Lesson 15-2: Regular Polygons and Schedules (Day 152)

Today I subbed in a high school English class. It's in my new district. Since it's a high school class that isn't math (and two of the classes have aides to boot), there's no "A Day in the Life" today.

One of the classes is seniors and the other two are juniors. The senior class is officially called ERWC -- I explained what exactly this class is, two years ago on the blog:

It's been some time since I've subbed in a Grade 12 ERWC class. It reminds me of a possible math equivalent to this class -- a senior "Algebra III" class proposed by Governor Brown. Well, that was five years ago, and nothing ever came out of that idea -- and now Brown is no longer governor. So that's the end of that.

In this class, the seniors are reading The Distance Between Us. This isn't a novel -- instead, it's the memoirs of Reyna Grande, a Mexican immigrant. It's all about the author's childhood in Mexico -- we finish today with Chapter 10, when Grande's mother (who already crossed the border) finally returns to visit Grande and her older siblings. The author is a few years older than I am, so the book obviously doesn't take place that long ago.

Meanwhile, the juniors are reading The Great Gatsby. This class isn't quite as far as the special ed class (in my first OC district). Today we read Chapter 7 -- Mrs. Wilson is killed by Gatsby's car. (In the other class, they've already reached Gatsby floating in a pool.)

As for the tutorial songs today, I choose different ones for the seniors and juniors. The seniors get "Sign of the Times," since the book that they're reading comes is set in Mexico -- oh, and because Square One TV aired in the 1980's, the decade where the book takes place. (That song is all about the letter "equis" or X, and not the tilde that's featured on today's Google Doodle. Today is Spanish Language Day, so either "equis" or "n-tilde" -- which I don't write in ASCII -- are worth celebrating today.)

For the juniors, I revert to "U-N-I-T Rate! Rate! Rate!" The regular teacher apparently earned her degree from UCLA, and I can't resist singing my fight song parody when I see Bruin flags on the wall. Based on the dates I see on her diploma -- 2003 -- the two of us were in Westwood at the same time.

Today is Tenday on the Eleven Calendar:

Resolution #10: We are not truly done until we have achieved excellence.

This is a tricky one today. Ordinarily, I would tell the students that they should finish the readings and the associated questions as soon as possible. But the assignments aren't due until Monday -- and there's a very specific reason why the due date isn't until then.

In my Wednesday post, I explained how all of my districts are expanding the number of days per week that students are in-person -- from one to two in LA County and from two to four in Orange County. As it turns out, many districts are scheduling an "asynchronous" day on the day of transition.

Yes, that's a weird term, "asynchronous," that can mean whatever the school wants it to mean. For example, from October to December, at my long-term middle school, the two weekly days that a cohort attended neither online nor in-person were called "asynchronous" -- even after that cohort had to log in to Zoom/Google on those days starting in January, some schools continued to call them "asynchronous" and assigned different work from the in-person students.

Now, schools are using that word to mean a day of transition -- students log in to just one class in the  morning, and then they must do work for all of their classes. Teachers must take attendance in all classes based only on the work completed on that off day.

Many districts are assigning asynchronous days when transitioning to more in-person days -- the reason is so that teachers can spend the day training for having more kids in the classroom and how to handle the new three-foot distancing requirements. (For example, recall the district that I attended as a young elementary student -- the one that reopened on Maundy Thursday despite being the day before spring break -- had three asynchronous days during that week -- Monday-Wednesday.) Here in my new district, the asynchronous day is Monday.

And so this is why the assignment isn't due until Monday -- if the students rush and finish it today, they'll have nothing to do on the asynchronous day. Indeed, the link to upload the assignment on Canvas won't even appear until Monday itself, since that's the link that will be checked for asynchronous attendance.

Thus the tenth resolution -- we're not done until we've achieved excellence -- makes sense on Monday when completing the asynchronous work, but not today.

This is what I wrote two years ago about today's lesson:

Lesson 15-2 of the U of Chicago text is called "Regular Polygons and Schedules." Just like Lesson 13-3 from last month, 15-2 naturally lends itself to an activity, so this will be considered the activity for this week.

The text defines a round-robin tournament as a tournament in which each competitor (or team) plays each other competitor exactly once. True round-robins are rare at the highest levels of sports. The first two rounds of the World Baseball Classic and first round of the World Cup (of soccer) divide the field into groups of four nations each, and each group is played as a round-robin tournament. After the first two baseball rounds or first soccer round, the tournaments become single-elimination instead.

On the other hand, round-robin regular seasons are fairly common in high school sports. During football season, every team in the league plays every other team. It might be easy to schedule the four-team round-robins by hand, but the school from which I graduated is in a seven-team league. So the U of Chicago text provides a method of complete a seven-team round-robin:

1. Let the 7 teams be vertices of an inscribed regular 7-gon (heptagon).

2. Draw a chord and all chords parallel to it. Because the polygon has an odd number of sides, no two chords have the same length. This is the first week's schedule.

3. Rotate the chords 1/7 of a revolution. This is the second week's schedule.

4. Continue rotating 1/7 of a revolution for each week. Because in a week no two chords have the same length, no pairing repeats. In a total of seven weeks, the schedule is complete.

Here is the resulting schedule as given by the U of Chicago text:

1st week: 7-2, 6-3, 5-4, 1 bye

2nd week: 1-3, 7-4, 6-5, 2 bye

3rd week: 2-4, 1-5, 7-6, 3 bye

4th week: 3-5, 2-6, 1-7, 4 bye

5th week: 4-6, 3-7, 2-1, 5 bye

6th week: 5-7, 4-1, 3-2, 6 bye

7th week: 6-1, 5-2, 4-3, 7 bye

In my alma mater's league, from one year to another, the home games become away games and vice versa, and then the schedule itself rotates so that the first week match-ups are contested the second week, the second week match-ups are contested the third week, and vice versa. Many of the stronger teams don't actually take the bye week -- instead they play special games against out-of-state teams.

If there are an even number of teams, then the extra team is placed at the center, and a radius is drawn in addition to the chords -- this replaces the bye week. For example, we can add an eighth team to the list above by having the eighth team play the team with the bye.

By the way, for Track season this year, my alma mater is following a round-robin for dual meets. The league has seven schools, but two of the smallest schools are combined into one team. Thus it's equivalent to scheduling a round-robin for six teams.

Today is an activity day -- and since this lesson naturally lends itself to an activity, I've traditionally always made Lesson 15-2 an activity day. The problem is that my old version of this activity isn't very pandemic-friendly at all. It requires cutting out and rotating the wheel that shows the team pairings.

Since this lesson is unique to the U of Chicago text, most Desmos mavens haven't created any online version of this activity. And most online references to round-robin tourneys are for generators where you enter the team names and the tournament is produced automatically -- no math or thought needed.

What might be interesting is if we could set up a round-robin game in Desmos. The game itself could be something simple (like tic-tac-toe) -- the actual task is for the students to predict who their next opponents will be by imagining a pairing wheel. But I'm not sure whether that's even possible using the current version of Desmos.

And so I have no choice but to post the activity from two years ago, even though I'm not sure whether it will work in a pandemic. Students will have to touch scissors and circles cut out of paper throughout this activity.

END