Today I subbed in a middle school special ed English class. It's in my first OC district -- and in fact, it's the same class that I subbed in a week and a half ago. I describe this class in my January 22nd post.
In January, it was a day when periods 4-6 met -- conference period, co-teaching, and then only one genuine class to cover. Today it's periods 1-3 -- so naturally all three classes are genuine. All three classes are doing a current event on an article about MLK's March on Washington.
Since exactly eleven days have passed since my last visit, it's Nineday again on the Eleven Calendar:
Resolution #9: We pay attention to math as long as possible.
That troublemaker from this class -- I mentioned him again in yesterday's post -- doesn't meet today, since he's in Friday's class, not today's. There is one guy in today's second period who keeps turning on music -- first on a phone, then a Chromebook -- that's loud enough for me to hear. Normally I don't mind if a student listens to music on a Chromebook using earphones, but he doesn't have any. There is no official aide today -- another adult passing through the classroom offers him earbuds. But he rejects them and asks for a restroom pass instead. Again, this takes us back to the original ninth resolution -- when a student is bored in class and loses control of a situation, he goes to the restroom instead.
If you'd rather think about my "one word" resolution -- homage -- OK, I paid homage to MLK today during today's lesson. (As I mentioned yesterday, it is Black History Month.)
I won't do "A Day in the Life" since I ordinarily don't for this class, even though there's no aide today. As I said earlier, several adults pass through the room. One of them is a tech guy -- my Chromebook is slow and I have trouble getting to Zoom.
Since it's no longer Palindrome Week, I don't sing that song today. Instead, I sing the song that I associated with February 2nd at the old charter school -- "Linear or Not."
LINEAR OR NOT
Exponent on the variable?
It's not linear!
Multiply two variables?
It's not linear!
Variable on the bottom?
It's not linear!
Everything else?
It's linear!
As I promised over the summer, I've made this song into a rap. Thus pitch doesn't matter here nearly as much as the rhythm. I repeat the line "It's not linear!" twice each time it comes up.
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.
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:
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