(Alternate Title: What is a Radian?)
What I wanted them to grok:
Discussed before they started: what is pi? Could you cut a string exactly pi inches long? Most of them said "no". They thought it was theoretically possible, but couldn't be measured precisely enough. I promised them that they'd walk out of class with a string of length pi in their pocket.
Materials: Circular object (collected from around my room: a CD, a water bottle, paper plates, cardboard mailing tube, etc etc)At the appropriate time we discussed this, and came around to "pi radii of the circle". Many of them still didn't like this and insisted it "wasn't really pi", so we had to have the discussion about operating in the perfectly abstract realm...some of them noticed that if their circle had a radius of exactly one centimeter, then the string would be pi centimeters...that seemed to help. Then I had them convert a bunch of degree measures to radians and back, and come up with formulas.
Strip of Paper (I purchased adding machine paper at big box office supply store)
1. Measure the diameter of
your circular object. Calculate the radius to the nearest 10th of a centimeter and record: r = _______ cm
2. Cut a strip of paper at least 8 times longer than your radius, and tape it to a flat surface.
3. With a marker, make a mark on the edge of your circular object. Make a corresponding mark near the end of your strip of paper. Line up the two marks, then roll your circular object one complete revolution. Make a second mark on the paper at exactly one complete revolution. What does the distance between the two
marks on the paper represent? Verify this value by using the radius to calculate
4. Cut a length of string the same length as the distance between the marks. Set it aside.5. Using one of the marks on your strip of paper as "zero", use your ruler and a pencil to mark off intervals along the paper of the same length as your radius. About how many radii before you get to the one-revolution mark? This value should be the same for all of the various circles. Why?
6. Take your length of string, fold it in half, and cut it exactly in half. This string is precisely pi long. You and your partner can walk around all day with pi in your pocket! The question is….
The scary thing was, most of them didn't recognize it from last year until they started writing down a formula. But I'm hoping that more of them actually know what the heck a radian is, and this makes the unit circle a slightly easier mountain to climb. We shall see.