Lesson 9-8: The Four-Color Problem (Day 98)

Today I subbed in a freshman English class. Believe it or not, not only is it in my old district, but it's even at the same school as yesterday. Thus it really is Day 98 at this school. It's ironic how after not subbing at this school for nearly five years, I make two visits on consecutive days.

Since it's a high school class and not math, there is no "A Day in the Life" today. But classroom management is important today, since freshmen are often immature. It's still a block day -- yesterday was odd periods, and so today is even periods.

Today the students begin with a Warm-Up assignment -- predict whether life will be better or worse 100 years from now. Then they must read, highlight, and annotate an article on Latino civil rights.

In second and fourth periods, two counselors enter the room and give a presentation about grades, college admissions, and the students' futures. Now that the first semester is complete, the freshmen get to see a copy of their first transcripts.

Today is Elevenday on the Eleven Calendar. There is no New Decade Resolution #11, and so my plan for Elevenday is to choose whichever resolution is relevant.

Well, I choose "No Drens" as my song incentive for the day (since I think of this as a late January song, when I first sang it at the old charter). This is an example of the seventh resolution on singing math songs (even though it's an English class) as well as the fourth resolution on inflating the bike wheels (which is what it means to avoid being a "dren"). But the most relevant resolution today turns out to be the ninth:

Decade Resolution #9: We attend every single second of class.

There are two violations of this resolution today. One is when the students violate it, and the other is arguably when I unwittingly violate it.

The problem begins in fourth period. Many of the students are talkative, and there are many students asking for restroom passes. One student is placed on the bad list for leaving without permission (since it's one student out at a time, and he leaves when another student is out).

Nine students in fourth period leave with a pass. An interesting question is, how many is too many? I notice that when the counselors arrive, not only are the students much better behaved, but no one asks for a restroom pass. This implies that at least for some of the students, asking for the restroom pass is something they do as part of the general misbehavior they show on sub days. If the regular teacher had been here today, at least some of those kids don't even ask for a pass.

And so during lunch, I make adjustment plans for sixth period. My ultimate goal is for the sixth period students to behave the entire block as equally well as fourth period did for the counselors -- especially considering that there is no counselor visit planned for sixth period.

At the time, I felt that part of this entails reducing the number of restroom passes. But once again, how many passes is reasonable? Certainly since it's a block schedule, more passes are reasonable than they would be at a school with a traditional six-period day.

There is a sign-out sheet for restroom passes, so this gives me an idea. I go back to Wednesday, the last time these classes met, and see how many students ask for passes that day, when the regular teacher was most likely present. Here's what I noticed about sixth period on Wednesday:

• The first student didn't leave until 2:02 (about an hour after the tardy bell).
• Only one student goes to the restroom or water fountain at a time.
• No student takes more than six minutes.
• A grand total of five students leave the room.

And so these sound like reasonable expectations for sixth period today. Moreover, setting these expectations means that I wouldn't be comparing these students to myself ("I went three years at a school with a block schedule without asking for a restroom pass once!") or fourth period. ("No one asked for a pass after the counselors arrived.") I've already seen that comparing students to either myself or other students never works unless it's a favorable comparison. Instead, I'm only comparing these students to themselves 48 hours earlier.

In the end, the plan doesn't work. At least three students use the word "emergency" in describing their bathroom needs today. It's theoretically possible that some students have emergencies today and not Wednesday, so I allow one guy and one girl to go before 2:02, and everyone else had to wait. Then at 2:02, so many students want to go that I can't keep it to only one student at a time. I let one student out at 2:02, and a second student leaves without permission. Then when they return, I allow someone else to leave, only to have a second student leave again.

By this point, that's already six students leaving, surpassing the Wednesday total -- thus my restroom plan is officially a failure. At this point, I give up and instead just keep track of how many students are taking advantage of the sub by going to the restroom. The grand total is ten, which I then write in my sub notes to the teacher.

How to the students justify their need for restroom passes, considering that the standard is how many students from the same class left on Wednesday? The excuse is that the teacher is more lenient than I'm making her to be, and that more than five students actually went on Wednesday. It's just that the other students without properly filling out the sign-out sheet. I give this claim a 50-50 chance of actually being true -- if true, it implies that the kids who left before 2:02 neglected to sign out, and then at 2:02 she reminded them to do so.

While I believe I did avoid yelling here, my discussion of the restroom policy has definitely crossed the line into an argument. And so I ask, what could I have done to avoid arguing?

Here's what I believe in the 50-50 event that the claim is false -- the teacher is somewhat strict regarding restroom passes. If a student asks to go right after lunch, she either says "You just had lunch!" in a strong teacher tone, or gives that student a strong teacher look, or both. Thus she doesn't let anyone leave until well after the start of the period, in this case 2:02. And soon after the school year begins, the students know not even to ask right after lunch, because she'll say no. (This is what happened at the old charter school. My eighth graders almost never asked for a restroom pass after September, and the seventh graders seldom did so. Only with the sixth graders did restroom passes become a problem.)

But even students who are used to a strict restroom policy from a teacher immediately forget it the instant they see a sub. The same students who know not to ask the teacher to leave right after lunch then ask the sub to go as soon as they see me, even before the tardy bell rings.

I estimate that twice as many students ask to go to the restroom each period on a sub day than on a regular teacher day. Thus a sub must say "no" that many times more than a regular teacher does in order to maintain the same low number of passes. And each time a sub says "no" is a potential argument waiting to happen.

Well, here are some alternatives to my arguing today. In any event, I should decide in advance how important it is to keep the restroom passes low. If I decide that it's important (especially if the regular teacher specifies to be strict in the lesson plan), then I should make restroom passes the main criterion for the music incentive. I sing the song if there are five or fewer total passes and refuse to sing if there are six or more. (Note: What I don't do is let the first five students go free and then write down the names of each subsequent leaver starting with the sixth. This is something I tried at previous schools, including the old charter school. The problem is that the first five might leave just because they feel like it, and then the sixth has a genuine emergency.)

Or perhaps I decide that the main criterion should be completing the Warm-Up (which I'd originally planned), but meanwhile I keep monitoring the restroom passes. Once again, in this case I only keep track of the number of students who leave, not their names. The one thing I should do differently is sing the song more often -- instead of waiting until the last few minutes to count how many students failed to finish, I sing every time five more students finish it. This especially would have helped me in the classes when the counselors arrive. Second period definitely deserves the full song today, but I don't finish it because of the counselors' unexpected arrival. Singing throughout the period means that they would have heard me sing before the presentation.

But then again, what do I really want in sixth period, deep down? What I want is for the sixth graders to behave as well as the fourth period students did when the counselors were there. Notice that the counselors never say anything about 2:02 or "no more than five total" -- they arrive, and no student even asks to leave. And the restroom passes are never the original problem -- instead, the passes are symptomatic of the fact that the students don't behave as well when I'm alone in the room. And of the six students on the bad list in fourth period, only one is for the restroom. The other five are for not completing the Warm-Up (and general talking and using phones instead of doing that Warm-Up).

So the real music incentive should have been based on behavior, not the Warm-Up. For example, perhaps every ten minutes that the students show great behavior, I sing the song. (If the students get tired of "No Drens," it's OK to switch to another song.) To keep students from abusing the restroom pass, I only sing when all of the students are in the room and no one is outside. I still insist that no more than one student leave at a time (since this is a genuine rule as confirmed by the student TA's in fourth period) -- if a second student leaves, it counts as bad behavior and no song for ten minutes. But in this case, I don't enforce 2:02 or "no more than five total."

Of course, you might point out that the counselors didn't need to sing to get the kids to behave. But in this case, I don't mind using my musical tone to make up for my lack of teacher tone.

Notice that during the first part of sixth period, many students have blank Warm-Ups, but most of them remain in their seats. This is an improvement from the start of fourth period -- I should have told them this (since it's a favorable comparison to another class). Of course, I didn't because I was too busy complaining about the blank papers and arguing about the restroom.

In fact, only three students fail to complete the Warm-Up. This should have been enough for me to sing the song (with about 30 total in the class, that's 90% compliance), with sixth period overall being better than fourth period.

But unfortunately, one incident ruins the period. Some students start playing with and throwing around a -- um, an adult object (that I don't wish to mention on the blog). Since this is completely inappropriate, I threaten to call Security unless they throw the object away, and they do -- but then someone takes out a second object. When the guy from security arrives, he makes the students apologize to me for playing with the objects.

I can't help but wonder, was this a backlash to my restroom arguments? Are some students so upset over the restroom policy that they get back at me by taking out the objects? In other words, if I never mention 2:02 or "no more than five total" today, do those objects appear in class at all? If so, then this is all the more reason I should have used a different incentive.

About half of the students violate the ninth resolution by asking for passes when they didn't really need to go. But earlier, I wrote that I may have violated the ninth resolution myself.

To see what I did wrong, you must remember the concept of "embedded support." In this district, each of the three two-hour blocks ends with embedded support time. Only students who have D's or F's, or fail to complete the day's assignment, must stay during that time. But here's the problem -- the counselors' presentations in second and fourth period extend into embedded support time. I remind one of them that it's embedded time, and she tells me that I'm free to leave.

But this, unfortunately, sends the wrong message, especially in fourth period. I write one student's name down for leaving the classroom, and then I leave the classroom while that same student is stuck there for the presentation.

It also reminds me that both embedded support and tutorial (in my new district) are loopholes in the sense that they count towards the State requirement for instructional minutes. Even if the students don't take these minutes seriously, the teachers should. Officially, a teacher never has to let the students out for embedded support even if their grade in the class is 100%. And if the students must stay for some reason such as the counselors' presentation, then the teacher (or I as the sub) should stay there too. The counselors say that I could leave, but counselors are not infallible. I should listen to the State, not the counselors -- and indeed, I shouldn't have even asked to leave. If the students start itching to leave as it's embedded time, then that's between them and the counselors, not me. (If it's possible to do so without arguing, I can back up the counselors as they tell the students to stay.)

Returning to sixth period, there are five minutes left after Security leaves before embedded time. I tell the students that in order to be released early, they must remain quiet and listen to me. I tell them about the article that they should be reading -- about the Latino civil rights movement. Back in the 1940's, some brave Mexican and Mexican-American students integrated the schools in California -- and this was a precursor to MLK and the black civil rights movement. (Yes, I know that I'm mentioning race here, but it's relevant to the lesson.)

Most of the students in this class are Latino -- and they are here only because of their grandparents' and great-grandparents' generation of civil rights leaders. Many of the students are here, at a school with real textbooks, air conditioning, and yes, even restrooms and indoor plumbing, because their forerunners fought for equal access in education. (When I mentioned this, one student also mentions Malala, who fought for equal access to education in Afghanistan.) They didn't sacrifice their free time in order for their grandchildren to play with adult objects in class all day. Instead, they should take advantage of the sub -- and all the regular teachers, counselors, and other adults -- to get a good education, to be more than just a "dren."

Lecture 14 of Michael Starbird's Change and Motion is called "The Fundamental Theorem at Work," and here is an outline of this lecture:

I. Recall that the Fundamental Theorem of Calculus relates the integral and the derivative.
II. We can systematically find antiderivatives of polynomials.
III. When students leave a calculus course, they often forget what the integral means; instead, they think it means antiderivative. But if the integral had no independent meaning in terms of sums, then it was be pointless to do the process of taking antiderivatives and subtracting. What would it mean?
IV. Why do we say "an" antiderivative rather than "the" antiderivative?
V. The trigonometric functions have interesting derivatives and integrals.

Starbird begins with a graph of a position function and its derivative, the velocity function. Then he reminds us that the derivative of a sphere's volume is its surface area:

(V(r + delta-r) - V(r))/delta-r
d/dr (4/3 pi r^3) = 4pi r^2

Then the professor shows us the graphs defining the integral the area under a curve. The main idea of this lecture is that the derivative and integral are related -- the Fundamental Theorem of Calculus:

integral _a ^b v(t) dt = distance traveled
= p(b) - p(a)
Thus integral _a ^b p'(t) dt = p(b) - p(a)

integral _a ^b F'(t) dt = F(b) - F(a) is the Fundamental Theorem of Calculus.

Starbird illustrates how the Fundamental Theorem can be used to solve problems:

d/dx (x^2) = 2x -- antiderivative of 2x is x^2
d/dx (cx) = c -- antiderivative of c is cx

The professor shows us the following derivative chart from a previous lecture:

f(x) = x^n, f' (x) = nx^(n - 1)

f(x)     f' (x)
x        1
x^2    2x
x^3    3x^2
x^4    4x^3
...       ...
x^n    nx^(n - 1)
sin x  cos x
cos x  -sin x

From this, we can find the antiderivative of any polynomial:

d/dx(x^(n + 1)/(n + 1)) = x^n -- antiderivative of x^n = x^(n + 1)/(n + 1)

Starbird now uses these antiderivatives and the Fundamental Theorem to calculate definite integrals:

F(x) = x^5/5  F'(x) = x^4
integral _1 ^4 x^4 dx = 4^5/5 - 1^5/5 = 1023/5

The professor now explains why a function can have many antiderivatives -- functions differing by a constant have the same derivative. He shows a picture of a road with two mileage markers -- one starting from his hometown (Austin), the other from Waco, 100 miles north of the Texas capital.

The final example involves finding the area under the sine curve:

d/dx (cos x) = -sin x
integral _0 ^pi sin x dx = -cos pi - (-cos 0) = 1 + 1 = 2

Lesson 9-8 of the U of Chicago text is called "The Four-Color Problem." This lesson doesn't appear anywhere in the modern Third Edition, because this is one of those "extra" lessons that we include mainly for fun.

In the past, I've mentioned several books and lectures which discuss the Four-Color Conjecture. One of these was David Kung's lectures. [2020 update: Let me snip out David Kung's lectures here, since we're now watching Michael Starbird's lectures.]

I wish to link to a member of MTBoS who actually teaches the Four-Color Theorem in class:

http://eatplaymath.blogspot.com/2015/10/the-four-color-theorem-and-pumpkin-time.html

Lisa Winer is the author of this post that is over four years old. She doesn't specify in what state she lives, nor does she make it easy for me to figure out what grade or class this is.

Anyway, in Winer's class, she uses the term "chromatic number" to describe the fewest number of colors required to fill in a map. The Four-Color Theorem, therefore, states that the chromatic number of any planar map is four. On a Mobius strip the maximum chromatic number is six, and of course on a torus the maximum is seven.

It's time to return to Euclid. Of course, he writes nothing about Four Colors or reflections across an axis, and so we proceed with the next proposition instead:

Proposition 11.

To draw a straight line perpendicular to a given plane from a given elevated point.

Propositions 11 and 12 are both constructions. Many of Euclid's propositions are constructions -- indeed, "The First Theorem in Euclid's Elements" (that is, Proposition I.1) featured in Lesson 4-4 is actually a construction.

Classical constructions are performed with a straightedge and compass, and David Joyce writes about the importance of actually proving constructions as theorems. But it's awkward to ask our students to perform a construction in three dimensions.

In this construction, we have a point A and a plane P, and we wish to construct the line perpendicular to P through A. How can our students do this? Is P the flat plane of the paper and A a point floating up in space?

It might be interesting to attempt Euclid's construction in the classroom. Here's how: We choose A to be a point on the ceiling and P is the plane of the floor. Thus our goal is to draw a point on the floor directly below A.

The key to this construction is to hang a rope from point A -- a rope that should be longer than the room is high. We can pull the rope at any angle and double-mark the points where the rope is touching the floor. I say "double-mark" because the point on the floor (where the rope touches) is marked (say with chalk), and then the point on the rope (where the floor touches) is marked (say with a piece of tape). The rope now can serve as a compass -- the point of the compass is at A, and the opening of the compass is set to the distance between A and the tape. The locus of all points on the floor that are the same distance from A as the point marked on the floor is a circle, and the locus of all points on a given line on the floor that are the same distance from A is a pair of points. So if we have a point (say B) drawn on a line on the floor, then we could find the unique point C on that line such that AB and AC are congruent.

All the lines on the floor can be drawn in chalk. There will be some plane constructions drawn on the floor as well, so we could use a large compass where the pencil has been replaced with chalk.

OK, so let's begin the construction. We start by drawing any line on the floor, and then we label any point on that line B. We now find C on this line exactly as given above -- we double-mark B on both the rope and floor, and then swing the rope to find C such that AB = AC.

Now we use the chalk compass to find the perpendicular bisector of BC. The midpoint is D.

Then we double-mark D with a second piece of tape, and then find the point on the last line we drew (that is, the perp. bisector of BC), to be labeled E, such that AD = AE. The second piece of tape must be higher up than the first since AD < AB, and so there's no danger of confusing which piece of tape is which.

Finally, we find the perpendicular bisector of DE. The midpoint is F. Euclid's G and H are any points on this last line -- their location doesn't matter. Only F is relevant here. AF is the desired line through A that is perpendicular to the plane of the floor, and F is directly below A.

Of course, this whole construction seems silly because of gravity. We can just hang a rope freely from A, label the point where the rope touches the ground F, and then we're done! The difference, of course, is that Euclid's three-dimensional space isn't physical space, and so there's no direction that's "favored" because of gravity or any physical force.

And so I'm not quite sure how David Joyce has in mind when he says he wants "the basics of solid geometry" to be taught better. Does he include Euclid's spatial constructions -- does he really want students to perform them? Or maybe he merely desires that students visualize the proofs in their minds while looking at the proof.

(Do you remember Euclid the Game, which is played on computers? Maybe in higher levels, players can make three-dimensional constructions that are difficult to perform in the real world!)

By the way, we can still modernize Euclid's proof:

Given: the segments and angles in the above construction.
Prove: AF perp. plane P

Proof:
Statements                              Reasons
1. bla, bla, bla                         1. Given
2. BC perp. plane (ED, DA)   2. Proposition 4 from last week
(Call it plane Q. In the classroom, Q is an invisible plane parallel to a wall.)
3. GH | | BC                            3. Two Perpendiculars Theorem (planar version)
4. GH perp. plane Q               4. Perpendicular to Parallels (spatial, Tuesday's Prop 8)
5. AF in plane Q                     5. Point-Line-Plane, part f (A, D, F all in plane Q)
6. GH perp. AF                       6. Definition of line perpendicular to plane
7. AF perp. plane (GH, DE)   7. Prop 4 (AF perp. DE is part of "Given")
8. AF perp. plane P                 8. From construction (both lines were drawn in plane P)

It might be tricky to reconcile this proof with the "rope" construction from above. In Euclid's construction, AD is designed to be perpendicular to BC, likewise AF is perp. to DE. Both of these perpendicular constructions technically occur in planes other than the floor -- yet earlier I direct you to perform perpendicular constructions on the floor -- which is the wrong plane.

But think about it -- given a point A and a line, how do we construct a line through A perpendicular to the given line? The answer is that, using the compass, we find points B and C on that line that are equidistant from A, and then find the perpendicular bisector (in that plane) of BC.

But technically, all we really need is D, the midpoint of BC. Then the line through points A and D is automatically the perpendicular bisector of BC in the correct plane. It doesn't matter how we obtain the midpoint D -- all that matters is that we find it. This includes finding the perpendicular bisector of BC in the wrong plane (that is P, the plane of the floor). This is why Euclid is able to assert and use statements like AD perp. BC in his proof, even though this isn't obvious from our ropes. (And as it happens, the perpendiculars in plane P appear later in the proof anyway, so we might as well construct these.)

In the end, let's just stick to the Four-Color Theorem and two-dimensional reflections. Here are the worksheets for today.