The Anatomy classes are mostly seniors, but one class has three juniors. This is significant because today is the infamous California Science Test -- the test that caused me many headaches when trying to set up my science classes last year. I haven't though much as much about the high school test as the middle school tests. I know that one grade (which could be any grade from 9-12) administers the test each year -- it makes sense that it would be juniors (since they test English and math as well).
Apparently, the California Science Test was supposed to be last week, but for some reason (computer failure?) it was delayed to this week. The three Anatomy juniors are sent to another classroom, presumably a science class with more juniors than Anatomy.
Meanwhile, today there is a science Google Doodle -- S.P.L. Sorensen, the Danish chemist who developed the pH scale. It's too bad that I didn't sub in a Chemistry class today (instead of Biology) in order to fit the doodle. The pH scale is often mentioned in Algebra II classes as an application of the common logarithm. (During tutorial, I am able to help a few students in Algebra II. One student has a logarithm problem for review, but it's not a pH problem.)
Apparently, the California Science Test was supposed to be last week, but for some reason (computer failure?) it was delayed to this week. The three Anatomy juniors are sent to another classroom, presumably a science class with more juniors than Anatomy.
Meanwhile, today there is a science Google Doodle -- S.P.L. Sorensen, the Danish chemist who developed the pH scale. It's too bad that I didn't sub in a Chemistry class today (instead of Biology) in order to fit the doodle. The pH scale is often mentioned in Algebra II classes as an application of the common logarithm. (During tutorial, I am able to help a few students in Algebra II. One student has a logarithm problem for review, but it's not a pH problem.)
Meanwhile, the Queen of the MTBoS has made her latest post. That's Fawn Nguyen, in case you don't recall who our queen is:
This is what she writes on Memorial Sunday:
There is something else that I do way better than teaching mathematics, even though teaching has been a 25-year plus career. That something is house cleaning.
Nguyen proceeds by listing ten steps required for house cleaning. I'll post only the first five steps here, since you can just click the link if you really want the whole list:
- Throw everything out.
- When done with step 1, repeat step 1 again bc we both know you really didn’t throw everything out.
- With remaining [ideally just 3] items, ask, “Is it really really pretty?” If so, it should be displayed in your home in a pretty spot. Ask, “Is it useful, like a wine-bottle-opener type of necessity?” If so, keep it in a drawer.
- Unless it’s a piece of furniture, a houseplant, or a 4-legged friend, forbid it from touching your floor.
- Counter space is only for items that do not fit inside a drawer/cupboard and are used almost daily — e.g., toaster, Nutribullet, knife block.
Then Nguyen compares these steps for house cleaning to those of lesson planning. I, of course, am currently thinking about lesson planning for my summer Algebra I course, so let's follow this closely.
The first two steps are to throw everything out. Well, last week's lesson planning was all about throwing things out -- the course as posted on Edgenuity is too long for a three-week course. But Nguyen goes on to explain that she often throws out questions in order to make some of her problems more open-ended. This I won't be able to do -- Edgenuity will ask the students questions, and it's difficult for a computer to evaluate open-ended answers.
Nguyen rejects the third step in her own classroom -- her students don't have time to make their work look "pretty." It does remind me of Foldable notes, where students take time to decorate their Foldables before the unit begins. The idea of Foldables actually comes from the Glencoe text, and I was considering having my students make them to take notes from the text. But Nguyen's right -- there won't be time in a summer course for Foldables, and besides, the main text for the course is Edgenuity, not Glencoe. In a computer-based class, there's no time for Foldable notes.
Steps 4-5 are connected to decorating the room itself. I don't know what my room will look like -- indeed, I probably won't have access to it until the Friday before summer school begins, since it's currently occupied by a regular school year class. So we'll worry about these two steps later.
It's time for our next musical scale. This week we'll look at 18EDL:
The 18EDL scale:
Degree Ratio Note
18 1/1 tonic
17 18/17 septendecimal chromatic semitone (Arabic lute index finger)
16 9/8 major tone
15 6/5 minor third
14 9/7 supermajor third
13 18/13 tridecimal semiaugmented fourth
12 3/2 perfect fifth
11 18/11 undecimal neutral sixth
10 9/5 minor seventh
9 2/1 octave
Degree Ratio Note
18 1/1 tonic
17 18/17 septendecimal chromatic semitone (Arabic lute index finger)
16 9/8 major tone
15 6/5 minor third
14 9/7 supermajor third
13 18/13 tridecimal semiaugmented fourth
12 3/2 perfect fifth
11 18/11 undecimal neutral sixth
10 9/5 minor seventh
9 2/1 octave
http://www.haplessgenius.com/mocha/
10 INPUT N
20 FOR D=9 TO 18
30 SOUND 261-N*D,4
40 NEXT D
This is a descending scale. To make the scale ascend, use:
20 FOR D=18 TO 9 STEP -1
We notice that this scale contains a new prime, 17. Indeed, 18EDL is the only 17-limit EDL, and it contains the entire 17-limit tonality diamond.
The name "septendecimal" for the 17-limit is a bit awkward. It reminds us of the name "hexadecimal" for base 16 (going back to my last post, on number bases). But notice that linguistic purity is violated, as "hexa-" is Greek but "septen-" is Latin.
Of course, this ordinarily isn't a problem. We know that number base enthusiasts don't even like odd bases, much less prime bases. But musical intervals are named for their prime limit. So it's unlikely we'll ever have to worry about confusing number bases with interval names. As a prime base, 17 will be ignored, so it doesn't matter whether it's "septendecimal" or "heptadecimal." And intervals such as 16/13 won't be called "hexadecimal" or "sexadecimal" neutral third, but "tridecimal" neutral third, after the prime 13.
(And besides, number bases often take Latin names anyway. There's been much discussion lately of base 6, called senary, not "hexary." And base 11, when it appears, is "undecimal," not "hendecimal.")
10 INPUT N
20 FOR D=9 TO 18
30 SOUND 261-N*D,4
40 NEXT D
This is a descending scale. To make the scale ascend, use:
20 FOR D=18 TO 9 STEP -1
We notice that this scale contains a new prime, 17. Indeed, 18EDL is the only 17-limit EDL, and it contains the entire 17-limit tonality diamond.
The name "septendecimal" for the 17-limit is a bit awkward. It reminds us of the name "hexadecimal" for base 16 (going back to my last post, on number bases). But notice that linguistic purity is violated, as "hexa-" is Greek but "septen-" is Latin.
Of course, this ordinarily isn't a problem. We know that number base enthusiasts don't even like odd bases, much less prime bases. But musical intervals are named for their prime limit. So it's unlikely we'll ever have to worry about confusing number bases with interval names. As a prime base, 17 will be ignored, so it doesn't matter whether it's "septendecimal" or "heptadecimal." And intervals such as 16/13 won't be called "hexadecimal" or "sexadecimal" neutral third, but "tridecimal" neutral third, after the prime 13.
(And besides, number bases often take Latin names anyway. There's been much discussion lately of base 6, called senary, not "hexary." And base 11, when it appears, is "undecimal," not "hendecimal.")
Let's check out the notes of 18EDL using Kite's color notation:
The 18EDL scale:
Degree Ratio Note
18 1/1 white D
17 18/17 umber D#
16 9/8 white E
15 6/5 green F
14 9/7 red F#
13 18/13 ocher G
12 3/2 white A
11 18/11 amber B
10 9/5 green C
9 2/1 white D
The 18EDL scale:
Degree Ratio Note
18 1/1 white D
17 18/17 umber D#
16 9/8 white E
15 6/5 green F
14 9/7 red F#
13 18/13 ocher G
12 3/2 white A
11 18/11 amber B
10 9/5 green C
9 2/1 white D
The new prime 17 requires two new colors -- "tan" for otonal and "umber" for utonal. Since EDL scales are utonal, the color "umber" appears in the scale, with "umber D#" the first note of that color.
The color "umber" is a odd choice, since it sounds so much like "amber." In fact, Mocha has a note at Degree 187 (Sound 74) called "umber-amber A#." But Kite, when choosing colors for his notation, was more concerned with the "u," not the "-mber" part. He wanted each color to start with a different letter, and he needed a color starting with "u," so "umber" it is. (His notation uses every letter of the alphabet except "z.")
Finally, you might ask why our first umber note is called D# rather than Eb. Well, there are two reasons for preferring sharps over flats. In utonal music, fourths are preferred over fifths (since the largest prime of 4/3 is in its denominator, unlike 3/2). If Degree 17 is umber D#, then Degree 51 (that is, 17 * 3) is umber G# and Degree 153 is umber C#. Then Degree 459 would be umber F#, and Degree 1377 would be umber B (even though these last two are out of Mocha range). So at least in theory, we can reach notes without a classical accidental by proceeding fourthward from D#. On the other hand, proceeding fourthward from Eb takes us into more flats before reaching double flats.
And here's another reason that Kite prefers sharps here. The interval 18/17 is 99 cents, while the interval 17/16 is 105 cents. Thus Degree 17, in cents, is closer to Degree 18 than to 16. And so we'd prefer 17 to have the same letter name as 18, which again means that D# is used over Eb. Both of these reasons generalize to higher primes, so if we have a choice between a flat name and a sharp name for a new prime, the latter is preferred.
What good are umber notes, anyway? The use of the other limits is obvious -- the 3-limit gives us fifths while the 5-limit gives us thirds. The 7-limit is used in jazz (including barbershop). Even the 11- and 13-limits can be used for neutral intervals, plus we might want harmonic 11th and 13th chords (following the pattern for harmonic 7th chords). But 17 seems to be just a useless prime.
Well, in Helmholtz-Ellis notation, the 17-limit can be indicated using a symbol for 256/255. This comma is so small that H-E calls it a "schisma" rather than a "comma." (At 6.8 cents, it's slightly smaller than the 225/224 comma that Kite calls the "sub.")
The 256/255 schisma is one of the smallest intervals playable in Mocha. Degree 256 (Sound 5) is called "white E," while Degree 255 (Sound 6) is called "umber-green E" (since 255 has the factor 5 as well as 17). In other words, the difference between "white" and "umber-green" is the schisma. As the color that cancels "green" is yellow, it follows that "yellow" differs from "umber" by the schisma.
In Mocha, otonal colors like "yellow" are unplayable, but utonal colors like "umber" are playable. So we can play major chords in Mocha using "umber" instead of "yellow." For example, an A umber major triad would be Degrees 192:153:128 (Sounds 69-108-133). Unfortunately, there still aren't enough umber notes to make a complete major scale (so for now, green Bb major remains the only playable major scale).
There's one more thing I'd like to say about the prime 17. Let's look at the 18EDL scale once more:
The 18EDL scale:
Degree Ratio Note
18 1/1 tonic
17 18/17 septendecimal chromatic semitone (Arabic lute index finger)
16 9/8 major tone
15 6/5 minor third
14 9/7 supermajor third
13 18/13 tridecimal semiaugmented fourth
12 3/2 perfect fifth
11 18/11 undecimal neutral sixth
10 9/5 minor seventh
9 2/1 octave
Hmm, that's a strange name for the 18/17 interval, "Arabic lute index finger." I'm very curious about the reason for that name -- and I wonder if there's an "Arabic lute index finger," are there also an "Arabic lute middle finger," "ring finger," and "little finger"?
First of all, a "lute" is a string instrument, similar to the modern guitar. On a modern guitar, we can place an index finger on the first fret to raise the pitch -- for example, fretting the D string at the first fret produces D#. So perhaps we can place an index finger on an Arabic lute to raise its pitch by the semitone-like interval 18/17.
Well, a traditional "Arabic lute" -- also known as an oud -- has no frets. Still, I can imagine raising the pitch of an oud string by using the index finger, thereby justifying the name.
The Xenharmonic website uses the name "Arabic lute" to refer to two intervals. One of these is 18/17, but the other is the "Arabic lute acute fourth," 49/36:
http://xenharmonic.wikispaces.com/49_36
But 49/36 is a septimal interval -- it has nothing to do with 17. And you'd think that as easy as it is to use Google to research ouds, no website seems to explain why 18/17 is "Arabic lute index finger."
Well, here's my conjecture -- 18/17 is actually part of a simple 18EDL system. The middle, ring, and little fingers can be placed on the strings to sound out 18/16, 18/15, and 18/14 -- that is, the fingers divide the string into 18 equal parts. But 18/16 isn't called "Arabic lute middle finger" because it reduces to 9/8, which already has a name ("major tone"). Likewise, 18/15 = 6/5 (minor third) and 18/14 = 9/7 (supermajor third), so these don't have Arabic names.
The next interval that could have an Arabic name would be 18/13 -- but by this point, we've run out of fingers! And besides, 18/13 is a sort of superfourth, while the interval between the strings are perfect fourths (just as they mostly are on a modern guitar). Thus once we reached the fourth, an oud player would just jump to the next string.
From the perfect fourth, it's easy to use the middle finger to add 9/8 to this to obtain a perfect fifth, and the octave is likewise reached on the next string. And within each string, we can play a sort of semitone (18/17), whole tone (9/8), minor third (6/5), and major third (9/7). In other words, from 18EDL we have the beginning of a twelve-tone scale, centuries before 12EDO was created.
Before we leave music, I like to point out that for one of the higher primes, Kite actually has a color called "fawn" -- which reminds us of Fawn Nguyen, the Queen of the MTBoS. We'll find out what prime the color "fawn" corresponds to in a later post.
We'll continue our discussion of 18EDL on Thursday.
Question 21 of the SBAC Practice Exam is on simplifying exponents:
Write an expression equivalent to b^11/b^4 in the form b^m.
Exponents are definitely a second-semester Algebra I topic. We know the Laws of Exponents, and the rule that to divide powers, we subtract exponents. Therefore the answer is b^(11 - 4) = b^7. The hardest part of students (provided they know the Laws of Exponents) is entering b^7 properly.
Question 21 of the SBAC Practice Exam is on the average rate of change:
The depth of a river changes after a heavy rainstorm. Its depth, in feet, is modeled as a function of time, in hours. Consider this graph of the function.
[The graph passes through many points, including (9, 18) and (18, 21) -- which, as you'll soon see, are the only two points that matter.]
Enter the average rate of change for the depth of the river, measured as feet per hour, between hour 9 and hour 18. Round your answer to the nearest tenth.
This is considered to be a first-semester Algebra I problem, but it's worded strangely. The phrase "average rate of change" confuses many students and teachers alike.
The first time I, as a young student, ever heard the phrase "average rate of change" was in an AP Calculus class. Our teacher fave an average rate of change problem, and polled the students whether they needed to find an integral or a derivative to find the solution. I forgot which answer I chose, but I remember that the correct answer is neither. Here's the reason why, in a nutshell -- the word "average" implies an integral (as in "the average value of a function"), while "rate of change" obviously implies a derivative. Thus in "average rate of change," the integration and differentiation cancel each other out, and so neither is needed (which had better be the case, otherwise this question has no business being on the SBAC).
In fact, the average rate of change of a function between is just the slope of the line passing through the two points. Here's somewhat of a proof, from Calculus:
"Average" means integral:
1/(b - a) times the integral from a to b of something dx
"Rate of change" means that "something" is derivative:
1/(b - a) times the integral from a to b of f '(x) dx
By the Fundamental Theorem of Calculus, the integral of f '(x) is f (x)
1/(b - a) times f (x), evaluated from a to b
1/(b - a) times (f (b) - f (a))
(f (b) - f (a))/(b - a)
which is indeed the slope of the line through (a, f (a)) and (b, f (b)). QED
Of course, Algebra I students don't deal with the proof. Instead, they're taught that the average rate of change through two points is simply the slope of the line passing through them. It's mentioned as a real-world example of slope and an instance of the Common Core Standards on modeling.
Oh yeah, let's solve the problem. The average rate of change, or slope, is:
(21 - 18)/(18 - 9) = 1/3
The directions ask students to round this to the nearest tenth, so the correct answer is 0.3 ft./hr.
SBAC Practice Exam Question 22
The color "umber" is a odd choice, since it sounds so much like "amber." In fact, Mocha has a note at Degree 187 (Sound 74) called "umber-amber A#." But Kite, when choosing colors for his notation, was more concerned with the "u," not the "-mber" part. He wanted each color to start with a different letter, and he needed a color starting with "u," so "umber" it is. (His notation uses every letter of the alphabet except "z.")
Finally, you might ask why our first umber note is called D# rather than Eb. Well, there are two reasons for preferring sharps over flats. In utonal music, fourths are preferred over fifths (since the largest prime of 4/3 is in its denominator, unlike 3/2). If Degree 17 is umber D#, then Degree 51 (that is, 17 * 3) is umber G# and Degree 153 is umber C#. Then Degree 459 would be umber F#, and Degree 1377 would be umber B (even though these last two are out of Mocha range). So at least in theory, we can reach notes without a classical accidental by proceeding fourthward from D#. On the other hand, proceeding fourthward from Eb takes us into more flats before reaching double flats.
And here's another reason that Kite prefers sharps here. The interval 18/17 is 99 cents, while the interval 17/16 is 105 cents. Thus Degree 17, in cents, is closer to Degree 18 than to 16. And so we'd prefer 17 to have the same letter name as 18, which again means that D# is used over Eb. Both of these reasons generalize to higher primes, so if we have a choice between a flat name and a sharp name for a new prime, the latter is preferred.
What good are umber notes, anyway? The use of the other limits is obvious -- the 3-limit gives us fifths while the 5-limit gives us thirds. The 7-limit is used in jazz (including barbershop). Even the 11- and 13-limits can be used for neutral intervals, plus we might want harmonic 11th and 13th chords (following the pattern for harmonic 7th chords). But 17 seems to be just a useless prime.
Well, in Helmholtz-Ellis notation, the 17-limit can be indicated using a symbol for 256/255. This comma is so small that H-E calls it a "schisma" rather than a "comma." (At 6.8 cents, it's slightly smaller than the 225/224 comma that Kite calls the "sub.")
The 256/255 schisma is one of the smallest intervals playable in Mocha. Degree 256 (Sound 5) is called "white E," while Degree 255 (Sound 6) is called "umber-green E" (since 255 has the factor 5 as well as 17). In other words, the difference between "white" and "umber-green" is the schisma. As the color that cancels "green" is yellow, it follows that "yellow" differs from "umber" by the schisma.
In Mocha, otonal colors like "yellow" are unplayable, but utonal colors like "umber" are playable. So we can play major chords in Mocha using "umber" instead of "yellow." For example, an A umber major triad would be Degrees 192:153:128 (Sounds 69-108-133). Unfortunately, there still aren't enough umber notes to make a complete major scale (so for now, green Bb major remains the only playable major scale).
There's one more thing I'd like to say about the prime 17. Let's look at the 18EDL scale once more:
The 18EDL scale:
Degree Ratio Note
18 1/1 tonic
17 18/17 septendecimal chromatic semitone (Arabic lute index finger)
16 9/8 major tone
15 6/5 minor third
14 9/7 supermajor third
13 18/13 tridecimal semiaugmented fourth
12 3/2 perfect fifth
11 18/11 undecimal neutral sixth
10 9/5 minor seventh
9 2/1 octave
Hmm, that's a strange name for the 18/17 interval, "Arabic lute index finger." I'm very curious about the reason for that name -- and I wonder if there's an "Arabic lute index finger," are there also an "Arabic lute middle finger," "ring finger," and "little finger"?
First of all, a "lute" is a string instrument, similar to the modern guitar. On a modern guitar, we can place an index finger on the first fret to raise the pitch -- for example, fretting the D string at the first fret produces D#. So perhaps we can place an index finger on an Arabic lute to raise its pitch by the semitone-like interval 18/17.
Well, a traditional "Arabic lute" -- also known as an oud -- has no frets. Still, I can imagine raising the pitch of an oud string by using the index finger, thereby justifying the name.
The Xenharmonic website uses the name "Arabic lute" to refer to two intervals. One of these is 18/17, but the other is the "Arabic lute acute fourth," 49/36:
http://xenharmonic.wikispaces.com/49_36
But 49/36 is a septimal interval -- it has nothing to do with 17. And you'd think that as easy as it is to use Google to research ouds, no website seems to explain why 18/17 is "Arabic lute index finger."
Well, here's my conjecture -- 18/17 is actually part of a simple 18EDL system. The middle, ring, and little fingers can be placed on the strings to sound out 18/16, 18/15, and 18/14 -- that is, the fingers divide the string into 18 equal parts. But 18/16 isn't called "Arabic lute middle finger" because it reduces to 9/8, which already has a name ("major tone"). Likewise, 18/15 = 6/5 (minor third) and 18/14 = 9/7 (supermajor third), so these don't have Arabic names.
The next interval that could have an Arabic name would be 18/13 -- but by this point, we've run out of fingers! And besides, 18/13 is a sort of superfourth, while the interval between the strings are perfect fourths (just as they mostly are on a modern guitar). Thus once we reached the fourth, an oud player would just jump to the next string.
From the perfect fourth, it's easy to use the middle finger to add 9/8 to this to obtain a perfect fifth, and the octave is likewise reached on the next string. And within each string, we can play a sort of semitone (18/17), whole tone (9/8), minor third (6/5), and major third (9/7). In other words, from 18EDL we have the beginning of a twelve-tone scale, centuries before 12EDO was created.
Before we leave music, I like to point out that for one of the higher primes, Kite actually has a color called "fawn" -- which reminds us of Fawn Nguyen, the Queen of the MTBoS. We'll find out what prime the color "fawn" corresponds to in a later post.
We'll continue our discussion of 18EDL on Thursday.
Question 21 of the SBAC Practice Exam is on simplifying exponents:
Write an expression equivalent to b^11/b^4 in the form b^m.
Exponents are definitely a second-semester Algebra I topic. We know the Laws of Exponents, and the rule that to divide powers, we subtract exponents. Therefore the answer is b^(11 - 4) = b^7. The hardest part of students (provided they know the Laws of Exponents) is entering b^7 properly.
Question 21 of the SBAC Practice Exam is on the average rate of change:
The depth of a river changes after a heavy rainstorm. Its depth, in feet, is modeled as a function of time, in hours. Consider this graph of the function.
[The graph passes through many points, including (9, 18) and (18, 21) -- which, as you'll soon see, are the only two points that matter.]
Enter the average rate of change for the depth of the river, measured as feet per hour, between hour 9 and hour 18. Round your answer to the nearest tenth.
This is considered to be a first-semester Algebra I problem, but it's worded strangely. The phrase "average rate of change" confuses many students and teachers alike.
The first time I, as a young student, ever heard the phrase "average rate of change" was in an AP Calculus class. Our teacher fave an average rate of change problem, and polled the students whether they needed to find an integral or a derivative to find the solution. I forgot which answer I chose, but I remember that the correct answer is neither. Here's the reason why, in a nutshell -- the word "average" implies an integral (as in "the average value of a function"), while "rate of change" obviously implies a derivative. Thus in "average rate of change," the integration and differentiation cancel each other out, and so neither is needed (which had better be the case, otherwise this question has no business being on the SBAC).
In fact, the average rate of change of a function between is just the slope of the line passing through the two points. Here's somewhat of a proof, from Calculus:
"Average" means integral:
1/(b - a) times the integral from a to b of something dx
"Rate of change" means that "something" is derivative:
1/(b - a) times the integral from a to b of f '(x) dx
By the Fundamental Theorem of Calculus, the integral of f '(x) is f (x)
1/(b - a) times f (x), evaluated from a to b
1/(b - a) times (f (b) - f (a))
(f (b) - f (a))/(b - a)
which is indeed the slope of the line through (a, f (a)) and (b, f (b)). QED
Of course, Algebra I students don't deal with the proof. Instead, they're taught that the average rate of change through two points is simply the slope of the line passing through them. It's mentioned as a real-world example of slope and an instance of the Common Core Standards on modeling.
Oh yeah, let's solve the problem. The average rate of change, or slope, is:
(21 - 18)/(18 - 9) = 1/3
The directions ask students to round this to the nearest tenth, so the correct answer is 0.3 ft./hr.
SBAC Practice Exam Question 21
Common Core Standard:
Rewrite expressions involving radicals and rational exponents using the properties of exponents.
SBAC Practice Exam Question 22
Common Core Standard:
Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
Commentary: The standard listed for Question 21 is the closest standard in the high school section -- the true standard is an eighth grade standard. Quotients of Powers appear in Lesson 9-7 of the U of Chicago Algebra I text. The phrase "average rate of change" doesn't appear in the text, but "rate of change" appears in Lesson 8-1, with "average" implied. Constant rates of change appear in the next lesson. Notice that the first eight chapters of the U of Chicago text correspond to the first five chapters of Glencoe and the first semester in Edgenuity.
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