As a warm up activity / pre-activity, each table was given several sheets of square paper and instructions for a simple origami dog and instructions for an origami crane. The students worked on folding these until we had enough students to start.
Since it was close to the beginning of the year we spent some time talking about the four rules of math circles:
1. Have Fun
2. Ask Questions
3. Make Mistakes
4. Help others have fun.
Each student was given a fresh piece of paper and their adult helper was given a chart to fill out. The students were instructed to fold the paper in half, then unfold the paper and count the number of rectangles. The adult helper recorded the number of rectangles in a chart. The students then refolded the paper and then folded it in half again. Some students quickly saw a pattern, number of rectangles to number of folds, but they were encouraged to continue folding and counting to get some information for the next part of the activity.
After most of the students were done folding we talked about the pattern that we saw. We then filled out a table on the board up to 10 folds. On the board we graphed the number of rectangles vs. the number of folds for up to 5 folds. I told the students that mathematicians call this an exponential curve. The students could see how when you double things they grow very quickly.
How quickly do things grow? We observed that every time the paper was folded it got thicker. The students were then asked how many times they thought they would have to fold the paper until it was as tall as the Y on the mountain (you could substitute the Statue of Liberty since it is essentially the same height). We then did a doubling exercise. I told them a piece of paper was .0001 meters thick and the Y was 94 meters tall. We then made a chart on the board with how thick the paper was after each fold. After 5 folds the paper is .0032 meters thick, After 10 folds the paper is .1024 meters thick or about four inches. After 15 folds the paper is 3.277 meters thick or about as tall as the ceiling. After 20 folds it is over 104 meters tall, or higher than the "Y" on the mountain. (We went through the first twenty folds on the board). Since the students were restless at this point I just mentioned that if the paper was folded 42 times it would reach to the moon.
We then talked about how many times they actually were able to fold the paper. The responses were 5-7. We talked about how they could fold the paper more times. We came up with two ideas, bigger paper and thinner paper. The students were given either very thin tracing paper or very large sheets of butcher paper and given the opportunity to try again. They got the same answers as before. We then tried to identify paper that was both thin and large... the answer toilet paper.
The students then broke into smaller groups and were given a roll of toilet paper and asked to fold it as many times as they could. This took a while. Most groups folded the paper about 10-11 times. We talked about how many rectangles would be on a single piece of toilet paper.
Then it was time for cookies and to depart until the next week.