Exponents are a fascinating topic to me. They rank up there with probability in the category of "unexpected answers that make no sense." With exponents students are amazed at how big numbers get and how fast they get there. I have a number of activities I do to hook them in, but the first one I do involves some rice...
I started with an ancient parable of a farmer in Ancient China. The parable has many different origins, most commonly in India, but I chose Ancient China because the students just finished a unit on this region and I felt it would be a great way to review the vocabulary and geography of the area.
The story goes like this...
Once upon a time a farmer found a way to vastly increase the amount of rice produced on his farm. He shared this information with his Noble who in turn told the Emperor. The Emperor was very happy, for growing rice was one of the ways to get wealthy in ancient times. It wasn't like they could just go to Kroger and buy rice, you know.
The Emperor insisted on an audience from the farmer, who showed the next day. The farmer, being only a peasant, approached his Emperor with his head bowed the entire time. The Emperor commended his subject and said as a reward he would grant him a reward of a pound of rice a day, every day, for a year. This was an astronomical amount of food.
We then measured what a pound of rice looked like. I always keep tupperwares of rice in the classroom - they come in handy for many different activities. I also got out our science beakers and digital scale. The pound of rice we measured had just more than 500 ml of volume. We also estimated there were between 7,000 and 10,000 grains of rice in that pound. Looking at a serving size, we learned that this would be between 8 and 10 servings of rice by today's standards.
Doing some more math revealed that at the end of 30 days the farmer would have about 300,000 grains of rice.
... Then the story continued...
Being a humble peasant, the farmer refused the reward. Instead, he said, he wished for merely one grain of rice, doubled each day for the 30 day period.
The Emperor laughed, but was impressed by his subject's modesty. He granted his wish.
Students then discussed if this was a good plan. Many of them had heard this story before, and so knew that the farmer made a good choice, but the challenge was to estimate how many grains of rice he would end up with after 30 days. Estimates ranged from "a few hundred" (from students that had never heard the story) to "about a million."
So to emphasize the deal I got out the rice again and said, "OK, day 1... ONE grain of rice" and I put the grain of rice on a random student's desk. "Whew... ok... day TWOOOO.... two grains of rice..." (another student) "day three... four grains! that looks SO FILLING!" (another student.) I do this through the first 5 days getting up to 16 grains.
Students were then challenged with proving which deal was better. To do this we decided to use chess boards and post its to help organize our thinking and planning. Students were put in groups of two or three students to record the results.
The numbers started small...
... but they started growing...
and kept growing...
Students were shocked at how large the final answer was (over 500 million grains of rice on day 30 alone!) At the end we did some reflection on exponents. Here are some responses from the closing:
- The numbers start off really small, but got big REALLY fast after like day 20.
- The emperor must have been really upset!
- It didn't seem like a good deal, but on day 19 he already had more than all 30 days from the other deal.
- That was WAY more than I thought it would be.
- How many pounds of rice is that???
The last part of the assignment was for them to write how the story ends. What happened to the farmer? Ideally they would use their knowledge of social structures to answer the question (would the Emperor kill the farmer? Honor the agreement? something else?)
It was a very fun lesson and really emphasized the power of exponents!