Today I subbed in a high school math class. In fact, it's the first day of a multi-day assignment, as I'm here the rest of the week.
This is only the second assignment I've had in my new district -- the first, of course, was the long-term middle school job that I just completed. So let me say a few things about high schools in this district.
First, notice that unlike K-8, the high schools were all distance for the entire first semester. Thus this is only the second week that the students have seen the inside of a classroom since last year.
At this school, the two cohorts attend Tuesday/Wednesday and Thursday/Friday. I mentioned this version of hybrid on the blog before, where I called it "Hybrid Plan #1," as opposed to "Hybrid Plan #2" at my middle school. And as I pointed out, this version is common at high schools that previously had an odd/even block schedule before the pandemic. Odd periods met Tuesdays and Thursdays and even periods met Wednesdays and Fridays before the coronavirus, so by having Cohort A meet Tuesday/Wednesday, the students continue to attend one odd day and even day per week -- and likewise for Cohort B meeting on Thursday/Friday.
As usual, Mondays are all periods online only. But the order of the periods is 1-3-2-4-5-6. I suspect this is because one teacher has odds and evens at different schools, just like my middle school. (Now I must wonder why my middle school didn't go 1-3-2-4-5 -- only two classes changing -- instead of 2-4-1-3-5.)
Meanwhile, this school doesn't have days off -- students must Zoom in on the other days. My middle school is requiring this next week, but this high school enforces it now.
And here's one more thing I must say about subbing today -- I didn't actually teach anything! You see, this class is set up so that the regular teacher continues to teach from home. My presence is only required to supervise the in-person students. In fact, the other two Math 8 teachers at my middle school (yes, including the "lead" Math 8 teacher) have this same setup -- they teach from home, while long-term subs supervise the students. (One of those long-term subs, in fact, is still there, while the other has a new sub to watch the classes.)
Thus even though I'm here all four hybrid days this week, I'm not doing much real work. I'm wondering whether I should even bother to do "A Day in the Life" since I did no real teaching. OK, I'll do "A Day in the Life" only because it's math -- but there's no need for me to do it all four days.
7:55 -- First period arrives. This is a Geometry class.
One thing I notice is that many more students have opted out of hybrid here. This is to be expected -- if both parents work, they're likely to send their young elementary school kids to school while allowing their high school teens to stay at home. Thus as the grade level goes up, so does the opt-out rate.
And so of the thirteen students on the Tues./Wed. cohort in this class, only five students have chosen the hybrid option. All five of them are present.
I try to log in to Zoom to see what the class looks like -- but the regular teacher rejected me. Perhaps she thinks I'm Zoombombing the class and doesn't realize who I am.
9:50 -- First period leaves for break.
10:15 -- Third period arrives. This is an Algebra I class.
Most students here are freshmen, as opposed to the sophomores in Geometry. And so more students have signed up for hybrid -- 11 out of 17 students are listed as in-person. But once again, there are only five students in the classroom. Five hybrid students log into Zoom, so there's only one true absence.
Of the five students in the classroom, all five are guys. But I'm sure whether we can make conclusions about which gender is more likely to stay at home even if signed up for hybrid -- at my middle school, more girls attended in person.
12:05 -- Third period leaves for lunch.
12:55 -- Fifth period is the teacher's conference period, but I'm assigned a sophomore English class to cover this period.
That is, I would have covered it if there had been any actual in-person students. No one shows up to class in person today. This time, the teacher (who is herself a long-term sub) lets me into Zoom. She explains that this class has only one or two in-person students anyway, so it's not surprising that I'd have a day without students.
1:50 -- Approximately halfway through the class, after the long-term sub has discussed Lord of the Flies, she tells me that I might as well go to the office and go home, since there are no students. And so my day of "teaching" ends here.
Of course, I did say on the blog how there was just one girl in some of my Math Skills classes in my final week of the long-term. But that class was always small anyway -- it had only 16 students, half on each cohort, and half of the Wed./Fri. cohort officially opting out of hybrid. Two girls had logged in online anyway, and one guy was truly absent, leaving one girl in my class.
But this class, with no students at all, takes the cake. And we've seen how the number of in-person students drops dramatically from Grade 7 to Grade 10 -- so we can only imagine how much smaller the junior and senior classes are.
Since one of the classes is Geometry -- and this is a Geometry blog -- let's at least look at the Geometry lesson in more detail. The students are currently working in APEX Unit 6. This unit is on circles, and so it corresponds roughly to Chapter 15 of the U of Chicago text:
- For as long as he can remember, the professor has loved games and puzzles, which ultimately led to his love of math.
- In the Game of 21, the first player picks a number between 1 and 3. The second player adds a number between 1 and 3 to create a new total. The first player adds a number between 1 and 3 to create a new total. Repeat until we reach 21 -- whoever gets to 21 first wins.
- The first player always wins by starting with 1, then reaching 5, 9, 13, 17, and finally 21. Thus the first strategy for successful game and puzzle solving is: work backwards from your goal.
- In the Game of 15, players take turns choosing distinct numbers between 1 and 9, and the winner is the first to choose three that add up to 15.
- It turns out that this game is equivalent to tic-tac-toe. Thus the second strategy for successful game and puzzle solving is: find a mathematical structure to represent your game or puzzle.
- In tic-tac-toe itself, X should start in a corner -- X can force a win unless O counters by going in the center.
- But in backwards tic-tac-toe where the first to get three in a row loses, X should actually start in the center. Then X can avoid losing by making the symmetrically opposite move from O,
- In the game of Cram, players take turns placing dominoes on a checkerboard. Whoever makes the last legal move is the winner. The second player has a winning strategy by making the symmetrically opposite move from the first player.
- In the game of Hex, Players Red and Blue take turns placing their own pieces on a honeycomb, and the winner is the one to make a bridge between two opposite sides. It's proved impossible for the game to end in a draw. Also, 4 * 4 * 4 tic-tac-toe can never end in a draw.
- In Twenty Questions, it's possible to find any word in the dictionary in 20 questions by using a binary search, or cutting the number of guesses in half.
- In Mastermind, the codemaker chooses a four-digit number (1-6). The guesser tries to guess the number, and the codemaker replies with how many digits are in the right position and how many are in the wrong position. The guesser can guess any number in five guesses. The best opening guesses are either 1122 (minimizes worst case) or 1123 (minimizes average case). This strategy was discovered by recreational mathematician Donald Knuth. (He is still alive and just celebrated his 83rd birthday this week!)
- And the third strategy for successful game and puzzle solving is: in most games, it pays to delay your gratification. For example, in Solitaire, we might delay putting a red 3 on the red Ace/2.
- In Ghost, the players take turns choosing letters in order to avoid making a word. The first player wants to make an even-length word, and the second player wants a word of odd parity. The first player should choose a word starting with H, J, M, or Z. A winning word for the second player is GHOST itself.
- The Tower of Hanoi consists of a stack of nine different sized discs and three pegs. We must move the disc from one peg to another without placing a large disc on a small disc. It's easier to try solving it with two or three discs first. The pattern is 1-2-1-3-1-2-1-big. A graph network that connects legal moves resembles Sierpinski's Triangle.
- Euler discovered graph theory by trying to solve the Bridges of Konigsberg problem. And probability was discovered by playing games of chance.
- When flipping four coins, the most common outcome is only one head/tail and three landing the other way.
- There are four types of games: Deterministic Intelligent (Chess, Checkers), Deterministic Predetermined (Mastermind, Minesweeper), Random Intelligent (Poker, Bridge), and Random Predetermined (Roulette, Craps).
- In the first few lectures, we will look at specific games of chance. We will eventually reach the Rubik's Cube, and Chess.
Lesson 8-7 of the U of Chicago text is on the Pythagorean Theorem, and Lesson 11-2 of the same text is on the Distance Formula. I explained yesterday that I will cover these two related theorems in this lesson.
The Pythagorean Theorem is, of course, one of the most famous mathematical theorems. It is usually the first theorem that a student learns that is named for a person -- the famous Greek mathematician Pythagoras, who lived about 2500 years ago -- a few centuries before Euclid. I believe that the only other named theorem in the text is the Cavalieri Principle -- named after an Italian mathematician from 400 years ago. Perhaps the best known named theorem is Fermat's Last Theorem.
It's known that Pythagoras was not the only person who knew of his named theorem. The ancient Babylonians and Chinese knew of the theorem, and it's possible that the Egyptians at least knew about the 3-4-5 case.
We begin with the proof of the Pythagorean Theorem -- but which one? One of my favorite math websites, Cut the Knot (previously mentioned on this blog), gives over a hundred proofs of Pythagoras:
http://www.cut-the-knot.com/pythagoras/
The only other theorem with many known proofs is Gauss's Law of Quadratic Reciprocity. Here is a discussion of some of the first few proofs:
Proof #1 is Euclid's own proof, his Proposition I.47. Proof #2 is simple enough, but rarely seen. Proofs #3 and #4 both appear in the U of Chicago, Lesson 8-7 -- one is given as the main proof and the other appears in the exercises. Proof #5 is the presidential proof -- it was first proposed by James Garfield, the twentieth President of the United States. I've once seen a text where the high school students were expected to reproduce Garfield's proof.
So far, the first five proofs all involve area. My favorite area-based proof is actually Proof #9. I've tutored students where I've shown them this version of the proof. Just as the Cut the Knot page points out, Proof #9 "makes the algebraic part of proof #4 completely redundant" -- and because it doesn't require the students to know any area formulas at all (save that of the square), I could give this proof right now. In fact, I was considering including Proof #9 on today's worksheet. Instead, I will wait until our next activity day on Friday to post it.
But it's the proof by similarity, Proof #6, that's endorsed by Common Core. This proof has its own page:
http://www.cut-the-knot.org/pythagoras/PythagorasBySimilarity.shtml
Here is Proof #6 below. The only difference between my proof and #6 from the Cut the Knot webpage is that I switched points A and C, so that the right angle is at C. This fits the usual notation that c, the side opposite C, is the hypotenuse.
Given: ACB and ADC are right angles.
Prove: BC * BC + AC * AC = AB * AB (that is, a^2 + b^2 = c^2)
Statements Reasons
1. ADC, ACB, CDB rt. angles 1. Given
2. Angle A = A, Angle B = B 2. Reflexive Property of Congruence
3. ADC, ACB, CDB sim. tri. 3. AA Similarity Theorem
4. AC/AB = AD/AC, 4. Corresponding sides are in proportion.
BC/AB = BD/BC
5. AC * AC = AB * AD, 5. Multiplication Property of Equality
BC * BC = AB * BD
6. BC * BC + AC * AC = 6. Addition Property of Equality
AB * BD + AB * AD
7. BC * BC + AC * AC = 7. Distributive Property
AB * (BD + AD)
8. BC * BC + AC * AC = 8. Betweenness Theorem (Segment Addition)
AB * AB
I mentioned before that, like many converses, the Converse of the Pythagorean Theorem is proved using the forward theorem plus a uniqueness theorem -- and the correct uniqueness theorem happens to be the SSS Congruence Theorem (i.e., up to isometry, there is at most one triangle given three side lengths). To prove this, given a triangle with lengths a^2 + b^2 = c^2 we take another triangle with legs a and b, and we're given a right angle between a and b. By the forward Pythagorean Theorem, if the hypotenuse of the new triangle is z, then a^2 + b^2 = z^2. (I chose z following the U of Chicago proof.) Thenz^2 = c^2 by transitivity -- that is, z = c. So all three pairs of both triangles are congruent -- SSS. Then by CPCTC, the original triangle has an angle congruent to the given right angle -- so it's a right triangle. QED
Interestingly enough, there's yet another link at Proof #6 at Cut the Knot, "Lipogrammatic Proof of the Pythagorean Theorem." At that link, not only is Proof #6 remodified so that it's also an area proof (just like Proofs #1-5), but, as its author points out, slope is well-defined without referring to similar triangles!
Now, my original worksheet included the Distance Formula as well, but this year, we're waiting until Lesson 11-2 -- which is where distance belongs in the test. So I decided to keep only the second worksheet -- which contains exercises but no proofs -- and include an activity I posted two days later, which gives the proof of the Pythagorean Theorem as given in the text. This is essentially Proof #4 from the link above.
And so two years ago, I began by posting the Common Core similarity-based proof and then the U of Chicago area-based proof two days later. This year, I wrote that I would stay true to the U of Chicago version, so that's what I'm doing today.
But during the year I taught at the old charter school, recall that that Pythagorean Theorem also appears in the eighth grade standards. And so I'm using the rest of this post to discuss how I taught -- or, as usual, how I failed to teach -- the Pythagoran Theorem that year.
Notice that even though the Common Core tells us to use similarity to prove Pythagoras in high school Geometry, similarity isn't mentioned in the eighth grade standard:
CCSS.MATH.CONTENT.8.G.B.6
Explain a proof of the Pythagorean Theorem and its converse.
CCSS.MATH.CONTENT.HSG.SRT.B.4
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
Then again, the Illinois State text suggests using similarity to prove Pythagoras in eighth grade. And so I tried to give the project in the text in which the connection from similarity to the Pythagorean Theorem is made. But this project failed for three reasons:
- The previous lessons on similarity and dilations were cancelled due to science lessons -- and arguments about science lessons. (See last week's Epiphany post and read under the section labeled "January 2017.")
- The day on which this Illinois State lesson was scheduled turned into "finish the extra credit project on Hidden Figures" -- also caused by a domino effect of poor previous lessons.
- The Illinois State project wasn't designed well, anyway. Students were supposed to form similar right triangles on Geoboards -- and somehow that would lead to their discovering the Pythagorean Theorem. But I fail to see, for example, how students would see what the hypotenuse of a right triangle with legs 3 and 4 is. Even building a square on the Geoboard with that hypotenuse as a side doesn't make it obvious that it's 5.
And "all components of it" includes badly designed projects such as the Geoboards. Of course, last year I wrote that I should have preempted all math projects with the much better science projects, but let's assume that we're writing about this current year -- if I had remained in the classroom, I assume that science wouldn't have been a factor.
The proper pacing plan is to teach one standard a week, beginning with 8.NS1. Skipping over short weeks, Benchmark Testing, and so on, it's possible to reach 8.G8 (the Distance Formula) before my old school gives the SBAC -- if not, hopefully at least 8.G6, when the Pythagoras is introduced.
That year, I began 8.G1 by teaching translations the first day, then reflections the second day, and then rotations the third day. This won't work if we follow the pacing plan -- where only one day is devoted to the traditional lesson, with other the days for projects, learning centers, and so on.
But I do see a loophole where I can teach the transformations one day at a time even with only one traditional lesson per week:
Week of 8.G1 -- reflections
Week of 8.G2 -- rotations
Week of 8.G3 -- translations
Week of 8.G4 -- dilations
Week of 8.G5 -- follow as written
And so the first week, I introduce my students to reflections -- and then in the Student Journal, we cover only the questions on reflections. Last year, I started with translations as these are the easiest for students to understand. But notice that translations lines up perfectly with 8.G3, since this is about the coordinate plane -- we like to perform translations with coordinates. Standard 8.G4 is about similarity and so it's the perfect time to teach dilations. Then we follow 8.G5 as written and teach Triangle Sum, Parallel Consequences, and AA Similarity.
When we reach 8.G6, we follow the sixth resolution and use similarity to teach Pythagoras. It's possible to create an activity that's just like the area-based project I posted today, yet is based on similarity rather than area.
Take a standard sheet of paper and cut it along its diagonal. This, of course, divides the paper into two right triangles. Now take one of the triangles and cut it along its altitude to the hypotenuse. Now there are three right triangles -- and notice that these are the three similar right triangles that appear in the similarity-based proof!
To make this proof easier for the students to understand, we'll begin by using numbers -- suppose the legs of a right triangle are 3 and 4, then what is the hypotenuse? We might as well use c for the hypotenuse, as usual. So even before we cut, we label the sides of the paper 3 and 4. (Notice that the sides of a standard sheet of paper, 8 1/2 * 11, are already nearly in a 3:4 ratio.)
After we cut, we label all three triangles with legs 3 and 4 and hypotenuse c. Of course, the triangles are merely similar, not congruent, so they can't all be 3-4-c right triangles. So instead, we draw a box (or blank) next to each 3, 4, and c, to be filled in by a constant for each triangle. Indeed, we anticipate this activity by emphasizing, during the Week of 8.G4, that we can multiply all three sides by the same constant (a dilation!) and obtain a similar triangle.
Notice that none of these triangles is considered the original 3-4-c triangle -- all three triangles must be multiplied by some constant. This is to avoid fractions -- for example, if the largest triangle were simply 3-4-c, the other two would have fractions in their lengths. The same would happen even if we chose the smallest triangle to 3-4-c -- we'll eventually find that the other lengths are fractions.
Small triangle: 3( )-4( )-c( )
Medium triangle: 3( )-4( )-c( )
Large triangle: 3( )-4( )-c( )
Now we start moving triangles and comparing their sides. We'll see that the short leg of the medium triangle is congruent to the triangle leg of the small triangle. The former is labeled 3( ) and the latter is labeled 4( ). Since 3 * 4 = 4 * 3, let's fill in the blanks with 3(4) and 4(3):
Small triangle: 3( )-4(3)-c( )
Medium triangle: 3(4)-4( )-c( )
Large triangle: 3( )-4( )-c( )
Recall that within each triangle, all the blanks must be labeled the same (otherwise the triangles aren't similar), so we write:
Small triangle: 3(3)-4(3)-c(3)
Medium triangle: 3(4)-4(4)-c(4)
Large triangle: 3( )-4( )-c( )
Notice that we won't actually multiply anything until we have to! The next thing we notice is that the hypotenuse of the small triangle c(3) equals the short leg of the large triangle, and the hypotenuse of the medium triangle c(4) equals the long leg of the large triangle. This suggests that the blanks in the large triangle should be filled with c:
Small triangle: 3(3)-4(3)-c(3)
Medium triangle: 3(4)-4(4)-c(4)
Large triangle: 3(c)-4(c)-c(c)
Now we move the pieces around to form the original rectangle. We now see that the short leg of the small triangle 3(3) and the long leg of the medium triangle 4(4) add up to the hypotenuse of the largest triangle c(c):
3(3) + 4(4) = c(c)
9 + 16 = c(c)
c(c) = 25
c = 5
This proof easily generalizes -- change all the 3's to a's and all the 4's to b's, and we instantly obtain the Pythagorean Theorem. Notice that in avoiding multiplication until it was necessary, the only numbers we had to multiply are a^2, b^2, and c^2.
And so this is how I keep the sixth [old] resolution -- we follow the basic framework of the Illinois State text while modifying the projects in order for them to make sense for the student. The students can keep the Illinois State textbook open while using a worksheet to complete the project.
Of course, I haven't addressed the elephant in the room -- the first [old] resolution. Students aren't learning anything unless they are quiet during the lesson. My eighth graders would talk the whole time regardless of whether I'm giving a project or a traditional lesson. Typically the Pythagorean Theorem is one of the easier lessons in the eighth grade curriculum, yet my students learned nothing. Not only did they forget the theorem by the time of the unit test or SBAC, but they would have forgotten it for that night's homework or the next day's Warm-Up, not even being able to do the first step.
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