Friday, April 17, 2015

PARCC Practice Test Question 4 (Day 149)

Last night I tutored my geometry student. Section 8-3 of the Glencoe text is on Special Right Triangles -- that is, the 45-45-90 and 30-60-90 triangles. I've already shown him my worksheet for Section 14-1 of the U of Chicago when these special triangles were inadvertently included on his Chapter 6 test (due to area and special right triangles being in the same chapter in a third text). Now, of course, he understands these triangles much better now.

I couldn't resist showing him my Pythagorean puzzle activity. To my surprise, he was actually able to figure out the puzzle using the c square much more easily than using the a and b squares -- most of my previous students much more easily solved the latter than the former.

Also, earlier this week PBS aired an episode of NOVA: "The Great Math Mystery" -- so naturally I must comment on this episode here on the blog. Featured on the program is Romanian astrophysicist Mario Livio. He is described as the author of the book Is God a Mathematician? which, just like this episode, discusses how accurately mathematics can describe the physical world. I am familiar with Livio as I've purchased his first book, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number, several years ago.

Actually, I've been meaning to buy another Livio book for a while now: The Equation That Couldn't Be Solved. On this surface, this book is about the quintic, or fifth-degree polynomial, equation -- for centuries, mathematicians wondered whether there was a Quintic Formula that solved any quintic equation the same way that the Quadratic Formula can solve any quadratic equation. In the 19th century, two young mathematicians proved that there is no Quintic Formula -- but along the way, they ended up developing much of group and ring theory. I've mentioned earlier on the blog that these abstract algebra theories are ultimately related to symmetry -- Livio emphasizes this in his book. And of course, Common Core Geometry is also based on symmetry. I can't believe that I haven't actually gotten the book yet -- but hopefully I will, soon enough!

There was a segment on the number pi, with mention of some of its mysterious properties far beyond anything that we mentioned on this blog. The Greek mathematicians Pythagoras and Plato also appeared -- we made a brief mention of the Platonic solids when discussing polyhedrons. (Dr. Franklin Mason discusses the Platonic solids in further detail.) Recall that just before spring break, I subbed in a classroom that discussed the Pythagorean Theorem. There's a connection to Plato here as well, as Plato knew of a formula that actually generated Pythagorean triples -- that is, sets of numbers that satisfy the Pythagoras formula. (There was also a Plato worksheet, but I didn't post it.)

But NOVA's mention of Pythagoras himself had nothing to do with right triangles. Instead, the show discussed the Greek mathematician's fascination with music. Indeed, I had been planning on posting an extra topic on the relationship between mathematics and music for some time now -- I'd wanted to include this as an extra topic during spring break, but I ended up posting other topics instead. Today I will use the NOVA episode as an excuse to post on music.

I've posted on music theory before on the blog -- after all, I've mentioned that I refer to half of an academic quarter as a "quaver" because this is a British term for an eighth note. But this topic is not about rhythm, but about pitch. Even though I've about to mention Pythagorean music, this topic isn't related to the Pythagorean Theorem at all. It fits better not in an Geometry class, but an Algebra II class, since it involves logarithms. But when I tutored Algebra II students -- especially those who have had some musical experience -- they are often fascinated when I tell them that it was Pythagoras who discovered why certain notes sound good together.

For example, musicians learn that notes are lettered A, B, C, D, E, F, G. But we are told that, for example, the notes C, E, G form what is called a "C chord." But this raises the question, why do we skip the notes D and F, and let the notes C, E, G be the "C chord"? As it turns out, this all started with Pythagoras.

The ancient Greeks played a stringed instrument called a lyre. It was Pythagoras who noticed that, while strings of different lengths produced different notes, the notes sounded best together when the string lengths exhibited a simple whole-number ratio. The simplest nontrivial ratio of whole numbers is 2:1 -- that is, when one string is exactly twice the length of the other. According to NOVA, this ratio is known as an octave. When two notes exactly an octave apart are played together, they end up sounding like the same note, except one's higher and the other's lower. Indeed, the note that's exactly an octave above C is also called C, and any two notes separated by a whole number of octaves are given the same letter. This is true for any stringed instrument -- notice that the twelfth fret of a guitar, the fret that produces a note one octave higher than the original note, is at the midpoint of the string. A violin appears in the episode.

The next simple ratio mentioned on NOVA is 3:2 -- that is, the longer string is half again as long as the shorter string. This interval is called a perfect fifth. The note a perfect fifth above C is G. So notice that we already have two of the three notes of the C chord, the C and the G.

The next simple ratio mentioned on NOVA is 4:3 -- that is, the longer string is a third again as long as the shorter string. This interval is called a perfect fourth. The note a perfect fourth about C is F. This gives us the three Pythagorean intervals -- the octave, the perfect fifth, and the perfect fourth.

Musicians are familiar with the circle of fifths. We begin at C and move up a perfect fifth to G. Then we move up another perfect fifth to D, and then to A, then to E, and so on. After twelve steps on the circle of fifths, we return back to C again -- or so we are told. That is, twelve perfect fifths are declared to be equal to a whole number of octaves -- seven, as it turns out.

But what exactly are seven octaves? Notice that one octave is 2:1, two octaves are 4:1, three octaves are 8:1, and so on. Since these are ratios, they must be multiplied, not added, just as performing a dilation with scale factor 2 thrice gives a figure that is eight times the original. So seven octaves are the ratio 128:1.

Now let's try twelve perfect fifths. Each fifth is 3:2, so twelve of them would be 531441:4096, which is not exactly 128:1. Cross multiplying these ratios would give 531441 = 524288, which is false. And we shouldn't be surprised that twelve fifths don't actually equal seven octaves. Since combining musical intervals amounts to multiplying, breaking them up amounts to factoring -- and we can't factor the same interval in two different ways (i.e., as seven octaves and twelve fifths) for the same reason that we can't factor the same number in two different ways -- this would violate the Fundamental Theorem of Arithmetic. The "equation" 531441 = 524288 states that a power of three equals a power of two, which is impossible.

But in music, we declare 531441 and 524288 to be equal. So the ratio 531441:524288 is considered to be equal to 1. In honor of the ancient Greek mathematician, we refer to the ratio 531441:524288 as the Pythagorean comma. And it's because it's the twelfth power of three that is approximately a power of two that our octaves consist of twelve notes -- and why it's the twelfth fret on the guitar that gives us the octave. (Notice that computer scientists have their own "comma." They declare 1024 = 1000 -- that is, a power of two equals a power of ten -- in order to justify the names "kilobyte," "megabyte," etc.)

The NOVA episode mentions only the octave, perfect fourth, and perfect fifth, since these were the only intervals with which Pythagoras was concerned. But notice that these give us the notes C, F, G, while the C chord is actually C, E, G. So where does the note E come from?

This was actually discovered about 500 years or so after Pythagoras, by the musician Didymus. We notice that Pythagoras stopped at the ratio 4:3. The next natural ratio to consider is 5:4. This ratio is now known as a major third, and represents the interval from C to E. Then the full C major chord is the extended ratio 4:5:6.

Notice that there's already an E on the circle of fifths. Declaring E to be a major third above C is the same as setting four perfect fifths to equal the ratio 5:1 -- which, as we know, is impossible. Four fifths would actually be 81:16, so cross-multiplying gives us 81 = 80. The ratio of 81:80 is considered to be another comma -- Didymus's comma.

We are now ready to produce the full C major scale. We start with the three intervals of Pythagoras, the octave, perfect fourth, and perfect fifth -- C, F, G. On each of these three notes, we build a major chord: the C chord (C, E, G), F chord (F, A, C), and G chord (G, B, D). Putting these notes in order gives us the full major scale -- C, D, E, F, G, A, B, C.

I remember taking a piano class the summer after kindergarten. As it was only a beginners' course, the only major scale taught was the C major scale, which was played on the white keys. But I wondered to myself why one couldn't play a major scale beginning on notes other than C. The following December (either for my birthday or Christmas) I received a small electronic keyboard as a gift, and naturally I tried playing other scales. beginning on D, then E -- but none of them sounded like the proper major scale Do, Re, Mi, etc.

But one of these scales sounded almost right -- the scale beginning on G. The scale G, A, B, C, D, E, F, G sounded correct except for the last few notes. But I didn't know how to make the last part sound like the major scale. Disappointed, I started playing around with chromatic scales -- where I included the black keys as well as the white keys. But as I was still tantalized by the G scale, I sometimes started out by playing the part of the G scale that sounded right -- G, A, B, C, D, E -- and then switched to the chromatic scale. The black key between F and G is called F sharp, or F#, so what I played was G, A, B, C, D, E, F, F#, G, and I often played around with this "scale" for awhile.

Then one day, I accidentally skipped the F note, so what I ended up playing was G, A, B, C, D, E, F#, G. And what I played sounded exactly like the major scale that I had been seeking! And I still remember to this day how excited my seven-year-old self was to "discover" the G major scale! After this, I quickly realized that I could make all the other scales (the D scale, E scale, and so on) sound right by including some of the black keys as well as the white keys on my keyboard.

The next ratio to consider is 6:5. This interval is called a minor third. To produce a minor chord, we take the major chord C, E, G and replace the E with the black key just below E, called E flat, often rendered in ASCII as Eb. So the C minor chord is C, Eb, G.

Why do minor chords sound "sadder" than major chords? Recall that a major chord has its notes in the ratio 4:5:6. Well, a minor chord has its notes in the ratio 10:12:15 -- that is, 5:6 is now the lower ratio and 4:5 is now the higher ratio. Since 4:5:6 is simpler than 10:12:15, the 4:5:6 major chord sounds brighter than the 10:12:15 minor chord. Many popular songs on the radio nowadays tend to be about one of two topics -- falling in love and breaking up. The former songs tend to be in major keys, while the latter songs tend to be in minor keys. Over the past few decades, popular hits in minor keys have increased.

The second keyboard that I owned, when I was in the second grade, had several songs built in, including the classic Greensleeves. This song was in the key of A, but I noticed that it was based on A minor rather than A major, as it contained the chord A, C, E (whereas A major would be A, C#, E). I soon learned that there were minor scales as well as major scales, and tried to deduce what the A minor scale was based on the song. Unfortunately, the version of Greensleeves that was built into my keyboard wasn't truly in A minor, but a scale called the Dorian mode. Instead, I was under the misconception that A, B, C, D, E, F#, G, A was the A minor scale -- that is, A minor was just like G major in that it contained F# rather than F natural.

It was not until the third grade until I took private piano lessons. My third grade classroom had a piano, and the teacher was a pianist. So she taught me piano one a week after school until she left on maternity midway through the year, after which she referred me to her mother to continue my study of the piano. This was when I finally learned that the A natural minor scale was actually the notes A, B, C, D, E, F, G, A -- that is, it has no sharps or flats, just like C major. This minor scale is formed by taking the three Pythagorean intervals (octave, fourth, and fifth) and building a minor chord on each note, rather than a major chord.

Here are a few more links discussing major and minor scales:

which in turn links to:

OK, I could talk about music forever, but that's enough for now. Let's get back to PARCC. Question 4 of the PARCC Practice Test is on a shape formed by rotating a rectangle in space:

A rectangle will be rotated 360 degrees about a line which contains the point of intersection of its diagonals and is parallel to a side. What three-dimensional shape will be created as a result of the rotation?

(A) a cube
(B) a rectangular prism
(C) a cylinder
(D) a sphere

This is the sort of question to which traditionalists might object. That word "rotation" might set them off, as they may assume that this is another transformation question. But we notice that the rectangle is being rotated about a line. If this were a 2D rotation, then its center would be a point, but since it has a line as its axis, it must be a 3D rotation. Indeed, what we are forming is a solid of revolution -- the figure whose volume is often found in BC Calculus. So an argument can be made that this is an excellent question to put on a Geometry test, as it prepares students to work with solids of revolution in BC Calculus.

One thing that may confuse students is "point of intersection of its diagonals." It might be clearer just to use the phrase "its center" -- but the problem is that the center of a rectangle doesn't have a rigorous definition in any geometry text. In general, "center of a polygon" is defined only if the polygon is regular, and a rectangle isn't regular (unless it's a square). There is a trade-off between using a phrase that is clearer for the students but is mathematically incorrect, and one that is confusing for students yet mathematically correct.

Once we get past the problem with the "center" of the rectangle and where the axis is, it remains only to tell what figure is the solid of revolution. Notice that the intersection of a solid of revolution and a plane perpendicular to the axis must be a circle, whereas neither a cube nor a rectangular prism can have a circular plane section. But the intersection of a solid of revolution and a plane containing the axis must be the original rectangle, whereas a sphere can't have a rectangular plane section. Therefore the solid of revolution must be a cylinder, which is choice (C).

Only one solid of revolution appears in the U of Chicago text, and I mentioned it a few weeks ago -- Section 10-7, Question 15. In this case, a right triangle is rotated -- its axis is the line containing one of its legs -- to form a cone. But this problem tells us that the solid is a cone -- the students' task is to find its volume, if the length of the other leg and the the hypotenuse are given. If the given figure had been a rectangle, then the solid would have been a cylinder rather than a cone. I wonder whether Question 4 on the PARCC would be more palatable to traditionalists if the dimensions of the rectangle were given and students had to find the volume of the cylinder.

Even though we're now reviewing PARCC questions, I still want to maintain my tradition of posting a weekly activity. The easiest possible activity here would be just to perform the rotation so that students can actually see the solids of revolution. For this activity, students should tape the rectangle to a pencil to represent the axis of rotation. My activity will provide for some of the students to cut the rectangle in half and rotate a triangle to form a cone. Then I ask the students to find the volume of the cylinder or cone, just as the U of Chicago does in that one question.

PARCC Practice Test Question 4
U of Chicago Correspondence: Section 10-7, Volumes of Pyramids and Cones
Key Theorem: none

Common Core Standard:
Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Commentary: The only question relevant to solids of revolution is Question 15, and it rotates a triangle to form a cone, rather than a rectangle to form a cylinder. Another U of Chicago section that may be relevant to the first part of the above standard is Section 9-4, which is on "Plane Sections" (cross-sections). On this blog, we jumped around Chapters 9 and 10, so Section 9-4 may not have been adequately covered.

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