Working on the Holiday

So, one of the oddities about teaching at an independent school is that days off that are taken fro granted most places are seen as prime days for campus visits here. So, we were in session today. No need to feel sad about this though as we have plenty of vacation time as well. Just another Monday.

But…it wasn’t. It was a terrific teaching day and I want to make sure that I make note of it even if only for my own pleasure.

1st Bell – Geometry. I was looking forward to returning their excellent tests from last Friday but first I wanted to dip into our new unit. I passed out paper, rulers, and protractors and gave them a simple task. Draw some polygons. I insisted on not defining what a polygon was and I did not reveal why they had protractors. I tasked each of them with drawing six polygons and I saw some pretty great stuff. Complex, crisply drawn concave polygons. Some students stuck to the middle of the road and drew squares, triangles, etc. Then the fun began. I started asking for definitions of polygons and I framed the question this way: Explain to my 11 year old son what a polygon is. Brainstorming began. I heard about the need for line segments as sides, I heard about the limitation that there had to be at least three sides, I heard conversations about polygons that pointed inside versus those that did not. Someone offered up the word concave and I pressed for a definition. Everyone seemed happy about the inside-ness of some points on a concave polygon. I pressed them not to use the word point for where the line segments met and they offered up vertex as a better alternative. We decided that each vertex needed two (and only two) line segments and one boy suggested that polygons that were not concave (we agreed to convex quickly) should have angles larger than 90 degrees. He backed off of that but I will definitely revisit this idea soon. When he tossed out that idea he was greeted with references to equilateral triangles. Well played. Then the highlight of the morning came for me. I drew a figure on the board that was made of line segments, that had two line segments at each vertex but it was clearly not a polygon. The reason why is that my figure was not closed. One of my students used that exact language and I pressed, again with my 11 year old son in mind, what we meant by closed. Miranda said ‘Imagine it has water in the boundaries. If it’s closed, the water can’t get out.’ I thought that this was a lovely image. I then closed my crazy drawing but blocked off access to some regions while doing so. I was quickly told that the water needed to be able to get everywhere. I hope that this image stays with my students the way that it is staying with me. As we wrapped up class in a blur of vocabulary about quadrilaterals one of my students said to her neighbor, ‘What a great way to start the week. We got to sit and draw.’ I’ll count this one as a success.

Bells Three and Six – AP Statistics. My senior heavy Stats class did not come back from winter break with much of a sense of urgency. I did not want to just launch right into a new chapter on the heels of the disappointing chapter test we had on Friday. My Computer Science colleague had recently shared with me information about Sicherman Dice which are two six-sided dice that are not standard dice but their sum has the same probability distribution as the sum of two standard dice. I presented my students with a  challenge. Describe two six-sided dice that replicate the probability distribution of two standard die. No other directions really. I have a rudimentary handout I gave them and you can grab it here. I fielded questions as they chatted about this problem with their neighbors. Can the die have negative numbers? Can the die have fractions? Can the die have zero? I kept replying in the reluctantly affirmative and checked on their progress. Most of them had a pretty logical attack where they would transform one standard die in a certain direction, say subtract two from every face and then transform the other one in the opposite direction. Not very sophisticated, but it was nice algebraic logic. One student was working on fractions trying to balance combinations of 1/3 and 2/3 so that she would always get integer answers. Overall, it was the most focused energy from this group that I have seen in two weeks. I hope that this is an omen for our next unit. They were pretty surprised by the reveal and I am curious, in retrospect, that they never asked if any die could have repeated values as this is necessary for the Sicherman Dice to work their magic.

Bell Seven – AP Calculus BC

We’re just getting ready to start exploring the magic of Taylor Series. We took baby steps today reminding ourselves of the language of arithmetic and geometric sequences and series. I always think that this material is such fun to untangle. Tomorrow we’ll play with GeoGebra and I will try to tease out of them the key ideas about how to make a polygon behave like the sine function. Nothing much else to report here.

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