Monday, February 29, 2016

Why Is There a Leap Day? (Day 112)

Today is February 29th, a very significant day on the Gregorian Calendar. It is the first time that February has had 29 days since 2012, and it won't have 29 days again until 2020.

Ever since my last calendar-tagged post on New Year's Day, I'd been planning to have a special calendar post for today. Here were my original plans:

-- Other Leap Units (Leap Seconds, Leap Months, Leap Weeks)
-- History of the Leap Day, from ancient times until today
-- Calendar Reform proposals with different Leap Day rules

Of course, today is still a school day, so I should be posting about Geometry today. But I knew that I could actually afford to waste a day on a Leap Day post. This is because last year, I only needed nine school days to cover my unit on perimeter and area (spanning Lessons 8-1 through 8-6 in the U of Chicago text). So I could begin Lesson 8-1 tomorrow and still be able to give the Chapter 8 Test on Friday, March 11th.

But today, February 29th, also has a personal significance. This morning was the funeral for my grandmother, who passed away last Tuesday. Ever since then, this blog has been in mourning -- I changed the background to all-black, and ever since that day all of my posts have either been brief or just cut-and-paste lessons from last year. Obviously, I haven't been in much of a mood to write my usual lengthy posts.

I was trying to decide whether or not to write my originally planned calendar post. Obviously, with today the day of the funeral I am still in mourning, but on the other hand, if I don't write a Leap Day post today, I'd have to wait all the way until 2020 to write it.

Notice that as much as I like to keep this blog religion-free, many of my posts with the Calendar label end up mentioning religion. This is because one of the main reasons for calendars throughout the ages was to set religious festivals. Our current calendar, the Gregorian Calendar, was named after a very religious man -- Pope Gregory XIII. My plan is to mention religion mainly in the Calendar-labeled posts (just as I mention politics mainly in the traditionalist-labeled posts), and keep the posts without the Calendar tag as secular as possible.

With today being the day of my grandmother's funeral, I can't help but dwell on the spiritual. And so I decided to write today's Leap Day post as planned. This allows me to justify writing about spirituality and religion here on the blog (as this post is Calendar-labeled), and also allows me to take a day off from writing about math (since I can still afford to start Lesson 8-1 tomorrow).

Here's the way I think about it -- we need a Leap Unit whenever the number of units in a longer period of time is not a whole number -- that is, the longer period of time cannot be evenly divided into the shorter units.

Leap Seconds, Leap Months, and Leap Weeks

Let's begin with the shortest Leap Unit of all -- the Leap Second. Why do we have Leap Seconds?

( ) the solar day cannot be evenly divided into SI seconds

The earth is slowing down in its rotation. So once in a while, a Leap Second needs to be added. The most recent Leap Second was on June 30th, 2015. Here's a link discussing the Leap Second:

By the way, here's one more reference to Leap Seconds:

( ) there can be negative leap seconds

Next, we look at Leap Months, which appear in lunisolar calendars. Why do we have Leap Months?

( ) the solar year cannot be evenly divided into lunar months

The two main Leap Month calendars in use are the Jewish and Chinese Calendars. In fact, the Jewish Calendar is currently in a Leap Month, called Adar I. Next month is Adar II, which is considered to be the actual Adar. Holidays that occur in Adar, such as Purim, are celebrated in Adar II, since Adar I is the Leap Month. Actually, I did some research, and I saw that there actually is a "Purim" celebrated in Adar I, called "Purim Katan." At the following link, a Jewish rabbi explains Purim Katan:

Meanwhile, in the Chinese Calendar, we are currently in the Year of the Monkey. The Year of the Monkey is special to me because this is the year that I will turn 36, or 3 times 12 -- therefore I must have been born in the Year of the Monkey.

According to this list, the last Chinese Leap Month was the ninth month of 2014, and the next such month will be the sixth month of 2017. Notice that whereas the Jewish leap month is always Adar I, the Chinese Leap Month varies -- the ninth month last time, the sixth month next time. Also, we see that unlike the Jewish calendar where Adar II is the "real Adar," in the Chinese calendar the first half of the Leap Month counts as the previous month, and the second half counts as the following month.

There's another lunar calendar in common use -- the Islamic Calendar. Unlike the lunisolar Jewish and Chinese calendars, the Islamic Calendar is purely lunar. There are no leap months, and so Islamic months vary through the seasons. This year Ramadan will start just before the summer solstice -- making the fasting days extra long. But in the year 2033, Ramadan will occur at winter solstice -- that is, it takes about 33 years for Ramadan to move all the way around the solar year. I provide a link to the Ramadan date:

It's also possible to have a Leap Week. Why do we have Leap Weeks?

( ) the solar year cannot be evenly divided into seven-day weeks

Existing Leap Week calendars are rare. One I know of is the Icelandic Calendar:

In the Icelandic Calendar, most years contain 52 weeks or 364 days. The calendar is similar to the Egyptian Calendar in that there are 12 months of 30 days each -- the difference is that there are only four extra days at the end instead of five, to reach 364 days. But some years contain an extra week, Leap Week, so that they contain 53 weeks or 371 days.

sumarauki (leap week): A leap week inserted after aukanætur just before miðsumar. 72 In the Gregorian version, it begins on 22 July, when that day is a Sunday, or on 23 July, when that day is a Sunday and the next Gregorian year is a leap year.

According to this rule, the last Leap Week occurred in 2012, but the upcoming Leap Week won't happen until 2018. So Leap Weeks occur less frequently than Gregorian Leap Days.

The Leap Day and Its History

Let's get back to Leap Days and February 29th. Why is there a Leap Day? On Google today, there is a special Google Doodle with the 29th rabbit literally leaping between the 28th and 1st rabbits. If you click on this Doodle, the search query "why is there a leap day?" appears. Well, by now we should know the answer to that question:

( ) the solar year cannot be evenly divided into solar days

The following link explains the origin of Leap Days and the Julian Calendar. It also explains the Doomsday Algorithm -- a method of determining the day of the week given its date, first devised by one of my favorite mathematicians, John Conway:

Julius Caesar introduced the concept of the Leap Year; i.e., the idea of adding an extra day to February in every year divisible by 4. His calendar is called the Julian Calendar, and it was used throughout the Western world until 1582. In 1582, Pope Gregory XIII decreed a modification of the Julian Calendar. He declared that Century years (that is years divisible by 100) are leap years if and only if they are divisible by 400. Thus 1700, 1800, and 1900 are not leap years, but 2000 will be a leap year. The resulting calendar is called the Gregorian Calendar.

The following link is, as of the time I clicked on the Google Doodle, these are the top news results to the query "why is there a leap day?"

I've incorporated information from all three links into today's post.

Here's a link to an old Mathforum post where Conway explains how to calculate the weekday of the assassination of Julius Caesar, the founder of the Julian calendar:

In the original Julian Calendar, 366-day years were called "bissextile" years. This name "bissextile" means something like "double-sixth," referring to the fact that the extra day was not February 29th but the sixth day before March 1st (the Calends of March, not to be confused with the Ides). If we count the Calends itself as the first day, then the repeated sixth day is February 24th. Until 1968, the official Leap Day was February 24th -- not until 1972 was February 29th made the actual Leap Day.

So the Julian Calendar is the simplest possible Leap Day calendar -- we simply add an extra day every four years, so each year would average 365.25 days in length. On your first birthday, you were either 365 or 366 days old, on your second birthday you were either 730 or 731 days old, and on your third birthday, you were either 1,095 or 1,096 days old. But on your fourth birthday, you were exactly 1,461 days old (according to the Julian Calendar). Because of this, I often refer to one's fourth birthday as the first Julian birthday. Then one's eighth birthday becomes the second Julian birthday, one's twelve birthday becomes the third Julian birthday, and so on.

The Telegraph link above mentions that in the 19th century play The Pirates of Penzance, Gilbert and Sullivan propose the idea that a person born on February 29th only has Julian birthdays. Therefore its main character, Frederic, must remain a pirate until his 21st Julian birthday -- when he is actually 84 years old.

Also according to the Telegraph link, a Maryland high school teacher once proposed that Leap Day should be a day off -- why should we work an extra day for free? As much as I wouldn't mind having an extra day off leading up to the Long March, I disagree with this -- especially for teachers. We work the same number of days every year -- 180 plus a few extra PD days -- regardless of whether there is February 29th or not. If our work schedule actually increased a day during years with a Leap Day, he'd have a point.

I myself was born in 1980. I did once meet a Leaper -- a girl from my school who was born on February 29th, 1980, which makes today her ninth Julian birthday. According to the 538 link above, Leapers are rare -- parents avoid scheduling C-sections on Leap Day because they want their children to celebrate a birthday every year, not just the Julian birthdays.

Now this is when we must consider religion in more detail. Pope Gregory XIII was the one who reformed the calendar -- but why would Protestants care about what the Pope has to say? For that matter, why would the Eastern Orthodox Church care about what the Pope has to say? Throughout this post, we must remember that there are three main branches of Christianity -- and these three branches observe the calendar differently.

Most Protestant countries switched to the Gregorian calendar by the 18th century (most notably Great Britain in 1752). But some Orthodox (most notably the Russian Orthodox Church) have never made the switch -- they are still using the Julian calendar. Other Orthodox Churches, such as the Greek, observe the Gregorian Christmas and the Julian Easter.

Throughout this post I am remembering my late grandmother. She was an observant Jehovah's Witness -- this is a branch of Christianity separate from the three main branches of Protestant, Catholic, and Orthodox. An American pastor named Russell created this religion in the 19th century, well after most Christian nations had switched to the Gregorian calendar.

There have been attempts to improve further on the Gregorian Leap Year, since even its average length of 365.2425 days is still slightly longer than the length of an average year. The Revised Julian calendar, accepted in some Orthodox churches, has Leap Days in 2000 and 2400 just like the Gregorian calendar, but then the next century Leap Days are 2900, 3300, 3800, and so on. The gaps alternate between 400 and 500 years. The average year length in this calendar is 365.2422... days.

The Dee Calendar

It must be pointed out that neither the Gregorian nor the Revised Julian Calendars actually accomplish the original goal of the Egyptian Calendar -- to make a day like the rising of Sothis (the summer solstice) appear on the same day every year.

The following link explains why the spring equinox can't occur on the same Gregorian date:

The vernal equinox cannot always occur on March 21 in the Gregorian Calendar (or on any single date) because in that calendar it is possible to have a leap year followed by seven non-leap years, e.g., 1796 (a leap year) followed by the years 1797-1803 (since 1800 was not a leap year). The true length of the seasonal year is about 6 hours longer than 365 days, so the vernal equinox occurs about 6 hours later from year to year. Over a period of seven years the last vernal equinox will occur about 42 hours later than the first, unless the date of the vernal equinox is made to be earlier by the insertion of an extra day in the preceding February. But if those seven years are all non-leap years, then this adjustment will not occur, so the 42 hours will always span at least two calendar dates.

The medieval Persian mathematician Omar Khayyam was one of the first to realize that an accurate solar calendar would usual have four years, sometimes five, between Leap Days, and so he decided to calculate how often a fifth year would be needed. He calculated that it would be about once every 33 years, according to the following link:

This calendar, used to calculate No-Ruz or the Persian New Year (at the spring equinox), is still used today in Iran.

About 500 years later, the British astronomer John Dee independently discovered the 33-year Leap Day cycle. This was right around the time of the Gregorian Calendar Reform -- in which England, as a Protestant country, didn't participate. Instead, Dee proposed to Queen Elizabeth I that she should adopt his calendar instead of Pope Gregory's calendar.

Well, any calendar with Leap Days would have to check this box:

( ) having months which vary in length from year to year is maddening

Here's one box that might be unique to the Dee Calendar:

( ) date arithmetic needs to be as easy as possible

Also, we haven't yet stated which eight years in a 33-year cycle have a Leap Day. As it turns out, we can choose the Leap Days in a way to make the arithmetic as easy as possible -- perhaps enough to avoid checking this box. Let's look at the following link:

The simplest implementation of the 33-year cycle, would continuously repeat, every 33 years, the first 8 leap-years, in the years 1 to 33 A.D, (nominally the years 4, 8, 12, 16, 20, 24, 28 and 32 A.D.). Long division would have been unnecessary to determine whether it is leap-year, since there is a short-cut using addition. Just add the century number to the number of years passed in the century. For example: for the year 2012 A.D., we add 20 to 12, and get 32 A.D. which is nominally a leap year in the traditional life of Jesus. This centurial-addition principle will on its first application generate a year-number in the first two centuries A.D. This derived year-number has the same remainder when divided by 33 (i.e. has the same position in the 33-year Anni-Domini Jesus-cycle) as the original year-number. If the result is in the second century A.D. rather than the first, then a second application of this additive principle will reduce this new year-number to a yearnumber in the first century A.D. (e.g. for 1996 A.D., we add 19 to 96 and get 115, a year in the second century A.D.; so then we add 1 to 15 and get 16 A.D.). Having used our principle to reduce the year to a corresponding year-number in the first century A.D., we may find that the resulting first-century year-number is greater than 33. In this case we then simply subtract 33 or 66 to bring it within the traditional life of Jesus (e.g. for the year 2017 A.D., we add 20 to 17 and get 37 A.D.; so then we subtract 33 and get 4 A.D.). One or two, double-digit additions, or, one addition and a subtraction, always suffice until 3498 A.D. After 3498 A.D., two additions and a subtraction or three additions may be necessary (but three additions and a subtraction would not be necessary until 340,099 A.D.). 

Yes, I did warn you that there will be references to religion in this post! Dee was trying to create a new Protestant calendar to rival the Catholic Pope Gregory's calendar, so we shouldn't be surprise with the reference to "the traditional life of Jesus."

Notice that according to this rule, 1981 to 2013 was a 33-year cycle (since 33 times 61 is 2013), and every fourth year in this cycle had a Leap Day, beginning with 1984. This implies that the Leap Days in the Dee Calendar are identical to those in the Gregorian Calendar during this cycle. But in the current cycle, from 2014 to 2046, the first Leap Day is not until 2017.

How well does the Dee Calendar fix the equinoxes and solstices? As it turns out, Dee, like Khayyam, was primarily concerned with the spring equinox. Khayyam wanted to calculate the date of No-Ruz, the Persian New Year, which occurs at the spring equinox, and Dee, a Protestant, was trying to calculate the date of Easter, which also depends on that equinox.

In Dee's day time zones didn't exist, but the relationship between time and longitude was known. So Dee knew that his Leap Day rule would fix the equinox for a particular longitude. If he was lucky, that longitude could be zero degrees -- the Prime Meridian running through Greenwich, in his home country of England. But unfortunately, it wasn't -- perhaps if it had been, we might actually be using the Dee Calendar today. Of course, he could have chosen a different Leap Day rule that did fix the equinox to March 21st in England. But that rule would have been more complicated -- resulting the "date arithmetic needs to be easy" box to be checked.

The first link above actually reveals the location of God's Longitude:

Simon Cassidy has proposed that John Dee, after formulating his calendar in the 1560s (1564-1568), considered the question of whether there is some longitude at which the vernal equinox always (or at least for several centuries) occurs on March 21, and that (after much observation, diplomacy and computation) he arrived at the correct answer: 77° west of Greenwich. At this longitude the vernal equinox will always occur on the same calendar date: March 21 in Dee's calendar (in accord with the ecclesiastical spring equinox date), or March 20 in the Dee-Cecil variant, during the 300 years from 1580 through 1879.

The existence of God's Longitude actually led an astronomer, Duncan Steel, to claim that the most important invention of the last 2000 years was, in fact, the Dee Calendar! Here's a link to this amazing story [edited for brevity -- dw]:

I choose to ask: "How did we get to where we are now?" The first step needed there is to define where we are, and the answer to that is: With the USA being the powerhouse of most of the rest of the world. Thus the branching point I look to is that which made the USA a reality. I do not mean the Declaration of Independence. I mean: What made the English first go and settle the Atlantic seaboard of North America?

John Dee and others (Thomas Harriot and Walter Raleigh amongst them) had secretly come up with a plan to implement a 'Perfect Christian Calendar' using the 33-year cycle (the traditional lifetime of Christ). To get the equinox to remain on one calendar day throughout the 33-year cycle one has to use as a prime meridian for time-keeping a longitude band which is just right, and quite narrow. It happened (in the late sixteenth century but with movement east since due to the slow-down of the Earth's spin) to be at 77 degrees west, which Cassidy terms "God's Longitude".

If you look down that meridian you will find that in the 1580s the settled areas (in the Caribbean, Peru, etc.) were under Spanish, hence Catholic, control. To grab part of God's Longitude and found a New Albion, enabling them to introduce a rival calendar — that Perfect Christian Calendar — and convert the other Christian states to the Protestant side, England mounted various expeditions which historians have since misinterpreted. In 1607 the choice of Jamestown Island seems bizarre from the settlement perspective — why not out on Chesapeake Bay, and away from the attacks of the local Algonquians led by Pocahontas' father Powhatan? — but makes sense from the paramount need to grab a piece of God's Longitude. But later utility/developments do not reflect the original purpose of the English coming to Roanoke Island and Jamestown Island any more than the Eiffel Tower was built to provide a mount for the many radio antennas which now festoon its apex.

If the English had never invented their non-implemented 33-year Protestant Calendar, then the USA as it is would not exist, and all of the scientific, technological and cultural development of the world over the past couple of centuries would be quite different. In view of this I nominate that calendar, due to John Dee, as the most important invention of the past 2000 years.

Notice that our capital, Washington DC, lies almost exactly on God's Longitude.

Back on New Year's Day, I introduced the Eleven Calendar, which was based on a post written by the Australian poster Wendy Krieger:

The shortest cycle for an exact calendar is 33 [years], giving dozenal 265;2XXXX.

Here she is actually referring to the 33-year cycle of Khayyam and Dee. Dozenal 265:2XXXX means the same as 365.24242... in decimal -- the average length of the Dee year.

The Usher Calendar

About six years ago, the American poster Andrew Usher also proposed a calendar with a new Leap Day rule. A long discussion of this calendar has been archived at Mathforum (the same place where some of the old John Conway discussions are archived):

Here is how Usher states his Leap Day rule:

3. That the leap year rule be changed to have a leap year occur every
fourth save that it be delayed when the leap year would start on a

At first glance, the Usher Calendar appears to be just another version of the Dee Calendar. Leap years usually occur four years apart -- occasionally five years apart. The only difference is that instead of the simple 33-year cycle of the Dee Calendar, the Usher leap year depends on the day of the week.

Here's the answer -- the Usher Calendar is actually a Leap Week Calendar in disguise! To see what's going on here, let's go back to another Leap Week Calendar -- the Icelandic Calendar:

sumarauki (leap week): A leap week inserted after aukanætur just before miðsumar. 72 In the Gregorian version, it begins on 22 July, when that day is a Sunday, or on 23 July, when that day is a Sunday and the next Gregorian year is a leap year.

But this doesn't explain that last part of the Icelandic Leap Week rule: or on 23 July, when that day is a Sunday and the next Gregorian year is a leap year.

This phenomenon occurs whenever we try to convert from a Leap Week Calendar (like the Icelandic) to a Leap Day Calendar (like the Gregorian). I give it a special name -- a Double Leap Year. Loosely speaking, a Double Leap Year is a year with both February 29th and a Leap Week. Here the year 1995-1996 (measured from the first day of the Icelandic "summer" in April 1995 to April 1996) contains both a Leap Week (in July 1995) and a February 29th (in 1996).

We can now see why Usher came up with his strange Leap Day rule -- the whole point of it is to avoid the Double Leap Year problem! Let's look at his Leap Week rule:

4. That the perpetual calendar can be made, by considering the first
day of the year of weeks to occur on the Sunday after the Assumption,
and if this is the first possible calendar day, it is called week 1,
and otherwise week 2, and every year runs through week 53. And this
calendar ensures that everything can be fixed to a day of a certain

Now the Assumption of Mary is a holiday celebrated in some Christian churches -- most notably the Orthodox churches. It falls on August 15th. So Usher's Leap Week Calendar actually begins on the following Sunday, which falls in the August 16th-22nd range. Usher writes that if this Sunday is the first possible calendar day -- that is, August 16th, it is Week 1, otherwise it is Week 2. We see that Week 1 is actually the Leap Week, since Usher numbers his weeks from 1-53 in years with a Leap Week and 2-53 in years without the extra week. We notice that August 16th, 2015 fell on a Sunday, so 2015 would contain a Leap Week in the Usher Calendar. The next Usher Leap Week is in 2020.

Notice that January 1st, 2004, fell on a Thursday. So Usher avoids the Double Leap Year problem simply by forbidding February 29th in years that begin on Thursday. Then whenever January 1st falls on a Thursday, the following August 15th must fall on Saturday, and so Leap Week must occur.

Let's compare the Leap Week Rules for the Icelandic and Usher Calendars:

Sumarauki (leap week) begins on 22 July, when that day is a Sunday, or on 23 July, when that day is a Sunday and the next Gregorian year is a leap year.

Usher Week 1 (leap week) begins on 16 August, when that day is a Sunday.

Usher doesn't need a special Double Leap Year case. He simply prevents the special Double Leap Year case from ever occurring by making sure that February 29th can never fall on Sunday (that is, a year with Thursday, January 1st can never contain February 29th).

We can see why Usher does this. Any Leap Week Calendar would have the following box checked:

( ) the solar year cannot be evenly divided into seven-day weeks
and possibly several other boxes as well. But the visible part of the Usher Calendar only changes which weeks have a February 29th -- and I already stated that the box that corresponds to Leap Days doesn't count since the Gregorian Calendar already has them. There might be less resistance to the Usher Calendar than to any other Leap Week Calendar since it makes no radical changes to the structure of the calendar, unlike calendars which have a visible Leap Week.

The most obvious differences between the Usher and Gregorian Calendars are the holidays. Usher places each holiday on a particular day in a numbered week (1-53) in the Leap Week Calendar.

So August 23rd is both the latest that Week 2 can begin and the earliest that Week 2 can end. That is, August 23rd is part of Week 2 no matter what. We can even define Week 2 to be the (Sunday to Saturday) week containing August 23rd. Likewise August 30th, exactly seven days after the 23rd, must be part of Week 3 no matter what, and September 6th must be part of Week 4 no matter what.

Here is a complete list of these anchor days that must belong to a particular week:

Week 2 -- Aug. 23rd
Week 3 -- Aug. 30th
Week 4 -- Sept. 6th
Week 5 -- Sept. 13th
Week 6 -- Sept. 20th
Week 7 -- Sept. 27th
Week 8 -- Oct. 4th
Week 9 -- Oct. 11th
Week 10 -- Oct. 18th
Week 11 -- Oct. 25th
Week 12 -- Nov. 1st
Week 13 -- Nov. 8th
Week 14 -- Nov. 15th
Week 15 -- Nov. 22nd
Week 16 -- Nov. 29th
Week 17 -- Dec. 6th
Week 18 -- Dec. 13th
Week 19 -- Dec. 20th
Week 20 -- Dec. 27th
Week 21 -- Jan. 3rd
Week 22 -- Jan. 10th
Week 23 -- Jan. 17th
Week 24 -- Jan. 24th
Week 25 -- Jan. 31st
Week 26 -- Feb. 7th
Week 27 -- Feb. 14th
Week 28 -- Feb. 21st
Week 29 -- Feb. 28th
Week 30 -- Mar. 7th
Week 31 -- Mar. 14th
Week 32 -- Mar. 21st
Week 33 -- Mar. 28th
Week 34 -- Apr. 4th
Week 35 -- Apr. 11th
Week 36 -- Apr. 18th
Week 37 -- Apr. 25th
Week 38 -- May 2nd
Week 39 -- May 9th
Week 40 -- May 16th
Week 41 -- May 23rd
Week 42 -- May 30th
Week 43 -- June 6th
Week 44 -- June 13th
Week 45 -- June 20th
Week 46 -- June 27th
Week 47 -- July 4th
Week 48 -- July 11th
Week 49 -- July 18th
Week 50 -- July 25th
Week 51 -- Aug. 1st
Week 52 -- Aug. 8th
Week 53 -- Aug. 15th

I like to refer to these anchor days as "Doomsday." The important thing to note here is that the Usher Doomsday matches the Conway Doomsday from March to August (or January to August, depending on how you interpret his "March 0th" rule). After the Week 53 Doomsday on Assumption Day itself, Usher Doomsday advances one day, so it's always the day after Conway Doomsday.

American Secular Holidays in the Usher Calendar

The easiest holidays to find in the Usher Calendar are Labor Day and Memorial Day. Usher begins by defining Labor Day to be the Monday of Week 4. We first note that the Doomsday for Week 4 is September 6th, and if this is a Sunday, Week 4 Monday is the next day, September 7th. If Doomsday is on Saturday, then Week 4 Monday is five days earlier, on September 1st. (We must remember that Usher weeks always begin on Sunday and end on Saturday for this whole discussion.)

Memorial Day is also easy to define in the Usher Calendar -- it is Week 42 Monday. We calculate that this is a Monday in the May 25th-31st range, so it agrees with the Gregorian definition. When I was a high school student, I noticed that as the first day of school was the Wednesday after Labor Day, the last day of school was usually the third Thursday after Memorial Day. This meant that the last full (five-day) week of school was the week in June between Memorial Week (with no school on Monday) and the last week of school (with no school on Friday). But in 2003-2004 -- a Double Leap Year -- the last day of school was only the second Thursday after Memorial Day. The last full week of school was in May, the week before Memorial Day. So even before I heard of the Usher Calendar, I knew that 2003-2004 had an exceptional school calendar.

Unlike Labor Day and Memorial Day, the other American secular holidays must be modified to fit the Usher Calendar. We begin with President's Day -- Usher places the holiday at Week 28 Monday. As it turns out, this means that the holiday occurs in the February 16th-22nd range -- one day later than the Gregorian 15th-21st range. In some ways, the 16th-22nd might be preferable -- notice that "President's Day" is not the official name of a federal holiday. The holiday is still called Washington's Birthday -- even though it can never fall on his actual Gregorian birthday, February 22nd. So the Usher Prez Day honors Washington because it can sometimes occur on his actual birthday.

Another holiday to consider is Thanksgiving. Usher places the holiday on Week 15 Thursday. We can calculate that this places it in the November 20th-26th range. We see that this range is two days earlier than the current Turkey Day as defined by FDR (which is itself two days earlier than the original Thanksgiving as defined by Lincoln.)

Although Usher doesn't specifically mention the following holidays, I feel that these all fit the pattern he establishes for the holidays that he does discuss:

Columbus Day: Week 9 Monday (October 6th-12th, Columbus's actual landing day on the 12th)
Martin Luther King Day: Week 23 Monday (January 12th-18th, MLK's actual birthday on the 15th)
Mother's Day: Week 40 Sunday (May 10th-16th, two days later than Gregorian range)
Father's Day: Week 45 Sunday (June 14th-20th, one day earlier than Gregorian range)

But now we wish to place Halloween on the Usher Calendar. Unlike all of the holidays we've seen so far that are tied to a particular day of the week, Halloween is assigned a fixed date on the Gregorian Calendar -- October 31st. So how should we place it on the Usher Calendar?

I've decided to place Halloween on Week 11 Saturday. The Doomsday for Week 11 is October 25th, and since Saturday is the last day of the week, it must fall on either October 25th itself or the following Saturday. So the range for Halloween is October 25th-31st. I think that this placement on the calendar is perfect -- if we had chosen one day earlier (Week 11 Friday), the range no longer contains the original Halloween (on the 31st), and if we had chosen one day later (Week 12 Sunday), Halloween would sometimes fall in November. Only Week 11 Saturday allows Halloween to occur sometimes on its original date and always in its original month. Furthermore, Saturday is a great day for Halloween anyway -- I've seen some people (who have never heard of the Usher Calendar) propose that we should change Halloween to a Saturday, for example, the following link:

Veteran's Day, at least here in California, is always on November 11th. We may wish to place it on a Monday like most of the other federal holidays. If we place it on Week 14 Monday, it would result in the range November 10th-16th. If this is considered too close to Thanksgiving, we might prefer to place it on Week 13 Friday, which results in the range November 7th-13th. This still results in a three-day weekend which includes November 11th in the range, but it might be considered bad luck for our soldiers to place Vets Day on Week 13 Friday (which might occur on Friday the 13th of November to boot).

Another holiday that may be interesting to place is Valentine's Day. I've pointed out before that the two biggest groups that enjoy V-Day are married females and schoolchildren -- in fact, I read this year that most recipients of valentines this year are teachers, not sweethearts! So I've decided to place Valentine's Day on Week 27 Friday (February 13th-19th range). Valentine's on Friday allows the two biggest V-Day celebrating groups to enjoy the holiday -- children during the daytime at school, and then women can go out on their dates at night.

As for St. Patrick's Day, placing it on Week 32 Sunday allows it to fall in the range March 15th-21st, giving it a simple rule "the third Sunday in March." Some people may prefer it to occur one day earlier, on Week 31 Saturday -- the range March 14th-20th implies that it is always within three days of the original March 17th date. I'm not sure which is better -- some cities (like Boston) hold St. Pat's Day parades always on Sunday, while others (like Chicago) hold them always on Saturday and never on Sunday.

Christian Religious Holidays in the Usher Calendar

Let's look at the middle Sunday in the Easter range, so that the holiday is neither early nor late. The third week of the Easter range is April 5th-11th. As it turns out, we see that Week 35 Sunday has this exact range -- Week 35 Doomsday is on April 11th, and since Sunday is the first day of the week, Week 35 Sunday must fall on either the 11th itself or the previous Sunday. These facts were enough to convince Usher to adopt Week 35 Sunday as Easter in his calendar.

Now let's move on to Christmas. Like Halloween and Valentine's Day, Christmas falls on a fixed date in the Gregorian Calendar -- December 25th. So we must begin by specifying which day of the week we want Christmas to fall. Usher chose Sunday for Christmas in his calendar. We see that Week 20 Doomsday is December 27th, and so Week 20 Sunday occurs on either this day itself or the previous Sunday, resulting in the range December 21st-27th.

So far in this post, I've mentioned two holidays that not all Christians observe -- the Assumption of Mary and Advent. In particular, most Protestants don't observe these. Catholics tend to observe more holidays than Protestants, and the Orthodox observe still more holidays than the Catholics.

Even though most Catholics no longer fast for Advent, many of them avoid putting up a Christmas tree until Christmas Eve, in order to respect Advent. This is especially common in many European countries, where children should not see the tree until Christmas Eve.

Some TV specials show kids in school on Christmas Eve, most notably Frosty the Snowman. In reality, no school is open on Christmas Eve. But some schools are open on December 23rd. This is actually more common outside of California -- in fact, I've seen schools open on the 23rd even when this isn't Friday -- in such schools, winter break ends up being less than two weeks!

The day before Christmas Eve has a special name -- Festivus. This name actually comes from a famous episode of the TV show Seinfeld. Here I will use the name Festivus to refer to the day that is two days before the Usher Christmas -- a Friday in the range December 19th-25th.

If we follow the Usher calendar, schools might be open all the way until Festivus (that is, the Friday two days before Christmas). Then if schools are out for two weeks, the last day of winter break will be the Friday twelve days after Christmas. To Christians, the Twelve Days of Christmas (as in the famous song) end on the day known as Epiphany, which is January 6th on the Gregorian Calendar and in the range January 2nd-8th on the Usher Calendar. So school lasting all the way until Festivus and then winter break lasting until Epiphany restores the Twelve Days of Christmas. So the Usher secular and religious holidays actually line up. Thanksgiving Thursday is followed by the first Sunday of Advent, then there are four weeks of school corresponding to the four weeks of Advent, and then we have winter break corresponding to the Twelve Days of Christmas.

Setting Easter to Week 35 Sunday (range April 5th-11th) results in Mardi Gras on Week 28 Tuesday, with the range February 17th-23rd. Notice that this is actually the day after President's Day! Even though only Louisiana (for obvious reasons) closes schools for Mardi Gras, I sort of like the idea of having a four-day weekend before the Long March. Then the Long March itself corresponds exactly to Lent, with both beginning on Ash Wednesday. Spring break would then be the week after Easter -- often called Easter Week or Bright Week. To Christians, the Easter season ends at Pentecost, which is seven weeks after Easter. This places Pentecost at Week 42 Sunday, range May 24th-30th. This just happens to be the day before Memorial Day. So we see that Usher unwitting has his secular and religious holidays line up perfectly -- President's Day to Mardi Gras and Pentecost to Memorial Day.

Two very obscure Christian holidays are Candlemas, the last day dated in reference to Christmas (being 39 days later) and Septuagesima, the first day dated in reference to Easter (9 weeks earlier). In the Usher Calendar, Candlemas falls on a Thursday, three weeks before Septuagesima.

If there is a Leap Day, then 15 weeks after December 25th is April 8th. We see that 2011-2012 was a year with both February 29th and Easter on April 8th. So we can look back to the 2011-2012 school calendar in each district to get an idea of what school calendars look like under Usher. All the Usher holidays in both 2011-12 and 2022-23 automatically line up, including Veteran's Day if we define it as Week 13 Friday (since if December 25th is on Sunday, November 11th must be on Friday).

The Usher Leap Rule

Now here's where I come in! I actually communicated with Usher six years ago and told him that there is a problem with his Leap Day rule, and I actually convinced him to change the rule. Therefore, until I posted the Eleven Calendar on New Year's Day, the Usher Calendar is the only calendar posted on the Internet for which I played at least a small part in its construction. This is why the Usher Calendar will always play a special role in my heart (and why I spend so much time on it).

Here's the problem with Usher's original Leap Day rule -- I've mentioned earlier that we can keep the current Gregorian calendar until 2032, a year beginning on Thursday. We must delay Leap Day until 2033, which is now a year beginning on Friday with a Leap Day. But notice that 2016 is also a year beginning on Friday with a Leap Day. So the 2033 calendar becomes identical to the 2016 calendar, and so the calendar repeats after only 17, not 29, years!

To fix this, Usher fixed his Leap Day rule. After seven such 17-year cycles, there is a special 5-year cycle with two Leap Days (and one Leap Week). The resulting 124-year cycle has a mean year length of about 365.241935... days, which is much more accurate.

Notice that the Usher Leap rule is equivalent to the Julian rule with a skipped Leap Day once every 124-year cycle. The Dee Leap rule is equivalent to the Julian rule with a skipped Leap Day once every four 33-year cycles, or 132 years. Some Calendar Reformers compromise between these two and have a Julian rule with a skipped Leap Day once every 128 years. (Notice that the Gregorian Calendar skips three Leap Days every 400 years, or once every 133+1/3 years.)

Implementing the Usher Calendar

Recall that at the Wendy Krieger link above, a Calendar Reform poll was taken, and a Leap Week Calendar ended up winning the poll (Sym010). Another Leap Week Calendar, Hanke-Henry, has been posted at various websites (including the Christian Science Monitor link above). Actually, the Usher Leap Week has more in common with Hanke-Henry than with Sym010, as both Usher and Hanke-Henry start weeks Sunday and place Christmas on Sunday. An old version of the H-H calendar placed Leap Week (which it called "Newton Week") in the summer, just like Usher.

The path to implementing Usher begins with football. Of all the major American sports, football has the most rigid schedule. The first preseason game is the Hall of Fame game, on a Sunday four weeks before Labor Day. Then after Labor Day there are 17 weeks of the regular season, during which each team plays 16 games with a bye week. Then there are three rounds of playoffs (Wild Card, Conference Semis, Conference Finals), followed by the Pro Bowl, and finally the Big Game is played 26 weeks after the Hall of Fame Game. Notice that this takes up exactly half of the year, as the Hall of Fame game is usually 26 weeks after the preceding Big Game.

Some people have proposed shortening the preseason by two weeks and adding these two extra weeks to the regular season. Notice that this already fits the Usher Calendar well -- the Hall of Fame Game would be played on Week 2 Sunday, and the first Sunday of the regular season is Week 5. The 19 Sundays of the regular season would be Weeks 5-23, with the playoffs Weeks 24-26, the Pro Bowl Week 27, and the Big Game Week 28. This is the day before President's Day -- which is popular because many people don't want to go to work the day after the Big Game. Here's a link:

Notice that 24 weeks after Labor Day is not Gregorian President's Day, but Usher Prez Day. So I can see what could happen -- the NFL moves to an 18-week season and fans enjoy having the President's Day holiday the day after the Big Game. Then suddenly the year 2021 hits, and President's Day is on February 15th, which is only 23 weeks after Labor Day. Suddenly, the NFL is faced with the choice of having the Big Game on February 21st with work the next day, or skipping the Pro Bowl week and having the championship game on February 14th -- Valentine's Day (which, if you've noticed, we haven't changed to Friday yet), and forcing male football fans to choose between watching the Big Game and treating their special ladies. The solution would be to convince Congress to change President's Day to February 22nd, Usher Prez Day. Then there would be 24 weeks from Labor Day to the Big Game, which would no longer be played on Valentine's Day.

Regarding the Christian holidays, Easter is more likely to be moved than Christmas. Many people have already considered changing Easter so that it falls during a fixed week -- for example, the British once considered defining Easter as the Sunday in the April 9th-15th range, which slightly overlaps the Usher April 5th-11th range.

Christmas would likely be the hardest holiday to change. December 25th is strongly entrenched in our culture as the big holiday. The NFL, which we used to encourage the President's Day and Thanksgiving changes, would be opposed to changing Christmas to Sunday. Notice that when the holiday falls on a Sunday, most games are played on Saturday, Christmas Eve.

There are some holidays which Usher doesn't mention and may be impossible to change. For example, when should New Year's Day be? We could consider Week 21 Sunday as this is exactly seven days after Christmas (range December 28th-January 3rd). But this is awkward -- the New Year ought to be when December, the last month of the year, ends and January, the first month, begins, no matter what day of the week this is. So New Year's Day doesn't really fit into a Leap Week Calendar. But if it's any consolation, at least January 1st always occurs before Epiphany (range January 2nd-8th), so it's always one of the Twelve Days of Christmas (and hence part of winter break).

Most people probably won't use the invisible Usher Leap Week Calendar to determine their birthdays or anniversaries. For example, I was born on Sunday, December 7th -- this is Week 18 Sunday in the Usher Calendar (range December 7th-13th). If I were to follow the Usher Calendar strictly, if my birthday is not on Sunday, I should celebrate it on the Sunday after my birthday. But in reality, if my birthday were on a Monday (as it was in 2015), I'm more likely to celebrate it on the following Friday or Saturday, or even the weekend before, than on Sunday the 13th. This is even more true for those who were actually born on a Monday.

Some people argue that the biggest problem with a perpetual calendar is that they like having their birthdays change every year rather than be stuck on a Monday! According to the 538 link above, birth by C-section is also rarer on weekends than on weekdays. This is the problem -- a birth is work that we prefer during the week, but a birthday is a party that we want to have on the weekend. But in any perpetual calendar, the day of the week is fixed, so all birthdays fall on the same day of the week as the birth itself.

Of course, some holidays like Pi Day are highly dependent on the digits 3/14 for March 14th. So we can't easily declare Pi Day to be Week 31 Wednesday or whatever. Although Week 31 Wednesday could be interesting -- Wednesday is the fourth day of the week, so it's Week 31 Day 4. Voila -- there are your digits 314, and the range March 11th-17th even includes the original Pi Day to boot!

Jehovah's Witnesses and the Calendar

Let me conclude this post by thinking about my late grandmother once more. As I mentioned earlier, she was a Jehovah's Witness. As it turns out, JW's celebrate fewer holidays than even most Protestants, as they observe neither Christmas nor Easter. Instead, JW's have only one observance during the entire year -- the Memorial of the Lord's Evening Meal, or Supper. Officially, this should be observed on the 14th day of the Jewish month of Nisan, which is Passover -- since after all, this meal is traditionally a Passover meal. This year, the Memorial will be observed on March 23rd.

A quick check of the Jewish Calendar shows that March 23rd corresponds to Adar II, not Nisan. So the Jews are actually still celebrating Purim, not Passover. I believe the discrepancy comes from the fact that the Jewish Calendar, as a solar calendar, is less accurate than the Gregorian Calendar (though more accurate than the Julian). The JW's use a version of the Jewish Calendar with a more accurate solar component, so that the Memorial is always a few days before Gregorian Easter.

My grandmother and several other members of my family are JW's, but actually there are several religions represented on that branch of my family. And so there are many ways for me to be spiritual as I reflect on my grandmother's funeral.

Today's worksheets are all about Leap Days and the Doomsday Algorithm. The description of the algorithm comes from last year's Easter post while the questions come from the following link:

Even though this morning was the funeral, this blog will remain in mourning -- that is, with a black background -- for two more weeks. Until then I'll slowly start writing more original math content -- fortunately most of what I will write for Chapter 8 is just cut-and-paste from last year. In two weeks I hope that this blog will be back to its normal self.

Friday, February 26, 2016

Chapter 14 Test (Day 111)

Today is approximately the end of the fifth quaver -- the midpoint of the third quarter -- so it's a good test day. This is what I wrote last year about today's test:

Today is the Chapter 14 Test. Here are the answers to my posted test:

1. DE = 32, EF = 16sqrt(3).

2. TU = 16, US = 8sqrt(3), SK = 8, TK = 8sqrt(2).

3. 3/4

4. 3/5

5. 0.309

6. 0.625

7. 1/2

8. sqrt(3) (Some people may consider this question unfair, since the above question and both corresponding questions on the practice had rational answers, leading students to believe that they can just use a calculator to find the exact value rather than use 30-60-90 triangles.)

9-10. These are vectors that I can't reproduce easily here.

11. BC/AC (or a/b, if the students learned it that way).

12. AB and AD

13. ACD and CBD

14. This is a vector that I can't reproduce easily here.

15. (9, 6)

16. 115 feet, to the nearest foot.

17. (1, 4)

18. (3, -3)

19. (3, 2). (I hope students don't get confused here and solve these three backwards!)

20. This is a vector that I can't reproduce easily here.

Thus concludes Chapter 14. Stay tuned -- we're jumping back to Chapter 8 on Area next week!

Thursday, February 25, 2016

Review for Chapter 14 Test (Day 110)

This is what I wrote last year about today's activity and review:

Today I finally post the vector activity that I've been planning this week. But today is supposed to be the review for the Chapter 14 Test. Well, that's no problem -- technically this activity counts as part of the test review.

There is also a page to cut out with 36 given vectors -- since as I mentioned earlier, I don't want the students to choose the vectors. Of course, cutting out the vectors is time consuming -- even if the teacher does it before the class -- and those tiny slips of paper are easily lost. Then teachers will have to cut them out several times throughout the day.

Another way to have the students choose vectors would be to have something larger represent the vectors, such as playing cards. The playing cards can be converted to vectors, as follows:

For the horizontal component:
Ace through 10 -- valued 1 through 10
Jack -- valued -2
Queen -- valued -1
King -- valued 0

For the vertical component, use the suit:
Clubs -- valued -1
Diamonds -- valued 0
Hearts -- valued 1
Spades -- valued 2

Eight of Diamonds -- (8, 0)
Ace of Clubs -- (1, -1)
Jack of Hearts -- (-2, 1)

Because they are larger, playing cards are less likely to be lost than the little slips of paper that I provide for this activity. But there are problems with using playing cards. First of all, the conversion from playing card to vector is another step that the first partner can get wrong -- and once again this may frustrate the second partner. (I've heard that some people don't even know the difference between clubs and spades!) Furthermore, vectors with large components, such as (8, 0) for the eight of diamonds above, become (30, -2) after performing the steps in Task Three -- and then they are asked to graph that vector (30, -2) in Task Four. So I leave it up to individual teachers whether to use playing cards or the slips of paper that I provide.

Wednesday, February 24, 2016

Lesson 14-7: Adding Vectors Using Trigonometry (Day 109)

This is what I wrote last year about today's lesson:

Lesson 14-7 of the U of Chicago text is on adding vectors using trigonometry -- and we can't skip it because it appears in the following Common Core standard:

Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

Notice that we are essentially converting the vectors from polar to rectangular form, adding their components, and then converting the sum back to polar form. And all of this is done without the students' even knowing what polar coordinates are.

In an actual trig course (as part of precalculus), we find out that we can avoid thinking about angles larger than 90 degrees by considering the reference angle -- the angle formed by the x-axis and the terminal side of the original angle. We see that even though the U of Chicago's Geometry text doesn't teach reference angles, the angles shown in the text are always formed using the x-axis -- that is, the west-east axis -- and never using the y-axis. So the U of Chicago, while not explicitly teaching reference angles, clearly has these reference angles in mind when writing this chapter.

The text states that one of the two components is found using the sine of the angle, while the other is using the cosine. But because of the way the angles are drawn, the horizontal component will always use the cosine and the vertical component will always use the sine -- just as they would be for polar coordinates in a trig class.

To convert from rectangular back to polar coordinates, we use the distance formula and the inverse tangent function. This is the only time that an inverse trigonometric function appears in the U of Chicago text -- although many other geometry texts discuss inverse trig in more detail.

Tuesday, February 23, 2016

Lesson 14-6: Properties of Vectors (Day 108)

"Always show common courtesy to others."
-- My grandmother (1935-2016)

This morning I learned that my grandmother passed away at the age of 81. She taught me one of my greatest lessons -- to be courteous to others. It is a lesson that I am still learning to this day. I know that I can be a better teacher by showing common courtesy to students, parents, and staff members.

This is what I wrote last year about today's lesson. Admittedly it wasn't much:

I post my originally planned lesson for Lesson 14-6, which contains many of those properties of vector addition from the Common Core Standards that I mentioned yesterday.

Monday, February 22, 2016

Lesson 14-5: Vectors (Day 107)

Today I subbed in a special education class. Second period was an eighth grade history class, but there were also three periods of math. (After lunch, during the conference period, I was actually called to cover two sections of drama!)

The math classes begin with a Basic Facts Test where students had about 40 simple addition, subtraction, multiplication, or division problems to answer in ten minutes. Then the main part of the lesson involved the multiplication of fractions. This is a fifth grade standard:

Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

The homework, meanwhile, was completely different -- dividing decimals by whole numbers. This is another one of these topics that appears twice in the Common Core -- in fifth grade using alternative algorithms and sixth grade using the standard algorithm:

Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

As we already know, traditionalists would prefer to move the sixth grade standard algorithm down to fifth grade and just drop the alternative algorithm altogether.

Oops -- here we go again with our third straight post to be labeled "traditionalists"! The problem is that I keep covering these special ed math classes where below grade-level students are working with the elementary math standards, and so the traditionalist debate keeps coming up.

Not only that, but on Facebook, I've seen a few parents commenting on Common Core standards again -- including the mother of a second- (now third-) grader I mentioned in an October 2014 post. I might or might not jump in with a comment -- if I do, I'll report on it here on the blog later this week in yet another traditionalist-labeled post.

Regarding today's topic, the multiplication of fractions, there actually is an issue with algorithm. It's been pointed out by some traditionalists that students rarely want to "cross-cancel" when multiplying two fractions. For example, when multiplying 7/8 by 4/3, I would want to cancel the factors of 4 to obtain 7/2 times 1/3, or 7/6 as the answer. I'm surprised that anyone would even want to multiply 7 by 4 to get 28 and 8 by 3 to get 24, obtaining 28/24 before simplifying to 7/6. Yet I've never had success teaching this cross-canceling -- instead I'm forced to teach 7/8 times 4/3 = 28/24 = 7/6.

Oh, and of course I played my usual game in order to encourage the students to participate!

As we already know, many students find fractions to be difficult. I suspect that if you ask adults to name the first math topic that they found hard, "fractions" may be a common response. Many teachers point out that not only can't many high school students, in classes anywhere from Algebra I to Calculus -- work with fractions, but neither can many college math students. Perhaps not until we get to a class of Ph.D. candidates in math can we finally assert that a majority of the students can add, subtract, multiply, and divide fractions.

This is what I wrote last year about today's lesson:

Lesson 14-5 of the U of Chicago text is on vectors. Much of physics deals with vectors. Force is a vector quantity.

I remember back when I was a high school senior taking AP Physics C, and our teacher wanted us to remember one thing about vectors:

Vectors operating at right angles are independent.

This means that if two vectors are perpendicular to each other -- most notably if one is parallel to the x-axis and the other to the y-axis -- then they are linearly independent. Motion along one vector has nothing to do with motion along the other. We notice this most clearly when one is throwing a ball, and we resolve the velocity into its horizontal and vertical components. The horizontal component of velocity is slowed down slightly by the force of air resistance, while the vertical component is slowed down -- and its direction is ultimately reversed -- by the force of gravity.

In fact, I remember learning about acceleration so many years ago, and I was confused as to why an object that is turning or slowing down is said to be "accelerating." After all, I thought that an object slowing down was decelerating. To find out the answer, we must look at vectors. If the velocity and acceleration vectors are perpendicular, then the object is turning. (They must be perpendicular because of the theorem from Chapter 13 (Lesson 13-5) that the tangent and radius of a circle are perpendicular.) So if the velocity and acceleration vectors are in opposite directions, then the object is slowing down. I was confused by the use of the word "acceleration" by non-technical English speakers, where it usually refers to an increase in speed without considering vectors at all.

As it turns out, we prove in Linear Algebra -- a college course beyond Calculus -- that any two vectors that aren't parallel are linearly independent. But the linear independence that we consider the most in physics involve right angles. We don't discuss this idea in Lesson 14-5, but we will look at both velocity and force vectors on the posted worksheet.

Vectors appear in the Common Core Standards, but not in the geometry section. Indeed, they appear in the "Vector & Matrix Quantities" section:

(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).

(The U of Chicago text uses the first option, a single boldface roman letter. So v is a vector.) The point is that the various domains of the standards don't line up exactly with courses -- not even for the traditional (as opposed to the integrated) pathway. So there are standards that appear in the geometry section that don't appear in a geometry course (i.e., conic sections), as well as vice versa (vectors).

The various Common Core Standards for vectors are spread out among the last three lessons of the chapter, 14-5 through 14-7. One standard that appears in today's Lesson 14-5 is:

Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

This is only partly realized in Lesson 14-5 -- component-wise addition will appear in 14-6. But adding vectors end-to-end and by the parallelogram rule do appear in 14-5, and the U of Chicago does give an example where the magnitude of a sum isn't the sum of the magnitudes -- indeed, there's an example where the sum is shorter than one of the vectors being added! The text explains that this is to be expected since one vector is the velocity of the boat and the other is the velocity of the current (and the boat is moving nearly upstream).

So in a way, we are beginning this standard today as well:

(+) Solve problems involving velocity and other quantities that can be represented by vectors.

The text points out that vectors are closely related to translations. Many texts tend to define the translation in terms of its vector. The U of Chicago does the opposite. Instead, translations are defined as compositions of reflections. The Two Reflection Theorem of Lesson 6-2 tells us how to find the magnitude and direction of a translation in terms of the two mirrors. Of course, some would say that the magnitude and direction of the translation really means the magnitude and direction of the translation vector. But the word "vector" can't appear in Lesson 6-2, since it isn't even defined until Lesson 14-5. Instead, we see the following theorem:

Two vectors are equal if and only if their initial and terminal points are preimages and images under the same translation.

This theorem, therefore, shows how to connect the U of Chicago definition of translation (two reflections) with the definition found in other texts (vectors). Notice that in both cases, "equal vectors" are defined to be vectors with the same magnitude and direction.

Finally, the text defines vector addition:

The sum or resultant of two vectors AB and BC, written AB + BC, is the vector AC.

David Joyce criticizes the use of the word "resultant" to refer to vector sum:

The section of angles of elevation and depression need not appear, and the section of vectors omitted. (By the way, who ever calls the sum of two vectors the "resultant" of the two vectors?)

But then again, U of Chicago rarely uses the word "resultant" and mainly uses the word "sum" -- the word that we normally use when referring to addition. Joyce would prefer to omit the section on vectors, but they are included in the Common Core Standards -- and as I mentioned, the U of Chicago devotes three sections to vectors, not just one!