Thursday, February 3, 2011

Trick for Teaching Basic Trig

During my middle-school teaching days I noticed that often kids would arrive in my 8th-grade class with a half knowledge of sine, cosine, tangent. There were two major problems they often had in solving for unknown sides in a right triangle using trig:

1. They couldn't visually distinguish opposite side vs. adjacent side. Many middle-schoolers I taught had a poor consistency (if any) with recognizing what "opposite" and "adjacent" meant in a diagram; it was just too abstract for them, even though I tried to explain how to look for the sides "across" the triangle, etc.

2. They couldn't figure out whether to use sine, cosine, or tangent in a given situation.

The first problem I solved successfully a few years ago, when I came up with an idea to start teaching kids to reach their hand out and to actually put their hand over the acute angle that is given in the problem. The "opposite" side (from the perspective of the given angle) is the ONLY side that their hand is not touching, since "opposite" implies "far away"; on the contrary, the "adjacent" side is the side (besides the hypotenuse) that their hand IS touching.

Trust me, if you're seeing the same problem in your classes, TRY THIS. It works like a charm. I taught sine, cosine, tangent from scratch today to my 9th-graders, and not a single person had trouble recognizing opposite / adjacent sides. (Granted, they were honors kids, but again, I've tried this with my regular 8th-graders in the Bronx. It had worked like a charm then, too!)

Issue #2 just takes a little bit of practice, but this year I found that I made a nice transition into this by having allowed a couple of days of pure similar-triangles proportions practice. Right from the start, kids really grasped the concept that in order to solve for x, you need a proportion that involved x as an only unknown, so it was super easy for us to transition to speaking about putting together x and the other known side inside the same trig ratio / proportion.

So, surprisingly, I taught all of basic trigonometry to my honors classes in one 75-minute period. (Inverse trig not included.) Seriously, those guys had never ever heard of SOH-CAH-TOA before today. Neat, eh? We'll see next week whether I can translate this success to my regular classes!!


  1. Oh this is just too awesome. I'll do basic trig on Monday with my honors juniors and this will be just right.

    But wow, 8th graders in the Bronx do trig? I was always really good at math in school, but when in ninth grade I took physics (normally just for seniors and juniors so teacher assumed everyone knew trig) I had to really struggle with trigonometry because I just couldn't "see" what sine, cosine etc referred to. Them being proportions was too abstract for me then, and I'm always impressed when kids just get it without struggle with abstraction. But then secretly I'm always wondering if they really do get it or if they are just placing lesser demands on their own understanding.

  2. I am glad you like it!

    And, if it helps... I also always introduce sine/cosine/tangent with a Do Now problem about solving for heights using shadow lengths, and then I talk to the kids about how we don't always necessarily have another object nearby that we can measure and have full information about, so mathematicians store the ratios inside their calculators. That seems to make the transition into trig more intuitive as well.

    PS. Pretty sure in all of NYC, trigonometry is a middle-school topic.... Grade 7 curriculum.

  3. You are too smart for your britches. :) Seriously though, that's pretty clever!

  4. I'm obviously not too smart, Ames, because I had to look up what britches meant. :( But now I understand: it's like you're saying I'm too sexy for my shirt. ;)

  5. hahahaha and funny

  6. Thanks everyone, your comments have helped me uderstand trig a lot more!
    I'm a teaching assistant living in the UK, I support Math in a local primary school, Yr 5 & 6 (9-11 yrs), and decided to re-take my GCSE Math to help in my job, exam coming up soon!!!

  7. >"a couple of days of pure similar-triangles proportions practice"

    What exactly did you do during those days? I'm starting trig tomorrow with my pre-calc students.

  8. Great idea. I will try this tomorrow when I introduce right triangle trig. to my lowest level sophomores in St. Paul, MN.

  9. So great! Thanks!!

  10. Well I'm a kid and you just kept me from failing thx so much!

  11. Very useful resources. Thank you.
    I also have a useful book related to this topic.

    Trigonometry For Dummies (2014).pdf - 7.2 MB

  12. thansk man...this really helps

  13. Thank you! I'll be trying this - decided a practice homework of working through a few triangles all flipped around, tell which sides are hypotenuse, opposite and adjacent a given angle.