Thursday, October 8, 2015

Review for Chapter 4 Test (Day 31)

Part III of Mandelbrot's The Fractal Geometry of Nature is called "Galaxies and Eddies," and consists of Chapters 9 through 11. In this chapter, Mandelbrot argues that many problems that appear in science can be solved by using fractal geometry.

For example, in Chapter 9, Mandelbrot discusses the distribution of stars in the universe. He writes about Olbers paradox, which poses the question, if the universe is uniformly full of stars -- that is, if there are stars everywhere -- then why is the sky dark at night? Mandelbrot -- who prefers to call this the "Blazing Sky Effect" rather than Olbers paradox -- proposes that it could be because the universe is not uniformly full of stars. Instead, the universe could be a fractal with dimension less than 3, and so the distribution of stars follows a fractal pattern.

In the ensuing chapters, Mandelbrot mentions other scientific ideas -- turbulence and fluid motion -- that could also be modeled using fractals. So we see that these fractals are hardly some abstract mathematical idea that has no practical significance, but instead have some scientific value.

Yesterday was the last lesson of Unit 2 on Reflections. Therefore, today I am posting a review for the Unit 2 Test.

Last year at this point I posted a Chapter 4 Quiz instead. Recall that so far, I'm reversing what I'm labeling a "quiz" and what I'm labeling a "test." And so I'm posting last year's quiz as a test. As this is a short quiz, it may be desirable to add another page of questions to it. Although in theory we could add some questions from the earlier part of Unit 2 (i.e., Chapters 2 and 3), I'd recommend adding some coordinate plane questions, especially since there are two extra coordinate plane lessons this year to consider as compared to last year. (Don't forget to add them to both the review and the test!)

It is also possible to add some of the questions from last year's "Chapter 3 Test" instead, since I'm changing that into a quiz next week. Of course, combining the quiz and part of the test may make for some weird question numbering. (Today I subbed in an English class, where the students had to work on problems numbered 1-14, then 21-40, and then 1-20 again!)

This is what I wrote last year about the upcoming test. Actually, I ended up spending more time discussing the school calendar and the significance of the mid-trimester point that I alluded to earlier:

This completes the first half of a 60-day trimester, for schools that divide the year into trimesters. Unlike quaver -- the name that I gave to half of a quarter and appears to be my own original name for this period -- half of a trimester already has an established name. It's called a "hexter."

The name hexter is interesting indeed. But to discover the origin of this name, we must first consider the origins of the words semester and trimester.

Where does the word semester come from? Some people might recognize a prefix semi- meaning "half" -- for example, in geometry a semicircle is half of a circle. Since a semester is half of an academic year, this seems logical -- but it's wrong. As it turns out, the word semester actually means "six months" -- it comes from Latin sex-, "six," plus mes- or mens-, "month." (Notice that in Spanish, the word mes still means "month.") But a semester can't possibly last six months, since then two semesters would be twelve months, the entire year, with no time for summer vacation. As it turns out, the word semester doesn't come directly from Latin, but passed through German. In German universities, the two semesters actually are six months long -- the winter semester lasting from October to March, and the summer semester lasting from April to September. There actually are breaks corresponding to our summer break, but they're actually included as part of the semesters! So semester means "six months," sex- plus mes-, but Latin speakers often drop the letter x when it appears right before the letter m, just as emigrate is really ex- (out of) plus migrate.

Therefore, a trimester actually means "three months" -- since it comes from tri-, "three," plus mes-, which we already identified as "month." It does not mean "one-third of a year." But since the school year is approximately nine, or three-squared, months long, one-third of the year just happens to be around three months. The term of a woman's pregnancy is also around three-squared months, and so some might believe that trimester means one-third of a pregnancy, but it still means "three months."

And so what about hexter? Now hex- is Greek for six (think hexagon), but is a hexter six of something, or one-sixth of something else? This word doesn't contain mes-, so it has nothing to do with six months or one-sixth of a month. On one hand, there are six hexters in a year, so this word, hexter, appears to be one-sixth of an academic year. But a hexter is also six of something -- it is close to six weeks in length, since there are approximately 36 or six-squared weeks in a school year! The answer is that we can't be sure, since the academic term hexter, while used at some schools, doesn't appear in a dictionary where we can discover its etymology.

Finally, notice that hex- is Greek while all the other numerical prefixes for academic terms are derived from Latin. Recall what I wrote about this lack of linguistic purity in geometry, where we have both hexagons (hex-, Greek) and nonagons (non-, Latin). To be linguistically consistent, we ought to use the Latin prefix sex- and call it a "sexter." The problem is that most schools using hexters are middle schools, and students at that age will assume that this has something to do with sexuality, even though the Latin sex, "six," has nothing to do with the Latin sexus, "sex." In order to avoid trying to explain to middle schoolers how "sexter" and "sexual" come from two completely unrelated Latin roots, the schools just throw linguistic purity out the window and use Greek-based "hexter" instead.

Notice that trimesters, and therefore hexters, appear mainly at the middle school level. High schools almost always use semesters instead, as this is what the colleges expect on the transcripts. But I have seen a few high schools give report cards three times per semester -- in other words, the progress report occurs at the end of every hexter. But I've never seen the word "hexter" used to refer to these thrice-a-semester progress reports. (I've once seen a high school use the name "triad" to refer to one-third of a semester.)

Here is the quiz review:

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