Thursday, September 3, 2020

Lesson 1-4: Points in Networks (Day 14)

There are a few things going on with the calendar in my new district. First of all, it's been announced that tomorrow is a PD day with no students.

The fall PD day in this district has moved around in recent years. It used to be near Halloween (depending on what day of the week that holiday falls), based on the assumption that many students won't want to go to school on Halloween (especially high school students -- elementary school students enjoy school on Halloween because they wear costumes to school). This year, with Halloween on a Saturday, there's less need for an extra day off near that holiday.

Then when the Early Start Calendar was announced, that PD day was changed to Columbus Day. As you may recall, Columbus Day is not a holiday in California. But as we've seen, many schools on the Early Start Calendar choose Columbus Day as a PD day. This includes my old district in LA County, and it almost included my new district in Orange County.

But then it was suddenly announced that the PD day would be tomorrow instead. In the LAUSD, tomorrow is Admissions Day -- while California became a state on September 9th, 1850, LAUSD observes it on the Friday before Labor Day to make a four-day weekend. Neither students nor teachers in LAUSD work tomorrow. But in my new Orange County district, it's a PD day -- students don't work tomorrow, but teachers do. (Still, except Orange County starting one day earlier, the LAUSD and Orange County district calendars are identical for students in the first semester.)

Let me fix that -- tomorrow, regular teachers and subs work. The district announces that there will be a second PD day for us subs tomorrow. So far this school year, I haven't picked up any sub work -- the only days I'm getting paid for are these training days. And for me, this marks the first time that I've actually worked on a PD day since February 2017 at the old charter school.

No, I'm not going to post on the blog tomorrow. I blogged on the first PD day three weeks ago, but that was technically a summer post, not a PD day post. Now that school has started, I'm only posting on the actual school days until vacation week. But I will blog on Tuesday if anything interesting happens at tomorrow's meeting.

Meanwhile, the district hasn't come up with a new start date for the Hybrid Calendar, now that the purple list has delayed our September 21st proposal. An email sent to employees mentions that schools could possibly open "at the end of September" -- suggesting that it could be just a one-week delay (to September 29th), rather than the two-week delay I mentioned in an earlier post. The board meeting is still scheduled for next week, and so a new proposed date might be mentioned then.

This also means that by changing the PD day from Columbus Day (after the schools reopen) to tomorrow (while it's still distance learning), this buys an extra day in the classroom for elementary students, but not secondary, since they won't be in the classroom on Mondays anyway. (The preliminary calendar for next year still has that PD day on Columbus Day -- we'll see how that goes.)

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

Lesson 1-4 of the U of Chicago text is called "Points in Networks." (It is combined with the old Lesson 1-1 to form the new Lesson 1-3 in the modern edition of the text.)

But there is a twist here. I actually taught the Lesson 1-4 activity three years ago as it was my first day of school activity for my middle school classes. And so I'm actually going to reblog the experience of my first day as a teacher from that year. Yes, this means that this will be yet another "crying over spilled milk" post that has dominated my blog these past few weeks. In this post, I'll only repeat what I wrote about my eighth grade class, as well as the commentary I wrote at the end.

11:25 -- My eighth grade class arrives. This is my smallest class, with only 12 students -- but there are only eight students present at the start of class. I begin the class the same way I start all my classes, with a Warm-Up question:

What is 2 * 2 * 2 * 2? (That is, 2 times 2 times 2 times 2.)

Most students answer correctly, although a few tried to add. A student or two is upset that the very first thing we do on the first day of school is multiply. I point out that the answer is 16 -- and that today is the 16th. I always go around to stamp correct papers -- many teachers point out that students enjoy getting stamps, and my students are no exception.

11:35 -- My student support aide arrives -- the English teacher and I are each assigned one. Actually, she arrives with the four missing students, all girls.

We move on to an Opening Activity -- the Konigsberg Bridge Problem. I've written about this problem previously on the blog and even suggested it as a first day of school activity -- well, now I'm finally giving the activity on an actual first day of school. This is a little of what I said about this problem here on the blog:

The Königsberg Bridge Problem is a famous math problem from nearly 300 years ago. Fawn Nguyen, a well-known math blogger and fellow Southern Californian -- she lives in Ventura County -- used this as an activity in her geometry class:

http://fawnnguyen.com/famous-bridge-problem/

As we all know, the Königsberg Bridge Problem is impossible to solve -- it has no solution. But I don't want to start the class with a problem that the students can't solve -- they're already frustrated enough with problems that do have solutions when they just can't find them.

The whole point of this lesson is to point out that students should look for patterns, and that sometimes it's just as important to know why something is impossible as it is to know why something is possible.

Let me complete this with a note on pronunciation. The U of Chicago text points out that the name Euler ends up sounding like "Oiler." But how does one go about pronouncing the name Königsberg? I once read that the o-umlaut ends up sounding like "uh," almost like "ur." 

I am posting this Numberphile video on Konigsberg. It was originally posted to Numberphile just 2-3 months after I taught the lesson at the old charter school.


[As I watch this video on Youtube, the Autoplay linked to another Numberphile video on a sort of "five-sided square." Actually, it's a hyperbolic pentagon with congruent sides and right angles. In spherical geometry there exists a "three-sided square" -- a triangle with each side a quadrant. But I digress, since this post isn't about non-Euclidean geometry.]

By the way. some students believe that they have a solution to the Konigsberg problem, but actually they are crossing one of the bridges twice (they start on island D, cross the bridge towards C, but then head back to D). I start to explain about Euler and why the problem is impossible -- and as I do so, the student who earlier complained about 2 * 2 * 2 * 2 figures out that the impossibility has to do with there being an odd number of bridges from each land! I'm impressed!

12:05 -- Because I know how tough the 80-minute block schedule can be on middle school students, I provide a music break. I get out my guitar and I play the following inspirational song:

The Dren Song -- by Mr. Walker

Along the way, I explain that a "dren" is a reverse-nerd -- a nerd is someone who's good at math, and a "dren" is someone who doesn't understand the basics of arithmetic. As it turns out, the student who complained about 2 * 2 * 2 * 2 enjoys this song and looks forward to my next song.

I show my students the September 2015 Boys' Life article about the mathematicians and scientists who work for NASA and the possible future of people traveling to and living on the moon. But as it turns out, eight of the 12 students in my class are girls, so I don't expect Boys' Life to motivate them.

Instead, I tell them about the movie trailer that was released just yesterday -- Hidden Figures, about the scientist Katherine Johnson who worked for NASA and the Apollo projects in the 1960's. For those of you who have read my blog before, it goes without saying that I plan on watching this movie, and I highly recommend that my students watch it in January as well.

12:15 -- I proceed with my next Opening Day activity -- Personality Coordinates. This activity comes from the King of the MTBoS, Dan Meyer:

http://blog.mrmeyer.com/2013/personality-coordinates-icebreaker/

Each person in a group picks a dot and writes her name next to it.
Now the group’s job is to label the axes. Physical attributes don’t require all that much thought and don’t reveal all that much, so don’t allow them.
That’s it. It requires a surprising amount of creativity and conversation. Happy first day of school, teachers.

12:30 -- My support aide leaves, and this is a good time to end the period with an Exit Pass:
If you don't know the answer, ..
The answer is "at least know where to find it," which is posted in a corner of the room. (I mentioned this in an earlier blog post.) Some wrong answers are "ask the teacher" and "you're a dren."
12:45 -- My eighth grade class goes out to lunch.

If there's anything I could change about the way I ran the class today, it would be to teach the entire class in reverse order. That way, the Exit Pass becomes a Warm-Up, a scavenger hunt to find the rest of the quote, Personality Coordinates occur earlier in the class, and the 2 * 2 * 2 * 2 question doesn't turn off students right at the start of the period.

I would also rewrite the Konigsberg worksheet. I'd already changed the worksheet to add more bridge problems, including some trivial ones. But now I'd number those trivial problems #1 and #2 (rather than #3 and #4, as they were numbered today).

Another problem I have has to do with explaining my directions clearly. I was hoping to create a seating chart directly from the Personality Coordinates worksheet (since the students are already seated in groups of four), but I couldn't because some groups randomly labeled the dots rather than place the student sitting northwest in the upper-left corner of the page. Also, some students wrote the Exit Pass on a separate sheet of paper rather than the back of the Warm-Up.

I remember explaining my directions to the students -- but I could be remembering my explanations to the 6th and 7th grade classes, not the 8th grade class. Anyway, I know that I don't always explain instructions clearly to my students from my days as a sub, so I must give the students the benefit of the doubt whenever I see them misinterpreting instructions.


OK, let's return to 2020. There are a few things that I want to say about my reflection -- from the perspective of it being four years later and I not having returned to that classroom.

First of all, that year I wrote that maybe I should have reversed the order of the first day. Then the opening Warm-Up question, 2 * 2 * 2 * 2, becomes an Exit Pass.

The answer to that question is 16, and the first day of school that year was August 16th. This was an idea that came from Rapoport -- making the answer to the Warm-Up question be the date. But, as we already know, this fell apart because the Illinois State Daily Assessment took over the Warm-Up.

Thus it would have been better for me to establish the Rapoport tradition by making the date be the answer to the Exit Pass, not the Warm-Up. In other words, I should have reversed my Exit Pass and Warm-Up not only that first day, but everyday. This also serves another purpose -- after Exit Passes, I often never gave the correct answer because some students are already walking out the door while others are still trying to correct their mistakes. By making the date be the answer to the Exit Pass, the students already know the answer, so never finding out the answer isn't an issue. Again, this then frees Warm-Ups for the Illinois State Daily Assessment, which we would then work out on the board.

That year, I wrote that I often had trouble explaining my instructions clearly. But in hindsight, I think this was part of my overarching management issue -- the students never stopped talking. I now believe that I often spoke quickly because I knew I needed to talk before the students did. And this applied not just to giving clear directions -- my math explanations suffered as well. I know that I could have taught math much better if I could count on the students being quiet during the lesson.

Obviously, at some point on the first day of school, I need to tell the students of the importance of being quiet. At the time, I actually didn't mind the students talking during the Warm-Up, but the problem is that they don't stop talking when I'm giving instructions, lessons, or tests. Therefore it's worth it to keep the kids quiet and in their seats during the Warm-Up as well.

By reversing the Warm-Up and Exit Pass, the new first day Warm-Up is the scavenger hunt to complete the phrase, "If you don't know the answer...." Unlike most Warm-Ups -- especially the Illinois State Warm-Ups (which would begin after that is set up online) -- this Warm-Up actually involves students moving around and talking -- so it's a bit awkward to show the importance of being quiet and sitting down at this point.

Of course, I could tell the students to be quiet before starting -- and then afterward, I tell the students to be quiet again. It's important not to accept any excuses for not being silent, including:
  • You're mean!
  • You're the only teacher who makes us sit down and be quiet.
  • You're unfair!
  • You're unreasonable!
  • You're weird!
  • Making us be quiet is juvenile.
  • I wasn't talking.
  • Why do we have to be quiet?
Oh yeah -- I haven't said much about the actual Bridges of Konigsberg worksheet yet. Well, that year I wrote that the two easiest questions should be numbered #1 and #2, not #3 and #4. So I decide to fix this for today's posting. The simplest way for me to do this is to make the old #1 and #2 into more examples, and so #3-8 are renumbered as #1-6. With the order of activities being reversed, the Bridges worksheet is given later in class, so it's good to shorten it to make sure that we get to #6 (the Konigsberg problem) and discuss its impossibility, and get to the Exit Pass. In this class, this will be the first time that they have to do any arithmetic in the math class.

By the way, earlier today I was fooling around with my guitar and practicing some songs. Earlier, I wrote that if I were called to sub a distance lesson from my house (where my guitar is), I might play some guitar songs for the class. But now it appears that there won't be many, if any, days of subbing during distance learning. So unless I play the guitar during tomorrow's training (hint: I won't), there is no opportunity for guitar music.

Still, when I first mentioned my nonstandard EACGAE guitar tuning, at the time I had no idea whether there'd be any subbing at all. So I'll still write a little about which songs I played. (Note: I'm not using the "music" label here, because I'm now saving it for an actual performance day -- whether playing the guitar online or singing in the classroom.)

The first song I tried was the "Mousetrap Car Song" -- a song I associate with the first week of September (when I originally sang it in my class). This song is in the key of E minor. We've seen that minor keys are tricky to play in EACGAE tuning (despite this tuning being compatible with EDL fretting -- EDL scales being more convenient for minor keys).

The opening chord of this song is Am, which I played as Am7 on the open strings. The C and G major chords (the latter I played as G/D) also appear early in this song. Like Am, Em itself is more easily played as Em7.

The real problem with the key of Em is the dominant chord. When I played this song at the old charter school (in standard tuning), I played a B7 chord -- the dominant chord of Em. But a B7 chord isn't playable in the new tuning. I've mentioned before how a Bm7 chord can be played simply by barring Am7 at the second fret. The tune of this song is mostly ambiguous between "natural" minor (with a minor seventh) and "harmonic" minor (with a raised seventh), and so it's possible to use Bm7 as a dominant chord in this song. But then it's difficult to jump from Bm7 (x22222 barre) to Em7 (022020 no barre).

The easiest chord to play here, of course, is Am7, so we could just have Am7 lead into Em7. Another chord that might sound good here is D7 (xx2232).

Also, when I played this song in class, I wanted this minor song to end on a major chord. There are two ways to do this -- one is to end on the parallel major G, and this is how I played the ending at my old charter school. But the other way sounds better to me today -- a Picardy third, where the song simply ends with E major. If we finger this as E7 (022120) and imagine this in 18EDL fretting, then this becomes a su (17-under) G#. But I'm not quite sure whether the su G# will sound good over the yellow E and B notes (recall that the open E in my tuning would be yo E). My prediction is that the interval yoE-suG# will be slightly dissonant. (Also, notice that D7-Em7 from earlier is easy to finger, but D7-E7 as written here isn't. Thus we must use Am7-E7 instead.)

Another song I tried to play today is the "Dren Song," which I mentioned earlier in this post. Like "Mousetrap Song," "Dren Song" is also written in the key of Em. But the tune of this song is more strongly harmonic minor, and so I'd really like there to be a B7 chord. I must decide whether I want to convert this to E natural minor (mostly likely with D7 instead) on the EACGAE/EDL guitar.

Other songs that I associate with early September include "Ratios" and "Another Ratio Song." Recall that I'm considering changing some of my old songs to raps. I think that "Ratios" might sound better as a rap, the "Ratio Rap." I'll keep "Another Ratio Song" as a consonant song, though I'm not quite sure of the key yet. I don't quite recall whether I intended "Another Ratio Song" to be minor, but I might make it major if it's easier to play on the EACGAE/EDL guitar.

My next post will be on Tuesday, after Labor Day Weekend.



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