It's a special ed class. In this district, late start days are on Fridays and then, just like most special ed teachers, I had periods of observation. My first actual class isn't until third period (two sections of Academic Enrichment and one section of senior English). Notice that most schools in this district have a block schedule, but one day a week all classes meet on a late day.
One of the periods I observe is a math class, Algebra I. The class is learning about statistics -- specifically box-and-whisker plots. Recall that California includes some of the Common Core standards for stats in Algebra I. As you can tell, most classes cover it at the end of the year (for example, stats is Chapter 12 of the Glencoe text).
This is actually the first time I've been in a math classroom in this district! And I still don't know what text this district uses -- all I see today are packets, with today's lesson considered part of Unit 9. A Google search doesn't reveal any particular curriculum where stats is Unit 9 of Algebra I (except for some school in Pennsylvania, but the other units don't match up). It might be the case that Algebra I here has no specific text, and so the "units" are simply based on taking the standards of California Algebra I in naive order (even though stats is actually the tenth set of listed standards).
Meanwhile, during the staff meeting (that I unwittingly attended because I wasn't sure of the late day in this district), I saw a Geometry text written by Ron Larson and Laurie Boswell. I did a quick Google search for the Larson Geometry text -- judging by its table of contents, it doesn't appear to be significantly different from the Glencoe Geometry text. Well, at least I finally know what Geometry text this district uses!
There is one classroom management incident today. In fourth period English, the students are working on a Personal Insight packet -- since they are just 12 school days away from graduating, they should be writing about their future plans. One guy asks for a restroom pass, and I allow it, since this isn't the class right after snack or lunch. But when he leaves, he takes out a cell phone, and a security guard catches him. She asks him for his ID (that he's supposed to carry at all times), but he's left it in his backpack that's still in the classroom. So he talks back to her -- and she ultimately escorts him back to my room.
Afterward, I decide to look at his Personal Insight packet. To the question "What do you want to do?" he's written a single word in reply: "Army." So now I'm thinking -- does he plan on talking back to his commanding officer the same way he speaks to security?
I have a discussion with him about the incident. Due to the personal nature of the conversation, I have nothing further to say about this on the blog.
Fifth period is the teacher's conference period, but this time I'm assigned to another class -- Guitar class. Yes, you guessed it -- this is now turning into yet another music post. After I planned on writing about music only on Tuesdays through Thursdays, this is the second straight Friday that I'm subbing in a music classroom.
(Change of plans -- music posts are now Tuesday, Thursday, Friday. This frees up Wednesdays, since that's when summer school training is.)
In today's Guitar class, a few students are practicing another scale -- the A minor pentatonic scale. Notice that 12EDL almost produces an A minor pentatonic scale (if we skip amber B) -- except that red F# is played where G is expected. (The red F# is the same reason that 10EDL is not a C major pentatonic scale.)
I ask a student what songs he's learned in this class. The first song in their music text is "Rumble," while Bob Dylan and Johnny Cash songs feature in the first few pages. The early songs are based on chords (rhythm guitar), but some of the students are jamming in lead guitar as well. Both acoustic and electric guitars are available for the students.
I notice that some simple music is written on the board. It takes me some time to realize that it's the twelve-bar blues. But as it turns out, the progression is intended for Band class, not Guitar!
Here is the chord progression:
Bb7-Eb7-Bb7-Bb7-Eb7-Eb7-Bb7-
I7 -IV7 -I7 -I7 -IV7 -IV7 -I7 -I7 -V7-IV7 -I7 -V7
Whole notes are written in all 12 bars. Only two parts are shown -- thus the entire seventh chords aren't being played. Here are the parts being played:
Bb7 bars: D-Ab
Eb7 bars: Db-G
F7 bars: Eb-A
So notice that the seventh chords have all been reduced to tritones. We know that the tritone is a prominent interval in 14EDL, so perhaps we might be able to play the 12-bar blues in 14EDL or something similar. It's ironic that musicians once called the tritone "the devil in music." Imagine what those musicians would say if they heard 12 straight tritones being played! In a seventh chord, there is a tritone between the third and the seventh.
But as I wrote earlier, the 12-bar blues on the board is set up for Band, not Guitar. We know that the Bb and other flat scales are more easily played on band instruments than the guitar. On the guitar, we might play E7 as the I7 chord, then IV7 becomes A7 and V7 becomes B7:
E7-A7-E7-E7-A7-A7-E7-E7-B7-A7-E7-B7
I ask a student to let me borrow his guitar, and then I played these 12 bars. I played the full E7, A7, B7 chords rather than the tritones as written on the board. But if I really wanted to play the tritones on the guitar, then how would I play them?
We notice that the first bar is D-Ab. The note Ab is enharmonic (in 12EDO) to G#, so we could have written this tritone as D-G#. But hold on a minute -- G# is the third of E7 while D is its seventh. Thus D-G# is already the correct tritone for E7! OK then, let's look at the tritone for Eb7, Db-G. Now Db is enharmonic to C#, so this is the same as C#-G -- which is already the tritone for A7. Likewise, if we take the tritone for F7, Eb-A, and change Eb to its enharmonic D#, the result is D#-A, the tritone for B7. And notice that the roots of the pairs Bb7-E7, Eb7-A7, and F7-B7 are themselves tritone pairs.
This idea is called "tritone substitution." Tritones and seventh chords are intimately related this way. In many even EDO's, including 12EDO, there are two enharmonic ways to write each tritone, and the two tritones appear as part of different seventh chords. (It goes all the way back to 2EDO, the first EDO where a tritone appears.) Because of this, we have have a triowhere two band players (saxophonists, say, since that instrument is strongly associated with the blues) can play the tritones as written on the board (Bb7, etc.) while a rhythm guitarist plays E7, etc., and no one would bat an eyelid.
But tritone substitution doesn't work in odd EDO's, nor does it work in just intonation (including EDL's). Typically, I consider the seventh chord of jazz and blues to be the 4:5:6:7 (otonal) tetrad. This means that the tritone that appears in this chord is the 7/5 small septimal tritone. Replacing one note with what appears to be enharmonic (in 12EDO) changes the 7/5 tritone to the 10/7 large septimal tritone.
Thus if we play the tritone progression as written on the board in just intonation, then we're playing a 7/5 tritone over the Bb7 chord, but 10/7 tritones over the Eb7 and F7 chords.
When I came up with the New 7-Limit Scale, I wanted to include a 4:5:6:7 chord. But the simplest EDL in which this chord is playable is 210EDL, since EDL's aren't designed to play otonal chords. And only one such triad (with root note Degree 210, a greenish G) is playable, so we can't play a full progression with I7, IV7, and V7. But perhaps if we only play the tritones rather than the full chords, then perhaps the progression can be Mocha-playable. The key here is 14EDL, since its fifth is indeed the 7/5 tritone.
Let's look again at all of the playable 14EDL scales:
Possible 14EDL root notes in Mocha:
Degree Note
14 red F#
28 red F#
42 red B
56 red F#
70 greenish D
84 red B
98 deep red G#
112 red F#
126 red E
140 greenish D
154 amber-red C#
168 red B
182 ocher-red A
198 deep red G#
210 greenish G
224 red F#
238 17esque-red E#
252 red E
We look at the red B (Degree 168), red E (Degree 126), and red F# (Degree 112). Above these three notes, we play 7/5 tritones to obtain green F (Degree 120), green Bb (Degree 90), and green C (Degree 80). Notice that all of these tritones consist of the third and seventh only -- the corresponding seventh chords are actually G7, C7, D7, with the roots G, C, D all greenish. The greenish G is at Degree 210, but the greenish C isn't Mocha-playable.
Here is a simple Mocha program to play the 12-bar blues using these tritones. Since we can't play harmony in Mocha, we play the two notes of the tritone as half notes (Length 8). There's no possibility of converting this to another key, and so we dispense with Degrees and just convert everything to Sounds.
10 FOR X=1 TO 24
20 READ S
30 SOUND S,8
40 NEXT X
50 RESTORE
60 GOTO 10
70 DATA 93,141,135,171,93,141
80 DATA 93,141,135,171,135,171
90 DATA 93,141,93,141,149,181
100 DATA 135,171,93,141,149,181
Question 9 of the SBAC Practice Exam is on irrational numbers:
Cheryl claims that any irrational number squared will result in a rational number.
Part A
Drag an irrational number into the first response box that when squared will result in a rational number.
Part B
Drag an irrational number into the second response box that when squared will result in an irrational number.
Here are irrationals that can be dragged: cbrt(2)/sqrt(3), sqrt(3)/sqrt(2), cbrt(2), sqrt(2), pi, sqrt(pi).
This is a tricky one to place. In the Common Core Standards, rational and irrational numbers appear in the eighth grade, and so it's arguably not a high school topic at all. If they do appear in an Algebra I text, it's likely to be in the context of quadratic equations, thus it's a second semester topic.
It's also the first question in the calculator section. Then again, calculators won't really help students with rational and irrational numbers.
Cheryl's idea that the square of an irrational number must be rational is an attractive one. After all, the first irrational numbers we encounter are numbers like sqrt(3), which when squared produces the rational number 3.
But, as Georg Cantor shows us (discussed in old posts), most numbers are irrational. Therefore, the square of most irrational numbers is still irrational. The list of choices includes cube roots, so their squares are still irrational. In fact, Cheryl's conjecture that the square of an irrational must be rational is just like a claim that doubling any fraction produces a whole number (presumably because the most commonly used fractions like 1/2 and 1 1/2 indeed have that property).
The choices involving pi are tricky. We know that the square of sqrt(pi) is pi, which is clearly irrational, and so sqrt(pi) would be dragged into the second box. But to which box should the number pi itself be dragged. I doubt that any high school text explains that pi^2 is irrational. We know that pi is transcendental, and so no integer power of pi can be rational -- but high school students wouldn't be expected to know this. Of course, students can forget about pi and just drag sqrt(pi) or one of the cube roots into the second box, since only one number needs to be dragged there.
So here is a complete answer: only sqrt(3)/sqrt(2) or sqrt(2) can be dragged into the first box. All other numbers are possibilities for the second box.
Question 10 of the SBAC Practice Exam is on building equations:
A train travels 250 miles at a constant speed (x), in miles per hour.
Enter an equation that can be used to find the speed of the train, if the time to travel 250 miles is 5 hours.
The guiding equation is d = rt, rate times time equals distance. The rate of speed is x, the time is 5, and the distance is 250. Therefore the equation is 5x = 250.
I consider this to be a first-semester Algebra I problem. While we might avoid the formula in middle school (or perhaps even mention dimensional analysis), by the time the students reach Algebra I, I should teach them the formula, the guiding equation.
Today is an activity day. As usual, my activities are the Exploration Questions from the U of Chicago text -- but this time it's the lessons from the U of Chicago Algebra I text. The lesson on rational numbers asks to find quadratic equations with rational solutions, while the lesson on d = rt asks for students to research similar-looking equations.
SBAC Practice Exam Question 9
Common Core Standard:
Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
SBAC Practice Exam Question 10
Common Core Standard:
Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
Commentary: Lesson 12-6 of the U of Chicago Algebra I text is on Rational and Irrational Numbers, and the Exploration Question there is on rational solutions of quadratics. Lesson 4-4 of the U of Chicago text is on Solving ax = b, with d = rt mentioned as an example. Students should have no problem with this question if they know the guiding equation. The Exploration Question in this section is on identifying other formulas.
Cheryl claims that any irrational number squared will result in a rational number.
Part A
Drag an irrational number into the first response box that when squared will result in a rational number.
Part B
Drag an irrational number into the second response box that when squared will result in an irrational number.
Here are irrationals that can be dragged: cbrt(2)/sqrt(3), sqrt(3)/sqrt(2), cbrt(2), sqrt(2), pi, sqrt(pi).
This is a tricky one to place. In the Common Core Standards, rational and irrational numbers appear in the eighth grade, and so it's arguably not a high school topic at all. If they do appear in an Algebra I text, it's likely to be in the context of quadratic equations, thus it's a second semester topic.
It's also the first question in the calculator section. Then again, calculators won't really help students with rational and irrational numbers.
Cheryl's idea that the square of an irrational number must be rational is an attractive one. After all, the first irrational numbers we encounter are numbers like sqrt(3), which when squared produces the rational number 3.
But, as Georg Cantor shows us (discussed in old posts), most numbers are irrational. Therefore, the square of most irrational numbers is still irrational. The list of choices includes cube roots, so their squares are still irrational. In fact, Cheryl's conjecture that the square of an irrational must be rational is just like a claim that doubling any fraction produces a whole number (presumably because the most commonly used fractions like 1/2 and 1 1/2 indeed have that property).
The choices involving pi are tricky. We know that the square of sqrt(pi) is pi, which is clearly irrational, and so sqrt(pi) would be dragged into the second box. But to which box should the number pi itself be dragged. I doubt that any high school text explains that pi^2 is irrational. We know that pi is transcendental, and so no integer power of pi can be rational -- but high school students wouldn't be expected to know this. Of course, students can forget about pi and just drag sqrt(pi) or one of the cube roots into the second box, since only one number needs to be dragged there.
So here is a complete answer: only sqrt(3)/sqrt(2) or sqrt(2) can be dragged into the first box. All other numbers are possibilities for the second box.
Question 10 of the SBAC Practice Exam is on building equations:
A train travels 250 miles at a constant speed (x), in miles per hour.
Enter an equation that can be used to find the speed of the train, if the time to travel 250 miles is 5 hours.
The guiding equation is d = rt, rate times time equals distance. The rate of speed is x, the time is 5, and the distance is 250. Therefore the equation is 5x = 250.
I consider this to be a first-semester Algebra I problem. While we might avoid the formula in middle school (or perhaps even mention dimensional analysis), by the time the students reach Algebra I, I should teach them the formula, the guiding equation.
Today is an activity day. As usual, my activities are the Exploration Questions from the U of Chicago text -- but this time it's the lessons from the U of Chicago Algebra I text. The lesson on rational numbers asks to find quadratic equations with rational solutions, while the lesson on d = rt asks for students to research similar-looking equations.
SBAC Practice Exam Question 9
Common Core Standard:
Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
SBAC Practice Exam Question 10
Common Core Standard:
Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
Commentary: Lesson 12-6 of the U of Chicago Algebra I text is on Rational and Irrational Numbers, and the Exploration Question there is on rational solutions of quadratics. Lesson 4-4 of the U of Chicago text is on Solving ax = b, with d = rt mentioned as an example. Students should have no problem with this question if they know the guiding equation. The Exploration Question in this section is on identifying other formulas.
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