## Some Strange Circles

I was trying to construct a simple, introductory intersection problem for the first day of Calculus class, so I started with a well-behaved circle:

This is a circle centered at (3,2) with radius 5. I picked two points on the circle, (0,6) and (7,-1), found the equation of the line between them, and put together my system of equations:

So I had successfully reverse-engineered my circle-and-line intersection problem with two nice solutions: (0,6) and (7,-1).

Unfortunately, I made a typo on the handout. At the end of the left side of the circle equation I wrote ” + 12″ instead of ” – 12″.

So all my work was for naught. Or so I thought. Turns out, at least two amazing things happen:

First, the mistake-circle still ends up having a nice radius, namely 1. What’s even more amazing is that the mistake-circle ends up having** two nice intersections with the given line,** (3,3) and (4,2)!

I wish my intentional work always turned out as well as this mistake!

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Ha, ha, ha! Sometimes the most outstanding progresses in math, science and art are discovered accidentally and by people not even in these fields. Still, it always amazes when it happens to oneself. This fits in nicely with Brown’s “what if” and “what if not” challenges. (I love these.)

If only our students could discover such magic in their mistakes. They would be more inclined to play with and learn more math and Math, and their disposition toward both would be more favorable if not anticipatory.

I’ve made plenty of fruitful mathematical mistakes, and I encourage students to do the same. A mistake is like a new problem to solve, one that you created–and the process of understanding

whyit is considered amistakeis, itself, a valuable mathematical and creative activity.