Today I subbed in a high school Precalculus class. It is in my first OC district. Since it's a math class, I'm definitely doing "A Day in the Life" today.
9:00 -- Today even periods meet. But second period just happens to be the teacher's conference.
Before today, I wasn't quite sure how I wanted to celebrate Pi Day this year. With the math holiday on a Sunday this year, we'd ordinarily observe it on the last school day before -- but "last school day" has a different meaning during the pandemic hybrid schedule. For the in-person students in Cohort A, today really is the last time they step on campus until the sixteenth, when it is already past Pi Day.
If this teacher's day didn't start with conference period, I likely buy no Pi Day food today. But since I had the luxury of time with the prep period, I go out and purchase snacks -- but which snacks? The traditional pie that is cut with a knife and shared isn't very pandemic friendly. Instead, I got some individual pies from 7-Eleven for $1 each.
10:10 -- Fourth period arrives. This is the first of two Precalc classes.
This class is studying the six trig functions, in the form y = a sin(bx + c) + d (where sin can be replaced with any of the other functions). These functions are in radians, and thus pi and its multiples are relevant just in time for Pi Day.
These students are preparing for a quiz on the next day this class meets, which is Friday (online for Cohort A and in-person for Cohort B). I hand out four pies today, one to each students who describes a graph or answers a Practice Test question on the board. I also hand out a pie to the student who helps me demonstrate a Putnam problem today --
-- oops, that right! The Putnam exam was delayed from December to February. But on the blog, I just blew right past the date without discussing our usual Putnam problem.
Let's fix this right now. I will go back and edit my February 23rd post and include the Putnam discussion that I usually give on the Tuesday after the exam. (I could just put it in today's post, but in future years I'll want to go back and reread my old posts for Putnam problems. It will be easier for me to find if it's in my post from Tuesday after the exam -- especially if I see the test was February 20th.)
OK, here is my edited February 23rd link:
https://commoncoregeometry.blogspot.com/2021/02/lesson-11-4-midpoint-formula-day-114.html
In this class, I discuss Problem B2 with the students. I find a chessboard in the room, and so I use pawns to represent pegs and chess squares to represent holes. I choose one guy to play the role of "Alice" while I play the role of "Bob." The "Alice" guy figures out quickly that he wins with odd number of pawns (k), but the case where k is even and n is odd is a little tricky.
There is a little time left after the Putnam problem, and so I sing my only song of the day. It is "Sing a Song of Pi Day" (a parody of "Sixpence" -- a song that also happens to mention a pie). I originally found the lyrics at the following website:
https://www.deviantart.com/chibimama/art/Pi-Songs-112578820
Sing a Song of Pi Day
(sung to the tune of "Sing a Song of Sixpence")
Sing a Song of Pi Day
Come celebrate with me
A holiday from math class
Relieves monotony
We've "rounded up" refreshments
An elegant cuisine
The party is approximately March fourteen!
If Pi Day's on the weekend
And we are not at school
We'll celebrate on Friday
A difference miniscule
We'll call it Pi Day Eve and
The party's just the same
With songs and snacks (all circular)
You'll be so glad you came!
11:05 -- Fourth period leaves and sixth period arrives. This is an honors/IB class.
Although this is also a Precalc class, these students are learning something different. Instead of trig, they are learning about conic sections, specifically hyperbolas. Since they just finished ellipses, I mentioned the ellipse area formula A = pi ab in order to draw a connection back to pi. Once again, I hand out four pies to students who help take notes on the board.
Since these students are more advanced, I do Problem A1 with them instead of B2. Unfortunately, I don't get to this problem until there are only about ten minutes left, and so I rush the problem. I truly believe I could have given the students a better feel for this problem if I had slowed down, but hyperbolas are the most difficult of the conic sections, and I needed almost all theperiod to give the hyperbola lesson justice. And there definitely is no time for me to sing anything.
12:05 -- Sixth period leaves for lunch. Ordinarily after lunch is academic support, but it appears that the regular teacher doesn't intend for me to do academic support today, seeing as she scheduled the Zoom meetings to be open to me for fourth and sixth period but closed for academic support. And so my day ends right here.
And this fits perfectly today. The point of showing the students Putnam problems is to demonstrate what exactly excellence in math looks like. Suddenly, their upcoming tests on trig and conic sections don't look as difficult when compared to the Putnam.
As for the Miller wager, I break even in fourth period with a dozen students (mostly seniors, a few juniors, each worth $2) in-person and two dozen online. But I lose money (that is, Miller wins) in sixth period since there are 18 in-person (mostly juniors, since it's honors) and 13 online.
Of the seven in-person students I see today in fourth period, only one is a girl, the rest guys. But of the ten in-person students in sixth period, only one is a guy, the rest girls. It could be that more girls sign up for the IB program, or it could be just that more IB girls on Cohort A opted into in-person learning while fewer non-IB girls on Cohort A did -- that's simply how the classes happened to be divided.
This is what I wrote last year about today's lesson:
Lesson 12-5 of the U of Chicago text is about similar figures. There is not much for us to change about this lesson from last year, except for the definition of similar itself. Recall the two definitions:
- Two polygons are similar if corresponding angles are congruent and sides are proportional.
- Two figures are similar if there exists a similarity transformation mapping one to the other.
The first definition is pre-Core, while the second is Common Core. The U of Chicago text, of course, uses the second definition. But that PARCC question I mentioned last week must be using the first definition, since it requires that we know what similar means before we can define dilations and ultimately similarity transformations.
In the U of Chicago text, the Similar Figures Theorem is essentially the statement that the second definition implies the first definition. We would actually need to prove the converse -- that the first definition (at least for polygons) implies the second. But the proof isn't that much different -- suppose we have two figures F and G satisfying the first definition of similar -- that is, corresponding angles are congruent and sides are proportional, say with scale factor k. Then use any dilation with scale factor k to map F to its image F'. Now F' and G have all corresponding parts congruent, so there must exist some isometry mapping F' to G. Therefore the composite of a dilation and an isometry -- that is, a similarity transformation -- maps F to G. QED
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