I’ve been at this high school math gig for a good long while now but I periodically have to remind myself of a couple of important facts. The most important one is that not everybody’s mind works like mine. Just because I like a certain way of thinking, or dislike a certain way of learning, I should not assume all my students will agree. In fact, I can be pretty certain that all of my students will not agree, there’s too many individuals for that to work.
When I studied Geometry I did not like physical drawings and constructions. In part because I am a bit inept when it comes to controlling something like a compass, but also because getting my hands engaged does not seem to fire too many of my neurons. So, when I wrote my Geometry book a couple of years ago I did not include much in the way of hands-on manipulations. The past couple of years of working through the text with our students has pointed out the weakness of this approach. So, I put my head together with one of my talented colleagues to try and make an activity that would trigger some neurons for those students who come to life when they get their hands busy. I had been using a pretty cool activity I ran across from Jennifer Silverman but I made pretty flimsy paper copies to work with on a pipe building activity where kids had to manipulate bent angle joints with different pipe lengths. It’s a great activity but using simple paper copies dragged the activity down. We invested in some packs of AngLegs this year and my colleague wrote a pretty cool activity modeled off of our pipe building activity. You can find his document here.
I was impressed as each of the seat groups in my class played with the AngLegs making some discoveries about combinations that worked and those that would not. We discussed, without naming it yet, the triangle inequality theorem to explain why some combos did not work. But the real fun, and the clever heart of my colleague’s activity, was when I asked one student from each group to come to the front of the room. When they left their group the remaining group members were given the following task – I slightly modified the original document on the fly – I asked them to make and measure a triangle. Find six measures, the three side lengths and the three angles. They then put the triangle away where it could not be seen. I sent the volunteers back and their teammate gave them three pieces of information. I left it to each group to decide what information to share. Once given three clues the volunteer student needed to manipulate the AngLegs to copy the triangle described. What ensued was a terrific conversation about what information is necessary to guarantee that I have to make the same triangle. We used this as a launching pad to discuss congruence theorems for triangles. I have some great links in the text to some wonderful GeoGebra activities up on the GeoGebraTube site but I know that many of my students do not do these explorations. I also know that some just need to get their hands dirty, so to speak. Some kids were able to recreate the triangle but admitted that it was a bit of luck. Some stumbled upon the ambiguous case of the Law of Sines without being told that this is what happened. Some realized that they had no choice but to create the correct triangle.
I was really pleased by the level of engagement and I am now thinking about ways to use the AngLeg sets again soon when we start talking about side and angle bisectors. I want to have tables create and draw their own triangles before we stumble into discoveries about concurrence of these bisectors. This will feel, I hope, a little more authentic than me just giving them a prescribed triangle which may feel a bit like I am just luring them into some pre-prepared trap. I think that this activity we ran benefited my students and we have referred to it on a number of occasions already. The grouping of three or four students together at a time helps and allowing them to get their hands busy has helped. Looking forward to loosening up a bit more and letting my students be more tactile in their approach to Geometry. I’ll still show them the GeoGebra and introduce them to Euclid the Game but I need to remind myself that they are not a bunch of mini Dardys in the room.