Agenda Item #1: Quiz on Pythagorean Theorem & Distance formula. Results were shockingly good.
Agenda Item #2: Discovery of the relationships in a 30/60/90 triangle.
Of all my "teacher led student discovery ideas" for this class, this one was the trickiest. I wanted student to be able to create a triangle on graph paper (as per the norm), so they can easily set the side lengths. But x and x*sqrt(3) don't lend to integers very well. Closest I could come was 11 & 19, which leaves an angle of ~59 degrees. Close enough.
Then I wanted students to be able to enlarge the triangle to test any ideas/gather more data. Oof. Finally settled on "short leg + 1" & "long leg + 2." Again, close enough.
Then I had students work the Pythagorean Theorem to solve for the length of the hypotenuse. I had been stressing that we shouldn't evaluate the radical, but I had to break the rule to make potential relationships a little easier to spot. So the hypotenuse for the first triangle is ~24, the next is ~26, then ~28 and so on.
Most students saw that "short leg * 2 = hypotenuse." Not bad. Some came up with the idea that "long leg + 3 = hypotenuse," which was true based on the data we had, so I quickly used GSP to show a MUCH larger triangle, worked the numbers, and showed that it didn't work. I love that they're trying ideas though.
I didn't really expect the whole "sqrt(3)" thing to pop out, but I wasn't sure how to point it out. By the 3rd iteration, I went with the idea that addition, "short leg + some number" doesn't work, because the number changes (2, then 3, then 4, etc). Which is ok, because when the number we're adding doesn't work, that's just multiplication. So there must be some number that we multiply by the short leg to get the long leg. But what?. 19/11 ~ 1.73, and test a few others. They should remember that sqrt(2) ~ 1.41, so that's not it. And sqrt(4) = 2, so it has to be something between 2 and 4. Why not 3? And that works.
After we've hammered out the ideas, I use either GSP or a protractor to measure the angles, because we have to figure out what was so special about this triangle in the first place. That was the easy part.