Introducing the Radian Measure

As I've posted before, I'm teaching grade 12 Functions and am working on making better use of a great outline put together by the OAME. I think this is good practice for me as a new teacher because it should help me be more efficient in my lesson planning, ensure that I hit all the curriculum requirements (because this outline was built based on them), and keep me from leaning on the textbook for examples and questions. One of the skills I want to develop in my students is to help them begin to use the textbook as a reference. It will be a much better supplement if the examples and descriptions in it are not word for word what the student has already seen in class and has in her notes.

I'm feeling good about this because I think I have a BIG improvement over the lesson for introducing radians. The lesson outline really just has students think of radians as an alternate unit for angles with an easy to develop pattern that leads into an easy to use formula for converting. Just like it's better to actually spend some time discussing the differences, pros and cons, etc between the metric system and the imperial system I think it's better to start students working with radians before they even know it and have them be useful because they are connected to other areas of mathematics, then get them to find the pattern for the easy conversion formula. Hopefully I'll also remember to talk to them about how any conversion formula relies on multiplying by 1 and "cancelling" units; in this case

I'm going to use the formula for circumference and unit circles. Oh, yeah, I'm planning on writing "Matt loves circles" on the board as my hook tomorrow. If I remember I'll be bringing my pi coffee mug too. I want to lighten things a bit right after the midterm and while I also set up for student conferences where I will make sure each and every student knows how they are doing at this point and I can get some 1-1 feedback and help them set realistic goals for themselves (unfortunately some came to me pretty unprepared and the goal might end up being "learn as much as I can so I get a better grade when I take this class again"). I'm going to ask them to spend some time trying to figure out why there are 360 degrees in a circle. I'm posting that question to twitter now too to see what other responses I can get. 

By using the unit circle, the circumference is 2pi and it shouldn't be a big jump to get students to tell me how far they would have to walk to get around half the circle and a quarter of the circle and 7/12ths of the circle. These being the arclengths associated with that ratio of the unit circle. I did a really quick modification to one of the included worksheets and am going to ask my students to tell me the arclength of the highlighted portion of the unit circle, as well as the angle in degrees between the terminal arm and the x-axis. (Next they'll get the formula). I tried using a highlighter at first but it didn't show up in the photocopy, so I reached for my favourite green sharpie and voila.

On a black and white photocopy the whole circle is still visible but the green is really obviously highlighted. If I keep teaching this course I might remake the page, doing something "nicer" on the computer for the same effect, but I also have some serious improvements to past portions of the course that I would want to make first. The instructions at the top of the page are not 100% accurate to what we're doing with the worksheet and in the future I will block that out to be more clear. I'll have them work on the new pairs I'm setting up for tomorrow - I've left a couple of students who need more attention at the back corner of the class for far too long already so they're going to move up and towards the middle starting tomorrow!

Today was the midterm, so tomorrow my students begin our unit on Trig Functions. It should last about 2 weeks at our accelerated pace (10 hours per week of class) which works out to about 2 of the outlined lessons per day, but I plan on putting some of them together and getting into a worked example of where a trig graph might come up (I'm thinking average monthly temperature for different cities with universities in Ontario and/or Canada where my students might want to go).

Almost forgot; the unit, lessons, and worksheets are available here.