Here is what I do, or try to do, and please, readers, I am asking you to criticize the hell out of it. I really want to do it better. (Due credit: this idea and the original document came from Dave Cox, who used to be a professor at Cornell, but I think he's retired.)
I give every student a protractor, ruler, and a 4-page packet. I usually have my desks arranged in pairs, and each pair of students is assigned an angle. Using the protractor, they draw a series of parallel lines that make that angle with a horizontal line on a given page with some axes provided. This creates several overlapping right triangles.
Then they use the ruler to measure all three sides of each right triangle. They record it in a table I provide. They use their calculator to compute the three relevant ratios, and enter them in the table. Here is the data sheet:They notice that the ratios for the two corresponding sides all come out to be the same, and we have a discussion about how similar figures have sides that are in proportion, and that's what it means for things to be in proportion. When you divide them, the quotients are equal. They average the five ratios to determine the "real" ratio.
Then they are supposed to use the ratios to solve this problem for "their" angle:Doing that successfully is probably the key part of the lesson, and it never goes well. They don't see right away exactly what to do, and give up. I end up doing a bunch on the board for them, using their angles and the ratios they calculated.
Next we are supposed to collect every pair's calculated ratios in a table, like this:
And they are supposed to copy them and use it to complete some problems for homework, that look like this:
I keep trying this lesson every year, because I really want it to work. I really think it should work. Part of the problem is that we run out of time. I can't get this whole thing done in 43 minutes, and there isn't really a good point to stop and pick up the next day. The next day, I just tell them "SOHCAHTOA" (*huge resigned sigh*) (at least I don't claim Sohcahtoa was a Native American princess) and start teaching procedure. And feel like I am committing malpractice, and stealing my paycheck.
Here is another link to the document.
Update: I revised this activity based on the comments and discussion on this post. Go here.