One problem that always arises on the school calendar is what to do about Halloween. For elementary students, Halloween is one of the best days of the year in the classroom, but many high school students would rather not go to school that day. And the next day, November 1st, isn't enjoyable for students at either age. After celebrating all night, students don't want to go to school -- and parents go along with it by not sending their students. So absences are up both October 31st and November 1st.
And so many schools solve this problem by scheduling PD either October 31st or November 1st -- and this is exactly what my new district is doing this year with no school today.
Which day is it better to take off, October 31st or November 1st? This is highly dependent on what day of the week Halloween happens to fall. With the holiday on Thursday this year, it's much better to take November 1st off -- if schools were to close October 31st instead, even students who don't celebrate Halloween might take today off anyway, since it's a lone day separating the closed day from the weekend. On the other hand, if Halloween is on a Monday, then it's better to take the 31st off for the same reason.
Last year, Halloween fell on a Wednesday. My new district responded by taking off Monday the 29th instead of the 31st or 1st. Some other schools might have closed on Wednesday the 31st instead -- not Thursday the 1st lest students also want Friday the 2nd off as well. In short, schools should close on whichever of the 31st and 1st falls on a Monday, Wednesday, or Friday.
Next year, Halloween will fall on a Saturday. This is the only situation in which both the 31st and the 1st fall on the weekend. Indeed, some people believe that Halloween should always fall on a Saturday (and some Calendar Reformers try to accommodate them).
I once asked on the blog, is it best to close on Columbus Day, Halloween, or All Saints' Day? In other words, suppose that your school could close only once between October 8th (the earliest possible date for Columbus Day) and November 1st. Which day would you choose?
Once again, it all depends on the weekday of Halloween. Next year, with the 31st and 1st on the weekend anyway, it's a no-brainer to choose Columbus Day. This year, my new district was open on Columbus Day while my old district was closed -- and in my new district, some students ask whether there will be school on Columbus Day (perhaps because either their parents, or their friends in other districts, have the day off). Sure, it's nice to have Columbus Day off, but if schools are open, students don't insist on taking the day off the way they do on the 31st or 1st.
By the way, personally I'd choose Columbus Day, only because Columbus Day is closest to the midpoint between Labor Day and Veteran's Day. Otherwise some schools might have no off days between Labor Day and Halloween. At least my new district also closes for one of the Jewish holidays between Labor Day and Halloween.
Of course, if you think that's awkward, notice that the teachers' strike in Chicago ended yesterday, and so the first day for students after two weeks off is today. It's the day after Halloween, plus a Friday after school was cancelled Monday-Thursday. Either of these occurrences is a reason for attendance to be lower than usual today, and in this case both of them occur today.
Today I subbed in a high school social studies class. Obviously, it's in my old district, since it's the only one of my two districts that's open. So today really is Day 56 in this district. Since it's a high school class and not math, there is no "A Day in the Life" today.
After all I wrote about attendance on the 31st and 1st, what's attendance like today? Well, as I suspected earlier, it's not very good. Of the 34 students in the zero period AP Psychology class, a dozen are absent, and many more are tardy -- maybe another dozen? If so, then this marks the first class I ever subbed where more students arrive after the tardy bell than before (though a few other classes were very close). At any rate it's probably the highest percentage of absences and tardies combined that I've ever experienced.
I have trouble logging into the regular teacher's laptop -- and since she left me her number, I call her after zero period to get the password. I tell her about the absences and tardies -- and inform her that in districts are closed (including an elementary feeder district to this high school district). She wonders whether this is a good idea -- if schools were closed on the 1st, students would act even wilder Halloween night than they currently do. (Of course, they'll all be wild next year on Saturday night!)
The last time I subbed at this school was on October 15th, the day after Columbus Day. This district was closed on Columbus Day (instead of today). Because of that closure and the PSAT, the block schedule was mixed up that week, with Tuesday an all-classes day instead of Friday. This week is also mixed up -- for some reason, there was an all-classes minimum day on Wednesday (having something to do with senior projects). Thus today was an odd day (as Wednesday would have been).
Two of the three block classes were sophomore World History, with the other being another section of the AP Psych class. The number of absences in each class were ten, three, and nine. The extra absences in the late class might be due to road sports contests, though.
I do sing in the sophomore classes. Since they are working on a packet today, I end up singing "The Packet Rap." This is after they watch the 1990's film Newsies, which is about paperboys. (President Teddy Roosevelt is a character.) The packets are about the Industrial Revolution (which is mainly during the 19th century, before TR's presidency.)
Chapter 13 of Ian Stewart's The Story of Mathematics is called "The Rise of Symmetry: How Not to Solve an Equation." Here's how it begins:
"Around 1850 mathematics underwent one of the most significant changes in its entire history, although this was not apparent at the time."
The title of this chapter should remind us of another side-along reading book -- the first I ever posted on the blog, years ago. It's Mario Livio's The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry. And indeed, it covers much of the same material. It all started with a new concept -- group theory:
"Today, group theory has become an indispensable tool in every area of mathematics and science, and its connections with symmetry are emphasized in most introductory texts."
Stewart opens with the hero of both this chapter and Livio's book -- the Frenchman Evariste Galois:
"He was a tragic figure living in a time of many personal tragedies, and his life was one of the more dramatic, and perhaps romantic, among those of major mathematicians. The story of group theory goes right back to the ancient Babylonian work on quadratic equations."
The author recaps the development of the Quadratic, Cubic, and Quartic Formulas. The next step would be a Quintic Formula, used to solve a fifth-degree polynomial equation:
"No doubt the formulas would be very complicated, and finding them would be even more complicated, but few seem to have doubted that they existed."
We return to the mathematician Lagrange. When he wasn't busy endorsing base 11, he was working on what happens when we switch the variables in a multi-variable expression:
"More interesting were expressions that only took on a few different values when the solutions were permuted."
In particular, he showed that the Cubic and Formulas were invariant when we switched the three or four solutions of the equation:
"But now Lagrange knew why those were the answers, and better still, he knew why answers existed to be found."
Gauss, meanwhile, believed that the Quintic Formula simply wasn't worth looking for:
"It is perhaps one of the few instances where his intuition about what is important let him down; another was Fermat's Last Theorem, but here the necessary methods were beyond even Gauss, and took a couple of centuries to emerge."
Stewart mentions Paolo Ruffini, who was the first to believe that a Quintic Formula is impossible:
"Unfortunately the main effect of this belief was to dissuade anyone from working on the problem. An exception was Abel, a young Norwegian with a precocious talent for mathematics, who thought that he had solved the quintic while still at school."
Although Abel was mistaken, he did stumble upon a trick:
"When it works, this trick is spectacular, and here it worked beautifully. It allowed Abel to reduce any hypothetical formula for solving the quintic to its essential steps: extract some sequence of radicals, in some order, with various degrees."
He called this a radical tower, and proved that if a Quintic Formula exists, it must be such a tower:
"This is called the Theorem on Natural Irrationalities and it states that nothing can be gained by including a whole pile of new quantities, unrelated to the original coefficients."
Abel eventually shows that the existence of such a tower leads to a contradiction -- one which involves switching the order of the solutions in a certain order. Thus there is no Quintic Formula:
"Ruffini and Abel had realized that an expression in the solutions did not have to be either symmetric or not."
It could be partially symmetric -- and Galois was the one who expanded upon this idea. He was the one who first came up the concept of a "group." This concept expanded into a full theory:
"The main architect of this theory was Camille Jordan, whose 667-page work Traite de Substitutions et des Equations Algebriques was published in 1870."
And Jordan's work also led to important results in our favorite subject, Geometry:
"But his work was a major step towards the understanding of Euclidean rigid motions, which are important in mechanics, as well as in the main body of pure mathematics."
Recall that "Euclidean rigid motions" are actually the Common Core isometries -- translations, rotations, reflections, and glide reflections. Stewart also mentions screw motions -- an isometry that exists in 3D, not 2D.
We've actually discussed some of Jordan's symmetry groups right here in the current Chapter 5 of the U of Chicago text. In Lesson 5-4, we learn that the symmetry group of a kite contains two elements (the identity and a reflection over its symmetry diagonals) while the symmetry group of a rhombus contains four elements (the identity, a reflection over its first diagonal, a reflection over its second diagonal, and the composite of these two reflections -- a rotation). In Lesson 5-5, we learn that an isosceles trapezoid has a two-element symmetry group, and a rectangle a symmetry group of order 4.
Returning to Jordan:
"He sets up the basic ideas of normal subgroups which are what Galois used to show that the symmetry group of the quintic is inconsistent with a solution by radicals, and proves that these subgroups can't be used to break a general group into simpler pieces."
Hence symmetry and group theory will be forever linked:
"Group theory also made it clear that negative results may still be important, and that an insistence on proof can sometimes lead to major discoveries."
On that note, Stewart ends the chapter as follows:
"If mathematicians had taken the easy route, and assumed the solution to be impossible, mathematics and science would have been a pale shadow of what they are today. That is why mathematicians insist on proofs."
Lesson 5-6 of the U of Chicago text is called "Alternate Interior Angles." The modern Third Edition of the text covers alternate interior angles slightly earlier, in Lesson 5-4 (along with Same Side Interior Angles, which aren't emphasized in the Second Edition).
This is what I wrote last year about today's lesson:
Today's lesson focuses on proving the Parallel Consequences -- that is, statements of the form, "if two lines are cut by a transversal, then ..."
This lesson will be set up almost exactly like Dr. Franklin Mason's Lesson 4.4. We wish to prove the converses of the Parallel Tests. We do so by using my favorite trick for proving converses -- we use the forward theorems along with a uniqueness statement. The uniqueness statement we need is the Uniqueness of Parallels Theorem -- in other words, Playfair.
It's possible to prove all of the Parallel Consequences by using the respective test plus Playfair -- so we'd prove the Corresponding Angles (CA) Consequence using CA Test plus Playfair, the Alternate Interior Angles (AIA) Consequence using AIA Test plus Playfair, and then Same-Side Interior Angles Consequence using that test plus Playfair.
But Dr. M only proves one of the consequences using Playfair -- he then uses vertical angles and linear pairs to derive the other consequences, as is traditionally done. We've seen that students should definitely be familiar with using one of the consequences to prove the others. Dr. M uses the Alternate Interior Angles Consequence to prove the others, but as we've discussed before, I'm changing this to the Corresponding Angles Consequence instead.
I looked back at last year's lesson and compared it to what I'm doing this year. In my first lesson after posting the Fifth Postulate last year, I posted some properties of two types of quadrilaterals, isosceles trapezoids and parallelograms. This year, we've already proved the isosceles trapezoid properties.
And now here's the activity I created -- a little something on spherical geometry (which is alluded to in Lesson 5-7):
Since I cut off the second part of the worksheet, I wish to replace it with something. I notice that in the U of Chicago and many other texts, students are given a taste of what would happen if we didn't have a parallel postulate. Yes, we'd have non-Euclidean geometry.
We've spent so much time on the blog discussing what's possible in neutral geometry and what requires a parallel postulate. The U of Chicago text introduces the students to spherical geometry -- as did I over the summer on the blog. But note that spherical geometry is not neutral -- neutral geometry includes only Euclidean and hyperbolic geometry. Technically, if we want to show students what impact the parallel postulate has on geometry, we should be showing them hyperbolic geometry, not spherical geometry. But since we live on a globe, spherical geometry is far easier for students at this age to understand.
Indeed, I often point out that the Fifth Postulate and its equivalents are, believe it or not, true in spherical geometry! This is because most of these statements begin with, "if two parallel lines..." or "if two lines are parallel." But in spherical geometry there are no parallel lines, so all Parallel Consequences are vacuously true! We might as well replace every occurrence of "parallel lines" with "unicorns," and all of the statements are still vacuously true:
Perpendicular to Parallels: If a line is perpendicular to one unicorn, it's perpendicular to the other.
Transitivity of Parallels: If a line is parallel to one unicorn, it's parallel to the other.
Parallel Consequences: If two unicorns are cut by a tranversal, corresponding angles are congruent.
Playfair: Using the version of Playfair given in Lesson 13-6 of the U of Chicago text doesn't work, but it does if we use Dr. Franklin Mason's version: "Through a point not on a given line, there's at most one line parallel to the given line." The phrase at most one allows for the possibility of zero.
Euclid's Fifth Postulate: If the two same-interior angles add up to less than two right angles, the lines will intersect on that side. This is true in spherical geometry because all lines (great circles) will intersect on both sides no matter what the angles add up to. Euclid's Fifth says nothing about what happens when the angles do add up to two right angles, only when they don't.
But still, I will mention spherical geometry as an example of a non-Euclidean geometry in which parallel lines don't work the way we expect them to.
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