The first week corresponds to Days 29-32. I already mentioned that on the blog, the first week of school will be treated like a second opening week, filled with activities. Well, following the digit pattern, this corresponds to Chapter 2 Review, Chapter 2 Test, Lesson 3-1, Lesson 3-2. I'll have to figure out how to make activities for this week that fit a concurrent (in-person and at-home going at the same time) hybrid schedule.
Meanwhile, it's been made official that subbing will be on-campus starting next week. As I said earlier, there might be more opportunities to sub then since some of the other subs are less willing to drive to campus, thus leaving more jobs for willing subs like me. As we've seen so far, I've gotten no work during this early stage when the subs work from home...
...until today, that is. On this final day of subbing from home, I finally pick up my first assignment. It is a high school special ed history class.
Ordinarily, this is the sort of class that I wouldn't do "A Day in the Life" on the blog. It's a high school class, it's not math, and it has special aides. But distance learning is so different from learning in-person that I want to detail my day using "A Day in the Life" format.
For distance learning, half of all classes meet each today. Since it's Friday, today is even periods:
10:00 -- Second period, which starts the distance day at 10:00, just so happens to be the regular teacher's conference period. She emailed me her lesson plans and suggested that I use this time to try logging in to the various websites I'll need, beginning with Google Classroom. And so I do just that -- I try both Google and the Aeries website used for attendance. Second period leads into a break, corresponding to snack/nutrition break on a pre-virus day.
11:10 -- Fourth period is a career guidance class for special ed students. Just before this class, I try logging into Zoom -- and suddenly, it doesn't work! I'm supposed to log in and then make it so I'm the "host" of the meeting, but it won't let me.
I keep calling the regular teacher and the tech department for assistance. The tech department tells me that the meeting is set up so that the regular teacher must start the meeting first and then I can make myself the host. But the regular teacher tells me that she had a sub yesterday, and that sub was able to log in without her having to start the meeting.
Ultimately, the aide tells the students to comment on Google Classroom, and I'm able to use the list of names for attendance. Exactly half of the sixteen students in this class are present. Then the students watch a video that they can access via Classroom.
12:00 -- Fourth period leaves and sixth period arrives. This is a US History class. Unfortunately, Zoom still isn't working for me.
12:30 -- Suddenly, about halfway through the period, Zoom starts working! I don't know whether the regular teacher does something on her end to make it work, or whether I unwittingly fix it on my end, or maybe there's something wrong with logging into Zoom on school accounts until now.
Because of the date, today's history lesson is on the September 11th attacks. Students watch videos from two athletes (a Yankees outfielder and a female marathoner) who comment on the importance of sports to get us through tough times such as 9/11. The students then answer questions about the videos and submit.
I see that 11 of the 18 students are logged in, and so I use these names for attendance. This class also has an aide, and he helps me find the link to a slideshow about 9/11 in Google Classroom.
1:00 -- Sixth period ends. One girl stays behind to ask about one of the videos -- she isn't able to find the one about the long distance runner. I can't find it either.
This is lunch break, followed by academic support time for students who need it. The regular teacher tells me that she usually holds this extra session on Tuesdays and Wednesdays only. And so my day ends here.
We expect teachers to struggle with Zoom when using it for the first time, and I'm no exception. And some of the Zoom problems are unavoidable. But can I truly say that I tried to log in to Zoom to the best of my ability? The answer is no, as I make two critical errors today:
- The first error is my failure to check Zoom during second period conference. The regular teacher suggested that I try logging in to all websites during the off period, but I logged in to all the needed websites but Zoom. I feared that if I logged in too early and no one was connected on the other end, I couldn't truly tell whether Zoom was working. So I wait until it's class time and I know that there's someone waiting on the other hand.
- The second error is that I wasn't logging in properly. During last week's meeting, it was mentioned which account we were to use for the log in. My problem is that when there's a long meeting when lots of info is given, I don't truly process what is said until it's time to use it. At this point I'll never forget what account I'm supposed to use to log in, only because of what I experience today. (Indeed, it was also stated at that meeting which codes we're supposed to use in Aeries for "present" and "absent" during distance learning, but I've forgotten them by the time I need to take attendance today. Fortunately, I'm able to figure them out quickly based on which codes are already in the system for attendance earlier this week.)
Once again, there's no guarantee that Zoom would have worked if I've avoided these two mistakes, but they certainly don't help me. This is the main reason why subs are expected to work on-campus starting next week -- if there's a Zoom problem, there are others on campus to help me, so I don't need to disturb the regular teacher.
Interestingly enough, today's school is actually the same school I subbed at on Pi Day Eve -- the last three days before the school closure. Both times I subbed special ed juniors, but the juniors I see today were sophomores then, so they can't be the same students. (One junior guy tells me that he recognizes me all the way from middle school -- the first year I subbed in this new district. He says that I gave him a detention that day.) The study skills class, on the other hand, contains some seniors who could have been the same students I had back in March -- but as that's the class where Zoom fails, I don't see them and they don't see me.
I blogged that I want there to be more positive interactions between the students and me as we all try to get through the tough times of the pandemic. I do praise two juniors in the US History class who help me find one of the video links, but I want there to be many more positive interactions. It was tough for me to stay positive when I'm struggling on Zoom -- but that's exactly when the students need me to be positive the most. Special ed students such as these really need the teacher to be positive as much as possible.
Before the pandemic, I'd given myself some resolutions to follow in the classroom. Today would have been the eighth resolution, "We sing to help us remember procedures," but I did write more recently that three of the resolutions are no longer active, including this one.
But nonetheless, I do sing one of my songs -- to the juniors, since that's the only class I end up seeing on Zoom today. And with subs returning to the classroom next week, today's the only day that I get to play my guitar (unless I wish to bring it on campus, which I won't).
Yes, I'm adding the "music" label to this post, since today is indeed a performance.
Of all the songs I mentioned in my July 31st post (the big music post), I decided to play my song "Plug It In" today. It's a relatively short and simple song -- and with all the trouble I have with Zoom today, the last thing I need is to attempt some complicated song.
As I wrote in my July 31st post, this song is based on an old 1990's jingle. I replayed that video beforehand in order to set up my new song.
And I did say that this song has a sort of "bluesy" feel to it. Actually, upon replaying it, I noticed that while the chords appear to be in the key of G major, the melody sounds like G minor, with the note Bb being prominent in the melody.
Let me recap how my esoteric music project evolved over the past few years. It all started when I stumbled upon the old Mocha emulator and was curious about its SOUND command. I learned that it was based on EDL ("equal divisions of length") scales, and then I imagined what a guitar fretted in equal divisions of length would sound like. Then last year, the tuning knob on my real guitar broke so that I couldn't tune the D string -- it was stuck on C. So then I came up with a new tuning, EACGAE, and saw that an EACGAE tuning is compatible with 18EDL fretting.
But let's think about it for a moment. Suppose I use Mocha to compose a random tune in the key of G, using the 18EDL scale. The note B in this scale is red (supermajor), 9/7 above white G. Then I get my guitar and try playing chords to go along with this scale -- but as I explained on July 31st, the G chord (actually G/D) uses a yellow (ordinary major) B, not a red B. Two notes with the same name but different colors will clash (dissonance), and so we can't play Mocha's red B over a G chord played on the guitar.
And so this music project will require some further thought. But ironically, "Plug It In" doesn't have this problem, since here we play a green Bb over the G yellow chord, not a red B. (Playing Bb over G major seems as if it would be dissonant, yet that's how the real 1990's jingle goes.) Then again, notice that while my guitar is really tuned to EACGAE, it's not really fretted to 18EDL, and so all of this discussion of 18EDL is hypothetical.
Here are the lyrics and the 1990's commercial whose tune I want the lyrics to fit:
PLUG IT IN
Plug it in, plug it in!
The variables are the letters.
Plug it in, plug it in!
That's where you put the numbers.
Use PEMDAS to simplify,
After you substitute.
To reduce the number of terms
You may need to distribute.
Plug it in, plug it in!
This is a song whose tune is lost, so I could invent a Mocha tune now for it. Even though I don't remember what I originally sang, I suspect I was influenced by this old 1990's commerical:
Plug it in, plug it in!
The variables are the letters.
Plug it in, plug it in!
That's where you put the numbers.
Use PEMDAS to simplify,
After you substitute.
To reduce the number of terms
You may need to distribute.
Plug it in, plug it in!
This is a song whose tune is lost, so I could invent a Mocha tune now for it. Even though I don't remember what I originally sang, I suspect I was influenced by this old 1990's commerical:
Here's how I did it -- the first and third lines "Plug it in!" match the tune: A-G-Bb-A-G-G. The second and fourth lines are played over background chords rather than a melody, and so I invented a melody for them (Bb-A-G-A for the second line and A-A-A-A-G for the fourth). I could have used some other notes, but I decided to keep the same three notes just to keep it simple.
The remaining lines fit over the "fresh for 30 days" part of the jingle, A-G-Bb-Bb-A. I actually only need two lines to fit this part of the jingle, but my original lyrics have four lines here. Therefore I can split the lines up into two verses:
PLUG IT IN (New Version)
First Verse:
First Verse:
Plug it in, plug it in!
The variables are the letters.
Plug it in, plug it in!
That's where you put the numbers.
Use PEMDAS to simplify,
After you substitute.
Plug it in, plug it in!
The variables are the letters.
Plug it in, plug it in!
That's where you put the numbers.
Use PEMDAS to simplify,
After you substitute.
Plug it in, plug it in!
Second Verse:
Plug it in, plug it in!
The variables are the letters.
Plug it in, plug it in!
That's where you put the numbers.
To reduce the number of terms
You may need to distribute.
Plug it in, plug it in!
Plug it in, plug it in!
The variables are the letters.
Plug it in, plug it in!
That's where you put the numbers.
To reduce the number of terms
You may need to distribute.
Plug it in, plug it in!
This song only needs three chords, G, C, and D7. Here are the fingerings for these chords:
EACGAE tuning:
G/D: xx2023
C: xx0030
D7: xx2232
In my July 31st post I also mentioned a G6 chord:
G6: 322020
This chord is difficult to finger, so I save it for the very end of the song, after the second verse. This corresponds to the final "Plug it in!" sung at the end of the jingle. The other chords are played at the ends of the lines: so in the first line, we play the G/D chord at the last "in!" The first two lines end with G/D, the next two lines end with C, the next two lines end with D7, and the last line ends with the G/D chord.
EACGAE tuning:
G/D: xx2023
C: xx0030
D7: xx2232
In my July 31st post I also mentioned a G6 chord:
G6: 322020
This chord is difficult to finger, so I save it for the very end of the song, after the second verse. This corresponds to the final "Plug it in!" sung at the end of the jingle. The other chords are played at the ends of the lines: so in the first line, we play the G/D chord at the last "in!" The first two lines end with G/D, the next two lines end with C, the next two lines end with D7, and the last line ends with the G/D chord.
Today the song proves to be tricky when I played it over Zoom. As expected, I'm so accustomed to the standard tuning that I keep forgetting how to play the various chords, especially D7. I play D7 with four fingers, but now I wonder whether this will be easier with a barre at the second fret. The question is, how easy will it be to jump from D7 barred to G/D?
But I won't be exploring this soon. I'll be subbing in the classroom next week and I won't be bringing my guitar, and so my songs will return to being purely vocal.
Meanwhile, since today is September 11th, I tell the students my own 9/11 experience. I've posted this on the blog the past two years when I subbed on 9/11, and since I subbed again today, I'll do likewise this year:
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.
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?"
Returning to 2020, I compared 9/11 to the coronavirus, just as I did on the blog a few weeks ago. In both 2001 and 2020, the resumption of sports give us something to cheer for. Then again, everything was mostly back to normal by November 2001, when the New York marathon was contested. But the virus closures are much more extensive -- we still won't be back to normal by this November.
And speaking of November, while my Orange County district will open for in-person learning on September 29th, my LA County district might not open until November, according to one county health official. (Unfortunately, she mentioned "election" in her statement, which leads to conspiracy theories involving the closures and Election Day. Here I wish only to report the date -- any political controversy surrounding the date is off-topic for this blog.)
Lesson 1-8 of the U of Chicago text is called "One-Dimensional Figures." (It appears as Lesson 1-6 in the modern edition of the text.)
This is what I wrote last year about today's lesson:
Lesson 1-8 of the U of Chicago text deals with segments and rays. The text begins by introducing the simple idea of betweenness. In Common Core Geometry, betweenness is an important concept, because it's one of the four properties preserved by isometries (the "B" of "A-B-C-D").
As I mentioned a few days ago, for Hilbert, betweenness is a primitive notion -- an undefined term, just as point, line, and plane are undefined. Yet the U of Chicago goes on to define it! It begins by defining betweenness for real numbers:
"A number is between two others if it is greater than one of them and less than the other."
Then the text can define betweenness for points:
"A point is between two other points on the same line if its coordinate is between their coordinates."
But Hilbert couldn't do this, because his points don't have coordinates. Recall that it was Birkhoff, not Hilbert, who came up with the Ruler Postulate assigning real numbers to points. Instead, Hilbert's axioms contain statements about order (Axioms II.1 through II.4), such as:
"II.2. If A and C are two points of a line, then there exists at least one point B lying between A and C."
Since we have a Ruler Postulate (part of the Point-Line-Plane Postulate), this statement is obvious, since points have coordinates and the same is true for real numbers -- between reals a and c is another real b.
I've seen some modern geometry texts mention a Ruler Postulate, but nonetheless leave the term betweenness undefined. Now as we mentioned earlier with point, line, and plane, if a term such as betweenness is undefined, then we need a postulate to describe what betweenness is. This postulate is often called the Segment Addition Postulate:
"If B is between A and C, then AB + BC = AC."
Notice that this statement does appear in the U of Chicago text. But the text doesn't call it the Segment Addition Postulate, but rather the Betweenness Theorem. As a theorem, we should be able to prove it -- and since after all, the text defines betweenness in terms of real numbers, we should be able to use real numbers to prove the theorem. Indeed, the text states that we can use algebra to prove the theorem, but the proof is omitted.
Following David Joyce's admonition that we avoid stating a theorem without giving its proof, let's attempt a proof of the Betweenness Theorem. We are given that B is between A and C. Now let us assign coordinates to these points. To make it easy to remember, we simply use lowercase letters, so point A has coordinate a, point B has coordinate b, and point C has coordinate c.
We are given that B is between A and C, so by definition of betweenness, we have either a < b < c, or the reverse of this, a > b > c. Without loss of generality, let us assume a < b < c (especially since the example in the book has a < b < c). Now by the Ruler Postulate (the Distance Assumption in the Point-Line-Plane Postulate), the distance between A and B (in other words, AB) is |a - b|. Since a < b, a - b must be negative, and so its absolute value is its opposite b - a. (To avoid confusing students, we emphasize that to find AB, we just subtract the right coordinate minus the left coordinate, so that AB isn't negative. This helps us to avoid mentioning absolute value.) Similarly BC = c - b and AC =c - a. And so we calculate:
AB + BC = (b - a) + (c - b) (Substitution Property of Equality)
= c - a (simplification -- cancelling terms b and -b)
= AC
The case where a > b > c is similar, except that AB is now a - b rather than b - a. All the signs are reversed and the same result AB + BC = AC appears. QED
Don't forget that I want to avoid torturing geometry students with algebra. And so I simply give the example with numerical values, with the variables off to the side for those who wish to see the proof.
The text proceeds to define segments, rays, and opposite rays in terms of betweenness. Notice that these definition are somewhat more formal than those given in other texts. A typical text, for example, might define a segment as "a portion of a line from one endpoint to another." But the U of Chicago text writes:
"The segment (or line segment) with endpoints A and B is the set consisting of the distinct points A and B and all points between A and B."
The definitions of ray and opposite ray are similarly defined in terms of betweenness.
The section concludes with the notation for line AB, ray AB, segment AB, and distance AB. But although every textbook distinguishes between segment AB and distance AB, many students -- and admittedly, many teachers as well -- do not. The former has an overline, but the latter doesn't. Unfortunately, Blogger allows me to underline AB and strikethroughAB, but not overline. For the purpose of the rest of this post, let's pretend that the strikethrough AB is really the overline for segment AB.
Now ifAB and CD are both of, say, unit length, then AB = CD, but we can't write AB =CD. After all, AB and CD are real numbers -- both are 1 -- and those numbers are equal. But the segments AB and CD can't be equal unless they have the same endpoints (that is,A and C are the same point, as are B and D, or vice versa). The numbers (the lengths) are equal, while the segments are congruent. But students and teachers alike confuse a segment with its length, and confuse equality with congruence.
To avoid confusion, in the following images I threw out Question 8 from the text, a multiple choice question which states thatAB literally equals BA (but ray AB is not the same ray as BA). Notice that I tried to draw all of the one-dimensional figures in red.
By the way, since today is Friday, you might wonder whether this is an activity day, or whether I'm rearranging the activity days to match the new district calendar. I've decided simply to keep all worksheets as written until the reopening on September 29th. Then I'll have several activities that week, and afterward I'll return to having a weekly activity.
But I won't be exploring this soon. I'll be subbing in the classroom next week and I won't be bringing my guitar, and so my songs will return to being purely vocal.
Meanwhile, since today is September 11th, I tell the students my own 9/11 experience. I've posted this on the blog the past two years when I subbed on 9/11, and since I subbed again today, I'll do likewise this year:
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.
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?"
Returning to 2020, I compared 9/11 to the coronavirus, just as I did on the blog a few weeks ago. In both 2001 and 2020, the resumption of sports give us something to cheer for. Then again, everything was mostly back to normal by November 2001, when the New York marathon was contested. But the virus closures are much more extensive -- we still won't be back to normal by this November.
And speaking of November, while my Orange County district will open for in-person learning on September 29th, my LA County district might not open until November, according to one county health official. (Unfortunately, she mentioned "election" in her statement, which leads to conspiracy theories involving the closures and Election Day. Here I wish only to report the date -- any political controversy surrounding the date is off-topic for this blog.)
Lesson 1-8 of the U of Chicago text is called "One-Dimensional Figures." (It appears as Lesson 1-6 in the modern edition of the text.)
This is what I wrote last year about today's lesson:
Lesson 1-8 of the U of Chicago text deals with segments and rays. The text begins by introducing the simple idea of betweenness. In Common Core Geometry, betweenness is an important concept, because it's one of the four properties preserved by isometries (the "B" of "A-B-C-D").
As I mentioned a few days ago, for Hilbert, betweenness is a primitive notion -- an undefined term, just as point, line, and plane are undefined. Yet the U of Chicago goes on to define it! It begins by defining betweenness for real numbers:
"A number is between two others if it is greater than one of them and less than the other."
Then the text can define betweenness for points:
"A point is between two other points on the same line if its coordinate is between their coordinates."
But Hilbert couldn't do this, because his points don't have coordinates. Recall that it was Birkhoff, not Hilbert, who came up with the Ruler Postulate assigning real numbers to points. Instead, Hilbert's axioms contain statements about order (Axioms II.1 through II.4), such as:
"II.2. If A and C are two points of a line, then there exists at least one point B lying between A and C."
Since we have a Ruler Postulate (part of the Point-Line-Plane Postulate), this statement is obvious, since points have coordinates and the same is true for real numbers -- between reals a and c is another real b.
I've seen some modern geometry texts mention a Ruler Postulate, but nonetheless leave the term betweenness undefined. Now as we mentioned earlier with point, line, and plane, if a term such as betweenness is undefined, then we need a postulate to describe what betweenness is. This postulate is often called the Segment Addition Postulate:
"If B is between A and C, then AB + BC = AC."
Notice that this statement does appear in the U of Chicago text. But the text doesn't call it the Segment Addition Postulate, but rather the Betweenness Theorem. As a theorem, we should be able to prove it -- and since after all, the text defines betweenness in terms of real numbers, we should be able to use real numbers to prove the theorem. Indeed, the text states that we can use algebra to prove the theorem, but the proof is omitted.
Following David Joyce's admonition that we avoid stating a theorem without giving its proof, let's attempt a proof of the Betweenness Theorem. We are given that B is between A and C. Now let us assign coordinates to these points. To make it easy to remember, we simply use lowercase letters, so point A has coordinate a, point B has coordinate b, and point C has coordinate c.
We are given that B is between A and C, so by definition of betweenness, we have either a < b < c, or the reverse of this, a > b > c. Without loss of generality, let us assume a < b < c (especially since the example in the book has a < b < c). Now by the Ruler Postulate (the Distance Assumption in the Point-Line-Plane Postulate), the distance between A and B (in other words, AB) is |a - b|. Since a < b, a - b must be negative, and so its absolute value is its opposite b - a. (To avoid confusing students, we emphasize that to find AB, we just subtract the right coordinate minus the left coordinate, so that AB isn't negative. This helps us to avoid mentioning absolute value.) Similarly BC = c - b and AC =c - a. And so we calculate:
AB + BC = (b - a) + (c - b) (Substitution Property of Equality)
= c - a (simplification -- cancelling terms b and -b)
= AC
The case where a > b > c is similar, except that AB is now a - b rather than b - a. All the signs are reversed and the same result AB + BC = AC appears. QED
Don't forget that I want to avoid torturing geometry students with algebra. And so I simply give the example with numerical values, with the variables off to the side for those who wish to see the proof.
The text proceeds to define segments, rays, and opposite rays in terms of betweenness. Notice that these definition are somewhat more formal than those given in other texts. A typical text, for example, might define a segment as "a portion of a line from one endpoint to another." But the U of Chicago text writes:
"The segment (or line segment) with endpoints A and B is the set consisting of the distinct points A and B and all points between A and B."
The definitions of ray and opposite ray are similarly defined in terms of betweenness.
The section concludes with the notation for line AB, ray AB, segment AB, and distance AB. But although every textbook distinguishes between segment AB and distance AB, many students -- and admittedly, many teachers as well -- do not. The former has an overline, but the latter doesn't. Unfortunately, Blogger allows me to underline AB and strikethrough
Now if
To avoid confusion, in the following images I threw out Question 8 from the text, a multiple choice question which states that
By the way, since today is Friday, you might wonder whether this is an activity day, or whether I'm rearranging the activity days to match the new district calendar. I've decided simply to keep all worksheets as written until the reopening on September 29th. Then I'll have several activities that week, and afterward I'll return to having a weekly activity.
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