8:15 -- It's the same middle school as Tuesday -- so once again, we know that homeroom leads directly into first period. This is the first of three seventh grade art classes. The students are sculpting a whistle out of clay.
9:20 -- And once again, we know that this school has a period rotation after first period. The rotation for Friday goes 1-6-2-3-4-5. Sixth period is the next of the seventh grade art classes.
10:15 -- Sixth period leaves for snack.
10:30 -- Second period arrives. As it turns out, only four of the five classes this teacher has is art -- the other class is P.E., believe it or not!
Yesterday, I mentioned that it was raining (during the false fire alarms). Today it's dry, but the other P.E. teachers decide that the ground is still too wet to hold P.E. outdoors. In the gym, the eighth graders have the choice to watch the movie Cool Runnings or play dodgeball.
11:25 -- Second period leaves. Third period is conference period, which leads directly into lunch.
1:05 -- Fourth period arrives. This is the last of the three seventh grade art classes.
2:00 -- Fourth period leaves and fifth period arrives. This is the only eighth grade art class. These students are to create a Google Doodle for this year's contest:
https://doodles.google.com/d4g/
I remember last year's winner being featured on Google last week. I assume that it appeared on Google on the same day that this year's contest is announced. I never imagined that I'd soon be subbing in a class where this contest is an assignment.
2:55 -- Fifth period leaves, thus ending my day.
By the way, notice that seventh period art lasts only for a trimester. It's part of what we often call an "exploratory wheel" of electives. Eighth grade art is for the entire year -- so presumably only committed artists would sign up for this class.
Let's return to our New Year's Resolutions. Here is the focus resolution for today:
3. Move on from past incidents instead of bringing them up with students or flouncing around.
Once again, I'm tempted to bring up past incidents today. In first period, one student starts throwing clay at other students. Then in sixth period, another boy sprays water at other students. Fortunately, I have enough willpower today to avoid mentioning the first period incident in sixth period.
Meanwhile, the added part of this relevant, "flouncing around," is actually relevant. It refers to unnecessary body movement that occurs when I get angry. I'm more likely to flounce when I'm standing up -- as I was in today's P.E. class -- than when I'm sitting down.
Only one incident in P.E. makes me want to flounce -- when taking attendance. The students can't sit on P.E. numbers outside, so instead they sit down in alphabetical order. At least they're supposed to be, but one boy whose last name starts with T is sitting in the middle of the alphabet instead. It takes me some time to figure out why my names seem to be off. I'm not quite sure why Mr. T is sitting out of order. He might be a troublemaker, or perhaps he transferred in the middle of the year (after P.E. numbers were assigned, so he just took the number of someone else who had moved out earlier). All I know is that the other teachers in the gym have finished attendance, while my students see this and are tired of sitting down. And all of this is because Mr. T is sitting out of order.
The reason I added "flouncing around" to the resolution this year is for the same reason that I want to avoid yelling. Students often get frightened when they see an unfamiliar person suddenly raising his voice or making strange body movements. Once again, I try to control my temper by squeezing a stress ball. As it turns out, the regular teacher already has such a ball in his classroom. This shows that I'm on the right track by using the stress ball -- other teachers clearly do the same.
The best class of the day is fourth period, since no one throws or sprays anything in this class. But I name the P.E. class as the second best. While one of the other P.E. teachers is explaining dodgeball, many of the other classes are talking loud, but my class is mostly quiet.
Lesson 9-4 of the U of Chicago text is called "Plane Sections." In the modern Third Edition of the text, plane sections appear in Lesson 9-6. The new edition of the text makes it clear that spheres are introduced in this lesson as well.
Indeed, let's start with Euclid's definition of a sphere and related terms:
Just as with cylinders and cones, Euclid's spheres are solids of revolution. In the U of Chicago text, a sphere is the set (or locus) of all points in space a fixed distance from a point. The one term we can't define for a general sphere is its axis, unless we have a particular rotation in mind (such as the earth).
David Joyce tells us that Euclid's sphere proofs aren't as rigorous as they could be. According to Joyce, Euclid hints at the proof that a plane section of a sphere is a circle in his Book XII. A full proof appears as Exercise 20 in our text, but I chose not to include the proof on our worksheet.
Oh yes -- plane sections, the other topic of this section. It turns out that both plane sections and Euclid's solids of revolution appear in the Common Core Standards:
CCSS.MATH.CONTENT.HSG.GMD.B.4
Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
We can keep this standard in mind by discussing both plane sections from today's U of Chicago lesson as well as Euclid's definition of cylinder, cone, and sphere. Naturally, I decided to slip this into the worksheet for today.
As usual, we'll look at the next proposition in Euclid:
We notice that once again, Euclid begins his proof with "For suppose it is not," which indicates that an indirect proof is needed:
Given:
Prove:
Indirect Proof:
Assume that the lines aren't coplanar -- that is,
By Point-Line-Plane, part g, the intersection of planes P and Q must be a line -- and that line can only be line EF. But now there are two lines through points E and F -- one that lies in plane P (the intersection line) and one that doesn't (the one through point G). This is a contradiction, since by Point-Line-Plane, part c, there is only one line through two points. Thus the assumption that the lines aren't coplanar is false. Therefore
According to David Joyce, Euclid assumes without proof that every line lies in a plane. Our version of the Point-Line-Postulate actually does prove that every line lies in a plane, as follows: By part b, every line contains at least two points (labeled 0 and 1). By part a, there is a (third) point in the plane not on the line. Finally by part f, through these three noncollinear points there is a plane. QED
Of course, the hidden assumptions that we all make are subtle. In fact, neither Euclid's postulates nor our Point-Line-Plane Postulate can refute the following statement:
"Space contains exactly one point (with no lines and no planes)."
This seems absurd -- space clearly contains infinitely many points. OK then, let's try to prove it by looking at each part of the postulate:
Point-Line-Plane Postulate:
a. Given a line in a plane, there exists a point in the plane not on the line. This can be written as "if a line is in a plane, then there exists a point in the plane not...." In other words, it tells what happens if a line lies in a plane, and makes no claims about what happens if there are no lines and no planes. Thus part a is (vacuously) true for single-point space.
b. Every line is a set of points that can be put into a one-to-one.... In other words, if there is a line, then it is a set of points. No claim is made if there are no lines. Thus part b is (vacuously) true for single-point space.
c. Through any two points there is exactly two line. In other words, if we have two points, then there is a line through them. No claim is made if there is only one point. Thus part c is (vacuously) true for single-point space.
And the same thing happens with parts d-g as well. There simply is no way to prove:
-- There exists two distinct points.
with no "if" or other precondition. The Point-Line-Plane Postulate has a one-element model. Yes, we did just prove that every line lies in a plane -- that is, "if there is a line, then it lies in a plane."
Notice that as soon as we have two points, then we have the intended model of Geometry. With two points, part c gives us a line passing through them. Then part b places infinitely many points on this line, one for every real number. Then from above, we know that this line lies in a plane, and so on.
This may seem like a big deal about nothing. But unless we can assert that at least two points exist, a "wise guy" student could challenge the entire Geometry course by answering every question with "point P" if it asks for a point, and "none" or "zero" if it asks for anything else (lines, planes, length, area, volume, and so on). Then the student can claim that he deserves 100% A+ in the course since neither the teacher nor the text ever refutes the statement "space contains exactly one point"!
It is Hilbert who assures us that two points exist, not Euclid or the U of Chicago. Indeed, Hilbert provides the following:
I.3 There exists at least three points that do not lie on the same line.
I.8 There exists at least four points not lying on a plane.
Hilbert specifically mentions that we don't need a line or a plane to exist in order for the trio or quartet of points to exist, since these sets aren't collinear or coplanar, respectively. Sometimes Hilbert's I.8 is written as:
"Space contains at least four noncoplanar points."
And we don't need to say, "if space exists, then ...." since space is defined as the set of all points. So space exists even if there's only one point, or even no points (the empty set). It's only for any of the undefined terms point, line, and plane where we can't automatically assume that any of them exist.
By the way, in some texts this is called the "Expansion Postulate." The Expansion Postulate guarantees that more than one point exists. Neither the U of Chicago text nor yesterday's Glencoe text contain an Expansion Postulate (so both are consistent with the idea that only one point exists).
Today is an activity day, and as I explained, this year I'm making new activity worksheets. It comes from the Exploration Question and looks at the more exotic plane sections of a cube.
As it turns out, one of these questions is, "Is it possible for a plane section of a cube to be a quadrilateral that is not a parallelogram?" The answer is no -- and the proof is implied by some of Euclid's propositions that we haven't reached yet. Specifically, Proposition 16 tells us that if a plane cuts two parallel planes, then their intersections are parallel. As the opposite faces of a cube are parallel, their intersection with any plane cutting the cube are parallel. Thus if the plane section is a quadrilateral, its opposite sides must be parallel.
Monday is the Martin Luther King Jr. holiday, and so my next post will be on Tuesday.
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