Today I received my second stimulus check -- er, Economic Impact Payment Card. Of course, I'm not as desperate for funds now as I was in the spring, now that the schools are open and I'm earning paychecks as a sub. Still, a little extra money is always nice.
Oh, and speaking of subbing, today I subbed in a middle school music class. Since it's middle school, I will do "A Day in the Life" today. It's in my new district, and so the bell schedule is quite similar to that at my school where I recently completed my long-term assignment.
8:45 -- First period is a Band class for Grades 7-8.
You may be wondering what Band looks like under hybrid. Well, I'm still wondering -- the regular teacher wants to avoid playing instruments with a sub, and so the students have another assignment instead. I hear that when hybrid first began in September up until last week, students played music outside. But recall that everything in middle school changed on January 19th -- students must log in to Google Classroom on the days their cohort doesn't meet. With at least half the students logging in from home everyday, does having practice outside still work?
The assignment today is on Edpuzzle, with which I'm already familiar. The students watch a video on the composer Johann Sebastian Bach and answer questions about him. (I've discussed Bach extensively on the blog, since he's the third person mentioned in the title of Douglas Hofstadter's book.)
The other new rule starting this week is that cameras must be turned on. But I have trouble figuring out how to get my own camera and mic working, and so I don't enforce this rule until second period.
Since I'm in a music classroom, of course I sing songs today. It's still Palindrome Week, and so I start out with the Palindrome Song. But from what I see written on the board, I deduce that the students are learning about the key of C major and its relative A minor. And so I also want to perform songs that are written in those keys.
For C major, I sing "Do-Re-Mi" -- a song which highlights the major scale. I could have performed the "Vote" parody of this song (semi-justified since yesterday was Inauguration Day), but I stick to the original "Do-Re-Mi" instead.
As for A minor, I perform my original song "Whodunnit." Technically, this song isn't in A minor but in the 12EDL scale, with two notes (12/11 neutral third and 12/7 supermajor sixth) diverging from minor. While 12/11 above A is close enough to B, the red 6th is closer to a minor 7th than a minor 6th. In order to avoid confusing the students, I sing this note as F anyway.
And I play these songs on the musical keyboard that is in the classroom. The Palindrome Song is in the key of G major, while the other two songs are in C major and A minor as indicated. It's been some time since I performed on the keyboard (or piano) for a class.
9:40 -- First period leaves and second arrives. This is a sixth grade Band class.
This marks the first time that I've taught sixth graders in years. After all, in my other Orange County district, sixth grade is considered elementary school, and my LA County district is exclusively a high school district. We must go all the way back to the year I taught at the old charter school to find a time when I taught sixth graders.
The sixth graders have the same Edpuzzle as the Intermediate Band class. By now, I've figured out the camera, and so for the first time I mark one student as non-participating for not having the camera on.
10:35 -- Second period leaves for snack break.
10:45 -- Third period arrives. This is also a sixth grade class, except that it's a one-quarter course as part of an Exploratory Wheel. It's also the only class with an aide -- I assume that some of the students here are special ed.
And instead of the Bach Edpuzzle, these students have an assignment on Beethoven. This isn't on Edpuzzle -- instead, they watch the film Beethoven Lives Upstairs and answer the questions on a worksheet posted in Canvas. (I actually watched Beethoven Lives Upstairs last month to celebrate the composer's semiquincentennial -- that is, the 250th anniversary of his birth. I had no reason to mention that on the blog until now.)
Since Beethoven Lives Upstairs is longer than the Bach Edpuzzle video, I have less time for singing any songs this period. I do catch two students with cameras off, so I mark them as non-participating.
11:40 -- It is now time for tutorial. At my long-term middle school, students are sometimes assigned different periods to tutorial, but here the aide tells me that the students always stay in third period. And so I do have time to sing "Palindrome Song," "Do-Re-Mi," and "Whodunnit" to this group after all.
12:05 -- Tutorial ends for lunch.
12:45 -- Fourth period begins. This is a Strings class for Grades 7-8.
Even though these students are playing different instruments from the morning classes, they have the same Bach Edpuzzle as the Band classes.
And they are also learning about the keys of C major and A minor. Notice that while C major, with no sharps or flats, is typically the first scale learned by a pianist, players of other instruments often learn other scales first. In particular, a flautist might start with Bb or another flat scale, while a violinist might start with A major or another sharp scale. Thus C major might be the third or fourth major scale that these students learn -- which explains why both Band and Strings wait until second semester to see it.
This is the class with the most cameras turned off -- I catch two students near the start of class and a third student later on. This is quite unexpected, since before the pandemic, Strings players are usually better behaved than Band players.
1:40 -- Fifth period begins. This is a sixth grade Strings class.
And then something different happens -- for some reason, the Bach Edpuzzle simply doesn't want to open up for this period. I, as the teacher, can't access any Edpuzzle today -- instead, I see only a message that something's not right about the "configuration." For all the other classes, the students don't see this message, but in fifth period they do. So instead, I continue to sing more songs.
I have the students choose these from my songbook. Here's what they picked: "Another Ratio Song," "Earth, Moon, and Sun," "Ghost of a Chance," and "Plug It In."
Of these, "Another Ratio Song" is one of my songs from the old charter school. I never performed it that often, and so its tune is lost. Over the summer, I wrote that I'd give it a new tune and would post it on the blog when I do -- so I might as well do so now. Here are the lyrics:
ANOTHER RATIO SONG
6th Grade:
What can we do with fractions?
What can we do with fractions?
As everyone knows
We write ratios.
You can do no worse
If you write the 1st one 1st.
In between write dots
Then the 2nd -- that's a lot
We can do with fractions!
7th Grade:
What can we do with fractions?
What can we do with fractions?
We see over there
With ratios we can compare.
The fractions to divide
Flip the 2nd & multiply.
Remember to simplify
And now you know why
We can use fractions!
8th Grade:
What can we do with fractions?
What can we do with fractions?
We can make them decimal
And that is not all.
Another major feat
Is that decimals repeat.
They go on forever.
Know that whenever
We can use fractions!
Over the summer, I planned to compose this song in 18EDL, so let me do so in Mocha:
https://www.haplessgenius.com/mocha/
2:35 -- Fifth period ends. Just like at my long-term middle school, sixth period is independent P.E. for everyone, and so students attend only five periods.
Lecture 7 of Prof. Arthur Benjamin's The Mathematics of Games and Puzzles: From Cards to Sudoku is called "Games You Can't Lose and Sneaky Puzzles." Here is a summary of the lecture:
- There are many games that seem innocent and fair, but where you actually have a big advantage.
- Nontransitive dice are four dice A, B, C, D such that A beats B, B beats C, C beats D, and yet D beats A, each with probability 2/3. Here are the faces of the dice: Die A has 6, 6, 2, 2, 2, 2, Die B has 3, 3, 3, 3, 3, 3, Die C has 4, 4, 4, 4, 0, 0, and Die D has 5, 5, 5, 1, 1, 1.
- In Roulette, the house advantage is 5.3 cents per bet. Yet if you bet on your favorite number 35 times in a row, you have a 61% probability of coming out ahead. That's because most of the time if you win once, your profit is only $1, but if you lose all bets, your loss is $35.
- In Bingo, some cards are more likely to win than others. If the numbers on your board are rarely on your opponent's boards, then not only are you more likely to win, but you have less worry of a tie/shared pot it you do win. Also, you're more likely to make a horizontal than a vertical bingo.
- According to Simpson's Paradox, it's possible for one player to have a higher batting average than another in each of two (or even three) years yet a lower average over those years. For example, David Justice beat Derek Jeter in 1995, 1996, 1997, yet Jeter had the better average overall.
Lesson 9-3 of the U of Chicago text is called "Pyramids and Cones." Our text refers to both pyramids and cones as "conic surfaces."
This is what I wrote last year about today's lesson:
This lesson is very similar to Lesson 9-2 in that the focus is on vocabulary. The difference is that today's lesson is not an activity, but a traditional worksheet.
Indeed, students are asked to calculate slant height in two of the problems on this worksheet. As we already know, this requires the Pythagorean Theorem. A right triangle can be formed with the slant height as the hypotenuse and the altitude height as one leg. The other leg is unnamed in the U of Chicago text, but notice that it's actually the apothem of the regular polygon base.
In fact, we discussed this in previous years -- many Geometry texts define "apothem," but not the U of Chicago text. Indeed, Lesson 8-6 explains the trapezoid area formula as follows:
"There is no known general formula for the area of a polygon even if you know all the lengths of its sides and the measures of its angles. But if a polygon can be split into triangles with altitudes or sides of the same length, then there can be a formula. One kind of polygon that can be split in this way is the trapezoid."
Well, another polygon that can be split in this way is the regular polygon. Indeed, all the altitudes and sides are of the same length, because the triangles are congruent. The common altitude of these triangles is the apothem of the regular polygon. (It has come to my attention that apothems do appear in the modern Third Edition of the text. This is in the new Lesson 8-7 on Special Right Triangles, since 30-60-90 and 45-45-90 triangles are used to find the apothems of equilateral triangles, squares, and regular hexagons.)
But let's get back on track with pyramids and cones. Again, we look at Euclid's definitions:
Notice that Euclid's definition of "pyramid" isn't that much different from the U of Chicago's. But Euclid's cones, like his cylinders, are solids of revolution. Therefore they are limited to right cones and right cylinders. The U of Chicago generalizes this with a definition of cone not unlike the definition of pyramid. This allows cones to be oblique as well as right, and pyramids and cones are examples of conic surfaces (the surfaces of conic solids).
Oh, and I also notice one more difference between the old Second and new Third Editions. The point at the top of the pyramid is called the vertex in the Second Edition and the apex in the Third. Euclid, meanwhile, never names the top point of the pyramid.
Let's look at the next proposition for us to prove:
This is basically the Two Perpendiculars Theorem of Lesson 3-5, except that the two given lines are perpendicular to the same plane, not merely the same line. The line version of Two Perpendiculars appears as the last step of the proof.
Let's modernize Euclid's proof:
Given: Line AB perp. plane P, line CD perp. plane P (B, D in plane P)
Prove: AB | | CD
Proof:
Statements Reasons
1. bla, bla, bla 1. Given
2. Draw DE in plane P such that 2. Point-Line-Plane, part b (Ruler/Protractor Postulates)
DE perp BD, DE = AB
3. AB perp. BD, AB perp. BE, 3. Definition of line perpendicular to plane
CD perp. BD, CD perp. DE
4. BD = BD 4. Reflexive Property of Congruence
5. Triangle ABD = Triangle EDB 5. SAS Congruence Theorem [steps 2,3,4]
6. AD = BE, Angle ABD = EDB 6. CPCTC
7. AE = AE 7, Reflexive Property of Congruence
8. Triangle ABE = Triangle EDA 8. SSS Congruence Theorem [steps 2,6,7]
9. Angle ABE = Angle EDA 9. CPCTC
10. DE perp. DB, DE perp. DA 10. Definition of perpendicular lines [steps 6,9]
11. Lines BD, DA, DC coplanar 11. Proposition 5 from Friday (DE perp. to all, steps 2,10)
12. Lines BD, AB, DC coplanar 12. Point-Line-Plane, part e (points A, B)
13. AB | | CD 13. Two Perpendiculars Theorem [step 2]
Some steps look strange here, such as Step 12. Here, Euclid uses the "proposition" (actually a postulate) that all triangles lie in a plane, so the plane containing two sides of Triangle ABD (namely BD and DA) must contain the third (AB). Here, we use our Point-Line-Plane Postulate part e. This is because Step 11 establishes that a single plane (not plane P, by the way -- we can call it plane Q if we want) contains the lines BD, DA, and DC -- that is, points A and B lie in plane Q. Thus by part e, the entire line AB must also lie in plane Q.
The reason the proof is so convoluted goes back to the original Two Perpendiculars Theorem:
If two coplanar lines l and m are each perpendicular to the same line, then they are parallel to each other. [emphasis mine]
Notice that key word coplanar. Most of the time we use the Two Perpendiculars Theorem, we take it for granted that the lines in question are coplanar. But in this proof, the bulk of the proof is just to establish that AB and CD are coplanar! After all, the two perpendicular statements (AB perp. BD, CD perp. BD) were established all the way back in Step 2, so we could skip directly from Step 2 to Step 13 if we didn't have to prove that the lines are coplanar.
On the other hand, it is truly necessary to prove that all three lines (the two lines and the transversal) are perpendicular. Otherwise, we could prove, for example, that DB | | DA following Step 10, which is absurd. But since three lines (AB, CD, and the transversal BD) were proved coplanar in Step 12 (they all lie in plane Q), we complete the proof that AB | | CD.
David Joyce points out that Euclid omitted a step here. It could be the case that the two lines meet the plane at the same point (i.e., B and D are the same point). But Euclid proves that this is impossible in a subsequent proposition.
Here are the worksheets:
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