Exponential Decay and Fraction Folding Strategies

Last week with One Grain of Rice, we learned about the extraordinary power of doubling starting with one. This one grain of rice increased to over one billion after 30 doubles and to over 18 quintillion after 64 doubles. This is called exponential growth.

This week, we looked at the power of exponential decay which is starting with large numbers and cutting them in half repeatedly. The children made conjectures (guesses) on how many times they could half a piece of standard 8.5 x 11 inch paper. Most of them guessed that they could do this over 15-20 times. They found that they could only do this 6 times. As they folded we counted on our fingers the fractions created by each fold. The first fold was 1/2, second fold 1/4, third fold 1/8, fourth fold 1/16, fifth fold 1/32, sixth fold 1/64. 

Many of the children saw the connection to last week's class where we counted powers of two with our fingers: 2,4,8, 16, 32, 64, 128, 256, 512, and 1024. The only difference is that the doubles created integers whereas the halves created unit fractions with a one on the top (the numerator) and the bottom (denominator) with the cooresponding power of two.

When they opened their 6 folds, they traced their pencil along the folds and saw evidence that they had split one whole (1/1) piece of paper into 64 equal sections. I showed them strategies for folding this in half a seventh time and even an eighth time.

We looked at other exponential decay examples. I asked them to make conjectures as to how many times I could walk towards the wall half way. Each time I would walk half way to the wall. Some students looking at my distance of 10-50 feet, made a leap to say 20-30 times. But many students recognized that the same rules of exponential decay would apply and guessed 6-8 times. Sure enough, I was within one inch of the wall after 7 half moves to the wall. The best moments of the week were when students would say that I would never actually reach the wall. This is a very high level abstract concept.

The most fun we had was taking a roll of toilet paper and stretching it all the way across the room and folding it in half repeatedly. Again, the maximum folds were 8.

The logical next step was to have the children look at the equal sections created by the folding. We talked about the numerator indicating the number of pieces (our focus was on unit fractions that always have a one in the numerator). The denominator represents the number of pieces that make up the whole. The connection we made is that the larger the denominator, the smaller the number. In fact, when we did our cheer at the end of class, we counted by powers of 1/2 and some of the children saw that we were approaching zero.

I gave the children the attached pdf with a list of fractions. Their goal was to fold paper using our 1/2, 1/3, 1/5, and 1/7 strategies and tracing along the folds and finally writing the fractions in each rectangle.  For example, if their goal was to create 1/16, they should be able to articulate that they need to fold the paper in half four times. If their goal is 1/6, they would day that they would first fold the paper in 1/3s and then 1/2, or first 1/2 and then 1/3. They began to see that this is really multiplication of prime factors. For example, if the goal is 1/36, the would first see that they need to multiply 2x2x3x3 so fold in half twice and in thirds twice.

They should spend their week, folding paper to create as many of these fractions as they can. I gave them at least 10 pieces of paper and when they run out they should take paper out of notebooks or printers.