Before we begin, Theoni Pappas has a full week of Geometry on her Mathematics Calendar 2018. I'll just say it now -- everyday this week I'll be posting a Pappas Geometry question.
Today on her Mathematics Calendar 2018, Pappas writes:
Suppose the point (1, 14) is reflected about the line y = x and then again about the line y = 0. For the new point's coordinates what is the value of x + y?
Well, we know that the reflection image of (x, y) over the line y = x is (y, x), and hence the image of the first reflection is (14, 1). The second reflection is tricky -- another name for y = 0 is the x-axis, since if you're stuck on the x-axis, your y-coordinate is indeed zero. We reflect points over the x-axis by switching the sign of the y-coordinate, and so the final image is (14, -1).
This second reflection may still be confusing, but notice that it generalizes to higher dimensions. In three dimensions, y = 0 is the equation of a plane -- and as we found out in Lesson 9-5, mirrors of spatial reflections are indeed planes. Anyway, the reflection image of (x, y, z) across the plane y = 0 is the point (x, -y, z) -- that is, the sign we switch is that of the variable in the equation. And let's also consider the one-dimensional case. A mirror on the number line is a single point -- indeed, the reflection image of P across mirror M is P' exactly when M is the midpoint of
OK, let's return to the problem. Our answer is a particular point on the coordinate plane, with both its x- and y-coordinates. A common Pappas trick in order to force the answer to be a single number is to ask for x + y, rather than x and y. Since the point is (14, -1), we find x + y = 14 - 1 = 13. Therefore our final answer is 13 -- and of course, today's date is the 13th.
Notice that in this problem, we are dealing with the composite of two reflections. Since the mirrors intersect at the origin, this composite is a rotation centered at the origin. To find the magnitude of the rotation, we notice that the angle between y = x and y = 0 is 45 degrees, so the magnitude of the rotation must be twice this, or 90 degrees. The direction of the rotation is the same as that from the first mirror to the second mirror, which is clockwise. We check this by notice that the point (1, 14) rotated 90 degrees clockwise about the origin really is (14, -1).
This type of question could appear on the PARCC or SBAC exams. On those tests, the second mirror is likely to be named as "the x-axis" rather than "the line y = 0." Other than that, this is a likely question on the Common Core tests. But unfortunately, the U of Chicago text doesn't fully discuss the images of reflections on the coordinate plane. We should therefore give our students extra practice reflecting points over the axes as well as y = x and y = -x. (The modern Third Edition of the text provides slightly more practice with these reflections on the coordinate plane than does my old text, the Second Edition.)
This is what I wrote two years ago about today's lesson. Here, I compared the treatment of the volume in two different texts, an old McDougal Littell California seventh grade text and the U of Chicago text:
The McDougal Littell text, in Lesson 10.7, demonstrates the sphere volume formula the same way that it does the cone volume formula. We take a cone whose height and radius are both equal to the radius of the sphere, and we find out how many conefuls of sand fill the sphere. The text states that two conefuls make up a hemisphere, and so four conefuls make up the entire sphere.
But of course, we want to derive the formula more rigorously. Recall that Dr. David Joyce states that a limiting argument is the best that can be done at this level -- but I disagree. Dr. Franklin Mason, meanwhile, enthusiastically gives another derivation of the sphere volume formula, and Dr. M's proof also appears in the U of Chicago text. Recall that Dr. M considers this day on which the sphere volume formula -- Lesson 12.6 of his text -- is derived to be one of the three best days of the year.
The U of Chicago text mentions that this proof uses Cavalieri's Principle. But it was hardly the mathematician Cavalieri who first proved the sphere volume formula. Indeed, according to Dr. M, this proof goes all the way back to Archimedes -- the ancient Greek mathematician who lived a few years after Euclid. (It's possible that their lives overlapped slightly.)
Here is a Square One TV video about Archimedes:
http://www.imdb.com/name/nm0146146/bio?ref_=nm_ov_bio_sm
The IMDB webpage describes Cathey as "an incisive personality noted for his uniquely expressive baritone voice." But Cathey, while playing the role of Archimedes, doesn't sing in the video above. In honor of Reg E. Cathey, I'm now posting a compilation of YouTube videos of Square One TV songs in which he does sing in his uniquely expressive baritone voice.
2. Nine, Nine, Nine (I'm counting Archimedes as Video #1!)
3. Round It Off
In my January 6th post, I wrote that "Nine, Nine, Nine" was one of the more popular songs I played in my class last year. Indeed, it was one of the special scholar's favorites. I sang it at the start of the year, for Back to School Night and the tenth anniversary of its uploading to YouTube. My intention was to play it again in time for the 9's Dren Quiz, but unfortunately, I left the classroom after the 6's Dren Quiz and never reached the 9's.
On the other hand, I never played "Round It Off" at all. There were some points in class when I could have sung it -- for example, when the eighth graders were learning how to approximate irrational numbers with rational numbers (i.e., round them off). I probably should have played it shortly after seeing how popular Cathey's other song "Nine, Nine, Nine" turned out to be.
It's interesting how Square One TV chose Cathey, a black man, to sing the two country songs. (Oops, I mentioned race in this school year post. But this is a part of Cathey's story.) So I'll also post some of Cathey's rap songs as well. For balance, notice that white guys rap along with Cathey in these songs:
4. Apple Rap
5. Fractions Rap
6. Rappin' Judge
I never sang any of these raps in class -- I wish that I'd tried at least one of these. At the very least, all three middle school grades had fractions at some point, so I could have used "Fraction Rap." The other two raps describe specific word problems -- rate, time, distance in "Rappin' Judge," and work problems ("How long will it take to do the job together?") in "Apple Rap." It's possible for teachers to try these two raps in high school Algebra classes as well.
According to the IMDB link above, Cathey passed away "never having had the chance to play a baritone jazz saxophonist, which would have been his dream role." Well, not so fast....
7. Count on It
In this video, Cathey plays the saxophone! But according to the commenter "T-Bone Hubbard" (on the actual YouTube comment thread), Cathey is playing a bass sax -- the actor next to him is playing the baritone sax. (I explained the difference between "bass" and "baritone" back in my Christmas Eve post -- apparently, bass/baritone is a distinction made for saxophonists, not just vocalists.) Still, imagine how the closest Cathey got to his bari sax dream role was on Square One TV.
I sang "Count on It" to my class last year on the third day of school. It was also one of the bigger hits with my class -- many students enjoyed it, including the special scholar.
8. Mathematics 'R' Us
This isn't a song. Instead, Square One TV had a skit called "Mathematics 'R' Us." Here Cathey plays Al Gorithm, a flamboyant "as seen on TV" advertiser.
Rest in peace Reg E. Cathey -- your songs made my class more enjoyable last year. I only wish that I'd played even more of your songs in class while I still had the chance.
OK, let's return to Archimedes and the volume of a sphere:
We mentioned earlier that Archimedes used polygons to determine the value of pi (also known as Archimedes' constant) -- hence the line in the song, "He was busy calculating pi." He was also famous for using the principle of buoyancy (also known as Archimedes' principle) to determine whether the king's gold crown was a fake -- and this is also mentioned in the song. Legend has it that the Greek mathematician was so excited when he discovered his principle -- he had been in a public bath at the time -- that he ran down the streets naked and shouted out "Eureka!" to announce his discovery. The Greek word eureka, meaning "I have found," is the motto of my home state of California.
But Archimedes himself actually considered the discovery of the sphere volume formula to be his crowning achievement -- to the extent that he requested it to be engraved on his tombstone. So let's finally derive that formula the way that Archimedes did over 2000 years ago. And no, he didn't simply drop a ball into water to determine the formula. Archimedes' sphere formula has nothing to do with Archimedes' principle of buoyancy.
We begin by considering three figures -- a cone, a cylinder, and a sphere. We will use the known volumes of the cone and cylinder to determine the unknown volume of the sphere -- thereby reducing the problem to a previously solved one.
Our cylinder will have the same radius as the sphere, while the height of the cylinder will equal the diameter (i.e., twice the radius) of the sphere. This way, the sphere will fit exactly in the cylinder.
Our cone, just like the cone mentioned in McDougal Littell, will have the its height and radius both equal to the radius of the sphere. Such a cone could fit exactly in a hemisphere. But we want there to be two cones, so that their combined height is the same as that of the cylinder. We set up the cones so that they have a common vertex (i.e., they are barely touching each other) and each base of a cone is also a base of the cylinder. The two touching cones are often referred to as a "double cone" -- Dr. M uses the term "bicone." (A bicone is also used to justify to Algebra II students why a hyperbola is a conic section with two branches. A hyperbola is the intersection of a bicone and a plane, such that the plane touches both cones.)
The focus is on the volume between the cylinder and the bicone. The surprising fact is that this volume is exactly equal to the volume of the sphere! Here is the proof as given by the U of Chicago:
"...the purple sections are the plane sections resulting from a plane slicing these figures in their middles. These purple sections are congruent circles with area pi * r^2. At h units above each purple section is a section shaded in pink. In the sphere, by the Pythagorean Theorem, the pink section is a small circle with radius sqrt(r^2 - h^2). The area of this section is found using the familiar formula for the area of a circle.
"Area(small circle) = pi * sqrt(r^2 - h^2) = pi(r^2 - h^2)
"For the region between the cylinder and the cones, the section is the pink ring between circles of radius r and h. (The radius of that circle is h because the acute angle measures 45 degrees, so an isosceles triangle is formed.)
"Area(ring) = pi * r^2 - pi * h^2 = pi(r^2 - h^2)
"Thus the pink circles have equal area. Since this works for any height h, Cavalieri's Principle can be applied. This means that the volume of the sphere is the difference in the volume of the cylinder (B * 2r) and the volume of the two cones (each with volume 1/3 *B * r).
"Volume of sphere = (B * 2r) - 2 * (1/3 * B * r)
= 2Br - 2/3 * Br
= 4/3 * Br
"But here the bases of the cones and cylinder are circles with radius r. So B = pi * r^2. Substituting,
"Volume of sphere = 4/3 * pi * r^2 * r
= 4/3 * pi * r^3." QED
The sphere volume is indeed the crowning achievement of Chapter 10. We began the chapter with the volume of a box and end up with the volume of the least box-like figure of all, the sphere. The start of Lesson 10-8 summarizes how we did this:
"It began with a postulate in Lesson 10-3 (volume of a box). Cavalieri's Principle was then applied and the following formula was deduced in Lesson 10-5 (volume of a prism or cylinder). A prism can be split into 3 pyramids with congruent heights and bases. Using Cavalieri's Principle again, a formula was derived in Lesson 10-7 (volume of a pyramid or cone). In this lesson, still another application of Cavalieri's Principle results in a formula for the volume of a sphere."
So take that, Dr. Katharine Beals! After all, she was the one who derided Cavalieri's Principle as progressive fluff that the Common Core tests on instead of actual math. But without Cavalieri's Principle, we'd be stuck finding the volumes of only boxes and their unions. Well, I suppose if we simply declared the volume formulas by fiat (i.e., as postulates) rather than actually deriving them, then Cavalieri's Principle is not needed. But if we want to prove them, then the Principle gives us an elegant proof of the sphere volume formula that was discovered over 2000 years before there ever was a Common Core -- a proof that, if mastered, should permit one to date a mathematician's daughter (as Beals mentioned on her website regarding the Quadratic Formula proof).
Sadly, we don't know whether Archimedes ever dated anyone's daughter, or whether he ever had daughters of his own. His life ended tragically, being captured by an enemy army. Legend has it that he was busy working on a geometry problem when the Roman army captured him. His last words before he was killed are said to be, "Noli turbare circulos meos" -- Latin for "Do not disturb my circles!"
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