Pi Approximation Day 2019

Today is Pi Approximation Day, so-named because July 22nd -- written internationally as 22/7 -- is approximately equal to pi.

It also marks five full years since my very first post here on the blog. Yes, I've declared Pi Day of the Century -- March 14th, 2015, to be the ceremonial start date of this blog. But the chronological start date was July 22nd, 2014. I like celebrating both Pi Day and Pi Approximation Day -- including eating pie both days -- and so I celebrate both of them as launch dates for the blog.

Last year, I used the anniversary to create a FAQ about my blog and myself. I do the same this year -- and this year's FAQ will reflect the changes that have happened to my career since last year. As usual, let me include a table of contents for this FAQ:

1. Who am I? Am I a math teacher?
2. Who is Theoni Pappas?
3. How did I approximate pi in my classroom?
4. What is the U of Chicago text?
5. Who is Fawn Nguyen?
6. Who are the traditionalists?
7. What's with the "line and its translation image are parallel" proof?
8. What's "Mocha music"?
9. Mocha Music for Pi Approximation Day
10. How should have I stated my most important classroom rule?
11. Who is Wendy Krieger, and what is the Eleven Calendar?

1. Who am I? Am I a math teacher?

I am David Walker. I'm trying to start a career as a math teacher, but I must admit that my career isn't off to as good a start as I'd like.

Three years ago was my first as a teacher at a charter middle school, but I left that classroom. By now, I fear that I won't be hired to teach this year at all. If I'm not hired, then I'll remain a substitute teacher. But this will make the launch of my teaching career that much more difficult.

By the way, in my last few posts, I announced that my old school was in danger of closing because the charter renewal petition was denied. But I wasn't completely sure whether the school will really be closed, since they could have appealed to the county or state. And so I wrote that in a few days, I'd find out once and for all.

Well, last last week I drove past my old school -- and guess what? There is now a completely different charter school at that site. That settles it -- the old charter school is closed. It truly now is the old charter school, as in no longer existing.

I'll still blog occasionally about how I could have taught better there. Perhaps if I'd been a better teacher, I could have still been teaching there, and the school might still be open. Or maybe the school might have closed anyway even if I had remained, since the primary reason for the denial was financial mismanagement, rather than any academic reason. Thus if I'd stayed, then I'd still be in the same boat -- having to look for a new teaching job. The only difference is that I'd have three full years of teaching experience -- and that's crucial when trying to get a leg up in this competitive job market for teachers.

So the answer to this question is no, I'm not currently a math teacher, but I'm hoping that I'll be one again soon.

2. Who is Theoni Pappas?

Theoni Pappas is the author of The Mathematics Calendar. In most years, Pappas produces a calendar that provides a math problem each day. The answer to each question is the date. I will post the Pappas questions for each day that I blog, provided that it's a Geometry question.

The question for July 22nd this year is trig, but it's simple enough that it might be taught in a Geometry class:

10sin^-1(sin 2 1/5 degrees)

We notice that since sin^-1 denotes inverse sine, this means that for any angle theta, we know that sin^-1(sin theta) is just theta. Thus sin^-1(sin 2 1/5) = 2 1/5. So the problem becomes:

10(2 1/5 degrees)

The answer is 22 degrees -- and of course, today's date is the 22nd.

By the way, I don't necessarily like the notation "sin^-1" to mean inverse sine. This is tough on students, where "x^-1" means "1/x," that is, the multiplicative" inverse of x, while "f^-1" means the composite inverse of f.

Some authors use the subscript "o" (same as the composite "o") with f^-1 to emphasize that this is the composite inverse of f, but this looks terrible in ASCII: f_o^-1(x) or sin_o^-1(x). One good thing about the o is that it also looks like the degree symbol, so that sin_o^-1(x) means "find out how many degrees the angle has whose sine is x." Of course, this then breaks down as soon as the students learn about radians.

Of course, there's always the notation "arcsin(x)," which is what I prefer. The only problem is that most TI calculators use the notation "sin^-1(x)."

3. How did I approximate pi in my classroom?

Since today is Pi Approximation Day, I should write something about approximating pi.

The dominant approximation of pi in my classroom was, of course, 3.14. I didn't quite reach the unit on pi in my seventh grade class -- Grade 7 being the year that pi first appears in the current Common Core Standards. But I did reach a unit on the volume of cylinders, cones, and spheres in my eighth grade class.

The main driver of my use of 3.14 as the only approximation of pi was IXL. I used this software to review the volume problems with my eighth graders. The software requires students to use 3.14 as pi, even though I provided them with scientific calculators with a pi key. For example, the volume of a cylinder of radius and height both 2 is 8pi cubic units. IXL expects students to enter 8(3.14) = 25.12, even though 8pi rounded to the nearest hundredth is 25.13. IXL will charge the students with an incorrect answer if they enter 25.13 instead of 25.12. And even on written tests, I didn't want to confuse the students by telling to do something different for written problems, and so I only used 3.14 for pi throughout the class.

But there was another problem with our use of calculators to find volumes. Some of the calculators were in a mode to convert all decimals into fractions. Thus 25.12 appeared as 628/25 -- and I couldn't figure out how to put them in decimal mode. This actually allowed me to catch cheaters -- only one of the calculators was in mixed number mode. Here was one of the problems from that actual test, where students had to find the volume of a cylinder of radius and height both 3:

V = pi r^2 h
V = (3.14)(3)(3)(3)
V = 84 39/50

so it displayed 84 39/50 instead of 84.78 or 4239/50. So any student who wrote 84 39/50 on the test -- other than the one I knew had the mixed number calculator -- must have been cheating.

Here's an interesting question -- suppose instead of catching cheaters, my goal was to make it as easy as possible on my students as well as my support staff member, who was grading the tests. That is, let's say I want to choose radii and heights carefully so that, by using 3.14 for pi, the volume would work out to be a whole number (which appears the same on all calculators regardless of mode). How could have I gone about this?

The main approximation of pi, 3.14, converts to 314/100 = 157/50. It's fortunate that 314 is even, so that the denominator reduces to 50 rather than 100. So our goal now is to choose a radius and height so that we can cancel the 50 remaining in the denominator.

We see that 50 factors as 2 * 5^2. The volume of a cylinder is V = pi r^2 h -- and since r is squared in this equation, making it a multiple of five cancels 5^2. It only remains to make h even to cancel out the last factor of two in the denominator. Let's check to see whether this works for r = 5, h = 2:

V = pi r^2 h
V = (3.14)(5)^2(2)
V = 157 cubic units

which is a whole number, so it works.

If we had a cone rather than a cylinder, then there is a factor of 1/3 to deal with. We can resolve the extra three in the denominator by making h a multiple of six (since it's already even). Let's check that this works for r = 5, h = 6:

V = (1/3) pi r^2 h
V = (1/3)(3.14)(5)^2(6)
V = 157 cubic units

Spheres, though, are the trickiest to make come out to be a whole number. This is because spheres, with the volume formula V = (4/3) pi r^3, have only r with no h. Thus r must have all of the factors necessary to cancel the denominator. Fortunately, the sphere formula contains the factor 4/3, and the four in the numerator already takes care of the factor of two in the denominator 50. And so r only needs to carry a factor of five (only one factor is needed since r is cubed) and three (in order to take care of the denominator of 4/3). So r must be a mutliple of 15. Let's check r = 15:

V = (4/3) pi r^3
V = (4/3)(3.14)(15)^3
V = 14130 cubic units

Notice that the volume of a sphere of radius 15 is actually 14137 to the nearest cubic unit -- that's how large the error gets by using 3.14 for pi. Nonetheless, 15 is the smallest radius for which we can get a whole number as the volume by using pi = 3.14.

Now today, Pi Approximation Day, is all about the approximation 22/7 for pi. Notice that if we were to use 22/7 as pi instead of 157/50, obtaining a whole number for the volume would be easier.

For cylinders, the only factor to worry about in the denominator is seven. Either the radius or the height can carry this factor. So let's try r = 1, h = 7:

V = pi r^2 h
V = (22/7)(1^2)(7)
V = 22 cubic units

For cones, we also have the three in 1/3 to resolve. Let's be different and let the radius carry the factor of three this time. For r = 3, h = 7, we have:

V = (1/3) pi r^2 h
V = (1/3)(22/7)(3^2)(7)
V = 66 cubic units

For spheres, unfortunately our radius must carry both three (for 4/3) and seven, and so the smallest possible whole number radius is 21:

V = (4/3) pi r^3
V = (4/3)(22/7)(21)^3
V = 38808 cubic units

This is larger than the radius of 15 we used for pi = 3.14. But notice that there are many extra factors of two around in the numerator -- 4/3 has two factors and 22/7 has one. This means that we can cut our radius of 21 in half. Even though 21/2 is not a whole number, the volume is nonetheless whole:

V = (4/3) pi r^3
V = (4/3)(22/7)(21/2)^3
V = 4851 cubic units

To make it easier on the students, I could present the radius as 10.5 instead of 21/2. (Recall that the students can easily enter decimals on the calculator -- they just can't display them.) There is still some error associated with the approximation pi = 22/7, as the volume of a sphere of radius 10.5 is actually 4849 to the nearest cubic unit, not 4851. Still, 22/7 is a slightly better approximation than 157/50 is, with a much smaller denominator to boot.

With such possibilities for integer volumes, I could have made a test that is easy for my students to take and easy for the grader to grade. I avoided multiple choice on my original test since I didn't want to give decimal choices for students with calculators in fraction mode (or vice versa). With whole number answers, multiple choice becomes more feasible. Of course, the wrong choices would play to common errors (confusing radius with diameter, forgetting 1/3 for cones).

By the way,, I point out that the two summer circle constant days (Tau Day and Pi Approximation Day) can be used as alternatives to Pi Day parties for summer classes. My district divides the summer into A Session and B Session. Tau Day, on June 28th, was the second Friday of A Session, hence it wouldn't have worked for a party (since there is no summer school on Fridays).

Today, Pi Approximation Day is on the Monday of the last week of B Session. Thus teachers could throw a party today.

4. What is the U of Chicago text?

In 1991, the University of Chicago School Mathematics Project published a series of secondary mathematics texts. I was able to purchase both the Algebra I and Geometry texts for two dollars each at a local public library. One thing I discovered was that the U of Chicago Geometry text is based on the same transformations that appear in the Common Core Standards -- even though the U of Chicago published the text nearly two decades before the advent of the Core.

The common denominator is NCTM, the National Council of Teachers of Mathematics. This group was behind both the U of Chicago text and the Common Core Standards. Still, this means that one can satisfy most of the Common Core Standards by teaching out of the U of Chicago text.

There are newer versions of the Geometry text than the one that I own. I've read that while my old version introduces reflections in Chapter 4 and the other isometries in Chapter 6, in the new version all isometries appear in Chapter 4.

The U of Chicago text is not the only book that I mention here on the blog, not by a long shot. Many posts here refer to the Illinois State text. This is mainly because I was a math teacher, and I used the Illinois State text when teaching at my old school. 

To supplement the Illinois State text (mainly when creating homework packets), I used copies of a Saxon Algebra 1/2 text and a Saxon 65 text for fifth and sixth graders.

To learn more about some of these texts, here are some links to a webpage by Cathy Duffy. Duffy's goal is to help homeschooling parents choose a textbook for their children. Here is her review of the U of Chicago text:

http://cathyduffyreviews.com/math/geometry-ucsmp.htm

and the Saxon series:

http://cathyduffyreviews.com/math/saxon-math-54-through-calculus.htm

5. Who is Fawn Nguyen?

For years, Fawn Nguyen was the only blogger I knew who was a middle school math teacher. Back when I was a middle school teacher, I enjoyed reading Nguyen's blog, but now that I've left, it's not as important for me to focus on her blog over all others.

Nonetheless, let's take a look at Nguyen's blog today. She hasn't posted in six months, and so the following link is to her most recent post:

http://fawnnguyen.com/jelly-beans-or-no-jelly-beans/

Nguyen begins with a Warm-Up question called a "Would you rather...?"

Would you rather have 364 jellybeans and give 188 to friends or have 281 jellybeans and give 137 to friends? Whichever option you choose, justify your reasoning with mathematics.

But in her class, she decided to change this to a different type of "Would you rather...?"

As a student, would you rather be given the problem on the LEFT (jelly beans) or the one on the RIGHT?

And here's the question on the RIGHT:

Which problem below, A or B, yields a larger difference?
A. 364 - 188 =
B. 281 - 137 =

She writes that a majority of her sixth graders, 71%, chose A. Moreover, when she posted this as a survey on her blog -- with most respondents presumably being teachers -- an even larger majority, 90%, chose A. But Nguyen herself would choose question B:

Well, I prefer the one on the right [that I’d typed up]. How did I get it so wrong? I’m normally not this lame. But, truth be told, I don’t love the jelly beans question. At all. Maybe the one on the right is the wrong “fix” for the left one. If I could retype the problem on the right, I’d remove the equal signs since the question is just asking which one yields a larger difference, not caring exactly what each difference is. I want to believe that anyone who spends 5 minutes with me learns that I love mathematics.

I hope Fawn Nguyen posts again on her blog soon, especially since it's been six months! (By the way, if she goes a full year without posting, I'll assume that her blog is no longer active, and then I'll remove her from the FAQ.)

6. Who are the traditionalists?

I use the phrase "traditionalists" to refer to those who oppose Common Core for mathematical -- not political -- reasons. The traditionalists tend to believe that many of the topics in the Common Core Math Standards are taught too late.

My own relationship with the traditionalists is quite complex. I tend to agree with the traditionalists regarding elementary school math and disagree with them regarding high school math.

For example, they believe that the standards in Grades 4-6 that require the use of the standard algorithm for arithmetic should move down at least one grade level. I tend to agree with the traditionalists on this point here -- most of the reasons for opposing direct instruction, rote memorization, and drill don't apply to the youngest students. I remember once learning about two nonstandard algorithms for addition -- the Left-Right Method and the Plus-Minus Method -- on an old 15-minute math show for sixth graders, "Solve It," that used to air on our local PBS station back when I was that age. For mental math, these nonstandard algorithms are great, but for a young elementary student first learning to add two-digit numbers I prefer the standard algorithm.

On the other hand, the traditionalists also prefer that high school seniors take AP Calculus in order to open the door to college STEM majors. In particular, some STEM colleges here in Southern California, such as Cal Tech and Harvey Mudd, require calculus for admission. Even colleges that don't formally require calculus are likely to send students rejection letters if they apply without having calculus on the schedule, especially if they apply to STEM majors. Therefore they want the classes leading up to senior year to prepare students to take AP Calculus -- including Algebra I in the eighth grade. They oppose Common Core because its eighth grade math standards do not comprise a full Algebra I class. I only partially agree because I don't want seniors who can't handle AP Calculus or eighth graders who can't handle Algebra I to be labeled failures. Just as AP Calculus can open the door to STEM, it can close the door to a non-STEM career if students are forced to spend too much effort trying to pass their high school math classes.

OK, so I support the traditional pedagogy with regards to elementary math and oppose it with regards to high school math. But what's my opinion regarding traditionalism in middle school math -- you know, the grades that I'll actually be teaching? In middle school, I actually prefer a blend between the traditional and progressive philosophies. And the part of traditionalism that I agree with in middle school is making sure that the students do have the basic skills in arithmetic -- since, after all, that is elementary school math, and with elementary math I prefer traditionalism.

I've referred to many specific traditionalists during my five years of posting on the blog. I'd consider anyone who chooses option B above to be a traditionalist -- which would make Nguyen herself a traditionalist! Actually, I'd say that Nguyen combines both traditionalist and progressive pedagogy in her class -- and if I were to return to the middle school classroom someday, I'll try to emulate her in this regard.

But the traditionalist who is currently the most active is Barry Garelick. Here is a link to his most recent post:

https://traditionalmath.wordpress.com/2019/07/18/out-on-good-behavior-dept/

Garelick also teaches middle school math right here in California. One of his frequent commenters is SteveH, a traditionalist in his own right. I consider SteveH to be a co-author of Garelick's blog since he posts there so often.

Let's look at Garelick's post in more detail:

I am currently writing a series called “Out on Good Behavior: Teaching Math While Looking Over Your Shoulder”  When the series is complete, it  will be published in book form by John Catt Educational, Ltd.

Before you ask, no, Garelick's book, when it's published won't be our next side-along book (unless for some reason I see it at a library book sale someday for $1).

I've actually discussed his Chapter 1 in an earlier post, so let's look at his Chapter 2 for this FAQ:

https://truthinamericaneducation.com/education-reform/an-espresso-based-job-interview-a-1962-algebra-book-and-procedures-vs-understanding/

(Note: Truth in American Education is an anti-Common Core website.)

In the remaining two weeks at my previous school, I applied for the few math teaching positions that were advertised. I had the typical non-responses except for one—a high school that specialized in problem-based learning. I had applied there out of desperation never expecting a response. I received an email saying they were interested in interviewing me. Despite my skills at making my teaching appear to be what people wanted to see, I knew that this one required too much suspension of disbelief on both sides of the aisle.

So as we can already tell, traditionalists such as Garelick don't like problem-based learning -- which I also used at the old charter school with the Illinois State text.

A few days later I was at the school for a 2 PM interview. I tend to get a bit logy in the afternoon so I thought I’d have an espresso prior to coming in. The principal, Marianne and assistant principal, Katherine, interviewed me and asked the usual questions: What does a typical lesson look like, what are my expectations and so on. My inner voice tried to keep me from extended caffeinated responses.  I emphasized how I leave time for students to start on homework in class, do the “I do, we do, you do” technique, and in my controlled ramblings managed to get across that I am, by and large, traditional.

Oh, and speaking of books that can be purchased cheaply, Garelick wrote that he sneaked copies of his favorite traditional textbook into his classroom -- and look at how little he paid for them:

“I bought about fifteen of them over the internet when they were selling for one cent a piece about four years ago. So I was basically paying for shipping. But now the prices increased because of Amazon’s supply and demand algorithm, so they’re selling for about $60 a copy last time I looked. Which tells me a lot of people are buying them.”

A more recent text that he was able to use in his classroom is JUMP Math:

“I used an alternative textbook, which my school let me use: JUMP Math. It was developed in Canada and broke concepts down into very small incremental steps. It scaffolds problems down to incremental procedures and builds on those.”

In the end, Garelick gets the job and begins teaching at St. Stevens, a Catholic school. I'm not quite sure whether he's still there now.

7. What's with the "line and its translation image are parallel" proof?

Both this year and in past summers, I devoted several posts out of the blue to the proof of a statement from Geometry, "a line and its translation image are parallel." The reason for this proof goes back to the goals of Common Core Math and the traditionalist debate.

The Common Core expects students to prove statements that traditionalists take for granted, such as the triangle congruence properties (that is, SSS, SAS, and ASA.) These statements are proved using transformations -- reflections, rotations, and translations.

Now it's possible to take this a step further. The parallel line properties (that is, Corresponding Angles and its converse) can also be proved using transformations. One of the first mathematicians to do so is UC Berkeley professor Hung-Hsi Wu. The two things to note about his proof are:

• Wu uses 180-degree rotations to map one parallel line to another. Therefore, his main proof is for the Alternate Interior Angles Test rather than Corresponding Angles.
• Wu delays using a parallel postulate as late as possible. In particular, he is able to prove the Alternate Interior Angles Test (that is, if ... then the lines are parallel) without any need for a parallel postulate at all. So his proof is valid in both Euclidean and hyperbolic geometry. On the other hand, the converse requires a parallel postulate (Playfair's).

Using Wu's proof, we can avoid the parallel postulate until a certain point when it's needed. Then later on, we can mention that there's another type of geometry called hyperbolic geometry -- and that all the proofs so far are valid in hyperbolic geometry up to the point where Wu first invokes Playfair.

Unfortunately, I find two problem with this approach:

• Wu's proof depends on 180-degree rotations, but for most students, translations are easier to understand than rotations. I once saw a website (now defunct) that demonstrated translating one parallel line to another to show that corresponding angles are congruent. It would be nice if I could convert that animation into a rigorous proof.
• Wu's proof is valid in hyperbolic geometry, but who cares, since we usually don't discuss hyperbolic geometry in high school math anyway. If we're going to mention non-Euclidean geometry at all, it would be spherical geometry -- after all, we live on a sphere. Some honors classes even introduce spherical geometry at the end of the year. It would be nice if I had a proof that, if valid in a second geometry at all, is valid in spherical, not hyperbolic, geometry.

Last year, I finally posted a valid proof of the Corresponding Angles Consequence. But this proof turns out to be very complicated. For example, I used the following theorem:

Three Perpendiculars Theorem:

If three lines are perpendicular to the same line, then the three lines are either parallel or concurrent.

This theorem is valid in both Euclidean and spherical geometry -- in particular, the "parallel" part is Euclidean, while the "concurrent" part is spherical. (In other words, the theorem is valid in "natural" geometry, which incorporates both Euclidean and spherical geometry.)

But Three Perpendiculars isn't the sort of theorem we want to teach in high school. And besides, we're trying to prove something about parallel lines, and there are no parallels on the sphere. Thus it's silly to prove these theorems in both Euclidean and spherical geometry, even if technically they're valid (for example, "if two parallel lines are cut by a transversal" is vacuously true on the sphere).

I completed the proof only because I wanted to finish what I started. But my final result is definitely inappropriate for a high school class. Perhaps instead of working on both goals, it might be better to focus on one or the other:

• We might replace Wu's 180-degree rotations with translations to prove statements that are valid and meaningful in both Euclidean and hyperbolic geometry. At the end of the year, we might say that there's a geometry called hyperbolic -- show the students the Numberphile video.
• We might mention spherical geometry at the end of the year, but only in connection to statements that are true in both Euclidean and spherical geometry (such as SSS, SAS, ASA), not parallel lines.

I won't pursue this any further in posts during the school year, but I will discuss it in posts during the summer. During the school year, my lessons are based on the U of Chicago text, where both Corresponding Angles and its converse are postulates.

If I ever teach in a classroom again, then it depends on the grade level. In eighth grade, where transformations are introduced but no formal proofs are given, we can informally show them that we can translate corresponding angles to each other -- as well as map alternating interior angles to each other via a 180-degree rotation. That way, the eighth graders can learn that these pairs of angles are congruent without having to teach them vocabulary -- "corresponding" or "alternate interior" angles.

In a Geometry class, it all depends on what text is being used. If parallel line postulates appear transformations, then I could show them how transformations can be used to prove the parallel line statements without delving too deeply with "Three Perpendiculars" or other nonsense. If parallel lines appear before transformations (as they do in the U of Chicago text) then just forget about it.

By the way, the statement I was trying to prove -- "a line and its translation image are parallel" -- doesn't even appear in the Common Core Standards. On the other hand, "a line and its dilation image are parallel" does explicitly appear in the standards. So this is something that we can keep in mind -- but then again, parallel lines usually appear well before dilations in most texts, so it's not as if we could use dilations to prove the parallel line statements.

So far this summer, I've written posts labeled "Natural Geometry Unit 1, 2, 3" that incorporate the ideas mentioned in this section. Today's post counts as "Natural Geometry Unit 4: Translations," and the two bullet points above illustrate the contents of such a unit. Translations are defined and used to prove properties of parallel lines in Euclidean geometry. Implicit is the fact that these theorems are valid only in Euclidean geometry, as opposed to results taught in the previous three units that are valid in both Euclidean and spherical geometry. Hyperbolic geometry will not be mentioned in the high school geometry class at all.

8. What's "Mocha music"?

In many recent posts, I refer to something called "Mocha music." This is a good time to explain what Mocha music actually means.

When I was a young child in the 1980's, I had a computer that I could program in BASIC. This old computer had a SOUND command that could play 255 different tones. But these 255 tones don't correspond to the 88 keys of a piano. For years, it was a mystery as to how SOUND could be used to make music. Another command, PLAY, is used to make music instead, since PLAY's notes actually do correspond to piano keys.

Last year, I found an emulator for my old BASIC computer, called Mocha:

http://www.haplessgenius.com/mocha/

When we click on the "Sound" box on the left side of the screen, Mocha can play sounds, including those generated by the SOUND command. So finally, I could solve the SOUND mystery and figure out how the Sounds correspond to computer notes.

I discovered that SOUND is based on something called EDL, equal divisions of length. We can imagine that we have strings of different lengths -- as in a string instrument or inside a piano. The ratio of the lengths determine their sound -- for example, if two strings are in a 2/1 ratio, then the longer string sounds an octave lower than the shorter string.

The key number for SOUND is 261, the "Bridge" (or end of the string). Mocha labels the Sounds from 1 (low) to 255, so we subtract these numbers from 261 to get a Degree ranging from 260 (long string) to 6 (short string). The ratios between the Degrees determine the intervals. I found out that the Degrees corresponding to powers of 2 (8, 16, 32, 64, 128, 256) sound as E's on a piano, with Degree 128 being the E just above middle C (that is, E4).

Let's say we were to play the following two notes on Mocha:

10 SOUND 51,8

20 SOUND 86,8

The second number 8 indicates a half note, since 8 is half of 16 (the whole note). But we want to focus on the first numbers here, which indicate the pitches (tones).

We first convert the Sounds to Degrees. Since 261 - 51 = 210, the first note is Degree 210. The Degree of the second note is 261 - 86 = 175. Now the ratio between these two Degrees is 210/175, which reduces to 6/5. This is the interval of a minor third, so the two notes are a minor third apart. As it turns out, the two notes sounds as G and Bb -- "rugu G" and "rugugu Bb."

Let's try another example:

30 SOUND 144,8

40 SOUND 196,8

Warning -- we don't attempt to find the ratio 196/144 (which is 49/36 by the way). We only find the ratios of Degrees, not Sounds. The Degrees are 261 - 144 = 117 and 261 - 196 = 65. Thus the interval between the notes is 117/65 = 9/5, a minor seventh. (Using Degrees instead of sounds makes a big difference, since 49/36 would be an acute fourth or tritone, not a minor seventh.) The names of the two notes played by Mocha are "thu F" and "thugu Eb."

Where do all these strange color names like "gu/green" and "thu" come from? Actually, they refer to Kite's color notation, and the colors tell us which primes appear in the Degree:

• white: primes 2 or 3 only
• green: prime 5
• red: prime 7
• lavender: prime 11
• thu: prime 13
• su: prime 17
• inu: prime 19

Kite's color notation also uses colors such as yellow, blue, and so on. But these are "otonal" colors, while EDL scales/lengths of string are based on "utonal" colors only.

The website where Kite explains his color notation is here:

https://en.xen.wiki/w/Color_notation

Actually, here's another link where Kite's color notation is explained:

http://www.tallkite.com/forum/index.php?topic=86.0

"Kite" (or "Tall Kite") formerly used different colors such as "amber" and "ocher," and so many of my old posts mention these colors.

9. Mocha Music for Pi Approximation Day

I keep saying that I should use these exotic Mocha scales for composing new music, not simply converting music in our usual scale (12EDO) to the new scales.

But on holidays, I'm in the mood for converting old music to the new scales. For Pi Approximation Day, I was hoping to convert songs about pi -- specifically songs that used to be posted on other websites that are now defunct. This includes "American Pi" and "Digit Connection" from the old Bizzie Lizzie Sailor Pi site, as well as Danica McKellar's old pi song based on "Dance of the Sugar Plum Fairy."

I would convert these songs if I had access to the sheet music, which I don't. Maybe I'll convert them some day, but until then, it's easier just to find YouTube videos of the songs on which these are parodies ("American Pie," etc.) and sing the pi lyrics loud enough to drown out the real words.

So instead, let's just code a pi song based on 16EDL, similar to the song we played for Tau Day:

NEW
10 N=16
20 FOR X=1 TO 32
30 READ A
40 SOUND 261-N*(17-A),4
50 NEXT X
60 DATA 3,1,4,1,5,9,2,6,5,3
70 DATA 5,8,9,7,9,3,2,3,8,4
80 DATA 6,2,6,4,3,3,8,3,2,7
90 DATA 9,5

As is traditional, I stop just before the first zero. Then digits 1-9 map to Degrees 16 down to 8, with the lowest note played on E (line 10, N=16). We can change the value of N to any value from 1 to 16 to change the key.

Here's an actual song converted to 12EDL, a simpler EDL scale, the Sailor Pi theme song:

NEW
10 N=13
20 FOR X=1 TO 26
30 READ D,T
40 SOUND 261-N*D,T
50 NEXT X
60 DATA 8,4,8,2,9,4,9,2,10,4,11,4,9,12
70 DATA 9,4,9,2,10,4,10,2,11,4,12,4,10,12
80 DATA 12,4,12,2,10,4,10,2,8,4,6,4,7,12
90 DATA 8,4,9,4,10,2,11,6,12,16

Only the main verse is coded here. The "bridge" part -- which is instrumental in both the original Sailor Moon and Lizzie's Sailor Pi song -- is too hard for me to code without sheet music.

Here are the lyrics for the first verse -- the part which we coded above:

Fighting fractions by moonlight
Perplexing people by daylight
Reading Shakespeare at midnight
She is the one named Sailor Pi.

10. How should have I stated my most important classroom rule?

This is what I wrote three years ago:

Rule #3: Respect yourself and others.

Students respect themselves and each other as well as me, the teacher, by following all rules and allowing others to learn the material.

This rule worked for Fawn Nguyen, the teacher from whom I got this rule. But it didn't work in my classroom at all. Of course, a student playing with a phone case isn't respecting the teacher, but I needed a rule that would require the student to put the case away immediately.

Here's a much better rule:

Rule #1: Follow all adult directions.

And so if I say, "Put the phone case away," the student couldn't counter that phone cases were against the rules, because my direction was to put the case away.

If I ever find myself in the classroom again, this will definitely be my first and most important rule.

By the way, Garelick also mentions that his main weakness is classroom management:

They asked about my classroom management techniques. In any interview or evaluation process, one has to have some weakness to talk about and I freely admitted that classroom management is not my strong suit. I mentioned that my seventh grade math class had behavior problems even though there were a total of 10 students in the class.

“How did you handle the problems?” Marianne asked.

“I had a warning system; two warnings and they got a detention.  I wasn’t too faithful in carrying that out though.”

“Why was that?”

“When I gave a detention, the two main troublemakers were really good at carrying on about it and crying.”

“They cried?”

“I hated giving detentions. I always got talk-back, like ‘But I wasn’t talking’. And then the crying. Which they did with all the teachers, I found out.”

But then Garelick quickly returns to discussing traditional pedagogy.

In my old class, many troublemakers claimed "But I wasn't talking," but very seldom did anyone cry to trick me into thinking that punishing them was unreasonable.

The lesson that both Garelick and I needed to learn was that we need to know outside of how the students are reacting whether our punishments are reasonable or not. Indeed, I see now that students who genuinely believe that they're being treated unfairly don't say "But I wasn't talking" -- instead, they do the opposite and give the teacher the silent treatment. (Indeed, this is what some people do when their spouses forget their birthdays.)

Who is Wendy Krieger, and what is the Eleven Calendar?

Wendy Krieger is an Australian who is interested in different number bases and innovative systems of measurement. She is a former member of the Dozens Online forum:

https://www.tapatalk.com/groups/dozensonline/

One day on this forum, Krieger mused about creating a calendar based on the number eleven. I quickly became interested, because I knew of no existing or proposed calendar based on 11.

If you were to choose a number and invent a calendar based on that number, your calendar will probably not be original -- someone else would have already thought of your idea. Consider the neighbors of 11 -- 10 and 12. Well, there's already 12 months in our current Gregorian Calendar, while the ancient Egyptian calendar used 10 days per week ("decans"). And suppose you were to look at the prime neighbors of 11 -- 7 and 13. Of course, the Gregorian Calendar has seven days per week, while the International Fixed Calendar has 13 months per year.

Despite all of this, no existing calendar is based on 11. And so, inspired by Krieger's post, I decided to create my own Eleven Calendar.

Suppose we want there to be eleven months per year. Then since a year contains 365 days, we divide 365/11 to obtain 33.18.... If we round this to 33 days, we notice that 33 is itself a multiple of 11. And so each 33-day month contains three weeks of 11 days each.

A simple version of the Eleven Calendar begins on March 1st, so that the first month contains the first day of spring (or autumn in Krieger's native Australia). Since 11 * 33 is exactly 363, we just tack in the extra days at the end of the year as blank days:

Month 1: March 1st-April 2nd
Month 2: April 3rd-May 5th
Month 3: May 6th-June 7th
Month 4: June 8th-July 10th
Month 5: July 11th-August 12th
Month 6: August 13th-September 14th
Month 7: September 15th-October 17th
Month 8: October 18th-November 19th
Month 9: November 20th-December 22nd
Month 10: December 23rd-January 24th
Month 11: January 25th-February 26th
Blank Days: February 27th, 28th, (29th)

So far, I haven't really given names to the eleven months or eleven days of the week yet.

Unfortunately, Krieger has been banned from the Dozens Online forum. I don't quite know why, but I suspect it was a dispute between her and the other members regarding number bases. As the forum name implies, most members of the forum like base 12. But Krieger's preferred base is actually 120, which she often calls "twelfty." Thus her banning might have been the result of her attempts to use twelfty on a dozenal forum.

Still, Krieger occasionally communicates with me via this blog. First, she came up with a clock to add to the Eleven Calendar. This clock divides the day into 22 hours, with 66 minutes in each hour -- and she also told me of an ingenious way to read a regular analog clock, based on 24/60, and convert it on the fly to 22/66 time.

A few days ago, Krieger wrote to me about dividing the world into 22 time zones to match the 22 hours on the clock. My problem was that it's awkward to divide the 360 degrees of our sphere into 22 time zones.

And so Krieger's latest idea is to convert each old degree into 1.1 new degrees. Thus there are now 396 degrees in a circle. She explains that Brisbane (presumably her hometown), which is at longitude 150E under the old system, becomes 165E in the new system.

In the past, I've often given my coordinates as 34N, 118W. As it turns out, in the new system, my longtitude is almost exactly 130W -- indeed, it's eerie how close my house is to 130W. Indeed, if I were to supply my latitude, you'd be able to find, or get within a block of, my house.

Of course, I don't want randos on the Internet to stalk me, so of course I won't post my latitude. To the nearest whole degree, my latitude in the new system is 37N. But the exact confluence 37N, 130W is in the Pacific, and so no one will be able to find my house using the information I supply here.

This is what I wrote as my response to Krieger:

Hmm, this is interesting. So now there are 360 * 1.1 = 396 degrees in a full circle. This reminds me of 400 "gradians" in a circle, since 396 is so close to 400.

My own longitude is extremely close to 130W under this system, so my time would be seven hours behind Greenwich, while your time would be nine hours ahead of Greenwich.

Thanks for an interesting sundial idea as well!

(For the rest of this post, there are 396 gradians in a circle. We don't care about the 400-grad circle.)

Krieger also mentions that we can call our 11-day weeks "decans" to reflect the ancient Egyptian weeks, even though our weeks are one day longer. She also mentions the ancient Chinese "ce" or "ke," originally 1/100 day but changed to 1/96 day to fit the 24-hour day. For her sundial, she divides each of the 22 hours into 11 "uncia," each six minutes long. (The word "uncia" really means 1/12.)

The concept of using 396 gradians for a circle for the Eleven Calendar is often mentioned on the Dozenal Forum and is called an auxiliary base. Another possible auxiliary base for 11 is 990 -- chosen due to its proximity to 1000 (so a semicircle is about 500 degrees, a quadrant 250 degrees, and so on). Under this system, my longitude becomes 325W.

But users of the Eleven Calendar might be using a purely undecimal (or "levimal") base, where everything is in base 11. Then the proximity of 990 to the decimal thousand means absolutely nothing to an undecimal user, so we might as well keep using 396 (330 in undecimal).

One problem with using 396 (or 990, for that matter) is that neither is divisible by eight, so that 45 degrees becomes 49.5 gradians. It's awkward to discuss 49.5-49.5-99 triangles in Geometry -- and this looks even worse in undecimal, where 49.5 becomes 45.555... gradians.

Oh, and since today is Pi Approximation Day, we notice that in the approximation 22/7, numerator 22 is divisible by 7, so it fits the our gradian system. Let's look at what happens here:

pi radians = 180 degrees = 198 gradians
22/7 radians = 198 gradians
1/7 radian = 9 gradians
1 radian = 63 radians

So using the approximation pi = 22/7, we find that one radian is approximately 63 gradians.

OK, here are some Pi Approximation Day video links:

1. Draw Curiosity

Notice that this video, from two years ago, actually acknowledges Pi Approximation Day.

2. Math Babbler:

The Math Babbler tends to post a video for Pi Approximation Day every year. In this one, he discusses how accurate the approximation 22/7 actually is.

3. Converge to Diverge

In this brand-new video from today, the speaker attempts to approximate pi just as Archimedes did it -- using a regular 96-gon.

4. Numberphile

No Pi (Approximation) Day is complete without a Numberphile video. In this video, we learn that pi is not just irrational, but is also transcendental -- that is, there doesn't even exist a polynomial f, with integer or rational coefficients, such that f (pi) = 0 (unlike sqrt(2), which is algebraic). This is even stronger than Legendre's result that pi^2 is irrational.

5. Sharon Serano

Well, I already gave ten facts about pi, and so this video is twice as good.

6. Vi Hart

No, I can't avoid another one of my favorite math YouTube posters, Vi Hart. Of course, in this video she argues that there's nothing special about pi at all

7. TheOdd1sOut

This video is specifically listed as a "Vi Hart rebuttal" to videos such as the previous video. 
8. A Song Scout

This is another pi song based on its digits. Unlike Michael Blake's song (listed below), it is in the key of A minor rather than C major.

9. Michael Blake

I like the idea of using music to teach math, so here's a favorite, Michael Blake's "What Pi Sounds Like." I tried to find a way to incorporate songs such as this one into the classroom.

10. tiradorfranco2

I'm surprised that I don't post this Square One TV video on Pi Approximation Day. The song is about the mathematician who first discovered that pi is approximately 22/7 -- Archimedes. The singer even mentions how Archimedes was "busy calculating pi" at around the 2-minute mark.

11. Coding Challenge #140

Today is Pi Approximation Day, and so this video is all about approximating pi.

Bonus: Numberphile

Earlier I wrote that Numberphile created some hyperbolic geometry videos. Here's one of the more interesting videos, about sports in hyperbolic space. This is just in case we decide we'd rather show our students hyperbolic than spherical geometry.

And so I wish everyone a Happy Pi Approximation Day. Notice, this is being posted shortly after midnight Pacific Time, so that it is still 22 July in Australia.