When I first started teaching trigonometry, I just introduced it the way the textbook did, sort of matter-of-factly and heavily reliant on SOHCAHTOA. But students were always asking things like, “Ok, but what *is* sin?” and thinking they could divide by sin to solve for x. Algebra is already pretty abstract, but the whys and hows of trigonometry made about as much sense to the students as the plot of *Interstellar*.

For the last few years, I’ve been introducing it differently and I feel like it sort of addresses the too abstractness and trick-oriented way I used to do it. Now I always first begin with an investigation into special right triangles. I do this with MYP4 (US 9th grade):

In it, students use their knowledge of Pythagoras’ Theorem and simplifying radicals to discover patterns in the relationships between the sides of a 45-45-90 triangle and the more complicated 30-60-90 triangle, write and check algebraic rules, and apply those rules in contexts of varying difficulty.

I do this as a summative assessment under test conditions, though if a student has gone totally off track (for example, by not correctly simplifying radicals), I sit with them to discuss their error and then give them a second go at it.

The idea is that students will see that if we know the angles of a special right triangle and we know one side, we automatically know all the sides and vice versa. My hope is that this primes them to see where trigonometry comes from. My follow-up lesson involves another investigation, but this one done informally in groups.

I ask the students to draw a few lines of varying steepness and investigate the gradient (slope) of the line and the size of the angle of elevation and find a pattern or rule. I’m doing this on Tuesday with my current group, but last year they were really clearly able to say that a gradient of less than 1 gave an angle of less than 45 (and the smaller the gradient, the smaller the angle) and a gradient of more than 1 gave an angle of more than 45 (and the bigger the gradient, the bigger the angle). I then tie it back to their discovery of the sides of a 45-45-90 triangle.

It is only then that I introduce tangent as the rule relating the angle of elevation to the gradient. I have them use the calculator to find the tangent of the angles of elevation that they measured and check them against the gradients that they had. Only after they feel comfortable with using tangent do I introduce the other ratios, again referring to the findings from the investigation on special right triangles.

I hope in this way that trigonometry is less of a black box and a procedure that the students do, but don’t know why they’re doing it. How do you introduce trigonometry? How do you mitigate its abstractness?