We started building polar graphs today in PreCalc. These graphs are often tedious to create by hand. I make my kids graph a few by hand so they understand how the graphs are generated, and then–DESMOS!
I love asking the class if they can visualize what the graph of r=theta looks like. A select few in each class can typically figure out that it would be a spiral. “Ok, what about r=2theta?”
“A fatter spiral?”
So, we graph it. And then, I replace the 2 with a slider, hit play, and watch the kids ooooh and ahhh. (Here–push play on the left and you can see the beauty of math, too!)
Then we go on to some more complicated graphs. One of the questions they’re given is to find the maximum value of |r|. I tell the kids, “This is the largest circle on your polar graphing paper that the graph will touch.”
The example we did was r=6-12cos(theta), whose graph looks like this:
They found the maximum |r| value to be 18. Correct.
And then it dawned on me…why I don’t I show them what they just found. So, I added the graph of r=18 to the screen:
“See how your graph fits inside this circle? In other words, 18 is your maximum |r| value.”
“Ooooooh!” You could just see the light bulbs go off.
Four years of teaching this topic and I just now thought of adding this simple step? As embarrassing as that is, it’s really fun for me to find different and better ways of explaining a topic, especially one that can be somewhat abstract and difficult to wrap your head around.
Thanks, as always, Desmos.