The Best Writing on Mathematics 2014 by Pitici Mircea;

Author:Pitici, Mircea;
Language: eng
Format: epub
Publisher: Princeton University Press
Published: 2014-07-14T16:00:00+00:00

FIGURE 1. Comparing volumes.

FIGURE 2. The surprising result.

Statistics: In a unit on confidence intervals for proportions, I use Goldfish® crackers to help students understand the capture-recapture process for estimating the size (N) of a population. Using a large bowl as a “lake,” we stock it with several packages of cheddar Goldfish®. Then we take an initial sample from the lake (the capture), find the number in the sample (M), “mark” them by replacing each (yellow) cheddar fish in the sample with a (brown) pretzel fish, and return the marked fish to the lake. After mixing thoroughly, we take a second sample of n fish (the recapture) and determine the number of marked fish (m) in the new sample. Using the proportion of marked fish in the sample, we get a point estimate for the proportion of marked fish in the lake from M/N = m/n and use that to solve for N, the total population. We can also use our point estimate to construct a confidence interval for the proportion and find a range of values for the true population. For extensions of this activity and discussion questions, see [10, pp. 126–130].

Multivariable calculus: An article [5] about measuring the volume of an angel food cake pan gave me the idea for a week-long writing project in Calculus 3. After we have discussed centroids and volumes, I bring a pound cake and a tube pan (Figure 3) to class and ask the students how to determine the volume of the cake. Students can measure the pan, but its middle tube is slanted, and the pan’s two-piece construction won’t hold water for a direct measurement of volume. In addition, though the cake’s straight sides conform to the pan’s dimensions, the top of the cake is uneven because of rising. I then give each student a slice of cake and some grid paper. Their task is to trace the slice (Figure 4), estimate its area and centroid, and apply the first theorem of Pappus [3, p. 1031] to approximate the volume (cross-sectional area times the circumference of the centroid’s path). They then document the methods they used in a short paper. Of course, they can eat the cake, too. (Another version of this problem is described in [4, p. 387]; it uses the shell method with heights along the parabolic cross section of a Bundt pan to find volume.)

Bridge course: The principle of complete induction is one of the more difficult concepts for students in a proof-writing course. There aren’t many easy examples that convey the necessity for the stronger induction hypothesis. I’ve found a tasty way, however, to explore the concept with Hershey® chocolate bars. The question is, “How many breaks (along grid lines) does it take to separate all the sections of a rectangular candy bar?” First, I distribute candy bars in different sizes and arrangements (e.g., 1 × 4, 3 × 4, and 3 × 6) and ask them to make a conjecture about the number of breaks required to separate n sections of a bar.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.