Inquiring into the Unit Circle, Part I

Little cute girl having a good idea on the blackboard - Learning
Photo Credit: Jack Moreh

I’ve really enjoyed teaching this last unit in my Grade 10 math class on trigonometry and the unit circle.  We’re finally moving away from SOHCAHTOA and into the real magic of trig.  That place where the kids finally say “Whoa–this IS cool!”  So many interesting connections there, and the best thing is that the kids really can figure much of this out on their own.

I’ve been teaching flipped-lesson style for about three years now.  By now, we all know how this goes: kids watch videos at home and work on problem sets in class.  It’s supposed to lead to more opportunities for inquiry and application.  So far, I’m struggling to find those opportunities.  Until now.  In fact, with this unit, I’ve unflipped it.

We started with this review of SOHCAHTOA and Pythagorean Theorem, leading into Special Triangles.  (A few kids had never been exposed to trigonometry before, so I had them watch Khan Academy videos on right triangle trig.)  Kids worked together to answer the questions , and though they struggled a bit with the special triangles, a few well-placed questions led them to make the connection between Pythagoras and trig.

The second lesson involved this investigation into the unit circle, in which students had the task of discovering the connections between radians and degrees, special triangles and the coordinates of terminal points, reference angles, and basic trig functions.  Students worked alone on Day 1, but had to come to me after every page to check their progress, as I didn’t want them to move on with incorrect information.  Without giving hints, only clarifying my instructions, I let kids know if what they had figured out was correct or not. While none of my students finished the investigation, every one of them discovered something on their own.

I was able to mark the investigation quickly, because of the way I structured it, so I returned it to the students during the next lesson.  I then grouped the students according to their progress on the investigation.  Those who were at reference angles worked together, while those who were still trying to figure out radians worked together to see if they could further their understanding.  All groups did, and at the end, students presented their understandings to the class to make sure everyone understood.  I was impressed with the kids for their willingness to continue these discussions, for their willingness to dig deep and wonder, and for the a-ha! moments they had.

We followed this up with a long set of problems, spending about three class days on these. Rather than drawn out lessons explaining how to find the cosine of 30 degrees or the tangent of 135, I referred the students to what we learned about these functions during the investigation.  Rather than watching a video about co-terminal angles, I gave students a few definitions.  Rather than long explanations on cotangent, secant, and cosecant, I asked students to google their definitions, and we discovered their relationships to the three main trig functions.  It was inspiring to see them unpack all this complicated information on their own.

The unit is not yet finished.  I did have the kids watch a video this weekend on solving trig equations, but this is the only video they will watch this unit.  We will spend a day solving trig equations and then move on to the graphs of trig functions.  Again, they will discover this on their own by un-wrapping the unit circle.

Their summative assessment of this unit will be to present their learning in the form of a video.  Students will choose from a menu of topics, create video-based lessons, and share them with the class as study tools.  I’m excited to see what the kids come up with and to share how it all went in this space.


4 thoughts on “Inquiring into the Unit Circle, Part I

  1. Wow Valerie, This is the perfect balance between student initiation and teacher encouragement! We so often forget as math teachers that the struggle is where students will learn the most. How do we teach this? How do we foster independent learners? You have taken a great first step towards that end! In my own teaching, I am frustrated with passive learning so my professional goal (we are required to have two) will center on the questioning process. Students questions will become as important as their answers! I have a lot of research to do. But your well structured unit here is helpful. And Unit Circle is a tough concept. Thanks for the inspiration!


    1. Thanks so much for the feedback, John. I’m still learning how to do this without telling the kids what to do and how to do it. It’s a journey for us all. Any shares you have are welcome.


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