# What's in a Circle?

Recently I had the good fortune to both attend #TMC13 and share my very own personal favorite lesson during, appropriately, one of the "My Favorite" sessions.  And now I'll share it with you.

I was getting ready to start a unit on conic sections with my Advanced Algebra kids.  I was planning to start, as I assume many of you do, with circles.  I imagined it going something like this, Lusto's 10 Steps to Circle Mastery:

1. Get somebody, anybody in the room, to spout the definition of a circle , hopefully including a phrase such as, "the set of all points a fixed distance from a given point..."
3. The distance formula.
4. Why the distance formula is fugly.  Remember the Pythagorean Theorem?  Of course you do.  It's awesome.
5. In the plane: $a^2 + b^2 = c^2$
6. In the Cartesian plane, centered at the origin: $x^2 + y^2 = d^2$
7. This distance, d, has a special name in circles, right?  Right: $x^2 + y^2 = r^2$
8. Appeal to function families and translations by <hk>.
9. $(x-h)^2 + (y-k)^2 = r^2$
10. Boom.

Based on my hopes for Step 1, and based on my need for like five uninterrupted minutes to take attendance, find my coffee mug, &c., I hastily scribbled an extremely lazy and unimaginative warm-up discussion question on the whiteboard.  Four simple words that  led to some surprising and amazing mathematical conversation.

What is a circle?

That's it.  My favorite lesson.  The whole thing.  And here's how it went.

I was walking around looking at/listening to all the different definitions the groups had come up with.  And they were nuts.  There were dubious claims about unquantifiable symmetry, sketchy sketches with line segments of indeterminate provenance, rampant appeals to a mysterious property known as roundness.  Most of the arguments were logically circular but, alas, mathematically not.  The word curvy appeared more than once.  It was a glorious disaster of handwaving and frustration.  I knew, deep in my reptilian brain, that this is what's known in the business as a "teachable moment."

Nailed it.

At this point I was basically just walking around being a jerk.  I was drawing all kinds of crazy figures that minimally conformed to what they were telling me a circle was, and getting lots of laughs in the process.  And then I had the thought, even deeper in my reptilian brain, that transformed the whole experience from an interesting activity into a bonafide lessonWhy the hell am I the one doing this?

So here's what the lesson eventually became, Lusto's 6 Instructions to Humans on the Brink of Amazing Mathematical Discussion: