Mathematics

Section 2.1: The Pigeonhole Principle as a tool for estimating using numerical reasoning is a useful skill in other chapters as well as this one. They also need to be reminded of how to use scientific notation to ease their estimations of very large and very small quantities. Many good homework exercises in this section, including (as an in-class, small group exercise) trying to estimate the largest number of hairs possible on a human being, as discussed in the text, but also I. 1-3, 6, 7.

As a bridge to Section 2.5, I suggest going over III, 4 and 5. This is also the point at which to familiarize them with prime numbers.

To be able to grasp the idea of different kinds of infinities, our students need to really understand the idea of using 1-to-1 correspondences between sets in order to talk about their different or similar cardinalities. Students will use the terms "cardinality" and

"1-to-1 correspondence" interchangeably, so they really need to be drilled on the distinctions.

Exercises I 3,4,9 are good for starters. Exercises II 1,2 make a nice reconnection with Chapter 2, as well as connect the cardinality of a finite set with the old notion of "the number of things in it." Many in-class examples of different pairs of sets seems to help, with and without the same cardinalities, with and without being infinite. Getting them to name specifically a 1-to-1 correspondence between two sets is the tough part; it will be attacked again in Sec. 3.2. Emphasize that even though one correspondence doesn’t work, it doesn’t mean there isn’t another.

Lots of good, easy examples of equinumerous sets and their 1-to-1 correspondences should be emphasized: I’d avoid the Ping-Pong Ball Conundrum until they seem ready for it. Make sure the equivalent cardinalities of the natural numbers, the integers, and the rationals is well-established: this sets them up nicely for the surprise about the reals in Sec. 3.3.

Good easy exercises are I2,3,4,6,8,9. Challenging and yet leading to fun in-class debate are the Hotel Cardinality exercises, I, 11-13. Problem I 14 is a beauty because it makes them review their estimating skills and use of scientific notation. They need some help getting started, though. I 17 makes a good point, but I’ve never got around to doing it in-class, though I think I should.

Here is probably the pinnacle of the course. The way Cantor’s Diagonalization Argument has already been introduced as the game Dodge Ball really works with our students. I think it is a wonderful achievement that liberal arts students in a terminal, lower-division math course can understand how a different kind of infinity was proved to exist.

Caution 1:

Caution 2:

A good journal assignment (and something that should be tested on a quiz) is that they see the connection between Dodge Ball and Cantor’s arguments. Assigning I 1 and 2 should help emphasize this. Exercise II 5 would also be a good type of journal, then quiz question.

The Pythagorean Theorem, proved, for once in their lives! That they all can do it

themselves using the manipulatives is a boon provided by this text. Now, of course, can you get them to connect this thing with the way they used the Pythagorean Theorem in all those countless Math 65 type problems! Assign I 3,5,9, 10 to get that point across. I keep assigning the Pythagorean Pizzas problem, II 4, as well, but they needs a lots of help in-class getting started.

This section also has some good points to make about triangles in general. Don’t pass up the opportunity to get them to prove to themselves that the sum of the angles of any old triangle (in Euclidean geometry, of course, you might point out) is 180 degrees by doing Problem I 1. I do it as in-class exercise, bringing in lots of paper towels from the bathroom down the hall, having them first make a bunch of arbitrary triangles, then exchanging them, then tearing off the corners as instructed in the exercise. Some students really liked seeing this idea for the first time!

The other good point is that not any three lengths can make up a triangle (the triangle inequality, but we don’t need to say that… or do we?)

Sec II 1.

Pull out your 3-D glasses and see the Phatonic Solids. Have them

1) Build them, using the kits provided

2) Fill out the chart on them, p.275

Work out what the dual of each solid would be (keep their noses out of the book and on just the definition of how to construct a dual).

On another day, probably with a handout to guide the process, have them work through the proof that there are only 5 Platonic Solids, using several constraints including the Euler Characteristic.

Assign I, 1,6,7

This is another good time to guide them, using a handout, say, through the process of building up knowledge about the unfamiliar by thoroughly understanding the familiar. Stepped exercises leading from 1 to 2, then 2 to 3 dimensions will help get them to make the step into 4.

Assign I 1,2,3 in-class and I 4-5, 6-7 otherwise. II 1, if you’re feeling adventurous…

For good in-class, small-group fun, get sets of 30 NEW pennies (otherwise, the edges aren’t beveled anymore), one for each group of 4 or 5 students and have them replicate the experiments described in the text. First put all the pennies on edge (we had to do it on the floor) and get them to fall down (the student enjoyed stomping up and down nearby to get this to happen) and collect the data. Have them record their data AND put it up on the board, so that all the small groups could compare numbers (and so that folks fudging the data got the idea that they couldn’t predict what was going to happen). Then get them to do the same thing with spinning pennies. I had introduced relative frequencies to them already for the purposes of this experiment. Sec 7.2 will reemphasize the difference between probability and relative frequency.

(sec. 7.1, continued)

Assign as journal work I, 1,2,3 so they will repeat the penny experiments at home.

Another great in-class exercise is to mimic the "Let’s Make a Deal" problem using 1 face card and 2 non-face cards, a la Exercise I, 4. I have them work in groups of two for this, each student taking a turn at being the dealer or the contestant. (By the way, for students who refuse to believe the probabilities are actually 1/3 and 2/3 for stick or switch strategies, I find that proposing that there were 100 doors originally to choose from, that after you pick one, Monty Hall opens up 98 other doors, carefully avoiding one particular door all that time, leads students to agree there would be a higher probability that the one remaining closed door is the right one.)