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.

Christmas Post (Yule Blog Challenge #5)

Table of Contents

1. Introduction

2. Yule Blog Prompt #7: Something New I Tried (or Still Want to Try) This Year

3. Analyzing 10EDL and 12EDL Tunes

4. Rapoport Question of the Day

5. Conclusion

Introduction

Today, of course, is Christmas Day. And tonight is also the last night of Hanukkah -- the night when all eight candles are lit. This combo is often referred to as "Christmukkah." Like Festivus, Christmukkah is generally attributed to a TV show, in this case The OC.

This also marks the second time I've ever posted on Christmas Day -- the first was two years ago. Both Christmas posts were driven by the Yule Blog Challenge and the desire to fulfill it by posting throughout the holidays.

Yule Blog Prompt #7: Something New I Tried (or Still Want to Try) This Year

Well, the one thing I definitely tried this year are my songs. While some of my songs come from previous years, some of them are brand new.

I'll post some of them today. Let's start with "Let's Get Mathematical," a parody of "Let's Get Physical." I wrote this to honor Olivia Newton-John, who passed away during the year:

LET'S GET MATHEMATICAL

Verse 1:

I'm sayin' all the things that you like me to know,

Makin' good education.

I gotta answer you just right,

You know what I mean.

I drew a coordinate plane for you,

Then two lines the axes to be.

The y-axis is up and down,

And x goes horizontally.

Chorus:

Let's get mathematical, 'matical,

I want mathematical.

Let's get mathematical.

Let me hear your math mind talk, your math mind talk.

Let me hear your math mind talk.

Let's get mathematical, 'matical,

I want mathematical.

Let's get mathematical.

Let me hear your math mind talk, your math mind talk.

Let me hear your math mind talk.

Verse 2:

I've been working, I've been good,

Tried to graph the points from the table.

It's gettin' hard, this graphin' lines,

You know what I mean.

I'm sure you'll understand my math homework,

Though I did my work mentally.

You gotta know that you're bringin' out,

The math person in me.

(to Chorus)

Not all of my songs are parodies. For Math III, an original song I wrote was "Transformation," about transforming parent graphs:

TRANSFORMATION

First Verse:

Parent graph? That's square root.

Orientation? Up to you.

Shift left four, then up one.

Stretch factor? Stretch by two.

Chorus:

Reflection.

Translation.

Compression.

Transformation.

Second Verse:

Parent graph? Hyperbola.

Orientation? Let's get down.

Shift right three, then down two.

Stretch factor? None to be found.

(to Chorus)

This song is written in a brand new scale -- the 10EDL scale. And in other songs I used 12EDL. These scales come from an old computer that I used decades ago. By composing in these scales, I'm able to come up with original tunes for these songs -- but these tunes are tricky to play on a standard guitar.

As the school year began, I wrote that I'd compose a few songs in the new scales first, and then discover which guitar chords sound good with these scales. Now that I'm done so, I'm ready to summarize the various chord riffs I played.

Analyzing 10EDL and 12EDL Tunes

Let's start with the 10EDL scale -- in the key of C, this scale goes C-D-E-F#-A-C'. The song that I listed above, "Transformation," is a simple 10EDL song. Here's the full melody:

F#-C'-F#-C-E-C' (repeated 4x for verse)

E-F#-A (repeated 4x for chorus)

And as I've written many times before, the 10EDL scale is very similar to the major pentatonic scale, except that it contains F# where pentatonic would use a G.

So what chords did I play for this song? Well, since the scale is so similar to C major pentatonic, of course I used a C major chord. But, you might point out, the C major chord goes C-E-G, but we changed the G to F#, so shouldn't C major be invalid?

Well, think about a true pentatonic song. We normally use the I, IV, and V chords -- in this case C, F, and G -- to accompany the song, even though there is no F in C major pentatonic (and even the G chord goes G-B-D, with no B in the pentatonic scale). Thus we're allowed to use notes in the chords even if they don't appear in the melody.

Now we need a secondary chord. We look at the scale again -- C-D-E-F#-A-C' -- and notice that there is a D major chord available, D-F#-A. And we can even add the C' to make it a D7 chord. Thus I've been using D7 often in my 10EDL songs. In fact, the only chords I used in "Transformation" were C and D7.

Of course, we don't use D7 in an ordinary C major song unless it's a double-dominant chord, V/V, that leads to the usual dominant chord G (or G7). But since there's no G in the 10EDL scale, I don't use the G (or G7) chords unless it's over a D in the melody. Only one of my 10EDL songs, "Growth Slope," had enough D's in the melody to justify playing a G7 chord. Therefore of all my 10EDL tunes, "Growth Slope" sounds the most like a traditional C major song.

As for 12EDL, this scale goes A-B-C-D-E-F#-A', so it closely resembles an A minor scale. Remember that the note I'm calling "B" here is a just ratio 12/11 above the root A, so it's about halfway between the notes Bb and B. On the guitar, I might play it as either Bb or B depending on the situation.

Here's an example of one of my simpler 12EDL tunes, "Linear Art," which I played in my Math I classes during that infamous linear art project:

LINEAR ART

First Verse:

Slope of line negative, graph it!

Don't be so negative, graph it!

Slope of line positive, graph it!

Let's all be positive, graph it!

Chorus:

Linear art,

Let's all take part.

Linear art,

We're all so smart.

Second Verse:

Here's a line vertical, graph it!

That one's horizontal, graph it!

Slope of line fractional, graph it!

Get into action all, graph it!

(to Chorus)

Here's the full melody:

F#-E-A'-C-E-F#-F#-D (4x for verse)

F#-F#-D-F# (4x for chorus)

Notice that unlike C major for 10EDL, a full A minor chord appears in 12EDL, and so I play Am chords in my 12EDL songs. For this particular song, there are so many D's and F#'s in the melody that I had to play yet another D chord.

This song avoids Degree 11, and so the issue of whether to play it as B or Bb doesn't arise. For some of the other songs, such as "Solve for y," I notice that a G chord fits the song the best (even though, once again, there is no G in the 12EDL scale). Then we can interpret Degree 11 as either Bb (to make a Gm chord) or B natural (to make a G major chord). Playing a G power chord (that is, with no third) also works in order to suggest the neutral third in the melody.

In a true Am song, we expect some sort of E chord as the dominant. During classical times, the usual choice would be E7, with a G# (harmonic minor), but in modern times, we might hear an Em chord with a G (natural minor). But once again, while there is an E in 12EDL, there's no G or G#. Once again, we could try an E power chord (but note that the B in the scale isn't a perfect fifth above this E).

By the way, I've gone back and edited the last few songs I posted. When I reflect upon these posts next year (assuming I'm teaching the same classes), I want the lyrics to reflect how I should have taught Math I Chapter 3 (to match my neighbors), rather than how I actually taught the chapter. Then when I see the songs, I'll teach it better next year.

The December 6th-7th song is now a transformation song, since this is the week that my neighbor teachers taught the transformation lessons:

SHORT TRANSFORMATION

Slide the translation,

Flip the reflection,

Turn the rotation,

Walk, glide reflection.

The melody is very simple (mainly due to the principal's observation that day) B-B-B-C-A'. Once again, I play some sort of G chord (major, minor, or power) over the Degree 11 notes, then end on Am.

The November 29th-30th song is trickier to change, since it now needs to be a Math III song on exponential and logarithmic functions:

A PRACTICAL USE FOR NONLINEARITY

First Verse:

How can I get, (putt-putt)

To reach my goal, (putt-putt)

To draw the graph, (putt-putt)

Exponential? (putt-putt)

Pre-Chorus:

Reflection, that's a flip,

Translation, that's a slide,

Compression, or a stretch,

Equation graphing pride.

Chorus:

Population growth,

There's actually,

A practical use,

For nonlinearity.

Second Verse:

How can I get, (putt-putt)

My memory to jog, (putt-putt)

To graph inverse, (putt-putt)

That's called the log? (putt-putt)

(to Pre-Chorus)

The changes cause a few problems, though. In particular, I had to replace the four-syllable word "Geometry" with the six-syllable word "nonlinearity." Also, I keep "Practical Use" in the title, even though only one practical use ("population growth") appears in the lyrics. (I might consider shortening the title to just "nonlinearity.")

The other songs I performed in December don't need to change. The "Fraction Busters" lesson should have been given on a Monday -- a non-block (and hence a non-singing) day, so there's no way to make the "Ghostbusters" parody song line up with the lesson. The other December parody, "Sweet Home Algebra," also remains where it is -- it actually fits that day's lesson (multiplying binomials) better.

Rapoport Question of the Day

Today on her Mathematics Calendar 2022, Rebecca Rapoport writes:

Find x.

Once again, we have a Geometry question where are the given information is in an unlabeled diagram, so I must provide all the labels. In Triangle ABC, we have D, E, F on AC, BC, AB respectively. Angle C = 130, AD = AF = BF = BE, Angle DFE = x, and we must find x.

At first I was tricked into thinking that Triangle ABC is isosceles -- but we don't know that, because we aren't told how CD and CE compare. Instead, the isosceles triangles are ADF and BEF (and that aren't necessarily congruent to each other, because we know only SS). So we must perform the necessary angle chase using only the two known isosceles triangles.

From isosceles Triangle ADF, we have Angle AFD = (180 - A)/2.

From isosceles Triangle BEF, we have Angle EFB = (180 - B)/2.

Since AFD + DFE + EFB = 180, this gives us (180 - A)/2 + DFE + (180 - B)/2 = 180, which then simplifies to 90 - A/2 + 90 - B/2 + DFE = 180, or DFE = (A + B)/2

Then from the original triangle, A + B + C = 180, and since C = 130, that gives us A + B = 50. So we obtain DFE = (A + B)/2 = 50/2 = 25. Therefore the desired angle is 25 degrees -- and of course, today's date is the 25th, Christmas Day.

Conclusion

On this Christmas Day, I'm making my fifth Yule Blog post, but the challenge leader Shelli is already on her eighth:

http://statteacher.blogspot.com/2022/12/review-games-i-want-to-try.html

Shelli's eighth post is on games she plays in her class just before a test. She links to a Balloon Pop game from Sarah Carter and states that the true origin of the game is from a decade-old post from a certain other blog. I've seen a version of Balloon Pop on the TV show Survivor, but it never occurred to me to play it in the classroom.

This concludes my post. I hope you were able to enjoy your Christmas -- or Hanukkah -- this year, and that your holiday is starting look like the old holidays we used to know pre-pandemic.

(By the way, an old joke is that Jewish people eat Chinese food on Christmas -- at least in years when it's not Hanukkah. So do they eat Chinese food or latkes today?)