Today is Day 105. It can be considered as the end of the fifth quaver, which we determine by noting that one-quarter of the way from Day 80 (the end of the first semester) to Day 180 is 105 -- and yes, at the school where I completed my long-term assignment, progress reports are being filled out.
Oh, and this is a good time to mention another calendar quirk. Both of my Orange County districts observe Lincoln's Birthday on the Friday before President's Day, so that there can be a four-day break in February.
(Four years ago at the old charter, by the way, students observed a five-day weekend, from Thursday to Monday. But the first two days had nothing to do with Lincoln's Birthday. Instead they were PD days -- in other words, we teachers had only a three-day weekend. President's Day itself is the only school-closing February holiday in the LAUSD.)
But what complicates matters is the hybrid schedule, where Monday is an online all-classes day and then each cohort meets two out of four days from Tuesday-Friday. When it's a Monday holiday, it's easy just to skip the online Monday. But when it's a Friday holiday, the schedule becomes unbalanced -- whichever cohort meets on Friday would have just one in-person day while the other cohort has two. As it turns out, different schools in my two districts are handling the holiday differently.
You might remember that my long-term middle school was closed as winter break started on a Friday -- and the solution was to treat the previous Monday as a Friday, with Cohort B attending periods 2, 4, 5 in person that day. And indeed, that school is doing the same thing this week, with Cohort B in person today (except that now all students attend all classes everyday).
But the high schools in my new district are doing something different from the middle schools. Yes, even periods meet today as if were a Friday, except that it's all online like a usual Monday. And each class is as short as it would be on a usual Monday despite only three periods meeting -- the rest of the day is declared to be "asynchronous" in order to add up to the state-mandated 240 minutes. Thus the week is completely unbalanced -- odd periods meet more minutes this week than even periods, and Cohort A is in-person twice while Cohort B is in-person only once this week.
Meanwhile, in my first OC district, the schools are simply pushing the whole week a day up. Thus today the students follow Tuesday's schedule, tomorrow Wednesday's schedule, and so on. Therefore Cohort A meets in-person Monday and Tuesday rather than the usual Tuesday/Wednesday, and first period meets Monday and Wednesday instead of the usual Tuesday/Thursday. This schedule is followed at both middle and high schools.
Today I subbed in a middle school special ed science class, in my first OC district. Ordinarily I don't sub Mondays during hybrid, since all students are online (and so a sick teacher works from home). But with Tuesday's students in the classroom today, my services are needed.
My last visit to this classroom was back in October 2019. Back then, she was a seventh grade teacher -- and the class was learning about the metric system during Metric Week (week of October 10th). This year, she has eighth grade classes (perhaps in order to "loop" with the special ed students -- they have the same teacher this year as last year).
And so these eighth graders are in the same unit as the Science 8 class I subbed a week ago (and by "week" I mean 11 days) -- the unit on forces and motion. This teacher uses Canvas and Edpuzzle -- I've seen these websites used so far in my new OC district but not my first district until today. But the Edpuzzle assignment contains the same Bill Nye video that was used in the previous Science 8 class back on January 28th.
In middle schools, periods 1, 2, 3 meet on the same day. This teacher has Science 8 first and second periods, followed by third period conference. Once again, aides take over the classes, and so there's no "A Day in the Life" today.
This is the week leading up to Valentine's Day, and so I plan on handing out holiday candy and pencils in all classes this week. And my song for today is Weird Al's "Polka Patterns" from Square One TV (a near contemporary of Bill Nye the Science Guy). I sang this song four years ago today at the old charter school, and I associate it with today since it was first posted to YouTube on February 8th, 2007.
Speaking of my old charter school, it's time for my usual comparison to the science that I should have taught at the old charter. February 8th, 2017 was a Wednesday, and so I did teach science to my eighth graders that day, but I was nowhere near where I should have been in the Illinois State text. I already mentioned in my January 28th post that if I'd followed the proper pace, I also would have been in the motion unit, perhaps a little farther along than today's class. The science project for February 8th, 2017 would have been related to the following standard:
The Illinois State text provides several projects in this lesson -- balancing a book on a table, dropping beads into a cup water, and estimating the weight of an apple/orange on other planets. (Once again, these are projects I should have done in 2017 -- we know that these are impossible in the pandemic era.)
Today is Fourday on the Eleven Calendar:
Resolution #4: We need to inflate the wheels of our bike.
This is my second straight Fourday in a Science 8 classroom. In both classes, I invoke this resolution by getting the students to solve a simple speed * time = distance problem. The problem for today is to find the distance traveled in five hours by an object moving 30 km/h. I guide the students to multiply this without a calculator, starting with 5 * 3 and then placing an extra zero.
As for "homage," the name Usain Bolt was mentioned again, this time to demonstrate how to calculate speed given distance and time. (I previously paid homage to Bolt during a P.E. class on the track.)
Lesson 10-5 of the U of Chicago text is called "Volumes of Prisms and Cylinders." In the modern Third Edition of the text, volumes of prisms and cylinders appear in Lesson 10-3.
This is what I wrote last year about today's lesson:
As I mentioned yesterday, we are moving on to Lesson 10-5 of the U of Chicago text, which is on volumes of prisms and cylinders. As we proceed with volume, let's look at what some of our other sources say about the teaching of volume. Dr. David Joyce writes about Chapter 6 of the Prentice-Hall text -- the counterpart of U of Chicago's Chapter 10:
There are a few things going on here. First of all, Joyce writes that surface areas and volumes should be treated after the "basics of solid geometry," but he doesn't explain what these "basics" are. It's possible that the U of Chicago's Chapter 9 is actually a great introduction to the basics of solid geometry -- after all, Chapter 9 begins by extending the Point-Line-Plane postulate to 3-D geometry, then all the terms like prism and pyramid are defined, and so on. Only afterward, in Chapter 10, does the U of Chicago find surface areas and volumes.
So I bet that Joyce would appreciate Chapter 9 of the U of Chicago text. Yet most texts, like the Prentice-Hall text, don't include anything like the U of Chicago's Chapter 9.
It's because most sets of standards -- including the Common Core Standards, and even most state standards pre-Common Core -- expect students to learn surface area and volume and essentially nothing else about solid geometry. Every single problem on the PARCC and SBAC exams that mentions 3-D figures involves either their surface area or volume. And so my duty is to focus on what the students are expected to learn and will be tested on, and that's surface area and volume.
Joyce writes that something called "Cavalieri's principle" is stated as a theorem but not proved, and it would be better if it were a postulate instead. Indeed, the U of Chicago text does exactly that:
Volume Postulate:
e. (Cavalieri's Principle) Let I and II be two solids included between parallel planes. If every plane P parallel to the given plane intersects I and II in sections with the same area, then
Volume(I) = Volume (II).
According to the text, Francesco Bonaventura Cavalieri was the 17th-century Italian mathematician who first realized the importance of this theorem. Cavalieri's Principle is mainly used in proofs -- as Joyce points out above, the volumes of prisms and cylinders is derived from the volume of a box using Cavalieri's Principle, and the U of Chicago also derives the volumes of oblique prisms and cylinders using Cavalieri.
The text likens Cavalieri's Principle to a stack of papers. If we have a stack of papers that fit in a box, then we can use the formula lwh to find its volume. But if we shift the stack of papers so that it forms an oblique prism, the volume doesn't change. This is Cavalieri's Principle.
Notice that we don't need Cavalieri's Principle if one is simply going to be handed the volume formulas without proving that they work. But of course, doing so doesn't satisfy Joyce. Indeed, the U of Chicago goes further than Prentice-Hall in using Cavalieri to derive volume formulas -- as Joyce points out, we can find the volume of some pyramids without advanced mathematics (Calculus).
But then the U of Chicago uses Cavalieri to extend this to all pyramids as well as cones. Finally -- and this is the grand achievement -- we can even use Cavalieri's Principle to derive the volume of a sphere, using the volumes of a cylinder and a cone! As we'll soon see, Joyce is wrong when he says that a limiting argument is the best that we can do to find the volume of a sphere. Cavalieri's Principle will provide us with an elegant derivation of the sphere volume formula.
Here are the Common Core Standards where Cavalieri's Principle is specifically mentioned
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.
(+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.
CCSS.MATH.CONTENT.HSG.GMD.A.3
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*
But not everyone is excited about the inclusion of Cavalieri's Principle in the standards. The following link is to the author Dr. Katharine Beals, a traditionalist opponent of Common Core:
[2021 update: The "Out in Left Field" blog is no longer active, and Beals has deleted all her posts. The only blog she regularly posts to is the Catherine and Katharine blog.
This sounds like yet another traditionalists' post. Recently, I didn't label this as traditionalists because Beals is no longer an active traditionalist. But our main traditionalist, Barry Garelick, has recently published a new book. Even though the publlication date was January 26th, both Garelick and Beals waited until after February 1st to blog about the new book, and so I missed knowing about the book in time for my last traditionalists' post. I'll wait to discuss Garelick's book in my next scheduled traditionalists' post on February 17th.]
Beals begins by showing a question from New York State -- recall that NY state has one of the most developed Common Core curricula in the country. This question involves Cavalieri's principle. Then after stating the same three Common Core Standards that I did above, Beals asks her readers the following six "extra credit questions":
1. Will a student who has never heard the phrase "Cavalieri's principle" know how to proceed on this problem?
2. Should a student who has never heard the phrase "Cavalieri's principle" end up with fewer points on this problem than one who has?
3. Should a student who explains without reference to "Cavalieri's principle" why the two volumes are equal get full credit for this problem?
4. Is it acceptable to argue that the volumes are equal because they contain the same number of equal-volume disks?
5. To what extent does knowledge of "object permanence," typically attained in infancy, suffice for grasping why the two stacks built from the same number of equal-sized building blocks have equal volume?
6. To what degree does this problem test knowledge of labels rather than mastery of concepts?
The question involves two stacks of 23 quarters -- one arranged as a cylinder, the other arranged not so neatly -- and asks the students to use Cavalieri to explain why they have the same volume.
Notice that in four of the "extra credit questions," Beals criticizes the use of the phrase "Cavalieri's principle," arguing that the label makes a simple problem needlessly complicated. But Cavalieri's Principle is no less a rigorous theorem than the Pythagorean Theorem. Imagine a test question which asks a student to use the Pythagorean Theorem to find the hypotenuse of a right triangle, and someone responding with these "extra credit questions":
1. Will a student who has never heard the phrase "Pythagorean Theorem" know how to proceed on this problem?
2. Should a student who has never heard the phrase "Pythagorean Theorem" end up with fewer points on this problem than one who has?
3. Should a student who explains without reference to "Pythagorean Theorem" what the hypotenuse is get full credit for this problem?
6. To what degree does this problem test knowledge of labels rather than mastery of concepts?
As for the other two "extra credit questions," yes, the volumes are equal because they contain the same number of equal-volume disks. We don't need Cavalieri's Principle to prove this, since the Additive Property -- part (d) of the Volume Postulate from yesterday's lesson -- tells us so.
The true power of Cavalieri's Principle is not when we divide a solid into finitely (in this case 23) many pieces, but only when we divide it into infinitely many pieces. In higher mathematics, we can't simply extract from finite cases to an infinite case without a rigorous theorem or postulate telling us that doing so is allowed. I doubt that the infant mentioned in "extra credit question 5" above is intuitive enough to apply object permanence to infinitely divided objects.
In fact, in the time since I wrote this, I've discussed the Banach-Tarski Paradox. That paradox tells us that we can divide a sphere into finitely many pieces and reassemble them to form two balls. I'd like to see someone try to apply an infant's intuition of object permanence to Banach-Tarski.
The Volume Postulate fails for Banach-Tarski because even though there are finitely many pieces, the pieces are non-measurable (i.e., they don't have a volume). The Volume Postulate fails for the oblique cylinder because we're dividing it into uncountably many flat pieces. In both cases we need something else to help us find the volume -- and in the latter case, that something is Cavalieri.
I wonder how Beals would have responded had the question been, "These two cylinders have the same radius and height, but one is oblique, the other right. The right cylinder has volume pi r^2 h. Use Cavalieri's Principle to explain why the oblique cylinder also has volume pi r^2 h." Nothing less rigorous than Cavalieri gives us a full proof of the oblique cylinder volume formula.
Furthermore, in another post a few weeks before this one, Beals tells the story of a math teacher -- the traditionalist Barry Garelick -- who would only allow those who successfully derive the Quadratic Formula to date his daughter.
I claim that deriving the sphere volume formula from Cavalieri's Principle in Geometry is just as elegant as deriving the Quadratic Formula from completing the square in Algebra I.
There's one more thing I wish to say about Geometry here. You're aware that I'm still following the math class from my long-term assignment (via the emails I get from Canvas). Now that Math 8 has completed Unit 4b on solving systems, we finally reach Unit 6, the first geometry strand unit.
While APEX does indeed reach the G strand after completing the EE and F strands, the G standards aren't taught in order. Last week the class studied the Pythagorean Theorem (G6 and G7). Today they learn the Distance Formula (G8), to be followed by volume (G9), including cylinder volume. And so I have another opportunity to match our U of Chicago lesson with a lesson in my long-term classroom.
The cylinder lesson begins on Wednesday with a worksheet posted to Canvas. That worksheet is not from Kuta, and it contains a copyright symbol, so I don't feel right posting it. It appears to come from an old textbook (and it's labeled "Chapter 10," which doesn't match APEX but ironically matches our U of Chicago text).
But I'm looking ahead to Thursday's assignment. It's labeled as a "coloring activity" but the project hasn't been posted to Canvas yet. Depending on what it is, I might consider posting it here on the blog that same day, even though we will have moved on to sphere volume by then.
By the way, notice that the Math 8 standards mention volumes of cylinders, cones, and spheres -- the volumes of other shapes having been taught in Math 7. While pre-Core texts would reteach these other volumes before starting cylinders, the Illinois State text (and most likely APEX) dive right into volumes of cylinders. Tomorrow night, the regular teacher will at least have them review circumference and area of a circle before starting cylinders.
This underscores the fact that at the old charter school, I should have reviewed something with the eighth graders instead of naively following the text and starting cylinders.
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