I taught high school math for 32 years, Algebra 1 to Calculus, plus a few electives. In this post, I will summarize my department’s approach to trigonometry. When I started there, we had a one-term trigonometry course, which became more engaging and comprehensible when we started using the now out-of-print *Trigonometry: A Guided Inquiry *by Chakerian, Stein, and Crabill. Yet, even though the course was interesting and coherent, it was not effective: while we enjoyed teaching it, and students enjoyed taking it, they retained next to nothing from it. Students were not accustomed to thinking in this domain, and nothing stuck.

To address this problem, we distributed the content across three courses: right triangle trig in Math 2 (a course consisting mostly of geometry); law of sines, law of cosines, and intro to the unit circle in Math 3 (our version of Algebra 2); and trig functions in our one-term precalculus course, Functions. One benefit of this approach is that it allowed us to extend student exposure to this important topic beyond a single term, in fact, beyond a single year. (See this article for the general argument about extending exposure.) In each round you can go in depth into that year’s topic, and review the previous years’ work as needed. This ended up working really well.

Our most significant departures from tradition were right at the start when we first introduced trigonometry. Here are the key ingredients of our approach.

Our Math 2 students mostly understand slope. To build on what they know, we point out that there must be a relationship between the slope of a line, and the angle it makes with the (positive) x-axis. “Slope angles” can be used to solve problems such as “how tall is the flagpole?”

Answering the question would require knowing what slope corresponds to 39°. Fortunately, this can be figured out with the help of the ten-centimeter circle: the radius of the circle is 10 cm, providing the run, and the rise in this case is 8.1 cm or so, so the slope is approximately 0.81.

In fact, with the help of a straightedge, the 10-cm circle allows us to find the slope that corresponds to any given angle, and vice-versa. Note that there was no need to say “tangent”, or “trigonometry”. The idea is to introduce the concept first: many right triangle trig problems can be solved with this tool. When students understand it and know how to use it, it is time to reveal that there are keys on the calculator that can replace the 10-cm circle, that we’ve been talking about what is usually called the tangent function, and that our right triangles can be drawn every which way: the legs need not be horizontal or vertical if we use the formulation “opposite over adjacent” to replace slope.

We introduce “ratios involving the hypotenuse” a few months later, once again using the 10-cm circle, and completing the soh-cah-toa set, or as I prefer to put it: “soppy cadjy toad”.

The payoff of this approach is a strong visual understanding of the trig ratios, and an early preview of the unit circle. You should try it! Find more information, and downloadable PDFs on my Web site.

— Henri

I love starting with slope (and also therefore starting with JUST tangent). I don’t remember any longer which of the things I do are the result of your influence, which are the result of influence from CME Project, and which I came up with on my own. But your influence is definitely part of the mix.

The thing that I really like doing is warming up class with a series of “which is steeper” puzzles when we’re starting trig. I’ll deliberately choose ramps that give height/width and ask students to compare them to ramps where the height/hypotenuse is given. In this way we get at the idea that steepness can be measured MANY different ways. Of course slope is one of them, and the angle of inclination is another. But the height/hypotenuse also tells you something about the steepness, and eventually I do introduce sine as another measure of steepness.

I love this material.

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I like this idea! Complicating things a little forces an interesting conversation up front, which can be revisited later. Very cool.

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