"The stone structures of the Renaissance showed a refinement of symmetry that relied on light and dark and solids and voids. With the discovery of new building materials, new mathematical ideas were adapted and used to maximize the potential of these materials."
We are still in the admittedly lengthy introduction to the math and architecture chapter. Pappas is in the middle of listing some historical examples of the use of math in architecture.
On this page, Pappas provides a photo of an unusually shaped house. This is to demonstrate how important our subject, Geometry, is to architecture. Here's the caption:
"Each of the floorplans of the three levels of this house is designed from two overlapping equilateral triangles. This triangle motif is carried out throughout the interior supports and windows."
And in case you're wondering how a house shaped like two equilateral triangles can even stand, it turns out that the house was carved into the side of a hill.
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.)
Unlike Lessons 1-1 to 1-3, which I've never covered before this year, we are definitely familiar with Lesson 1-4 and the Bridges of Konigsberg from previous years. It is, therefore, the first lesson this year for which I can just reblog the worksheets and comments from last year.
But there is a twist here. I actually taught the Lesson 1-4 activity last year 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 last 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." A Google search reveals a ten-second video in which this name is pronounced:
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
[OK, I'll skip the song since I already reblogged it in a recent post.]
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 2017. There are a few things that I want to say about my reflection -- from the perspective of it being a year later and I not having returned to that classroom.
First of all, last 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 last year was August 16th. This was an idea that came from Pappas -- 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 Pappas 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.
Last 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, last 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.
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