1. Draw an irregular quadrilateral ABCD with one right angle A; use the corner of your white paper to get an easy right angle.
2. Measure AB and AD only.
3. Measure all 4 angles of ABCD.
4. Draw a dashed line from B to D.
Then, I had the kids fold the quadrilateral back along the dashed lines, completely solve for everything inside triangle ABD (all angles and all sides), and then open it up and completely solve for all remaining angles / sides, and finally to find the quadrilateral area! (I had to remind them that earlier we had found the area of a scalene triangle to be A = (1/2)(a)(b)(sinC). Other than that they were uber-independent and got the whole thing quite brilliantly. The activity was again self-checking using protractors and rulers, so it helped to build their confidence in their own rather elaborate calculations.)
After this, we worked a little bit on Kristen's awesome trig project problems. Since my honors kids are sharp, I didn't need to elaborate further on how to divide up quadrilaterals besides emphasizing that we had seen how right triangles make up scalene triangles; now we were going to see how scalene triangles make up quadrilaterals as well, and to use that as our basis for analysis.
Day 1 of the project was a bit slow. I didn't need to help them too much, but they were working slowly in groups, trying still to grasp when to apply which formula. They were tentative about discussing the problems, and mostly worked individually. I saw this as a sign that no one was sure-footed enough to pipe up in a group, and it made me wonder whether my kids were really fully ready to venture into quadrilaterals on their own just yet. So, I decided that what they needed was another scaffolding activity, just to ease the transition a bit.
So, on Day 2, I devised another Do Now that furthered this idea of cutting quadrilaterals into smaller pieces and applying Laws of Sines / Cosine to analyze the individual pieces. This time, I wasn't so interested in them actually carrying out all of the calculations. I just wanted them to make a plan of attack, so that they can begin to see the pattern/big picture:
1. On a sheet of white paper, draw an irregular quadrilateral ABCD. This time, don't use right angles.
2. Measure all 4 sides of ABCD.
3. Measure only angle A.
4. Describe, step by step, how you would solve for all angles of ABCD.
5. Also describe how you would find the area of quadrilateral ABCD.
Seems innocuous enough, but the kids had to work hard for this one. Only a few kids were able to run with it completely on their own. Most figured out how to cut the quadrilateral into two appropriate triangles. After waiting a while, I started to give the class hints along the way, one hint every few minutes. (I picked kids from the class to share how to do the next step, every few minutes.) They struggled through this, but the process of thinking about it was very worthwhile. After this Do Now, the kids worked on the remainder of the project packet with much more fluidity and confidence.
Today, we more or less wrapped up the whole project. I had given them 5 problems from Kristen's packet, and asked them to solve 4 (and had given them time in class to work on 4 problems), the fifth one serving as extra credit. By the end of today, a bunch of kids were on -- or had finished -- their fifth problem. I wasn't checking their answers any time during the project; I asked them to compare with their partners, and then to compare with another group if they still weren't sure. I asked them not to round whatsoever, so that in the end I can verify that everyone's answers are absolutely accurate (down to 12 sig figs, or whatever TI-89 allows). And it was glorious. As a contrast to how timid they had been at the beginning of the project, kids were LOUD. I had to shush them because some kids were working on it during my Study Hall (like an advisory period) and their exuberance was getting in the way of me helping other kids with other mathy things. (I normally have 12 kids in my own Study Hall; today I had about 30. A good 20 of them were honors kids working together on their trig projects, and the rest were regular Geometry kids asking me for help with studying for their test.) Every kid devised their own way of arriving at the same answers, even within the same group. Even my weakest honors kids were able to call me over, walk me through (in shocking clarity) what they had done to solve the majority of the problem and to ask me very specific questions about the remaining steps. The whole groupwork thing was so glorious that I took a mental picture of them working in groups today and really, really wished that I had an actual camera. It was truly one of the greatest days we had had this year, in terms of complexity, how great kids felt about their work and each other, and everything else. Afterwards, when we started reviewing for the upcoming trig test, I heard various kids whisper, "Now it all seems so easy!"
In the end, I told them how proud I was of them. I told them that I had taken this project from an 11th-grade teacher, and that I hadn't been totally sure that they could handle it but that they had handled it fabulously. And all of it is so true!