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!

[NBI] Modeling Temperatures with GeoGebra

This post is part of the new blogger initiative.

The more that I grow as a teacher, the more I believe in teaching modeling with mathematics.

That might sound somewhat silly. I've always known, in some sense, that modeling is a good idea. They certainly encouraged it in my prep program and it's a significant part of the Common Core. However, when you're actually making instructional decisions for the day or week, modeling can be tough to choose. It takes more class time, and in some ways, can seem "softer".

But every time I have my students complete a good modeling activity, those thoughts just seem ridiculous. I see real struggle. I see connections get made and broken. I see students conjecture. I see them contextualize and decontextualize. Beyond the truly deep learning, I haven't encountered a better assessment tool.

Monthly temperatures in Wisconsin

One of the better lessons that I taught during my first year of teaching was a modeling lesson. We were finishing up a unit on trig functions and I *thought* that my students had learned the transformations. I wanted them to see that what we learned really does show up in the world, so I grabbed the last five years of monthly temperatures in Wisconsin from the Online Climate Data Directory. I then entered that data in the spreadsheet of GeoGebra, created a list of points, and distributed the GeoGebra file to my student. (I would love for my students to be able to handle the data, but I made an executive decision based on time-constraints).

I know that it's supposed to look like this, but it still kind of amazes me. 

For the activity itself, I distributed the following halfsheets to guide students through the techy aspects of the activity:
Temperature Directions

Basically, I wanted students to find an equation that fits the data as accurately as possible, interpret the different parts of their equation in the context of monthly temperatures, and present their work in a professional looking document. As we had done very little work of this type, I created a template with Google Docs to use.
Temperature in Wisconsin

Results

It wasn't perfect, but it was good. The insights into students' thinking were amazing. For example, it turned out to be very difficult for my students to interpret the period as one year, or that the midline represents the average temperature. I had a number of very good conversations trying to help students make these connections.

Additionally, I really enjoyed watching students while they were fitting equations to their data. I had considered setting up the equation for them with sliders, but I'm glad that I didn't. There was value in having them manually type in and change constants in the appropriate places in their equation. Students were checking their notebooks to figure out how to change the amplitude, or just tinkering to see if it worked. Really, either was fine with me. 

In the end, I hope trig was a bit more concrete, and I had a much better idea of what my student understood.

Next time

Instead of ending the unit with this, I will start with it. Trig would then come from the real world and students would be playing with all of the transformations before I teach the vocabulary. We could then discuss the ideas of trig transformations with a solid context to refer to and draw comparisons to other places in the world - "How do you think the amplitude of the monthly temperatures in Wisconsin compares to Cuba's? Why? What about the period?"

Next time.