Thursday, August 7, 2014

Section 1-4: Points in Networks (Day 1)

Today is the first day of school -- at least for this blog. I calculated this date as follows: a school year is 180 days, so a semester is 90 days or 18 weeks. I wish to have 90 days before Christmas, which falls on a Thursday this year, so the last day is Friday, December 19. Since many schools now take the entire week of Thanksgiving off, I count back 19 weeks from this date. Finally, since most schools have Labor Day and Veterans Day as holidays, I count back two more days, to include Thursday and Friday of this week.

Also, I begin with Section 1-4 from the U of Chicago Text. Notice that the first five sections of the text consist mainly of descriptions of a point -- these are omitted from most texts and could be considered a waste of time when there are 15 chapters to cover. But the first day or week of school traditionally consists of an opening activity, and it might be possible to make a suitable activity out of these sections. Now some sections, like Section 1-3 on ordered pairs, are certainly important. But many students find graphing to be a huge turn-off -- certainly not what most teachers want out of an opening activity.

But Section 1-4 -- now there's something. The Königsberg Bridge Problem is a famous math problem from nearly 300 years ago. Fawn Nguyen, a well-known math blogger and fellow Southern Californian -- she lives in Ventura County -- used this as an activity in her geometry class:

Notice that Nguyen taught this lesson after her class had "just finished with the State tests" and that this was her "geometry lesson for the entire week!!" Here, I suggest this as an opening activity, but it can still span the entire first week of school. On this blog, though, the first week consists of only Thursday and Friday.

Nguyen was disappointed when her textbook, published by McDougal Littell, simply gives away the answer to the problem. Fortunately, the U of Chicago text doesn't do this. It instead gives Example 1, with network diagrams, and Example 2, asking which of the networks in Example 1 are traversable.

As we all know, the Königsberg Bridge Problem is impossible to solve -- it has no solution. But I don't want to start the class with a problem that the students can't solve -- they're already frustrated enough with problems that do have solutions when they just can't find them. Because of this, I decided to create the following images. The example is actually Example 1, Network III from the text. Problem 1 is Network I from the same example, and Problem 2 is actually Exercise 12 -- this one's a bit more challenging, but the network is traversable. Problem 3 is actually Königsberg.

The second image below is actually the same as the first, but with the maps drawn as networks. This is designed to be given on the second day. Just as Nguyen suggested in her own class, here is where the teacher begins discussing "vertices" (called nodes in the U of Chicago text), "edges" (arcs), and "valence" (even and odd nodes).

My suggestion is that the teacher have the student count the arcs at each vertex for the pentagon (all are two) and the pentagram (all are four). Both of these are easily traversable. The other diagrams all have some nodes with three arcs, and these are what make the networks harder to transverse. Eventually, the students will see that a network can't have too many "threes." Then the students can try drawing some of their own networks -- possibly with some "ones" and "fives" as well -- to find out that it's the odd nodes and how many of these there are that make the difference.

The whole point of this lesson is to point out that students should look for patterns, and that sometimes it's just as important to know why something is impossible as it is to know why something is possible.

Let me complete this with a note on pronunciation. The U of Chicago text points out that the name Euler ends up sounding like "Oiler." But how does one go about pronouncing the name Königsberg? I once read that the o-umlaut ends up sounding like "uh," almost like "ur." A Google search reveals a ten-second video in which this name is pronounced:

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