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.
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