This was a small thing that sent lightbulbs off. I should have done this years ago.
My kids are reasonably comfortable with radians, expressed as fractions of pi, and we do a lot of measuring with radian protractors, so they can even do things other than special angles. (More on that another day.) If I give them an angle, they can tell me what quadrant it is, and they can often explain their reasoning.
But then we hit inverse trig today. I don’t know what your calculator does, but our calculators spit out decimals when they evaluate inverse trig functions. And every year, my kids get confused. “Wait, but I thought this was supposed to be an angle in radians? It’s a decimal! Where’s the fraction? Where’s pi?” And then they get confused about whether they have an x-coordinate or an angle and they switch up arccosine and cosine (an admittedly easy thing to do) and the whole thing is a mess.
Using the radian protractors helped. I have one set that has fractions of pi on it and another set has decimals on it. Things got a little better.
We solved some equations like cos(theta) = 0.3 and got things like 1.26 and 5.02.
But then, right at the end of class, I decided to throw up a coordinate axis and label it: 0, pi/2, pi, 3/pi/2, 2pi.
“Guys, what’s pi as a decimal?”
And I labeled the negative x-axis as 3.14.
“What about pi/2?”
“You have a calculator….
“Oh!” (furious typing) “1.57!”
And I labeled the positive y-axis as 1.57. We kept going and ended up with something like this:
Lightbulbs went off all across the room. “So what quadrant are our answers in?”
“I and IV!”
“And does that make sense?”
The bell rang, and we didn’t get to the last question, but it helped them so much just for me to write down what pi, pi/2, 3pi/2, and 2pi were as decimals. They could see where the answers they were getting belonged. I should have done this years ago.