A circle is a set of points in a plane that are equidistant from a fixed point called the center. A radius of a circle is a line segment drawn from the center of the circle to any point on the circle. A chord of a circle is a line segment whose endpoints are on the circle. A diameter of a circle is a chord that contains the center of the circle. So far, there is no problem in teaching definitions involving circles. The students know what a circle looks like; they started seeing them on Sesame Street.
Tangents and secants are a little trickier. A tangent to a circle is a line in the plane of the circle which intersects the circle at one and only one point. The point at which the tangent intersects the circle is called the “point of tangency.” A few years ago, one of the rubber companies had a commercial which contained the lyrics, “where the rubber meets the road.” I use this to teach that a tangent only touches the circle at one point. I pick a light student and say how if he got on a ten speed the tire would only touch the ground at one point, especially if we filled the tire to a high pressure. But, if I got on the bike that the tire would flatten out and touch in many places. That is, if it didn’t burst.
A secant is a line which intersects a circle in two points. A simple enough definition, but if you start drawing the secant outside the circle and stop drawing the secant inside the circle and ask the students, “What kind of line is it?” They will respond, “A tangent.” I then point out that a line has no endpoints, so that if we continued the secant that it would come out the other side of the circle. And besides, where does the rubber meet the road?
The circumference of a circle is the length of the circle expressed in linear units. I mention how “circum” is Latin for “around”, so circumference means the distance around a circle. So far, there are no problems. The textbook then moves to the formula for circumference. I mention “pi” and I may as well be speaking Greek. To quote Piaget, we have gone past concrete knowledge into abstractions.
Pi is the ratio of the circumference of a circle to the length of its diameter no matter what the size of the circle. Bring in the bicycle and other wheels and have the students measure them. Then compare the ratios. The first time I did this I did not have a measuring tape so I used lengths of yarn. Guess what? Yarn stretches. Our results were a little off. However, the students have now physically met pi. When they encounter pi in the formula for circumference, they will not be overwhelmed by the symbol.
My students have found problems like, “How far will a wheel with a five foot diameter travel in two revolutions?” very difficult. They seem to have difficulty translating spinning around to a linear distance. Again, before trying to solve the problem on paper, let’s use the bicycle. Wrap a piece of tape around the tire to mark a starting position. You could also use the valve stem as a position marker. Start with the tape on the ground and roll the tire one complete revolution. Now have someone measure the circumference. The two numbers should be the same. Once the student sees the relationship between the circumference and one revolution, you are now ready to roll the tire more than one revolution. Have the student make a chart with multiples of the circumference and compare the computations with the measured distances.