Just a quick post before I try this out tomorrow. Just starting the big trig unit, and I hate the way I've been doing it all these years, so I'm going to try it differently, in reverse order almost, by starting with something that I used to save for the end. At the same time, I want to weave an important concept into the mix right at the get-go, and that is the relationship between time and angle.
I'm giving them this virtual ferris wheel, and this pdf to fill in.
They will move the little green car around the ferris wheel, and geogebra will give them its height above the ground as they do so. I expect they will make it go to the easiest places at first, that is, the points of the compass, which correspond to 15 seconds, 30 seconds, and 45 seconds, where it's easy to get the car in the right place just by eyeballing it. They'll be able to fill in the table with these ordered pairs pretty fast:
(0, 1), (15, 6), (30, 11), (45, 6), and (60, 1).
My evil plan:
But then they will have to fill in the times in between, like 5 seconds, 35 seconds, and what I'm hoping is that someone will at some point ask me, "Mrs., wouldn't it be easier if we could see the angles around the circle?'
"WHAT?!?!?" I shall say. "Well, if you insist, I suppose there's no harm in it. Click on the tiny little checkbox at the lower right corner, if you really think that knowing where the angles are will help...."
This is what they will then see:
Now they will know exactly where to put the little green car for all those times in between North, South, East, and West.
And BOOM! They will, for the rest of this unit, know that time is glued to angle. I hope.
I'd like them to share and compare their graphs, so I've made a gdoc for them to upload to. Hopefully by this time tomorrow, there will be a few there.
I will also get them to talk about other things they notice about the function, like the periodicity, the symmetry, the total absence of an asymptote, the restricted range, whatever!
I already made this for the answers, but I had so much fun doing it, maybe I should make them do it themselves, at the end, when they'll know a thing or two about trig functions.
Ideas for the future:
I'm also thinking that as I add new parameters to the basic functions, I'll keep coming back to this and changing, for example, the radius of the wheel, or the speed at which it's turning, or the position at which the car begins. If you can think of anything else, please feel free. I'm sure someone else has already thought of this and done it for years, but oh well, better late than never. Fingers crossed!