On Thursday afternoon we took a break from the lab to participate in a shared activity presented by Paul Myers. We did two mini-lessons, both related to probability and statistics, and both with the aid of a TI-92 calculator hooked up to an overhead projector.
First Paul handed out index cards with a 9 by 7 grid printed on one side. Each of us drew our "favorite triangle," connecting three vertices of the grid. Paul collected the cards, shuffled them, and then redistributed them, face down. We were instructed to poke 50 holes randomly in the back of this second card. The idea was that counting the number of "random" hits that fell inside the interior of the triangle would be a way to estimate the area of the triangle. Clearly, the greater the number of pokes, the better the estimate, but we were all pleasantly surprised with the accuracy achieved with only 50 pokes. Here's the calculation:
[(number of triangle hits) / 50] * [total area of grid] = estimated area of triangle
Cereal Box Problem
The second activity Paul presented is known as the Cereal Box Problem. A cereal company is offering six different prizes in its cereal boxes. How many boxes would one expect to have to buy, on average, in order to collect all six prizes?
Rather than giving us a lot of time to compute probabilities and expected values, Paul asked us to make a quick guess and then construct a histogram of all our guesses on the board. Our guesses were substantially lower than those Paul usually finds in his classroom, due probably to our relative experience in this type of problem. The mean of our guesses was somewhere around 20.
We then discussed how students might "do" this problem, to test out their hypotheses. Short of (or in fact better than - see discussion below) going to the store and buying a lot of cereal, the best choice is to use dice. We all conducted two trials, recording how many roles it took to get all 6 numbers on the die. With these results we built another histogram on the board. Now the mean was closer to 15.
That's pretty close to the theoretical result. In order to see this, you need to understand the relationship between the probability of getting a certain prize and the expected number of trips it takes to do so. Think of the dice. Since the chance of getting an odd number is 1/2 on each roll, you expect to have to roll an average of twice before seeing an odd. In general, the expected value is the reciprocal of the probability. On the first trip to the store, you're happy with any of the six prizes since the probability of getting a desired prize then is 6/6. The expected value is also 6/6, or 1. On the second day, you need 1 of 5 prizes. The probability of getting a desired prize is 5/6, and the expected value is 6/5.
The total expected number of trips, then, is the sum: 6/6 + 6/5 + 6/4 + 6/3 + 6/2 + 6/1. This works out to be about 14.7, or 15.
The discussion of this interesting and well-received problem was quite lively. It focused on assessing the pedagogical value of the exercise, and on potential extensions in the classroom, and quickly turned to the issue of how "real life" this problem is. This is clearly not a new question, and mention was made of the ongoing and recurring thread on the NCTM-L mailing list. Sarah Seastone pointed out that reality for a student did not often coincide with the assumptions of the cereal box problem, and she asked Paul what further applications he could suggest to make it even more realistic. We all agreed that most problems in classrooms, and in mathematics in general, are somewhat contrived, but that they are not without value because of this. At the least, the setting and props for the problem are familiar to us.
Steve spoke of the role a problem like this one can have in helping to develop people's mathematical intuition. Probability and statistics, he said, are areas of math in which students' intuition is particularly lacking. By working out a problem like this one, people can hone their estimation skills by orders of magnitude, and this is in itself a help in the real world. The rest of the group agreed, and added that discussing the discrepancy between the model and reality would be a good exercise for students. In particular, Ruth Carver suggested an interdisciplinary unit in which the students could write letters to the cereal company explaining their calculations and experimental results.