Today I subbed in a middle school math class. It's in my new district. Since I really do teach today (as opposed to watching the students see their teacher on Zoom), it's worth doing "A Day in the Life" today.
8:45 -- My day begins with -- fifth period? Actually, this school has a schedule similar to those at the other middle schools in the district (including my long-term school). But the periods are numbered differently -- Cohort A (early alphabet) has periods 0-4, while Cohort B (later alphabet) has periods 5-9. Zero and fifth periods correspond, as do first and sixth periods, second and seventh, and so on. (They could have easily numbered the periods 1-5 for both cohorts like most other schools.)
And fifth period isn't a math class, but an ASB elective. Some of the students create posters and hang them all around the school.
Taking attendance is confusing today -- but it's not because in-person students are in "fifth period" while at-home students are in "zero period." It's really because some students are enrolled in a second elective -- and there aren't enough periods in the day for students to attend both of them everyday. I believe that there are some rules that determine which one these double-elective students are supposed to attend each day, but I can't figure out what these rules are.
Moreover, zero and fifth periods together have a total of 45 students -- the largest class by far that I've taught in the pandemic era. Roughly a third of these students each are on Cohort A, Cohort B, and opting out of hybrid. These leaves about 15 students packed into the activity room. Fortunately, some students help create posters outside, and so the room isn't actually that crowded.
9:40 -- Fifth period leaves the activity room, and sixth period arrives at the math room. This is the first of three eighth grade classes, and the only class with an aide.
As it turns out, Math 8 is still in APEX Unit 4b, on solving systems of equations. The pacing guide always had this unit spanning all of January. I taught the first week of this unit while still at the long-term position, and now they are in the last week of the unit, preparing for the test on Friday. When the students finish the worksheet, they also have a practice test on Quizizz.
One thing I notice about today's review worksheet is that there are questions on solving systems by graphing or substitution, but not by elimination. And -- since you know that I'm still keeping up with what my long-term kids are learning now -- they aren't doing elimination either. I'm not sure whether APEX actually omits elimination, or the teachers reached the consensus to eliminate elimination.
In some ways, this makes me feel a little less guilty for not reaching elimination with the eighth graders at the old charter school four years ago. But then again, elimination appeared in the Illinois State text, so technically I was supposed to teach it.
As I've been doing all week, I sing the Palindrome Song to these students.
10:35 -- Sixth period leaves for snack break.
10:45 -- Seventh period arrives. This is the second of three eighth grade classes.
11:40 -- Seventh period leaves and eighth period arrives. This is the last of three eighth grade classes.
By the way, since I wrote how I'm still following my long-term class, let me reveal what they're doing this week. Instead of worksheet, they have a Google Slides activity on solving systems. They must create five slides -- introduction, what a system is, graphing, elimination, and a pizza problem. The students must complete this assignment before their test, which is also on Friday.
12:35 -- Eighth period leaves for lunch.
1:15 -- At this school, after lunch is tutorial. Just like at my long-term school, the students are assigned to go to a different period each day -- and today, it just happens to by fifth period ASB.
But this leads to confusion. First, I start to pack up my Chromebook and sub folder and head over to the activity room -- only to have some students show up outside my door. They tell me that they're not quite sure whether to go to the math room or ASB room on tutorial days. The last time fifth period was assigned to tutorial was last Tuesday, but that was a Cohort A day -- and the last time before that was before the schedule changed at all middle schools. Thus today was the very first day that these particular kids were on campus on a day that tutorial was tied to fifth period -- and so they genuinely don't know where to go. Well, since there are kids outside my door anyway, I let them in and hold tutorial in the math room.
But then there was a problem with double-electives again. Students are told that if they don't attend one of their electives during the day, then they should attend it during tutorial instead. And so I see several in-person students who are absent earlier in the day. And, as it turns out, some of these kids are on my zero period roster -- meaning that they're in Cohort A, which shouldn't even be on campus today!
So there might have been fifteen students in the classroom, maybe slightly more -- but there aren't 15 desks in the room, since some were removed during the pandemic. And so some students end up sitting on the floor.
And moreover, my last class of the day is Math Skills, which counts as an elective -- and for some students, it's their double-elective. And so some of these students try to come in during tutorial, since they won't be able to come during the last period! I tell them that my classroom is full, and so they should just attend their other elective. But I don't know how to mark this on attendance.
The students in this room can't work on posters, and so I just sing the Palindrome Song again. Then two girls put on their own dance performance. And with a few minutes left in tutorial, I have them choose an extra song for me to perform -- they choose "Count on It," also from Square One TV.
1:40 -- Tutorial ends and ninth period begins. This is a Math Skills class.
Just like at my long-term school, Math Skills is the last class of the day. It's also the smallest -- there are only three in-person students, all girls, today. Some students have opted out of hybrid and are attending class online, while I may have chased some double-elective students away during tutorial. Also, the ninth period cohort is much smaller than the corresponding Cohort A class (fourth period).
And before you ask, no, these students don't do 60 minutes of ST Math or Dreambox in Math Skills the way we did at my long-term school. Instead, the regular teacher directs me to go over the worksheet and Quizizz just as I do in the other classes. The three girls tell me that they already finished both assignments in their real Math 8 class, and so they work on English or other assignments.
2:35 -- Ninth period leaves, thus ending my day of subbing. (Just as the other middle schools, the students have one period of independent study PE.) Of course, I head directly to the attendance office to explain all the problems I had with attendance, electives, and double-electives.
OK, so today my schedule was very confusing. A few years ago, I subbed an entire week in a class that was very similar to today's -- the regular teacher also had five classes, including three eighth grade math classes (albeit Algebra I, not Math 8), a Math Skills class, and an ASB class. Of course, that was both in a different district and before the pandemic. (You can read more about that class on the blog -- I was there in mid-November 2018 and then again on Pi Day 2019.)
At that school there was also several double-elective students. But over there, the solution to the double-elective problem is to assign such students a zero period -- as in a real zero period that starts an hour before the rest of the students arrive, not today's so called zero period (that was really first period).
While yesterday's high school had a genuine zero period (and indeed, my long-term middle school also had a real zero period), today's middle school likely wanted to avoid having students attend an extra hour at school. So instead, they came up with this confusing schedule where students go to different electives on different days.
I'm not quite sure how the school could improve this schedule (other than offer the extra period before school starts, like most other schools). Perhaps the tutorial schedule could be improved -- notice that there are five possible periods for tutorial and five days of the week. Yet the tutorial periods don't correspond directly to days of the week -- instead, they rotate every fifth day. Thus holidays and monthly minimum days (like the one we had a few days ago on Monday) disrupt the correspondence.
Perhaps instead, tutorial should be tied to Periods 0 and 5 every Monday, Periods 1 and 6 every Tuesday, and so on. If there's a holiday, then that class simply doesn't have tutorial that week. Then ASB only has tutorial on Mondays -- and since Mondays are fully online, the problem of which room to go to for tutorial and having enough seats in that room rarely occurs.
Lecture 11 of Prof. Arthur Benjamin's The Mathematics of Games and Puzzles: From Cards to Sudoku is called "Mathematics and Chess." Here is a summary of the lecture:
- In this lecture, the professor plans to talk about one of the world's oldest games, namely, the game of chess. Despite not requiring many calculations, the game is still very mathematical, because mathematics is the study of patterns.
- Two of the greatest grandmasters of all time -- Emanuel Lasker and Max Euwe -- also held doctorates in mathematics. And some of the greatest mathematicians -- Euler and Gauss -- were greatly interested in Chess puzzles.
- The king can move one square in any direction. The rook makes either horizontal or vertical moves of multiple squares. The bishop makes diagonal moves. The queen makes horizontal, vertical, or diagonal moves. The knight makes an L-shaped move.
- The Knight's Tour is a special problem -- can we move a knight around a chessboard so that it visits every square in as many moves? One possible solution: divide the chessboard into four quadrants, and divide each quadrant into four diamonds (up, down, right wheel, left wheel).
- Each player starts with eight pawns. Pawns move only forward. The first move can be one or two squares, but thereafter it moves only one square. It captures diagonally. If it reaches the eighth rank, it can be promoted to a queen.
- If the king is threatened, it's said to be in check. The next move must be to get it out of check, either by moving the king or blocking/capturing the checking piece. If it's impossible to get out of check, it's called checkmate, and the checking player is the winner.
- Here's where math comes in: a pawn is worth 1 point, a knight or bishop is worth 3 points, a rook is worth 5 points, and a queen is worth 9 points. The king is infinite. This is used to determine whether to exchange pieces -- so don't sacrifice a rook (5 points) for a bishop/pawn (4 points).
- Nowadays, computers can analyze millions of moves in a split second. Since Deep Blue in 1996, computers have grown in power. No human has beaten the top computer program since 2005.
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 2021, 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. AD | | BE, BE | | CF 4. Definition of parallelogram
5. AD = BE, BE = CF 5. Parallelogram Consequences, part b
(opposite sides of a pgram are congruent)
6. AD | | CF 6. Transitivity of Parallels (Prop 9 from yesterday)
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