Lesson 5-2: Types of Quadrilaterals (Day 52)

Today I subbed in a continuation school English class. As it turns out, three of the classes are for English learners, and so they actually have an aide. Thus there is no "A Day in the Life" today.

Two of the classes are reading for the EL students. There are also two sections of senior English (one of which is with the EL aide) and one of junior English. The reading classes basically have silent reading and a reading log for the entire period. The seniors have a pre-write for a essay on whether college is necessary, while the juniors fill out a chart on the characters of John Steinbeck's Of Mice and Men.

Since it's finally Halloween week, I have candy and pencils to give away the entire week. I also decide to sing a song for the two unaided classes -- and it's Square One TV's "Ghost of a Chance," my Halloween week go-to song.

There are two classroom management issues in the unaided classes. One of them is in the junior class, where one guy has only filled in one (out of five) characters on his chart, but decides to steal some reward candy while I'm singing the song.

The other is in the senior class. As a sub, there are two times when I enforce a seating chart:

• when there is a quiz or test, and
• when the regular teacher explicitly mentions to enforce the chart.

Clearly there is not a quiz or test today, but the latter does appear in the teacher's lesson plan. In one of the reading classes, one girl convinces the aide to let her sit in another seat. After I see that she and the girl she sits next to both fail to complete the reading log, I know that I must be stricter in the two classes without an aide. I write on the board that any student not in the assigned seat will be written up with a referral.

This takes me to the unaided senior class. As the students enter the class, I greet each one -- a subtle reminder that the seating chart is in force. (I mentioned on the blog last week that this is something that I should do, and I indeed do so today.) But one girl moves a few minutes after class starts (and she can't claim that the teacher just forgot to change it on the roster, because she's the third student sitting at a desk meant for two). She says that she's been absent and thus needs some help, so I tell her that she can stay for a few minutes.

But then she claims that she's a slow worker and needs the entire period to get help. I tell her that she can stay at the other seat for fifteen minutes, then she must move. Fortunately, the seat she chooses is close to the teacher seat, so I can monitor how many of the fifteen minutes she's working. My estimate is at most five of the fifteen minutes -- the rest of the time she has non-academic discussion with the others. When time is up, she asks to go to the restroom, spends ten minutes there, and then returns to the incorrect seat. And so I must write up the referral.

But what makes this tricky is that in the other senior English class, the aide allows five students to sit in different seats. (Seat switching is often a domino effect -- one student moves to another seat, but the original owner of the new seat must now move to a third seat, the original owner of which must move to a fourth seat, and so on.) This is what makes my referral decision tricky -- yes, the girl in my solo class is defiant and should have returned to her correct seat, yet the aide lets several students move in her classes.

Anyway, this whole incident gives me hints regarding what to look for. The student claims that she needs more time to work because she's slow -- and I've already written about the dangers of equating intelligence with speed. But as I see today, a student can claim that she's slow when what she really wants is to sit next to her friends and have a non-academic conversation.

Therefore teachers must say no and require their students to work quickly -- not because intelligence equals speed, but in order to close gaps of time that students can fill with non-academic talk.

In fact, if I were a young student returning from an absence, I'd only need a few seconds at most for another student to tell me how to do the assignment. Or more likely, I could figure out the lesson without speaking to anyone at all -- simply by reading the packet, I could determine that the whole lesson is a pre-write to write an essay, with evidence coming from the passages in the packet. Thus I have little sympathy for students returning from absences who need to spend fifteen minutes talking to another student to catch up. My assumption should be that anyone needing to speak to another student to catch up just wants non-academic talk. (I don't say any of this to the girl, though, since there's no way to do so without arguing.)

By the way, one of my favorite math bloggers, Fawn Nguyen, has posted:

http://fawnnguyen.com/an-update/

The young nurse asks how my day is going as she wraps the blood pressure cuff around my arm. I pretend to be relaxing on a yacht to yield low numbers, and the machine beeps 124/74. Dammit, I’ve never been above 120 — the yacht is sinking and death is near. She tells me that the procedure I’m about to undergo is quick. Math practice number six beckons and I ask, “What is quick?… Like five-seconds-kinda-quick or…?” She smiles, “About fifteen minutes.”

Get well soon, Fawn! Oh, and many condolences for the loss of your brother.

Meanwhile, she also writes about her new job as a math coach:

I now have the privilege of observing different classrooms, modeling a number talk or task, designing a lesson and co-teaching, working with younger students, facilitating PD, creating slide decks and docs that might be helpful. There are three of us TOSAs in the district: English, ELD, and Math. I don’t get to see much of these two smart, strong, caring women outside of meetings, but they make me laugh and have my full admiration. There’s something special here with personnel. I liken it to the DNA that Oregon Ducks’ head coach Cristobal often speaks about, the DNA of each player that collectively makes up the team’s DNA. The culture is good here. My bosses are passionate and grounded, their roots are strong within the community because they are part of the community; their history is their present. It’s a cool place to be, and I feel very fortunate to be a part of an incredibly hard-working and caring network.

Chapter 9 of Ian Stewart's The Story of Mathematics is called "Patterns in Nature: Formulating Laws of Physics." Here's how it begins:

"The main message in Newton's Principia was not the specific laws of nature that he discovered and used, but the idea that such laws exist -- together with evidence that they way to model nature's laws mathematically is with differential equations."

This chapter is all about applying mathematics to the physical sciences -- and as the above implies, this required Calculus and solving differential equations. The solution to a differential equation is not a number, but a function -- a formula:

"In a way, this was unfortunate, because formulas of this type usually fail to exist, so attention became focused on equations that could be solved by a formula rather than equations that genuinely described nature."

Stewart's example is the motion of a pendulum, where "theta" is the angular displacement of the pendulum as measured from the vertical:

"The main disadvantage is that the solution fails when theta becomes sufficiently large (and here even 20 degrees is large is we want an accurate answer)."

The author describes two types of differential equations -- ordinary differential equations (ODE) whose solution is a function of a single variable, and partial differential equations (PDE) whose solution is a function of several variables. (Partial derivative appear in Multivariable Calculuis -- the next class after Calc BC. Recall from last week's post that a few high school seniors might attend Multivariable Calc classes at the community college.)

Here Stewart refers to to motion of a violin string, first studied by John Bernoulli in 1727 and later expanded upon by Jean d'Alembert two decades later:

"Starting from Bernoulli's results, which were based on Newton's law of motion, and making some simplifications (for example, that the size of the vibration is small), d'Alembert showed was led to the PDE d^2 y/dt^2 = a^2 d^2 y/dx^2 where y = y(x, t) is the shape of the string at time t, as a function of the horizontal coordinate x."

That PDE looks bad in ASCII. The "d" symbol should actually be a partial differential symbol, often written as a backward "6." It's often pronounced "del," but sometimes pronounced "partial" as well.

Stewart tells us that this PDE is now known as the wave equation. Euler worked with it as well:

"D'Alembert did not recognize the possibility of discontinuous functions in Euler's sense. Moreover, there seemed to be a fundamental flaw in Euler's work, because trigonometric functions are continuous, and so are all (finite) superpositions of them."

Notice that it's the wave equation that leads to one of my favorite subjects -- music, and why string lengths in simple ratios produce harmony:

"The ancient Greeks knew that a vibrating string can produce many different musical notes, depending on the position of the nodes, or rest points."

But, as Stewart explains, that's not all the wave equation is used for:

"Waves arise not only in musical instruments, but in the physics of light and sound. Euler found a three-dimensional version of the wave equation, which he applied to sound waves."

The author now moves on to the application of PDEs to gravitational attraction. Colin Maclaurin (the namesake of the "Maclaurin series" of Calc BC) studied spheroids -- the ultimate shape of an object that rotates, such as the earth (which is why it's not a perfect sphere):

"His main result was that if two spheroids have the same foci, and if a particle lies either on the equatorial plane or the axis of revolution, then the force exerted on it by either spheroid is proportional to their masses. In 1743 Claraut continued working on this problem in his Theorie de la Figure de la Terre."

We now proceed with heat and temperature. The main scientist who studied heat in the 19th century was Joseph Fourier:

"However, his work was widely criticized for its lack of logical rigour and the Academy refused to publish it as a memoir."

But that's not the end of that story -- Fourier ultimately got the last laugh:

"In 1824 Fourier got even: he was made Secretary of the Academy, and promptly published his 1811 paper as a memoir."

The Fourier series is named for him -- it's sort of like a Maclaurin series, but with sines and cosines:

"Fourier considered it obvious that any initial distribution of temperature could be so expressed, and this is where the trouble began, because a few of his contemporaries had been worrying about precisely this issue for some time, in connection with waves, and had convinced themselves that it was much harder than it seems. Fourier's argument for the existence of an expansion in sines and cosines was complicated, confused, and wildly non-rigorous."

(By the way, here's an aside. Back when I was a young student taking Calc BC, our teacher showed us a lesson on modeling that she had found on the internet, since she wanted us to learn more about mathematical modeling. But she had unwittingly given us a lesson on Fourier series -- which is well beyond anything that we study in Calc BC! Naturally, the entire class was confused with the HW, and the lesson was eventually thrown out.)

Stewart now moves on to fluid dynamics:

"Equations for a viscous fluid were derived by Claude Navier in 1821, and again by Poisson in 1829."

I once had a T-shirt that I'd purchased from a museum. On it was printed, "What part of (complicated PDE) don't you understand?" If I'm not mistaken, this is the sort of PDE that was printed there:

"In 1845, George Gabriel Stokes deduced the same equations from more basic physical principles, and they are therefore known as the Navier-Stokes equations. We close this section with two far-reaching contributions to the use of ODEs (ordinary differential equations) in mechanics."

One was by Joseph Lagrange (yes, the base 11 fan), who described the motion of a pendulum:

"The first innovation is to pair off the coordinates: to every position coordinate q (such as the angle of the pendulum) there is associated the corresponding velocity coordinate q' (the rate of angular motion of the pendulum)."

(By the way, q' is the derivative of q in ASCII. Lagrange used Newton's notation -- a dot about q, to indicate the derivative with respect to time.)

The other was by William Hamilton, who expanded upon Lagrange's idea:

"Theoretical work in mechanics generally uses the Hamiltonian formalism, which has been extended to quantum mechanics as well. Newton's Principia was impressive, with its revelation of deep mathematical laws underlying natural phenomena."

But it's differential equations -- both ODEs and PDEs -- that describe these laws:

"Many of the most important technological advances, such as radio, television, and commercial jet aircraft depend, in numerous ways, on the mathematics of differential equations."

On that note, Stewart concludes the chapter as follows:

"It is fair to say that Newton's invention of differential equations, fleshed out by his successors in the 18th and 19th centuries, is in many ways responsible for the society in which we now live. This only goes to show what is lurking just behind the scenes, if you care to look."

And our hope, of course, is that our students sitting in our classes care to look at the math behind many of the things that they enjoy.

The sidebars in this chapter are on how Fourier series work, a biography of Russian mathematician Sofia Vasilyevna Kovalevskaya, what differential equations did for them, and what differential equations do for us.

Lesson 5-2 of the U of Chicago text is called "Types of Quadrilaterals." In the modern Third Edition of the text, quadrilaterals appear in Lesson 6-4.

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

Lesson 5-2 of the U of Chicago text covers the various types of quadrilaterals. There are no theorems in this section, but just definitions. The concept of definition is important to the study of geometry, and in no lesson so far are definitions more prominent than in this lesson.

The lesson begins by defining parallelogram, rhombus, rectangle, and square. There's nothing wrong with any of those definitions. But then we reach a controversial definition -- that of trapezoid:

Definition:
A quadrilateral is a trapezoid if and only if it has at least one pair of parallel sides.
(emphasis mine)

Just as with the definition of parallel back in Lesson 1-7, we have two extra words that distinguish this from a traditional definition of trapezoid -- "at least." In other textbooks, no parallelogram is a trapezoid, but in the U of Chicago text, every parallelogram is a trapezoid!

To understand what's going on here, let's go back to the first geometer who defined some of the terms in the quadrilateral hierarchy -- of course, I'm talking about Euclid:

http://aleph0.clarku.edu/~djoyce/java/elements/bookI/defI22.html

Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

Of course, the modern term for "oblong" is rectangle, and a "rhomboid" is now a parallelogram. The word "trapezia" is actually plural of "trapezium." In British English, a "trapezium" is what we Americans would call a trapezoid, but to Euclid, any quadrilateral that is not a parallelogram (or below on the quadrilateral hierarchy) is a "trapezium." But the important part here is that to Euclid, a square, for example, is neither a rectangle (oblong) nor a rhombus. He makes sure to say that a rectangle (oblong) is "not equilateral," and that a rhombus is "not right-angled." And of course, neither a rectangle nor a rhombus is a parallelogram (rhomboid).

These are called "exclusive" definitions. For Euclid, there was no quadrilateral hierarchy -- each class of quadrilaterals was disjoint from the others. But since the days of Euclid, more and more geometry texts have slowly added more "inclusive" definitions.

One of the first inclusive definitions I've seen was the definition of rectangle. It was mentioned on an episode of Square One TV, when a Pacman parody character named Mathman was supposed to eat rectangles, and then ate a square because "every square's a rectangle." I would provide a YouTube link, but I haven't found the link in years and it doesn't come up in a search. (I even remember someone posting in the comments that just as for me, his first encounter of the inclusive definition of rectangle was through watching that clip when it first aired so many years ago!)

But many of my family members were also teachers, and one relative gave me an old textbook that still mentioned some exclusive definitions. In particular, it declared that a square isn't a rhombus. A little later, my fifth grade teacher then taught the inclusive definition of rhombus. I then blurted out that a square isn't a rhombus, then actually brought the old text to school to prove it! She replied, "Wow!" but then, if I recalled correctly, told me that this definition was old, and that by the new definition, a square is a rhombus. And so all modern texts classify the square as both a rectangle and a rhombus, and that all of these are considered parallelograms.

So we see that there is a tendency for definitions to grow more inclusive as time goes on. (We see this happening in politics as well -- for example, the definition of marriage. But I digress.) And so we see the next natural step is for the parallelogram to be considered a trapezoid.

One of the first advocates I saw for an inclusive definition of trapezoid is the famous Princeton mathematician John H. Conway. He is best known for inventing the mathematical Game of Life, which has its own website:

http://www.conwaylife.com/

But Conway also specializes in other fields of mathematics, such as geometry and group theory (which is, in some ways, the study of symmetry). Twelve years ago, he posted the following information about why he prefers inclusive definitions:

http://mathforum.org/kb/message.jspa?messageID=1081135

The preference for exclusive definitions arises, I think, from
what I call "the descriptive use". Of course, one wouldn't DESCRIBE
a square table as "rectangular", since that would wantonly use
a longer term to convey less information. So in descriptive uses,
there's a natural presumption that a table called "rectangular"
won't in fact be square - in other words, a natural presumption
that the terms will be used exclusively.

But the descriptive use is unimportant to geometry, where the
really important thing is the truth of theorems. This means that we
should use a term "A" to include "B" if all the identities that
hold for all "A"s will also hold for all "B"s (in the way that
the trapezoid area theorem holds for all parallelograms, for
instance).

You might worry about the consistency of switching to the
inclusive use while other people continue with the exclusive one.
But there can be no consistency with people who are inconsistent!
I've seen many geometry books that MAKE the exclusive definitions,
but none that manage to USE them consistently for more than a few
theorems.

Indeed, Conway advocated taking it one step forward and actually abolishing the trapezoid and having only the isosceles trapezoid in the hierarchy! After all, there's not much one can say about a trapezoid that's not isosceles -- just look ahead to Lesson 5-5. There's only one theorem listed there about general trapezoids -- the Trapezoid Angle Theorem, and that's really just the Same-Side Interior Angle Consequence Theorem that can be proved without reference to trapezoids at all. All the other theorems in the lesson refer to isosceles trapezoids. In particular, the symmetry theorems in the lesson refer to isosceles trapezoids. (Recall that Conway's specializes in group theory -- which as I wrote above is the study of symmetry.) I suspect that the only reason that we have general trapezoids is that they are the simplest quadrilateral for which an area formula can be given.

This is now another digression from Common Core Geometry, so I'll just provide another link. Notice that here, Conway also proposes a hexagon hierarchy based on symmetry. There's also a pentagon hierarchy, but there are only three types of pentagons -- general, line-symmetric, and regular -- just as there are for triangles. It's easier to make figures with an even number of sides symmetric.

http://mathforum.org/kb/message.jspa?messageID=1074038

Another advocate of inclusive definitions is Mr. Chase, a Maryland high school math teacher. (And no, in yesterday's post and today's I'm referring to two different Maryland teachers.) I see that he is so passionate about the inclusive definition of trapezoid that he devoted three whole blog posts to why he hates the exclusive definition of trapezoid:

http://mrchasemath.wordpress.com/2011/02/03/why-i-hate-the-definition-of-trapezoids/

http://mrchasemath.wordpress.com/2011/02/18/why-i-hate-the-definition-of-trapezoids-again/

http://mrchasemath.wordpress.com/2013/08/12/why-i-hate-the-definition-of-trapezoids-part-3/

One reason Chase states for using inclusive definitions is that it simplifies proofs:

When proving that a quadrilateral is a trapezoid, one can stop after proving just two sides are parallel. But with the exclusive definition, in order to prove that a quadrilateral is a trapezoid, you would have to prove two sides are parallel AND the other two sides are not parallel.

Regarding some of the others to whom I refer regularly, Dr. Wu uses the inclusive definition:

"A quadrilateral with at least one pair of opposite sides that are parallel is called a trapezoid. A trapezoid with two pairs of parallel opposite sides is called a parallelogram."

while Dr. Mason uses the exclusive definition:

"A trapezoid is by definition a quadrilateral with precisely one pair of parallel sides."
(emphasis Dr. M's)

So which definition should I use for trapezoid? Well, this is a Common Core blog, so the definition favored by Common Core should have priority over all other definitions. The following is a link to the information that will appear on the PARCC End-of-Year Assessment for geometry:

http://www.parcconline.org/sites/parcc/files/ES%20Table%20Geometry%20EOY%20for%20PARCC_Final.pdf

And right there in the column under "Clarifications," it reads:

i) A trapezoid is defined as “A quadrilateral with at least one pair of parallel sides.”

And that plainly settles it. The PARCC Common Core assessment uses the inclusive definition of trapezoid, and so it's my duty on a Common Core blog to use the Common Core definition. Of course, we notice that this is the definition given by PARCC -- but so far I've seen no information on what definition Smarter Balanced is using. It would be tragic if PARCC were to use one definition and Smarter Balanced the other. But as I can't say anything about Smarter Balanced, I will use the only definition that's known to be on a Common Core test, and that's the inclusive definition. The fact that this definition is already used by the U of Chicago is icing on the cake.

There is one problem with the inclusive definition of trapezoid, and that's when we try to define isosceles trapezoid. The word isosceles suggests that, just as in an isosceles triangle, an isosceles trapezoid has two equal sides -- the sides adjacent to the (parallel) bases. But in a parallelogram, where either pair of opposite sides can be considered the bases -- the sides adjacent to these bases are also equal. This would make every parallelogram an isosceles trapezoid. But this isn't desirable -- an isosceles trapezoid has several properties that parallelograms in general lack. The diagonals of an isosceles trapezoid are equal, but those of a parallelogram in general aren't. But the diagonals of a rectangle are equal. So we'd like to consider rectangles, but not parallelograms in general, to be isosceles trapezoids.

According to the old Moise text I mentioned above, all parallelograms are isosceles trapezoids while no kite is a rhombus. Otherwise his definition of trapezoid matches U of Chicago's.

This dilemma is mentioned in the comments at one of the Chase links. It's pointed out that there are two ways out of this mess -- we may either define isosceles trapezoid in terms of symmetry, as Conway does, or we can use the U of Chicago's definition:

"A trapezoid is isosceles if and only if it has a pair of base angles equal in measure."

Some don't like this definition, because it violates linguistic purity -- the word isosceles comes from Greek, and it means "equal legs," not "equal angles." But as it turns out, it's a small price to pay to make the quadrilateral hierarchy and other theorems work out. And besides, any geometer who calls a nine-sided polygon a nonagon should just shut up about linguistic purity!

Note: Mr. Chase made only one post in 2018 (and none in 2019). It's all about using pure Geometry to prove identities in Trig.