(No, I am not in Alaska. Yes, I am overly fond of alliteration.)
My trig lessons were going pretty well, when I noticed that a good number of kids were having trouble identifying the opposite or adjacent side of the angle. This always seems trivial when you’re teaching a big, complicated subject. The kid gets SOHCAHTOA, gets the ratio, gets the concept of inverse, and just has a little bit of trouble figuring out which angle is which. Big deal, right? Lots of concepts grasped and a glitch on implementation. But from a testing and demonstrated knowledge standpoint, the implementation is the ball game.
It suddenly dawned on me that I was constantly guiding students on one point, and one point only, despite a lot of time spent explaining different ways to find opposite or adjacent sides. And that this seemed awfully familiar, a memory from teaching this two years ago. I couldn’t assume they’d finally pick it up, or that this was non-trivial. I realized I needed a different approach.
And so, I gave them a Kuta handout, magic markers, and an explanation:
“Suppose I ask you to find the trig ratios for angle A. You guys have shown me you know the ratios of SOHCAHTOA. But how many of you are still not quite sure which is the opposite, the adjacent, or the hypotenuse?” (Half the class had their hands up before I finished the last sentence.)
“Yeah, that’s what I thought. So we’re going to try this. In your notes, create a right triangle like this one. Using a marker, trace the two sides of angle A.” (I can see all the kids, not just the usual worker bees, busily creating triangles and marking angles, which is encouraging. They want this method; it’s clearly a hitch in their understanding they want fixed.)
“Now, here is an important concept. The angle is created by the hypotenuse and the adjacent. The opposite is not a part of the angle.” (I repeat this several times. I can hear several kids already saying “Oh, I get it!”)
“So, which side is NOT a part of angle A, Corinne?”
“B and C.”
“Okay, remember that you describe segments by saying the two points together, like this: BC. And yes, you’re right. Segment BC is NOT a part of angle A. Sandy, what did I just say about a side NOT being being part of the angle?”
“The opposite. You said the opposite is NOT a part of the angle.”
“Awesome. So Marco, which side is NOT a part of the angle and so must be the OPPOSITE?”
“Great. So everyone mark BC as the opposite, like this:”
“Now. You’ve got a foolproof way of finding the opposite. Just trace over the angle, and then label the side you haven’t traced as the opposite. Now, what about the angle itself? What sides form the angle, class?”
“The adjacent and hypotenuse.”
“But how do you tell the difference between them?” asks a student.
“It’s opposite the right angle,” pops in another.
“True, and that’s a really helpful method. But if you’re worried about what “opposite” means, just do this. Take your pencil, start at the right angle. Draw a straight line through the interior, the inside, of the triangle, until you hit a side. That side is the hypotenuse.”
(Again, I can see that all the kids are doing this. It’s really friggin’ awesome to to realize how much the kids wanted to close this loophole and to have a method be so instantly useful.)
“And now, what’s left?”
“There you go. That’s how you identify the sides. Helpful?”
“Yes!” (big response.)
We practiced it several more times and then I let them loose on the handout. I knew I was on the right track when one of my weakest students showed me his entire handout labelled correctly, but told me he wasn’t sure what to do next. We went through SOHCAHTOA and the light dawned. He worked every problem correctly.
From there, the next big teaching challenge was inverse, when I finally let them use calculators. Yes, up to now, they’d been using a trig table. I wanted them to fully internalize the fact that each trig value is a ratio. Their familiarity with the trig table really aided their comprehension of the inverse.
Next up was solving right triangles and figuring out what ratio to use. But this was much easier because they were now close to 100% accurate in identifying sides, so they weren’t compounding errors.
This quiz is by another math teacher, and much easier than my quizzes. The kids nailed it. The mode–the MODE–was 100%. The mean was 83%. Only four kids failed the test, and one of them failed because he wasn’t trying.
To reward them for their great job, I still graded the quiz on a curve (meaning one of the Fs converted to a D), and weighted the quiz as 100 points instead of 50.
Then, the kids spent two days moving back and forth between the three methods of solving for sides: Pythagorean Theorem, special rights, and trig.
Once, when I was up front reviewing the results, a kid asked “So why can’t I just use trig instead of special rights?”
Another one chimed in. “Yes, I really don’t like the ratios.”
I said, “I’m fine with you using trig. But remember, you are expected to know the trig ratios for special rights, although we don’t call them by the same names. So if you really like using trig, take a little time to memorize certain values.” And then I went through the trig values for sin(30), cos(30), sin or cos(45).
“So just memorize these three, and you’re fine. And of course, you should never forget the tangent of a special right triangle.”
“ONE!” said a good number of the class. Yay.
We’re moving onto polygons, which will allow me to revisit trig when finding areas.
I’m psyched. If only I had a job next year!