Student Directed Curriculum (Day 2)

Day 2 of my experimental student curriculum has come and gone, and I must say, it went quite well. Here's how it went down: On the first day I gave them this paper and gave them 5 minutes to calculate as many angle measures as they could as well as find any polygons that jumped out at them.

Obviously, everyone found the right angles, the majority were able to do the vertical angles, and a couple went as far as making the connection to corresponding angles. After the five minutes, I asked for a volunteer to put something they found on the board. Both of my classes started with angles, so I went with that. We started with the angles the students gave me, I asked them how they knew what the measures were, and we connected it to other types of angles and found examples. I was greatly impressed with the discussions we had and reasoning I got from students. This was the first real lesson we've had where I've asked them to explain themselves and they went for it. It was awesome! That was day one.
On day 2, we came back to this same paper, only this time I asked for polygons since we already exhausted all of the angles. This is where it got interesting. I had a student in both classes come up and highlight a few polygons. Both classes pretty much gave me the same list: rectangle, triangle, trapezoid, parallelogram, rhombus, pentagon. I asked where they wanted to start. My second block class voted for the trapezoid; we dove in. I showed them a trapezoid in comparison to an isosceles trapezoid and they commented on what they thought were the differences. We talked about consecutive angles between the bases and related it to corresponding angles. There wasn't a single aspect of trapezoids that we didn't discuss. For the most part, they were able to toss out the ideas and then we explored them; I did very little nudging. We ended the day talking about the reflection symmetry of an isosceles trapezoid, and the relationship to the perpendicular bisector of the bases. I thoroughly enjoyed it.
Then block three came, and oh boy it just got better. Same routine as above, however, they wanted to start with pentagons instead. Game on. The student that highlighted the pentagon was unsure if it really was one because it wasn't regular (only he didn't say regular). Discussion started, we came to a consensus that it doesn't have to be regular to be a pentagon. Great. So I drew a regular pentagon on the board and said, "What's the measure of each of these angles?" *confused/thoughtful looks/blank stares* "Ok, how many degrees are in a pentagon?" 360? 540? 720? Awesome, these students have an idea of where I'm going with this. I drew a triangle. "180 degrees!" someone said. I added a triangle to it and showed them the quadrilateral formed. "360 degrees.?" someone hesitated. So far so good. I add another triangle to form the pentagon. The pattern? "540 degrees!" confidence building. I kept going: another triangle, six sides, 720, another triangle, seven sides, 900, another triangle, eight sides, 1080. What's the pattern? "Oh, I get it!" "How many degrees in a polygon with 100 sides?" "You would take 98 times 180." "Why 98?" "Because the number of triangles is always two less than the number of sides."

"Sah-weet! So back to my original question. What is the measure of each angle in the regular pentagon?" "108. 540 divided by the 5 equal angles." Needless to say, I was excited. Everyone was paying attention. Multiple students were participating (the above conversation included many students). We discussed a few more pentagonal items, and then I guided them toward the symmetrical nature of the polygon. Reflection symmetry was discussed in a similar matter as my previous class, but here I had time to move on to rotational symmetry. No problem. They were good to go even though this was a relatively new concept for them. The only issue I had (or think I'll have) is the phrasing, 'five-fold rotation symmetry.' My students always have trouble with that, knowing what it means, using it correctly, etc. I have yet to find a good way to explain it. Any ideas?
So, yeah, I'm excited about this whole freedom of curriculum concept so far. After my third block class, I realized its going to get tough to keep track of what I taught from day to day. I guess its time I figure out some kind of organization skills past post-it notes.
The students seem to be with me too. Everyone was listening, involved, engaged. Questions were being asked both by me and them, and answers were the same. We're both learning, and I don't think it gets much better than that.
Here's the kicker: I only have this plan 'planned' for these first two units on quadrilaterals and triangles. After that, I'm still trying to figure out how to get it to work. My next units are area/perimeter, surface area/volume, similarity, reflections, proofs, and circles. Theoretically, it would be easy to mix and match units and make all kinds of connections, however, that would almost require me to planned for the ENTIRE SEMESTER since I'll never know what's going to come up in class. Maybe in the spring? Ugh... that'll be tough. We shall see.