Lesson 10-7: Volumes of Pyramids and Cones (Day 107)

Today I subbed at a middle school. Like last Tuesday, it's for a special ed English/history teacher. Since three of the classes are co-teaching and the other two have a special aide, there's no need for "A Day in the Life" today.

Today is also 2/7 -- that's right, it's e Day. The digits of e, the base of the natural exponential function, is 2.718281828459045...., and so today is e Day. I first mentioned e Day was coming back in my Phi Day post on January 6th.

This is my second visit to this class. My first was three weeks ago, and thus I wrote more about this class in my January 15th post. No, I didn't mention e Day to the students -- I might have if these had been high school students who have studied the number e, but these clearly aren't.

The two classes I have in the resident teacher's classroom are both eighth grade English. Like last Tuesday's gen ed English 8 class, these students are learning about arguments and claims. Most of the management problems occur in the sixth period class -- the larger of the two classes (about 24 students) and is right after snack on today's period rotation. The three aides (yes, an extra aide is assigned to this class only) help control this class.

Well, at least I assume the third guy is an aide. Maybe he's there just to place blue tape on the whiteboard to set up an agenda section. Students copy the agenda into their planners, and they receive stamps just as they would for a Warm-Up. So far, the class seems to be slightly better behaved than they were on January 15th. Hey -- perhaps this goes to show that teachers can improve behavior simply by establishing the simple routine of starting class with writing the agenda into a planner, as long as this routine doesn't take too long (cf. my fourth resolution on lengthy Warm-Ups).

Indeed, this reminds me of two years ago back at the old charter school. It was in fact around this time of year, when SBAC Prep was established for sixth and seventh grades. The eighth graders during this time were placed into Study Skills with the history teacher, and he had the students write down daily agendas. He wanted me to tell them to write math (and science) assignments into those same planners.

Stamping the agendas is a good way to encourage the students to complete them, just as today's teacher and aides do -- and I tried to do likewise two years ago. But this confused the students -- should I have stamped something that technically the history teacher assigned? At one point I told them that I'd stamped my Warm-Up page if they completed the planners. But then they assumed I was meaning for them to copy the agenda right there on my Warm-Up page.

(What should I have done that year? Perhaps I should have asked the history teacher whether it was OK for me to stamp his assignment pages. Then not only would the eighth graders have worked on them, but then the history teacher would see the stamp and get confirmation that they actually wrote down the assignments correctly!)

By the way, this is the same class that played the Meghan Trainor parody "All About That Claim," which in turn inspired me to write my own parody "All About That Base (and Height)." By now we've taught enough of Chapter 10 to be able to play this song in full. Then again, that song only mentions prisms/cylinders, not today's pyramids/cones. We could add a third verse to emphasize the factor of a third in the volume formula if we wish.

The seventh grade history class I co-teach is still working in the same unit as they were back on January 15th -- feudal Japan. But the seventh grade English classes have started to read a brand new book -- I Am Malala. Of course I didn't read this book as a young seventh grader myself -- back then the Pakistani author of this book hadn't even been born yet, much less had begun her activism that led to her Nobel Peace Prize.

This English co-teacher informs me that not only will she be out tomorrow, but so will the teacher I'm covering for. That will make tomorrow's class tricky -- it's a class with two co-teachers, and both are out tomorrow. It might have been nice if I could have been retained as a sub for tomorrow (so at least one adult in the class knows what's going on), but unfortunately I'm not.

This means that students might try to take advantage of the two subs tomorrow. For example, some students might choose to listen to Malala on Chromebooks with headphones, but they're supposed to sign out the headphones by writing their names on the front whiteboard. I can easily see them not writing their names and giving some flimsy excuse to the subs why they shouldn't -- and then one of the headphones turns up missing. At least if I were one of the two subs, I could enforce the rule using a teacher look. (I know that you're supposed to write your name to get headphones, so don't even try to convince me otherwise!)

Lesson 10-7 of the U of Chicago text is called "Volumes of Pyramids and Cones." In the modern Third Edition of the text, volumes of pyramids and cones appear in Lesson 10-4.

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

Lesson 10-7 of the U of Chicago text is on the volumes of pyramids and cones. And of course, the question on everyone's mind during this section is, where does the factor of 1/3 come from?

The U of Chicago text provides two ways to determine the factor of 1/3, and these appear in Exploration Questions 22 and 23. Notice that without the 1/3 factor, the volume formulas for pyramid and cone reduce to those of prism and cylinder, respectively -- so what we're actually saying is that the volume of a conic surface is one-third that of the corresponding cylindric surface. So Question 22 directs the students to create a cone and its corresponding cylinder and see how many conefuls of sand fill the cylinder. The hope, of course, is that the students obtain 3 as an answer. This is the technique used in Section 10.6 of the MacDougal Littell Grade 7 text that I mentioned in last week's post as well. (By the way, I just realized that I mentioned four different math texts in last week's post!)

But of course, here in High School Geometry, we expect a more rigorous derivation. In Question 23, students actually create three triangular pyramids of the same base area and height and join them to form the corresponding prism, thereby showing that each pyramid has 1/3 the prism's volume. But this only proves the volume formula for a specific case. We then use Cavalieri's Principle to show that therefore, any pyramid or cone must have volume one-third the base area times height -- just as we used Cavalieri a few weeks ago to show that the volume of any prism, not just a box, must be the base area times height.

I decided not to include either of the activities from Questions 22 or 23. After all, there will be an activity tomorrow and I wish to avoid posting activities on back-to-back days unless there is a specific reason to.

2019 Update: As is usual for Thursdays this year, the second page is a new worksheet of exercises.

Let's get back to e Day. Here is a post from this year mentioning e Day:

https://www.ipwatchdog.com/2019/02/07/in-support-inaugurating-february-7-world-e-day/id=106100/

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:

http://www.haplessgenius.com/mocha/

70 N=7
80 FOR X=1 TO 44
90 READ A
100 SOUND 261-N*(18-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 2,6,6,2,4,9,7,7,5,7
160 DATA 2,4,7,0


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 18EDL, with zero as the tonic. I included 44 digits of e so that the song ends on the tonic. Oh, and using N=7 places that tonic on red E. A song about the number e needs to be in the key of E. (The fundamental tonic for 16EDL would be E, but 16EDL contains only nine notes from tonic to tonic when we need ten.)

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. 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)

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. Some people would suggest 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.