In Algebra II class in high school, I confused and confounded the class with a 2==1 problem (like those posted by jordanh or gmpassos). My class mates were saying stuff like "cool" and "wow". My teacher looked at for 2 or 3 minutes. Then a smile crossed her face and she said something like "Nice try! You had me going for a minute there. But you can't divide by zero."

Of course, someone asked why you can't divide by zero. After a brief explaination that it can't be done she moved on with the lesson plan. So like any good geek, I stayed after class to discuss my ideas with her.

I was not nearly so precise as you have explained it. I said something like "Any number multiplied by zero results in zero. So any number divided by zero results in an answer that contains the infinite series." I saw a light go on above her head; a sparkle appeared in her eye. It was then that I realized that I had taught my teacher something. That is one of the best feelings in the world!