Tuesday, October 3, 2017

Lesson 3-4: Parallel Lines (Day 34)

This is what Theoni Pappas writes on page 276 of her Magic of Mathematics:

"Simeon Poisson (1781-1840) had difficulty finding a career that suited him. His family urged him to pursue medicine or law, but he seemed to lack the talent and/or desire."

Pappas goes on to describe how Poisson stumbled upon a logic puzzle and realized that his ultimate calling was mathematics. He went on to study the subject, and there are concepts named after the French mathematician, such as Poisson distribution and Poisson kernel. This is why the title of this page is "How a Puzzle Can Be a Turning Point."

Now Pappas ends the page with a puzzle similar to the one that Poisson solved so easily:

"The milkman had two 10 quart pails of milk. Two customers want two quarts each in their containers. One has a five quart container and the other has a four quart container. How does the milkman solve the problem? This version of the problem was created by Sam Loyd."

(Oh, and of course the famous puzzler Sam Loyd didn't actually write the problem for Poisson. In fact, Loyd was born nine months after Poisson died.)

I definitely had trouble with this one. I was able to fill one of the containers with two quarts using a trick that I posted earlier on this blog, and then I had to look up the answer in order to figure out how to fill the other container.

As usual, I'll post the full answer tomorrow. But let's look at the trick that I mentioned earlier on the blog in order to provide you with a hint.

There is actually a program written in Logo that can solve this problem. Three years ago -- that is, during the 2014-15 school year -- I wrote about this problem during Lesson 13-8, since the U of Chicago text teaches Logo in this lesson. Two years ago, I ripped up Chapter 13 (just like the new Third Edition of the text), and I never formally cover Lesson 13-8. Last year, after I left my old school, I covered Lesson 13-8 and gave several Logo programs, but not the one that we want now.

And so this is what I wrote three years ago about this problem:

"You are at the side of a river. You have a three-liter pitcher and a seven-liter pitcher. The pitchers do not have markings to allow measuring smaller quantities. You need two liters of water. How can you measure two liters?"


So we can change the sizes of the pitchers to 5 and 4 instead of 3 and 7. (Well, we should also change liters to quarts, but a quart is only slightly smaller than a liter.) The difference of course is that our liquid isn't water from a river, but milk from two pails. Well, we can at least try the problem out with pitchers of size 5 and 4 and see how to get two quarts out of this.

If I had a computer that runs Logo, I could input the program from the above link and type in:

pour [5 4] 2

But unfortunately, I only have access to a BASIC emulator, not a Logo emulator. Then again, I notice that Brian Harvey, the Berkeley Logo author who wrote the above link, mentions a certain equation that we can try to solve:


"In this equation, x represents the number of times the three-liter pitcher is filled and y represents the number of times the seven-liter pitcher is filled. A positive value means that the pitcher is filled from the river, while a negative value means that it's filled from another pitcher."

And so to solve our problem, we must solve the following equation:


Notice that negative solutions are allowed, but they must be integers. According to Harvey, an equation with integer-valued variables is called a Diophantine equation. And hey -- that definitely sounds familiar. Didn't we read something about Diophantine equations from Ogilvy two weeks ago?

We can solve our Diophantine solution by inspection. We immediately notice that 5-4=1, and so if we had x = 1 and y = -1, the right-hand side would be 1. Since we want 2 instead of 1, we just double our solution to obtain (2, -2).

According to Harvey, this means that we fill the five-liter container twice from the "river" (which means one of the pails), and then we fill the four-liter container twice from "another pitcher" (which means the five-liter container). So we know that we have at some point:

?. Pour from the first pail into the five-liter container.
?. Pour from the first pail into the five-liter container.
?. Pour from the five-liter container into the four-liter container.
?. Pour from the five-liter container into the four-liter container.

In the end, we will have two liters in one of the containers, but what about the other? Unfortunately, neither the Diophantine equation nor the Logo program can help us now. But since the equation and program are based on their being only one "river" (or pail), it follows that we must use the second pail in order to get two liters into the other container.

That's all you get for hints. The full solution will be posted tomorrow.

\Meanwhile, yesterday I wrote about Monty Hall's passing and the Monty Hall Problem. As it turns out, the BUZZR channel had its tribute to the game show legend today. All day today, the channel aired both Let's Make a Deal and his other game show, Beat the Clock, in two-hour blocks. I watched one full block from each program, for a total of four hours. And to be clear, no, the classic Monty Hall Problem with three doors, two Zonks, and a car doesn't actually appear on the show, not even when Monty himself hosted it.

But before we let Monty Hall rest in peace, here's an article from today's Chicago Tribune that uses the host's passing as a reason to write about the Monty Hall Problem again:


OK, let's leave the Chicago newspaper and move on to the (U of) Chicago text.

Lesson 3-4 of the U of Chicago text is called "Parallel Lines." (It appears as Lesson 3-6 in the modern edition of the text.)

This is clearly one of the most important lessons in the entire text. And yet, believe it or not, I never actually covered Lesson 3-4 in any of the three years that I've been blogging!

The reason is that I was always unsure how I wanted to teach parallel lines. Over the years, I kept changing the way I presented this topic, for various reasons. The U of Chicago text teaches parallel lines the way it's done in most Geometry books, with two postulates:

Corresponding Angles Postulate:
If two coplanar lines are cut by a transversal so that two corresponding angles have the same measure, then the lines are parallel.

Parallel Lines Postulate:
If two lines are parallel and cut by a transversal, corresponding angles have the same measure.

But Dr. David Joyce, an education critic, wrote that there should only be one parallel postulate. In fact, we know that not only does Euclid have only one Parallel Postulate (his famous fifth postulate), but he doesn't even need the postulate it in his proof of the "Corresponding Angles Postulate." Instead he uses something called the Triangle Exterior Angle Inequality, or TEAI. On the other hand, his proof of the "Parallel Lines Postulate" is based on his fifth postulate.

Another education writer, Dr. Hung-Hsi Wu of Berkeley, also proves the Corresponding Angles Postulate without need of any parallel postulate. Instead, Wu uses Common Core transformations -- specifically 180-degree rotations -- to attain this result.

Three years ago, I used Wu's approach and based my parallel lessons on these half-turns. But after that, I felt that this was unsatisfactory for two reasons:

-- The Corresponding Angles results are more apparently related to translations than rotations -- after all, if two parallel lines are cut by a transversal, then a translation clearly maps one corresponding angle to another. Indeed, I often wonder why we use the same word "corresponding" to refer to angles formed by a transversal and to refer to parts of congruent (or similar) triangles. In the case of congruent (or similar) triangles, there is by definition an isometry (or similarity transformation) mapping one to the other. Well, in the case of angles formed by a transversal, there is a translation mapping one to the other. This justifies use of the term "corresponding angles."

-- The idea of delaying the Parallel Postulate until it is needed comes from the idea of "neutral geometry," based on theorems proved using only the other four postulates of Euclid. But we found out that there are two types of neutral geometry -- Euclidean geometry (where the fifth postulate is true) and hyperbolic geometry (where it is false). If we're going to mention non-Euclidean geometry at all, I'd prefer spherical geometry -- as in the spherical earth -- over hyperbolic geometry. But unfortunately, hyperbolic geometry isn't neutral. At least one of the first four postulates fails in spherical geometry -- whereas the fifth postulate, ironically, does hold.

Should I attempt to solve these two problems? In the end, these issues don't matter unless they actually affect teaching in the classroom. The second problem, with neutral geometry, matters only if we want, say, to introduce our students to non-Euclidean geometry at the end of the year -- and if were to do that, we'd want to show them spherical, not hyperbolic, geometry. (I've mentioned before how some high schools actually do this.) It would be nice if there was a clean break between "natural geometry" (that is, Euclidean+spherical geometry) and Euclidean geometry so we can say, "these Euclidean results still hold in spherical geometry while those don't" -- just as there's a clean break between "neutral geometry" (that is, Euclidean+hyperbolic geometry) and Euclidean geometry (simply by introducing the parallel postulate).

But does the first issue really matter? Last year, I wanted to use translations to demonstrate the Corresponding Angles Postulate in my class. Since this was eighth grade and not high school Geometry, we don't have to worry about proofs at all, and we can just show the results. It might be easier for an eighth grader to visualize the translation mapping corresponding angles than it is to remember what "corresponding angles" are. Ironically, this actually allows us to avoid using terms like "corresponding angles" and "alternate interior angles" in the eighth grade classroom, so instead the students can just focus on identifying the congruent angles.

I really did try to use translations to show corresponding angles to my eighth graders, as well as Wu's half-turns to demonstrate alternate interior angles, without using those terms. Unfortunately, I can't say that I was successful -- but once again, this ultimately goes back to classroom management, with students spending too much time talking in class and not enough time learning.

Last week, another math teacher, Education Realist, blogged about classroom management. I choose not to link to Education Realist as this blogger often writes about controversial topics and indeed, the author avoids self-identifying with a name or even a gender for this reason. (The author often identifies with the name "Ed," but this is short for "Education Realist," not Edward. Still, I admit it's tempting to use masculine pronouns with the male-sounding name "Ed.")

But what Ed writes about classroom management isn't controversial. I'll compare what Ed writes to how I actually ran my classroom, but only after I finish writing about parallel lines.

The point of all this is that if I'd had a strongly-managed class, I could have determined whether teaching parallel lines via translations helps eighth graders master the concept -- and if this does help them, then this justifies proving parallel lines via translations in high school. On the other hand, if translations don't help in eighth grade, then we shouldn't bother with parallels via translations in high school. Instead, we can use rotations (as in Wu), TEAI (as in Euclid), or just stick to two separate postulates (as in the U of Chicago).

Unfortunately, resolving both issues (translations and "natural geometry") often conflict. For example it's awkward to use translations in spherical geometry depending on how they're defined. Of course, the Corresponding Angles Postulate is false in spherical geometry anyway (since there are no parallel lines at all). Technically speaking, the Parallel Lines Postulate is true in both Euclidean and spherical geometry, albeit vacuously true in spherical geometry (since again, there are no parallel lines).

So what is my final decision regarding how I'll teach parallel lines this year on the blog? Earlier, I wrote that I don't want to change anything from the U of Chicago text at all this year. And so that's my final decision. Corresponding Angles and Parallel Lines are both postulates, and there is no attempt to prove either of them using translations, rotations, or "natural geometry" at all.

Also, I'll keep the names "Corresponding Angles Postulate" and "Parallel Lines Postulate" as the names of these postulates. In the past, we sometimes called the former "Corresponding Angles Test" and the latter "Corresponding Angles Consequence," but no more. In each case, the name of the postulate refers to the given information, so in the former we're given that the corresponding angles are congruent, and in the latter we're given that the lines are parallel. The U of Chicago text usually avoids the names of these postulates anyway, instead abbreviating them as "corr. angles = => | | lines" and "| | lines => corr. angles =" respectively. (By the way, in the new edition of the text, there is just one Corresponding Angles Postulate with both parts. Somehow, I doubt David Joyce would count this as being just "one postulate for parallels.")

This lesson also introduces slope and the slopes of parallel lines. This is another topic that's often difficult to teach rigorously, as both David Joyce and the Common Core Standards for Grade 8 suggest that slope should be taught after similarity. Again, I'm now following only the text. This is a good time to introduce Sarah Carter's famous "Slope Dude."

Notice that on this brand new worksheet, one of the questions actually refers to the fact that the Corresponding Angles Postulate is false on the earth (that is, in spherical geometry).

I also included the Exploration question as a bonus. This question refers to Desargues' Theorem. As it so happens, I mentioned this theorem in my March 28th post, since Pappas writes about it on page 87 of her book. Notice that students only need to "verify" the theorem with a few examples, rather than actually attempt to prove it. As we found out on March 28th, a full proof goes well beyond high school Geometry (and certainly beyond Lesson 3-4 of high school Geometry).

OK, let's get to Ed Realist and classroom management. Ed writes:

One of the most valuable pieces of advice I received, from two different teachers in two different years (student teaching, first year), was that a new teacher had to know what “quiet” is.  If kids wouldn’t shut up, then kick them out until finally, the teacher experiences….silence. Without that baseline, a new teacher has no gauge to assess the ambient classroom noise.

This was definitely a problem during my first year. What I needed was a silence-dominated classroom, but I didn't know what silence really was. The only time the class was truly silent was when my support staff member forced them to.

As a mentor, I always advise new teachers to err on the side of excess with disruptive students. If they have an entire class out of control, ask for help. If they have a few students misbehaving, toss them out after a warning. Screw fair.

Recall that many students told me that I was "unfair" when I told them to stop talking. Well, here is Ed's response to being labeled unfair -- "Screw fair!"

New teachers are often fearful of  sending students out. They worry that administrators will judge them. They’re right to worry. Administrators often notice.

I've always believes that the ideal manager doesn't need to issue many referrals, and so I definitely didn't want to become overly reliant on sending students out. At my old school, the administrators at least wanted me to send some work along with the misbehaving student. This, of course, requires my class to be more worksheet-dominated -- and my second big failure was when I didn't give my students worksheets at critical times.

Ed also gives a few tough management questions followed by answers:

“I’m teaching an Algebra 10-12 class, and the kids start packing up their stuff with fifteen minutes to the bell. Does that ever happen to you? What do you do to prevent that?”
I tell them to unpack their damn books and get back to work. Right now. And if they don’t start moving right away, oh my goodness, pop quiz.

In my old class this never happened, because I gave Exit Passes. Since students know that there will be an Exit Pass, at 15 minutes to go they were more likely to ask "When's the Exit Pass?" rather than start packing up. Only once in my career do I ever recall students packing up that early -- and it was just last month, at the recent subbing assignment I described on the blog.

“I’ve been having so much trouble with kids using cell phones constantly in class, not paying attention at all. What do you do?”
I take their damn cellphones away, giving myself extra points if I can swipe it from under their nose without signaling intent. Students who can’t keep off their phones lose them until the end of the day instead of the end of class. And they don’t dare complain, because I can always hand it over to the administrators, whose penalties are far more stringent.

My biggest fear with just "swiping phones from under their nose" is that we may accidentally break the phone in the process. The students complain to the parents, who are then angry at the teacher -- understandably, they're more concerned with having to replace a phone for hundreds of dollars than that their child was breaking the rules.

“I have these two kids who constantly talk to each other, but when I try to separate them, they insist on sitting together. It’s so frustrating.”
Why the hell do you give them a choice? Tell them where to sit. In fact, tell everyone where to sit.

My fear was that students would refuse to sit where I tell them to -- and this would be at beginning of the year, before I learn their names. I wanted to let them sit wherever they wanted at least until I learned their names. After this, I moved them to where I needed them to sit.

“I tell the kids not to bring food to the class, but what do you do when they’ve just bought lunch?”
You take the lunch away and tell them they can enjoy it cold later.

At the start of the year this was a big problem. Later on, students didn't eat in class as often, which of course is good.

“I’ve tried taking away phones/telling them where to sit/taking their lunch but they refuse to give it over, and I don’t know what to do.”
You call and have them removed from the class.
“What? For something so minor?”
Listen well, little teachlings. Defiance of a teacher is not minor.

Of course, if I called the office every time a student defied me last year, I would have been calling the office every single day. But Ed continues:

But well over half the time, simply picking up the phone has results[...]

In my class, merely picking up the phone didn't have the results I wanted. But notice that I'd created a vicious cycle here. The first few times students were defiant, I didn't want to call the office. So students just grew more defiant, to the point that I had to call sometimes. But they were never sure whether I'd actually call or not, and so touching the phone wasn't a threat. Ed, on the other hand, called the office a few times at the start of the year, and this was sufficient to get them to behave so that defiance was rare in Ed's class the rest of the year.

So you might be reading all this saying, wow, Ed’s a tyrant. Which is hysterical, because I’m one of the loosest teachers you’ll ever run into.

I see what Ed's getting at here. The first few times the teacher seeks out silence, the students think that enforcing silence is cruel. But once they are accustomed to the fact that silence is expected, they no longer find it to be cruel -- but we must get past those first few times. My problem was that I never got past those first few times, because I never truly knew what silence really was.

This is the new worksheet on parallel lines:

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