Today I subbed in a high school special ed class. It's in my first OC district -- and in fact, it's the class I've covered several times lately, most recently last week in my April 28th post. Since today is Tuesday, there is a Business Math class, and so I will do a quick "A Day in the Life" today:
9:00 -- Third period arrives (since "first period" = zero period). This is a senior English class.
Now that The Great Gatsby is complete, it's time to start the final novel of the year -- Siddhartha, by Hermann Hesse. Although the author is German (and both Gatsby and Siddhartha were written a few years apart), this novel takes place in ancient India during the Buddha's time.
Unlike The Great Gatsby, I really did read Siddartha as a young high school student. It was one of the novels I read when I had two English classes during my senior year. (Thus I might have been reading this book in-person in the COVID-97 universe.)
I don't remember much about Siddartha except that the title character eventually meets a woman named Kamala -- significant only because we now have a Vice President Kamala Harris. (Yes, it's well-known that the VP's first name is Indian.)
Oh, and by the way, today is the day of the AP Calculus exam. Even though this is a class of seniors, these are special ed students, and so it's doubtful that any of them would be in Calculus.
9:55 -- Third period leaves for break.
10:10 -- Fifth period arrives. This is the Business Math class.
The class is now learning about auto insurance. I just paid for my insurance premium last week, and so I connect my own experience to what the students are learning now.
This is the only class in which I sing a song -- that's mainly because there's just barely enough time to get two chapters of Siddhartha in during the English classes, with no time for songs. This time, I'm already looking ahead to Cinco de Mayo, and so I sing "Sign of the Times" from Square One TV.
11:05 -- Fifth period leaves and seventh period arrives. This is a junior English class.
Once again, both grades are reading the same books, and so they read Siddartha as well.
12:05 -- Seventh period leaves, thus completing my day since the regular teacher almost always covers academic support for this class.
This one is tricky today, especially in the English classes, since we just barely get through the reading with no time for answering the questions.
This is what I wrote two years ago about today's lesson:
Lesson 15-9 of the U of Chicago text is on "The Isoperimetric Theorems in Space." These are the 3D analogs of the theorems we discussed yesterday.
Isoperimetric Theorem (space version):
Of all solids with the same surface area, the sphere has the most volume.
Of all solids with the same volume, the sphere has the least surface area.
We don't even bother trying to prove these theorems. As we've seen, the 2D proofs are very difficult, so imagine how much more so the 3D proofs would be.
This is the final lesson in the U of Chicago text. Here is how the U of Chicago closes the text:
"The Isoperimetric Theorems involve square and cube roots, pi, polygons, circles, polyhedra, and spheres. They explain properties of fences, soap bubbles, and sponges. They demonstrate the broad applicability of geometry and the unity of mathematics. Many people enjoy mathematics due to the way it connects diverse topics. Others like mathematics for its uses. Still others like the logical way mathematics fits together and grows. We have tried to provide all these kinds of experiences in this book and hope that you have enjoyed it."
Well I for one have certainly enjoyed this text, and I hope you, the readers of this blog, have as well.
This lesson mentions the ancient Carthaginian queen Dido. I wrote about her last year as well:
"According to one of the legends of history, Dido, of the Phoenician city of Tyre, ran away from her family to settle on the Mediterranean coast of North Africa. There she bargained for some land and agreed to pay a fixed sum for as much land as could be encompassed by a bull's hide."
"Her second bright idea was to use this length to bound an area along the sea. Because no hide would be needed along the seashore she could thereby enclose more area."
We know that the solution to the Isometric Problem is the circle -- the curve that encloses the most area for its length. We've also seen questions in which we are to maximize area by building a fence along a river to enclose a rectangular area -- the answer is a rectangle whose width is exactly half of its length. Combining these two ideas, we can solve the Dido problem:
"According to the legend, Dido thought about the problem and discovered that the length of hide should form a semicircle."
So we see that without water, the largest area is a circle -- with water, it's a semicircle. If we restrict to rectangles, without water the largest area is a square -- with water, it's a semi-square (that is, half of a square, or a rectangle whose width is half of its length).
"[Dido's new lover] Aeneas was a man on a mission, and he soon departed to found a new civilization in Rome. Dejected and distraught, Dido could do no more for Aeneas than to throw herself on a blazing pyre so as to help light his way to Italy...Rome made no contributions to mathematics whereas Dido might have."
By the way, let's tie this back to Pappas and ask, what numerals would Dido have used? Carthage is actually derived from the Phoenician culture. Notice that the Square One TV video "The Mathematics of Love" seems to imply that the Phoenicians used our (current) numerals (in contrast with the Roman), even though Phoenician has nothing to do with Hindu-Arabic. So in the end, I don't know what numerals Carthaginians might have used.
Since the test is tomorrow, let me post the Chapter 15 review questions as well. My new first page includes two questions each from Lessons 15-3 and 15-4 and one each from 15-1, 15-5, 15-6, and 15-7.
Last year's Page 2 contains some questions from Lessons 15-3 and 15-8. But there are also some questions from different sections.
Questions 9 through 11 are on the equation of a circle, Questions 14 and 16 are on the volume/surface area of a sphere, Question 15 is on the Pythagorean Theorem, and Question 17 is on the relationship between radius and tangent.
The new version of the test still contains these questions from four other chapters, but now most of the questions are from Chapter 15.
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