Every year in Pre Cal I approach right triangles differently. This year is no different. The number of times you could say I "deliver a lesson" is starting to dwindle. I design tasks. The students spend class time doing those tasks, and we find the interesting questions within. Here's a short primer on how we progressed through right triangles.
Task 1: Make Some
I passed out protractors and some paper and told them to make a bunch of right triangles. Specifically, make a bunch of 30/45/60 right triangles. Side measurements are up to you. I had them make nine, it should probably be half that because the process took FOREVER. Useful as a reminder on how protractors work.
Task 2: Humor Me
After the long process of making triangles, we labeled them with opposite, adjacent, etc. For each triangle they computed the decimal value of opposite/hypotenuse and adjacent/hypotenuse and recorded the results. No explanation as to why, just entertain me for a minute here.
I found this website helpful during my discussion of the results. In one class not every kid got this far, again because TOO MANY TRIANGLES, MAKE FEWER NEXT YEAR. But we worked through it. Lots of interesting stuff here. The ratios seem to be close no matter the triangle, no matter the kid that made them. And what's that? Repetition? Is there some kind of pattern here? But what are these goofy decimals about?
At this point I did a rare moment of lesson delivery, aka talking. Recall of special right triangles from geometry and the ratios present. While making the triangles, many kids asked why we had to bother with the 60 degree ones when duh there's a 60 degree angle in my 30 degree one. This part. This part right here.
We spend roughly a week with the unit circle and the six trig functions and the ratios and all of that. Important difference here is that I made an effort to talk about the unit circle as a collection of triangles. We always defined the trig functions in terms of triangle sides (not x and y), making note that in this very special collection of triangles, the hypotenuse is 1 and organizing the adjacent and opposite sides as a series of coordinates is helpful. An hey, in all cases the adjacent side runs horizontally (x) and the opposite runs vertically (y). Weird.
So often the unit circle is treated as a stand alone thing that we must do because the book told us to. But the unit circle is everything! Vectors, polar coordinates, and trig graphs make so much more conceptual sense if you can convince students to get the unit circle.
Task 3: Measure Stuff
Now to branch out a bit. Right triangles that are not necessarily special. We wandered out into the school and found some objects that were tall. They measured a hypotenuse length and an angle.
That covers finding unknown sides when one side an angle are known. But what about unknown angles?
Task 4: Compute Stuff
Armed with new knowledge of trig functions and that oh hey, the calculators has those buttons, I gave them a table to complete. Values of sin, cos, and tan from 0 to 90 in increments of 5.
I hand out a series of triangles where the corner (non-90) angles are unknown. Two sides are given. Students determine which trig ratio works the best based on given information and approximate the value of their angles. A few remember the inverse feature of the calculator and I was totally find with them using it. However, the idea here is to reinforce the idea that trig is big patterns. An observant student noted that the sin and cos tables seem to be going in opposite directions. Weird.
Task 5: All the Things
In conclusion, I had them create some artwork for their notebook. Display the four quadrants, orient a triangle in each one, make the sides whatever you want, show the values of the six trig functions for the reference angle, and finally, approximate the value of the reference angle.
In the end we covered right triangle trig faster than ever. And other than the unit circle, the kids spent more time listening to each other than listening to me.