Here’s how I taught students how to solve trigonometric equations in our grade 12 advanced functions class.
Started with this Ferris wheel problem
From 101qs.com find it here
What has been working well is starting our “math” at a very low level…..like on a dial…..then we slowly turn the dial up….adding more “math” in. Read more about the Math Dial from a comment on Dan Meyer’s blog here.
Starting with this video the math on the “math dial” is very low.
I asked: What questions do you have after seeing this….
Answers:
How fast is it spinning?
What’s the radius?
What’s the period?
Where will the red dot be after 3 min?
And that last one is the question we studied.
Act 2:
From 101qs.com
Almost all kids solved this problem using proportions! They kept the dial in the low position still! They realized that it takes 5 seconds to travel from dot to dot. Therefore it takes 40 seconds to go all the way around. They divide 3 minutes up into 40 second sections and get 4.5 rotations. The dot will end at the top of the Wheel!! But the Trigonometry in me was screaming to get out……I asked, “Did anyone create a trig equation to model the height?” — cue crickets!
So we cranked the math dial up a tad!
I said:
When I go on a ferris wheel I always look for my house.” We talked about how high that might be in relation to Dan’s problem….we settled on about maybe 40 feet.
My question: How long will it take to get to that height?
Guesses? Will it be a nice number? No? Why not?
Crank it up a bit more …
Let’s create an equation for the height in terms of time (we had already learned how to do this and it was no problem for the class) .
Now, to solve our question we have to solve this equation!
Student: That looks super hard!
Me: It does doesn’t it!
Let’s make that our goal!
We don’t want the math dial going up too quickly!
Let’s start with the easy ones, like this:
Gotta keep the math dial low for a bit more…
We solve this as a class, then another, and another, slowly building up our skills; slowly bringing the dial up. We stop at the end of the class. I assign a few more like the ones above. “Let’s get good at these so we can do the super hard one… Practice these for homework….”
Next day:
We take up the assigned questions then get back on track! We then solve these:
We have a discussion on how many solutions there are here… and plop down a graphical solution in Desmos
The math dial is getting up there…
Me: “Are you ready to try the big one?”
We do it! And everyone is into it….they have been waiting two days to see the answer! And the dial is pretty far up there!
After:
One student says: “That was pretty awesome! ”
That was my highlight of the day! Best compliment for a teacher!
We then show the graphical solution in Desmos. 
Oh…..and we started class playing Pictionary (It’s our Wednesday thing) there was a tie and we have a good o’l match of Rock, Paper, Scissors to declare the winner. It was Intense!!!
