## Thursday, March 3, 2011

### Best Group Work Ever!

Continuing with my "This was a trig Do Now that I had used that worked well for me" thread, I used two Do Now exercises to illustrate to my Honors Geometry kids how to break down quadrilaterals without ever lecturing. First one looked something like this:
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.