I had a pretty good lesson recently that I wanted to share. It was at the end of my quadrilaterals unit, and so we were working on coordinate proofs. I love coordinate proofs because you can get so much information from just a pair of coordinates, which lends itself to lots of different ways of solving the same problem. Add to that how many different ways there are to prove something is a square, and we have the start of something good.
I gave the students the above sheet, starting off with some noticing/wondering about the graphed figure. Then I assigned each table a different method to prove that the quadrilateral is a square. Each group was off to their whiteboards to get started.
It was really great to see each group discussing the problem so intently, and it reminded me how easy it is to facilitate discussion when up at the vertical whiteboards. Afterwards, the students went around in a gallery walk to compare their proofs to the other methods. They analyzed how they were similar, how they were different, and thought about which method they might prefer in the future. (Some comments included things like preferring method 2 because it only involved slopes, even though it involves more lines.)
The whole lesson went so smoothly and had tons of intra- and inter-group discussion. Need to use the structure again.
My math coach gave me this idea as we were planning my Circles unit. I think it went fairly well, so I’ll share it here. The idea is that we have, essentially, three basic objects that we’ve combined in different ways in geometry: circles, lines (including segments), and angles. So, as an opening activity to the unit, the task was this:
“Think of as many ways as possible to combine those three objects.”
First they brainstormed individually, as I reminded them that they can use multiple lines or angles or circles if they wanted. Then they went up to groups and made a master list per pair or group, eliminating ones that were “pretty much” the same. I gave them some vocabulary based on what I saw they drew, and they had to use that vocabulary to describe what each drawing had. Finally, they chose one example and created one neat, fully correct example, in color that we combined into class posters. (I approved what they chose, to ensure a variety of possible layouts.)
Between my two classes, they came up with almost every scenario I could think of that we would learn in the unit, with the exception of Tangent Line & Radius, which I drew and put in myself. Now they are hanging in the classroom, acting as a guide for our journey into circles.
Yesterday I introduce radians to my students for the first time. I started out by asking why they thought a circle had 360°. There were a few good answers – four right angles makes a circle, so 4*90 is 360; a degree is some object they measured in ancient Greece, and so a circle was made of 360 of them; something to do with the number of days in the year. All good answered, but I told them it was completely arbitrary based on the Babylonian number system.
Once we decided that it was arbitrary, I asked them to come up with their own method of measuring a circle. I would classify their responses into three categories
- Divide the circle up into 200 “degrees” (most common)
- Divide the circle up into 100 “degrees”
- Divide the circle up into 2 “degrees” (least common)
I was expecting 100 “degrees” to be the most common, so I was very surprised to see that most of the students want to split the triangle into two sections, each with 100 parts.
I have been a proponent of tau for a while, as I thought it was natural to think of radians as pieces of a whole circle, but my students were clearly thinking of the circle as two semicircles right off the bat.
I pushed the students who came up with the third way in a whole class discussion. If this whole semicircle is one student-name-degree, what would you call this section? And so we got to using fractions of those degrees.
That made a pretty easy transition into radians. I went a little into the history; instead of using a degree, some mathematicians decided to use names based on the arc length – and so that semicircle’s angle was 1 π radians, instead of 1 student-name-degree. And the fractions we used were the same.
This almost made me doubt my tau ways – maybe π was more natural. But then, as we started converting angles from degrees to radians, some students kept complaining that, for example, 90° was 1/2 π instead of 1/4, since it was clearly a quarter-circle – so maybe I can stay a tau-ist.
Back in grad school, instead of one big thesis we had to do two research projects – one math research and one classroom research. I don’t think my math research is particularly noteworthy (it was about automating the instrumentation of a harmony given a melody, in the style of John Williams), but I did like my classroom research. Towards the end of the program I was talking with my of my classmates and we said how we both enjoyed it and could see going back into a program to do research in the future. I don’t know if that’s still true for me, but the future holds many possibilities.
The focus question for my research was, “Given the three standard methods of solving systems of equations, which methods do students prefer and why?” I had some ideas going into it that held true, some that were thrown out, and some interesting other ones. For example, some of the students preferred a certain method just because it was the one they learned first, even if they knew it wasn’t the best one. Visually-inclined students did not prefer graphing, as I expected, but rather elimination, because of the way the numbers lined up. Some students changed which method they used in order to avoid something – one student had trouble solving equations with x on both sides, so the method they used was the one that didn’t lead to that scenario.
You can see more of the findings, and the whole paper, below if you are interested. And yes, if you read it, you’ll notice that the pseudonyms I chose for the students were all based on Doctor Who companions.
At the beginning of yesterday’s lesson, I threw up this monster of a problem:
I told my students that, by the end of the lesson, they would be able to solve it. They flipped and freaked out. “No way, Mr. Cleveland, not going to happen.”
In all 4 sections, 1 hour later, every student correctly solved the problem. And they were all so proud of themselves for doing so. There’s no better feeling than that.
So on the first day of math class, I gave the students this little analogy:
“Math is like cooking. You don’t need to know how to do it to live your life, but if you don’t you need to always rely on someone else to do it for you, and it will wind up costing you more money. Most people know how to do the very basics, enough to get by, but those who really understand the concept make their lives richer and more enjoyable on a daily basis.”
I also told them math was like a language, a pretty familiar analogy. But then I asked them to come up with their own, and they created a poster based on the different answers.
Here’s some they said:
“Math is like your parents: sometimes you just don’t understand them, but they’re just trying to look out for you.”
“Math is like a wave: sometimes it’s big, sometimes it’s small, but it never stops.”
“Math is like the subway: you can read the map and think you know where to go, but you don’t really know until you’re there.”
“Math is like time: there’s a new number every second.”
“Math is like climbing a mountain: it’s really hard, but you feel great when you get to the top.”
“Math is like HIV: it never goes away.”