New Year's Eve Post (Yule Blog Challenge #8)

Table of Contents

1. Introduction

2. Yule Blog Prompt #12: What I Learned in 2022...

3. Quinters on the 10-Day Calendar

4. Quinters on the 11-Day Calendar

5. Links to Other Yule Bloggers and Juneteenth

6. Rapoport Question of the Day

7. Conclusion

Introduction

Today is New Year's Eve, so this marks my final post of 2022. It's only my eighth Yule Blog post though -- but with eight posts in two weeks, I remain on pace to finish the twelfth post in my third week of break.

When 2022 began, I made the following prediction (on the Stats blog):

But the pandemic didn't end in 2021, and now many consider 2021 to be a Murphy's year too. Case in point, when legendary actress Betty White died yesterday, many people tweeted that it was just one last bad thing that had to happen in 2021.

That being said, why should we believe 2022, the third year (n + 3) of the pandemic, to be any better than the first two years? After all, the year begins with omicron raging more strongly than ever. As far as I know, some celebrity will die on December 31st, 2022, and the response will be that it's one last bad thing to happen in 2022 -- a year to be just as horrible as its two predecessors.

Well, I suppose that a celebrity did die today -- Pope Emeritus Benedict XVI. And yesterday a celebrity died -- journalist Barbara Walters. And the day before that a celebrity died -- soccer legend Pele. Here's an article about all three passings:

https://www.who2.com/last-minute-deaths-of-2022-pele-a-pope-and-barbara-walters/

So it appears that 2022 was a bit like what I predicted -- a Murphy's year (though perhaps not quite as Murphy-ish as 2020 or 2021) that ends with significant figures passing away.

Anyway, speaking of looking back to 2022, the year we're about to complete...

Yule Blog Prompt #12: What I Learned in 2022...

There is one lesson this year where I can truly say that I learned as much as my students. It was October 24th, the day of the "slope walk" in my Math I class. Because it was on a Monday (a non-posting day), I only briefly mentioned it in my post that Tuesday:

By the way, yesterday's Math I lesson was interesting. The TOSA set up a calculator-based laboratory in the library, and invited the Math I classes to go on a "slope walk" -- it was my turn yesterday. The students must walk according to a given linear equation -- so for y = 2x + 1, for example, they must begin at the 1-foot mark and walk 2 feet per second. A motion sensor captures the movement and sends the data to the calculator, and the kids get to evaluate each other on their accuracy.

I'm a bit surprised that with so many MTBoS posts about the use of technology in the classroom, I don't see much mention of calculator-based laboratories on the MTBoS.

Anyway, the activity worked as described -- the calculator and motion detector were placed on a desk, and several yardsticks were placed on the floor (with the labels changed so that 1, 2, 3, 4, 5, and so on, feet were marked). As the student walked along the yardsticks, the TI-84 would keep track of the position (in data list L2), velocity (L3), and acceleration (L4) of the student at each unit of time (stored in list L1).

The teacher on special assignment (TOSA) completely set up the calculator and activity for my second Math I class that day, helped me during fourth, then left me on my own for sixth. So as the day went on, I learned more about how to set up the lesson. In second period, he began by calling on one student volunteer to perform the slope walk, and then called other students to guess the equation that the first volunteer had walked.

And so in fourth period, I tried to begin my class the same way. I used my own TI calculator (not the one set up for the slope walk, but the one with my random name generator) to choose the student to make the first slope walk, and then to choose the kids to guess the equation. But unfortunately, the first few kids refused to take a guess, which got me upset that they weren't participating -- likely either talking to each other or playing on phones, not paying attention. And so I began to argue with them.

I admit that the argument was a continuation of a dispute that had happened earlier in that class, but I don't quite remember what it was. It was either a group of students who left their Warm-Ups blank, or who had been caught talking on the previous Friday's quiz and refused to write down their standards as a consequence of cheating.

At any rate the TOSA called me out on arguing here. He pointed out that once a guess has been made, the other students are more likely to participate and correct the first student's guess once they see the guess and the data graphed on the TI calculator. So the best idea is just to get a first guess up there -- perhaps calling on a stronger (not random) student to take the first guess, or even have the students discuss it with each other and graph the first equation I overhear. Indeed, it's better to have a weaker student be the one to make the first walk, not the first guess -- a weaker student might not know how to come up with linear equations, but definitely knows how to walk at a more or less constant speed. This is a situation where choosing students at random isn't the best idea.

And this was the more valuable lesson for me to learn that day -- how to avoid arguments in the class and get students to participate, not just how to set up the TI calculator. Indeed, from that moment on, I avoided most arguments in fourth period, which quickly became my best behaved Math I class. I'm glad that the TOSA was the one who criticized me for arguing, rather than the principal -- who would go on to observe that period a month and a half later for my evaluation.

Instead, most arguments involving Math I occur in either second or sixth periods. I still need to find a good way to get those students to follow my instructions without resorting to arguments.

Quinters on the 10-Day Calendar

Let's get back to Calendar Reform and quinters. We will move on to a calendar with ten-day weeks and figure out how to divide the 180-day school year into five 36-day quinters.

First, we need a ten-day calendar. A simple one I found is the 6 * 6 * 10 calendar:

https://calendars.fandom.com/wiki/6*6*10_regular_calendar

It's just like the Sexagesimal Calendar, except with six ten-day weeks rather than ten six-day weeks. As the author doesn't state the names of the days of the week, we might call them Oneday to Tenday, or perhaps Zeroday to Nineday (since the counting in this calendar starts with zero).

Suppose we let Zeroday to Fiveday be the school week, and Sixday to Nineday be the weekend. Then each month would contain six weeks of six days to give us 36-day quinters -- but then there's no week left over at the end of the year for a vacation week. That's the cost of having four-day weekends.

Also, there's no relation stated here between months and seasons. We might assume that the first month (Zeromonth?) starts on the winter solstice (like the Sexagesimal Calendar). Then Threemonth starts on the summer solstice and counts as summer break -- the lone long break of the year. The other five months are the school quinters.

Here's a link to another ten-day week -- the Metric Week:

https://zapatopi.net/metrictime/week.html

This site proposes dividing two straight years into 73 weeks, with one week split between the years -- even years start on Zeroday and odd years on Fiveday. The only blank day needed is Leap Day. The site also proposes having either seven weekdays (Zeroday-Sixday) or five weekdays, but neither five nor seven divides the 36-day quinter evenly. So I might prefer having six weekdays (like 6 * 6 * 10 above).

But the site says nothing about months (or even how to label the weeks if there are no months). As far as we know, the author intends to use Metric Weeks with the Gregorian year.

There are a few ways to expand this into a full calendar. Since the ten-day week is called a "Metric Week," the natural idea is to have ten weeks per month, or 100 days each. Then there are three long months and one short month of 65 (or 66) days. If we align the short month with the summer vacation, then the resulting school year is functionally equivalent to the 6 * 6 * 10 calendar (except that the long months correspond to trimesters, not quinters).

It's also possible to use 10 as the number of months per year, not the weeks per month. But each month would need to have 36 or 37 days, which doesn't align with the 10-day week easily.

What sorts of months align with the weeks then? We could have either nine 40-day months or twelve 30-day months. Nine 40-day months align with the Modern Calendar and appear in the 10-10 Calendar:

https://calendars.fandom.com/wiki/10%E2%80%9310_calendars

But as we've seen, quinters don't work very well with nine 40-day months.

Twelve 30-day months appear in the Tenstrong Calendar -- and it's noted that this idea is an old one, dating back to the French Revolution:

https://calendars.fandom.com/wiki/Tenstrong_Calendar

But each 30-day month would essentially be half of a 60-day month in the 6 * 6 * 10 calendar, and so the resulting school calendar once again would be similar.

Quinters on the 11-Day Calendar

Of course, the ten-day week isn't our target week length. My ultimate goal is to divide my original calendar, the Eleven Calendar, into quinters.

So far, we've seen that as we moved from nine- to ten-day weeks, the fourth weekend day forces us to eliminate all breaks except summer break. An eleventh day -- making six school days and five weekend days per week -- would cut into summer break even more.

Indeed, we see a very easy way to divide the year into quinters. Each quinter would contain two months with six weeks each. That fills ten of the eleven months, so the eleventh month is summer break. So this would be just like 6 * 6 * 10, except the summer break is now an even smaller fraction of the year -- one-eleventh, rather than one-sixth.

In past posts, I divided the three-week summer break into three one-week breaks, and then spaced those weeks equally throughout the year. This would give us three short breaks instead of a long one -- and trimesters rather than quinters.

What we'd really like to do is keep the quinters, but have shorter weekends so that the 36 days within each quinter take less time to complete, leaving us with a break at the end of each quinter. We can still combine this with the eleventh month to provide us with a summer break.

Hmm -- the Eleven Calendar contains 11 months of 33 days each. Hey, that sounds familiar -- yes, it reminds of the 352/384 Calendar, which has 11 months of 32 days each. So perhaps we should just modify that calendar -- add a 33rd day of to each month, and voila! So the resulting calendar would have eight-day weeks instead of eleven (with the 33rd as a blank day).

And there's a way to keep 11 days in this calendar as well. Instead of having holidays ten days apart (the so-called "new," "half," and "full moon" days that have nothing to do with the actual lunar phases), we have holidays that are eleven days apart. Indeed, this was one idea I had with the original Eleven Calendar -- the first three days of the week in the calendar are "Muslim Sabbath," "Jewish Sabbath," and "Christian Sabbath."

Suppose we had a calendar with holidays on the Christian Sabbaths -- the 3rd, 14th, and 25th. We can then divide the 33 days of each month into weeks -- say Threeday to Eightday, with Oneday and Twoday as the weekend. In this case, the school week would be six days, except that if one of those days is a Sabbath, then the week is reduced to five days.

So the first school week would go from the 3rd to the 8th. The 3rd is a Sabbath, so the first school week contains five days, from the 4th to the 8th. The second week, from the 11th to the 16th, has a midweek Sabbath on the 14th. The third week goes from the 19th to the 24th. While the 25th is the Sabbath, we can observe this Sabbath on the 24th, so we again have five days. The only week with a full six-day week is the last week, from the 25th to the 32nd. It's the longest week, but we make up for it by adding the blank day on the 33rd to the weekend.

The the next three weeks are the 4th-8th, 11th-16th (sans the 14th), and the 19th-23rd days in the second month. This gives us seven weeks with five days each except for one six-day week, so it's a grand total of 36 days in the quinter. The holiday at the end of the quinter runs from Sabbath to Sabbath, from the 25th to the 3rd of the next month.

We do end up with a working Quinter Calendar, though it's a bit complex. The school days are based on an eight-day week, while the Sabbaths follow an eleven-day week. It's been pointed out that any calendar that breaks the seven-day week is troublesome because followers of the Abrahamic faiths will continue to follow the seven-day Sabbath cycle. Thus if we're going to overlay two different week lengths anyway, one of them should be the seven-day cycle. In fact, as we'll soon see, it's the 11-day week that's dead weight here -- overlapping the seven- and eight-day weeks makes a workable calendar.

Suppose that we have overlapping seven- and eight-day cycle. The seven-day cycle contains the standard day names, and suppose that you're a Christian who want to take Sundays off. The eight-day cycle contains eight days, with six consecutive days as school days.

Each quinter, as we've seen above, contains seven school weeks. It then follows that exactly one of these seven school weeks starts on Monday, exactly one on Tuesday, one on Wednesday, and so on though we don't know which day, say, the first week begins (and it will vary from quinter to quinter).

The six-day school week that begins on Monday goes Monday-Saturday -- it contains no Sabbath, and so all six days are school days. All the other weeks contain a Sabbath, and so each of those weeks contain five school days. So once again we have seven school weeks -- one with six days, the rest with five -- again giving us 36 school days in the quinter.

You might ask, wouldn't that blank day on the 33rd mess things up? Yes it will -- so instead of having every month be 33 days, let's alternate between 32 and 34 days. The first month of 34 days is the summer month, and then each quinter consists of a 32- and a 34-day month, with the only blank (with respect to the eight-day week) days on the 33rd and 34th of the long months.

And by going 34-32-34-...-32-34 days, the entire year contains not 363 days, but 364 days -- which is a multiple of seven, so it lines up with the seven-day weeks. And since 364 days is too short for the shorter year, we must eventually add -- you guessed it -- a Leap Week. which can be made to line up with the Leap Week in any other type of Calendar (Usher, Pax, etc.).

Of course, this calendar isn't without its problems. One week in each quinter begins on a Saturday, so we'd have a lone school day before the Sabbath (and the temptation to take that day off too). And a week that starts on Wednesday would end with a lone day on Monday -- another day that many students will try to take off (so you better hope that's not finals week).

And by this point we've long since diverged from the Eleven Calendar as I originally planned it. The only thing eleven-ish about this calendar is that there are eleven months. The number of days per week is no longer eleven, and the number of days per month is no longer a multiple of eleven.

Finally, the one thing about the Fixed Festivity Week Calendar (and any "Fixed Festivity" calendar listed on the Calendar Wiki) is just that -- the "festivities" (holidays) are fixed. The original purpose of the calendar was to avoid having fixed holidays -- days when airlines, amusement parks, and other businesses raise their prices.

This is why I originally wanted a hybrid week -- weeks when different shifts of students are attending school at different times. But during the pandemic, hybrid schedules became a reality -- and as we found out, the hybrid schedule turned into chaos. So I no longer wish to have hybrid on my calendar.

But still, we might have different school schedules implemented at the district level. Then it will be harder for businesses to jack up their prices since many kids are off at different times. The only requirement will be that there should be one month off for summer, the other ten months divided into five quinters, and 36 days of school in each quinter.

We'll keep the 3rd, 14th, and 25th as off days -- but let's just call them Holidays instead of Sabbaths, in order to avoid the temptation of overriding these with a seven-day Sabbath cycle. Some of these Holidays might correspond to Christian (or other religious) holidays, so many people might wish to go to church on this Holidays anyway.

On my Eleven Calendar, the months are labeled March-January. My March starts at the same time as on the Gregorian, so the last 2-3 days of Gregorian February (including Leap Day) are blank days. March is mostly aligned in the two calendars (except the last two days of my March), but each of my months start progressively later in the corresponding Gregorian month. My January only lines up with the last week of Gregorian January, plus the first 26 days of February (before the blank days).

So now we need 33 holidays, three in each of the eleven months. I was considering using many of the holidays from the Fixed Festivity Day (not Week) Calendar:

https://calendars.fandom.com/wiki/Fixed_Festivity_Calendar

There are already 29 holidays here, so we need just four more holidays. One reason for taking the Day version rather than the Week is that the FFW version combines several holidays into one week. For example, the spring holiday week goes Shrove Monday, Fat Tuesday, Ash Wednesday, Maundy Thursday, Good Friday, Holy Saturday, Easter Sunday (so holidays from the start and end of Lent are combined into a single week). The FF version combines the early Lent days into a single February holiday and the late Lent days into a single March holiday -- in other words, it takes into account that the holidays occur in different weeks, not a single week like the FFW version.

The FF holidays come from differing traditions -- for example, "Independence" is in July to celebrate the US Fourth of July, but "Thanksgiving" is in October to match the Canadian holiday, and "Labor" is in May to match International Labor Day. Instead of US Labor Day, "Children/School" is at the start of September to mark the start of the school year (like Liber on the Sexagesimal Calendar).

So let's create our holiday schedule. One thing notable about many of the US federal holidays is that they fall slightly more than a month part -- consider Labor-Columbus-Veterans Day in the fall, as well as MLK-Presidents in the winter, and Memorial-Independence in the summer. In fact, these holidays are close to be 1/11 of a year apart (instead of 1/12), so they fit on an Eleven Calendar.

So let's place Memorial Day on May 25th in our calendar. The Independence Day works out to be June 25th (which corresponds to July 2nd, aka Adams Day, in our calendar). And the other holidays also fall on the 25th of their months, although since my months start later, the holidays are in the named months preceding their Gregorian months, so Labor Day is August 25th, Columbus Day is September 25th, Vets Day is October 25th, MLK Day is December 25th, and Prez Day is January 25th.

I've already named a date for Christmas -- December 3rd, which matches Gregorian December 25th and is hence the right day for Christmas. July is the summer month, and so August become the first month of the school year. Yes, this is before Labor Day, but it's later than the start of Gregorian August. And recall that marker of the school start is Children/School Day, not Labor Day, so we can place that holiday on August 3rd.

Of course, some people might object that Independence Day ought to be the Fourth of July -- and perhaps rearrange the calendar so that holidays occur on the 4th, 15th, and 26th of their respective months, and even align July (rather than March) to the Gregorian year, so that the Fourth of July really is the Fourth of July (thus making June the skipped month rather than February). This plan has the advantage of placing all the above-named federal holidays in their correct months (from Labor Day on September 4th, up to Memorial Day on May 4th), but throws off Children/School Day and Christmas.

I will keep my original plan with March aligned and holidays on the 3rd, 14th, and 25th. Then I'll use the holidays from Fixed Festivity to fill the rest of the calendar, starting from our new Christmas holiday on December 3rd.

Here is the calendar that I came up with. Three holidays per month (on the 3rd, 14th, 25th) are listed, most coming from the Fixed Festivity (FF) Calendar, with commentary:

December: Christmas; Epiphany/Baptism; MLK (New Year's has been moved, since my New Year is in March. Christmas to Epiphany form the Twelve Days of Christmas using inclusive counting, where both Sabbath holidays are included.)

January: Candlemas/Equality; Valentine/Groundhog; Presidents (FF is illogical here -- "Candlemas" and "Groundhog" ought to be the same, as both are Gregorian February 2nd. The middle holiday should be Candlemas/Groundhog -- this is 40 days after Christmas using exclusive counting, where all Sabbath holidays are excluded. Shrove/Lent goes with Presidents' Day, just like Usher Calendar.)

March: New Year; Spring/Lady/Annunciation; Good/Holy/Maundy (Actually I prefer Palm here, since it's a week before Easter. Lady/Annunciation ought to be March 25th in both calendars, but FF places it early in order to avoid Holy Week. Cesar Chavez Day fits here as well.)

April: Easter; Father/Men; Remembrance (No quinter break lines up with Easter. FF lists Labor Day here, but we use US Labor Day instead. And some nations use "Remembrance" to refer to US Veterans Day here instead. The newest LAUSD holiday fits here --- Armenian Genocide Remembrance Day.)

May: Mother/Women; Ascension/Parents; Memorial (Ascension is 40 days after our Easter, excluding Sabbath holidays. This is also Month 3, Day 14, so it can serve as a new Pi Day, although it's now a Sabbath holiday rather than a school day. Pentecost goes with Memorial Day, just like Usher Calendar.)

June: Corpus Christi/Family; Midsummer; Independence. (Juneteenth fits with Midsummer, just like Gregorian Calendar, and places it on a June fourteenth.)

July: World/Unity; Assumption/Transfiguration; Freedom/Liberty. (This is summer break.)

August: Children/School; Ruler/Region; Labor. (This is the first month of school.)

September: Peace/Rest; Thanksgiving; Columbus. (Since "Thanksgiving" here means Canadian, it ought to be the same as Columbus.)

October: Community/Civic; Saints/Hallows/Reformation; Veterans. (The Reformation was on October 31st, which is why it's placed with Hallows here.)

November: Carnival/Joy; Dead/Souls; Advent/Nicholas. (FF actually places Mexican Day of the Dead with US Veterans Day -- also a day of the dead, as in war dead.)

Links to Other Yule Bloggers and Juneteenth

Shelli, the leader of the Yule Blog Challenge, made her twelfth Yule Blog post yesterday. It's part of her series "My Favorite Friday":

http://statteacher.blogspot.com/2022/12/myfavfriday-favorites-of-2022.html

So Shelli is the clear winner of the Yule Blog Challenge as the first to a dozen posts. Moreover, she wasn't satisfied with twelve, and so she makes it a baker's dozen today, with a post about preparing for the new year:

http://statteacher.blogspot.com/2022/12/preparing-for-2023.html

In this post, Shelli writes:

I report back to school on Monday for meetings and collaboration time, then kids come on Tuesday.  Like most teachers over the break, I've been thinking about what to do on the first day of the semester.  I like this time as a reset and definitely plan to do some non-curricular tasks and to revisit our classroom norms.

In other words, Shelli must work on Monday, even though it's the legal federal holiday (namely New Year's Observed). But at least the students don't have to attend on the federal holiday.

But this makes it seem OK to have a Quinter Calendar (for the usual Gregorian Calendar) with the last teacher day on June 18th, even if this is Juneteenth Observed. Having teachers works on observed federal holidays (January 2nd, June 18th) isn't equivalent to having them work on the actual holidays (January 1st, June 19th). I doubt Shelli's school will make her report on Monday, January 1st, 2024, since that is the actual New Year's Day.

This reminds me of the UCLA calendar -- I'm surprised I haven't checked out to see how my alma mater is observing the new federal holiday:

• In 2021, the new holiday was just barely announced a few days in advance. So UCLA observed it a week late -- on Monday, June 28th.
• In 2022, the holiday was observed on Monday, June 20th -- the observed federal holiday. It did mean that the first day of summer school was a Tuesday rather than the usual Monday.
• In 2023, a Leap Week was added to the calendar. So Juneteenth, June 19th, falls during the week off between spring quarter and summer school, rather than the first week of summer school.

Recall that the UCLA Leap Week actually corresponds to another ethnic holiday -- Cesar Chavez Day, observed on the last Friday in March. In 2022 this is on its earliest possible date (the 25th), while in 2023 it will be on the 31st.

From 2024-2027 this pattern continues -- Chavez Day falls during spring break (the week between winter and spring quarters), and Juneteenth during summer break (between spring quarter and summer).

But then in the posted calendar for 2028, three strange things happen:

• Winter Quarter begins on Tuesday, January 4th, instead of its usual Monday. (I'm not sure why January 3rd is to be avoided, especially considering that the quarter began January 3rd, 2022.)
• Spring break is the week before Chavez Day, rather than the week of Chavez Day.
• Summer break is the week before Juneteenth, rather than the week of Juneteenth.

Each of these three anomalies results in the respective quarter being a day shorter than usual. And what makes it even weirder is that if a Leap Week is inserted before Winter Quarter 2028 (that is, if the quarter begins on January 10th rather than the 3rd), then all three problems vanish. January 3rd, March 31st, and June 19th would all fall during break weeks, and so none of them result in an extra day off.

The resulting calendar still isn't perfect though. In 2033, Chavez Day will fall on its earliest possible date, March 25th and Juneteenth on its latest possible date, June 20th. This is the opposite problem from the K-12 Quinter Calendar concerning early Juneteenths. But we have until 2033 to figure that one out (for example, Chavez Day could be observed on April 1st -- the day that it will be observed in LAUSD that year -- instead of March 25th). The students who will attend UCLA in 2033 are mostly in elementary school now.

Rapoport Question of the Day

Today on her Mathematics Calendar 2022, Rebecca Rapoport writes:

How many papers did Ron Graham and Paul Erdos write together (additional co-authors allowed)?

This doesn't even count as a research problem -- a Google search tells me about these two famous mathematicians (both of whom have been mentioned on this blog before), but no website anywhere tells me how many papers they co-authored. Both are prolific authors so it doesn't surprise me that they co-authored many papers, but no source reveals that they've written exactly 31 papers together -- which must be the answer, since today's date is the 31st.

Here's an easier question -- what is Ron Graham's Erdos number? Well, since they've co-authored at least one paper, Graham's Erdos number must be 1.

Conclusion

In my next post, I'll return to my COVID What If? stories. But here's one thing I do want to say about my past as a student, and an unexpected connection to the present day.

A month ago, I got in touch with one of my old friends from the district we attended for K-8. I first met him in the first grade, and the last class we had in common was eighth grade History.

The reason we got in touch is, as it turns out, he has a son who's now a freshman at the very school where I currently work! That's how he found me -- he discovered my LinkedIn page, saw where I'm now working, and contacted me to tell me about him.

My friend's son isn't in any of my Math I classes, though. I actually spoke to his Math I teacher recently since we have the same conference period -- she informed me that my friend's son is one of the top students in her class. (I reckon she taught the slope walk lesson to him correctly.) And I saw the young son myself a couple of times since this discussion -- he plans on joining the Track team in the spring.

Anyway, I wish everyone a happy new year, and for a less Murphy-ish 2023 than we had in 2022.

Waxing Half Moon Post (Yule Blog Challenge #7)

Table of Contents

1. Introduction

2. Yule Blog Prompt #10: 3 Good Things From 2022

3. The School Year on the 8-Day Calendar

4. The 352 or 384 Day Calendar

5. Quinters on the 9-Day Calendar

6. Rapoport Question of the Day

7. Conclusion

Introduction

There aren't any major holidays to acknowledge in today's post, so I'll instead refer to the moon phase -- the waxing half moon, also known as "first quarter." I'll be returning to Calendar Reform today, including a lunar calendar on which the half moon is relevant.

Yule Blog Prompt #10: 3 Good Things From 2022

Whenever I respond to a "three good things" prompt, I like to acknowledge the entire calendar year starting in January, not the current school year starting in August. Here are three good things -- three days on which I especially enjoyed teaching in 2022.

My first good day was Friday, February 11th, when I had just returned to my old magnet school from testing positive for COVID. On Fridays, the block schedule had both of my Ethnostats classes meeting.

Taking a cue from my predecessor Ethnostats teacher, I played a video for the class -- 13th, named after the constitutional amendment. It's definitely relevant to an Ethnostats class, as the documentary discusses the statistics regarding prisoners of various races (particularly African-Americans) in this country. I'd chosen that particular date for several reasons -- it was during February (Black History Month), it was close to the thirteenth of the month (as the 13th was on a Sunday), and we were just about to start the 13th chapter of the Stats text. This chapter was on probability, and so I was able to tie conditional probability to the video content, such as P(black | prisoner), the probability of being black given being a prisoner.

And I believe that the students learned much from this lesson. This is one of those projects that I included in their interactive notebooks that year -- video notes, questions from the video, and even some of their own research regarding ethnicity and imprisonment.

My second good day was Friday, October 14th. In my Math I classes, I regularly give a quiz every Friday, but this week I decided to give a different sort of quiz for the first time, a "Hero Quiz."

The idea of a Hero Quiz goes back six years -- all the way back to the old charter school. From time to time, I would assign a multiplication quiz in order to help my middle school students remember their times tables. I continued this tradition during my long-term assignment at another middle school, which is where I came up with the term "Hero Quiz," as strong math students are my heroes.

We all know that even high school students can use more multiplication practice, but I thought that students at this age might consider multiplication quizzes to be beneath them. So as the school year began, I was still debating in my mind whether to implement Hero Quizzes this year. In mid-October, we started Chapter 2. I wanted to give a weekly quiz, but as the material on graphing linear functions was new, I didn't think the kids were ready for a true quiz. So instead, I decided to give a quiz reviewing equations on Chapter 1 and called it a Hero Quiz. The quiz consisted of two questions.

I actually required students to get both questions correct -- all phones and electronics must be put away until they fix their answers. Then I stamped their quizzes with an A+ stamp, worth 10/10 points. One reason for giving the Hero Quizzes is that for many students, math is a struggle -- they're not used to getting A's on math quizzes. So at least for one day, my students feel that they are math people.

My third good day was Monday, November 28th. It was a tough day to teach -- Cyber Monday, the first day after Thanksgiving break. But it was a lesson that I'd been looking forward to -- transformations, the first Geometry lesson.

I already wrote about this lesson during the Yule Blog Challenge. I passed out a reflection worksheet -- the students figured out how to draw the reflection images, but they struggled with coordinates. While I enjoyed teaching the lesson, and it was largely a successful lesson -- if I'd waited until December 6th, I could have obtained a better worksheet from my neighbors.

Oh, and one more thing -- just before she taught the lesson on December 6th, my neighbor teacher discovered some tiny mirrors in her classroom that helped her out with the lesson. If we're both still at this school next year, I might borrow her mirrors and use them with this lesson. So while this lesson wasn't the best, I know have materials that will help me teach it better next year.

The School Year on the 8-Day Calendar

So far this week, we looked at implementing quinters -- that is, dividing the school year into five terms rather than the usual two or four -- on calendars with six- and seven-day weeks. Thus the next step is to try eight-day weeks.

I've posted a few eight-day calendars on the blog before. Most of them were versions of the Modern Calendar, linked below:

https://calendars.fandom.com/wiki/Modern_Calendar

Let's use the Modern Calendar as described at the above link. The eighth day of the week is called "Remday," and it counts as a new weekend day between Friday and Saturday. The calendar contains nine months of five weeks or 40 days each, with December as the tenth month of 5-6 days.

It becomes apparent that quinters won't work on this calendar -- at least not as easily as they did on the Sexagesimal Calendar. For one thing, each quinter would need to contain 180/5 = 36 days. The school week goes Monday-Friday (just as in the Gregorian Calendar), so that's five days per week. But five doesn't divide 36 evenly.

It makes much more sense to divide the year into quarters instead. Each quarter contains 180/4 = 45 days, and so that would be nine weeks. Every month contains five weeks, so that would be ten weeks for every two months. So we can have a vacation week very two months. According to the link, every holiday is on a Remday, so we don't need a week full of holidays at the end of each quarter, but we can still use the vacation time. The four quarters span eight months, leaving one full month of vacation.

We might let July be the summer vacation month. Then the school quarters are August-September, October-November, January-March, and April-May. (According to the link, there is no February or June, and December is a short holiday month.)

Since two quarters (rather than two quinters) must occur before Christmas, the school year must start in August (which according to the link starts "12 days early" -- that is, in Gregorian July). Thus those who are opposed to summer break ending early will not appreciate this version of the school calendar.

All that means, of course, is that we need a different 8-day calendar on which to place our school year.

The 352 or 384 Day Calendar

When I was looking for eight-day calendars, I noticed that a new calendar was posted at the Calendar Wiki -- the "352 or 384 Day Calendar":

https://calendars.fandom.com/wiki/352_Or_384_Day_Calendar

The 352 Or 384 Day Calendar is a calendar that has 32 days in each month, and 11 or 12 months in the calendar. There are 352 days in a normal year, but every 8 years, there is an extra month, making the calendar 384 days long.

Let's figure out the average year length. With seven 352-day months and one 384-day month, the average length is (352 * 7 + 384)/8 = 356 days, which is much shorter than a tropical year. In other words, this calendar isn't very accurate as a solar calendar.

Ah, but that's not all -- apparently this calendar has a lunar component as well:

This calendar is also a lunar calendar. In every month, the 1st month is a New Moon, the 11th day is the half moon, 21st day is the full moon, and then the 32nd day is the last day before the New Moon.

This doesn't sound right either. Today, if you recall is a half moon, so it would have to be the 11th day of some month. Then the new moon would have been ten days ago -- December 19th -- but that wasn't a new moon. And the full moon would have to be ten days from now -- January 8th -- but that won't be a full moon.

In fact, not only would this calendar fail on our planet, there's no hypothetical planet on which it could possibly work. There might be an exoplanet with a 356-day year, but it can't have possibly have a moon that's waxing for 20 days (from the first to the 21st) and waning for only 12 days. (Notice that there's mention of the waxing half moon or first quarter, but not the waning half moon or third quarter.)

I don't know who created this calendar, but it's filled with errors. So let's see if we can fix it so that it actually fits the solar and lunar cycles.

Let's start with the solar component. We can still have 352- and 384-day years, but we need a different combination of them. As it turns out, the shortest possible reasonable cycle contains not eight, but 29 years, with seventeen of them short and twelve of them long.

The average year length is (352 * 17 + 384 * 12)/29 = 365 + 7/29 days. The fraction 7/29 indicates that this is equivalent to a Leap Day Calendar except with seven Leap Days every 29 years. A similar calendar, the Dee Calendar, has eight Leap Days every 33 years. As it happens, 8/33 is a bit more accurate than 7/29, but the 33-year Dee Cycle isn't a multiple of eight days, while 7/29 is.

There's still so much missing from the solar calendar. There are eleven or twelve months per year, so the twelfth month serves as a Leap Month. We could name the regular months January-November and the Leap Month December, or perhaps March-January with February as the Leap Month (since we're already associate February with leap units anyway).

But the link above names only the Leap Month -- "Additiacius" (from "additional," I presume), with no indication of what season any month is supposed to occur (as you'd expect in a solar calendar). So we would have to work this out ourselves.

Also, it's stated that the first day of the week is Sunday, but there's no indication of what the names of the other seven days should be. For simplicity, let's just follow the Modern Calendar and use the seven Gregorian names plus Remday between Friday and Saturday.

Now let's try to figure out the lunar component. The long year of 384 days is notable for being close to a whole number of lunar months -- but it's thirteen months, not twelve 32-day months. The short year of 352 days is fairly close to 12 lunar months (around 354 days) -- but that's not close enough. No respectable lunar calendar is off by two days after just one year.

We might begin with the 384-day year, since it's almost exactly 13 lunar months. There are a few links that will help us here -- the Goddess Lunar Calendar, which is based on 13 lunar months or 384 days:

https://www.fractal-timewave.com/mmgc/mmgc.htm

and the Hermetic Lunar Week Calendar, one of the few that incorporates new, half, and full moons:

https://www.hermetic.ch/cal_stud/hlwc/hlwc.htm

Unfortunately, there's a problem -- just because a year contains 384 days, there's no guarantee that the first day of any 384-day year is a new moon. Indeed, the first new moon of a year need not fall on a Sunday, much less the first of the month. According to the Goddess Calendar, there should be seven lunar months of 30 days each and six 29-day months in the 384-day year.

Instead of referring repeatedly to "new/half/full moons," let's call them by the name that the Hermetic Lunar Week Calendar calls them -- "Moonday." In that calendar, Moondays can be anywhere from six to nine days apart. (For example, my calendar shows a full moon on December 7th, third quarter on the 16th, a new moon on the 23rd, and the first quarter today. Then December 8th-16th are a nine-day week on the Hermetic Calendar, while December 24th-29th is a six-day week.) But for simplicity, we'll assume that all lunar weeks are seven or eight days.

On the Hermetic Calendar, Moonday is a separate day of the week. But for us, Moonday will fall on one of the eight established days of the week, including Remday. If Sunday is a Moonday, then the next Moonday might be Saturday (if it's an seven-day lunar week) or Sunday (if it's eight days). Then as the months progress, Moonday will gradually regress through the week, from Sunday to Saturday to Remday and so on.

Let's assume that there are two Moondays on the same day of the week, and then it moves back one -- so we'd have two Sundays, two Saturdays, two Remdays, and so on. This is equivalent to assuming that every lunar month has 30 days. If instead, our sequence of Moondays contains only one Saturday, one Thursday, and one Tuesday, and two each of the other five days, then this forms a sequence of 13 Moondays in 96 days -- exactly one-fourth of our 384-day year. Thus four repetitions of the sequence will fill the entire long year.

We can continue this sequence throughout the short 352-day year as well, so that shows us how to assign lunar phases to the short years. Of course, this assumes that 13 lunar months = 384 days is an exact relation -- but of course, it isn't. The Goddess Calendar must skip one day every decade (and that's ten Goddess years -- almost eleven solar years). We might try to squeeze some similar sort of cycle into our calendar, but note that this cycle almost certainly not line up with the 29-year Leap Month cycle of solar years (which, by the way, isn't equal to a whole number of lunar months).

Clearly, this calendar is a work in progress. I need more time to work it out -- but that's what happens when someone claims to have an accurate lunisolar calendar when in reality it isn't.

Can we still try to place a school calendar on it? I was considering declaring each Moonday (as opposed to each Remday) to be a holiday, in order to salvage the lunar component of the calendar. (Otherwise in what way is the calendar lunar at all?) Then most school weeks will contain five days (including Remday if it's not Moonday), unless the Moonday falls on Saturday or Sunday, then it's six days.

If each quinter contains one six-day week, then we can obtain 36 days in seven weeks. So every two months would contain seven weeks of school and a holiday week. That fills ten months. The eleventh month will be the summer vacation. Additiacius is a vacation month in the years in which it falls.

If there are too many holidays, then we might drop the third quarter/waning half moon holiday, since the author never mentions it. This is also similar to the ancient Roman Calendar -- the kalends, nones, and ides (as in "Ides of March") used to correspond to the new, first quarter, and full moon phases, with no holiday corresponding to the third quarter. (So today would be the "Nones of December.")

In fact, we might even choose just to forget about the moon and just place the holidays on the first, 11th, and 21st of the month -- the so-called Moondays according to the author. Oops -- the first is a Sunday, so let's make it the second, 12th, and 21st of the month instead (which work out to be Monday, Wednesday, Thursday). The last week of each month contains no holiday, so the quinters contain 36 school days. (The last week of every other month, between the quinters, is a vacation week.)

By the way, the idea of placing a lunar structure on a solar calendar is from "One Day Before":

https://www.hermetic.ch/cal_stud/palmen/1db4.htm

except that since our months have 32 days, it's more like two or three days before. I can't help but notice that the One Day Before Calendar is based on a 11600-year cycle, and 11600 is a multiple of 29. That link also leads to another calendar, the Annuary Calendar:

https://www.hermetic.ch/cal_stud/palmen/anry.htm

which also has a eight-year subcycle -- the same cycle that the 352/384 calendar claimed to have. So I wonder whether the 352/384 author was trying to incorporate one of these two calendars into his own.

Quinters on the 9-Day Calendar

On the nine-day calendar, we might have six school days per week plus a three-day weekend. Then this is more convenient for quinters, since the six-day week divides the 36-day quinter evenly.

Our calendar might contain ten months of 36 days and four weeks each (plus five extra days, perhaps similar to the Modern Calendar). Each quinter requires only six weeks, so every two months can have two full weeks off.

But all these extra weeks off come at a cost -- the five quinters span all ten months, so there wouldn't be a long summer vacation. Then again, if we follow the quinter plan given in my last post, then the fifth quinter can be for more enrichment and "fun" activities, and we can place that quinter in the summer.

Still, we easily came up with a Quinter Calendar for the nine-day week. All that remains is what to name the nine days of the week and ten months of the year to make it into a full solar calendar.

Rapoport Question of the Day

Today on her Mathematics Calendar 2022, Rebecca Rapoport writes:

Find x.

Once again, this is a Geometry question, and all the given information is in an unlabeled diagram, so I must provide the labels. Quad ABCD is inscribed in Circle O, with Angle A = 29, Angle B = 125. Also, CB and CE are opposite rays, with Angle DCE = x.

This is a case of the Inscribed Angle Theorem. As Angle A = 29, we conclude that Arc BCD = 58. Then Arc BAD must be 360 - 58 = 302, so Angle BCD = 312/2 = 151. Then BCD and DCE form a linear pair, so DCE = 180 - 151 = 29 degrees. Therefore the desired angle is 29 degrees -- and of course, today's date is the 29th.

Notice that in this problem, the opposite angles of the inscribed quadrilateral are supplementary (in this case 29 and 151 degrees). This can be generalized into a proof that for any quadrilateral inscribed in a circle, the opposite angles are supplementary. (Therefore any inscribed trapezoid must be isosceles.)

Conclusion

We still have work to do in establishing quinters for the eight- and nine-day calendars. In my next post, we will continue working out quinters for weeks of different lengths.

Kwanzaa Post (Yule Blog Challenge #6)

Table of Contents

1. Introduction

2. Yule Blog Prompt #9: A Tool or Strategy from 2022 That I Will Continue to Use in the Future

3. Dividing the School Year into Quinters

4. Calendar Reform and Quinters

5. Quinters: Other 7-Day Calendars

6. Quinters: 6-Day Calendars

7. Conclusion

Introduction

Today is the second day of Kwanzaa, a holiday celebrated by African-Americans. Technically, Hanukkah and Kwanzaa overlapped this year, but only in the daytime hours (as Jewish days start at sunset) -- thus the last Hanukkah candle was lit one night before the first Kwanzaa candle.

This also marks my first Calendar Reform post of the season. It's my tradition to write each year between Christmas and New Year's about various calendar-related topics. Although most Calendar Reforms are pure fantasy, the issue that I wish to discuss today involves the very real calendar at my current school.

Yule Blog Prompt #9: A Tool or Strategy from 2022 That I Will Continue to Use in the Future

Last year when I responded to this prompt, I wrote about interactive notebooks. In my Ethnostats classes, I had a notebook of sorts, but I didn't let the students take them home. At the time, I was worried about whether touching and passing out notebooks during a pandemic was a good idea -- and I knew that many students would refuse to bring the notebooks to school each day.

This year, I decided to start the interactive notebooks anyway. I did so after seeing the lone other math teacher at my school (my partner teacher), and some of my fellow Math I teachers are using them too.

So how are notebooks going so far this year? Well, just as I predicted, there are many students who don't bother to bring notebooks to school regularly. And the worst class by far is sixth period Math I.

Indeed, I did the Chapter 3 Notebook Check on the day of the final exam. Since it takes me so long to grade the notebooks, I decided to simplify the last check before grades are due. Each notebook would be scored as full credit, half credit, or no credit. In order to receive full credit, no more than five pages should be missing or incomplete. This sped up the grading -- as soon as I counted the sixth missing page, I marked it as half credit.

The total number of students who received full credit for their notebooks in sixth period is -- one. The other Math I totals were disappointing as well, but none so as much as sixth.

In this class, there are at least two intentional non-learners -- guys who have no intention of doing any work in my class whatsoever. So nothing I can do with notebooks will make them work. But there are also kids who seldom (as opposed to never) bring their notebooks. Since the Warm-Ups and Exit Passes are in the notebooks, they refuse to do any work during the first few and last few minutes of class -- and I suspect this contributes to the general misbehavior of this class throughout the period.

I don't know whether the other Math III teachers are using notebooks, but since I'm using them in Math I, it was an easy decision to do so in Math III as well. One thing about the Math III grade weighting is that homework and classwork are each worth 20% of the grade (as opposed to Math I, where homework and classwork together total 25%) -- and most of my classwork component is the notebook. Thus a student who never turns in a notebook must be perfect on everything else to get an 80% in the class -- the highest realistic grade such a student will get is 79%, a C+. Fortunately, my Math III students are better than Math I as far as turning in notebooks is concerned.

As for technology in the classroom, I regularly use DeltaMath for homework and assessments and Desmos for some of the lessons. One of the other Math III teachers had created a single Desmos for all of Chapter 3 (on solving equations and inequalities). Thus I counted Desmos as a classwork grade (a boon to those few students who never turn in notebooks).

Dividing the School Year into Quinters

Page 1 of the notebook was a table of contents for the entire year. I assigned it the second full week of school, which was in mid-to-late August, since the first day of school was on the tenth. The last page of the first semester was Page 67, which was a review page for the final exam. I assigned it during finals week, namely the week of December 12th-16th.

And this leads to my next topic -- the calendar, specifically the school calendar. We know that school must start in early-to-mid August in order to end the first semester before winter break -- a calendar that I've often referred to as the Early Start Calendar. But many teachers are tired of having to return to school so early in August. And so the union took a survey of teachers to decide upon a new calendar for next year.

As it turns out, the winning proposal is to start school a mere one week later. We were told that we couldn't shorten the first semester (which was 85 days this year), since 85 is less than half of 180. But we could start school one week later, since we could then have the semester end one week later -- if the calendar is approved, the last day of the semester will be December 22nd, 2023.

My worry about all of this is attendance on finals days. Recall that the Math I teachers decided to give their finals on the 13th and 14th because they feared low attendance on the 16th (Las Posadas). While that ended up not being much of a factor this year, the 22nd is not the 16th. Many parents might not send the students to school on the 22nd, or for the last few days leading up to the 22nd.

I've pointed out that in some locations such as New York City, there is regularly school on December 22nd, and even the 23rd (including this year). But NYC starts school after Labor Day, and so first semester finals (or "Regents," as they're called there) aren't until January. I don't know what attendance was like in NYC on Festivus this year, but that doesn't matter. My worry is what will happen at a school that schedules final exams on the 22nd and 23rd.

As it turns out, some Orange County districts this year -- including the one where I used to sub from 2018-20 -- had finals on the 23rd this year. My other OC district regularly ends the semester on a Thursday rather than Friday, so the last final in that district was the 22nd. This is mainly a problem in years when Christmas falls on a Sunday -- last year, when the holiday was on a Saturday, the last day of school was either Thursday the 16th or Friday the 17th.

In school districts with a two-week break, in years when Christmas is on a Sunday, the last day of school before the break is Friday the 16th. But school can't resume on Monday, January 2nd, because that is a legal federal holiday, New Year's Day Observed. So instead, they open on Tuesday the third -- and then there would be only 179 school days in the year unless that day is made up at some point. So having school until December 23rd (and resuming on Monday the 9th) avoids that problem -- the school year is a clean 180 days with no need to make any other changes.

No matter what, we must choose between starting the first semester in early August and ending it with final exams just days before Christmas. We can't start school later -- say after Labor Day -- because we can't fit half the school year, 90 days, between Labor Day and Christmas. But we might be able to fit about two-fifths of the year between September and December.

Six years ago on the blog, I introduced the concept of quinters -- dividing the school year into five terms instead of four quarters. Then two of these quinters fit comfortably between the Labor Day and Christmas holidays. The first semester would consist of the first two quinters, and then Quinters 3-4 make up the second semester (approximately January-April).

The fifth quinter (May-June) would be for other activities -- AP exams, SBAC or state tests, field trips, and other enrichment activities or projects. Indeed, a few MTBoS teachers have written about the concept of a "Winterm" right after winter break:

https://ispeakmath.org/2017/01/08/student-blogging-class-2017/

My fifth quinter would be similar, except it would last most of May and June (as opposed to a single week or two of Winterm). Students who fail classes can repeat them during the fifth quinter (if not during summer school). It would remove the need to have eight period days (as we've seen some local schools implements in recent years).

Let's see what a possible Quinter Calendar might look like. Let's consider the worst-case scenario -- when Labor Day is on September 7th, the latest possible date -- and see whether we can squeeze in 72 days between Labor Day and Christmas.

This will follow the old calendar that I followed as a young student -- back when school in November lasted until Thanksgiving Eve, and there were just two weeks of winter break. In particular, it closely follows my calendar for senior year (a year when Labor Day fell on September 7th):

September 8th (teacher prep day, no students)

September 9th-11th (Days 1-3)

September 14th-18th (Days 4-8)

September 21st-25th (Days 9-13)

September 28th-October 2nd (Days 14-18)

October 5th-9th (Days 19-23)

October 12th-16th (Days 24-28)

October 19th-23rd (Days 29-33)

October 26th-30th (Days 34-38)

November 2nd-6th (Days 39-43)

November 9, 10, 12, 13 (Days 44-47)

November 16th-20th (Days 48-52)

November 23rd-25th (Days 53-55)

November 30th-December 4th (Days 56-60)

December 7th-11th (Days 61-65)

December 14th-18th (Days 66-70)

January 4th-8th (Days 71-75)

January 11th-15th (Days 76-80)

January 19th-22nd (Days 81-84)

January 25th-29th (Days 85-89)

February 1st-5th (Days 90-94)

February 9th-12th (Days 95-98)

February 16th-19th (Days 99-102)

February 22nd-26th (Days 103-107)

March 1st-5th (Days 108-112)

March 8th-12th (Days 113-117)

March 15th-19th (Days 118-122)

March 22nd-26th (Days 123-127)

March 29th-April 2nd (Days 128-132)

April 12th-16th (Days 133-137)

April 19th-23rd (Days 138-142)

April 26th-30th (Days 143-147)

May 3rd-7th (Days 148-152)

May 10th-14th (Days 153-157)

May 17th-21st (Days 158-162)

May 24th-28th (Days 163-167)

June 1st-4th (Days 168-171)

June 7th-11th (Days 172-176)

June 14th-17th (Days 177-180)

June 18th (teacher check out day, no students)

So we notice that there are 70 days before winter break In some years there might be an extra week before Christmas, bringing us to 75. Our goal is to have 72 days in the first two quinters -- and 72 lies right between 70 and 75.

There is one problem with this calendar, though. This post might be labeled "Kwanzaa Post," but the African-American holiday that matters to us the most is the newest federal holiday, Juneteenth. On this calendar, Juneteenth would fall on June 19th, a Saturday -- so the observed federal holiday would be on Friday the 18th instead.

But on this schedule, the school is open on that holiday (that clearly wasn't a federal holiday in my senior year.) Sure, it's just for the teachers and not the kids, but still, you have teachers having to work on a federal holiday. There are a few ways to avoid this -- just have the last day of school be June 17th with no extra teacher check-out day, or eliminate Lincoln's Birthday (which is not a federal holiday) so that the last day for students is on June 16th, with teacher check-out on the 17th.

This problem only occurs in years when Labor Day is on September 7th. In all other years, Day 180 and the teacher check-out day both occur before the federal holiday. (It's just unlucky that the latest possible Labor Day, September 7th, corresponds to the earliest possible Juneteenth, June 18th.)

We can check to see when the other quinters will end. Day 36 is on Wednesday, October 28th, Day 108 is Monday, March 1st, and Day 144 is Tuesday, April 27th. Since this is based on a late Labor Day, these represent the latest the quinters can began and end. So for example, the first quinter would end on the Wednesday on or before October 28th (that is, the fourth Wednesday in October). 

OK, so this is one way to implement a Quinter Calendar. If you want to have the entire week off for Thanksgiving or three weeks for Christmas, then you must either start the year before Labor Day or end it after Juneteenth.

By the way, even before the federal holiday Juneteenth was created, the "do not pass" date at the end of the year would be the summer solstice, around June 21st. There usually isn't a significant difference between the 19th and the 21st, so we might as well take the federal holiday to be the "do not pass" date (though the example calendar above is the one where 19th vs. 21st is significant).

Calendar Reform and Quinters

Since I'm promoting a Quinter Calendar, I ought to apply quinters to the various reform calendars that I've posted on the blog. But we shouldn't blindly apply quinters just for the sake of doing so. The original reason to apply quinters (rather than quarters or trimesters) is because they fir the holidays in our calendar better. If a reform calendar has differently-spaced holidays, then applying quinters to it might be counterproductive.

For example, consider the following calendar created by James Colligan. This calendar divides the year into 13 months of 28 days each, plus a possible Leap Week called "Pax." Although Colligan didn't name his calendar, it's usually called the "Pax Calendar," named for its Leap Week:

https://calendars.fandom.com/wiki/Pax_Calendar

The extra month is called "Columbus" and is placed between November and December.

Now let's try to implement a school year on this calendar. We can assume that Labor Day is the first Monday in September (which works out to be September 2nd) and that Christmas is on December 25th (which works out to be Wednesday).

It's obvious that the existence of the extra month Columbus means that we can squeeze more days between Labor Day and Christmas. Indeed, we can easily place 85 days -- the same as the calendar in my current district -- between those two holidays.

My district began on a Wednesday, with teacher days Monday and Tuesday. Starting from Labor Day, the teacher days are Tuesday and Wednesday, so the first day for kids is Thursday, September 5th. The other non-student days are a teacher day on the first Monday in October (the 2nd), Veteran's Day (November 11th, a Wednesday) and Thanksgiving week (the last week of November), with the semester ending on Friday, December 20th.

So we've magically solved the problem by inserting an extra month. There's no need for quinters on this calendar, since our current semester fits between Labor Day and Christmas. While it works, the time between Thanksgiving and Christmas might drag on. Since we've placed the holidays in the same months as the Gregorian Calendar, the new month Columbus has no holidays, and so the holiday stretch would feel more like a Big March. (Perhaps we should place a federal holiday in the new month -- the most logical would be the holiday of the same name, Columbus Day. Then again, my district doesn't close for Columbus Day, so it would make no difference.)

But we must think back to why so many teachers object to starting school in August at all. If it's because a month named "August" should be a vacation month, then this Pax Calendar solves it. But some people might not want to work in August because of the temperature. All the Pax Calendar really does is rename the dates -- it doesn't change the weather on that day.

We should check to see what date on the Gregorian Calendar corresponds to Pax September 5th. As it's a Leap Week Calendar, the dates don't correspond exactly. But according to the link above, the Pax year 2023 doesn't began on the same day as the Gregorian year (despite both beginning on a Sunday) -- instead, Pax year 2023 begins on Gregorian December 25th. Then Pax December 20th works out to be nine days earlier -- Gregorian December 16th.

So the last day of the semester is the same actual day in both calendars. And since both calendars have the same number of holidays in the first semester, it follows that they begin at the same time. Thus Pax September 5th is the same as our actual first day of school, Gregorian August 10th (actually the 11th, since the Pax school year starts on Thursday, not Wednesday, after we moved Labor Day). In other words, Pax September has the same weather as Gregorian August -- that is, it's equally hot. If the goal is to avoid having school on the hottest days of the year, we have failed.

Of course, we can create our own Pax year such that September lines up with Gregorian September. It would then, as a result of the extra month Columbus, push Christmas back to Gregorian January. This is the only real way to include a full semester after the current Labor Day -- delay Christmas. If we wish to keep Christmas where it is, then we must go to quinters.

Quinters: Other 7-Day Calendars

One of my favorite calendars to promote on the blog is the Usher Calendar. Like the Pax Calendar, the Usher Calendar is a Leap Week Calendar. But this Leap Week Calendar is invisible -- instead, it redefines holidays and Leap Days (that is, February 29th) to fit the invisible Leap Week Calendar.

This calendar was first posted to Usenet over a decade ago by its creator, Andrew Usher. So it's not posted on the Calendar Wiki website, though the following calendar is quite similar:

https://calendars.fandom.com/wiki/North_American_Weekday_Holiday_Act

For example, Presidents' Day is the same on both this NAWHA Calendar and the Usher Calendar -- it lies in the February 16th-22nd range. It matches the Gregorian Prez Day unless it's February 15th, when the new calendars switch it to the 22nd.

Anyway, since the Usher Calendar resembles the Gregorian Calendar, applying quinters to Usher is similar to doing so to the Gregorian. In fact, we can mostly use the same calendar as above, except with Presidents' Day on February 22nd.

Usher created his calendar before Juneteenth became a federal holiday. As I've stated above, I like the idea of placing Juneteenth on a Saturday between the 13th and 19th (so that the holiday actually falls on a June --teenth), but observe the federal holiday on the following Monday, not the previous Friday. (It works out to be the third Monday in June, exactly three weeks after Memorial Day.) Then the last day for teachers can indeed be June 18th as listed above, since the federal holiday would be the 21st.

Oh, and there's one more thing to fix -- the date of Christmas. On his calendar, Usher defined Christmas to be the Sunday between December 21st and 27th. Notice that this would mean that there would always be school on Christmas Adam -- and this could be late as December 25th. Of course, in years when this happens, Christmas itself wouldn't be until the 27th, but still, it seems wrong to have school open (with final exams, no less) on a day that's been a holiday for so long.

And indeed, winter break is still only two weeks, with only Thanksgiving and Black Friday off in late November as well. It's possible to start school in August to accommodate both of these breaks, but how early should we go?

Recall that in the invisible Leap Week Calendar, Usher numbered the weeks from 1-53. In his calendar, Labor Day is Week 4, so the two weeks before Labor Day are Weeks 2-3. Week 1 is Leap Week, which runs from Sunday-Saturday, August 16th-22nd -- in years without Leap Week, the weeks are numbered from 2-53 (strange, yes, but that's how Usher numbered them).

Usher intended the school year to be fully contained within Weeks 2-53. Thus this is the earliest the school year should begin -- two weeks before Labor Day. When Labor Day is on its earliest possible date (September 1st), the school year begins on August 18th.

So we can extend winter break by moving the last week before Christmas up to Week 3 (and this solves the Christmas Adam problem right away). And the three days Monday-Wednesday of Thanksgiving week can be moved up to Wednesday-Friday of Week 2, with teacher days Monday and Tuesday.

Notice that there's still a teacher day on the day after Labor Day. We can keep it on this day (thus giving the kids a four-day weekend) or switch it to a different day (perhaps a Monday in early October, matching my current district).

I was hoping to make the quinters exact -- 72 days before winter break, 108 days after. But as it turns out, it's difficult to squeeze in 108 days between Epiphany (the 12th day after Usher Christmas -- a Friday in the January 2nd-8th range) and Juneteenth (the third Monday in June).

This stretch contains 23 weeks. Assuming one of these weeks is spring break, that leaves us 22 weeks -- in other words, 110 weekdays. So there are just two days to spare -- but we already know we need to place at least three federal holidays -- MLK Day, Presidents' Day, and Memorial Day. And that's not to mention other days we might wish to place -- Lincoln's Birthday, Good Friday/Easter Monday (depending on when spring break is), and so on.

Of course, you might wonder, why are we bound between Epiphany and Juneteenth anyway? (After all, there's been school on January 6th before, and Juneteenth is new.) The answer is that this calendar is meant to be perpetual (and perhaps implemented on the Gregorian Calendar someday, even without switching to Usher). Starting school before Epiphany in some years would mean starting before January 1st, and ending school after Juneteenth in some years would mean ending after the summer solstice. (I assume that even Usher can't change New Year's Day -- it must be when December becomes January -- and likewise he can't control the solstice.) So Epiphany and Juneteenth are Usher dates that stand as proxies for New Year's Day and summer solstice.

It's possible to have 108 days if we have a bare minimum of holidays -- MLK Day, Memorial Day, and Ski Week (including Presidents' Day) instead of spring break. But this isn't desirable -- the Big March would extend all the way to the end of May with no days off.

Instead, the quinters are only approximately equal. There are 75 school days before Christmas and 105 days afterward. This is the best that can be done on the Usher Calendar (and perhaps on the Gregorian Calendar as well).

Before we move on, note that there exist calendars that preserve the seven-day week, yet move all the holidays around. The most extreme example is the Fixed Festivity Week Calendar:

https://calendars.fandom.com/wiki/Fixed_Festivity_Week_Calendar

All the holidays are squeezed into four weeks of the year -- one holiday week near each solstice and equinox on the calendar. Then, as the link above points out:

Employees do not get paid holidays in any other week.

(In particular, New Year's Eve or "Sylvester" is the Sunday of holiday week, but January 1st, a Monday, isn't a holiday at all.)

Each season contains 12 weeks containing 60 weekdays. Notice now that it becomes very easy to set up a school calendar. Instead of quinters, this calendar naturally suits trimesters corresponding to the seasons of fall, winter, and spring. The winter and spring holiday weeks become winter and spring break, dividing the year into trimesters that total 180 days.

And summer break would be longer. It's a Leap Week Calendar so the correspondence isn't exact, but if it's set up so that the summer holiday week always contains the summer solstice, then the fall holiday week would end up taking the fall equinox. Then kids could have a true summer break, spanning the entire season of summer, from solstice to equinox,

But then all three trimesters would become three Big Marches. (It would be just like having no holiday between Ski Week and Memorial Day in the Usher Calendar above, except times three.) Once again, I'm not quite sure whether this is desirable.

Quinters: 6-Day Calendars

Before we attempt to apply quinters to calendars with six days per week rather than seven, we must remind ourselves that we shouldn't apply quinters at all unless they naturally fit the calendar.

Of all the possible six-day calendars, the Sexagesimal Calendar is best suited for quinters:

https://calendars.fandom.com/wiki/Sexagesimal_calendar

http://www.sexagesimal.org/en_propos.php

The year is divided into six "sixths" of 60 days each. The first sixth, "Frigee," starts on the day of the winter solstice -- as a result, the fourth sixth, "Granee," starts on the summer solstice. Then Granee could become a summer break, with the other five sixths becoming the five quinters of the school year.

Indeed, instead of starting after Labor Day, the school year starts after "Liber Day." The extra five days of the year become holidays placed between the sixths. The calendar author most likely chose Liber for the August holiday because Liber is the god of books -- as in school books for the first day of school.

Each sixth contains 10 weeks of six days, with four weekdays each. Since each quinter requires only 180/5 = 36 days, there are nine weeks of school and one week of holidays.

These holiday weeks can follow the adventitious days already listed. For example, the author already lists Bacchanal day as subsuming "lover's day" (Valentine's Day) and "last day of Carnival" (Mardi Gras), so these can be expanded to a full week similar to the Fixed Festivity Week Calendar.

Like Fixed Festivity Week the holidays become concentrated into one week, but instead of having to endure twelve five-day weeks before getting a break, we only have to endure nine four-day weeks. And of course, four-day weeks fit odd/even block schedules better than five-day weeks.

There are other calendars similar to the Sexagesimal Calendar. Double Rainbow is the closest:

https://calendars.fandom.com/wiki/Double_Rainbow_Calendar

But its seasons don't line up with holidays as well as Sexagesimal -- for example, Violet season corresponds to December-January, so Christmas would be right in the middle of the season. And neither Yellow nor Orange lines up with the desired summer break.

The Primavera Calendar is essentially the Sexagesimal Calendar, except with each sixth divided into two months of conventional length:

https://calendars.fandom.com/wiki/Primavera_Calendar

And the Raenbo Calendar is nearly identical to Primavera, except that the names of the months and days look like their Gregorian counterparts, and the dates are written in dozenal/base 12. (The link to the Raenbo Calendar no longer exists.)

Conclusion

As Calendar Reform week continues, we'll look more at how to implement quinters in calendars with longer weeks.

Small Business Saturday: Revisiting What If?

Table of Contents

1. Introduction

2. A COVID-92 What If?

3. Middle School Pandemic Letter Grades

4. Possible COVID-92 Report Cards in 7th Grade

5. A What If? That Matches My Students' Struggles

6. Possible COVID-93 Report Cards in 8th Grade

7. Revisiting the Long-Term Sub Position

8. The Eight-Period Block Schedule

9. Rapoport Question of the Day

10. Conclusion

Introduction

I knew that Thanksgiving was truly over once I checked my email yesterday and found an avalanche of late assignment notifications from Google Classroom and other questions about grades.

Today is Small Business Saturday, the day  when customers are asked to support local stores rather than the big box retailers of Black Friday or the large online companies of Cyber Monday. I think of this day as State Meet Saturday, the day when Cross Country champions are crowned here in California.

It is my tradition to make my annual viewing of McFarland USA on State Meet Saturday. And I'm also watching the Christmas comedy Elf today, only because TBS is airing a marathon of it today. I've discussed both films on the blog before -- in particular, the accuracy of their calendars. The dates in Elf work out if we assume that Buddy arrives in New York on Monday, December 20th. But the timeline in McFarland USA are completely wrong -- in particular, the State Meet is depicted on December 12th, when the race has always fallen on the Saturday after Thanksgiving.

(On second thought, I wonder whether the reason for placing the race on December 12th is that it is Guadalupe Day. An image of the Lady appears throughout the film, including on the Whites' wall when the family moves into the house. So it's only fitting to put the climax of the movie on her feast day.)

Three years ago on the blog, I wrote about my Cross Country career on State Meet Saturday. And since then, I also wrote some of the COVID What If? stories during Thanksgiving break -- in other words, what if the pandemic had occurred not in 2019-2020, but back when I was a young student? How would such a pandemic have affected my high school (including my Cross Country) career?

I wasn't sure whether I wanted to extend the What If? stories this year. The What Ifs are supposed to end when the pandemic does, but many people believe that the pandemic is now becoming endemic, just like the flu -- a disease that never actually ends. This year, COVID-19 continues to affect us -- my students are still missing time due to COVID. And with cases creeping up again, COVID testing kits have been distributed for students and staff to use twice -- once tomorrow night before returning to school, and once during the upcoming week.

But more importantly, the COVID What Ifs force me to put myself in my students' shoes and understand what they've been going through over the past few years. And since I'm still trying to avoid arguments and make better connections with my kids, some empathy on my part definitely can't hurt.

And so I'm continuing the COVID What If? stories in this post.

A COVID-92 What If?

So which year should I choose for the pandemic year in today's What If? Well, I'm supposed to base it on my current students -- first I should figure out what grade my kids were in when the schools were shut down, then place the pandemic in the year when I was in that same grade as a young student.

I'm currently teaching Math I and III, so my students are mainly freshmen and juniors. Since three out of my five classes are Math I, the focus should be on the freshman. They were sixth graders when the schools first closed, and so I go back to my own sixth grade year, which was 1992-93. This means that today's What If? should be COVID-92. The date COVID-n refers to the December 31st of the year n, when the virus was first identified. The schools would then close the following March.

But there are problems with a COVID-92 What If? story. First of all, COVID-92 would clash with some of the previous What Ifs that I've written -- in particular COVID-91. I like this story because the dates work out better -- March 13th was a Friday in both 1992 and 2020, and so the schools can close on the exact same date in both COVID-91 and COVID-19. (And that's not to mention the simplicity of just reversing the digits in COVID-19 to obtain COVID-91.)

And more importantly, it's more difficult to compare a sixth grade closure at the K-6 elementary school that I attended to a closure at the 6-8 middle schools that my kids attended. For them, losing sixth grade just meant losing the last part of the first year of middle school, but for me, it would mean missing sixth grade graduation and other activities such as sixth grade camp -- which my kids would have attended in fifth grade at their K-5 elementary schools, just before the pandemic.

Then again, on the original timeline, I didn't go to sixth grade camp (although my friends did). The last field trip I attended as a sixth grader was to the Fullerton Arboretum (that is, a tree-place) in Orange County -- I don't remember the exact date, but I keep thinking it was in February or March. So I believe that we would have barely made it on this trip before COVID-92 shuts down the schools.

Two years ago during my long-term assignment, I had a choice whether to blog about COVID-92 (to match my then-seventh graders) or COVID-93 (for the eighth graders). And I chose the latter, just to avoid the "6th = elementary or middle school?" problem. And thus I have both COVID-91 and 93 What Ifs on the blog, crowding out a possible COVID-92 story.

Let's continue to explore the COVID-92 What If? anyway -- and perhaps in doing so, we can find out whether another year might match my current students' experiences and allow me to empathize better.

Middle School Pandemic Letter Grades

For sixth grade, my grades would been frozen at what they were at the time of the closure. These would have been straight A's, except for a C grade that I was earning in health. Seventh grade would have been a different issue -- I would have needed to earn my grades during distance learning.

This week I was reading about some parents who were complaining about their students' grades during the pandemic closures. While the grades earned from March-June 2020 didn't count, the parents don't want the grades during the 2020-21 school year to count either.

As I read the article further, it appears that the creator of the petition is the mother of a current eighth grader who's trying to apply to magnet high schools in their district. And the district won't even look at the application unless a certain GPA is obtained in Grades 6-7. The child earned straight A's in all classes that were held in-person before the pandemic closure (fifth grade) and afterward (seventh grade) but struggled just to scrape by with C's in sixth grade in 2020-21. Thus the parent wants the grades that year not to count, so that her kid's GPA for the purpose of high school entrance would become 4.0.

(By the way, though this petition is in some Southern California district, many articles have been written about New York and its high school application process. The difference is that the Big Apple only looks at seventh grade marks, not sixth grade as in California.)

It's tough for me to decide whether I agree with this mother or not. But since we're having a COVID -92 What If? fantasy here, let's look to see what grades I would have earned during my first year of middle school (in my case seventh grade), and whether I would have struggled as much as the students who are the subjects of the petition.

On the original timeline, I took Algebra I as a seventh grader, after having studied Pre-Algebra independently from Grades 2-6 at my elementary school. My teachers would send my work to the 7-12 school, and the math teacher there would grade it and assign new work for me to complete. In the seventh grade, I finally met the teacher who was grading my work all those years, and she became my new Algebra I instructor.

Anyway, her policy was that grades would be based on quizzes and tests, except that homework would add a plus or minus to the grade. So if your grade based on tests was a B, then it would become a B+ if enough homework was turned in and a B- if there's not enough work.

At the end of the first quarter, my grade was an A-. In other words, I had earned an A based on the tests alone, but didn't submit enough work to avoid the minus. Of course, an A- in math isn't bad, nor is the S for citizenship. The problem was the N that I received for work habits -- along with a comment stating that I hadn't turned in enough homework.

Why was I slacking in homework that quarter after having faithfully turned in assignments to that same teacher ever since second grade? Most likely, the independent work led to bad habits on my part -- I'd only turned in work whenever my elementary teachers were ready to drop them off at the 7-12 school, which might have been once every couple of weeks. But once I was at that school myself, I was expected to turn in work each and every day -- and I wasn't prepared to do so.

After that embarrassing N on my first quarter report card, I never had problems turning in work to Algebra I class again. For the other three quarters, my grades were all A+'s -- that is, A's on the tests and sufficient homework turned in. (And yes, A+ was a real grade at that school.) Of course, this all refers to the original timeline (where the pandemic doesn't happen until 2019-20), not COVID-92.

Recall that for our COVID What Ifs, distance learning is based on the technology that was available during that actual year -- so there is no Zoom school in the COVID-92 world. My belief is that had a pandemic occurred in the 1990s, distance learning would have consisted of packets to be completed at home and submitted to the teachers each week. That is, it would have been similar to the independent Pre-Algebra work that I had for Grades 2-6, except that it's for all students (not just me) and all teachers, and that the students must go to the school to turn in and pick up new work themselves.

This means that for an Algebra I class, the entire grade would be based on homework -- and again, my first quarter grade on the original timeline was A-. If I recall, the threshold to get a plus instead of a minus was 75%. So if my COVID-92 grade is based on homework, then we must assume that my grade would have been no higher than 74% -- a C grade.

And this meant that -- just like the petitioners' children in the real world -- I would have earned a C in my best class, a class where I normally get A's, due to the pandemic.

By the way, there was nothing like applying to magnet high schools when I was in middle school -- my school spanned grades 7-12, so I was already at the school I'd attend as a freshman. The closest counterpart there would be the advanced classes -- eighth graders are assigned to advanced classes if they earn a 3.0 throughout the year in the corresponding seventh grade classes, and then they can remain in those classes for high school if they can maintain a 2.0 in the advanced classes.

Since middle school Algebra I is already an advanced class, I'd only need a 2.0 in order to make it into the Geometry class the following year, so that C wouldn't disqualify me. But now we must figure out whether I could get into the other advanced classes in eighth grade under COVID-92.

Possible COVID-92 Report Cards in 7th Grade

The COVID What Ifs are based on Homer Simpson's idea of crisitunity -- the pandemic is a crisis, but out of that crisis comes an opportunity. But so far, most of my stories focused on opportunity -- for example, in COVID-93, I avoid getting a C in science the fourth quarter of Science 7 since my grade would be frozen at a B once the schools close down for COVID that spring. That would then qualify me for Advanced Science 8 (which I didn't qualify for on the original timeline). And in eighth grade, I avoid getting suspended for hitting my PE teacher (due to a dare) because of distance learning, and so my grades don't drop in second quarter for missing a week of school.

And when I wrote about the COVID-96 What If? to line up with my seniors at the magnet school last year, my focus was on making the Varsity Cross Country team (because enough faster runners quit the team during the pandemic and open up the last alternate Varsity spot for me). Thus in both of those stories, I gain opportunities due to the pandemic taking place.

But the purpose of today's COVID What If? is to focus more on the crisis -- for me to empathize with my current students who are struggling in my class. Showing how my grades would have been higher under COVID isn't showing empathy. I must write about how my grades would have suffered.

And so I decide to use my TI random number generator to make up possible grades for the seventh grade under COVID-92. I believe that no matter what, I would have avoided D's and F's, and so I randomly assign an A, B, or C grade to each class. (For simplicity, we ignore pluses and minuses here.)

Here's what I came up with:

First Quarter:

1st period: C

2nd period: B

3rd period: A

4th period: C

5th period: A

6th period: C

Second Quarter:

1st period: A

2nd period: A

3rd period: A

4th period: B

5th period: B

6th period: A

Third Quarter:

1st period: A

2nd period: B

3rd period: C

4th period: B

5th period: B

6th period: A

Now let's match these up with my classes and find out in which classes I earn which grades -- and compare them to my actual grades on the original timeline.

For the first quarter, one of my C's is in fourth period Algebra I. This fits what I wrote above -- I don't turn in enough work, and since my grade is based only on homework, it drops to a C. The other two C's are in first period Art (exploratory wheel) and sixth period History 7. On the original timeline, I get a C+ in Art anyway, after having an F at the quaver progress report due to slacking off. So it's reasonable that under distance learning, I receive that very same C+. My History grade on the original timeline was a B, so it drops to a C under COVID-92. This quarter would mark the only time during K-12 that my GPA would be below 3.0, and thus I'd fail to make the honor roll.

The second quarter is my best quarter on both the original and COVID-92 timelines. In real life I earn straight A's except for a B in first period Shop (exploratory wheel). On COVID-92 I earn an A. In some ways this makes sense -- on the original timeline, I likely struggled a little with something I had to make out of wood or metal, so my grade was a B. But that difficult project would likely not be required under distance learning, and so without my project, my grade would be A. Meanwhile, after getting a C in Algebra I first quarter, I turn in enough work second quarter to get a B instead,

In third quarter, the lone C is in third period PE. I already discussed the independent study PE that I had to do on the original timeline after my suspension. I can easy see myself slacking in PE during this quarter under COVID-92 -- third quarter spans February and March, the rainy season in California, and so if it rained a lot, I'd fall behind on my required PE activity. Meanwhile, I earn an A in the first of two quarters of first period Science 7, an improvement from the B on the original timeline. I know that I was distracted a lot in real life with talking the other students, and there was a tricky dissection project during that quarter. With neither other students to distract me nor a dissection project during distance learning, getting an A in that class is now plausible.

The schools would reopen on April 1st (which was when the state encouraged schools to reopen). In 1994, April 1st was Good Friday -- not just the last day before spring break, but of the third quarter. I point out that in 2021, April 1st was Maundy Thursday and the last day before spring break as well (now that Good Friday is a school-closing holiday in that district) -- but the school reopened on that date anyway. So it's not a stretch to have the schools reopen after COVID-92 on April 1st, 1994 -- but it is awkward to have the kids attend their third quarter classes for just one day (especially since it's hybrid, with only half the school attending that one day).

And thus we'll declare April 1st to be the first day of the fourth quarter instead -- and with schools  open, my fourth quarter grades will match the original timeline. This means that I still get the C that I earned in Science 7 fourth quarter -- but due to the third quarter A, my average is 3.0 and so I still get into Advanced Science 8 under COVID-92 (as opposed to real life). Indeed, all of the other C's in core academic classes are balanced by A's in other quarters, and so I end up qualifying for advanced classes in all subjects for eighth grade. And this makes sense -- if I earned any C during distance learning, I'd be more motivated to work harder and get a A the next quarter.

So this means that this What If? still doesn't match my current freshmen's struggles. I still find a way to qualifying for all the same advanced classes as the original timeline and even sneak in an extra class, Advanced Science 8, to boot.

Before leaving this timeline, I point out that just as I avoid getting suspended in COVID-93, I also have no suspension in eighth grade of COVID-92 as well. That's because in real life, I first met the kids who dared me to hit the teacher in May of sixth grade -- by which time the schools have already been closed on the COVID-92 timeline. And in real life, I meet them again in my regular Science 8 class -- but I qualify for Advanced Science 8 under COVID-92. I'll still see those kids in PE class, of course -- but we'll be too busy playing sports for PE and will most likely avoid interaction and any double-dares.

A What If? That Matches My Students' Struggles

Again, the whole purpose of this What If? is to see how the pandemic closure -- from March of their sixth grade year to the entirety of their seventh grade year -- affected my student's learning. But I can't imagine a pandemic during my own years in Grades 6-7 -- or any other year, for that matter -- would result in my struggling with, say, slope and linear equations, as much as they are now. Math was always my strongest subject -- that's why I became a math teacher. My math grade might drop to a C because I forget to turn in work, but I'd always understand how to find slope no matter what.

This is why I often like to compare my student's math struggles to my own problems in subjects other than math. Perhaps my kids are having as much trouble in math as I had in, say, English, or History, or even science. More precisely, I should say Biology -- the physical sciences are more allied with math, and so anyone who succeeds in math should have no trouble with them. (Indeed, as you already know, I've had trouble with life science both as a student and as a teacher at the old charter school.)

But those other subjects aren't as cumulative as math is. A student who struggles in Biology (say because of a pandemic and distance learning) can still excel in Chemistry. A student who struggles in World History can still excel in US History. A student who struggles reading Romeo and Juliet one year can still excel reading Julius Caesar the following year.

There is, however, one class that is as cumulative as math. It's a subject that, if I struggle on it during a pandemic year, I really would have trouble catching up with it. And it's the only subject to which I can compare my own struggles to those of my math students.

That class is foreign language.

At the 7-12 school that I attended as a young student, eighth graders were allowed to take a high school foreign language class provided that they also qualified for Advanced English 8 -- which of course I do, on both the original and all What If? timelines. Like most schools, both Spanish I and French I were offered, and I ultimately selected French I -- but why? After all, there are many Spanish speakers here in California, and so Spanish would be much more useful than French.

It's because back in fourth grade, one of the teachers at the elementary school was a French speaker, and so she offered an introductory French class before school to advanced students in Grades 4-6. Even though it was for just once a week for one quarter of one year, I was intrigued by learning a new language, enough for me to want to learn more French four years later.

On the original timeline, I earned A's and B's throughout French I in eighth grade. But in French II and III, I consistently earned only B's in that class. At the end of my sophomore year, I was given the choice whether to continue into the next level of French -- AP French Language. I knew that the AP exam was quite demanding -- every question was written in the target language, and for the free response question I'd have to give an entire conversation in French.

And so the French teacher and I agreed that AP wasn't for me. Instead, I took French IV -- which was basically an independent study class. (The class was actually French III and IV combined -- the teacher would actively instruct the III students while checking to see how we IV students were doing.) Due to the structure of the class, it was an easy A -- which the AP class certainly would not have been.

OK, so this is all the original timeline. If we wish to make this into a COVID What If? then we must decide when to place the pandemic. Since I took French I as an eighth grader, we should place the pandemic in March of my seventh grade year. Then the year corresponding to 2020-21 would be the year of eighth grade French I.

This means that we should go right back to COVID-93 -- the What If? that I first explored two years ago at the long-term middle school. Even though the year corresponding to the present would now be my sophomore year as opposed to my kids' freshman year, it matches in that my students graduated from their K-5 schools before COVID-19, just as I graduate from K-6 before the COVID-93 closure.

What would French I look like under 1990s-style distance learning? I can see a situation where we're given long lists of French vocabulary words, and perhaps even a cassette recording in French. Then when we make our weekly visits, we must read something in French and speak to the teacher in the target language. In other words, it would have been like the AP exam every week -- and that's something that I would have definitely struggled in, as much as much as my students are in math.

Possible COVID-93 Report Cards in 8th Grade

We're throwing away the COVID-92 What If? in favor of COVID-93, so now we must figure out what my eighth grade report cards would look like under distance learning. Recall that under COVID-93, my third quarter B in Science 7 extends into fourth quarter (thus getting me into Advanced Science 8), and I avoid hitting my PE teacher -- but the focus here is on French, not science or PE.

Though I'm throwing away COVID-92, I won't let those COVID-92 report cards go to waste. We'll just use those as COVID-93 report cards and apply them to my eighth grade classes instead.

So once again, I get first quarter C's in first, fourth, and sixth period, but these are different classes. In particular, sixth period is Geometry, and first period is Science 8 (now Advanced). I might get a C in Geometry for slacking with the homework (just as I did in COVID-92), and while Science 8 is based on physical science which is more to my liking, I still get a C for the same reason as Geometry. What matters the most here is the C in fourth period French -- and that's exactly the grade I expect to get here.

Second quarter is my best quarter once again. On the original timeline, second quarter is my worst quarter, but that's mainly due to the suspension which is eliminated under COVID-92. But notice that my C's in science and math become A's second quarter, while my French grade only rises to a B. Again, this makes sense -- I work harder in French after getting a C, but I still can't memorize all the words or speak enough French to my teacher, so the grade is a B.

Third quarter is a bit tricky here. On the original timeline, the fallout from the suspension results in my switching from third to first period PE at the semester and becoming a library aide third period. In past posts, I suggest that for COVID-93, I keep third period PE and take first period Keyboarding as my extra elective. Thus the C grade here is for PE (and for the same reason as COVID-92 -- rainy weather and laziness in February and March).

Once again, the schools reopen for fourth quarter. But this time, let me run my random grade generator once more for fourth quarter rather than rely on the original timeline for the grades:

Fourth Quarter:

1st period Keyboarding: A

2nd period History: B

3rd period PE: C

4th period French: C

5th period English: C

6th period Geometry: B

And there we have it -- even though I'm back in person, I'm still having trouble with French. I fall behind earlier in the year, plus it's still hybrid, so some work at home is still needed. Notice that the grades in other class are also suffering, perhaps as stress over French class. Here it happens that the closer a class is to French, the lower the grade is.

But recall that there was also an odd/even block schedule during hybrid, so this might not have happened in real life. It might make sense to make periods 2/6 the other C's, but I don't want to give myself another C in Geometry. At any rate, the stress I feel during French class is starting to equal that my kids feel during math class.

We can extend this What If? into freshman (2021-22) and sophomore (current) years. I move from my 7-12 school to my new high school in November of my freshman year. On the original timeline I earned only A's and B's throughout high school, but in COVID-92 I continue to earn C's in French II and III. I was considering even sneaking in some D's and declare that I don't advance all the way to III, but here we'll say that I advance, just as my kids moved from middle school math to my Math I class.

Since it's still State Meet Saturday, I wonder what my Cross Country career would look like under this COVID-93 What If? I believe that there's enough time between the pandemic and my XC career that XC would match the original timeline -- unless we wish to assume that stress over my French class is so strong that it affects my ability to run. Then again, if I hear that math is affecting the ability of the athletes in my class to compete, I might change COVID-93 to reflect that.

Recall that for COVID-96, I place myself on the Varsity team as a senior. This would correspond to freshman year under COVID-93 -- but this was before I moved, so I was still at the 7-12 school. Our XC team was so small that I could see only seven total runners even being on the team, so any runner would be a Varsity runner by default. Yes, this would make me a Varsity runner, but it would be hollow, since our team was unlike to qualify even for CIF Prelims -- and I wouldn't have made even to League Finals, as that was the week when I moved to the new school.

(And that's assuming that I'd even make the XC team in COVID-93. I was only inspired to join XC when my second semester PE teacher saw me run the mile and recommended me -- but if that was third quarter and hence still distance learning, no one is there to see me run. Let's assume that I run the mile in fourth quarter after the schools reopen, and then the PE teacher recommends to the XC coach.)

Revisiting the Long-Term Sub Position

There's another way to think about what my math classes look like from my students' perspective without considering COVID What Ifs. After all, recall that the reason the COVID-93 What If? from two years ago matches today's story is that these are the same cohort of students. The seventh and eighth graders I taught during that long-term sub position are now freshmen and sophomores, the same age as my current students.

Of course, they aren't literally the same kids. The long-term subbing was in Orange County, while my current position is in LA County. Moreover, the kids from Orange County are currently attending a school that offers traditional Algebra I and Geometry, not Integrated Math I -- and back then, OC had a year of mostly hybrid while LA County was at home the entire year. Nonetheless, both groups were the same age at the time -- and so if I want to know what it's like for kids to have their school close in the middle of sixth grade, I should look at those OC kids.

Let me reblog one of my old posts from Fall 2020. This one is a Math 8 lesson (current sophomores), but it's on slope, which is a key topic in Math I as well:

In the eighth grade classes, I continued with the lesson on comparing functions. This is an introduction to the idea of slope as the rate of change, including finding slope using the rise and run on a graph.

There's no sugarcoating this one -- this lesson is a struggle. It takes the entire block period today to get through the lesson in first period. Fortunately, since students stay in first period for today, I have them attempt Quiz 3.1.5 today. Most of the students who try it need two or three attempts just to pass it by getting three out of five correct. Indeed, three students proceed to Lesson 3.2.1 on slope-intercept form even though I tell them not to (once they start it, they must finish it). Of those three, two of them get perfect scores on Quiz 3.2.5 while the other at least matched his Quiz 3.1.5 score (and this is without getting the actual lesson).

I suspect that this is because today's lesson involves graphs, which are always tricky. Of course, graphs are also related to slope-intercept form, but this is more about plugging in values to the equation y = mx + b, whereas today's rise and run questions involving inspecting the graph directly.

Recall that on Monday, the eighth grade teacher leader suggested an Edpuzzle activity and a graphing worksheet that might help the students out. I'm still trying to figure out how to assign these, so I didn't do so for first period -- but after seeing the students struggle, I might make more of an effort to set up these assignments to help fourth period practice the graphs.

On one hand, I'm hoping to get through Lesson 3.1.x today and Lesson 3.2.x on Monday, thus setting up the rest of the week for another lesson suggested by my colleagues. I don't want to give it away in today's post, but it has something to do with Halloween (and graphing lines in slope-intercept form). Of course, what matters the most is making sure the students are learning. Rushing through a difficult lesson just to get to an assignment in time for Halloween is not teaching.

So these are the students I'm teaching now -- kids who had trouble getting slope from a graph two years ago, and are still having trouble with it now. Except for those three students who rushed through the lesson and had no trouble figuring out slope-intercept form, everyone else struggled. That year, we had to come up with an extra Edpuzzle to help the students understand it.

That Halloween lesson reminds me of what I did for that holiday this year -- dive into Chapter 3 on transformations, even though we were still having problems with Chapter 2. I wonder whether in both cases, Halloween lessons were set up based on the belief that we'd be further in the text, but once we slowed down due to students' lack of understanding, we had to jump over material just to reach the Halloween lesson by Halloween.

Here's a slightly later Math 8 lesson on solving two-step equations, just before Thanksgiving break:

Well, in the eighth grade class, I attempt to go over some more two-step equations after singing the "Solve It" song for music break. But unfortunately, we do only two more equations.

The problem I have today is with the third quaver progress reports. I completed them on Aeries over the weekend, but the actual deadline for submitting them is noon today. And so I tell the students that they should make up any missing work by noon today in order to raise their grade from a D or F in order to avoid getting a progress report.

There are many more students getting progress reports in eighth than in seventh grade -- and this is true for several reasons. Much of it is because of APEX -- in order to advance to the next quiz, APEX is set up to require eighth graders to earn a score of 60%, but seventh graders must earn at least 70%. And since most quizzes have only five questions, "at least 70%" really means 4/5 or 80%. I force the seventh graders to keep retaking the quizzes and tests until they pass them.

Thus it's impossible for seventh graders to earn less than 70% if they pass the test -- and it's impossible for them to earn more than 59% unless they pass it. Therefore there are no seventh graders with D's -- and only two students in each period missed the test and have F's.

On the other hand, more eighth graders get D's on all the quizzes and the test, and so there are eighth graders with D's, in addition to the ones who missed the test and have F's. And the cohort with the most progress reports is today's first period class -- and it also happens to be the last full class meeting before the noon deadline.

And so I spend the entire time after music break checking and resetting APEX tests. This is for students who missed the test as well as those with D's who wish for me to reset the tests in APEX. I'm glad that many of them are able to raise their grades to C's today.

Once again, we recognize many problems from this class creeping up in my current classes -- a lesson is ruined because most of the students are doing quiz corrections, right when grades are due.

And just after Thanksgiving break, I tried teaching this same lesson to the seventh graders:

Since it's Tuesday, this is a seventh grade post. The title of this post is "Solving Two-Step Equations," but in reality, today wasn't anywhere nearly as simple as doing the Texas two-step.

For starters, we had to squeeze in the Performance Task portion of the Benchmarks today. So once again, it's a balancing act between giving the Benchmarks and teaching the new lesson on APEX.

In the end, I decided to go 30-40-30 for consistency. That is, the first half-hour of class is for last-minute review to make sure that the students understand what they are to do on the Benchmarks. Then the middle 40 minutes of class are for the Benchmarks proper. The last half-hour of class is the time into which I must squeeze the entire APEX lesson. (In Canvas, the test is set to unlock and relock at these particular times.)

You might think that 30 minutes would be enough time to show the students two-step equations -- especially considering that I already introduced two-step equations a few weeks ago in order to prepare them for the multiple choice Benchmarks. But the problem is that the APEX Lesson 4.3.1 isn't just about two-step equations. First, it discusses how to convert real-world phrases into math (the usual "total," "ten less than," and all that). Second, problems are solved by working backwards rather than traditional algebraic manipulation of equations. Third, two-step equations are actually solved. And finally, equations of the form p(x + q) = r are solved by first dividing by p, not by distributing p.

Time is wasted when APEX decides to slow down -- for some reason, at certain times it doesn't let me advance to the next page of the lesson. Then I almost tore my hair out trying to decide which parts of the lesson to cover and which parts to throw out.

Of course, I'll fix these problems and improve the lesson throughout the week. But this isn't necessarily fair to the Tuesday cohort. They always get the short end of the stick with an unpolished lesson as I figure out the best things to teach as the week proceeds. I suppose that even before the pandemic, this has always been a problem with secondary teachers -- first period gets the rough version of the lesson, while the last class gets the polished version as teachers adjust throughout the day. (To this end, period rotations are useful because the same group of students isn't always first.)

And indeed, I'm still trying to figure out what to teach tomorrow. The last section of the lesson, on solving p(x + q) = r, will certainly be thrown out. (By the way, the APEX method is influenced by the way it appears in the Common Core Standards.) But then the department head tells me that I should replace this with solving equations px = r where p is a fraction -- this isn't adequately covered by APEX at all. And so there's still lots for me to cover. One way to save some time will be to have the students fill out the chart for the first part (listing all the addition phrases, subtraction, etc.) as they individually finish the test.

Now we see another lesson that the students struggle with -- and once again, it's because they're something else going on that day, namely District Benchmarks. We also see our online text that year, APEX, trying to squeeze everything and anything into a single lesson. I notice that our current CPM text sometimes puts too much material in a single lesson too.

The important thing to notice is that my students are struggling the way I taught them back then -- and this same year-group of students is still having trouble with my teaching methods. After having placed myself in my students' shoes, I can see that I need to adjust the way I teach, as soon as possible.

The Eight-Period Block Schedule

One thing I've noticed lately about some of the local high schools is that a number of them are starting to adopt an eight period day. In particular, both of the secondary schools that I attended as a young student now have eight periods, as well as a school I subbed at during the earliest years of the blog. But what exactly is that eighth period used for?

First of all, let's revisit what block schedules look like at most high schools. I once wrote that if a school were to convert a six-period day naively to a block schedule, each block would be two hours -- and that's too long for most students and teachers. Thus most schools add something else to the schedule in order to avoid two-hour blocks. For example, the high school I attended used to have a seventh period that met each and every day. Athletes would have seventh period sports, while most other students would attend one of their other classes for Tutorial. The school where I worked last year did the same thing, except the class that met all days was zero period Advisory. Mondays were all-classes days.

My current high school has four blocks meeting on each block day, with each block at 90 minutes. As there are six periods, there are three block days, with each period meeting two blocks per week. Then the remaining two days (Monday and Friday) are all-classes days.

Thus the eight-period day looks like my current four-block schedule, except that odd periods meet on one block day and even periods meet on the other day.

But most students have only six classes, occasionally seven. So for the extra period, most students end up doing -- nothing. They are "unscheduled" for that period. Most unscheduled periods are either first, second, seventh, or eighth periods -- that is, the first or last block of the day. Then those with Period 1 or 2 unscheduled don't need to arrive at school until Period 3 or 4, and those with Period 7 or 8 unscheduled can leave after Period 5 or 6. From the teacher's perspective, each teacher would have six classes, with two conference periods. Usually one prep period would be odd and the other even so that three classes are taught per day. Also, at least one prep period would usually be 1/2/7/8 since fewer students are scheduled these periods.

Notice that the eight-period day has nothing to do with the COVID What Ifs. I added a block schedule to the COVID-93 (and earlier) What If? because the block schedule simplifies the hybrid plan. But I don't blindly add eight periods to any What If? because the pandemic has nothing to do with adoption of the eight-period day. Still, it might be instructive to look at what my own schedule as a young student would have looked like if the eight-period day had existed back then. 

First of all, I was a Cross Country athlete. Most coaches like to have practice on all days, and so sports are usually scheduled for both seventh and eighth periods. Thus my schedule would have looked not much different from that on the original timeline, with my "period 1-6 + 7th period sports" being replaced with "period 1-6 + 7th/8th period sports." The only change is that when I was in Grades 9-10, I had a fourth period "sports tutorial" to replace seventh period tutorial. On the eight-period day, most likely my second period class would be moved to fourth, and then I'd have second period unscheduled.

Another thing about the eight-period schedule is that there are no all-classes days. For one thing, with eight periods, each period would be short. Moreover, if all periods 1-8 were to met in order, then someone (like me with my hypothetical schedule above) would have to attend first period, and then have nothing to do during the unscheduled second period. Unscheduled seventh would also cause problems on all-classes days, with nothing to do while waiting for eighth period.

Thus eight periods result in a pure block schedule -- all five days of the week are block days. There is no longer any connection between day of the week and the schedule -- students must check the schedule to find out whether odd or even periods meet that day. (In my hypothetical scenario, I'd have to know whether to wake up for first period or sleep in until fourth period.)

Rapoport Question of the Day

Today on her Daily Epsilon of Math 2022, Rebecca Rapoport writes:

2^(2x - 5) 4^(3x - 4) = 8^(2x + 13)

To solve this, we might take the base-2 logarithm of both sides (here abbreviated "lg"):

lg(2)(2x - 5) + lg(4)(3x - 4) = lg(8)(2x + 13)

2x - 5 + 2(3x - 4) = 3(2x + 13)

Recall that my Math III students will be learning about logs next week in Chapter 5. But it will be only an intro to logs -- they won't be using logs to solve any equations until the next chapter.

Ironically, it could be my Math I students who will solve this equation in December. That last section of Chapter 3, Section 3.3, is on solving complex equations of this types. Of course, they won't be using logs to solve it -- instead they'll use the Laws of Exponents to rewrite both sides as a power of 2, and then set the exponents equal to each other. After doing this they'll reach the same equation as before:

2x - 5 + 2(3x - 4) = 3(2x + 13)

This is a linear equation that is solvable in Math I. Notice that it's an equation that requires all five verses of the "Solve It" song to sing, or all five parts of the "Don't call me after midnight" mnemonic:

2x - 5 + 2(3x - 4) = 3(2x + 13)

2x - 5 + 6x - 8 = 6x + 39 ("Don't" = distribute)

8x - 13 = 6x + 39 ("call" = combine like terms)

2x - 13 = 39 ("me" = move the variable)

2x = 52 ("after" = add/subtract)

x = 26 ("midnight" = multiply/divide)

Therefore the desired solution is 26 -- and of course, today's date is the 26th. Of course, I'm really hoping that Math I won't get to Section 3.3 at all since this Math III equation has no business being taught in Math I. (But I'll have to teach it if it appears on a Chapter 3 Test or final exam.)

Meanwhile, tomorrow's Rapoport question is also related to logs. It's simpler, so we might be able to solve it in the current Math III chapter:

If y = -log_3((log_8 2)^3), what is 3^y?

As usual, we should work from the innermost parentheses outwards:

y = -log_3((log_8 2)^3)

y = -log_3((1/3)^3)

y = -log_3(1/27)

y = -(-3)

y = 3

3^y = 27

It's possible to skip a few steps -- here -log_3(1/27) can be transformed to log_3(27) if we remember what it means for a log to be negative. Then since y = log_3(27), 3^y must be 3^log_3(27), and since the exponential and log are inverses each other, we're left with just 27. In either case, the correct answer is 27 -- and of course, tomorrow's date is the 27th.

Yes, tomorrow is the 27th, and the next day is the 28th -- back to school after a long vacation, which is always tough. My next post will be the day after that -- the 29th, a Tuesday.

Conclusion

I was thinking back to COVID-93 again and my decision to take French as my foreign language. As difficult as learning a language is, I knew that I needed it in order to get into college. And I enjoy learning about languages -- it's just not easy to do so.

There was once a girl who lived next door to me. I recognized her from some of my classes, but we rarely interacted with each other. But during the lonely isolation of COVID-93, I could easily see us speaking to each other on those long days when there was nothing else to do.

And one of the things we might have discussed is our upcoming schedules for eighth grade. I might even have asked her for advice on what language to take -- and if I had done so, she almost certainly would have recommended Spanish. After all, she was of Hispanic descent, and most likely spoke Spanish herself -- if not, she'd have a relative who spoke the language. So of course she'd suggest Spanish as the language worth learning.

And if I'd taken her advice and chosen Spanish, I would've had the benefit having a Spanish speaker living next door -- someone who might have helped me learn that same language during 1990s-style distance learning. While I doubt that receiving help from a fellow teenager would have raised me all the way from a C to an A, perhaps I'd at least gotten a B just by talking to her in Spanish. And that would have given me confidence to take Spanish II and III in subsequent years.

I've stated that I've been having problems communicating with my students lately. Part of the problem might be the language barrier -- Hispanics are in the majority, with many Spanish speakers. In fact, on Friday during the Hero Quiz when I asked about project suggestions, one guy asked whether I was Hispanic/a native Spanish speaker. The implication, of course, is that if I could have given some of the directions in Spanish, he and the others might have understood the project better.

So perhaps I could extend the COVID-93 What If? all the way to the present day -- today, nearly thirty years after the pandemic, my students won't have survived a pandemic as I did. But they will speak Spanish to me -- and having decided during COVID-93 to learn Spanish, a more useful language of French, I'll be back to reply to them in Spanish. And that will improve our communication.

But alas, that's the world of What If? In the real world, I must imagine ways to incorporate Spanish into my daily lessons -- either by learning a few words today or using Google Translate on the lessons. And meanwhile, I will get ready to make my annual viewing of McFarland USA -- a movie that's all about Californian students of Mexican descent who do speak Spanish throughout the film.

I suspect that the real Coach White ultimately learned some Spanish in order to communicate with his runners more effectively. And if Coach White learned the language, then so can I. It won't be easy to learn a new language, but it's definitely worth the effort.