A New Look at the Unit Circle’s First Quadrant

I saw something a few weeks back at my AP Summer Institute at Arkansas State that I’d like to share.

The Unit Circle has never been a real stumper for me, personally, and, thus far, I’ve done a fairly decent job of teaching it in my Trig classes.  That said, I continually enjoy staring at a completed unit circle and searching for new patterns.  I’ve never seen this one before.

What a unique, and even better, simple way to see the first quadrant sines and cosines!  The denominators are all equivalent (2), and the numerators are just the succession of integers of 0 to 4 or 4 to 0 under a radical.  I think it’s wonderful.

I’m wrapping up my Precalc review mini-unit in CalcBC and today I started class my splitting the students into two groups and giving each of them a completely blank unit circle (by the way, embeddedmath.com has a great downloadable pdf of a completed and blank unit circle).  The teams were to compete against each other to fill it out quickly and accurately.  The exercise took about 20 minutes (too long, imo), but I was shocked at just how closely the two teams finished (within 30 seconds of one another).

The good news was that both teams each turned in perfectly completed unit circles.  That’s a victory, I believe.  For these seniors, it has probably been a year and a half since they’ve been asked to do that.

What was great was all the group discussion that was occurring.  Like we all know, the unit circle is full of patterns, and everybody has their own.  I got to hear students explain their own methods of completing the circle to each other.  That’s another win.  I was just a passive observer – unneeded in any form.

We debriefed the activity once both groups were finished, and we discussed some of the patterns we knew, some we learned, and those which made for quick and efficient unit-circle-filling-out.


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