Today I subbed in a middle school English special ed class. It is in my first Orange County district, and indeed, I've subbed in this classroom before. My last visit to this class was back on 9-17-19. (Yes, my last visit was during Palindrome Week 2019. Maybe I should look back and see where else I subbed that week in 2019, and predict those to be my next assignments for the rest of Palindrome Week 2021.)
Of course, that was before the pandemic -- so once again, everything's different now. This is one of the middle schools that used to have a period rotation. But that's been scrapped for hybrid. Instead, three periods meet each day, although the split is different from middle schools in my new district. Here Cohort A meets Tuesday/Wednesday and Cohort B on Thursday/Friday. And the period that meet are 1, 2, 3 on Tuesdays/Thursdays and 4, 5, 6 on Wednesdays/Fridays.
(In some ways, this is more logical than odd/even because it allows a teacher to work part-time at two different schools during hybrid. If my long-term middle school had used this split, they wouldn't have needed to rearrange the periods this month when ending the block schedule.)
As it turns out, fourth period is conference period while fifth period is co-teaching, and so the only real class I must teach is sixth period. (So I won't do "A Day in the Life" -- instead, I'll briefly mention the other periods and discuss the one real class in some detail.) Actually, during fourth period the fire alarm goes off, and so students must evacuate to the field and meet with their first period teacher -- I end up meeting students during this period anyway.
In fifth period English 8, the students are reading My Brother Sam Is Dead -- their lesson is connected to their U.S. History class, as the novel takes place during the American Revolution. The resident teacher, himself a long-term sub, likes to start the first few minutes of class with an IXL Warm-Up. I, of course, am already familiar with IXL from my time at the old charter school.
That takes us to sixth period. These students are reading Boy in the Striped Pajamas, a novel set not during the Revolutionary War, but during World War II. Their assignment is to go to Google Classroom and answer questions from the first few chapters of the book.
But there's a problem. Usually the regular teacher must allow me access to her Google Classroom, and I'm expected to create my own Zoom session and place the link in Google Classroom. By the way, when teachers do add me as a sub to their classes on Google Classroom (or Canvas), they usually don't remove me after the assignment is over. Thus I continue to get email warnings about what yesterday's music class is doing, as well as Wednesday's history class, and so on. By the end of the year, my Google Classroom and Canvas pages will be littered with links to classes that I've spent only one day in.
Unfortunately, today's regular teacher neglects to give me Google Classroom access. This class has an aide, and she contacts another special ed teacher, who comes in to help me with Zoom, Google Classroom, and the in-person students. (In this district, students always had to log in to Zoom on the days when they don't attend face-to-face.)
One in-person student does cause trouble today. When I'm trying to take attendance, I keep telling him to be quiet, but he refuses and says, "I don't want to." And when I go to my usual song incentive, he says, "I don't want to hear your boring song." I remind him that yes, learning is boring, but I'd much rather be bored and educated than entertained knowing nothing.
The student doesn't know how to respond to this, so he turns to the more experienced special ed teacher, who is still in the room. When he tells the student to be quiet, the youngster tells him to shut up. And so the other teacher ultimately kicks him out of class and sends him to the office.
Since the start of the new year, I've been meaning to return to my old habit of posting one of my New Year's Resolutions and discussing how I applied it each day -- and I used my own Eleven Calendar to determine which resolution to discuss. Well, today is Nineday on the Eleven Calendar:
Resolution #9: We pay attention to math as long as possible.
Except for the "math" part, this is essentially what I tell the troublemaker today -- we pay attention to school as long as possible, even if it is boring. Oh, and connecting this to the pre-pandemic version of Resolution #9, he does ask for a restroom pass (and this is the class after snack break).
Ironically, what he describes as boring is my song (before I sing it) -- even though the reason I sing in class is to break up the boring parts of class when learning takes place. Then again, this is likely a "sour grapes" reaction -- he's already upset with me for telling him to stop talking, and so he'll claim that anything I do is boring, even performing a song. I do sing "Palindrome Song" today -- after this guy is sent to the office, of course.
Today on her Daily Epsilon on Math 2021, Rebecca Rapoport writes:
What is 44 times sine of the complement of 60 degrees?
Hmm, "sine of the complement" -- sine-co -- hey. that's cosine. So in other words, we're asking:
What is 44 times the cosine of 60 degrees?
And we know that cos 60 = 1/2, and so 44 cos 60 = 22. Therefore the desired answer is 22 -- and of course, today's date is the 22nd. Even though this question is about trig, it's simple enough to be taught in Lesson 14-4 of the U of Chicago text (thereby making this a Geometry question).
Lecture 8 of Prof. Arthur Benjamin's The Mathematics of Games and Puzzles: From Cards to Sudoku is called "Solving 'Impossible' Puzzles." Here is a summary of the lecture:
- In this lecture we will explore some classic puzzles, many of which have been driving people crazy for centuries. Each one depends on odd and even numbers in their analysis.
- The Fifteen Puzzle was the Rubik's cube of the 19th century. In its classic form, the numbers 1 to 15 are placed randomly in a 4-by-4 square. The player slides squares into the empty space, and keeps going until all the numbers are placed in order.
- A good strategy is to place all the numbers in order by pairs -- first 1 and 2, then 3 and 4, 5 and 6, 7 and 8, and then 9 and 13. A champion can complete the puzzle in under half a minute. Our professor can solve it in under a minute.
- In the 19th century, Sam Loyd started from the position where 1 to 13 are in the right place, but 14 and 15 are switched. He offered $1000 to the person who could solve it. As it turned out, Loyd's money was safe, because the puzzle had no solution.
- If we have three cups numbered 1, 2, 3, these can be placed in three different orders, which are called permutations. Some permutations, such as 1, 3, 2, can be solved in one swap or any odd number of swaps. Others, such as 2, 3, 1, can be solved in two or more swaps as long as it's even.
- Loyd's arrangement is an odd position, so it requires an odd number of swaps. But the blank square must move an even number of swaps, since every left swap is balanced by a right swap, and every up swap by a down swap, to bring it back to the corner. So Loyd's puzzle is unsolvable.
- Another puzzle begins with a 3 by 3 set of lights. We can label these lights 1 to 9. Some of the lights are on at the start of the game. The goal is to turn off all the lights.
- Pressing a corner button (1, 3, 7, 9) toggles (switches on to off or vice versa) it and the three lights closest to it (so pressing 1 toggles 1, 2, 4, 5). Pressing an edge button (2, 4, 6, 8) toggles it and the corner lights closest to it (so pressing 2 toggles 1, 2, 3). Pressing 5 toggles 2, 4, 5, 6, 8.
- In mod 2 arithmetic, 0 and 1 are the only numbers. Addition mod 2 is like regular addition except we have 1 + 1 = 0 (that is, odd + odd = even). In this puzzle, a position is a vector mod 2. Button 2 corresponds to the vector (1, 1, 1, 0, 0, 0, 0, 0, 0). Pressing buttons is like adding vectors.
- The order in which the buttons are pressed doesn't matter, exactly because vector addition mod 2 is commutative. Also, no button needs to be pressed more than once, since the vector sum of any vector with itself is the zero vector. Thus finding a solution counts as linear algebra.
- To solve the problem, our strategy is to press the edges first, then the corners, then 5. We must press an edge button if the numbers of lights currently lit inside the rectangle adjacent to it (for example, lights 1-6 for edge button 2, lights 1, 2, 4, 5, 7, 8 for edge button 4) is odd.
- Another puzzle is called Peg Solitaire. There are 32 pegs, placed on a pegboard like a cross with an empty hole in the center. The only legal move is to jump a peg across or down over a peg into an empty hole, removing the peg jumped over. The goal is to have only one peg left in the center.
- A winning first move is bottoms up. Then the following mnemonic shows which pegs to move next: "right up left, then move it up." Do this four times, rotating 90 degrees each time. ("Move it up" consists of two moves, the net result being that one peg is moved up one spot.")
- There should now be eleven pegs left, shaped like a house. The third step is a grand tour of that house -- the peg in the middle of the house moves around the house, leaving only five pegs left in an upside-down T position. The fourth and final step is to solve that T-formation, and we win.
- We can prove that most of the time, the last peg standing will be in the center. To show this, we label the holes x, y, z as follows -- the three pegs on the top and bottom rows are xyz. Below or to the right of any x is y, below or to the right of y is z, and next after z is x. The center peg is y.
- Here's how to add symbols -- the sum of any two letters is the third. The sum of any letter with itself is 0. The sum of any letter with 0 is itself. (This is the Klein 4-group.) For any position, its signature is the sum of the occupied squares. For any legal position, the sum is always y (center).
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: AB | | CD, E on AB, F on CD
Prove: AB, CD, EF are coplanar
Indirect Proof:
Assume that the lines aren't coplanar -- that is, AB and CD lie in plane P (as parallel lines, by definition, are coplanar) while EF lies in another plane Q. That is, EF contains a new point G such that G is in plane Q, not plane P.
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 AB, CD, EF are coplanar. QED
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. So far this year, I've simply reposted the old activities that I've done on the blog every year, without regards to whether the activity is doable in the pandemic era. For example, last week I posted my favorite Pythagorean Theorem and pi activities, even though these involve cutting up pieces of paper. Well, today that will all change. This time, I'm posting an activity that makes sense in current times.
One interesting thing about today's Lesson 9-4, on plane sections, is that it also appears in the Common Core Math 7 standards, in the geometry strand:
CCSS.MATH.CONTENT.7.G.A.3
Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
Recall that I completed my long-term assignment just as Math 7 was starting the G strand. And I wrote earlier in this post that I still have access to previous classes I subbed for in Canvas -- which clearly includes the long-term assignment. Thus I know exactly what the seventh graders are studying this week -- and just take a wild guess what lesson it is.
You guessed it -- plane sections!
And so I can set up an activity to match what my seventh graders (sorry, they aren't my seventh graders any longer) are learning this week. Here's what their week looks like. (Recall that this is the first week that they must log in on their off days.)
Well, I won't link to APEX here on the blog -- and while I can still access Canvas, I don't have access to that Quizizz. So that leaves us the Khan Academy as today's activity:
I'll continue to monitor my long-term class for possible activities to post on the blog. I also still have access to last week's Geometry class, so that's another possible source of activities.
I'll keep last year's Guided Notes. The intent is for this to be followed by Khan Academy.
I won't post the Khan Academy as a video, but I will post a link to Square One TV's "One Billion Is Big" in honor of tonight's Mega Millions jackpot. (But no -- Arthur Benjamin doesn't even bother to analyze the lottery in his course, since the expected value is around -50 cents per dollar wagered.)
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