# Day 47: Polar to Unit Circle, Oh My!

I am so thrilled with how the change to using polar coordinates to build the unit circle worked so well.  Unlike in past years, students this year seem to move easily around the circle and can locate both positive and negative angles relatively easily.  Its so fun to see their eyes track around their imaginary circle in their mind to figure out what quadrant they are in.  I opened today’s class with the question, “if we superimpose the rectangular coordinate system over the polar coordinate system, how would you determine the x- and y-coordinates that represent the polar coordinates [1, 30°], [5, -135°] and [-1,315°]?” and then set them free to explore the question.  About 10 minutes later, the groups had answers and then I played “Vanna” as they talked the class through determining the rectangular coordinates.   The obvious next step was to see if they could generalize the conversion.  AND THEY DID.  They even wrestled and then included what to do if the radius was not 1.  It was awesome and awe-inspiring to watch these mathematicians SOAR: seeking to make sense of their observations, organizing their results to look for patterns, analyzing to quantify patterns for a variety of cases, and reason through to the generalization for all cases.  Whoo-hoo!  This was why I teach 🙂  Once we had their conversion rules, they practiced using Quizlet.  How do you help students to understand the relationships for the special angles on the unit circle?