Inquiring into the Unit Circle, Part II

A few weeks ago I was mid-way through a trig unit, and shared my thoughts on how it was going in this space.  Having now finished the unit, I am ready to write about the rest of the unit.  At this point, I feel this is one of the best units I have ever taught, and as my colleague Valerie Kasper and I reflected last night “It was great, because we didn’t teach them ANYTHING.”  Indeed, they figured it all out for themselves and then taught each other.

To quickly recap, the kids used their prior knowledge of SOHCAHTOA and Pythagoras to learn about Special Triangles.  We followed this up with an investigation into the Unit Circle, where they discovered radians, and what sine, cosine, and tangent really mean.  We then put their new knowledge to the test by applying it to problem sets.

Next the kids learned how to solve trig equations.  I did ask the kids to watch a video in advance of the lesson, though to be fair, a substitute teacher may or may not have mentioned the video to the kids and I forgot to put it on the school’s VLE, so I don’t think many of them actually watched it.  No big deal, because they understood enough from our previous work to be able to solve these with a 5 minute chalk-and-talk lesson.  They understand why we get two answers by using the sign of the function to determine the quadrant of the angle.

Next we moved on to graphs of trig functions, which we again learned in an inquiry-based way.  Using this packet, the kids worked jig-saw style in groups to discover the graph of sine and cosine functions on their own.  They started in 4 groups each working on one particular graph and the accompanying questions.  They then returned to mixed groups, and took turns teaching what they learned with their mixed groups.

For the summative assessment we  had the kids teach trig lessons.  They chose topics from a menu of different options, each choosing a main course and a dessert.  The main course was something we had learned in class; the dessert was always an extension–something they had to figure out on their own or research.  And this, wow, was a fantastic experience for me.  Letting go of control and seeing what they could do with it was, well, just wow!

They selected their partners and topics and went with it.  They needed no help with the technical side of things, though I did one of our tech teachers to check in with the kids. They told ME how we should share the videos: Google Drive, of course. They planned, they collaborated, they figured out roles, and taped their videos.  Call me impressed; most of them just knocked this one out of the park.  Check it their work for yourself: this one about the tangent function or this one about the relationship between the unit circle and radians or this one about special triangles and sine/cosine values.  These kids shone as the experts, spoke articulately and with confidence.  I dare say they learned more in preparing these videos than they would have in taking a test.

I wrapped this up by having them share the videos with each other through their blogs.  Each student had to watch at least two videos and comment on them.  Initially, they all wanted to comment on improvements that could be made or where they made mistakes.  I shared with the kids that I thought this was MY job, while their job was to praise their classmates for their good work; I wanted to use these videos as a way to help build community.  They were to write about these things: 1. Something that they liked; 2. Something they found interesting/helpful/or something they learned; 3.  A question they still had.

Some of their comments are here.  So touching and so kind:

“I really liked how your video was like an actual lesson. You guys were super clear and I understood almost everything you said. You did well in choosing a different real life example when you couldn’t find one for tangent as well 🙂 the layout of your video was good because you could draw on the pictures and show your working out.”

“I thought your video was pretty good and covered radians, degrees and the unit circle well. You started with the basics and the kept building on that. In the end I felt the video was a little fast but I do like you approach to the task. The video enhanced what I already know about radians, degrees and the unit circle and helped me revise for my exam.”

“Your video is very descriptive and detailed and throughout you sound like you understand what you are talking about, therefore it seems more reliable. The language you used was easy to understand so it was suitable for a math lesson. One think I took away from it was that sin of theta is = cosine of theta. Really good job guys :)”

I call that a major win.  They learned a ton in the last few weeks.  They did most of without me telling them how to do it.  They leaned together and taught each other.  They really enjoyed it.  They supported each other.  In the end, I think they understand trig on a deeper level than they would have if I had just taught them everything.

There’s still so much to learn with trig, but the school year has come to an end.  Hopefully, I’ve inspired enough curiosity that they will want to keep learning more on their own.


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