Tuesday, March 12, 2019

Lesson 12-8: The SSS Similarity Theorem (Day 128)

Today I subbed in another high school English class. It's at the same school as yesterday, but in a different classroom. Once again, there's no reason for "A Day in the Life" today.

Yesterday's teacher had only three periods. Today's teacher has six periods -- in other words, he has no conference period. And today's schedule is the opposite of yesterday's -- on Monday I had regular juniors and AP English Language sophomores. Today I have two classes of regular sophomores and four classes of AP English Literature juniors.

Well, that answers my question from yesterday's post. What class do AP English Lang sophomores take as juniors? The answer is AP English Lit. Of course, now I wonder what English they'll take next year as seniors if they've already completed both AP English classes.

Today the juniors are reading Arthur Miller's The Crucible while the sophomores are reading Ernest Hemingway's A Farewell to Arms. There are assignments on both paper and Chromebooks in each grade level.

I name fifth period as the best sophomore class of the day. This is mainly because I must write one name on the bad list for the other class -- a student asks for a restroom pass five minutes after class starts, and per the regular teacher's instructions I have him sign out for the pass. He ends up spending 22 minutes in the restroom before returning, which is much too long. That's why I write his name -- the regular teacher may assign him three days of lunch detention.

The best AP class of the day is seventh period. All classes are hardworking, and so I give it to seventh based on cleaning up the Chromebooks at the end of class.

Today on her Mathematics Calendar 2019, Theoni Pappas writes:

The area of the black region is (9 - 2.25pi) sq. m. A square circumscribes the two congruent semicircles. What's the square's perimeter?

[Here's the additional given info: the "black" region is the area between the square and semicircles.]

Well, the area of the two semicircles equals the area of a circle with the same radius/diameter. And the diameter of this circle would be the side of the square. Thus the given black area should be the area of a square minus the area of a circle.

The given area is 9 - 2.25pi square meters. Just by looking at this, it's easy to say that the area of the square is 9 m^2 and the area of the circle is 2.25pi, so that the square minus the circle is 9 - 2.25pi.

Since the area of the square is 9, its side length is 3, and so its perimeter is 12. Therefore the desired perimeter is 12 meters -- and of course, today's date is the twelfth.

Technically this is a fallacy -- just because S - C = 9 -2.25pi, we can't assume S = 9 or C = 2.25pi. But pi is a transcendental number, so assuming that the side/diameter is a natural (or rational) number implies that the "whole number part" is the area of a square. We should actually write something like 4r^2 - pi r^2 = 9 - 2.25pi and solve for r.

By the way, let me at least mention the elephant in the room -- the college cheating scandal (since it is related to education). I'm ashamed to admit that Southern California is related to the scandal, with both my alma mater UCLA and our rival USC accepting bribes. (See the Joanne Jacobs link below.)

Lesson 12-8 of the U of Chicago text is called "The SSS Similarity Theorem." In the modern Third Edition of the U of Chicago text, the SSS Similarity Theorem appears in Lesson 12-6.


This is what I wrote last year about today's lesson:

Let's get to our Geometry lesson. We are now working on the AA~ and SSS~ theorems, which complete our study of similarity. There are several ways we can prove these at this point. We can use the original dilation proofs given in the U of Chicago text (Lessons 12-8 and 12-9), or we can use the one similarity theorem we already have (SAS~) plus the corresponding congruence theorems (ASA and SSS, respectively). In the past, I reversed Lessons 12-8 and 12-9 but this year I'm preserving the original order, so no, we can't use SAS~ to prove SSS~.

I don't need to make any changes to last year's lesson. But unfortunately, today the pictures/images on Blogger suddenly aren't working for me. It's slow to upload any worksheets, and even those already posted on my blog don't want to appear on my computer.

I have no idea what the pictures look like on computers other than my own. I'm hoping that this will be only a temporary problem, and that everything will be back to normal soon -- especially by Friday, when I want to create and post a new activity page.

Edit: As it turns out, once I finally loaded last year's worksheet, I realized that it was based on a method that I no longer wish to teach (see last week's posts for more info). And so I create and post a new instruction worksheet based on the U of Chicago method.

Let me also add a link to the Joanne Jacobs website, to an article describing the college scandal:

https://www.joannejacobs.com/2019/03/rich-bribe-cheat-to-get-kids-into-elite-colleges/

Traditionalist Bill writes in the comments:

Bill:
This is wrong on so many levels, but it will be interesting what the plea deals will come to, and how many people were denied entry due to these scams…

College admissions are so competitive these days. Indeed, now I'm wondering whether the reason my school gives AP English in Grades 10-11 rather than the usual Grades 11-12 is so that it looks better on college applications. This way, both years of AP English are completed, and the scores have been received for both years before college applications are due.

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