Lesson 12-7: Can There Be Giants? (Day 127)

Today I subbed in a high school math class. It's in my old district, and so it really is Day 127 at this high school. And since it's a math class, there will be "A Day in the Life" once again.

By the way, this school is out early on Mondays. Other high schools in this district and my new district have Late Start days on Mondays.

7:00 -- This teacher does have a zero period (and it really is called "zero period" here). The early class is AP Calculus BC.

This class is working on parametric equations. I try as hard as I can to answer questions on the board, since I did once earn a "5" on the BC test. I still remember that the derivative dy/dx of a parametric equation is (dy/dt) / (dx/dt), but I couldn't recall how to find the second derivative until I must look it up online. I should have known that it ultimately goes back to the Chain Rule -- we're trying to find the second derivative d/dx (dy/dx) where dy/dx is a function of t, so the Chain Rule tells us that it must be d/dt (dy/dx) (the t-derivative of the first derivative that we already found) over dx/dt (which we already found in the process of finding dy/dx).

(By the way, if we're thinking back to the Michael Starbird lectures, this is related to his Lecture 20, which I described in my February 3rd post. But Starbird really doesn't discuss how to find dy/dx of a parametric function where x and y are given in terms of t.)

Believe it or not, there are five juniors in this class. Any seniors who took Calc AB last year and this Calc BC class this year are already working at Bruce William Smith level. Thus the juniors in this class are beyond Bruce William Smith. I ask one of the juniors about his plans for math next year. He replies that he might take AP Stats. Recall that according to some traditionalists, we can't fully understand Stats until we've already had Calculus anyway.

Naturally there are no behavior problems in this advanced class.

7:55 -- Zero period leaves. First period is the teacher's conference period.

8:50 -- Second period arrives. This is the first of three Algebra I classes. And this class is only for freshmen who have failed the first semester of Algebra I.

The class is learning about graphing quadratic functions. The students are taught how to identity the parabola direction, y-intercept and vertex.

There's actually a song for today that's been in my mind for a while. Since this is the unit on functions of degree two, Quadratic Weasel immediately suggests itself. But I've been meaning to add a sort of bridge (over the "a penny for a spool of thread" part) that gives some other properties of parabolas that might be useful during this unit.

And so here is the resulting song:

Quadratic Weasel

Main Verse:
x equals negative b,
Plus or minus the square root,
Of b squared minus 4ac,
All over 2a.

Bridge:
If a's negative the curve opens down,
It's up for positive a.
Vertex x is negative b,
All over 2a.

Notice that the last line of both the verse and bridge are identical -- and the tune is playing the same notes (G-D-F-E-C, assuming the key of C major). This highlights the fact that the Quadratic Formula and Vertex Formula are similar. Indeed, in her notes, the regular teacher refers to the Vertex Formula as a "hidden" part of the Quadratic Formula.

Naturally for a freshman class, some students are talkative and disturb each other. I do issue out several warnings, though I don't actually write any names on my bad list.

Meanwhile, one guy spots an error in the regular teacher's notes -- a ball is thrown with an initial vertical speed of 45 ft./sec., but the coefficient b in the equation is 34, not 45 -- and then the teacher plugs in b = 48 to find the maximum height!

9:40 -- Second period leaves for snack.

10:00 -- Third period arrives. This is a Statistics class, though not AP Stats. Like zero period, most of these students are seniors, with a few juniors thrown in.

The class is learning about p-values and rejecting the null hypothesis. I never took Stats in high school, though I did take one college course at UCLA. Unfortunately, most of that class isn't part of my bicycle of things to remember.

And so I'm learning as I go along. I read the relevant chapter of the Stats text. I see a picture of a comic that I immediately recognize as coming from XKCD:

https://xkcd.com/882/

XKCD describes itself as "a webcomic of romance, sarcasm, math, and language." Thus many math ideas often show up in this comic. In this case, it describes the fallacy of p-hacking -- repeating an experiment until the desired results show up (in this case, a putative link between eating green jellybeans and getting acne).

By the way, this also reminds me of a scene from Square One TV, specifically Mathnet. A magazine article shows a study that links acne to eating, not green jellybeans, but berries (a specific fictional type called "chuckberries"). The study is flawed because the control group is mostly middle-aged while the berry eaters are mostly teenagers (who are likely to get acne anyway).

10:50 -- Third period leaves and fourth period arrives. This is the second of three Algebra I classes and the first of two with Grades 10-12 students who failed Algebra I as freshmen.

But this class is for the most part working in the same unit as the second period freshmen -- graphing quadratic functions. The notes and assignment worksheets are slightly different in the two classes.

11:40 -- Fourth period leaves for lunch. Fifth period is another off period. This makes up for the fact that the teacher had a zero period, although most zero period teachers prefer to take sixth period off rather than fifth. (Notice that on odd block days, the teacher has four hours off and has only the advanced classes, but pays for it by teaching eight straight hours on even days, including all of the Algebra I classes.)

1:10 -- Sixth period arrives. This is the last Algebra I class. This class has an aide, suggesting that some students here are special ed.

2:00 -- Sixth period leaves, thus ending my day.

Today is Nineday:

Resolution #9: We attend every single second of class.

I do watch out for excessive tardies and restroom breaks, especially on a Monday when all the classes are shorter (so there should be less need to go to the restroom than on a block day).

But there's a problem with the way that I enforce this as well as other rules. Recall that I wrote these rules to avoid arguments -- they reminds me what I should say in various situations.

Last week, several students arrive late or ask for restroom passes right after snack. But I didn't say "We attend every single second of class," because it wasn't Nineday. Instead, I got upset and ended up arguing -- and I stopped the arguments only by singing.

This week, today's Nineday, so I remind students of the importance of attendance. But in fourth period, one guy refuses to sit in his assigned seat, and another throws paper airplanes. Yet I'm still worrying about tardies and restroom passes. Instead of dealing with the seat and paper airplanes, I ended up arguing.

And the guy in the wrong seat replies by saying "I'll do whatever you want in order for you to sing -- except return to my seat." He never returns, and so I don't sing Quadratic Weasel (until the final minute before lunch, just so they can hear it once).

This is my problem with having focus rules. Just because it's not Nineday, it doesn't mean that I should argue when someone tries to miss class time -- and conversely, just because it is Nineday, it doesn't mean that I should argue when someone breaks another rule. The resolutions tell me what to do and say in a variety of situations, regardless of what day of the week it is.

It's good that I'm focusing on the different resolutions in order to remember them. The next level up is for me to keep the nine non-focus rules in mind whenever I react to student behavior.

Sixth period misbehaves, but it goes more smoothly thanks to the aide's presence. I remind the students in advance to go to the correct seat. But somehow, I end up missing that one girl isn't in her assigned seat -- until I randomly call her name to identify the value of x for the vertex.

Ironically, the students who answer the vertex question -- perhaps the toughest question that I ask any student to answer today -- are the ones in the wrong seat in each class! The fourth period guy remains on the bad list but with a mitigating note about answering the vertex question, while the sixth period girl is on the good list since I'd never written her name for the bad list (as she'd managed to sit in the wrong seat without getting caught for so long).

By the way, speaking of tardies, there are plenty of students late to zero period today. But this is the one day of the year when tardies are at least semi-justified -- the day the clocks spring forward.

I welcome you to my biannual Daylight Saving Time post. As usual, this post is all about the progress of California towards making Year-Round DST -- Proposition 7 -- a reality.

There are still two more things that must happen before Year-Round DST can begin. The state legislature must approve the time change, and then so must Congress. So far, a bill has passed unanimously the State Assembly. But it has stalled in the State Senate. Here is a link to the bill, first proposed by Assemblyman Kansen Chu (the author of Prop 7):

https://leginfo.legislature.ca.gov/faces/billTextClient.xhtml?bill_id=201920200AB7

Recall that Florida was the first state whose legislature approved Year-Round DST. A senator and a congressman from the Sunshine State introduced bills to Congress as well. If the bill passes, the president has indicated that he'd sign the bill.

Once again, here's my own opinion of this issue -- I have no problem with the biannual clock change, but of the two clocks, I prefer DST to Standard Time. Thus I promised on the blog that if there was the opportunity to vote for Year-Round Standard Time I'd oppose it, but if it was for Year-Round DST then I'd vote for it. Originally, Chu supported Year-Round Standard Time, but later he changed it to Year-Round DST instead. Since this is the clock I said I preferred, I kept my promise and voted for the proposition.

My favorite plan for a single clock year-round is the Sheila Danzig plan. On this plan, some states would have Year-Round DST and others would have Year-Round Standard Time. This would reduce the number of time zones in the Lower 48 to just two, separated by two hours.

Unfortunately, Florida -- Danzig's home state -- is headed towards Year-Round DST, even though her plan recommends Year-Round Standard Time for her home state. The Panhandle, which is currently on Central Time, would keep its clocks forward while the rest of the state would keep them back. The proposed bill instead places the whole state, including the Panhandle, on Eastern DST. This might lead to especially late winter sunrises in the Panhandle.

I assume that the rapid progression towards a single year-round clock is being driven by the desire to have DST year-round, with more sun in the evenings. Notice that any state could choose to have Year-Round Standard Time even without the approval of Congress, but few (only Arizona and Hawaii) actually do so. But it's the allure of Year-Round DST that's driving the current legislation, with at least half the states proposing Year-Round DST bills.

If Year-Round DST passes nationally, I assume some states might still prefer Standard Time. For example, Arizona already has a single year-round clock, and it will probably keep it. The net result will be that Arizona would essentially now be in the Pacific Time Zone -- the same clock at the West Coast year-round. Certain other states on the west side of their respective time zones might also keep Year-Round Standard Time.

An interesting case is in the Northeast. Massachusetts is considering Year-Round DST. If MA chooses DST, then so should Connecticut (since suburbs of Boston are in that state). If CT chooses DST, then so should New York (since suburbs of NYC are in CT). If NY chooses DST, then so should New Jersey (since suburbs of NYC are in NJ). If NJ chooses DST, then so should Pennsylvania (since suburbs of Philadelphia are in NJ). But Pennsylvania is far enough west in its time zone that Year-Round Standard Time is preferred in Pittsburgh, not Year-Round DST -- and in fact, there is currently a Year-Round Standard Time bill proposed in the PA legislature.

It's actually more important to keep metro areas, not states, in the same time zone. If Pittsburgh prefers Year-Round Standard Time and Boston prefers Year-Round DST, then at some point a state must be divided in order to keep metro areas together -- either PA (between Pittsburgh and Philly), NJ (between Philly and NYC), or CT (between NYC and Boston). Of course, the entire nation could just adopt the Danzig plan, which keeps the eastern half of the country in one large time zone.

But once again, the Danzig plan isn't the current proposal in Danzig's home state, FL. Even though Danzig herself would prefer Year-Round Standard Time to Year-Round DST in her home state, I'm sure that she would rather have Year-Round DST than the current biannual clock change.

Meanwhile, our California Senator, Kamala Harris, has introduced a bill in the Senate. No, it's not a Year-Round DST bill, but a proposal to keep schools open from 8AM to 6PM:

https://www.cnn.com/2019/11/07/politics/kamala-harris-extend-school-day-bill-trnd/index.html

(For non-Californians who recognize the name Harris, yes, she was also a presidential candidate. But she dropped out before any states began voting. By the time I visited the "I Side With" website, she was no longer a candidate, and thus she didn't appear on any of my Super Tuesday lists.)

Hmm -- open schools until 6PM, you say. Well, in the winter, it's usually dark by 6PM, and we wouldn't want kids out at school after dark -- unless, of course, there's Forward Time. That's right, the Harris bill is compatible with Year-Round DST. (Ironically, the Senator proposed the bill just after the fall back change to standard time, when it began to get dark early.)

I haven't heard much about the Family Friendly Schools Act since its proposal. But at any rate, I favor the bill -- since its passage increases the likelihood that Forward Time might pass. (Of course, no teacher is in favor of the bill if it means that we teachers are the ones who must staff the schools these extra hours.) Likewise, I also favor the movement to make the tardy bells at schools ring later (at least no earlier than 8AM, the start time mentioned by Harris), since this is also compatible with Year-Round DST.

Actually, Senator Marco Rubio (from Florida -- the other state that approved Forward Time) has proposed an actual Year-Round DST bill.

As for CA and Prop 7, we didn't mind changing our clocks forward to DST this weekend because DST is what we want to be year-round. The only question is, will the Assembly and Congress cooperate by November 1st so that we won't have to change the clocks back to Standard Time?

Lesson 12-7 of the U of Chicago text is called "Can There Be Giants?" This is one of those "fun lessons" that we can cover if there's time, but in the past we bypassed it to get to 12-8 and the all-important SSS Similarity.

This is what I wrote last year about today's lesson:

As I wrote above, Lesson 12-7 naturally lends itself to an activity. The whole idea behind it is that while dilations preserve shape, they don't preserve stability. This is because of the Fundamental Theorem of Similarity -- a dilation of scale factor k changes lengths by a factor of k, areas by a factor of k^2, and volumes by a factor of k^3. Weight varies as the volume, or k^3, while strength varies only as the area (as in surface or cross-sectional area), or k^2. Therefore, the answer to the question in the title of the lesson is no, there can't be giants because their k^2 strength, couldn't be strong enough to carry their own k^3 weight.

Notice that the monomial worksheet from today's lesson somewhat fits this lesson. Some of the problems ask the students to find the area of a square or volume of a cube -- with a particular monomial as one side, so that the students can practice squaring and cubing monomials.

Today is not supposed to be an activity day. But it's almost impossible to give this lesson without making it an activity. Indeed, it's just like Lesson 9-8 on the Four Color Theorem. We just happened to luck out that Lesson 9-8 landed on an activity Friday, but we aren't so fortunate with today's Lesson 12-7.

By the way, here's one of my favorite questions to ask about this lesson. Today's lesson provides an indirect proof that giants -- Brobdingnagians -- don't exist.

Prove: Brobdingnagians don't exist.

Indirect Proof:
Assume that Brobdingnagians exist. In each of the three dimensions, a Brobdingnagian is 12 times as large as a human. So each bone of a Brobdingnagian would have to carry 12 times as much weight as a human bone. Thus a Brobdingnagian standing is like a human carrying 12 times its own weight. But no human can carry 12 times its own weight, a contradiction. Therefore Brobdingnagians can't possibly exist. QED

But there's a problem with this proof -- it can be used to prove that humans don't exist:

Prove: Humans don't exist.

Indirect Proof:
Assume that humans exist. In each of the three dimensions, a human is 12 times as large as a Lilliputian. So each bone of a human would have to carry 12 times as much weight as a Lilliputian bone. Thus a human standing is like a Lilliputian carrying 12 times its own weight. But no Lilliputian can carry 12 times its own weight, a contradiction. Therefore humans can't possibly exist. QED

The proof is clearly invalid, since humans do exist. My question is, which step is invalid?

Answer: "But no Lilliputian can carry 12 times its own weight, a contradiction."

This step is invalid, because in the real world, Lilliputians can carry 12 times their own weight -- indeed, much more than 12 times their weight. (Real-world Lilliputians are often called "ants.")

Here are the worksheets that I've created for this lesson: