Today is Day 108 -- at least in my old subbing district. It can be considered as the end of the fifth quaver, which we determine by noting that one-quarter of the way from Day 85 (the end of the first semester) to Day 180 is 107.75. Of course, in my new district today is only Day 99. Thus it's not yet the end of the fifth quaver in my new district, seeing as it's only the second week of the semester.
Oh, and this is a good time to mention another difference between the two district calendars. In my old district, Lincoln's Birthday and President's Day are on two separate Mondays. But my new district does something different -- this one and several other districts observe Lincoln's Birthday on the Friday before President's Day, so that there can be a four-day break in February.
(Two years ago at the old charter, by the way, students observed a five-day weekend, from Thursday to Monday. But the first two days had nothing to do with Lincoln's Birthday. Instead they were PD days -- in other words, we teachers had only a three-day weekend. President's Day itself is the only school-closing February holiday in the LAUSD.)
What does this mean for the blog calendar? Well, I'm already committed to the calendar of the old district on the blog. This means that Lesson 10-9 won't be posted until Tuesday.
Meanwhile, today I subbed in a middle school special ed class. This is my third visit here -- the first was back on November 9th. As usual, you can refer to my November 9th post to learn more about this class.
This is a self-contained class where the same six or seven students (all boys) stay in our classroom for most of the day. The middle school rotation today starts with second period, which happens to be math. That's right -- this is the school where all the periods rotate, and the rotation has nothing to do with the day of the week.
Once again, there's no "Day in the Life" because there are special aides in charge. Apparently, this is the time of year when all the special ed teachers have their meetings, since every class I've subbed for this week is special ed.
The fact that the rotation starts with second period benefits me, since I'm a math teacher. If you recall from my November 9th post, the students often have free time during the last period on Fridays. And on Friday, November 9th, the rotation started with third period and ended with math. (That's what happens when the rotation isn't tied to the day of the week!) Today, it's first period science that gets the short end of the stick.
And thus I do get to see some math today. The students have five problems to solve -- three multiplication problems, one long division, and one fraction addition. Here is a sample of some of the problems the regular teacher left on the board:
1. 16 * 90 =
2. 4/8 + 3/24 =
3. 11 * 60 =
4. 7929/67 = (long division)
5. 629 * 84 =
(All of these problems are made-up except the fractions, which is the exact problem she left.) Two students do fairly well on these problem. One boy struggles -- mainly because his writing is so messy that he can't keep track of the numbers. He also tries to simplify 4/8 = 1/2 and thinks he's done without realizing that he's supposed to add the fractions. (It's too bad that the teacher didn't originally pose the question as 3/24 + 4/8 instead. If the boy simplifies 3/24 = 1/8 first, then all he'd needs to do is add 1/8 + 4/8 = 5/8.)
I wish I could have helped this student more. I wonder whether an alternate method from the Number Talks book would have helped -- perhaps setting up the multiplication in boxes (near the end of the multiplication chapter).
But then a traditionalist would counter that what he needs isn't another method -- he needs to learn how to print more neatly overall so he can do the standard algorithm correctly. (Later on in English, his printing is still messy.) And besides, there's no guarantee that he would draw the boxes neatly. (I wouldn't recommend the lattice method that many of my old charter students favored, since that requires even more neatness than the standard algorithm.)
As it turns out, one of the aides is also out today -- and his sub is also strong in math. He tells me that unlike me, he hasn't earned his teaching credential yet.
By the way, try guessing what the students are writing in English. Yes, you guessed correctly -- a persuasive essay! And this one is also about violence in the media, just like yesterday's class. So obviously this is a district-wide assignment. (Checking back to my subbing posts from last year, yes, the students were writing persuasive essays in February as well.)
Actually, there is a girl in the class today -- but she's only scheduled here for homeroom. She also returns just before 11:30, when her parents check her out for the day. (There's another girl on the roster for first period only, but she's absent the whole day.)
It's approaching Valentine's Day, so you know what that means -- pencils and candy. I give two eighth graders a Valentine's pencil -- including the girl, just before she leaves (while the seventh graders are in another room). At the end of the day, I give four seventh graders a pencil plus some chocolate that I bought during my long lunch/conference period (while eighth grade is in another room). Because of the schedule, the eighth graders miss out on the holiday sweets.
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
The exterior angle of a 16-sided regular polygon is (pi/?) radians.
At first, this seems to be a straightforward application of the Exterior Angles of a Polygon Sum Theorem, which doesn't appear in the U of Chicago text until Lesson 13-8. The angles add up to 360, so all we should have to do is divide 360 by 16.
That's until we realize that we need to give the angle in radians which don't appear in most Geometry texts at all. Provided we know that 360 degrees is 2pi radians, it's simple -- instead of 360/16, our answer is 2pi/16, which reduces to pi/8. The question already provides the pi, so all we need to do is fill in the ? with 8. Therefore our answer is eight -- and of course, today's date is the eighth.
We might have missed science today, but let's not miss the science Google Doodle today. Friedlieb Ferdinand Runge was the 19th-century chemist who researched the caffeine molecule. (Missing science -- hey, that reminds me of the problems I had at the old charter school.)
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.)
Here is a Square One TV video about Archimedes:
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. It's based on this lesson's Exploration question:
25. Unlike a cone or cylinder, it is impossible to make an accurate 2-dimensional net for a sphere. For this reason, maps of the earth on a sheet of paper must be distorted. The Mercator projection is one way to show the earth. How is this projection made?
Notice that the correct answer to this question is quite complex. Here's a link that describes both a misconception and the correct answer:
https://www.math.ubc.ca/~israel/m103/mercator/mercator.html
There is some controversy regarding the Mercator projection. The following link describes the problems some people have with this map (including a clip from the TV show West Wing).
Remember that Monday is Lincoln's Birthday in my old district (and on the blog calendar), and so my next post will be Tuesday.
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