We (along with most of the area, it seems) headed back to school today. We have 3 days of regular classes to finish up the semester and then exams next week. You know how kids have trouble remembering things over a weekend? Try a 2-week break and then an exam over a semester's worth of material. Sometimes it gets ugly.
With three days this week, I didn't want to give my precalc kids that much free time to work on exam reviews. We'd finished up before our break with law of sines/cosines, so I'm taking two of these days to work on area of triangles. It's a nice way to ease back into school, I think.
I started out by giving them these two triangles and asked them to find the area.
They whipped through the first one easily and most kids automatically drew an altitude for triangle #2, found its length, and used that for the area of the triangle.
In 3rd period (my second time through) I then gave them this and asked them to find the area. (Assuming that A, b, and c were known values)
Despite having just done this (with numbers) some of them were a little unsure. I advised them to repeat what they'd done in #2 but leave the letters. I have to admit - I'd never done this myself before to derive the area formula. I'm embarrassed at how easy it was and how I should've made them do this years ago!
If you work through it, the altitude has length of csinA, so when you plug it in to the 1/2bh formula, you get an area of 1/2 bcsinA. Voila!
The second area formula (Heron's) isn't so easily derivable (though I tried on my own). But at least we got one in!
I usually like to spend a few days working on some area problems after we talk about the formulas, but with my limited time for the next couple of days I can't do that this year. Instead, I'm going to give the kids the 6 problems I normally hand out for them to work individually and have them work them as table groups. I don't want to take a lot of time to grade them, so I copied a spinner with 7 sections (because I threw in a toughie at the end) and am going to have them write their answer in the 7 blocks. That way I'll be able to see all of their answers at a quick glance.
I'm wondering if I can get even lazier and make this even easier on myself. :)