Math Teacher Ninja - The Unit Circle

I have been trying to teach the unit circle for 5 or 6 years. Each year I think that I've tweaked the process to make it clear(er). Each year, at some point in the process, I have been met with blank stares of "what in the world are you talking about?"

This year I finally had success! In my 9th grade Geometry class of all places.

The Setup:
Earlier in the year I introduced trig through similar triangles. Despite some success, the intro had left many students with murky feelings of "so what" or "seems hard." So a few weeks ago I began planning a second trip through trig-land. I drew heavily from the Exeter Math problem sets and (contrary to Exeter's intentions) put together 3 problem sets each focused loosely around a different trig function.

• Tangent - I loved starting here to make connections to slope/gradient
• Sine
• Cosine

The students were coming around and felt better using and choosing trig ratios to solve problems. I loved the mix of the standard "how far/long is _____" combined with questions seeking deeper conceptual understanding. 

Feeling good about right angle trig I shifted the focus of class towards circles, in particular I wanted/needed to look at the equation for a circle. To start, I gave the students the equation:

In groups they found points that satisfied the equation (using any method they liked) and then used Geogebra via my computer and projector to plot the points. As a class they slowly watched the circle take form as more points were added. We then used (previously learned) geometric construction tools/concepts to find the center and the radius of the circle. I made no mention of how the radius or center were related to the equation, that would come later. 

Wanting to continue the exploration I reached out, once again, to the Exeter problems for inspiration and pulled together a bunch of problems on Circles. We spent 2-3 days working, solving and sharing solutions. Some students grabbed graph paper, some fired up Geogebra and others tackled the algebra head on... 

Today - The Unit Circle:

Then today rolled around. It's the day before spring break, we're all fried - teachers and students. I had my doubts, but I went all in and laid my cards on the table. I started my 65 minute class by putting the following image on the board or at least a hand drawn rendition of this image.

I described the circle as centered on the origin and with radius 1. As a group they all chanted the equation. I described the line segment, that it started at the origin, angled up at 30 degrees then ending when it intersected the circle. I posed the question "what are the coordinates of the point where the circle and line segment touch?" I mentioned that I could think of  3 ways to solve it and that there are probably more... I put them in groups and let them have at it. 10 minutes later 2 ways to solve the question were presented both with correct answers. 

Still, I made no overt connection to trig. At this point I emphasized that the students could use the equations for the line and circle (one group correctly pointed out that the tangent of 30 degrees is the slope of the line) and their GDC to find the intersection point. 

I put up the following spreadsheet and explained that I wanted them to repeat the process with more angles. Each group had a color that corresponded to different angles (see spreadsheet). When a group had an answer they quickly added it in on the spreadsheet. When all the results were up and the resulting patterns discussed and explained I quickly added the sin(x) and cos(x) columns to the spreadsheet. You could taste the learning that happened in those few moments. 

Now for the hard part... 

Graphing. This one took some work to explain. The idea of graphing "x" on the vertical axis made some heads spin. We talked about going around the circle more than once or going backwards and what that meant in terms of the graph... Here the GDC helped out wonderfully.

Class ended with the "non-mathy" students declaring, "I love it when you work hard and then it all makes sense" or "I really like it when I understand things." I felt like a freaking math teacher ninja! 

Reflections:

To be honest I don't know where the stage for this success was first set (what I described took 3 weeks - 8 to 9 hours in class). While having all the mathematical tools needed (circles, constructions and right angle trig) is necessary I don't think it's sufficient. Two more pieces were needed:

1. Students feeling the freedom to tackle a problem with different methods, thus allowing them to see problems in the context that is most natural to them.
2. Students being trained to solve new tough problems not simply repeating steps that the teacher has demonstrated on a whiteboard.