Today I subbed in a high school math class. It's in my first OC district, and it's my second visit to this classroom -- I describe this class in my February 25th post. Since it's a math class, I will definitely do "A Day in the Life" today.
9:00 -- Notice that February 25th was a Thursday. While both Thursday and Friday in this district are Cohort B, Thursday is odd periods while Friday is even. So we begin with second period. This is her only Algebra II class -- I didn't see any Algebra II students last month.
This class is beginning a unit on exponential functions. The students are to watch a video, take notes, and then take a photo and submit those notes on Canvas. The video introduces exponential growth and decay. Recall that under the Common Core, this is now taught in Algebra I. In fact, the song I sing for this period is the exponential verse of "U-N-I-T Rate!" -- just as I performed for Algebra I in February.
Of course, Algebra II will explore this topic much more deeply -- the speaker in the video makes it clear that the students will soon learn about logarithms, which is purely an Algebra II topic.
(Speaking of Algebra II, I'm still getting emails for Algebra II classes in my new district, even when I sub for them for just one day. In that district, these students are about to wrap up the Algebra II part of the course and move on to Trig. They'll take an Algebra II final -- more like a "midterm" -- next week!)
9:55 -- Second period leaves for snack break.
10:10 -- Fourth period arrives. This is a Geometry class.
These students are taking a Go Formative quiz on Lessons 10.1-10.3 -- and since Geometry is our favorite subject, of course I'm going to describe this quiz in more detail.
I assume that these lesson numbers refer to the Glencoe text, the curriculum used in the district. Chapter 10 of the Glencoe text is on circles. Thus it corresponds to Chapter 15 of the U of Chicago text. But the first two sections are on circumference and arc length -- these are taught in Lesson 8-8 of U of Chicago.
Thus it appears that these students had a lesson on pi close to Pi Day last week. Of all the Geometry texts I've seen, the Glencoe text is the best at placing a pi lesson in a chapter that's taught in March. The text has 13 chapters, and so Chapter 10 is reached around the end of the third/start of the fourth quarter.
(In fact, when I was shuffling around chapters during the early years of this blog, I was following the U of Chicago text but strongly influenced by the Glencoe text. I wanted to follow the Glencoe order of topics in order to get the pi lesson close to Pi Day, so I covered Lesson 8-8 on Pi Day and then jumped to Chapter 15, just as these students are doing the equivalent of now.)
For this class, I revert to "Benchmark Tests" -- a song that I often perform on quiz/test day, even if it's not a Benchmark test. The message of that song -- that the students should attempt every problem and that there's still time to grow -- applies to quizzes as well. Since circumference was on the test, I might have been justified in singing Pi Day songs -- but I don't, as the math holiday was five days ago, and I need to stop finding excuses to sing them.
I also tell the students that I'm considering writing a new math parody song. A year or two ago, the performer Post Malone created a song called "Circles" -- and with a title like that, it's begging to be made into a math parody. There's a good chance that someone else already wrote such a parody -- that's how much of a no-brainer it is. I hope to find or write a parody soon, while these students are still in the circles chapter. If I must write the song, I'll write about the non-pi parts of a circle (Chicago Chapter 15 and Glencoe Chapter 10 stuff), since I already have so many pi songs. (And yes, I still need to compose tunes for my other songs, but it's much easier to write parodies of songs that are already composed.)
11:05 -- Fourth period leaves and sixth period arrives. This is an Algebra I class.
These students also have a Go Formative quiz. It's on Lessons 8.1-8.3 -- assuming that this is the Glencoe Algebra I text, this is on polynomials.
Two students definitely struggle on the quiz. With only a few minutes left in class, one guy is still on the first few questions, while another has finished it quickly -- but when I ask him whether he believes he passed it, his answer is no.
And so I ask him to play a short game with me. The game is played with three pegs in eight holes -- and I tell him that for this game, his name would be Alice while my name is Bob...
You guessed it -- I mention this year's Putnam B2 problem. OK, so I've finally managed to let Pi Day go, but I apparently I still can't let the Putnam go. In reality, I was planning to mention the Putnam today anyway because I should have discussed it to the classes on February 25th -- that was the first math class I subbed for after the exam was given on the 20th. Unfortunately, at the time I still hadn't remembered to look up the Putnam problems.
I was hoping that there would be some extra time after today's quiz to discuss the Putnam, since the February 25th students finished their Chapter 7 Test that day with time to spare, while today's assessment is merely a quiz on three sections. But today's quiz is more challenging -- it goes up to multiplying polynomials (including, it appears, a binomial times a trinomial). The rest of Glencoe Chapter 8 is on factoring, and so multiplication is reached in 8.3. And that's certainly why this particular guy (and his friend) are having trouble with it.
Here's what I've found to be the best way to present the problem -- start with three pegs and eight holes (a case where Alice can win). Then add a fourth peg and show that Bob (the teacher) wins, and ask Alice (the student) to guess what combination of pegs and holes can produce an Alice win. At this point, reduce it to just one peg (where the winning strategy is obvious). Finally, try two pegs, and challenge Alice to see in which cases Bob's strategy works, and in which cases we can use Bob's strategy against him to produce an Alice win.
Once again, today's "Alice" quickly figures out that it has something to do with odds and evens. At the end, I tell him that he's able to answer a problem from the Putnam, the world's hardest math test -- and if he can figure out the Putnam, surely he can pass a mere Algebra I quiz.
The guy tells me that he might attend academic support someday -- perhaps next week when the regular teacher returns. I'm glad that my Putnam question motivates him -- and this is why I like to present Putnam problems, especially those like 2020 B2. It's much easier to say "let's play a game with me" to a student who's just failed a math test than it is to say "let's do a math problem with me."
12:05 -- Sixth period leaves for lunch.
1:00 -- Academic support begins. There's no need for me to discuss it here.
Today is Eightday on the Eleven Calendar:
Resolution #8: We follow procedures in the classroom.
Two guys keep on trying to talk during the Algebra I quiz -- yes, these are the same guys who struggle with it (which explains why they're talking). I tell them that I'd separate them if they talk on the quiz -- and shortly thereafter this leads to the Putnam game anyway.
Since this is a high school class, it's worth doing the Miller wager today. In second period, there are two in-person sophomores and nine juniors versus a score (20) of online students, so I break even. In fourth period, there are eight in-person underclassmen and one junior versus 22 online students, giving me a dozen dollars. In sixth period, there are ten in-person underclassmen and one junior versus 25 online students, giving me a profit of $13. Of course, the only reason I break even in Algebra II and make money in the other classes was my arbitrary decision to pay double for in-person juniors.
Once again, some students have signed up for four days of in-person rather than two-day hybrid. Notice that in Darren Miller's recent blogposts, he complains about the hybrid schedule that will be starting at his school very soon. So perhaps a more accurate Miller wager would have been to pay $1 for each student who opts in to hybrid and $2 for each student who opts in to all four days (instead of basing it on grade level), while still gaining $1 for each online student. Then Miller can win this bet if he can show that students prefer going to school more than two days per week (since his argument is more days in the classroom is better than hybrid).
But even if I simply double my stake (that is, pay $2 for each four-day underclassman and $4 for each four-day junior), it's still a profitable day for me. There are a total of nine four-day students -- three in each class. Of these nine, three are juniors. My winning margin for the day drops from $25 to $13, so I'm still in the black.
Let's finally get to today's U of Chicago lesson.
Lesson 13-2 of the U of Chicago text is called "Negations." In the modern Third Edition of the text, negations appear as part of Lesson 11-2 (which is the old Lessons 13-1 and 13-2 combined).
This is what I wrote two years ago about today's lesson. (That's right -- due to the pandemic, it's been two years since I've taught Chapters 13-15 on the blog, not counting the special "Shapelore" posts that I wrote during the pandemic.):
that is, he used the the Substitution Property of Equality from the hypothesis 1=2.
In today's lesson, the U of Chicago text introduces the symbol not-p for the negation of p. In other texts, the notation ~p is used, but I have no reason to deviate from the U of Chicago here.
Before leaving this site, let me point out that this [Metamath] site gives yet a third way of writing the "not" symbol used in negations:
http://us.metamath.org/mpegif/wn.html
Today I wish to post a pandemic-friendly activity. Here is a Desmos lesson I found -- created by Jennifer Abel, it's a negation/inverse/contrapositive card sort:
https://teacher.desmos.com/activitybuilder/custom/579735108bea723b4b91c4c6
I'll combine this with the single worksheet that I posted for this lesson two years ago.
END
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