Wednesday, February 7, 2018

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

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.

Today is also 2/7/18 -- that's right, it's e Day of the Century. The digits of e, the base of the natural exponential function, is 2.718281828459045...., and so today is the special e Day.

I first mentioned e Day was coming back in my Phi Day post on January 6th. It's amazing how Pi Day, Phi Day, and e Days of the century are all within a few years of each other. A high school freshman on 3/14/15 becomes a senior in time for 1/6/18 and 2/7/18.

I also pointed out that of all the special days of the century, only e Day falls on a school day. Both Pi and Phi Days of the Century fell on Saturdays. Therefore, I'm hoping that many Algebra II classes will be celebrating e Day of the Century today.

(Interestingly enough, today's special e Day post is the 666th of my blog. This proves that "e" must stand for "evil.")

A Google search reveals a few e Day links. Amazon is selling an e Day t-shirt:

Meanwhile, a Reddit user suggests eating Yule Logs on e Day -- except it should be "Oil Logs" instead, as in "Oiler" (Euler), the mathematician who named the number e. Also, "logs" remind us of logarithms, as in "ln," the log to base e.

On the MTBoS, the famous math blogger Sara Vanderwerf includes both Phi and e Days of the Century in a list of math holidays for the year:

On the Twitter side, Luke Walsh and a tweeter who goes only by "Maths Ed" are making several tweets under the e Day label this week.

No math holiday is complete without music. Apparently e is a magic number. (Well, the singer does use e in the magic formula e^(i pi) + 1 = 0, so I guess that's magic.)

Here's another e song -- and this one actually explains why the magic formula works:

Unfortunately Michael Blake has written songs based on the digits of pi and Phi, but not e. Here is a link to Allan Costa, who does use the digits of e to write a song:

Of course, even if Blake doesn't have an e Day song, we can write it ourselves in Mocha. I've taken the songs we coded for Phi Day and changed them to the digits to e below:

10 DIM S(9)
20 FOR X=0 TO 9
30 READ S(X)
50 DATA 72,180,160,144,135
60 DATA 120,108,96,90,80
70 N=1
80 FOR X=1 TO 39
100 SOUND 261-N*S(A),4
110 NEXT X
120 DATA 2,7,1,8,2,8,1,8,2,8
130 DATA 4,5,9,0,4,5,2,3,5,3
140 DATA 6,0,2,8,7,4,7,1,3,5
150 DATA 2,6,6,2,4,9,7,7,5

Notice that whereas we can avoid the zero problem in pi, zero appears early in e. Blake uses a rest for the zero in his song, but I decided to use a major tenth for zero instead, since the 0 key is to the right of 9 on the keyboard. (Also, notice that the notes start with C on 1, whereas Costa starts with C on 0. If you prefer Costa's version, drop 72 from the start of Line 50 and add it to the end of Line 60.)

Once we have completed the song for the major song, there's not much effort needed to change the song to make it fit the minor scale:

50 DATA 30,72,64,60,54
60 DATA 48,45,40,36,32

or the New 7-Limit Scale:

50 DATA 54,105,96,90,84
60 DATA 81,72,70,63,60

or the Bohlen-Pierce Lambda Scale:

50 DATA 21,63,53,49,45
60 DATA 38,35,29,27,23

This is why my New 7-Limit Scale contains ten notes -- I had digit songs like the e song in mind when I created it.

Numberphile, of course, has an e video. (The continued fraction for e -- which I mentioned in my December 28th post -- appears in the video as evidence that e is irrational.)

One person who once wrote an e song is Bizzie Lizzie -- that is, Elizabeth Landau. Her old Sailor Pi page used to link to her e song, but unfortunately it disappeared years before the pi pages did, and so I never wrote down the lyrics.

Of course, now that I found Landau's LinkedIn page, we could attempt to contact her via LinkedIn and ask her to post the lyrics somewhere. (See my January 26th post for more information.) The other alternative would be for me to drive today to the city of La Canada Flintridge to find Landau. (Yes -- this city of 20,000 is where her workplace, JPL, is actually located -- not Pasadena. Caltech, on the other hand, really is in Pasadena proper.)

Bizzie Lizzie's e song was a parody of "Sugar, Sugar," by the Archies. I think I recall the refrain:

e (2.718)
Ah, number number (281828)
You are my natural log,
And you got me calculating.

e (2.718)
Ah, number number (281828)
You are my derivative,
And you got me calculating.

But alas, I can't remember the rest of the song. Well, since it's e Day, let me supply extra lines. Some of these lines are from my faint memories of Landau's original song -- I made up all the lines that I couldn't remember.

1st Verse:
I just can't believe the loveliness of graphing you.
I can't believe you're more than two.
I just can't believe the loveliness of graphing you.
I can't believe you're more than two. (to Refrain)

2nd Verse:
I just can't believe your digits go forever now.
As long as a number can be.
I just can't believe your digits go forever now.
As long as you're the number e. (to Bridge)

Put a little cash in the bank, money.
Put a little cash in the bank, baby.
I'll make more next year, yeah, yeah, yeah!
Put a little cash in the bank.

100% interest on my money.
Compound it continuously, baby.
I'm gonna take the limit now, yeah, yeah, yeah!
My cash is multiplied by you, e. (to Refrain)

The bridge is mostly mine -- Landau didn't mention anything about money in her song. I chose to include money since it rhymes with the original Archies lyrics ("honey") as well as retell the story of Jacob Bernoulli's discovery of this constant.

Landau also had a Phi song -- but it was merely a version of her "American Pi." Her "American Phi" was nearly identical to the pi version, with the following as one of the changes:

American Pi:
I just want to see the numbers 3.1415.

American Phi:
I just want to calculate the square root of 5.

I personally like the pi version better. (The original song is "American Pie," not "American Fie" or anything like that!) And so I don't consider "American Phi" to be a true Phi Day song.

By the way, let's think back once more to e Day itself. The Reddit link above suggests that in deference to Europeans and other little-endians, e Day should be 2nd July, not February 7th. For that matter, Phi Day should be 1st June, not January 6th. (Instead of 1/6, we could also use 16/1, or 16th January, for little-endian Phi Day.) Unfortunately, 2nd July is during summer break and too close to the Fourth of July in this country. Though 1st June is on a Friday, it may be too close to the last day of school to make it a celebration day in math classes.

Some Phi Day celebrants suggest observing it on June 18th, merely because they are thinking about phi = 0.618... rather than Phi = 1.618... as the number to consider. Another suggestion is for Phi Day should be the date that divides the year into a golden ratio, which works out to be either August 13th or 14th each year. The former date, August 13th, has the advantage of serving as Phi Approximation Day as well, since 13/8 is approximately equal to Phi (in the same way that July 22nd is Pi Approximation Day). Then Phi Day of the Century would approximately divide the century in a golden ratio. Perhaps we could continue taking digits of phi, but in big-endian format (year first) --observe it on 61/8/03 (that is August 3rd, 2061 -- or maybe August 13th, 2061 for the pattern). Notice that this works better if we consider the century of the 2000's (2000-2099) rather than the 21st century (the official definition of which is 2001-2100).

Meanwhile, e Approximation Day works out to be July 19th as 19/7 = 2.7142857... is approximately equal to e. So e Approximation Day would be just three days before Pi Approximation Day.

One last constant I wish to celebrate on the calendar is closely related to e -- eta. The number eta is used in tetration (which I explained in a June 2016 post). We define eta as e^(1/e), because if we define the sequence:

a_0 = 1
a_(n + 1) = eta^(a_n)

then the limit of a_n as n approaches infinity is e. But if we replace eta with any greater base -- even eta + epsilon for small epsilon -- the sequence diverges. The tetrater Gottfried Helms was the first to use the symbol eta for e^(1/e).

eta = e^(1/e) = 1.44466786...

Thus Eta Day would be January 4th -- yes, just two days before Phi Day. Little-endians might observe Eta Day in April -- on either the 1st (1/4) or 14th (14/4). As for Eta Approximation Day, we can clearly approximate eta by the rational number 1.444..., which is 13/9 for September 13th. We are more than two decades away from Eta Day of the Century.

On her Mathematics Calendar 2018, Pappas gives us a second straight Geometry problem:

(We are given two concentric circles. Points A, B, C, and D lie on a chord common to both circles -- AD is the chord of the larger circle and BC a chord of the smaller circle. The perpendicular dropped from the center to this chord has length 3, and the smaller radius is 5.)

BD = 15, find AB.

Even though we haven't reached Chapter 15 on circles yet, we can solve this problem now. First of all, notice that the given length 3 is the length of a right triangle with 5 the hypotenuse, and so by the Pythagorean theorem, its leg has length 4. This length is half that of BC, which is thus 8. (That the perpendicular from the center to the chord bisects the chord can be proved using HL.)  Since BC = 8 and BD = 15, we conclude that CD = 7. And CD = AB (provable using HL again), so AB = 7. The desired length is 7 -- and of course, today's date is the 7th.

This is what I wrote two years ago 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

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.
(+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.
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:

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:

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.

I wanted to make this post at an appropriate time for e Day. I choose 7:38 AM (Pacific time) in order to represent the number e^2.

No comments:

Post a Comment