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

Today I subbed in a middle school special education class. This is a self-contained class with several aides who run the class, and so there's no reason to do "A Day in the Life."

It marks one full year of subbing in the new district. My first day in this district was in a middle school English class, with a self-contained high school special ed class the second day.

It's the second time I subbed in this class. The first was about two months ago, and so I wrote about this class in my December 4th post.

But first I must cover another computer class for a late-arriving sub. It's the same multimedia class that I've covered twice before -- first while filling in for another late sub and second as the actual sub for that class. I wrote about this class two weeks ago, in my January 24th post.

Yes, the multimedia teacher tends to be out a lot, and some subs wish to avoid picking it up. As I wrote in my January 24th post, some of these classes are notorious for their bad behavior. But today, all I basically cover is homeroom -- I show the announcements and take attendance, and then the actual sub finally arrives. So I leave for my actual class, the special ed class.

By the way, whenever I see a class that misbehaves as much as the multimedia classes, sometimes I wonder whether it's due to poor classroom management on the part of the regular teacher -- and then I feel guilty. As you already know, my own management two years ago at the old charter school wasn't exactly strong, either. If that had been at a public school and I'd needed a sub often -- even if it's just to attend meetings at the district office -- subs might try to avoid my class too. (But I can see why multimedia would be a difficult class to manage effectively.)

I also assist some students with some math, including adding simple dollar amounts. I keep saying that I'm going to use Number Talks tricks to help these students out, and I never do -- the students end up just using the standard algorithm. One student does use ten-frames to add single-digit numbers -- ten-frames being a staple of Common Core. Another student is supposed to use the standard algorithm to multiply a pair of two-digit numbers.

For the rest of the day, the students read and watch videos on several celebrations going on this week or month -- Chinese New Year (which is today!), Groundhog Day, and Black History Month.

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, and I myself basically skipped over most of it and went straight to Chapter 10. Why did I do this? [2018 update: Of course, this year I finally covered Chapter 9 as well as some of Euclid's theorems.]

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). 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

[2019 update: This sounds like yet another traditionalists' post. Last year, 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 year 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.