Lesson 10-1: Surface Areas of Prisms and Cylinders (Day 101)

Today I subbed in a self-contained middle school special ed class. There are several aides the entire day, and thus there is no "A Day in the Life" today.

I've subbed in this room before, but not since last year. There is in fact a new regular teacher for this class, while many of the eighth graders return to this class from last year.

The aides lead the other lessons, but of course I make sure to lead the math lesson. This year, the seventh and eighth graders have been separated into two math classes -- while the eighth graders have math the seventh graders have English, and while seventh graders have math, the eighth graders are in another classroom altogether. (Hence it's not quite a self-contained class.)

In any case, both grades are studying two-step equations, so my "Solving Equations" song fits here. I notice that not all the students enjoy this song, so when I help them out, I switch to SADMEP. (There are no parentheses here, so the traditionalists' post from yesterday about whether to distribute or use SADMEP is irrelevant today.)

While the seventh graders have English, I take the eighth graders to the center room and have each of the three students solve an equation on paper. In seventh grade I have the luxury of using the front whiteboard, so I have each of the three students solve three equations on the board. Furthermore, since there's no risk of disturbing the eighth graders who are gone, I add another song -- what's become my late January song, "No Drens."

Ordinarily, I like to give out pencils and candy to this sort of small class. But I didn't pick up this job until this morning, before I had a chance to purchase any goodies. In this case I'd go out and get it during any break I had. But today was the rare combination when it's Wednesday, an early out day with only one class after lunch, and that one class is my conference period. Thus by the time I get a long break, the students have already left the classroom for the day!

For the students, P.E. is their final class today. I decide to go out and watch them. (As it turns out, many kids from yesterday's subbing class also have P.E. at this time.) The P.E. teacher introduces them to the high jump, a Track and Field event. It delights me, as a former Track and Field athlete (albeit the distances) to see the high jump in a P.E. class for the first time.

Today is Fiveday on the Eleven Calendar:

Decade Resolution #5: We treat the ones born in 1955 like heroes.

And so I mention our three heroes -- Gates, Jobs, and Berners-Lee -- to both grades. With seventh grade, I connect them to the "No Drens" song. In eighth grade, I mention them in English. The students are reading "Flowers for Algernon," and the protagonist, Charley, fantasizes about raising his IQ from 68 to 200. In order to give the students an idea of what those IQ's mean, I quickly look up the scores of the three heroes. Their IQ's have been estimated as around 160.

Oh, there's one more thing I'd like to say about this fifth resolution. I refer to the 1955 trio as members of a "generation," but of course, one year does not a generation make. It's obvious that the fifty-fivers would be considered part of the Baby Boom Generation -- indeed, since this generation is almost always given as 1946-1964 (as WWII ended in 1945), 1955 would be considered the exact midpoint of the generation. So if anyone is a Baby Boomer, the fifty-fivers are.

I bring this up because lately, a certain phrase has become more common to disparage the members of the Baby Boom Generation. That phrase is "OK Boomer."

The meaning of the phrase is that the Boomers' thinking is outdated. But what exactly makes their thinking outdated? Well, I'd say new technology, for one thing. And who exactly invented all that new technology? That's right -- Gates, Jobs, and Berners-Lee. In other words, Boomers have become outdated because of very (three specific) Boomers!

In following the fifth resolution, I won't use the word "Boomers" to refer to the three heroes. Instead, I'll only use 1955 or "fifty-fivers," or perhaps refer to their high school grad year of 1973. (Once again, "seventy-three'ers" doesn't sound right, so I'll say "Class of '73" instead.) This is to avoid the possibility of anyone saying "OK Boomer" to refer to this suddenly-disparaged generation.

In many ways, the traditionalists' debate reminds me of "OK Boomer." It's possible that many traditionalists are Boomers, especially since they often bring up the Golden Age of Textbooks (whose dates approximately line up with the generation, 1946-1964). But once again, I won't refer to them as Boomers -- I'll just call them traditionalists. I have no reason to insult the entire generation just because I disagree with the traditionalists (since three members of that generation are heroes).

Lecture 17 of Michael Starbird's Change and Motion is called "Real Numbers and the Predictability of the Continuous." Here is an outline of this lecture:

I. The number line is considered from kindergarten to graduate school.
II. Filling all the gaps in the number line creates the real numbers.
III. Every point on the number line is a real number.
IV. 0.99999... equals 1.
V. Accept the limit.
VI. Discrete versus continuous views of the world.
VII. The idea of continuity inherently involves the idea of the dependency between varying quantities.
VIII. We can examine the idea of continuity graphically.
IX. Continuity represents a valuable viewpoint about nature.

Starbird's lecture is on the set of real numbers. The set of real numbers is first taught in eighth grade under the Common Core, but we don't truly know what they are until Calculus.

Of course, he begins by first describing the natural, integer, and rational numbers. He mentions the Pythagorean Theorem and how it shows that the diagonal of a unit square is the irrational sqrt(2). It can be shown that the Real Numbers = all infinite decimal numbers. All real numbers have infinitely many digits, even 1.1, which can be written as 1.100000....

The professor shows us that these real numbers resolve the Runner Paradox (that is, the one involving Achilles and the Tortoise). We see that the Achilles catches the Tortoise at 1.1111..., and he tells us how to find its value:

n = 0.1111...
10n = 1.1111...
-n      -0.1111...
9n = 1
n = 1/9

Thus the warrior catches up to his reptilian friend at point 1 1/9. But now Starbird moves on and uses this same argument for another number -- 0.9999...:

n = 0.9999...
10n = 9.9999...
-n      -0.9999...
9n = 9
n = 1

Therefore 0.9999... is exactly 1, not merely close to 1. In particular, there can't be any number strictly between 0.9999... and 1, such as the average of 0.9999... and 1. We formalize this using limits:

1 = lim{.9, .99, .999, .9999, ...}
1/9 = lim{.1, .11, .111, .1111, ...}

The professor now introduces the idea of continuity -- if we know our position at every instant except noon, then we can figure out where we must have been at noon.

Starbird shows us graphs of position functions that are continuous and some that have a jump instead. The continuous functions can be drawn without lifting our pencil. The area of a circle varies continuously as the radius, since circles with nearly equal radii have nearly equal areas.

The professor concludes his lecture by showing why the real numbers are complete -- if we divide the rational numbers into two sets (a left and right set), then there must be a real number in between. He refers to these as Dedekind cuts. (As I once stated on the blog, I learned about these cuts in a college math course, although they weren't referred to as "Dedekind" cuts.)

Chapter 10 of the U of Chicago text is on surface areas and volumes. Measurement is usually the focus of the three-dimensional chapters in a Geometry text, not Euclid's propositions that we've been discussing the past two weeks.

Lesson 10-1 of the U of Chicago text is called "Surface Areas of Prisms and Cylinders." In the modern Third Edition of the text, surface areas of prisms and cylinders appear in Lesson 9-9.

I don't have much to say about today's worksheet. I will say that I include the Exploration Questions as a bonus. One of them is open-ended -- don't let traditionalists see that problem, as they'll complain it's ill-posed. Everything else I have to say about lateral and surface area I mentioned in my parody song "All About That Base (and Height)."

Even though we're in Chapter 10 now, we might as well continue with Euclid. After all, David Joyce implies that he wouldn't mind teaching only "the basics of solid geometry" and throwing out surface area and volume altogether.

Proposition 14.

Planes to which the same straight line is at right angles are parallel.

This is another version of the Two Perpendiculars Theorem. Earlier, in Proposition 9, we had two line perpendicular to one plane, and now we have one line perpendicular to two planes. In all three theorems, two objects perpendicular to the same object are parallel.

Euclid's proof, once again, is indirect.

Indirect proof:
Point A lies in plane P and point B lies in plane Q, with AB perpendicular to both. Assume that planes P and Q intersect in point K. By the definition of a line perpendicular to a plane, AB is perp. to AK, and for the same reason, AB is perp. to BK. Then Triangle ABK would have two right angles, which is a contradiction since a triangle can have at most one right angle. (This is essentially Triangle Sum -- the two right angles add up to 180, so all three angles would be more than 180.) Therefore the planes P and Q can't intersect -- that is, they are parallel. QED

Euclid mentions a line GH where P and Q intersect, but Joyce tells us it's not necessary. Joyce also adds that Euclid forgot to mention the case where K, a point common to both planes, is actually on the original line that's perpendicular to them both. He doesn't tell us how to prove this case, but here's what I'm thinking -- consider a plane R that contains the original line. Planes P and R intersect in some line through K, and planes Q and R intersect in another such line, since the intersection of two planes is a line. By definition of a line perpendicular to a plane, these two new lines are both perp. to the original line. So in a single plane R, we have two lines perpendicular to a line through the same point K on the line, a contradiction.

Here is the worksheet for today's lesson: