8:25 -- At this school all periods rotate, but today the rotation actually starts with first period. This is one of the two Algebra I classes.
As promised, I perform the "Compound Interest Rap" today. I hope that it will inspire these students to remember the formula y = P(1 + r/n)^(nt).
I look back to some of what I taught three years ago at the old charter school, and I realize that I actually taught my seventh graders the simple interest formula that year. (Thus we see that under the Common Core Standards, simple interest is a seventh grade lesson while compound interest is now an Algebra I lesson.) And indeed, it was right around this time of the year when I taught the seventh grade interest lesson.
But I didn't write much about it on the blog at the time -- instead, I focused more on the eighth grade lessons as well as the Hidden Figures field trip that week. Indeed, I didn't write a song for the seventh graders that week -- otherwise, I might have already had a song about simple interest (so that I'd only have needed to add a verse on compound interest and then sing it today).
Like exponential growth, compound interest appears in the U of Chicago Algebra I text -- this time it's in Lesson 9-1. But the only formula given there is T = P(1 + i)^n -- in other words, only annual compounding is studied here (and not biannual, quarterly, monthly, or daily compounding).
9:15 -- First period leaves and second period begins. This is the first of three Math 8 classes.
The students take the quiz on solving systems of equations. One student doesn't have time to finish and asks to complete the quiz at lunch. I'm unsure whether to let him do so -- normally I do want to go out and eat. So I tell him that I'll inform the regular teacher, who can then decide whether to grant him the extra time.
Otherwise, this class goes fairly smoothly. I name this class the best Math 8 class of the day.
Since I've already mentioned teaching interest at the old charter school, I might as well discuss how my lesson on solving systems went three years ago.
Admittedly, solving systems was a mess. Part of this is due to the Illinois State text and its insistence that a single Common Core Standard be taught per week. Also, according to the intended lesson order, all EE standards must be taught before any F (functions) standards are covered. This means I couldn't teach solving systems by graphing, since the students learn how to solve systems before they learn how to graph.
So during the week that I covered Standard EE 8b, my intention was to teach both substitution and elimination -- the two methods that don't require graphing. I began with substitution, and since the students were confused, I ended up spending the whole week on substitution. At first, I thought that this wouldn't be a problem, since the following week is EE 8c on applying systems. Thus I could squeeze in the elimination while solving these word problems.
But then suddenly I received word that I needed to teach more science (for obvious reasons). And so EE 8c was scrapped so that I could spend an entire week on science. And, as you might recall, the rest is misery -- I ended up leaving the charter school shortly thereafter.
10:10 -- Second period leaves for snack, which leads into third period conference.
11:15 -- Fourth period arrives. This is the second of three Math 8 classes.
I begin the class by announcing that I would send any students who talk during the quiz to the center room (in between the classrooms in this building). This seems to work, as I don't need to send anyone out of the room. (This makes today much go better than the infamous December 6th class.)
But now a few more students have trouble finishing the quiz. So I decide to send these students to the center room during lunch to finish the quiz.
12:05 -- Fourth period leaves for lunch. I get my lunch and eat it in the center room so that I can watch the students as they complete the quiz.
12:50 -- Fifth period arrives. This is the last of three Math 8 classes.
The class after lunch begins with silent reading. (Yesterday, silent reading was cancelled since extra minutes were added to the sixth grade tour classes.) Today, there is also a special reward for students with all C's or better and satisfactory behavior. The co-teacher for this class stays outside to supervise the quick activity while I stay in the classroom for silent reading.
Once silent reading ends, there's just enough time for me to sing "x's and y's" (as I didn't sing it for this period yesterday) before the co-teacher returns and the quiz begins.
More students ask for extra time to finish the quiz. This time, the co-teacher informs me that actually, I should not grant the extra time. He allows a few special ed students with accommodations for extra time to finish during sixth period, but informs the others that they won't get any more time. (In second period, an aide allows a few students with limited English skills to continue working into third period -- but once again, these probably count as special ed students.)
2:00 -- Fifth period leaves and sixth period arrives. This is the other Algebra I class.
In addition to singing the "Compound Interest Rap," I help this class with two questions from the homework, both of which are trick questions. If something triples, then it increases by 200%, not 300% as some news articles claim. Also, if something like tickets sold doubles every hour and the event sells out in 12 hours, then 1/4 of the tickets are sold after ten hours, not three. Typically this trick question is asked about bacteria:
https://www.grandforksherald.com/news/government-and-politics/4868213-Cyber-attacks-against-North-Dakota-increased-300-last-year
https://www.algebra.com/algebra/homework/Rate-of-work-word-problems/Rate-of-work-word-problems.faq.question.137122.html
The students behave well, and so I end up naming this class the best class of the day.
2:55 -- Sixth period leaves, thus ending my day.
Today is Fourday on the Eleven Calendar:
Decade Resolution #4: We need to inflate the wheels of our bike.
I tell this to the students in the Math 8 classes just before they take the quiz. If they're having trouble remembering things like how to graph or how to work with negative numbers, then they should add this to the math that they remember as they approach high school.
Unfortunately, I'm not sure how well this works out today. As I glance at the quizzes, I see that many students struggled on the graphing problem where the lines are in standard form, to be graphed using intercepts, despite my singing the "x's and y's" song about graphing via intercepts. These students still have work to be done before they can claim graphing lines as part of their bicycle.
One problem that's come up recently when I sing songs is that some students wish to take out their cell phones (in violation of school rules) and record my performance. I should watch out for this, especially at middle schools where many students are too immature to use the phones properly. (High school students, of course, make trouble on their phones as well, but they usually have the sense at least to avoid making bad pictures or videos on campus.)
This problem has a simple solution -- if a middle school student takes out a phone as I'm about to sing, then I stop singing until the phone is put away. If the song is part of a reward for good behavior, then I can easily say that the reward hasn't been earned (since taking out the phone is considered part of misbehaving).
Lecture 10 of Michael Starbird's Change and Motion is called "Circle, Pyramids, Cones, and Spheres," and here is an outline of this lecture:
I. Greek mathematicians had a keen sense of integral-like processes. Here is an ancient process for discovering the formula for the area of a circle [omitted here].
II. The derivative gives a dynamic view of the relationship between the area of a circle and its radius.
III. The area of a triangle is dependent only on the height and base, not whether it is leaning.
IV. We can determine the volume of a tetrahedron (a pyramid over a triangular base) by thinking of sliding its parts as we did with the triangle.
V. The volume of a cone is easy to compute once we know the volume of a tetrahedron.
VI. The surface area of the sphere can be computed by breaking that surface into small pieces.
VII. These examples show ancient ideas that resemble the modern idea of the integral. The ancients did not have a well-defined idea of what happens at the limit, but their arguments are persuasive and can not be made mathematically rigorous.
Notice that today's Starbird lecture applies integral calculus to Geometry, and indeed many of its ideas apply to the 3D figures that we are currently studying in Chapter 9 of the U of Chicago text.
We begin with the area of a circle, pi r^2. The professor divides the circle into sectors -- a process I recently mentioned in my Lesson 8-9 post a week ago.
Starbird's next example is a triangle. He begins with a right triangle, just as Lesson 8-5 does (even though we skipped over this lesson). To generalize this to other triangles, he "shears" the right triangle so that it forms other triangles with the same base and height. He argues that this doesn't change the area of the triangle (using a computer animation to bolster his argument), and so all triangles with the same base and height have the same area.
The professor's first example of a 3D volume is a tetrahedron (a triangular pyramid). He now argues that all tetrahedra with the same base and height have the same volume. And the way he demonstrates this is by showing us a model of a tetrahedron and noting that even as he moves the vertex around, the plane sections (today's Lesson 9-4!) in each plane remain congruent. (As we will learn coming up in Lesson 10-5, this idea leads to the much-maligned Cavalieri's Principle. And technically speaking, Starbird's argument about the area of a triangle is simply the 2D-analog of Cavalieri's Principle as we look at the "linear sections" of each triangle.) Afterwards, he completes the argument that a prism can be divided into three congruent tetrahedra (just as the U of Chicago does in Lesson 10-7).
Starbird quickly shows us the volume of a cone by considering its circular base to be approximated by polygons, so that the cone itself is approximated by polygonal pyramids. (The U of Chicago text tells us that Cavalieri's Principle directly applies the pyramid volume result to the cone.)
The professor now moves on to the surface area (not volume) of a sphere. His idea here is much different from the method shown in Lesson 10-9 of the U of Chicago text. He begins by inscribing the sphere in a cylinder of the same radius and height (that is, the height of the cylinder equals the diameter of the sphere). The sphere and the cylinder intersect at the Equator of the sphere. He now has several planes parallel to the plane containing the Equator. But now we are not considering plane sections of the sphere -- instead, we consider the surface area of the part of the sphere that lies between any two of the planes. (Legendre referred to this as a "zone" of the sphere.) Now he makes a grand claim -- the surface area of any zone on the sphere equals the surface area of the corresponding zone of the cylinder! As he puts it, the smaller radii for latitudes near the Poles are exactly compensated for by the slantiness at those higher latitudes.
To prove this claim, Starbird begins by letting r be the radius of the cylinder (and sphere), and s be the larger "radius" of the zone. He now draws an actual plane section that is perpendicular to the plane of the radius. From this perspective, there is a right triangle with radius r and leg s joining the center of the sphere, a point on the surface of the zone, and the foot of the perpendicular from this point to the radius drawn at the equator. There is also a near-triangle with hypotenuse delta-s and leg delta-h, where delta-s runs along the surface of the sphere from the bottom to the top of the zone. This plane also cuts the cylinder, and the distance between the planes along the cylinder is also s from the Equator to the nearer plane and delta-h between the two planes marking our zone. He points out that this diagram is identical to the one he uses to prove that the derivative of sine is cosine -- the larger and smaller triangles are similar. So r/s = delta-s/delta-h. Thus r delta-h = s delta-s, and multiplying both sides by 2pi gives 2pi r delta-h = 2pi s delta-s. On the left side we have the surface area of a cylinder with radius r and height delta-h, and the right side is the surface area of a cylinder with radius s and height delta-s (which the zone approximates). Therefore the area of the sphere equals the lateral area of the cylinder, which is its height 2r multiplied by its circumference 2pi r -- so the surface area is 4pi r^2.
This is what I wrote last year about today's lesson:
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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