Monday, September 14, 2020

Lesson 1-9: The Triangle Inequality (Day 19)

Today I subbed in the same special ed history class as Friday. Indeed, this teacher tells me that she'll be out indefinitely -- eventually, a long-term sub will be needed for this class. After today, I'll be in the class for at least one more day -- possibly two.

And as I wrote in my last post, today's the first day that subs are reporting to campus. Again, the hope is that if Zoom doesn't work, there'll be someone there to help me.

So, does this second day of subbing go more smoothly than Friday? We'll find out in today's "A Day in the Life" -- which I am doing today because this has now become a multi-day assignment. Indeed, my last subbing assignment before the closure was at this very school, and it was also a three-day assignment in a special ed class.

So without further ado, here is today's "A Day in the Life":

9:00 -- Today is Monday -- the day when all classes meet as the "Launch Day" for the week. But first period really means zero period in this district, and we know from Friday that second period is the teacher's conference period. So my day doesn't start for some time now.

After Friday's fiasco, I start out by making sure that I can connect to Zoom, as well as checking the Google Classroom and attendance links as well.

With so much extra time before the first class -- and knowing that this is the school where I lost my old songbook six months ago -- you might wonder whether I try looking for it today. Well, the answer is no, I don't even bother. Substitute teachers are working from a special room set up inside the library, and the area where the classrooms are is blocked off. Even if the classrooms weren't blocked off, most likely the room where I left my songbook is locked -- and it's just as likely that the book was thrown away months ago. Once it was announced that the schools wouldn't reopen before summer, I assumed that the book was lost forever. (That's why I made reblogging my old songs one of my summer projects.)

10:20 -- Third period begins now. Many special ed teachers have periods of co-teaching, and this regular teacher is no exception. Actually, she has two co-teaching classes -- it's just neither of those classes met on Friday, so I didn't write about them until today.

And as it turns out, both co-teaching classes are English, despite her being a history teacher. The third period class is a senior English class. The students will soon read Born a Crime, the autobiography of talk-show host Trevor Noah.

The resident teacher begins the day with a Warm-Up, where she gives the students a topic and they respond using the Zoom "chat" (which really means "typing") feature. Today's topic is about the upcoming hybrid schedule and the students' feelings about it. She really does the word "concurrent" to describe this schedule, where some students are in class while the others watch on Zoom. Thus it's official -- the concurrent hybrid schedule is indeed coming in two weeks.

Some students ask whether they are Cohort A (in-person Tues.-Wed.) or B (in-person Thurs.-Fri.), and the teacher replies that it will most likely be alphabetical by last name. Others ask why exactly there needs to be a hybrid schedule in the first place -- and the teacher herself doubts that this is the best possible schedule. Yes, concurrent is tricky for both students and teachers -- once again, when I recommended hybrid here on the blog, I didn't mean concurrent hybrid.

10:50 -- Third period ends for snack break.

11:10 -- Fourth period begins. This is the career guidance class.

The good news is that the Zoom video is working now. The bad news is that the microphone on my Chromebook isn't, and so the students have trouble hearing me. It doesn't help that one guy is playing music in the background -- it's not too loud, but when my voice is also quiet, it's a problem.

I've heard that Zoom has a "mute" feature where the teacher can silence the students, but I've never known how to use it. (It doesn't help that the special aide who was here on Friday is out today.) But once again, the students still might not have heard me even if I knew how to use the mute.

And so only a few students know how to access the video that they're supposed to be watching -- it's a short video of the career counselor showing them how to fill out a job application.

11:40 -- Fourth period ends and fifth period begins. This is the second English co-teaching class -- this time, it's sophomore English.

The resident teacher introduces a famous speech to the students -- the Whiskey Speech. First given by Judge Noah Sweat in 1952, the speech both attacks and defends the repeal of prohibition in the state of Mississippi. "If by whiskey you mean the devil's brew...then certainly I am against it. But if by whiskey you mean the oil of conversation...then certainly I am for it."

The students' task is then to write their own "whiskey speech" about some topic. If they do it right, it should be difficult to determine which side of their chosen issue the writer is on.

Hey, I can do my own "whiskey speech" right now, about the hybrid schedule:

If by hybrid schedule you mean the concurrent plan where teachers must teach to students in person and on Zoom at the same time, with these poor teachers unable to focus on either group and forces them to multi-task, then certainly am I against it. But if by hybrid schedule you mean a plan where the at-home group is doing asynchronous work while the teachers are truly devoting all their time to the smaller in-person, socially distanced group, and then the at-home group can Zoom in with the teacher later on to get extra help if they need it, then certainly I am for it. This is my stand. I will not retreat from it. I will not compromise.

12:20 -- Fifth period ends for lunch.

1:05 -- Sixth period begins. This is the first of two US History classes.

The Monday Launch Day for the history classes involves reading a current event. This week's current event is all about -- what else -- distance learning all around the country. The main history lesson (about slavery -- one of the first topics of the year in a junior history class) will be given the rest of the week.

But unfortunately, the students still can't hear me. The aide for this class decides that I have no choice but to find another Chromebook with a better mic. This I do between sixth and seventh period.

1:35 -- Sixth period ends and seventh period begins. This is the second of two US History classes.

Well, no one complains about being unable to hear me, so I figure that the mic is working now. For some reason, I can't see the link to the current event questions, but the aide for this class reassures me that the students can see them, even if I can't.

Now that my mic is working, I decide to sing a song to this class. (Actually, I attempt the song in sixth period but I don't know whether anyone heard it. I didn't even bother singing to fourth period since I knew no one could hear me.) But with the uncertainty with the sound, I didn't want to waste time preparing one of my longer songs, so I choose "Benchmark Test Song," a shorter song associated with the start of the school year.

(Actually, if we had still been subbing from home, "Benchmark Test Song" would have an excellent song to try on the EACGAE-tuned guitar. This song is in the key of C, and one of the G chords from last time can be played as well. The melody is ambiguous whether it ends Dm-G-C or D7-G-C, but I already mentioned a D7 chord in my last post.)

Oh, and I think I'll avoid the "music" label in today's post. Yes, this counts as a performance, but I didn't make any material changes since the last time I sang it, last year. I'm saving that label for when I perform some of the new tunes that I wrote (or started to write) over the summer.

2:15 -- Seventh period ends, thus ending my day.

I still haven't completely thought about what resolutions I'm following, but today would have been the "eleventh" resolution, meaning the millennium resolutions from 20 years ago. The focus of those resolutions was communication.

Well, due to the microphone problems on Zoom, I definitely struggle to communicate today. I did try to remain positive when speaking to the students on Zoom chat in fourth and sixth periods and then vocal Zoom in seventh period.

One thing I notice about the room I sub in is that one wall is basically a whiteboard, but there are no markers for me to use. So now I know to carry some markers with me to class tomorrow. Thus if the mic fails again then I can write what I want to say on the whiteboard. If the mic is working, I can also ask the students for their answers to tomorrow's history questions so I can write them on my board -- and in the name of keeping positive, I can orally praise the students for their answers.

My communication resolution also extends to communicating with teachers and aides. Unfortunately, the only time I speak to them today is when I'm asking them for help with Zoom and the mic.

And indeed, I don't speak to any of my fellow subs today because I don't see any of them. With no students in-person, subs continue to get less work than usual. I myself might get more jobs since some other subs might not wish to work in person, but I don't expect there to be many subs on campus on the same day until hybrid begins.

Lesson 1-9 of the U of Chicago text is called "The Triangle Inequality." (It appears as Lesson 1-7 in the modern edition of the text.)

Over the years I've had several problems with Lesson 1-9, and here's why. Six years ago -- by which I mean the 2014-15 school year -- I noticed that Lesson 1-9 presents the Triangle Inequality as a postulate, when it's in fact provable using the theorems of Lesson 13-7. And so I decided to delay Lesson 1-9 until after 13-7, so we could prove the Triangle Inequality Theorem.

But then five years ago -- the 2015-2016 school year -- I juggled Chapter 13 around again. I ended up covering other lessons in Chapter 13 at various times, but never 13-7. And because I never posted Lesson 13-7, I'd never post 1-9 either. (Recall that the new Third Edition of the text no longer has our version of Chapter 13.)

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

And that takes us to the topic of today's worksheet -- the Triangle Inequality. In the U of Chicago, the Triangle Inequality was given as a postulate, yet it can be proved as a theorem. Many texts, including the Glencoe text, do prove the Triangle Inequality as a theorem, and this is what we will do.

The proof of the Triangle Inequality begins in Glencoe's Section 5-2, where we must prove two theorems, which the U of Chicago calls the Unequal Sides and Unequal Angles Theorems. My student told me that he had no problem understanding these two theorems -- he wanted just a quick review of Indirect Proof in Section 5-3 before moving on to the Triangle Inequality in 5-4. (This is why I'm squeezing in the Triangle Inequality now, rather than prove only the Unequal Sides and Unequal Angles Theorems today and save the Triangle Inequality for next week.)

Dr. Franklin Mason also proves these theorems. In many ways, Dr. M's Chapter 5 is similar to the same numbered chapter in Glencoe, except that Dr. M saves the concurrency results for a separate chapter, Chapter 10. Both Dr. M and Glencoe follow the same sequence of theorems, in which each theorem is built from the previous theorem in the list:

[2020 update: No, "concurrency" here has nothing to do with the hybrid schedule. If by concurrent you mean hybrid then I'm against it, but if by concurrent you mean three lines intersecting in a single point then I'm for it.]

Exterior Angle Theorem (abbreviated TEAE in Dr. M)
Exterior Angle Inequality (TEAI)
Unequal Sides Theorem (TSAI)
Unequal Angles Theorem (TASI)
Triangle Inequality (too important to be abbreviated!)
SAS Inequality (Hinge)

The U of Chicago follows the same pattern, except that the Unequal Angles Theorem is not used to prove the Triangle Inequality. Instead, the Triangle Inequality is merely a postulate. And since Unequal Angles isn't used to prove the Triangle Inequality, the U of Chicago didn't have to wait until
Chapter 13 to state the Triangle Inequality. Instead, the Triangle Inequality Postulate is given in Chapter 1, and the SAS Inequality, which depends on that postulate in its proof, is given in Chapter 7, still well before Chapter 13.

My blog attempted to restore the Dr. M-Glencoe order by delaying the Triangle Inequality. But I screwed up by not delaying the SAS Inequality as well. This is why I plan on delaying SAS Inequality, so that the full logical sequence is given. Of course all I did that year was make things worse!

But the first four theorems in the list are proved in U of Chicago's Section 13-7. Since I briefly mentioned the Exterior Angle Theorem (TEAE) at the end of the first semester, and the TEAI follows almost trivially from TEAE, my worksheet skips directly to the Unequal Sides Theorem. Its proof is given in the two-column format. Here I reproduce that proof, starting with a Given step:

Unequal Sides Theorem (Triangle Side-Angle Inequality, TSAI):
If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the larger angle is opposite the longer side.

Given: Triangle ABC with BA > BC
Prove: angle C > angle A

Proof:
Statements                                     Reasons
1. Triangle ABC with BA > BC     1. Given
2. Identify point C' on ray BA       2. On a ray, there is exactly one point at a given distance from
    with BC' = BC                               an endpoint.
3. angle 1 = angle 2                       3. Isosceles Triangle Theorem
4. angle 2 > angle A                       4. Exterior Angle Inequality (with triangle CC'A)
5. angle 1 > angle A                       5. Substitution (step 3 into step 4)
6. angle 1 + angle 3 = angle BCA  6. Angle Addition Postulate
7. angle BCA > angle 1                  7. Equation to Inequality Property
8. angle BCA > angle A                  8. Transitive Property of Inequality (steps 5 and 7)

The next theorem is proved only informally in the U of Chicago. The informal discussion leads to an indirect proof.

Unequal Angles Theorem (Triangle Angle-Side Inequality, TASI):
If two angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger angle.

Indirect Proof:
The contrapositive of the Isosceles Triangle Theorem is: If two angles in a triangle are not congruent, then sides opposite them are not congruent. But which side is opposite the larger angle? Because of the Unequal Sides Theorem, the larger side cannot be opposite the smaller angle. All possibilities but one have been ruled out. The larger side must be opposite the larger angle. QED

My student told me that he wanted to see one more indirect proof before showing him the Triangle Inequality, so why not show him this one? The initial assumption is, assume that the longer side is not opposite the larger angle. Since the angle opposite the longer side is not greater than the angle opposite the shorter side, the former must be less than or equal to the latter. And these are the two cases that lead to contradictions of Isosceles Triangle Contrapositive and Unequal Sides as listed in the above paragraph proof.

Now finally we can prove the big one, the Triangle Inequality. This proof comes from Dr. M -- but Dr. M writes that his proof goes all the way back to Euclid. Here is the proof from Euclid, where he gives it as his Proposition I.20:

http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI20.html

Here is the two-column proof as given by Dr. M. His proof has eight steps, but I decided to add two more steps near the beginning. Step 1 is the Given, and Step 2 involves extending a line segment, so that it's similar to Step 2 of the Unequal Sides proof. Indeed, the proofs of Unequal Sides and the Triangle Inequality are similar in several aspects:

Triangle Inequality Theorem:
The sum of the lengths of two sides of any triangle is greater than the length of the third side.

Given: Triangle ABC
Prove: AC + BC > AB

Proof:
Statements                                     Reasons
1. Triangle ABC                            1. Given
2. Identify point D on ray BC       2. On a ray, there is exactly one point at a given distance from
    with CD = AC                               an endpoint.
3. angle CAD = angle CDA           3. Isosceles Triangle Theorem
4. angle BAD = BAC + CAD         4. Angle Addition Postulate
5. angle BAD > angle CAD           5. Equation to Inequality Property
6. angle BAD > angle CDA           6. Substitution (step 3 into step 5)
7. BD > AB                                    7. Unequal Angles Theorem
8. BD = BC + CD                          8. Betweenness Theorem (Segment Addition)
9. BD = BC + AC                           9. Substitution (step 2 into step 8)
10. BC + AC > AB                         10. Substitution (step 9 into step 7)

To help my student out, I also included another indirect proof in the exercises. We are given a triangle with two sides of lengths 9 cm and 20 cm, and we are asked whether the 9 cm side must be the shortest side. So we assume that it isn't the shortest side -- that is, that the third side must be even shorter than 9 cm. This would mean that the sum of the two shortest sides must be less than 9 + 9, or 18 cm, and so by the Triangle Inequality, the longest side must be shorter than 18 cm. But this contradicts the fact that it is 20 cm longer. Therefore the shortest side must be the 9 cm side. QED

Notice that the U of Chicago text probably expects an informal reason from the students. A full indirect proof can't be given because this question comes from Section 1-9, while indirect proofs aren't given until Chapter 13.

Returning to Lesson 1-9, let's post the worksheets. First of all, since I'm now following the U of Chicago order, students are no longer responsible for a proof of the Triangle Inequality, so I only post the questions that don't depend on a proof.

On the other side, I post a review for the Chapter 1 Test. Recall that the Chapter 1 Test must be given on Day 20, or tomorrow, since Day 21 is Lesson 2-1. If there are eight or fewer lessons in a chapter, then there's a separate review day, but if there are nine lessons in a chapter, then the ninth lesson falls the day before the test.


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