Motivating trig through modeling

I'm starting to love teaching trig. In the past few terms, we've ended the unit with a GeoGebra project where we model monthly temperatures in Wisconsin. The project is a great way to see if students have connected the ideas to the real world and the students actually end up thinking it's a little cool.

However, I don't like having to do a week or two of abstract, context-free trig before really getting to students to see why it matters. So, to try something a bit different, I put together an intro task where they look at some more real data - in this case, a fellow teacher's monthly gas bill. You can find that task here. I give them this graph and a few questions:

1. How many years of data is represented on the graph? How do you know?
2. Circle the point associated with the most expensive bill.
1. How much did he pay?
2. Which month of the year do you think that was? Why?
3. Put a box around a point associated with an “average” bill.
1. How much did he pay?
2. Which month of the year do you think that was? Why?
4. On the graph, sketch a curve that fits the data.
5. Let’s convert that graph into a different form. Create a table for months 6 - 29.
6. How would you describe the pattern of the gas bills?
7. Predict what Mr. Bruns’ gas bill will be in the 40th month. Show or explain your reasoning.
8. Predict what Mr. Bruns’ gas bill will be in the 100th month. Show or explain your reasoning.

Hmm, it would be awfully nice if we had a function that could do some of the work for us...

From here, I'm thinking that we would go on to develop a "cost" function, connect it to a circle, and then get to a point where there is a motivation to develop the unit circle, sine, and cosine:

I'm going to try this out this week - we'll see how it goes!