In Geometry, we’re in the middle of our introduction to trigonometry. (If you want to hear more about that gentle, conceptual introduction, read up here.) Up until now, kids have been using this right triangle book to figure out missing angles and side lengths. The big idea is that every triangle in the world is similar to one of the triangles in this book… so we can use that similarity to find side lengths and angles.
And finally we did the magic transition where kids saw how ratios could be used to find the labeled angle… This is the key turn, from sides of triangles to ratios of sides.
Initially all students used the pythagorean theorem to find the hypotenuse and then they “scaled the triangle down” to find the similar triangle in the book with hypotenuse one. From there, they could find the angle.
But then I asked: How could you find the angle without using the Pythagorean theorem? They could still use the triangle book, and the basic functions on their calculators (kids obviously at this point don’t know about sine/cosine/tangent). Some were stuck, some saw it right away. But eventually everyone recognized that they were looking in the triangle book for a triangle which was similar to the triangle given. And we know that proportions of corresponding sides in similar triangles stay constant… so they used guess and check to find triangles in the book with a vertical/horizontal ratio that matched 2.2204/0.8082.
Okay, great. They hated that. They had to divide the side lengths on a bunch of different triangles in the triangle book. So annoying. It was much worse than just using Pythagorean theorem and scaling down.
So here’s where we paused. I said: “okay, fine, agreed. This is annoying and horrible, and the Pythag approach is much nicer. Let me ask you this… What if I put the ratio of the vertical/horizontal leg on every page in the book. So you had the ratio. Which way would be more efficient to use then?”
Everyone said the ratio. Why? Given a triangle, you simply take the ratio of two sides, and then flip in the book until you see the same ratio. Then you can immediately read off the missing angle. One division, that’s all. (With Pythag, you have to first calculate the hypotenuse without any error, and then scale that triangle so the hypotenuse is one, and only then flip in the book! And that might lead to more error.)
So I showed them trig tables. Of course they don’t have sine/cosine/tangent yet on them. And I let them use it on a few problems.
And then… FUN! I asked kids to just look at the table and just “notice” patterns.
They came up with some great things, which I then started playing with on the fly. I told them to call the three columns “Ratio 1,” “Ratio 2,” and “Ratio 3”:
- As the angle increases, Ratio 1 starts close to 0 and goes close to 1.
- As the angle increases, Ratio 2 starts close to 1 and goes close to 0.
- Whoa, wait, the numbers in Ratio 1 and Ratio 2 are “reversed”! Reading Ratio 1 from the top-down, and Ratio 2 from the bottom-up is exactly the same.
- As the angle increases, Ratio 3 starts close to 0 and gets higher… dramatically higher at higher angles!
- The numbers in the Ratios didn’t seem to be going up “proportionally”
While they were looking for patterns, I noticed no one had taken out their calculators, so I told ’em to see if their calculators could help them figure out any additional patterns.
- Ratio 1 divided by Ratio 2 is the same as Ratio 3.
They will be exploring some of these ideas later, and class was coming near to a close, so we didn’t explore everything we could have. But we did talk about a few things.
(1) We briefly discussed why Ratio 1 will never equal 1. (The hypotenuse of a triangle can’t ever equal a leg of a triangle! You wouldn’t have a triangle, but a segment.)
(2) We saw in a triangle why Ratio 1 divided by Ratio 2 yields Ratio 3.
And finally, I most wanted to capitalize on the observation that I hadn’t anticipated… but discussing it would combat a great question kids don’t really grok well in higher grades… What is the shape of the sine curve? Usually they think it is linear from 0 degrees to 90 degrees. That there is a linear relationship between angles and the ratios. So here’s what I did:
I told students that I would be plotting on the x-axis angle number, and the y-axis Ratio 1. If this was a line, then if you pick any two points on this line and calculate the slope, the slope should be constant. 
Each kid chose two different angles, and looked at the associated Ratio 1 numbers, and calculated the slope. While they did that, I was doing a little magic in Geogebra to show the data graphed.
Kids were getting different slopes. So they knew it wasn’t a line. But many slopes very close to each other! Curious.
A kid saw the graph and said “Hey, it looks linear at the beginning” and that explained why so many slopes were similar but not the same. Kids were mainly ch0sing angles from the first page of the ratio table! Ha! Love it! Last year teaching geometry, I didn’t ever show them a sine curve. But this came up so naturally that I had to!
This was a bit on the fly and haphazard, but this discussion of whether the ratios were linear or not was one of my favorite things I’ve done recently! I should find a way to formalize it and build it into the curriculum in more solid way.
UPDATE: OMG I am an idiot. I forgot to mention something crucial. I want kids to recognize that if they have a trig table with only Ratio 1 (aka Sine), they can generate the entire trig table. We have an abundance of information! And this discussion of their noticings seems perfect for that. This is the follow up I used last year, and I will again use it this year.
The key point I’m getting to: the truth is we don’t need sine, cosine, and tangent. We only need one of them. For example, if I know sine, then cosine can be defined as and tangent can be defined as . So why do we have all three? Life is easier. Look at triangle (g) above in this post. Try using a table with only Ratio 1 to find the missing angle. It is more work than if we had a table for Ratio 3.
 Okay, yeah, so afterwards, I realized I could have just asked if the ratios had a constant difference. But my more complicated approach led to something interesting! Also: if I had more time, I would have asked kids to develop a way to decide if the ratios were growing linearly or not. I bet some would have said common difference, some would have said find the slope, and some would have said graph!