Sunday, July 29, 2018

Ratios, Subtraction, and Traditionalists

Table of Contents

1. Pappas Question of the Day
2. Barry Garelick: Ratios and Subtraction
3. Summer School Final Grades
4. Graphing Linear Equations in the Common Core
5. The Importance of Similarity
6. Music: Tall Kite's New Color Notation
7. Music: From Xenharmonic to Tall Kite
8. Conclusion: Lunar Eclipse and Mars Opposition

Pappas Question of the Day

Today on her Mathematics Calendar 2018, Theoni Pappas writes:

What is the heading of vector v from the harbor?

I know what you're thinking -- finally Pappas has included spherical geometry on her calendar! The givens include the coordinates (longitude and latitude) of both the starting and destination harbors, and then we have to set up the Spherical Law of Sines or Napier's Laws to find the initial heading....

Uh, that's not exactly what this question is. Instead, the givens include only that the heading is 61 degrees, as measured counterclockwise from the positive x-axis (as we normally do in trig). But in navigation, headings are usually measured clockwise from north, not counterclockwise from east. So we calculate 90 - 61 = N 29 E, or 29 degrees east of north -- and of course, today's date is the 29th. I guess we should have known that Pappas would never really put spherical geometry on her calendar!

Barry Garelick: Ratios and Subtraction

This is a traditionalists post. Here's a link to what Barry Garelick posted yesterday:

Garelick flashes back to his days as a student teacher:

During a lesson on ratios for my seventh grade math class, I introduced the concept by saying how when we compare things we sometimes use subtraction, like comparing heights.  That led into how ratios is a comparison by division. There are times when subtraction is appropriate and times when ratios are a better measure of comparison. This is a standard introduction in most math books–it certainly was in mine, and I’ve seen it in many.
My teacher criticized me for talking about subtraction in my lesson, feeling it detracted from (rather than setting up the discussion for) ratios. The retired math teacher agreed.  I said nothing, but I felt and still feel that they were dead wrong.

In past posts, I've written about the concept of an "anticipatory set." Back during the 2016 Blogging Initiative (MTBoS), some other math teachers wrote about how they began their classes with an anticipatory set -- a introductory lesson that gets the students thinking before the main lesson. And anticipatory sets were emphasized during my own days as a student teacher (high school Algebra I).

Anyway, apparently Garelick wanted to mention subtraction as an anticipatory set to introduce the seventh grade ratios lesson. But his master teacher and supervising teacher didn't accept this, even though "it certainly was in" his textbook. He should have been protected by the presence of this anticipatory lesson in the text, and yet he wasn't. (Then again, he could mean that it's in the text he teaches with now, not the text he used when student teaching.)

I decided to compare this to the U of Chicago seventh grade text (Transition Mathematics). Lesson 11-5 of that text is called "The Ratio Comparison Model for Division," but there's no real discussion of why we'd want to divide rather than subtract.

Neither was there any discussion of subtraction vs. division in the Illinois State text that I used during my one year of teaching at a charter middle school. Notice that by the time I started teaching there, all thoughts of "anticipatory sets" have been long since forgotten, since the anticipatory activities for Illinois State are the projects. When rates and ratios are mentioned in the projects, students are simply asked to divide -- for example, consider the opening project on mousetrap cars (calculate the speed) or the first main seventh grade project "Orienteering" (including draw a map to scale). So at no point are the students directed to compare division to subtraction.

Technically, Warm-Ups are not anticipatory sets. But it's possible for me to make a Warm-Up into an anticipatory set -- for example, before the first ratio lesson, just ask a subtraction question. I was required to use Illinois State daily assessments as Warm-Ups (and there were no subtraction questions there), but I suppose I could have asked a subtraction question on the first day of the unit. (They can't answer the Illinois State ratio questions because they haven't learned ratios yet, so there's a little leeway there.)

Garelick continues:

I’ve taught several seventh grade classes at this point, and I never hesitate to start off with talking about subtraction as the springboard into a discussion on ratios. As a way of illustrating how the difference between what subtraction and ratios measure, (and the appropriateness of each) I show a video of the old comedy team Abbott and Costello which no student has ever heard of.

Well, let me actually embed the Abbott and Costello video to which he refers here, which is called "You're 40, she's 10":

I've heard of Abbott and Costello, because of the Square One TV parody "Cabot and Marshmallow":

There's also a parody of the famous "Who's on First" routine. Cabot has set Marshmallow up on a blind date with a girl. What is her name. (Notice the period there.)

Anyway, it might even be possible to create a project based on subtraction vs. division. For example, students might have to draw a picture of themselves -- but the human body can't fit on a page. So they can either subtract or divide to make themselves fit on the page. A student's head might be seven inches tall, and so if she divides by seven, then she can draw her head to be one inch on the page -- but if she subtracts seven, then her picture will have no head at all (while her legs still can't fit).

This project, while not an Illinois State project, fits Garelick's subtraction vs. division idea. But of course, Garelick is a traditionalist, so he's opposed to any projects at all in math class.

Neither SteveH nor any other traditionalist has commented on Garelick's latest post yet as of now. I see only one "comment" listed there -- a pingback to a "Nonpartisan Education Group." (I'm not quite sure how pingbacks work -- after all, I'm linking to Garelick's site right now, but this post will never be "pingbacked.")

Summer School Final Grades

Thursday was the last day of the B session, so let's check out those final Algebra 1B grades. Once again, these grades don't include a written district final exam -- and again, the final exam might be graded, but the results will never be posted to Edgenuity, so I will never know them.
  • The top student from 1A has a final 1B grade of 91%.
  • This isn't the top student in 1B -- he was beaten by two girls with grades of 95% and 92%.
  • The bottom student from 1A has a final 1B grade of 48%.
  • This isn't the bottom student from 1B -- one student earned a dismal 34%.
I decided to look at the bottom 1B student in more detail. He earned a 65 on his first test, Exponents and Radicals, but all his other test scores are in the 20's and 30's. He was not enrolled in either of the 1A classes, so presumably he passed 1A in the fall and failed only 1B in the spring.

Meanwhile, the 48% student earned D's on three of the first four tests and a 50 on the other. But he fell off a cliff on the last two tests, with scores of 16 and 20 on the two quadratic tests. Since this is the bottom student from 1A, I go back to check those scores as well. He earned a B on the first test on Solving Linear Equations (right in the middle of the pack with 80), but then failed all the other tests with scores of 40, 55, 50 -- when graphing linear functions begins.

Once again, the Algebra II class actually has the final exam grades posted on Edgenuity. One guy actually earns a perfect score on the final -- his overall 2B grade is 94%. The second highest score belongs to a girl who missed who question and earned a score of 96, but she has the highest overall grade in the 2B class with a grade of 97%.

On the low end, one student earns a 50 on the final. His 2B grade of 58% is the lowest among all students (excluding those who attended none or fewer than half of the class sessions). He passed only one test (Rational Functions, 68) and failed all the others with scores between 36 and 55.

This is a traditionalists post. I decided to include the summer school grades in this post since after all, this is what the traditionalists' debate is all about -- what's the best way to teach math?

Come to think of it, it's interesting that the only test that the bottom Algebra 2B passed would be Rational Functions, considering what I wrote in my last post about simplifying rational expressions being the one task that correlates the most with success in college math classes.

Returning to Algebra I, consider how an Algebra I class is divided into semesters, 1A and 1B. We see that 1A is more or less the "linear semester" and 1B is the "quadratic semester." This year, we've seen slight differences among certain topics such as systems of (linear) equations -- the Glencoe text places them in Chapter 6, which is the end of the first semester (since the text has twelve chapters, the natural semester division is Chapters 1-6 and 7-12). On the other hand, systems of equations were taught during the B session this summer. Otherwise, the first/second semester division is consistent over all Algebra I classes.

It goes without saying that the quadratic 1B semester is harder than the linear 1A semester -- which is why there were more 1B classes than 1A this summer. But within 1A, we see that a unit students always seem to struggle on more than others is graphing linear equations -- and this is exactly what we observe by looking at the grades on Edgenuity.

Furthermore, consider the bottom 1A student who failed his graphing linear equations tests. In 1B, he did more or less OK on the first few tests, but failed miserably on the first quadratics test, which presumably contains graphing quadratic equations. We can't expect a student who doesn't know how to graph lines to be able suddenly to graph parabolas. When I was a student teacher covering the first semester of Algebra I, my students had trouble with graphing linear equations.

Much of the big traditionalists' debate concerns whether Common Core is a good idea -- that is, can the Common Core method actually raise achievement? It would be a huge triumph for the Common Core method if it can be used to make graphing linear equations easier for students to master.

Graphing Linear Equations in the Common Core

As I just wrote, we can cut an Algebra I text in half, with each half corresponding to one semester of the class. The first half is linear, while the second half is quadratic.

I mentioned in previous posts that this division roughly explains the algebra content of the first two years of Integrated Math. The linear half is taught in Integrated Math I, while the quadratic half is taught in Integrated Math II. (In Math II, we may go a step further and introduce complex numbers.)

It also explains the algebra content of Common Core Math 8, with the linear half of the Algebra I text covered in eighth grade. Systems of equations are definitely mentioned in the eighth grade standards, and so for the purposes of this discussion, systems are an Algebra 1A topic.

I've said before that while an Algebra I text can be cut cleanly into halves, with the first half equal to both Common Core 8 and Integrated Math I, the Geometry text can't be cut in half as cleanly. It's true that most of the eighth grade geometry content does appear in the first half of the U of Chicago Geometry text, one major topic (formal proofs) is in Geometry A but not Math 8, while another major topic (similarity) is in Math 8 but not Geometry A.

On the other hand, the Geometry A/B split does correspond to the geometry content of Integrated Math I/II respectively. This means that Common Core 8 and Integrated Math I aren't quite identical courses -- they agree on the algebra content but not the geometry content. Still, the fact that the courses are almost the same implies that a school district on the integrated pathway might attempt to accelerate students towards AP Calculus by letting them go directly from Common Core Math 8 to Integrated Math II as freshmen.

Anyway, similarity (the one topic in Math 8 but not Geometry A) is the key. The Common Core Math 8 Standards clearly connect similarity to graphing linear equations:

Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

But we ask, do most Common Core Math 8 texts really teach linear equations this way? I can only write about my own experience with the Illinois State eighth grade text. That traditional text was set up so that all standards are studied in naive order. This means that all EE (Expressions & Equations) standards are taught before any G (Geometry) standards are. Yet this standard listed above (8.EE6) is clearly intended to be taught after similarity (8.G4).

I wonder how many other eighth grade texts out there -- even newer Common Core-compliant texts -- teach all EE standards before any G standards. Such texts can't possibly follow the spirit of the Common Core Standards, where similarity is introduced in 8.G4 and used in 8.EE6.

And so it does mean that there is much potential for improvement here. If we could find a way to teach the eighth grade standards 8.G4 before 8.EE6, will this improve the students' understanding of linear equations -- so that by the time they reach Algebra I, we can make that grade dip during the graphing unit disappear?

By the way, I wonder whether it's possible to organize a high school Geometry class so that it can be divided elegantly into two halves -- with the first half matching the geometry content of Math 8, just like Algebra I. This itself may be a little tricky, since standard 8.G9, on the volumes of cones, cylinders, and spheres, doesn't appear until Chapter 10 of the U of Chicago text. A Geometry A course based on Math 8 would require us to cover all of the first four chapters, then parts of Chapter 5 (alternate interior angles), 6 (translations/rotations), 8 (Pythagoras), 10 (volume) -- and of course, Chapter 12 on similarity.

I probably wouldn't want to make this many changes, but at the very least we'd want to move similarity up into Geometry A if we want it to match Math 8.

The Importance of Similarity

With all of this discussion of similarity, we're now wondering, how easy is similarity a topic for students to grasp, compared to other Geometry topics? If students stumble with similarity, then having a linear functions unit based on similarity will definitely be unhelpful.

Well, let's check out the summer school grades for Geometry. First of all, I notice that to my surprise, Edgenuity actually includes similarity as part of Geometry A. Here is the list of Geometry A units:
  • Foundations of Euclidean Geometry
  • Geometric Transformations, Part I [reflections, translations -- dw]
  • Geometric Transformations, Part II [rotations, compositions -- dw]
  • Angles and Lines
  • Triangles
  • Triangle Congruence, Part I [SAS, ASA, AAS -- dw]
  • Triangle Congruence, Part II [SSS, HL -- dw]
  • Similarity Transformations
  • Applying Similarity Concepts
  • Quadrilaterals and Coordinate Algebra
With similarity included in this course, this seems like a strong framework for both Geometry A and Math 8 (with similarity used to prepare students for 8.EE6). But this is only if the grades are strong and students are actually learning similarity.

Let me post the scores of a few Geometry A students at random. We'll include the grades for some of the earlier units, and then the one on similarity transformations:

Student     Overall     FoEG     GTII       Triangles   Similarity
#1             72             80           64           88              76
#2             77             72           76           60              68
#3             76             76           76           64              64
#4             74             72           68           92              52
#5             72             80           84           84              52
#6             85             72           96           96              72
#7             54             68           28           40              24     

We see that Student #1 earned a higher grade on similarity (76) than the overall grade (72), Student #2 improved from the previous test to the similarity test (from 60 to 68), and Student #3 remained flat (both at 64). But for the other four students, there was a dramatic drop from the other tests to the similarity test. This includes Student #6, the top student in the class, who went an amazing 96 on the triangles test to a mere 72 on similarity. Student #7 was one of the lowest scorers, and that student dropped from an already dreadful 40 to an outright terrible 24.

When I see so many low scores on the similarity test, this makes me prefer to follow the U of Chicago Geometry text (and my first instincts) and keep similarity in the second semester. It surely doesn't inspire me to want to emphasize similarity during the linear functions unit in eighth grade. I must ask, how would it help students do well in a unit on which they struggle (linear functions) by emphasizing another unit on which they struggle (similarity)?

But let's return to the original topic of this post -- ratios in seventh grade math. There is, in fact, a Common Core Math 7 standard that relates ratios to both similarity and graphing linear functions:

Explain what a point (xy) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

Hmm, so I wonder whether it's possible to emphasize this standard more strongly. We can have the students learn and master the graphs of y = rx, as mentioned in this above standard. Perhaps we can even sneak in some extra notation -- change the r to m without actually introducing slope. We might even have the students graph y = rx (or y = mx) for some negative values of r (or m).

Let's compromise between Garelick's traditionalism and project-based learning here -- we motivate ratios using subtraction, but it's up to the teacher to decide whether to do it his way (that is, without a project) or our way (using the drawing project mentioned above).

If it becomes necessary to tie this to similarity (for the eighth grade standard), then we can just draw in the right triangles with (0, 0) at one vertex. We can't escape dilations here -- but it's directly stated in the Common Core that a line passing through the center of the dilation is invariant. Since the line y = rx (or y = mx) passes through the origin, it is invariant.

Finally, one more transformation is needed. To get from y = mx to y = mx + b, a translation of b units upward (if b is positive) is used.

I can't guarantee that students will learn these tricky topics better and avoid those grade dips, but I hope that this can put them on the right track.

Music: Tall Kite's New Color Notation

In my Pi Approximation Day post, I mentioned that Kite's color notation is changing. I'd stumbled upon his new forum website that day, but my post was jam-packed with other information and I didn't have time to peruse the entire thread in which he introduces his new colors.

First, the colors white, yellow, green, and red are not changing.

As for blue (7-limit otonal), Kite wants to keep this color, but there's just one problem with it. The color "blue" starts with the letter "b" -- but the letter "b" is also used to represent "flat" in ASCII. And even on the musical staff, the letter "b" for "blue" and the flat symbol look alike.

For example, notice that the link above doesn't take you to the first page of the thread -- instead, it's the fifth page. I chose this link because this page was the fourth result of a Google search for Kite's color notation. But notice that the first thing you see after clicking the link is a song written by Kite himself, "Without You" Piano. Midway through the second full measure, we see the the eighth note "blue D-flat." It's notated with the letter "b" for indicate "blue," followed by a flat symbol. In ASCII, we would write this note as blue Db, or even bDb.

In this song, all of the blue notes are also flat. But later in this song, there is a flat note that isn't blue in the last line. The song is in the key of B-flat major and thus all B's and E's are flat, but in the last line, a reddish E-natural appears. Later on in the same measure, the flat symbol restores the (white) E-flat -- but a musician might mistake this for a blue symbol and play blue E instead of white E-flat.

And therefore, because of possible confusion between "blue" and "flat," Kite is considering changing blue to a different color. One color under consideration is "azure," since it's another name for blue and contains the letter "z" -- the one letter that doesn't already have a meaning in this notation.

It's a shame, though, since Kite first selected the color blue because 7-limit otonal intervals often appear in blues music (the harmonic or Barbershop 7th). And besides, "flat" doesn't even start with the letter "b" -- it's just that the flat symbol looks like a "b." Since no one is going to change the flat symbol to "f" instead of "b," Kite's only other option is to change "blue" to "azure" and "z."

Notice that blue/azure is an otonal color. Mocha EDL music is based on utonal notes, and so most of the time, the blue/azure issue won't arise. But then again, even if we won't play absolute blue/azure notes, some scales might contain relative blue/azure intervals -- most notably 14EDL, which contains a blue/azure 3rd, 6th, and 7th. Also, we've seen that 7-limit otonal notation might be significant in Atari music, which will also force us to use blue/azure.

More significant for Mocha will be Kite's changes to the higher-limit colors. All of our Mocha scales contain 11, and many include 13, 17, and 19 as well. I admit that it's often tricky to have both "amber" and "umber" as the colors for 11 and 17.

First of all, notice that Kite's original notation stopped at the prime 19. For other primes, he originally used the suffix "-ish" for otonal and "-esque" for utonal. Thus Degree 23 (Sound 228) in Mocha would be labeled "23esque A#." Anyway, Kite wants these names to be easier to remember, so he seeks to replace "-ish" and "-esque" with "-o" and "-u," for "otonal" and "utonal." Therefore the new name for Degree 23 will be "23u A#" or "twenty-thu A#."

Moreover, Kite wants to use colors only up to the 13-limit, since 17 and 19 are hard to remember. So the new names for "umber D#" and "khaki C#" are "17u D#" and "19u C#." The only problem is how to pronounce "17u" and "19u." On one hand, Kite wants to shorten 17 and 19 to "s-" and "n-" (as 7 already has its own colors and 9 isn't a prime, so there's no chance of confusion). But if we pronounce "17u" as "su" and "19u" as "nu," these sound like other words, "su" and "nu." Kite suggests saying "sunda" and "nunda" (as in utonal or "undertone") instead. Meanwhile I wonder why we can't just say "seventu" and "ninetu," since the original idea was to pronounce the prime with an "-o"/"-u" ending (just as with "twenty-thu").

So this leaves primes 11 and 13. Actually, Kite's vision is to use the "-o" and "-u" abbreviations for all primes, including 5 and 7. This allows us to avoid mixed colors like "greenish" (red-green), which we might struggle to remember ("greenish" = "green" plus what?), and say "rugu" instead (since both red and green are utonal).

But if we keep "amber" for utonal 11, then the abbreviation would be "au," which isn't easy to say. So the new colors would begin with consonants. For 11, Kite wants t- for otonal and f- for utonal. Then the syllables for 11 would be to- and fu- (as in meat substitute, "tofu"). We could keep the colors tan and fawn (formerly used for 17 and 19), but Kite also likes "topaz" and "fuchsia," with the advantage of containing the vowels "o" and "u" already.

Only the 13-limit remains. Kite likes the letters "j" and "k," for "jo" and "ku." He hasn't settled on any colors yet. Perhaps he could recast "jade" and "khaki" for the 13-limit. It's too bad that we don't know any colors starting with "jo-" or "ku-." Hmm, from a list of colors used by computers, I see "jonquil" (a shade of yellow) and "kumquat"(a shade of orange) as valid colors. Perhaps I'll use those names for 13, so that we can clearly distinguish between "khaki" (old 19) and "kumquat" (new 13).

So let's look at a list of old and new color names. For Mocha, the emphasis is on the utonal:

Sound     Degree     Old Name           New Name
250         11             amber B              fuchsia B
248         13             ocher G              kumquat G
244         17             umber D#           17u D# (seventu D#)
242         19             khaki C#            19u C# (ninetu D#)
239         22             amber B             fuchsia B
238         23             23esque A#        23u A# (twenty-thu A#)
236         25             deep green Ab    gugu Ab
235         26             ocher G              kumquat G
228         33             amber E              fuchsia E
227         34             umber D#           17u D#
226         35             greenish D          rugu D
90           171           khaki B               19u B
71           190           khaki-green A    19ugu-A (ninetu-gu-A)
66           195           ocher-green Ab  kugu Ab
61           200           deep green Ab    gugu Ab
53           208           ocher G              kumquat G
31           230           23q-green F#      23ugu F# (twenty-thu-gu F#)

Kite is also concerned with what these names sound like in other languages. For example, we see that Sound 66 (Degree 195) is "kugu Ab," while Sound 61 (Degree 200) is "gugu Ab." In some languages, "gugu" and "kugu" might sound alike. And so he might change 13 to a different color, depending on what comments he receives from world language speakers. (So much for "kumquat" then!)

Not only were the color names for 11 and 13 debated (in two long threads, both at Kite's own website and on Facebook), but the issue over correct intervals for 11 and 13 also came up. Should Degree 11 be spelled "fuchsia B" or "fuchsia Bb?" Should Degree 13 be spelled "kumquat G" or "kumquat G#"?

The names that I've chosen (which are equivalent to "topaz 4th" = P4 and "jonquil 6th" = M6) result in simpler commas (33/32 between "fuchsia E" and "white E," also 27/26 between "white G" and "kumquat G" as well). But Kite himself points out that all spellings for 11 and 13 have advantages and disadvantages -- spelling t4 = P4 and j6 = m6 results in 13/11 (kG-fB) being spelled as a major 3rd, even though it's played as a minor 3rd.

Hold on a minute -- 33/32 and 27/26 are ratios, which is the main topic of this post! I've stated before that it's possible to motivate students to learn math using music. For example, if students ask where ratios are used in real life, music is a possible answer!

Let's consider a simple 6EDL "scale" -- which is really just a minor arpeggio:

Degree     Note
6               white A
5               green C
4               white E
3               white A

We can play other 6EDL arpeggios using ratios -- for example:

Degree     Note
216           white G
180           green Bb
144           white D
108           white G

Now is a good time to show the difference between subtraction (since "Degrees" are lengths of string, subtraction means cutting off the same length from each string) and division. If we divide each string above by the same number, then we get something that sounds like another minor arpeggio:

Division by 2:
Degree     Note
108           white G
90             green Bb
72             white D
54             white G

Division by 3:
Degree     Note
72             white D
60             green F
48             white A
36             white D

Division by 4:
Degree     Note
54             white G
45             green Bb
36             white D
27             white G

But subtraction doesn't produce another minor arpeggio:

Subtraction of 100:
Degree     Note
116           29u F#
80             green C
44             fuchsia B
8               white E

Here's a quick Mocha program that demonstrates this:

20 SOUND 261-A,4
30 SOUND 261-B,4
40 SOUND 261-C,4
50 SOUND 261-D,4

If we enter the first four combinations listed above (216, 180, 144, 108 down to 54, 45, 36, 27) then we hear a minor arpeggio, but the one we found by subtraction (116, 80, 44, 8) sounds awful.

Music: From Xenharmonic to Tall Kite

To me, it's amazing that so many posters are participating in the Kite and Facebook threads. After all, most people aren't musicians, and most musicians play in the standard 12EDO scale, so Kite's colors don't matter to them. Apparently, Kite's website allows users to download software called "alt-tuner" that allows them to play these exotic colorful scales. I'm also using software to play alternate scales, except our software is called "Mocha."

The Xenharmonic website is set to disappear in two days. I was considering making many more music posts throughout July ahead of the site disappearance, but I didn't. I've already devoted so many recent posts to scales anyway, and I didn't want to tie up even more posts with music. I wanted to cut-and-paste more information from Xenharmonic to preserve it, but that's also misleading -- most Xenharmonic pages are on EDO scales, while Mocha plays EDL scales, not EDO's.

My original idea was to follow the Pappas pattern -- whatever the date was, I'd cut-and-paste the corresponding EDO from Xenharmonic. But that wouldn't have worked anyway -- for example, my first post this month was on the Fourth of July. But 4EDO is just a subset of our usual 12EDO scale -- and besides, that day I wanted to post patriotic music (that's not written in 4EDO). And my third post this month was on the 12th. Why in the world would I post information on our usual 12EDO scale?

Anyway, I consider Tall Kite's website to be the successor website to Xenharmonic. So there's no need for me to preserve info from Xenharmonic anyway, as the site lives on with Kite.

Some of the Xenharmonic EDO pages mention Kite's colors (and I already cut-and-pasted some of those color charts). At first, Kite's colors should have nothing to do with EDO's, since Kite's colors are based on exact ratios (just intonation), which EDO's aren't. But sometimes Kite's colors can be used to distinguish between notes in an EDO.

These are the EDO's for which Kite's colors are used:

17, 19, 22, 24, 26, 27, 29, 31, 41, 46, 53, 72

For the lower EDO's, Kite's colors seem unnecessary to name the notes. In 17EDO and 19EDO, we just use standard note names, except that notes such as G# and Ab are no longer enharmonic. And the notes of 24EDO are readily named using quarter-sharps and quarter-flats.

But consider the highest EDO on this list, 72EDO. I already explained how Kite's colors can be used to name all 72 notes in this scale:
  • First, the twelve notes 72EDO that are in 12EDO are all white. Here it doesn't matter whether we call a note "white G#" or "white Ab," since these are enharmonic in 72EDO. (In other words, 72EDO tempers out the Pythagorean comma.)
  • One step above white is green, and one step below white is yellow. This gives the 5-limit.
  • Two steps above white is red, and two steps below white is blue/azure. This gives the 7-limit.
  • All that's left are the notes halfway between white notes. The 11-limit is sufficient to name these notes, though we must deal with the "topaz 4th = P4 or A4?" problem. If we follow our usual "topaz 4th = P4" notation, then the note halfway between "white E" and "white F" can be name either "topaz E" or "fuchsia F." (Using "topaz 4th = A4," these would have to be switched to either "fuchsia E" or "topaz F.") If we wish to avoid the "P4 or A4" problem, then we can just stick to the 7-limit and name this note "greenish (rugu) E" or "yellowish (zoyo) F." (But keeping 11 isn't as bad here, since the problems occur only when we use 11 and 13 together.)
Anyway, Mocha is designed to play EDL scales, not EDO scales. And higher EDO's are even more inaccurate than lower EDO's. Even though Mocha can play (more than) 72 notes in an octave, most of those notes don't correspond to 72EDO at all.

But for the lower EDO's, Kite's colors can help us approximate the EDO in Mocha. For example, let's look at the lowest EDO for which Kite's colors are given, 17EDO. From the old Xenharmonic site:

Combining ups and downs notation with color notation, qualities can be loosely associated with colors:

qualitycolormonzo formatexamples
minorblue{a, b, 0, 1}7/6, 7/4
"fourthward white{a, b}, b < -132/27, 16/9
midjade{a, b, 0, 0, 1}11/9, 11/6
"amber{a, b, 0, 0, -1}12/11, 18/11
majorfifthward white{a, b}, b > 19/8, 27/16
"red{a, b, 0, -1}9/7, 12/7

Chord Names

All 17edo chords can be named using ups and downs. Here are the blue, jade and red triads:
color of the 3rdJI chordnotes as edostepsnotes of C chordwritten namespoken name
blue6:7:90-4-10C Eb GCmC minor
jade18:22:270-5-10C Ev GC~C mid
red14:18:270-6-10C E GCC major or C

Again, this is the old website, so we see "jade" and "amber" instead of "topaz" and "fuchsia." We also note that "yellow" and "green" are missing, since 5 is not consistently represented in 17EDO.

I once wrote a program in Mocha to calculate the best EDL to approximate an EDO. For 17EDO, the best scale is 140EDL. But Degree 140 is greenish (rugu) D. Kite's scales are usually based on having a white note as the tonic. And the chart above is based on C as the root. So instead of Degree 140, we can choose white C (Degree 162) as the tonic.

Starting from 162, we divide by the 17th root of two and round off to the nearest integer to find Degrees for the other notes:

162, 156, 149, 143, 138, 132, 127, 122, 117, 112, 108, 103, 99, 95, 92, 88, 84, 81

According to the chart above, blue and (fourthward) white are enharmonic in 17EDO, just as jade and amber are also enharmonic. Since Mocha is utonal, we focus only on red and amber (fuchsia), which corresponds to multiples of 7 and 11.

Some of the notes listed here are already red or amber (fuchsia). Step 5 is Degree 132, which is correctly labeled as fuschia E. Fuschia A (Degree 99) and fuschia B (Degree 88) are also properly included in the scale. Red notes already on the list are red F# (Degree 112) and red B (Degree 84), while in addition to white C, white G (Degree 108) is also listed.

But some notes can be adjusted. For example, step 3 is Degree 143, but white D corresponds to the next note, Degree 144. In our calculation, the Degree to two decimal places is 143.35, which is closer to 143 than 144. In other words, 162/143 is closer to three steps of 17EDO than 162/144 is, but 162/144 is exactly equal to the interval (9/8) which 17EDO is designed to approximate. So using Degree 144 instead of 143 results in more consonant music.

Another white note to include is Degree 96 (white A) instead of Degree 95. Finally, Degree 127 can be replaced with either red E (Degree 126) or white E (Degree 128). To me, red E is preferable, since there are already four white notes but only two red notes in the scale -- and besides, red E allows us to play the red triad mentioned in the chart.

All the other notes have no white, red, or fuchsia note available, so we keep them as they are. The resulting scale is:

162, 156, 149, 144, 138, 132, 126, 122, 117, 112, 108, 103, 99, 96, 92, 88, 84, 81

Let's try this for one more scale -- 29EDO (hey, today is the 29th after all). Here are the charts given at the old Xenharmonic site:

Combining ups and downs notation with color notation, qualities can be loosely associated with colors:

qualitycolormonzo formatexamples
downminorblue{a, b, 0, 1}7/6, 7/4
minorfourthward white{a, b}, b < -132/27, 16/9
upminorgreen{a, b, -1}6/5, 9/5
"jade{a, b, 0, 0, 1}11/9, 11/6
downmajoramber{a, b, 0, 0, -1}12/11, 18/11
"yellow{a, b, 1}5/4, 5/3
majorfifthward white{a, b}, b > 19/8, 27/16
upmajorred{a, b, 0, -1}9/7, 12/7
All 29edo chords can be named using ups and downs. Here are the blue, green, yellow and red triads:
color of the 3rdJI chordnotes as edostepsnotes of C chordwritten namespoken name
blue6:7:90-6-17C Ebv GC.vmC downminor
green10:12:150-8-17C Eb^ GC.^mC upminor
yellow4:5:60-9-17C Ev GC.vC downmajor or C dot down
red14:18:270-11-17C E^ GC.^C upmajor or C dot up

The first chart mentions colors for 11 (jade and amber) but the second chart doesn't, which indicates that 11 isn't as well-represented in 29EDO as the 7-limit. Also, even though the second chart is based on C, there was another chart on the old site based on D. So we'll start this scale on white D (144):

144, 141, 137, 134, 131, 128, 125, 122, 119, 116, 113, 111, 108, 106, 103, 101, 98, 96, 94, 91, 89, 87, 85, 83, 81, 79, 77, 76, 74, 72

Because 29EDO has a very accurate perfect fifth, all of the white notes (E at 128, G at 108, A at 96, and C at 81) are already placed correctly on the scale. But a few changes must be made if we wish to add red or green notes:
  • Change 134 to 135 for green Eb.
  • Change 125 to 126 for red E.
  • Change 119 to 120 for green F.
  • Change 111 to 112 for red F#.
  • Change 89 to 90 for green Bb.
  • Change 83 to 84 for red B.
  • Change 79 to 80 for green C.
Here is the resulting scale:

144, 141, 137, 135, 131, 128, 126, 122, 120, 116, 113, 112, 108, 106, 103, 101, 98, 96, 94, 91, 90, 87, 85, 84, 81, 80, 77, 76, 74, 72

The chart for 41EDO contains the color red, blue, and gray for every note. Actually, "red major" means "red" and "blue minor" means "blue," but "blue major" means "yellow" and "red minor" means "green," and "gray" means "white." I don't repeat that chart, since 41EDO is probably too many notes to keep accurately even for this method of conversion to EDL.

Also, 29EDO's next-door neighbor 28EDO doesn't have Kite colors, because its fifth is inaccurate. I wrote about 28EDO for my Easter song, but I didn't use Kite colors that day. If we seek to add more just intervals to the 28EDO conversion, it would be better to base it on 5/4, a just major third (very accurate in 28EDO), instead of 3/2.

Conclusion: Lunar Eclipse and Mars Opposition

Let's conclude this post with a number of celestial events in the recent news. First of all, it's eclipse season, and on Friday, there was a lunar eclipse. Not only was it a blood moon, but it's said to be the longest lunar eclipse of the 21st century.

Lunar eclipses always occur at the moment of the full moon. Friday's full moon was at 1:20 PM Pacific Time, and so the eclipse was visible at any time zone where this is nighttime (which was nowhere in the Americas).

Recall that during every eclipse season, there's always at least one lunar and one solar eclipse. As it turns out, there are two new moons, and hence two solar eclipses, during this season. Both of these solar eclipses are partial. The July new moon occurred on the 12th at 7:47 PM Pacific Time, and the partial eclipse was visible in the ocean between Australia and Antarctica. The August new moon will occur on the 11th at 2:57 AM Pacific Time, and the partial eclipse will be visible in many of the same countries where the lunar eclipse was seen -- that is, Northern Eurasia, not the Americas.

Meanwhile, there is another event that is currently visible around the world -- Mars opposition. This means that the sun and the red planet are directly opposite each other in the sky, so that Mars is much closer to Earth than usual. Occurring almost once every two Earth years (or almost once every Martian year), the opposition occurred on July 27th, the same day as the full moon. Mars should be especially bright in the sky the morning of July 31st at around 12:50 AM Pacific Time.

A previous especially close approach of the red planet was in 2003, and the next such approach won't be until 2035. The 2035 opposition has often been suggested at the date when human beings will first arrive on Mars. The year 2035 is mentioned as a particularly elegant year for our arrival on Mars, since just as 66 years separated the Wright brothers' flight in 1903 to the moon landing in 1969, so should 66 years separate the 1969 moon landing from the 2035 Mars landing.

Each year in late December, I make my annual Calendar Reform posts. Even though Calendar Reform will make timekeeping easier everywhere on Earth, many people believe that it will never happen, since the Gregorian and other world calendars are so firmly entrenched in our minds. Thus Calendar Reform will never happen anywhere on Earth -- if we want a new calendar, we'd have to get ourselves another planet.

The basic unit of a Martian Calendar is the sol, or Mars day. A Mars sol is slightly longer than an Earth day -- about 24 hours 40 minutes. Meanwhile, a Martian year is about 687 Earth days, or 668.6 Mars sols. Thus two Earth years is about one Mars year (another ratio! -- 2/1).

Since a Mars year is about twice as long as an Earth year, most proposed Mars calendars are of one of two types. One, the "Stretched Gregorian," is so-called because it stretches the twelve Gregorian months so that they're about twice as long as their Earth counterparts. The other, the "Darian"-type calendar, has 24 months instead of 12, so that each Martian month could be about the same length as an Earth month.

When I was a young elementary school student and my school introduced the "Path Plan," I once joked to myself that "paths" are what a Martian school would call their grade levels, since the normative time spent in each path is two (Earth) years -- or about one Mars year.

Sunday, July 22, 2018

Pi Approximation Day 2018

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

It also marks four 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?

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.

Two years ago was my first as a teacher at a charter middle school, but I left that classroom. And this summer, I was just about to start teaching a high school summer class, but I was told that summer enrollment at the school is declining, and so they decided not to hire me as a teacher.

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.

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 Algebra, and so normally I won't post it. But for the sake of today's FAQ, I'll post the Pappas question:

If a != b/3 (ASCII for "if a does not equal b/3),
(-11/0.5)(3a - b)/(b - 3a) = ?

To solve this problem, we notice that (3a - b)/(b - 3a) is just -1. Interestingly enough, the Pappas question for two days ago (July 20th) featured a similar simplification to -1. The a != b/3 restriction is there just to make sure that the denominator isn't zero.

So now we have (-11/0.5)(-1) or 11/0.5. Dividing by 0.5 (or 1/2) is the same as multiplying by 2 -- if you don't see this, then set up the decimal long division -- dividing 0.5 into 11 is equivalent to dividing 5 into 110. Either way, we obtain 22 as the answer -- and of course, today's date is the 22nd.

Two years ago, as I was about to begin my first year of teaching, I came up with the idea of starting each class with my own Pappas question -- a Warm-Up question whose answer is the date. But this quickly fell apart when I found out that our school curriculum provided a "daily assessment" question that I was required to give as the Warm-Up. Of course, the answers to these questions weren't just the date. In hindsight, I should have used the Pappas question as an Exit Pass at the end of the class, rather than a Warm-Up at the start of class. In fact, if I ever get my own class again, I might use Pappas questions as Exit Passes. 

3. How did I approximate pi in my classroom?

Since today is Pi Approximation Day, I should write something about approximating pi. I won't let my disappointment -- of thinking I'd be hired at a new job only to be denied -- get in the way of celebrating the special day.

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, with the summer school class I almost taught still on my mind, 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 Thursday of A Session, hence it was a perfect day for a party (especially since there is no summer school on Fridays).

Today, Pi Approximation Day is on the Sunday between the second and last weeks of B Session. Just as we would if Pi Day itself were on a Sunday, a Pi Approximation Day party would have been held on the previous school day -- Thursday, July 19th. Notice that this day is itself a day to celebrate a constant -- e Approximation Day, since 19/7 = 2.714... while e = 2.718.... Indeed, if this had been an Algebra 2B class (in which exponential functions are studied), it might have been more appropriate to call it an e Approximation Day party than a Pi Approximation Day party.

Next year, Tau Day falls on Friday (remember, no summer school on Fridays) and Pi Approximation Day falls on a Monday. Thus I'm actually hoping to teach B Session next summer. Of course, the decision to teach summer school is out of my hands since it all depends on how many students enroll.

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:

and the Saxon series:

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 two weeks, and so the following link is to her most recent post:

Nguyen writes about her upcoming visit to her childhood home in Minnesota. In doing so, she writes about growing up as a young Vietnamese immigrant in a new country. (The move must have been tough -- she leaves Vietnam, where the climate is tropical, and goes to Minnesota, where the climate is, um, not tropical.)

Nguyen begins:

I don’t remember the landing. It’s been a very long flight. Nor do I remember walking through the airport. We have no luggage anyway, like none.
My first memory of America is sitting in the back seat of TuAnh’s uncle’s car — an Oldsmobile wagon with wood panel trim. I’m almost eleven and a half years old, and this is the second time I’m in an automobile, a car car, which is much smoother than a bus or a van, and you’re not squished between strangers. The Oldsmobile is taking us straight home, not having to make a million stops along the way like my last bus ride from Saigon to Mũi Né.
So Nguyen must have been around the sixth grade when she moved -- and now she's a sixth grade teacher.

Nguyen writes:

I will sit and watch the news with the uncle. I have no idea what they are saying, but I just like seeing white people’s faces and listening to how fast they talk. The best part is there’s always something on TV, there’s no curfew. I have two favorite shows, The Price is Right and Happy Days. You don’t have to understand very much English to watch The Price because prices are numerical, and English numbers look the same as Vietnamese numbers, except Americans are weird to write $50 instead of 50$. They claim to read from left to right too.

Hey, The Price Is Right is one of my favorite game shows, too! But Nguyen's story takes place during the show's earliest seasons in the 1970's, when Bob Barker (with dark hair) was still the host. Nguyen writes that she doesn't need to know much English to understand numerical prices or math, while Common Core often requires students to explain their answers, which can be difficult for English learners like the young Nguyen.

She continues:

I like Happy Days because it’s a show with cute boys, Chachi and Fonzie. (My family calls me Fawnzie. My name morphed from Phương to Fawn to Fawnzie. More recently, my son Gabriel probably sensed that I was stressed in our conversation and said, “Mom, I need you to be Fawnzie right now.” And I knew what he meant.)

That's interesting -- so her name "Fawn" isn't really a color, but is actually the Anglicized version of her birth name Phuong.

Fawn Nguyen -- um, Phuong Nguyen -- concludes:

I get to visit St. Cloud this August; it’ll be my first time back since I left in 1979. I’ll be facilitating a full-day workshop, and St. Cloud will just be 70 miles away. I’m flushed with nostalgia and gratitude — going back to my first home in America.

I hope Fawn Nguyen posts again on her blog soon!

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 four years of posting on the blog. The traditionalist who is currently the most active is Barry Garelick. Here is a link to his most recent post:

Garelick 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:

There are different levels of understanding. While there are some concepts that a student may not understand, there are still connections that students make to previously learned material and concepts which serve to inform a recently learned procedure—and ultimately may lead to further understanding. In freshman calculus, for example, students learn an intuitive definition of limits and continuity which then allows them to learn the powerful applications of same; i.e. taking derivatives and finding integrals. It isn’t until they take more advanced courses (e.g., real analysis) that they learn the formal definition of limits and continuity and accompanying theorems. Does this mean that they don’t understand calculus?

As Garelick explains, the problem is with basic math, such as the times tables. Traditionalists want students to reach the point where students can answer a one-digit multiplication problem in one second, so that they can move on and use multiplication to solve harder problem. But opponents of the traditionalists -- in this post, Garelick calls them reformers (including proponents of the Common Core) -- insist that students should have a deeper "understanding" of multiplication. To the reformers, the ability to multiply single digits in one second shows only that a student is a "math zombie" -- someone who can give the correct answer without truly understanding why it is correct.

Garelick concludes:

The reform movement has succeeded in foisting its beliefs upon ever growing populations of new teachers who believe this is the only way. It is so entrenched, that even teachers who adamantly oppose such views feel guilty when teaching in the traditional manner so reviled by well-intentioned reformers. Given that today’s employers are complaining over the lack of basic math skills their recent college graduate employees possess, the math reform movement has created a poster child in which “understanding” foundational math is often not even “doing” math.

That line about "lack of basic math skills" explains why Garelick disagrees with the reformers. The reformers are so concerned that the students "understand" multiplication that in the end, they aren't able to do any actual multiplication at all.

Garelick provides an example of a traditional problem that, to him, also exhibits understanding:

One proxy that teachers use for understanding and transfer of knowledge, is how well students can do all sorts of problem variations. A student in my seventh grade math class recently provided an example of this. As an intro to a lesson on complex fractions, I announced that at the end of the lesson they would be able to do the following problem:
Complex fraction
The boy raised his hand and said “Oh, I know how to solve that.” I recognized this as a “teaching moment” and said “OK, go for it”.  He narrated what needed to be done: “You divide the two fractions on top, by flipping the second one and multiplying, and since one is negative you’ll get a negative answer. You get -5/4.  Then you do the bottom, and you have to convert  to a fraction by multiplying 2 by 3 and adding 1, so you get 7/3 , and you multiply and you cross cancel and get -7/4 .”  Now you divide  by -7/4  so you flip the -7/4  and multiply. You have two negatives so your answer is positive. And you can cross cancel. You get 5/7.”
One commenter, SteveH, responds to this problem:

Your complex fraction problem does require a whole lot of mathematical understanding to solve correctly. Even if students could (properly) prove why “invert and multiply” works, they would still need a lot more understanding to solve the problem. What if you had ((-3)/5)) in the numerator or (3/(-5)). There is never any reason to prove everything before you use it or understand it at a deep level. Also, understanding a proof does not give one the flexibility of understanding achieved from doing lots of P-sets. Astoundingly, one of the core ideas for many in education is the belief that understanding (conceptual or whatever) allows for less individual homework. In K-6, they claim hegemony over content, skills, and understanding even though their only academic turf is education (process). However, they then do little as a guide on the side. Student, teach thyself!

"Guide on the side" is a reform-minded teacher. A traditionalist teacher is a "sage on the stage."

By the way, SteveH and other traditionalists criticize the need to "explain everything." To this extent, someone like Fawn Nguyen would agree (recall her statements about being a young English learner and not needing to know English to understand numerical prices). On the other hand, Nguyen and SteveH strongly disagree on homework -- Nguyen doesn't believe in grading HW or assigning a lot of it, while to SteveH, HW (in the form of "P-sets" or problem sets) are crucial.

But the commenter I want to focus on today is Wayne Bishop:

Wayne Bishop:
A friend who, for years, worked with ETS on the CSU-wide ELM (Entry Level Mathematics exam, now defunct) observed that a single item on the exam correlated so well with passing the exam and subsequent success in the student’s choice of non-remedial college math that the rest of the test could be ignored. What was it? Simplification of a modestly complicated complex rational function (algebra fraction). Without verbalizing anything, recalling the arithmetic properties of fractions along with the (always arranged to be trivial) factoring well enough to complete the task correctly is enough for a reasonable promise of mathematics success at any CSU campus. Now we use the SBAC (Smarter Balanced Assessment Consortium of Common Core) exam for admission math competence. No such item.
A while ago, someone (clearly not a traditionalist) criticized the appearance of rational expressions on the CSU (California State University) math test. (As so often happens, I read a website and don't immediately blog about it -- then when I do want to blog about it, I can't find the link!) Anyway, whoever this person was wrote that rational expressions are useless in real life, yet their appearance on the ELM was torture and blocked bright students from obtaining college degrees.

It could have been the following 2015 post from the famous math blogger Dan Meyer, except that this link doesn't specifically mention the ELM:

Anyway, Bishop's post implies that not only should rational expressions appear on the test, but they should be the only questions on the test! Rational expressions might be useless in real life, but their presence on the test guarantees that only smart students -- those who are likely to get A's in classes such as Calculus and above -- are admitted to STEM majors.

Even though I didn't end up teaching summer school, I was given access to the Edgenuity software that these classes use -- including classes that I was never expected to teach, such as Algebra 2B. I can report that on the "Rational Functions" quiz, one girl earned a perfect score, while three students earned 88 on the test (high B/B+). On the flip side, one student failed with a 52, while another student scored 60 (low D/D-). Following Bishop, this quiz should be the ELM. Instead, the ELM has been replaced with a Common Core test that doesn't emphasize rational expressions -- much to the delight of the blogger who compared them to torture.

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.

Let's go back to the comment made earlier by the traditionalist SteveH:

There is never any reason to prove everything before you use it or understand it at a deep level. Also, understanding a proof does not give one the flexibility of understanding achieved from doing lots of P-sets.

Here SteveH was discussing the proof of basic multiplication of integers, or perhaps the rule for division of rational numbers. I agree that "proofs" of these statements are unnecessary. But there's one math class where proofs are expected even in traditionalist classes -- Geometry.

But the Common Core takes Geometry proofs a step further. 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.
On July 12th, 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.

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. On the blog, 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.

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:

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 -- "greenish G" and "greenish-green 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 "ocher F" and "ocher-green Eb."

Where do all these strange color names like "green" and "ocher" 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
  • amber: prime 11
  • ocher: prime 13
  • umber: prime 17
  • khaki: 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 set to disappear at the end of July. Actually, here's another link where Kite's color notation is explained:

Oops, I just stumbled upon this today when I was trying to find a link that isn't about to disappear at the end of July. Apparently, "Kite" (or "Tall Kite") is considering changing up the colors. So you might want to forget everything I said about colors until this debate is resolved.

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:

10 N=16
20 FOR X=1 TO 32
40 SOUND 261-N*(17-A),4
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, the simplest EDL scale, the Sailor Pi theme song:

10 N=13
20 FOR X=1 TO 26
40 SOUND 261-N*D,T
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 two 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.

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.

Bonus: Charlie Lindelof

This video has nothing to do with Pi Approximation Day. But while I was searching for pi videos I stumbled upon the above video. It goes back to what I was saying about the traditionalist Wayne Bishop and college placement exams. This is all about a rational expression question on the Accuplacer. It appears that some California community colleges still use the Accuplacer as a math placement exam, even if the Cal States are using the SBAC instead.

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.