Today I subbed in a high school English class. It is in my first OC district. Since it's a high school class that's not math, there's no need for "A Day in the Life" today.
Odd classes meet today, including first period, which really means zero period. This is a senior class that is reading Jon Krakauer's Into the Wild, a book about a young man who died in the Alaska wilderness back in the 1990's. This is followed by third period, a junior class that is also reading The Great Gatsby. Fifth period is the teacher's conference, and teachers with a first period don't usually have a seventh. Thus my teaching day is over by 10:00 (but there is still academic support in the afternoon).
Both the juniors and seniors have a writing assignment where they must take an object from the book that they're reading (or a creature in the case of Into the Wild) and create an "instruction manual" -- they should describe how to use it and keep it in working order (or alive).
Today is Nineday on the Eleven Calendar:
Resolution #9: We pay attention to math as long as possible.
This doesn't really come up today. One thing I do want to say about timing is that I was confused as to whether this teacher had a first (zero) period today, which resulted in my setting up the Zoom late, nearly halfway into the period. Some students quickly notice the message I put in Google Classroom for today's Zoom link, while others don't see it until class is almost over.
One girl doesn't figure out the link until after first period is over -- later on, she claims in a Google Classroom message that since I have the same first name as one of her other teachers, she thought that the link was for that teacher and not first period. Since so many people (including myself) have had trouble with Zoom, I'm inclined to believe her.
Since this is a high school class, I want to perform the Darren Miller wager. I mentioned this in last Wednesday's post -- some traditionalists like Miller want high schools to reopen for in-person learning, yet I've seen that when schools reopen, high school students opt out. So for this wager, I lose $2 for each junior and senior who opts in to in-person learning and gain $1 for each student who opts out.
Here are the stats for today's classes -- in first period, ten seniors opt in and two dozen opt out. This gives me a profit of $4. (Of those ten, four are in Cohort A, and only two of those four are present.) In second period, a dozen juniors opt in and 22 opt out, for a loss of $2. But if I were wagering with the actual traditionalist Miller, he'd point out that the junior class has one TA (most likely a senior) who attends in person, and so he should get an extra $2 for her. So I win $4 for first period and he wins $4 for third, making it a break-even day.
Notice that one of the first period seniors is listed as an in-person student but attends online today. But (as Miller) would point out, the wager is based on the roster, which lists him as in-person. Thus I lose money for that student.
Of course, we break even only because I raised the stakes to $2 for in-person juniors and seniors. If I'd kept it at 1:1 then I win easily. The point I was making was that only about a third of the upperclassmen attends in-person, so it's not as critical to reopen high schools as quickly as Miller insists.
The song for today is "Packet Rap." This is the second time I've performed it this month, despite there being no printed packets during the pandemic. I just like the song.
Lesson 12-4 of the U of Chicago text is on proportions. I wrote a lot about comparing the U of Chicago approach to those of other theorists, and I retain some of that discussion in today's post.
Let's start with Hung-Hsi Wu. His "Fundamental Theorem of Similarity" actually consists of some of the properties of dilations that appeared in yesterday's Lesson 12-3. If the image of PQ under a dilation with scale factor k is P'Q', then P'Q' | | PQ and P'Q' = k PQ.
Wu proves this in cases based on what sort of number the scale factor k is. For natural number k, Wu proves it using induction on k. This initial case, k = 2, is based on a special version of the Midpoint Connector Theorem of Lesson 11-5. The inductive case from k to k + 1 involves repeating the Midpoint Connector Theorem argument over and over.
For rational number k, if k = 1/q (q natural number), then Wu notes that a dilation of scale factor 1/q is the inverse of a dilation of scale factor q. And if k = p/q (p, q natural numbers), then Wu notes that a dilation of scale factor p/q is the composite of two dilations, with scale factors p and 1/q.
All that's left is to extend the argument to irrational k. Wu hand-waves over this by using what he calls the "Fundamental Assumption of School Mathematics" -- many theorems of pre-college math that apply to all rational numbers also apply to all real numbers. (This assumption also appears in Algebra II when defining what it means to raise a number to an irrational power.)
Even though Wu gives this proof, it's not the sort of proof we expect high school students to figure out easily. In the years since I posted it, I've regretted it. But every year since then, when I return to Lesson 12-6, I keep the Wu proof and change other parts of the worksheet!
As I wrote in the comment above, the EngageNY curriculum is based on the Wu proofs. But there's no way in the world EngageNY would use Wu's "Fundamental Theorem of Similarity" proof.
Instead, they replace it with a simpler proof based on the area of a triangle. I've mentioned the idea before how in proofs, area and similarity are often interchangeable -- this is why the Pythagorean Theorem has both area and similarity proofs. EngageNY's area proof removes the need for Wu's induction on k and subsequent extension to rational and real values of k.
But there's one problem here -- similarity is a "Module 2" topic, but area is a "Module 3" topic. This is the naive order suggested by the Common Core Standards -- since they mention similarity before area, all similarity lessons must appear before any area lessons. Once again, EngageNY justifies this by having students recall the triangle area formula from eighth grade (or earlier).
Notice that in the U of Chicago text, area (Chapter 8) appears before similarity (Chapter 12). Thus the U of Chicago could validly follow the EngageNY sequence of proofs -- except that it doesn't.
Many traditionalists attack the Wu/EngageNY plan from not following Euclid's geometry. But ironically, Wu/EngageNY actually follow Euclid more closely than traditional texts do! His Book VI, which teaches similarity, begins with Proposition 1 (ratios between areas of triangles). This is followed by Proposition 2 (Side-Splitter Theorem), from which both Wu/EngageNY derive their first similarity theorem (Proposition 4, AA~).
Of course, there's also the idea of deriving the similarity theorems "classically" -- that is, by assuming AA~ as a postulate -- and then ultimately deriving the properties of dilations. This is the method used by the old PARCC test, which should be irrelevant since California isn't a PARCC state, and backlash against the Core has led to most PARCC states dropping that test. But I still end up posting old PARCC questions from time to time -- including the Lesson 12-3 worksheet from yesterday.
Here is today's worksheet.
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