We were about to start right-angle trig and I didn't want to just tell them the ratios without any reason for all of this. Naturally, I went over to Andrew Stadel's 3-Act Catalog and came upon Boat on the River. It was perfect!
Here is the video for Act I. Go on, watch it. It only takes 12 seconds. So cool! There were lots of questions flying around the room, then we got down to one:
We talked about the word "trigonometry" and how trig is used (Canadarm2 on ISS) and then looked at the ratio of sides of similar triangles (they each created their own triangle, similar to mine).
We talked about how one of the ratios represents the slope and how it changes as you change the angle but that it does not change if you scale up/down the side lengths.
I made a trig table for tangent, but called the columns "angle" and "slope" so they could find the angle that corresponds to any slope. Then I told them that the slope ratio has another name: tangent. We looked at some examples from the table and talked about that magical 45 degree angle.
After going over sine and cosine and working through a number of examples it was time for act II. They quickly came up with what we needed to know:
And that's what they got. Sort of. Many of them laughed when they saw 6". "That's not 6 inches!".
They didn't notice the bottom left corner:
This leads nicely to another good conversation about similar triangles.
They worked out the angle proudly using their new knowledge...
And then we got to see if the captain had done the math before attempting to go under the bridge - Act III. The video is here (54 seconds - with just the right music too).
This gave what we needed to do purpose. It engaged them and they HAD to see whether the boat made it. Another huge thank you to Mr. Stadel - you rock!