As far as classroom management is concerned, I use teacher look to get the students to be quiet during roll in both fourth and fifth periods. Of these two classes, the teacher look is much more effective in fourth period, which is considered to be the best behaved class of the day.
As for eliminating restroom passes in fourth and sixth periods, no one asks for a pass in fourth period (another reason why it's the best class of the day). In sixth period, one guy asks for a nurse pass. This, of course, is a tricky issue. Sometimes I ask the student the reason for the pass, and many times the student wants to tell no one other than the nurse (patient-doctor confidentiality). But I always fear that students will lie about needing to go to the nurse -- especially when I've already told them that I won't give them a restroom pass (as I do today). Fortunately, I'm able to use the attendance loophole again today -- I allow him to go to the nurse provided that he drops the rosters off at the attendance office along the way. (Two girls properly wait until the midpoint of the period to go to the restroom.)
In each class, I connect to the students by asking a third of them to describe their artwork. (The plan is to ask the rest of them the next two days.) In the AP class, one girl shows me a picture of her male friend that she's doing in charcoal, rather than paint or color pencil. She tells me that charcoal is rather expensive, but fortunately the regular teacher has a class supply.
On the opposite end, one girl in a Fundamentals class tells me that she's not good at drawing. I reassure her that she's much better than I am at drawing. On her paper, she's sketched a background, but not much more. The assignment is to draw pictures or symbols representing herself -- this is called the "Me" project. I tell her that I hope she succeeds on this assignment and in the class -- I suspect that she's a college-bound student who needs this art class to satisfy the A-G requirements.
A few students do finish their "Me" projects. These students start working on another assignment, which involves drawing hands in various positions. This means that if by Thursday many other students start claiming "I'm done," I can tell them to do the hands assignment.
At snack, there is a special ceremony to commemorate September 11th. There was a speech from the father of a girl who had just graduated in 2001 but tragically died on 9/11. In her honor, a scholarship for current students has been established. The father pointed out that his late daughter had a 4.5 weighted GPA, but there are many current students with even higher GPA's.
Surprisingly, I've never discussed "where I was on 9/11" on the blog. Two years ago was the 15th anniversary, but it fell on a Sunday. I somewhat recall telling the students at my old charter school where I was that day (possibly on the 12th, a Monday, in between coding classes). But let me write the full story here on the blog.
In the year 2001, I was an undergrad student at UCLA. I had just completed my second year and decided to take two classes over the summer. (These summer classes, along with my AP credits, would allow me to complete my bachelor's degree in just three years.)
Notice that since UCLA is on the quarter system, summer classes extended into September. We see that the fall semester typically begins in late August so that an entire semester is completed before Christmas (the same reason that high schools also now have an Early Start Calendar). But at quarter schools, only one quarter (not one semester) needs to be completed by winter break, and so they can afford to start later. The fall quarter at UCLA began the last week of September, and so summer classes start and end later.
Summer at UCLA was divided into two halves, called "A Session" and "C Session." Each session was six weeks. (Officially, "B Session" refers to a few special one- or two-week seminars only in certain departments, such as art.)
If you check the UCLA website nowadays, you'll see that math courses could be offered either A Session or C Session. But back in 2001, summer math courses were eight weeks long -- the six weeks of A Session plus the first two weeks of C Session. The class I took in Summer 2001 was MATH 132, Complex Analysis. (I still have a copy of a test I took that summer -- a perfect 300/300 score.)
The second course I took that summer was Geography 5, "People and Earth's Ecosystems," which I used to fulfill my general ed requirements. This was a C Session course that met twice a week, on Tuesdays and Thursdays. So by September, I had finished Complex Analysis, but was still attending the Geography 5 course.
Meanwhile, during my years at UCLA, I earned money by working part-time at the library. During the summer, we worked a fixed schedule for A Session and C Session. Since my math and geography classes overlapped for the first two weeks of C Session, my library hours for that session would have to accommodate both classes. If I recall correctly, my schedule was like 8-10 for Geography 5, then noon-2 for math, and afterwards I worked at the library from 2-6. Once the first two weeks of C Session had passed, I was left with a long gap between geography class ending at 10 and work not starting until 2 for the final four weeks.
Just like this year, September 11th, 2001 fell on a Tuesday -- and just like this year, it's the sixth and final week of C Session. And so I had to wake up early for Geography 5. I commuted a long distance to UCLA back then -- I woke up around 5-something in order to leave by 6-something.
When I first turned on the news that morning, I heard that a plane had struck one of the Twin Towers in New York City. Originally, I assumed that it was an accident. I took an early morning shower -- and by the time I came out of the shower, a second plane had hit the other tower. I knew that the probability of two planes having accidents about a half-hour apart, with each plane hitting the World Trade Center, was infinitesimal. The plane crashes were clearly intentional!
I began the long commute to UCLA. On the way there, I hear about the events of the East Coast on the radio. I reach the campus in time for my 8:00 Geography 5 class. I arrived expecting a long lecture followed by review for the final -- to be held two days later, on Thursday.
As the title of the class implies, the subject material of the class is all about what effect humans have on the planet and its ecosystems -- indeed, how we cause the ecosystems to change over time. And so this is what the professor said to open the class:
"This summer, we've learned how the world can change over a period of many years. But today, we see that the world can change a whole lot in a single day."
He dropped the lecture format for that day, and instead allowed the class to discuss what was happening in New York. Some students had grown up on the East Coast, and they were undoubtedly fearful of what was going on. Not until the last few minutes of class did the professor remind us of what would be on Thursday's final.
Class ended, and it was time for my long break between class and work. I walked to the library (the same one where I worked) and studied a little for Thursday's final. Then I went to the computers and decided to play around on the Internet. It was only then when I realized exactly how much the world had changed that day.
You see, there was a huge bank of computers on the first floor of the library. But on the second floor, there was a lone computer in walking distance of a restroom. I knew about the computer mainly because I worked there -- so most of the time this computer was open. Because of its proximity to the restroom, I liked to use that computer. Yet I was afraid that by the time I'd return from the restroom, someone would take that computer. So I had a bright idea -- I'd leave a window open on some website and leave my backpack behind as I went to the restroom. Hopefully, other patrons would get the hint that the computer was taken.
When I returned from the restroom, a security guard was standing at the computer waiting for me. He asked me, "Is that your backpack over there?" When I nodded, he continued, "Someone had called in a bomb threat, so we had to make sure that it was yours."
I knew that the "bomb threat" was the general events occurring on the East Coast. In other words, because it was 9/11, the security guards were afraid that my backpack contained a bomb and that I'd walked away before it exploded. Only then did I know that the world had changed -- and it was a world full of fear, uncertainty, and doubt.
After the encounter with the security guard, I was no longer in the mood to use the computer. It was approaching lunch time anyway, and so I left the library and walked to the student union. As I ate, I saw the news continue to cover the events of New York -- and Washington DC. I recall reading the scroll bar on the bottom of the TV -- all sporting events had been cancelled, not just the games in the two cities that had been attacked.
I thought about the people -- the people on the planes, the people in the towers. I began to shed tears for all of the victims.
When it was time for work, I returned to the library. It goes without saying that my boss and student coworkers were discussing the tragic events. After work, I took the long walk back to the car. I glanced at the TV in the student union along the way and saw burning buildings on the screen -- and I was afraid that downtown LA had been attacked as well. It wasn't until I listened to the radio along the drive home that I learned that the West Coast had been spared after all.
On Thursday, September 13th, I took the Geography 5 final and passed the class. During the long break between class and work, a group of students were gathering in front of Royce Hall. A moment of silence was held for all the victims who had died two days earlier -- just as a moment of silence is held at the school where I sub at today.
And that's my answer to the question, "Where was I on 9/11?"
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. Four 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 three 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:
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 2018, right now I'm thinking about a book that was published today. No, I'm not referring to Fear. Instead, another new book is on my mind -- and its author is Eugenia Cheng.
I've discussed Eugenia Cheng's first two books as side-along reading books -- How to Bake Pi and Beyond Infinity. I was hoping that she was planning a third book -- and I'm pleasantly surprised that not only does she indeed have a third book, but that it's coming out now in 2018. I'd assumed that her third book wouldn't come out until 2019 at the earliest, if not 2020.
Cheng's third book is titled The Art of Logic in an Illogical World. We already know that math and logic are closely related, especially when it comes to Geometry. After all, I just mentioned Chapter 13 of the U of Chicago text a few lines above -- "Logic and Indirect Reasoning." But her title mentions "an illogical world" -- this implies that Cheng will apply logic to more than just Geometry or math in her book.
I just ordered a copy of Cheng's third book from the public library, and I'm hoping that it will arrive in time for me to pick up this weekend. If it arrives, then expect me to begin this book for side-along reading on Monday. If not, then it should get here by the following weekend.
I enjoy reading about female mathematicians like Eugenia Cheng because they remind us that women can do math too -- a lesson that the girls in our classes sorely need. In yesterday's post, I wrote again that sometimes I have trouble inspiring the girls in my classes to do math. (Hopefully, I am able to inspire that one reluctant girl to do art in today's class. I'll check up on her tomorrow and Thursday.)
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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