Wednesday, May 6, 2015

PARCC Practice Test Question 17 (Day 162)

Chapter 8 of Mario Livio's The Equation That Couldn't Be Solved is titled, "Who's the Most Symmetrical of Them All." In this chapter, Livio shows how symmetry can be applied to other fields beyond mathematics and physics.

The title of the chapter is an allusion to the question asked by Snow White's stepmother, "Who's the fairest of them all." The point Livio makes here is that "fairest" -- in the sense of "most attractive" -- really means "most symmetrical." According to him, a preference for symmetry has been hard-wired into humans by the forces of evolution.

I once read about a math teacher who wanted to motivate her high school students to learn geometry. I must admit that this was over a year ago, and I don't remember who this teacher was -- otherwise I would provide a link. Anyway, when her class reached the lesson on symmetry and reflections, this teacher motivated her students by discussing the link between symmetry and attractiveness. The point was that these are high school students, at an age where one is confused about one's own sexuality. So these students may be interested in whether or not they are attractive -- and the teacher took advantage by telling them how attractiveness related to symmetry, and therefore to geometry. This was especially effective with the female students. (Still, it's a shame that I couldn't find the link.)

Livio moves on to symmetry in music. We've already discussed the relationship between music and mathematics after I watched the NOVA episode -- the same episode in which Livio appeared (and ultimately led me to reading his book). The author repeats how Pythagoras is credited with the discovery that notes that sound good together have frequencies in simple ratios -- 3:2 for the perfect fifth, 4:3 for the perfect fourth, and so on. Anyway, Livio links this back to group theory, in that if we represent the notes by numbers, so that C is 0, C# is 1, D is 2, and so on up to B as 11, we repeat the first note, C, an octave higher -- so after 11 is 0. He points out that the same think happens on the clock, where 12 o'clock is also equal to 0 (to be followed by 1 o'clock). Here is a link to a Square One Video that discusses this clock arithmetic:

Both musical arithmetic and clock arithmetic are one and the same -- and Livio refers to both of these as addition modulo 12. This is an example of what Livio calls a cyclic group -- and for good reason, since we can see easily how a clock cycles around the hours. Musical notes cycle as well -- I had "discovered" of the G major scale at age 7 simply by playing by ear, but in reality, the G major scale is formed by adding 1 or 2 notes (semitones and tones) in a specific pattern modulo 12, beginning with the note G or 7. Groups theorists often symbolize the additive group modulo 12 as Z/12Z, where Z stands for the usual additive group of integers.

Livio also refers to atonal music, or music based on a twelve-tone row. I've already mentioned the mathematician/musician Vi Hart, who complained when someone tried to copyright the number pi (converted to music). Well, she has produced a 30-minute video based on the twelve-tone row that Livio mentions in this chapter:

(Note: the conversion of pi to music is not generally based on a twelve-tone row. But there was a video that I had posted earlier that used the digits of pi in base 12 so that all twelve tones are used.)

The octave is divided into twelve tones because so many powers of the twelfth root of two are equal to Pythagorean ratios. As Livio points out, 2^(7/12) is 1.498 (close to 1.5, or 3:2) and 2^(3/12) is 1.189 (close to 1.2, or 6:5). A question that sometimes comes up is, is there any other division of the octave other than 12 that produces ratios that are near the simple ratios of Pythagoras? In other words, can music be based on a group other than Z/12Z?

Some musicians believe that the next natural group to consider is Z/19Z. Just as some people have an esoteric interest in calendar reform, others have an esoteric interest in musical reform. According to the 20th century musician Joseph Yasser, 19 is the next natural step after 12:

"A 19-tone keyboard based on the traditional keyboard would have pairs of black keys (Db and C#, Eb and D#, etc.), and an additional black key between the semitone intervals E-F and B-C."

Another musician, Jonathan Glasier, has produced three YouTube videos on 19-tone music. I don't want to link to all three of these, but a search on YouTube for "Jonathan Glasier 19" suffices. We see that 2^(11/19) = 1.494, which isn't as close to 3:2 as 2^(7/12) is. So, as Glasier points out, we no longer have a perfect fifth, but just a "fifth." But 2^(5/19) = 1.2001, which is much closer to 6:5 than 2^(3/12) is, so we now have what Glasier calls a "perfect minor third." Its inverse ("inversion," according to Livio), the major sixth, also becomes a "perfect major sixth."

Some of the new notes correspond to new intervals. The next ratio to consider after 6:5 is 7:6. There is no note in 12-tone that gives the ratio 7:6, but in 19-tone, we have that 2^(4/19) = 1.157 which, while still a bit off from 7:6 = 1.167, is still better than anything in 12-tone. Ratios based on seven in music are often called "septimal" and sometimes sound good in blues music. So we call the new ratio a "septimal minor third" or "blue third" and is denotes by the interval from C to D#, where D#, as is written above, is not the same note as Eb.

Here's a link to the flutist Anne La Berge, who plays a song based on 19 tones. Let's find out whether this sounds more "bluesy" when one uses 19 tones instead of 12:

With some teenage girls in our classes, we can start talking about attractiveness and symmetry and they will keep discussing it forever. Well, I can keep talking about mathematics and music forever -- and I can't let music take over this post when I need to get back to the PARCC. Question 17 of the PARCC Practice Exam is on trigonometry:

An archaeological team is excavating artifacts from a sunken merchant vessel on the ocean floor. To assist the team, a robotic probe is used remotely. The probe travels approximately 3,900 meters at an angle of depression of 67.4 degrees from the team's ship on the ocean surface down to the sunken vessel on the ocean floor. The figure shows a representation of the team's ship and the probe.

How many meters below the surface of the ocean will the probe be when it reaches the ocean floor?

Give your answer to the nearest hundred meters.

In this question, the figure helps us immensely. We can form a right triangle where the angle of depression is 67.4 degrees, and the hypotenuse is 3,900 meters, and we wish to find the side opposite the given angle. This is a job for the sine function:

sin(67.4) = x/3900
x = 3900 sin(67.4)
x = 3600.5

Rounded off to the nearest hundred meters, the answer is 3,600 meters. Notice that the answer is already fairly close to a multiple of 100 meters. Indeed, the test writers probably had a 5-12-13 right triangle in mind when coming up with this problem.

The trig ratios appear in Sections 13-3 and 13-4 of the U of Chicago text. I've stated earlier that most angle of elevation problems require the tangent since typically either a horizontal or vertical distance is given and we wish to find the other. But here, 3,900 is clearly the hypotenuse, and so we must use the sine, not the tangent. Common errors include using the wrong trig function as well as having the calculator in radian mode rather than degree mode.

Livio wraps up the chapter by returning to group theory. He describes that some groups are simple -- that is, they can't be broken up into smaller subgroups. All cyclic groups containing a prime number of elements, such as the musical Z/19Z, are simple. But there exists groups that contain a composite number of elements yet aren't simple. One of these contains 60 elements and is the group that Galois discovered as blocking the quintic from being solved. But it wasn't until 1980 that one of the largest simple groups was discovered. It contains a mind-boggling 8 * 10^53 elements -- so many that it has been dubbed the "monster." Yet the "monster," as large as it is, is just simple at heart.

PARCC Practice EOY Exam Question 17
U of Chicago Correspondence: Section 14-4, The Sine and Cosine Ratios
Key Theorem: Definition of sine and cosine

In right triangle ABC with right angle C,
the sine of Angle A, written sin A, is leg opposite Angle A / hypotenuse
the cosine of Angle A, written cos A, is leg adjacent to A / hypotenuse

Common Core Standard:
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Commentary: Most angle of elevation and depression problems require the use of the tangent, but interestingly enough, Question 14 in Section 14-4 -- the only appearance of the phrase "angle of depression" in the U of Chicago text -- requires the use of sine instead. If we change Question 14 so that it asks for the height of your eyes above the ground and not the distance between you and the object, the question becomes analogous to the PARCC problem.

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