After spending the summer engaging in some Makeover Mondays and doing one live at TMC, I decided to spread the love during district staff development.

I played with a simpler version of this idea last year in Algebra II just to test the waters.

### The Task

First, they were given a copy of this task from our Pre-Cal book:

I asked for good and bad things about this task. There wasn't a lot to like but there were some wonderful observations. I was actually glad the overall tone was negative. Textbooks use doesn't seem to equal textbook love. The experience level in the room ranged from 1 yr to 40 years.

Main points:

- data is contrived
- when plotted, the data is a nearly perfect correlation
- when plotted, it could be determined that it's modeled by a quadratic
- extrapolation (to make it appear more like a trig function) is cloudy business, who says business is always predictable?
- the questions give away the answers, students are sent towards trig and aren't doing the thinking

### The Fix

I was excited that the room was not a fan of the data. "It's not real" they said. Prepared for this, I sent them looking for some. I put six US cities on the board and had them find climate data in their Wikipedia articles. We used the average high in ºF. You could do average lows. For an extension you can compare and contrast northern or southern hemisphere cities. Or edge cases like San Francisco, where it's mid-60s and partly cloudy all the time. Record the data and plot it in desmos (I chose Charlottesville, VA for completely random reasons):

Depending on when you introduce this or how you wandered here conceptually, the student might decide this is trigonometric. Though having the quadratic debate would be good. Proving this is more trigonometric requires less stretching. If you use average highs, it would give you room to extrapolate to the next year and add on to the data table.

Then you let them fiddle with the model (by the time we get to this, I expect to have taught some built-in desmos literacy):

There will be more than one way to match the data. Some of the cities won't follow the curve so nicely (a lot of cities will have 3-4 outliers at the end of the curve). Which can bring the opportunity for some interesting extension questions:

- what's represented by the vertical translation term
*d*? - what's the meaning of the amplitude?
- can the period be an arbitrary value, or is that known?

This activity is worth it just for the science and geography connections. And it fits well in lots of places. I can't decide if this is a Unit Circle intro, a graphs of trig intro, or a graphs of trig post-game project.

Thanks, textbooks.