At some schools, the Big March doesn't start until next week, because students get the entire week of President's Day off from school. This is true in New York City, and last year I linked to a Northern California teacher whose school observes a week of February break. Actually, recently I found a Southern California district that has a mid-winter break as well. But that district clearly isn't one of the two districts where I'm employed as a sub.
In fact, sometimes I wonder whether things would have gone better for me three years ago if my old charter school had a February week off. Many of the problems that I described earlier on the blog actually occurred on the first four days after President's Day.
After that first week, I wrote that I'd stop blogging for the rest of the Big March. If I were teaching full-time this year, I might have taken off the entire Big March again this year, just to make sure that I work hard to survive it rather than spend so much time posting on a blog. Since this year I'm just a sub, I'll continue to blog throughout the Big March.
Indeed, today I subbed in a middle school music class. While I definitely avoid "A Day in the Life" for middle school P.E., it's difficult for me to decide whether to do it for music, which is also unrepresentative of the math I'd like to teach someday. Since there's so much else going on in today's post, let's skip "A Day in the Life," but of course I'll say a little about today.
Today's teacher has four instrumental music classes and one vocal music class. Of the four classes with instruments, two are band and the others are strings.
In general, string classes are better-behaved. In fact, today the regular teacher doesn't even trust the seventh grade band to perform -- he declares today to be a study hall instead. The other instrumental classes are allowed to perform. Among the songs I hear performed today are:
Band 8: "Avenue Swing," "Celtic Art & Dance"
Strings 7: "Viva los Conquistadores," "Fantasy on a Japanese Folk Song," "Fiddles on Fire"
Strings 8: "Dance of the Tumblers," "Echelon"
Among the classes that perform today, the only class with a behavior issue is in Strings 7, where some students argue with the designated performance leader over the song tempo. Ordinarily a metronome is used to keep time, but the teacher's metronome has been locked away. I try to keep time by rapping a pencil over the wooden piano, but this isn't enough. The seventh grade girl in charge ends up yelling at the students (but at least I don't yell today).
On the Eleven Calendar, today is Sunday, the third day of the week:
Resolution #3: We remember math like riding a bicycle.
Of course, there's not much math today. Two girls work on math during the Band 7 class where they're not allowed to perform. I tell them to remember math like riding a bike -- including formulas related to graphing and slope.
The Valentine's pencil and candy incentive ought to be done by now. But I know that while many middle school students enjoy the pencils and candy (as opposed to the high school seniors, as I found out last week), I recently experienced a middle school drought. Not only did I not sub in any middle schools during Valentine's Week (February 8th-14th), I didn't sub during Valentine's Fortnight (February 1st-14th), perhaps the longest stretch of time when giving away Valentine's goodies may be considered acceptable. The last middle school class I subbed in was on Friday, January 31st -- and it was for P.E., where giving away pencils and candy is awkward regardless of the date. The last time I subbed for middle school where giveaways might be convenient was January 28th. Someone could have asked why I was giving out valentines on the 28th of January.
Anyway, I had about ten extra pencils left over for today -- and I told myself that if I finally subbed for middle school on the very first day, I'd give them away. The teacher's schedule is also convenient because he starts with conference period, so I had an opportunity to add candy as well. (And of course, the candy was half-price.) The tricky part is to decide who earns the goodies -- after all, everyone is working together. I attempt to give them away to those who clean up (put away chairs and music stands) at the end of class, but this is still awkward.
Meanwhile, the song incentive for today is of course "The Big March." This year I sing it to seven verses, the length of the Big March in this district. The incentive comes in handy during the two periods where the regular teacher enforces a strict restroom policy -- Band 7 and Choir. As I mentioned in an earlier post, if the teacher specifies a strict policy, then I enforce it by telling the students that they must avoid restroom passes if they want the song. No one asks for a pass during the Band 7 class. But in Choir, one student goes -- a special ed student with a one-on-one aide. She tells me that the regular teacher is already aware that this student may need to restroom more often, and so I don't penalize the entire class.
That vocal music class is quite interesting. While Choir, like Band 7, has a strict restroom policy, at least they're allowed to perform today. This time, the teacher asks them to sing a "Spring Song" -- which, as it turns out, means to create and perform an original song in groups. This is obviously tricky -- some students have come up with lyrics but need to add a tune.
And so it's time for me to inspire the students with my own original songs. Today's "Big March" is not an original song, since it's based on "The Ants Go Marching In." There are currently five songs in my rotation that are purely original:
- The Dren Song (First Day of School)
- The Benchmark Test Song
- The Mousetrap Car Song
- Solving Equations
- Whodunnit?
And indeed I sing some of these to them. (Note to blog readers: yes, I wrote some other original tunes for songs I blogged during the charter year, but I don't like most of them. One of these, "Packet Song," I transformed into a rap, and I might consider doing the same to a second song. This was all before I discovered the Mocha emulator, so I might one day run Mocha and create tunes of some of the other songs that I blogged.)
This would actually be the perfect time to bring in Mocha music. I could run the program, have it generate some tunes, and then let the students adapt the tunes to their music. But unfortunately, the Chromebooks in this room aren't connected to any speakers, and so I couldn't play Mocha music.
Last year, I subbed for music at another school and actually played some Mocha music. It was close to Mother's Day, and so I created some "Mother's Day music" (that is, springtime music). Unlike Halloween, when minor-sounding scales such as 12EDL might work, for springtime we'd probably prefer some major-sounding scales. That day, I might have actually passed 14EDL off as a springtime scale, with 14/11 as the "major third." Of course, 18EDL contains both a (super)major third 9/7 and a minor third 6/5, so playing random notes is just as likely to sound minor as major.
If I was really trying to use Mocha to create springtime music for a class like today's, the scale I'm most likely to use is 10EDL. This is basically a pentatonic scale, except with 10/7 (a tritone) in place of the perfect fifth. Then I can just tell the students to sing it as if it were a perfect fifth. Pentatonic scales are wonderful for generating random music, since almost any sequence of notes in this scale sounds good. (The four-note Fischinger scale is a subset of the pentatonic scale, which is why Google used it to allow non-musicians to generate consonant music.)
It's too bad that I'm not returning to this class anytime soon. I would have loved to run Mocha on my computer tonight, generate some tunes, and then bring them to the Choir class tomorrow.
Today on her Daily Epsilon of Math 2020, Rebecca Rapoport writes:
Find x.
[All of the given info is in the diagram. We are given a polygon whose angle measures are 82, 110, 150, and x degrees.]
This problem seems straightforward. We simply write an equation:
82 + 110 + 150 + x = 360
x + 342 = 360
x = 18
Therefore the desired angle is 18 degrees -- and of course, today's date is the eighteenth. The main theorem used is the Polygon-Sum Theorem of Lesson 5-7.
But there's one huge problem here -- the given polygon is not a quadrilateral! It's a pentagon, with four of the angle measures as 82, 110, 150, and x. The fifth angle isn't stated, but it definitely appears to be a reflex angle of greater than 180 degrees (in other words, the pentagon isn't convex).
And notice that the answer we obtain by ignoring the fifth angle -- 18 degrees -- actually does match today's date. Indeed, since the sum of the angles of a pentagon is (5 - 2)180 = 540 and one of the angles is greater than 180, it follows that the the sum of the other four angles is less than 360, so that the last angle can't possibly be 18 degrees.
Thus what really happens is that Rapoport has made an error here -- by giving only four angles whose sum is 360, she intends the polygon to be a quadrilateral, yet she draws a pentagon instead. And in fact, the three given angles appear to be much smaller than their labels -- the "82-degree angle" looks much less than 82 degrees, and the "110-degree angle" is actually at best a right angle and perhaps even slightly acute. (Maybe the "110-degree angle" is really 82 degrees!)
But today's error pales in comparison to the mistake that Rapoport wrote during the long President's Day Weekend:
The sketch is of equally spaced railroad ties drawn in one-point perspective.
Notice that perspective appears in Lesson 1-5 of the U of Chicago text, as well as Lesson 0.8 of the Michael Serra Geometry text. But neither of these texts equip us to solve this quantitative problem about perspective. The two railroad ties in the sketch are of apparent lengths 25 and 20, and we're asked to find the length of the next tie.
This is one of Rapoport's research questions, one where we're supposed to look up something about perspective (or Projective Geometry, the branch in which it appears), and then some exact formula tells us the answer. At first. but I was unable to find such a formula.
It's evident that the ties (of equal length in the real-world 3D space) appear to shrink as we approach the vanishing point. Since the two given ties are of lengths 25 and 20, a first easy guess for the length of the next tie is 15. In other words, our conjecture is that the tie lengths form an arithmetic sequence.
Well, this is obviously wrong, because the date of this problem wasn't Saturday the fifteenth -- instead, it was Sunday the sixteenth. Then this leads to another conjecture -- we notice that 25, 20, 16 form a geometric sequence with common ratio 0.8. So we expect there to be some theorem which asserts that the tie lengths in perspective form a geometric sequence.
Of course, this is one flaw with calendar math problems -- it's so tempting just to look at the date and go backward from there to search for the theorem. Since the date was the 16th and 25, 20, 16 form a geometric sequence, obviously there must a theorem involving perspective and geometric sequences.
So when I performed a Google search for perspective and geometric sequences, I stumbled upon the following link:
https://math.stackexchange.com/questions/2337183/one-point-perspective-formula
At the above link we see the answer for another perspective problem using matrices. It helps now to step back and see what's going on here.
The perspective mapping from 3D to 2D is called a projection. Projections are transformations, just as reflections, translations, and so on are. The key difference between projections and the Common Core transformations is that the latter map 2D to 2D (and any space to space of the same dimension) while projections decrease the dimension by 1.
Now both projections and Common Core transformations are associated with matrices. We learn in the U of Chicago Algebra II text that if we multiply the transformation matrix by a vector for the point (pre-image), the product is the vector for the image.
The U of Chicago provides matrices for reflections in certain mirrors passing through the origin as well as certain rotations centered at the origin. But matrix multiplication clearly doesn't work for translations -- it's easy to see why. Any matrix multiplied by the zero vector is the zero vector, and so any transformation with a matrix must have the origin as a fixed point. Translations (other than the trivial one) have no fixed points, least of all at the origin.
But there's a trick to find matrices for translations as well. We embed the 2D plane in 3D space -- the plane of equation z = 1. Then the pair (0, 0) becomes the triple (0, 0, 1) -- and since that's not the origin, we can multiply it by matrices without obtaining the zero vector. Then what appears to be a translation in 2D is actually a "transvection" (or "shear") in 3D. Likewise, a translation in 3D must be written as a transvection in 4D.
We find out that 2D isometries have 2 * 2 matrices and 3D isometries have 3 * 3 matrices. It turns out that projections from 3D to 2D have 2 * 3 matrices, so that when we multiply them by a 3D vector (a 3 * 1 matrix), the product is a 2D vector (a 2 * 1 matrix). The specific projection mentioned at the link above does not map the 3D origin to the 2D origin, and so we do the translation/transvection trick mentioned above to obtain a 3 * 4 matrix (which is what we indeed see at the link).
For the specific projection mentioned at the link above, the formula given for the vanishing point at (3, 2) turns out to be:
((1 + 3mu)/(1 + mu), 2mu/(1 + mu))
where for our railroad track problem, we can simply assume that mu = 1 for the first railroad tie, then mu = 2 for the second tie, mu = 3 for the third railroad tie, and so on.
Of course, we don't know that the vanishing point is (3, 2) for the Rapoport problem. But the key here is to observe the difference in coordinates between (3, 2) and the calculated values above. For example, looking at the y-coordinates, we obtain:
2 - 2mu/(1 + mu) = (2 + 2mu)/(1 + mu) - 2mu/(1 + mu)
= 2/(1 + mu)
This is clearly not a geometric sequence, as we had conjectured. Indeed, we notice that the reciprocal of this sequence is an arithmetic sequence, (1 + mu)/2. Such a sequence is a harmonic sequence.
And we find that the x-differences also form a harmonic sequence. And since the (parallel) ties and sides of the rail form similar triangles, it follows that the lengths of the ties must likewise form a harmonic sequence.
This generalizes to the Rapoport problem. Thus the next tie length must follow 25, 20 in a harmonic sequence -- that is, their reciprocals form an arithmetic sequence. The reciprocal of 25 is 0.04 and the reciprocal of 20 is 0.05, and the next number after 0.04, 0.05 in arithmetic sequence is 0.06. And its reciprocal is 16 + 2/3 -- which must be the length of the next tie, not 16.
Therefore Rapoport makes another error here, since 16 + 2/3 wasn't Sunday's date. She can't even claim that she was rounding, since 16 + 2/3 rounds to 17, not 16.
So that makes two errors on the Rapoport calendar in one week. I enjoy her calendar, and so I hope there won't be any more errors in the near future.
Let's get to today's lesson. I've written above that one of the most difficult units always seems to begin right around the start of the Big March. Many students have trouble with graphing throughout Chapter 11, and furthermore, today we learn the equation of a circle, which just a few years ago was part of Algebra II! (Meanwhile, English classes tend to read Shakespeare during the Big March, for example.)
Lesson 11-3 of the U of Chicago text is called "Equations for Circles." In the modern Third Edition of the text, equations for circles appear in Lesson 11-6.
This is what I wrote last year about today's lesson:
The first circle lesson is on Lesson 11-3 of the U of Chicago text, on Equations of Circles. I mentioned that I wanted to skip this because I considered equations of circles to be more like Algebra II than Geometry. Yet equations of circles appear on the PARCC EOY exam.
Furthermore, I see that there are some circle equations on the PARCC exam that actually require the student to complete the square! For example, in Example 1 of the U of Chicago text, we have the equation x^2 + (y + 4)^2 = 49 for a circle centered at (0, -4) of radius 7. But this equation could also be written as x^2 + y^2 + 8y = 33. We have to complete the square before we can identify the center and radius of this circle.
In theory, the students already learned how to complete the square to solve quadratic equations the previous year, in Algebra I. But among the three algebraic methods of solving quadratic equations -- factoring, completing the square, and using the quadratic formula -- I believe that completing the square is the one that students are least likely to remember. In fact, back when I was student teaching, my Algebra I class had fallen behind and we ended up skipping completing the square -- covering only factoring and the quadratic formula to solve equations. And yet PARCC expects the students to complete the square on the Geometry test!
I also wonder whether it's desirable, in Algebra I, to teach factoring and completing the square, but possibly save the Quadratic Formula for Algebra II. This way, the students would have at least seen completing the square in Algebra I before applying it to today's Geometry lesson.
Here are the worksheets for today:
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