Today I subbed in a high school special ed English class. This is in my old district. And yes, the last time I subbed for this teacher was on Pi Day Eve -- the day before the pandemic.
Of course, I won't do "A Day in the Life" today, but I will say a little about this class. With even periods meeting today, second period is juniors, fourth period is seniors, and sixth period is conference. Both grades are listening to an audio recording of The Great Gatsby, Chapter 3.
This is the first time that I've subbed juniors and seniors since the very last time I've subbed for this same teacher, before the pandemic. We already know that sophomores tend to opt out of hybrid, and special ed classes are smaller anyway. So I expect there to be very few students in person -- and there aren't. In fact, there are no juniors in-person today. The aide tells me that there's only one student on Cohort B who hasn't opted out of hybrid. But he's enrolled in two classes, including one of the odd periods -- and he didn't attend yesterday's class, so it's not surprising that he doesn't attend today's class.
The senior class does have one in-person guy. He, along with some of the online students, remember the pizza party we had on Pi Day Eve.
Now that I've taught all secondary grades. I can't help but think about the debate about whether schools should be reopened. In particular, some traditionalists believe that the schools should be reopened, and this includes high schools. But so far, we've seen that if schools are open, most students in Grades 10-12 will opt out anyway, so is there any reason for higher grades to reopen?
Some people bring up private schools in the debate. I wonder whether students in Grades 10-12 whose parents are paying big bucks for tuition are as eager to opt out as those in public schools. For example, Orange County is home to a famous private school, Mater Dei High School. I know that there's a hybrid schedule there since when I click on their school calendar, I see the word "hybrid" all over it.
Of course, Mater Dei is famous for its athletic program -- and I also know that many people who are clamoring for schools to reopen mention high school sports as a reason. I can't help but notice that the only guy I see in-person at my school today is on the volleyball team. Since sports practices are awkward to hold online, I wonder whether athletes are more likely to attend in-person than others
Again, I'm grateful that my schools are open now because I'm getting sub jobs. Just keep in mind that most of my assignments are for Grades 7-9, since so many older students attend online only.
Today is Sevenday on the Eleven Calendar:
Resolution #7: We sing to help us learn math.
And speaking of older students, ordinarily I don't sing in such classes, but since these students remember me from last year, I do sing today. I couldn't let Valentine's Day pass by without singing "Mathematics of Love" from Square One TV. In the senior class, there's enough time for me to sing a second song, and so I choose "Count the Ways," which I've been using as a backup Valentine's song.
I will post the "Mathematics of Love" video again today:
The Mathematics Of Love
Lead vocals by Larry Cedar
Backup vocals by Reg E. Cathey, Cris Franco, Luisa Leschin, and Beverly Mickins
Featured vocals by Arthur Howard
Two hearts two hearts were overflowing
Three words hit like a bolt from above
Bum bum bum
Four arms four arms were hugging tightly
Five times five times I kissed you lightly
So goes the mathematics of love
The mathematics of love
I’ll keep on counting the ways
One thousand nights I’ll hold you
And love you all of my days (and love you all of my days)
Two hearts two hearts were intertwining
So goes the mathematics of love
The mathematics of love
I’ll hold the memory of
The one night two hearts thundered
The mathematics of love
Great, Tony! You got it!
One two three forever
The mathematics of love
One more time!
Do I pay homage to anyone today? I didn't, but Google did. A famous Mexican composer, Maria Grever, was featured with a Google Doodle. Notice that her heyday was in the 1920's and 1930's -- around the time that The Great Gatsby is set (and so it's theoretically possible that Gatsby, or a real person living at that time, would have played some of her music at his parties).
Oh, and speaking of music, you might notice that I've finally made it to the teacher whose guitars and ukeleles I often played. Well, I'm disappointed that he no longer has those guitars in his room. Note that he's in a new classroom now -- I subbed in his old room last week, which is now a self-contained class (and I did find my lost songbook there).
So I guess I can stop talking about those guitars or playing them in the classroom. I won't be performing guitar in the classroom until I decide to bring my own to class again (and which I'm unlikely to do unless it's another long-term).
Once again, I was holding off posting a music topic because I knew that I'd be subbing here today (ever since the last time I was at this school), and I was hoping that there would be guitars to play. I will say a little about guitars, ukeleles, and Arabic fretting (that is, 18EDL).
Recall that the ukelele has only four strings, not six. From my last visit to this teacher's old classroom, the four strings are tuned to GCEA. If I recall, the C string is tuned to middle C on a piano, and the E string is the same as the high E on the guitar. The open G on the ukelele is actually tuned to the G that's above middle C -- that is, the strings are not tuned low to high.
Suppose we had a four-stringed instrument with Arabic fretting. Recall that the interval 18/17 is called the Arabic lute -- not guitar or ukelele -- index finger. And so this ancient lute was just as likely to have four strings as six. (Actually, some research reveals that the Arabic oud probably started out with only three strings, and then one string at a time was added over the centuries.)
OK, so let's assume Arabic fretting with four strings tuned the same as the modern ukelele, GCEA. It makes the most sense to make the G and C white and the E and A yellow. Then the four open strings would produce a just C6 chord, and fingering the A string at the third fret makes a just C major chord.
Notice that while EACGAE -- the guitar tuning I was using when I had the broken D string tuner -- is now obsolete, it lines up perfectly with the GCEA ukelele tuning. Thus we can convert all of those EACGAE songs from my long-term subbing assignment into GCEA tuning on the fly.
(By the way, during my long-term, there was a eighth grade girl who played the ukelele. Because she had opted out of hybrid, she often played the uke from her bedroom when she finished her work.)
Even as I return to standard tuning EADGBE for the guitar, I'm still working on completing my "Another Ratio Song" in 18EDL. It might be easier to come up with playable chords with four strings rather than six, and so I might take the four-string GCEA tuning into consideration.
Recall that the standard D, G, Em7, and Bm7 chords all become just with Arabic fretting. We can almost play many songs in the key of D major with these chords, but we're missing a dominant chord (A in the key of D minor). The standard A chord is dissonant since it contains two different A's and two different E's (one white/wa, the other yellow/yo). One A chord that might be playable is xx2223. It is an A7/E chord, with the C# colored yo and all other notes colored wa.
I'll continue working on "Another Ratio Song" using these chords, and I'll post it when it's ready.
Lesson 10-8 of the U of Chicago text is called "The Volume of a Sphere." In the modern Third Edition of the text, the volume of a sphere appears in Lesson 10-6.
This is what I wrote last year 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.)
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 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!"
Today is an activity day. I'm still trying to figure out how to incorporate the coloring activity that the regular teacher is assigning at my long-term school, without violating copyright.
For starters, here's a link to the website which has copyrighted the material:
https://www.maneuveringthemiddle.com/
I notice that while Kuta has both free and paid worksheets, this website appears to have only paid worksheets and assignments. If it were free, then I could post it without problems, but if it costs money, then I can be sued for posting copyrighted material and losing money for the company.
There are many interesting things on this website (including a blog with several posts on classroom management -- and many of these entries are dated 2016, the year I was at the old charter school). But I couldn't easily find any free worksheets. Therefore I can't post this worksheet without possibly violating a copyright.
Because of this, here's what I'll do. I state that the suggested activity for Day 108 is titled "Volumes of Cylinders Solve and Color," found on the Maneuvering the Middle website linked above. If you're able to access or pay for the lesson, then do so. Otherwise, come up with another activity yourself. I can't post the worksheet because it belongs to Maneuvering the Middle, not me.
Our regular teacher suggests folding lined paper into eighths, showing the work for all 16 problems on this paper, and then scanning and posting this paper. The picture to be colored is worth extra credit -- the students can use colored pencils, markers, or online tools.
I will post the guided notes from last year -- it's my own worksheet, so there's no copyright issues here.
Remember that tomorrow is Lincoln's Birthday on the blog calendar and Monday is Presidents' Day, and so my next post will be Tuesday.
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