But there are a few things of note about this class. First, this is in my old district, and so today really is Day 97 in this district.
Of all the high schools in this district, it's the last one that I'd had yet to revisit since leaving the old charter school three years ago. And indeed, my last visit was in April 2015. I didn't write much about that visit on the blog because back then, my rule was that I didn't mention subbing at all unless it was a math class. (Just to show how long ago that is, on the day I subbed there, I blogged about the student I had tutored the previous night! I stopped tutoring after one year of subbing here.)
Today I'm assigned to cover various teachers as they go out for special ed meetings. This is not the lone school in the district with a traditional six-period day, and thus there were three blocks today -- two of the blocks are Study Skills, while the other is a sophomore English class. One of the Study Skills classes is only for a half-hour during the teacher's meeting (so that I had a "conference period" of sorts during the rest of that block), while the others are for the entire period.
Today is Tenday on the Eleven Calendar:
Decade Resolution #10: We are not truly done until we have achieved excellence.
I mention this resolution during the two full-period classes. The purpose of this resolution, if you recall, is to convince the students not to rush through the assignments just so that they say they're "done" and get free time. In each of those two classes, some student starts playing around, and that leads me to bring up this resolution to them.
In the English class, one guy charges his phone and keeps checking to see whether the charge is complete, as well as talk about unrelated topics, when he should be answering questions about a poem (which is worth 25 of the 60 grade points available today). And in one of the Study Skills classes, one student tries to throw a football in class. At no point do I ever see him working on any class assignment (though at least he helps put away Chromebooks at the end of the day). The aides help to keep the students stay out of any further trouble.
In that same class, a freshman asks for help on his Algebra I assignment. As it turns out, this is the same Big Ideas Algebra I text as the middle school I subbed at last week -- and indeed, these students are working in the same chapter on exponential functions. (One difference is that these freshmen must learn about exponential decay, while last week's eighth graders apparently only needed to learn about growth. Oh, and today's assignment is on the Big Ideas website, while last week's eighth graders had only written assignments.) In order to get through most of his missing assignments, we end up staying 15 minutes past the dismissal bell, until a family member arrives to pick him up.
Since aides manage all the classes today, I don't sing any songs today. It might have been nice to sing last week's "Compound Interest Rap" to the Algebra I guy today, but he hadn't reached the interest lesson yet -- and besides, we're rushing through assignments, so there's no time for any singing. Thus there's no song today.
By the way, the fact that I didn't write about non-math subbing during the earliest years of this blog turned out to be a huge mistake. I could have learned more about classroom management and how to become a better teacher simply by reflecting upon my non-math subbing days. After I left the old charter school, I started blogging about my non-math days -- but once again, the time to do so was before I started at the charter.
So let me correct this wrong now. Five years ago today was the last day of a multi-day assignment -- the longest assignment I had since creating this blog. For nine days (two full weeks minus MLK Day), I subbed at a continuation school. It was a special day class for students who had made lots of trouble at traditional high schools -- and outside the schools as well. In fact, let's just say that many of the students there had criminal records. The class consisted of about a dozen guys and one girl. I repeat that at the time, I made very little mention of it on the blog -- on the days when I felt that there was an important math lesson.
Even though there were plenty of aides here, along with a team teacher who taught English and history, many subs might be intimidated by the prospects of teaching such a class. Yet five years ago today, not only had I survived the two-week assignment, but the other adults and students told me that I'd been the best sub that class ever had.
How was I able to make such an impact on them during the two weeks? No, it wasn't because I sang songs to them (as I didn't start this until the charter school). It wasn't even because I played "Who Am I?" with them (which I had previously established, but never had time to play it in this class).
It was because 2-3 times per week, P.E. was the first subject. Most of the time, P.E. was in the weight room, but once a week it was basketball. The aides and other teacher participated with them -- and so did I. After all, I needed to keep in shape just as much as they did, especially since there was a scale in one of the rooms and I'd checked my weight. (Once I left that school, I used that measurement for "Guess my weight!" in the Conjectures game.)
And I believe that I'd gained the students' respect by participating in the P.E. class. To this day, I believe that if I'd sat on the side during P.E., the students would have tormented me as much as they'd tormented their previous subs.
Two years later at the charter school, I wouldn't outright participate in P.E., but I did high-five the students as they walked laps around the field. I believe that this did enable me to make deeper connections with some of the students -- but I think that I could have taken it a step further. Perhaps I could have talked with some of the students outside of the classroom -- especially those who had taken issue with the way I ran my own classes. Perhaps I could have defused arguments simply by asking some students to stay in my room during P.E. just to talk.
And in fact, this week three years ago was when P.E. was replaced with SBAC Prep. Once that happened, I'd lost my chance to make further connections with my students. Perhaps if I'd blogged more about the continuation school in 2015, I would have realized that making connections during P.E. was a wonderful idea -- and with fewer arguments and yelling, perhaps I could have survived my first full year at the school.
Lecture 13 of Michael Starbird's Change and Motion is called "Abstracting the Integral -- Areas, Volumes, and Dams." Here is an outline of this lecture:
I. Recall how the integral is used to computer the distance traveled given the speed of the moving car.
II. Computing areas bounded by curves is one of the most natural applications of the integral.
III. Computing volumes is another natural application of the integral.
IV. We can measure the areas and volumes created in this fashion using geometry and integrals. Let's consider a cylinder.
V. Let's consider a cone in which the height is 4 and the radius of the base circle is 3.
VI. The integral is important because the process of summing that the integral is performing is precisely what we need to do to solve various problems in various settings.
Since today's Starbird lecture is indirectly related to Geometry, let's do this one in more detail. He begins with a semicircle of radius 3, and the points have coordinates (x, sqrt(3^2 - x^2)). We find the area under this curve using an integral:
integral _-3 ^3 sqrt(3^2 - x^2) dx = area of 1/2 circle
To find the area of the whole circle, we can double this:
integral _-3 ^3 2sqrt(3^2 - x^2) dx = area of circle
= pi(3)^2 (known area formula)
Thus the professor concludes that the value of this integral is 9pi.
Starbird now moves on to volume and considers the pyramid:
height = 200 ft
base = 200 * 200 sq ft
We divide this into approximate plane sections -- squares with thickness delta-h. The professor then finds the volume of all of them using an integral, noting that the square that is distance h from the top has a side length of 2h:
slice vol = (2h)^2 delta-h
integral _0 ^200 (2h)^2 dh
Starbird's next example is a cylinder of height 4 of radius 3. He divides it into plane sections -- circles with thickness delta-x:
slice vol = pi 3^2 delta-x
integral _0 ^4 pi 3^2 dx
The professor moves on to a cone of the given height and base. The plane sections are still circles, but now we must work to find the radius of the circle at a distance x from the vertex:
slice vol = pi(3/4 x)^2 delta-x
integral _0 ^4 pi (3/4 x)^2 dx
Starbird's final example has nothing to do with Geometry. The "dam" mentioned in the title is 40 feet tall and 100 feet wide. We ask, what is the total pressure on the dam?
The professor tells us that at depth h, the water pressure is about 62.5h pounds per square foot. So we can divide the wall into thin rectangular strips of width 100 and height delta-h:
pressure on thin rec = (62.5h)(100 delta-h)
pressure on dam face = integral _0 ^40 (62.5h)100 dh
Lesson 9-7 of the U of Chicago text is called "Making Surfaces." In the modern Third Edition of the text, making surfaces appears in Lesson 9-8.
This lesson is all about making nets that can be folded to form polyhedra and other surfaces. Some figures have much simpler nets than others.
25. A regular polyhedron is a convex polyhedron in which all faces are congruent regular polygons and the same number of edges intersect at each of its vertices. There are only five regular polyhedra; they are pictured here.
a. Determine the number of vertices of each regular polyhedron.
b. Determine the number of edges of each regular polyhedron.
Ah -- we've seen these before. The five regular polyhedra are also called the Platonic solids. I've mentioned these in previous posts -- three summers ago we explained why there are only five of them, and two years ago we discovered that there are six regular polytopes in four dimensions. The Platonic solids are the tetrahedron, hexahedron (cube), octahedron, dodecahedron, and icosahedron.
The Exploration section continues thusly:
In 26-30, use cardboard and tape to construct a model of the regular polyhedron from the net provided. The patterns below should be enlarged. Cut on solid lines, fold on dotted lines.
Many teachers have given Platonic solid lessons in their classes. Since I don't want to try to create the nets myself, I link to previously made lessons. The first page, based on Question 25 (counting the vertices and edges) comes from the following link -- an elementary school in Washington State:
http://wilderptsa.ourschoolpages.com/Doc/Math_Adventures/Platonic_Solids.pdf
Question 26-30, the nets themselves, come from the following link:
https://www.math-drills.com/geometry/net_platonic_solids.pdf
The Math Drills link provides two nets for the dodecahedron. I chose the second one, since it more closely resembles the net in the U of Chicago text. On the other hand, their icosahedron net is very different from ours in the U of Chicago text.
Several members of the MTBoS have had Platonic solid activities in their own classes. Let's link to some of them:
Our first link is to Pamela Lawson, a Maine charter high school teacher. She taught her class about the Platonic solids about four years ago today:
https://rawsonmath.com/2016/01/26/how-do-we-know-that/
https://rawsonmath.com/2016/02/07/more-3d-geometry/
I’m teaching this 12 week geometry class focusing on 3-dimensional figures. It’s a brand new class, like many at Baxter Academy, so I get to make it up as I go. Since our focus is on 3-dimensional figures, I thought I would begin with some Platonic solids. So I found some nets of the solids that my students could cut and fold. Once they had them constructed, there was a lot of recognition of the different shapes and, even though I was calling them tetrahedron, octahedron, and so on, many of my students began referring to them as if they were dice: D4, D8, D12, D20. Anyway, I must have made some statement about there only being 5 Platonic solids, and they now had the complete set. One student asked, “How do we know that? How do we know that there are only 5?” Great question, right?
(She's teaching a 12-week Geometry course? That's right -- hers is one of the rare high schools that uses trimesters!) Of course, I'd already give a full explanation here on the blog, just after Independence Day in 2015. Let me repeat parts of that post here:
Legendre's Proposition 357 states that the sum of the plane angles that make up a solid angle must be less than [360 degrees]. He proves this essentially by "flattening out" the solid angle -- he takes a plane that intersects all sides of the solid angle and uses the previous Proposition 356 (which we've already proved here on the blog) to show that each plane angle of the solid angle is less than the same angle projected onto the new plane. A good way to visualize this is to imagine that the solid angle is formed at the vertex S of a pyramid -- the points A, B, C, etc., mentioned Legendre can be the vertices of the base of the pyramid, and the point O can be any point in the plane of the base -- for example, the center of the polygonal base.
I won't take the time to show the full proof of Proposition 357, but I will mention an application of this theorem. Suppose we want to figure out how many Platonic solids there are. Recall that a Platonic solid is a completely regular polyhedron -- all of its faces are congruent regular polygons. As it turns out, we can use Proposition 357 to find all of the Platonic solids.
We start with the equilateral triangle, with each angle measuring 60 degrees. Now each vertex of our Platonic solid forms a solid angle. We need at least three plane angles to form a solid angle, but there is an upper limit to how many plane angles there can be. Proposition 357 tells us that the plane angles must add up to less than 360 degrees, and since each angle is 60 degrees, there must be fewer than six of them (since 6 times 60 is 360). So there can be three, four, or five 60-degree plane angles. The Platonic solid with three 60-degree plane angles is the tetrahedron, with four is the octahedron, and with five is the icosahedron.
If we move on to squares with their 90-degree angles, we can have three 90-degree plane angles, but not four (since 4 times 90 is 360). Three 90-degree plane angles gives us the cube. Regular pentagons have 108-degree angles. Again, we can't have four of them (since 4 times 108 is more than 360), and three 108-degree angles gives us the dodecahedron. Regular hexagons have 120-degree angles, but 3 times 120 is already 360. Since each solid angle must contain at least three plane angles, we are done, since increasing the number of sides in the polygon only increases the angle. Therefore, there are only five Platonic solids -- tetrahedron, cube, octahedron, dodecahedron, and icosahedron.
Returning to 2020, let's go back to Euclid, who gives definitions of the Platonic solids:
We notice that the tetrahedron is missing. According to David Joyce, Euclid refers to the tetrahedron merely as a triangular pyramid. In Book XIII, he also proves that these are the only five Platonic solids -- and there, he refers the tetrahedron simply as "pyramid."
Since I don't wish to jump to Book XIII of Euclid, let's look at the next proposition here in Book XI:
As usual, let's modernize the proof:
Given: l, m intersect at B, n, o intersect at E, l | | n, m | | o (lines not all coplanar)
Prove: The angle between l and m is congruent to the angle between n and o.
Proof:
Statements Reasons
1. bla, bla, bla 1. Given
2. Choose A, C, D, F on l, m, n, o 2. Point-Line-Plane, part b (Ruler Postulate)
so that AB = DE, BC = EF
3. ABED, BCFE are parallelograms 3. Parallelogram Tests, part d
(one pair of sides is parallel and congruent)
4.
5. AD = BE, BE = CF 5. Parallelogram Consequences, part b
(opposite sides of a pgram are congruent)
6.
7. AD = CF 7. Transitivity of Congruence
8. ADFC is a parallelogram 8. Parallelogram Tests, part d
(one pair of sides is parallel and congruent)
9. AC = DF 9. Parallelogram Consequences, part b
(opposite sides of a pgram are congruent)
10. Triangle ABC = Triangle DEF 10. SSS Congruence Theorem [steps 2,2,9]
11. Angle ABC = Angle DEF 11. CPCTC
We can't help but notice that the six points A, B, C, D, E, F are the vertices of a triangular prism. And indeed, we see that the translation that appears in the U of Chicago definition of prism is the same translation that maps Triangle ABC to Triangle DEF.
Of course, this requires us to show that if two lines are parallel, then a translation must map one line to the other. I've alluded to the proof of this in posts from previous years, but I no longer include it as part of our curriculum.
No comments:
Post a Comment