Do Kids Really Understand Trigonometry once Sine/Cosine/Tangent are Introduced?

This year I’ve been doing a lot of work with my geometry kids to get them to build up a deeply conceptual understanding of trigonometry. Right now we’re still in the part of the unit where the the terms sine/cosine/tangent haven’t been introduced, and kids are building up their understanding by thinking of ratios in specific triangles. But soon we are going to introduce the terms, and I’m afraid they are going to go to their calculator and use it blindly, and forget precisely what sine, cosine, and tangent really mean.

For my kids, at this level, I want each term to be a ratio generates a class of similar triangles — which all look the same, but have different sizes. And I want kids to conjure that up, when they think of \sin(40^o)=0.6428. But I fear that 0.6428 will stop losing meaning as a ratio of sides… that 0.6428 won’t mean anything geometric or visual to them. Why? Because the words “sine” “cosine” and “tangent” start acting as masks, and kids start thinking procedurally when using them in geometry.

So here’s the setup for what we’re going to do.

Kids are going to be placed in pairs. They are going to be given the following scorecard:


They will also be given the following sheet, with a clever title (the Platonic part refers to something we’ve talked about before… don’t worry ’bout it) (.docx form). This sheet has a bunch of right triangles, with 10, 20, 30, … , 80 degree angles.


Then with their first partner, on the front board, I project:


The kids will have 3 minutes to discuss how they’re going to figure out which two triangles/angles best “fit” these trig equations. (I’m hoping they are going to say, eventually, something like “well the hypotenuse should be about twice the length of the opposite leg, so that looks a lot like triangle C in our placemat” for the first equation.)

They write down their answers. If they finish early, I have additional review questions from the beginning of the year that will be worth some number of points — to work on individually.

When time is up, they move to a new chair (in a particular way) so that everyone has a new partner. I throw some other equations up. And have them discuss and respond. Then they move again, and have new equations up.

I’ve scaffolded the equations I’m putting up in a particular way — so I’m hoping they lead to some good discussions. And I’m hoping as soon as a few people catch onto the whole “let’s compare side lengths” approach, the switching will allow for more discussion — so soon everyone will have caught on.

At the end of the game, we’ll have some discussion, and through those discussions we’ll reveal the answers. And of course, the student with the most correct answers will win some sort of fabulous prize.

The questions I’m going to ask are here:

The discussion questions are here:


I’m super excited to try this out on my kids next week sometime.



  1. Totally jealous of how creative and thoughtful your documents are. I wrestled with using the names of the trig functions this year and I am convinced I used them too quickly.

    1. I’d imagine working things out this way would help. But I admit I don’t remember when I started to understand trigonometry. It certainly wasn’t anytime soon after the terms sine and cosine and tangent were introduced, though.

    1. Haley! I asked Ting the other day what she remembered about geometry and I got ‘SOHCAHTOA, I have no idea what that means’. This week I taught trig to my geometry class, and really hesitated to introduce SOHCAHTOA. I ended up mentioning it on the third day, after a student brought it up to me. It really is a nice way to remember which ratio belongs to which function, but now when I ask what sine is some kids say SOH- opposite and hypotenuse, not even saying divided by. Uh-oh!

      1. Hi Ellen! I taught trig without SOH CAH TOA until the *very* end. They had learned about it in physics (ugh) but didn’t truly understand it… And I made them read aloud and sign a pledge telling them they can’t use those terms (or sine, cosine, tangent) until the ban has been lifted. It worked well!

  2. Hm, I think sometimes when I use sin, cos, and tan I am not really thinking what they mean on a triangle either, especially when we get into calculations that seem like a series of equations where you need to apply rules. I like your plan for making students think about these relations. How did the game structure and rotation work out for you?

    1. It worked out perfectly. Not only did the rotation allow the kids to share their ideas (if they didn’t quite get it early on), but they had to use critical thinking skills for ones that weren’t “nice” numbers. On the assessment that happened the next day, with similar questions to these, most kids got the answers right!

  3. I actually want to comment on your desmos central park post, but for some reason can’t, so I’m commenting here. I really liked that activity. It reminded me of a program that someone in my research group was developing called Giant Steps, which posed problems around a giant taking a certain number of steps and then going a certain number of feet in order to hide treasure. This game seems more relevant to real life than a giant hiding treasure, but also maybe less exciting. It seems like there could be multiple productive ways to represent algebra to beginning students through computer simulations or games.

    1. I don’t remember writing a post about Central Park (?!?) – but I love the activity. I don’t teach at a level that I’ve gotten to use it at (yet), but I know it would go over well.

  4. I just wanted to say thank you. I’m reviewing before taking the MCAT exam for medical school, and while I can regurgitate the rules for when to use these values mathematically, I struggled to conceptualize them visually. The MCAT is a timed exam, and I have been wasting precious time pausing to remember the rules and do the calculations. I knew a deeper conceptual understanding would help me move more quickly and with fewer mistakes. Stepping myself through this lesson helped me finally grasp the relationship of ratios, and has helped me tremendously. Thank you so much. You’re my hero right now.

    1. Ahhh, this is so awesome to read. Thank you so so so much for taking the time to comment on this. It makes me feel great to know that this was helpful to someone outside of my classroom!!!

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