theta), asin(b*theta)
Provided these instructions:
How did I make these graphs? Terms a and b are integers no larger than 10.
The old way, I gave too much in the instructions about things like a = b, a > b, and a < b. I wrote out all these discussion questions and wanted them to write down answers. It was nothing but confusion while they were working. This time the task was simpler: solve my puzzle, write down what you think works.
For the most part, silence. Students plucking away at desmos trying to solve the mystery. Sheer joy when they crack one. Frustration when the x-intercept matches but the y-intercept is OH SO CLOSE. And then, pure euphoria when they crack the system and decipher what the intercepts give away about the value of a and b.
Finally, the discussion. I first solicited answers from the crowd for the eight functions provided. This step is intended to see if there's disagreement. I starred any that weren't unanimous. I pulled up the answer key and typed in their suggestions to check for matches.
We used our results to classify the equation types into three groups: cardioids, circles, and roses.
First, the rose curve discussion. Lots of fascinating theories about what determines the petal count. Often two students would take opposing views, thanks to my accidental genius inclusion of 8cos(4*theta). It has 8 petals. In almost every class, a student would argue the leading "8" is why. Another would counter argue it's the 4 and that petal count is that term multiplied by 2. In one class, a student agreed with the multiplied by 2 theory, but that it only worked with cos. The looks on their faces when "8" became "7," "4" became "5," and their prior assumptions broke.
I just, I mean, you guys, it was THE BEST moment.
MANY students figured out how limaçons are created (b>a) and acted like it was such a "duh" discovery. MANY worked out that the value of "a" tells you the x/y intercept (depending on orientation) and that combining the values of "a" and "b" via addition and subtraction yield minimum and maximum radii. MANY felt empowered that they even if they didn't notice both those properties, they were proud to have caught on to one of them. More importantly, EVERY kid found this task achievable. Two finished early and on their own just started experimenting.
Lastly, I animated a couple rose curves on top of each other as the children reach for their phones to record it:
Look, maybe you read about 3ACTS and think, ok cool, but I don't teach middle school, these lessons are too simple. EVERY subject in math benefits from the idea behind those movements. Create a puzzle. Ask a question. Make them beg. The theory of 3ACTS enabled a group of seniors to argue with each other about polar equations and hash out the mechanics entirely on their own. There's no way a worksheet produces the same results.