I teach Precalculus, which means that my students have seen sin, cos, and tan twice: once each in geometry and algebra II. So I decided to see how many of the students would recognize the functions from the input and output. Spoiler: very, very few.
I showed students that when you drag a corner of a right triangle around, the angle and the ratio seem to change. So I created two triangles, showed them how I measure the angle, write it down, and measure the two sides next to that angle (I try to avoid using the words “adjacent” and “hypotenuse” so they don’t get tipped off to the trig functions that way), and divide them to find the ratio. I then typed these into a spreadsheet, which I shared with them all through Google Classroom.
I showed them how to type “=50/71.2” so the spreadsheet would calculate the ratio for them. I then instructed them to run along and “find” (mostly create) right triangles of their own with wacky angles. They could work in groups, but they needed 2 triangles (and therefore 4 rows) per person in the group. They typed these into the spreadsheet themselves. I also showed them how the units divided out, so they could use whatever units they wanted to for this activity, as long as they were consistent within the ratio.
I then typed “=radians(A2)” for them and showed them how I could drag the formula down.
Unfortunately this took the entire day. I thought that drawing 2 right triangles, measuring 4 angles, and measuring 6 sides would take roughly 15 minutes. Between my introduction (10 minutes), them making + measuring the triangles, and showing off what spreadsheets can do, it took the entire class period. I was a little bummed, but looking back I think it was worth it.
I started off by showing them how awesome Desmos is. I copied the radians and ratio columns (see image above) and pasted it into Desmos. I told them to do the same and find a function to fit the data. I think I reminded them to label their axes.
My instructions were “find a function that fits this data”.
What (shouldn’t have) Surprised Me
Student comments: “Oh, you mean like a line?”
“Can we use the squiggly line thing you showed us in Desmos?” “Yes, but you need to pick the parent function for that.” “Oh…”
“This looks like a parabola!”
Many of the groups found complicated formulas, such as (see image below).
In the end only 3 of the 7 groups found that it was cos(x), but I have a hunch that 2 of the 3 groups overheard the first one talking about it, and in that pair, I think one of the two students had the idea of trying cos(x) and the second though “yeah, I guess that might work”.
Pause for Teacher Reflection
On the one hand I’m really excited about the parabola above (there were 3 other parabolas that fit the data just as well, if not better) because it means they’re (1) recognizing shapes of functions and (2) able to manipulate the functions to fit any data. On the other hand, this means they really had no idea what the trig functions meant the first time they learned them.
I think that students are too often handed these functions, perhaps told what the input and output are, but mostly just memorize steps of how to use the buttons on their calculators and don’t have any deeper understanding of why we even have sin and cos. If they actually understood this function, they should have started from the very start of the introduction and said “Why are we plotting these, isn’t this just cos?” The fact that cos(x) surprised these students with how well it fit just goes to show that they don’t understand the functions. For that reason I think that every introduction of the trig functions should start with an activity like this and build up the need for the functions instead of spoon-feeding students the method solving for x on a right triangle when you have a side and an angle.
Now that my rant is over, here’s the beautiful graph with cos(x).
Extension with Students
Since it was a block day, we had lots of things we could do. We talked about how the size of the triangle doesn’t matter, since it’s a ratio. I foolishly made the off-handed comment of “we could make this triangle fill the basketball gym” at which point they said “yeah, let’s go do that!!” How could I say no?
The triangle ended up being incredibly accurate (to within 5 hundredths of an inch when checked it with the Pythagorean Theorem!) and fell right onto our function.
I showed them some more excel magic, moving the columns around, finding the complementary angles with “=90-A2”, and getting data to graph sin(x) instead.
The last 20 minutes, students spend practicing finding the sides, given a side and an angle. Hopefully by this point they’re thinking of it in terms of “this trig function of this angle gives this ratio” rather than “follow these steps”. I need to come up with some quiz questions that can distinguish between the two.
Summary & Thanks
This lesson was the first one this semester that I hadn’t flipped. I don’t know how to do this kind of intro over a video, and really wanted students to struggle with “what is the best fit for this function?” so they see that sin is a function created for a specific purpose. I was very glad that I didn’t flip this activity and I hope that I can identify future lessons which would benefit from not being flipped.
I wanted to thank Alex Overwijk (http://slamdunkmath.blogspot.com/) for giving me the idea while he was sharing a similar activity at TMC15 this past summer. He even gives the students a table of ratios for sin and cos and they prefer using that all year instead of typing it in a calculator!
 I used the 45-45-90 triangle and the 30-60-90 triangle since we had just done these in class the day before.
 A side goal of today’s lesson was showing them how awesome spreadsheets are and how powerful they can be. Originally I even made the table in the spreadsheet so they’d see the graph, but between days I remembered how awesome Desmos is.
 No, you don’t have to even create a table, Desmos is that awesome. Oh, and it will adjust the domain and range automatically if it’s larger than the default window. I didn’t know this, otherwise I might have just gone with degrees instead of changing it to radians. But radians works better since Desmos defaults to that and typing in “cos(x)” gets the perfectly fitting graph immediately.
 There are almost too many ways to share it. I could have shared the Desmos graph, but (a) they already had access to the spreadsheet and (b) I wanted them to experience how awesome Desmos was and what it could do.