## Back in the Saddle

I would love to hear any advice/questions/concerns about these HW assignments. Please reach out by commenting here or through twitter where I am @mrdardy

## Trying to Help My Students Help Themselves

I’d love to hear any words of advice about reinforcing these habits of work and habits of mind. You can drop comments here or over on twitter where I remain @mrdardy

## My Students are Making Some Smart Guesses

This conversation was a wonderful way to end our day on Friday. I am delighted that my students are comfortable enough to make these guesses out loud and even more delighted that they are making such good guesses right now. I pointed out how helpful it is to play with GeoGebra to check these guesses and I hope (I hope hope hope!) that some of my students are making a habit of this.

## A Delightful Conversation

Last week in my Geometry class we had a fantastic conversation about a homework problem. Here is the problem in question –

I wish that I could take credit for having written this, but I am certain that I ‘borrowed’ it from somewhere. Likely from the fantastic resources shared with me by Carmel Schettino (@SchettinoPBL)

If you recognize the above problem as your own, feel free to claim it and let me know. Know in advance that I am very grateful for such a rich problem to tie together ideas of distances, slopes, line equations, properties of squares, and triangle congruencies all into one tidy package!

## Hands-On Geometry

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.