Differentiation pt. 1

Pardon me while I continue to idolize Sam Shah. He’s inspired me quite a few times recently.

A couple days back, Mr. Shah posted a project he was planning to use for his students to foster extracurricular exploration of math. Quick summary: There are TONS of cool math sites and blogs and newsletters on the internet which highlight progressive, artistic, fun, puzzling math, and he put together a project for students to choose several things to investigate.

As I read it, I LOVED it. I replied to him on Twitter and soon realized that another favorite math teacher blogger had been his inspiration. SO MUCH COLLABORATION!

As it is my first year teaching all the classes I am teaching, I am trying to hold off on making a huge investment in forming projects at least until I have a chance to get through a first year and make it to the summer to have a chance to reflect. I began to think of ways I could still use these ideas in my class.

Enter tension 1: In a recent observation, my observer challenged me to go all in with differentiation. Its always something I struggle with, as it tends to make me think I have to plan separate lessons for each level student in my class. I have no more time for that. However she introduced the idea of just offering choices in how students show mastery. No doubt its a lot easier to give a problem set and get all students to work through it, but she suggested a choice board. This one is from an art class, but the point is to create options that engage students from the different learning styles and preferences.

Enter tension 2: I usually teach to the majority of low level learners I have. I have not been happy with how I am (not) challenging my high level learners. I feel like the common advice was to turn them into peer tutors when they finished their work. I know they can benefit from that. I am learning a ton about statistics by teaching it for the first time this year. I’ve always felt a bit uneasy about that idea. I want them to be excited to finish other work early so they can continue to challenge themselves with a new and exciting topic.

Tension 1 + Tension 2 = My Current Idea

I am trying on using most of Mr. Shah’s mini-exploration activities as an opportunity for choice and for my students to continue to push themselves. So far, I’ve had a only few students I’ve asked to participate. I ask them to go to our class website and read the directions and pick an activity to complete. See the work in progress. I’m still working to break it down into clearer chunks. Vi Hart’s doodling has been pretty popular so far, as well as a couple of the games on Math Munch. Still looking forward to see how it works out as time goes on and as the routine continues to settle in.

Introducing basic trigonometric ratios

I take no credit for the idea, but I am absolutely sharing the results. Thanks to Mr. Shah, a rock-star teacher blogger, and his Virtual Filing Cabinet, I had a wealth of ideas at my disposal when planning a trigonometry unit for the second time this year.

Last year, I started with special right triangles. We then for some reason detoured into arc/sector measurements before returning to me telling them the patterns that these “special triangles” could make in the unit circle. YAWN!  I asked my students to memorize the unit circle, which I am now not convinced that it is that important anyway (Although Ms. Gruen’s blog gave me a great memory tool to help students remember unit circle values). Then we used a great activity from TI Education to build the sine and cosine waves. I stand behind this one, although I already know this year it will make SO MUCH MORE SENSE to them. 

The reason I am optimistic about the TI activity is now all rooted in the new approach I took this year. Courtesy of Mr. Lark, my students were able to find their own patterns and shortcuts. Mr. Lark’s activity takes a lot of the heady, abstract nature of trig and returns to a bunch of simple calculations. Students measure angles, draw radii, and write the x- and y-coordinates of the points at which those radii end. By the time they get really tired of doing this over and over again, they can start to see patterns that they can latch on to. I introduced my version of the activity (see below) with a very explicit exhortation to use the patterns when you find them. After a couple days, we discussed and students suggested patterns that could be later tied into:

  • reference angles
  • why sin, cos, and tan are positive and negative in different quadrants
  • symmetry between the various quadrants

I’m drooling at this point. My students are thinking like mathematicians and all while working on an activity that really only required knowledge of how to use a protractor and how to read the coordinates of a lot of points. I then make the big reveal: “I know you’ve heard how miserable trigonometry is, but YOU’VE BEEN DOING IT ALL ALONG!” The stares I get back say, “Who is this crazy person?” but then I explain how the x-coordinate they were finding over and over is cosine, and the y-coordinate is sine, and the ratio of y/x is tangent (which I somehow didn’t realize until this year is the slope of the radii in the unit circle formed by that angle!).

We moved forward and derived the SOH CAH TOA general trig ratios by showing how any right triangle was proportional to a unit circle triangle. I am really impressed with how well my students have latched onto the sin, cos, and tan since then.

It’s also pretty cool when you hear “I didn’t know SOH CAH TOA actually meant anything! I thought it was just something people said…”

Now that I am getting to introducing the sine and cosine waves this week, I’m looking forward to lots of A-ha! moments from students remembering that cosine is the x-coordinate and sine is the y-coordinate on the unit circle.