Today a student came in at the end of the day and said, "Mr. Meyer...wanna see why pi equals 4?" I was intrigued. He started me off with this image:
and then said, "The perimeter of that square is 4. But what if we pulled the corners in so that they touch the circle? We wouldn't change the perimeter of the square. Like this..."
"and we could just keep doing that..."
"and keep doing that..."
"until pretty soon it would look exactly like a circle even though its a bunch of really small steps."
"So wouldn't pi be equal to 4?"
"Oh yeah, and ignore the snail I drew."