Lesson 10-5: Volumes and Prisms of Cylinders (Day 105)

Today I subbed in a high school P.E. class. This is in my old district, and so it really is Day 105 here at this school.

It goes without saying that there's no "A Day in the Life" today. In fact, today's subbing schedule turns out to be very light today. This isn't the one district school with a traditional schedule, and so today is a block day, with even periods. But of the three blocks, second period is the conference period, while sixth period is off-season football, with an assistant coach taking over. Thus the only real class for me to cover is fourth period.

This high school is so small that there's only one other P.E. teacher, but she has a student teacher. The two of them help me out with the Warm-Up -- walk six laps on the track (a mile and a half). After that is free play. I walk the laps with them and high-five some of them as they finish them, especially the special ed students.

Even this light day isn't without its problems. During the laps, one guy decides to climb another guy's back, then falls off, slightly hurting himself. Then later on, a girl vomits. The female P.E. teacher summons a nurse, who then escorts the girl to the office.

Although it's been a full year since I subbed here, one guy recognizes me from another school in the district where he transferred from, and so he asks for a song. Once again, I don't always sing during P.E. class, but this time I choose Square One TV's "Angle Dance." Again, it's a song that I associate with this time of year (after winter break at the old charter school), and the dancing part is loosely related to P.E. class.

(Notice that the students walk laps today but don't stretch, while last week the middle school students had to stretch first. An argument can be made that I should have switched "Angle Dance" and last week's "Count the Ways." Then "Angle Dance" fits the stretching part -- "To start bend your knees 45 degrees" -- while the heartbeat in "Count the Ways" is more relevant after completing six laps.)

Today is Elevenday on the Eleven Calendar. My rule is that on Elevenday, I look for any opportunity to fulfill any of my new rules, though it's tricky on a day like this. I loosely followed the eighth rule on singing to help remember procedures, though this would have worked better if I could have actually performed it during stretching (so that the song describes a stretching procedure -- that is, bending the knees 45 degrees).

Elevendays are also great days to look back at my "New Millennium's Resolutions" (since after all, it's still the new millennium). These rules instruct me to have more conversations with my fellow teachers (as well as connect better with students). Today I talk to the female P.E. teacher and her student teacher about the use of the gym for the upcoming primary election. In California, the primary election now lasts for one week -- and by "one week," I mean eleven days. (See -- my new Eleven Calendar is changing how I think about weeks!)

Lecture 21 of Michael Starbird's Change and Motion is called "Physics, Music, and the Planets," and here is an outline of this lecture:

I. The planets go around the sun in ellipses. But why?
II. Newton deduced Kepler's laws from his law of universal gravitation.
III. Designing instruments for observing the heavens makes use of calculus. A parabola is an effective shape for a telescope, and calculus shows us why.
IV. In resonance to the ancients' concept of allying astronomy with music, in their case through the idea of the music of the spheres, we now apply calculus to music.
V. Other applications to physics include mechanics, fluid dynamics, waves, thermodynamics, electricity and magnetism, optics, innumerable applications to technology of all sorts, and more.

Starbird begins with Johannes Kepler (1571-1630) and his discussion with Tycho Brahe (1546-1601) regarding the motion of the planets around the sun. The ancients believed that the planets' orbits (around the earth, of course) were either circles, or epicycles (circles upon circles). Kepler's first law is that the planets' orbits are ellipses. His second law is that planets move faster when they are closer to the sun (in their elliptical orbits) and slower when they are farther from the sun (in the sense that it "sweeps out equal areas"). His third law is that planets nearer the sun orbit it faster than the distant planets. He points out that while Kepler was correct about his three laws, he was wrong in his hypothesis that the distances between the planets are related to the five Platonic solids.

Now the professor returns to Isaac Newton. As it happened, Newton was able to show why Kepler's laws were true -- it's all because of his own theory of gravity. He tells an interesting story about baseball -- if the earth were a point mass and the batter hit the ball, it would orbit the earth in an ellipses and land right back at home plate at the same speed it was hit. Therefore an outfielder can just wait behind home plate for the ball.

Vectors can be used to demonstrate Kepler's laws. A planet is moving around the sun at a certain velocity, and the force towards the sun causes it to change direction. Then the planet will continue to sweep out equal area. (A full explanation is difficult to post here without the diagram.)

Starbird proceeds with his telescope example. If a telescope is shaped like a parabola f(x) = x^2, then any light approaching it vertically will reflect off the parabola and land at the focus (0, 1/4). To prove this, the only Calculus needed is that the derivative of f(x) is 2x, then the rest is all Geometry. If the light approaches the point (x, x^2), then bouncing off the parabola is equivalent to bouncing off of the tangent line at that point. The slope of this tangent line is 2x (the derivative), and so by Point-Slope, we find that the tangent line passes through (x/2, 0).

Now we consider two triangles -- one with vertices (x, x^2), (x/2, 0), and (0, 1/4), and the other with the same first two vertices but the last point at (x, -1/4). We now claim that these two triangles are congruent. To prove this, we use Distance Formula/Pythagorean Theorem:

sqrt(x^2 + (x^2 - 1/4)^2) = x^2 + 1/4

So the first sides are congruent. The second sides are congruent by Midpoint Formula -- (x/2, 0) is clearly the midpoint of (0, 1/4) and (x, -1/4). And the third sides are congruent by Reflexive. Thus by SSS, the triangles are congruent, and therefore the angles (of incidence and reflection) must be congruent by CPCTC.

The professor's final example applies Calculus to music. He sings two notes that are a perfect fifth apart, and draws their combined wave oscillations. He tells us that Fourier transforms -- a type of epicycles -- can be used to analyze harmony. The lecture ends as several formulas related to Calculus and Physics appear quickly on the screen.

Lesson 10-5 of the U of Chicago text is called "Volumes of Prisms and Cylinders." In the modern Third Edition of the text, volumes of prisms and cylinders appear in Lesson 10-3.

This is what I wrote last year about today's lesson:

As I mentioned yesterday, we are moving on to Lesson 10-5 of the U of Chicago text, which is on volumes of prisms and cylinders. As we proceed with volume, let's look at what some of our other sources say about the teaching of volume. Dr. David Joyce writes about Chapter 6 of the Prentice-Hall text -- the counterpart of U of Chicago's Chapter 10:

Chapter 6 is on surface areas and volumes of solids. It is strange that surface areas and volumes are treated while the basics of solid geometry are ignored.

There are 11 theorems, the only ones that can be proved without advanced mathematics are the ones on the surface area of a right prism (box) and a regular pyramid. Most of the theorems are given with little or no justification. The area of a cylinder is justified by unrolling it; the area of a cone is unjustified; Cavalieri's principle is stated as a theorem but not proved (it can't be proved without advanced mathematics, better to make it a postulate); the volumes of prisms and cylinders are found using Cavalieri's principle; and the volumes of pyramids and cones are stated without justification. The only argument for the surface area of a sphere involves wrapping yarn around a ball, and that's unlikely to get within 10% of the formula. Finally, a limiting argument is given for the volume of a sphere, which is the best that can be done at this level.

In summary, there is little mathematics in chapter 6. Most of the results require more than what's possible in a first course in geometry. Surface areas and volumes should only be treated after the basics of solid geometry are covered. Alternatively, surface areas and volumes may be left as an application of calculus.

There are a few things going on here. First of all, Joyce writes that surface areas and volumes should be treated after the "basics of solid geometry," but he doesn't explain what these "basics" are. It's possible that the U of Chicago's Chapter 9 is actually a great introduction to the basics of solid geometry -- after all, Chapter 9 begins by extending the Point-Line-Plane postulate to 3-D geometry, then all the terms like prism and pyramid are defined, and so on. Only afterward, in Chapter 10, does the U of Chicago find surface areas and volumes.

So I bet that Joyce would appreciate Chapter 9 of the U of Chicago text. Yet most texts, like the Prentice-Hall text, don't include anything like the U of Chicago's Chapter 9.

It's because most sets of standards -- including the Common Core Standards, and even most state standards pre-Common Core -- expect students to learn surface area and volume and essentially nothing else about solid geometry. Every single problem on the PARCC and SBAC exams that mentions 3-D figures involves either their surface area or volume. And so my duty is to focus on what the students are expected to learn and will be tested on, and that's surface area and volume.

Joyce writes that something called "Cavalieri's principle" is stated as a theorem but not proved, and it would be better if it were a postulate instead. Indeed, the U of Chicago text does exactly that:

Volume Postulate:
e. (Cavalieri's Principle) Let I and II be two solids included between parallel planes. If every plane P parallel to the given plane intersects I and II in sections with the same area, then
Volume(I) = Volume (II).

According to the text, Francesco Bonaventura Cavalieri was the 17th-century Italian mathematician who first realized the importance of this theorem. Cavalieri's Principle is mainly used in proofs -- as Joyce points out above, the volumes of prisms and cylinders is derived from the volume of a box using Cavalieri's Principle, and the U of Chicago also derives the volumes of oblique prisms and cylinders using Cavalieri.

The text likens Cavalieri's Principle to a stack of papers. If we have a stack of papers that fit in a box, then we can use the formula lwh to find its volume. But if we shift the stack of papers so that it forms an oblique prism, the volume doesn't change. This is Cavalieri's Principle.

Notice that we don't need Cavalieri's Principle if one is simply going to be handed the volume formulas without proving that they work. But of course, doing so doesn't satisfy Joyce. Indeed, the U of Chicago goes further than Prentice-Hall in using Cavalieri to derive volume formulas -- as Joyce points out, we can find the volume of some pyramids without advanced mathematics (Calculus).

[2020 update: Of course, we just learned about Cavalieri last week in our Starbird lectures.]

But then the U of Chicago uses Cavalieri to extend this to all pyramids as well as cones. Finally -- and this is the grand achievement -- we can even use Cavalieri's Principle to derive the volume of a sphere, using the volumes of a cylinder and a cone! As we'll soon see, Joyce is wrong when he says that a limiting argument is the best that we can do to find the volume of a sphere. Cavalieri's Principle will provide us with an elegant derivation of the sphere volume formula.

Here are the Common Core Standards where Cavalieri's Principle is specifically mentioned

CCSS.MATH.CONTENT.HSG.GMD.A.1
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.

CCSS.MATH.CONTENT.HSG.GMD.A.2
(+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

CCSS.MATH.CONTENT.HSG.GMD.A.3
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*

But not everyone is excited about the inclusion of Cavalieri's Principle in the standards. The following link is to the author Dr. Katharine Beals, a traditionalist opponent of Common Core:

http://oilf.blogspot.com/2014/11/math-problems-of-week-common-core_14.html

[2020 update: This sounds like yet another traditionalists' post. Recently, I didn't label this as traditionalists because Beals is no longer an active traditionalist. Still, I know this appears that I keep sneaking in traditionalists to my posts.]

Beals begins by showing a question from New York State -- recall that NY state has one of the most developed Common Core curricula in the country. This question involves Cavalieri's principle. Then after stating the same three Common Core Standards that I did above, Beals asks her readers the following six "extra credit questions":

1. Will a student who has never heard the phrase "Cavalieri's principle" know how to proceed on this problem? 

2. Should a student who has never heard the phrase "Cavalieri's principle" end up with fewer points on this problem than one who has? 

3. Should a student who explains without reference to "Cavalieri's principle" why the two volumes are equal get full credit for this problem? 

4. Is it acceptable to argue that the volumes are equal because they contain the same number of equal-volume disks? 

5. To what extent does knowledge of "object permanence," typically attained in infancy, suffice for grasping why the two stacks built from the same number of equal-sized building blocks have equal volume?

6. To what degree does this problem test knowledge of labels rather than mastery of concepts?

The question involves two stacks of 23 quarters -- one arranged as a cylinder, the other arranged not so neatly -- and asks the students to use Cavalieri to explain why they have the same volume.

Notice that in four of the "extra credit questions," Beals criticizes the use of the phrase "Cavalieri's principle," arguing that the label makes a simple problem needlessly complicated. But Cavalieri's Principle is no less a rigorous theorem than the Pythagorean Theorem. Imagine a test question which asks a student to use the Pythagorean Theorem to find the hypotenuse of a right triangle, and someone responding with these "extra credit questions":

1. Will a student who has never heard the phrase "Pythagorean Theorem" know how to proceed on this problem? 

2. Should a student who has never heard the phrase "Pythagorean Theorem" end up with fewer points on this problem than one who has? 

3. Should a student who explains without reference to "Pythagorean Theorem" what the hypotenuse is get full credit for this problem? 

6. To what degree does this problem test knowledge of labels rather than mastery of concepts?

As for the other two "extra credit questions," yes, the volumes are equal because they contain the same number of equal-volume disks. We don't need Cavalieri's Principle to prove this, since the Additive Property -- part (d) of the Volume Postulate from yesterday's lesson -- tells us so.

The true power of Cavalieri's Principle is not when we divide a solid into finitely (in this case 23) many pieces, but only when we divide it into infinitely many pieces. In higher mathematics, we can't simply extract from finite cases to an infinite case without a rigorous theorem or postulate telling us that doing so is allowed. I doubt that the infant mentioned in "extra credit question 5" above is intuitive enough to apply object permanence to infinitely divided objects.

In fact, in the time since I wrote this, I've discussed the Banach-Tarski Paradox. That paradox tells us that we can divide a sphere into finitely many pieces and reassemble them to form two balls. I'd like to see someone try to apply an infant's intuition of object permanence to Banach-Tarski.

The Volume Postulate fails for Banach-Tarski because even though there are finitely many pieces, the pieces are non-measurable (i.e., they don't have a volume). The Volume Postulate fails for the oblique cylinder because we're dividing it into uncountably many flat pieces. In both cases we need something else to help us find the volume -- and in the latter case, that something is Cavalieri.

I wonder how Beals would have responded had the question been, "These two cylinders have the same radius and height, but one is oblique, the other right. The right cylinder has volume pi r^2 h. Use Cavalieri's Principle to explain why the oblique cylinder also has volume pi r^2 h." Nothing less rigorous than Cavalieri gives us a full proof of the oblique cylinder volume formula.

Furthermore, in another post a few weeks before this one, Beals tells the story of a math teacher -- the traditionalist Barry Garelick -- who would only allow those who successfully derive the Quadratic Formula to date his daughter:

http://oilf.blogspot.com/2014/11/my-daughter-can-now-date-barrys-daughter.html

I claim that deriving the sphere volume formula from Cavalieri's Principle in Geometry is just as elegant as deriving the Quadratic Formula from completing the square in Algebra I.