Tuesday, June 5, 2018

Algebra I on the MTBoS (Day 179)

Today I subbed in a high school special ed math/English class. I've subbed in this class before -- in fact, I wrote about the class in my April 20th post. You and I alike can reread that post so that I can compare the two visits to the class. Hopefully, I've improved in the past month and a half. We'll find out how much I improved in today's "Day in the Life." As it turns out, this is also the same school that will host the summer math classes this year, including my summer Algebra 1A class.

(Of course this is in my new district, where today is only Day 173. That's why there's no mention of finals in this "Day in the Life." In my old district, today is Day 179 and finals week.)

7:55 -- As I mentioned back on April 20th, second period is Algebra 1B, the second of two years that special ed students spend in Algebra I. (As usual, "first period" = zero period.) The students are now working in Chapter 9 of the Glencoe text, which is on quadratic equations. Specifically, the students are learning how to solve equations using the Quadratic Formula.

I've already shown some improvement since April, since I avoid the attendance mistake. In April, I waited to ask for a roster (since the computer still isn't working), but today, I send a student to the office right away. When the special aide arrives at 8:15 (the real aide, not a sub as in April), she shows me how I can use one of the Chromebooks for attendance.

Lately, I've been writing about music and how I'll use new scales to create new music that I can sing during math lessons. Of course, today's lesson already has a well-established song associated with it, namely "Pop Goes the Weasel" (with the lyrics replaced by the Quadratic Formula). It goes without saying that I sing this song in class today. (It's too bad I didn't bring a guitar to play it on, though.)

One guy doesn't want to do the work. He tells both the aide and me that he prefers working with the regular teacher, and how he fears that our explanations will confuse him. He reluctantly agrees to work with the special aide. On the other hand, he definitely enjoys my singing of "Pop Goes the Weasel," perhaps more than any other student in this class.

8:50 -- Second period ends. Like all special ed teachers in this district, this regular teacher co-teaches another class. For third period, I travel to a junior English class. The main teacher in this class is having her students work on a research project.

9:45 -- It is now time for tutorial. I return to my classroom. Some students come in for help on the quadratic worksheet, and the aide and I help them. This includes the second period student who prefers the regular teacher's assistance. The aide convinces him to let me help him, and he agrees -- as long as the aide remains close by.

She has him work on an old worksheet, which is on completing the square. He is confused, but I believe our explanations help him out. I notice that he prefers avoiding subtraction completely, replacing it with addition of negatives. (In April, I wrote that the aide similarly avoided subtraction, but that was a sub, not the usual aide who's in the classroom today.)

10:25 -- Tutorial ends and it's time for snack -- except for the kid from second period. He asks me why there is more than one way to solve a quadratic equation. (By this time the aide has left too.) In fact, I take out four whiteboards and show him all four ways of solving quadratics -- graphing, factoring, completing the square, and the Quadratic Formula. On all the whiteboards I show him the equation x^2 + 4x + 3 = 0 -- and he's fascinated when he notices that I chose the exact same equation that his teacher did in his notes on solving by graphing.

10:40 -- As snack ends, the student apologizes for complaining about my help. He realizes that I can assist him just as well as his regular teacher does. After working through the first part of the day, I enjoy a well-needed break during the teacher's conference period.

11:40 -- Fifth period begins. This is a junior English class. They are reading Of Mice and Men, and the regular teacher hopes that I play the audio version of the book on YouTube while the special ed kids follow along. But as I wrote earlier, the computer has failed -- and the Chromebook on which I take attendance doesn't connect to any speaker system. And so I read most of Chapter 3, but allow students to volunteer to read parts of it. Because the students have to do this unexpected reading and do so well, I consider this to be the best-behaved class of the day. I leave the names of good readers for the regular teacher (though one girl who doesn't read tries to write her name on the list anyway).

12:30 -- It is time for lunch.

1:15 -- It is now sixth period and the second Algebra 1B class. In April, I wrote that there is supposed to be a second aide for this class. He didn't come that day in April and he's a no-show again today.

This is the worst-behaved class of the day. Many of the students talk during the entire lesson, and some guys near the back start throwing rulers at the other students. I grow frustrated because I can see that some of the students need extra help with the Quadratic Formula, but I can help them since whenever I turn my back, someone is throwing something. I almost write a referral, but it's difficult to tell who is throwing objects. (It's clear that more than one student is involved.)

2:10 -- It is now seventh period. As I explained in April, the regular teacher is the tennis coach, and by now all sports seasons have completed. The tennis players come in and use this class essentially as another tutorial. One player claims that he's going to the restroom for a few minutes -- but he spends the whole period in another classroom instead. I mark him absent.

2:50 -- A few minutes before seventh period ends, the regular teacher returns from her meeting (which is all about curriculum development for English). I tell her all about the students whose names I left for her, as well as the sixth period situation.

As I reflect upon this class, I wonder how I could have taught the class better. In April, my focus resolution for this class was:

3. Move on from past incidents instead of bringing them up with students.

I believe that I follow this resolution well enough today -- as tempting as it might have been to mention the students' April behavior again today.

But I do wish to reflect about how I could have managed the sixth period class better. I notice that sixth period appears to enjoy the song "Pop Goes the Weasel" better than second period -- to the extent that many students who are talking the whole time stop to listen to me sing. Then they beg for me to repeat the song again -- though I know that they want to enjoy the song itself rather than do any work or try to memorize the formula. As soon as I stop, I know they'll go right back to talking and throwing objects.

Perhaps I could have used the song as an incentive. For example, every time the students are quiet and attentive, I give the class a point. Each time a student throws a ruler, I deduct a point. Then at the end of class, I sing "Pop Goes the Weasel" as many times as the students have points. I have no problem with singing such a short song multiple times.

But some readers might object to this. It sounds too much like giving the students a reward just for doing what they're supposed to be doing. Yet keep in mind that I'm going to sing the song anyway, since I consider the song to be a part of my Quadratic Formula lesson. I'm just using a song in a way to maximize its effectiveness -- including its use as part of classroom management.

Perhaps another idea is just to sing the song each time I set up a quadratic equation. This means that I'll sing the song more if the class gets through more problems -- which means that the students should be quiet and let us get through more problems if they want to hear the song more. Now the song is no longer a reward -- it's a natural part of the lesson. The students now get what they want -- more renditions of the song -- if we can complete more of the lesson.

Speaking of music, it's time for our next EDL scale. This week's new EDL scale is 20EDL:

The 20EDL scale:
Degree     Ratio     Note
20            1/1         tonic
19            20/19     undevigesimal chromatic semitone
18            10/9       minor tone
17            20/17     septendecimal augmented second
16            5/4         major third
15            4/3         perfect fourth
14            10/7       large septimal tritone
13            20/13     tridecimal semiaugmented fifth (ratwolf fifth)
12            5/3         major sixth
11            20/11     small undecimal neutral seventh
10            2/1         octave


20 FOR D=10 TO 20
30 SOUND 261-N*D,4

This is a descending scale. To make the scale ascend, use:

20 FOR D=20 TO 10 STEP -1

The 20EDL scale introduces a new prime, 19. It's the first of two 19-limit EDL scales -- the other such scale is 22EDL. Of course, 20EDL is the only EDL that is 19 odd-limit. It contains the entire 19 odd-limit tonality diamond.

In Latin, we use "undevigesimal" to mean "one less than twenty," or 19. We know that the ancient Romans often counted downward from the next multiple of ten, just as XIX in Roman numerals is 19 (one less than XX). (In number bases, prime bases like 19 are rare. On the other hand, base 20 is often mentioned, and it's indeed called "vigesimal.")

As for Kite colors, we need new colors for 19. The otonal color for 19 is "fawn." (Yes, Fawn Nguyen, I told you that your namesake color was coming up soon!) The utonal color for 19 is "khaki" -- and as usual, it's the utonal colors that appear in Mocha scales:

Let's check out the notes of 20EDL using Kite's color notation:

The 20EDL scale:
Degree     Ratio     Note
20            1/1         green C
19            20/19     khaki C#
18            10/9       white D
17            20/17     umber D#
16            5/4         white E
15            4/3         green F
14            10/7       red F#
13            20/13     ocher G
12            5/3         white A
11            20/11     amber B
10            2/1         green C

We see that the new khaki note is named C# (rather than Db) for the same reason that the umber note is named D# -- sharp names for new utonal primes turn out to be better in the long run.

Just as with umber notes, we might wonder what musical purpose the new khaki notes serve. Once again, we return to Helmoltz-Ellis. There is a symbol that looks like a backslash, \, which means to lower a note by the 19-limit schisma, 513/512. This schisma, at 3.4 cents, is smaller than any comma available in Mocha, since Degrees 512 and 513 are beyond 260.

But if Degree 512 existed, it would be white E (as are all degrees that are powers of two), while Degree 513 would be khaki E (three perfect fifths below khaki C#). Thus khaki notes differ from white notes by this small schisma -- and are perhaps audibly indistinguishable from them.

In other notes, we can use khaki notes where white notes are unavailable in Mocha, just as we can use umber notes as a substitute for yellow notes. But at first glance, this seems useless -- yellow notes are otonal and hence unplayable in Mocha, whereas there are playable white notes in Mocha.

But recall that last week, I attempted to build a major scale using umber rather than yellow notes. In general, a just major scale has a yellow 3rd, 6th, and 7th, with all the other intervals white. The first three playable umber notes in Mocha are D#, G#, and C#. These are exactly the yellow notes needed for a major scale on white E. But unfortunately, E major requires two white notes -- F# and B -- that are unplayable in Mocha.

Well, white F# and B are unplayable, but khaki F# and B are playable. In fact, the first three playable khaki notes are C#, F#, and B. This means that we can combine umber and khaki notes to create a playable major scale on white E:

The white E major scale (using umber/khaki):
Degree     Ratio     Note
256          1/1         white E
228          64/57     khaki F#
204          64/51     umber G#
192          4/3         white A
171          256/171 khaki B
153          256/153 umber C#
136          32/17     umber D#
128          2/1         white E

The only true major scale in Mocha starts on green Bb -- making the root note green allows us to play white notes that lie at yellow intervals over the root. The two root notes white E and green Bb differ by a sort of tritone, so that most songs in a major scale can be raised/lowered by a semitone/whole tone (at most a minor third) to fit either the E or Bb major scales.

But today's topic is supposed to be about 20EDL, not the E umber/khaki scale. We do notice that just like the aforementioned Bb scale, the fundamental root note of 20EDL is green -- and in fact, a major third can be played over the root. It's the first EDL scale (beyond 10EDL, that is) that contains a major third, so we can call 20EDL the major EDL. Unfortunately, 20EDL lacks a perfect fifth, and so we must go to 30EDL if we want a full major triad on the root. (The first full major scale in an EDL is of course 180EDL, starting on green Bb.)

Notice that 20EDL (and in fact every EDL starting with 16EDL) contains the 15:12:10 triad, which is a true major triad. The only problem is that 15 can't be the root of an EDL since there's no octave. In the case of 20EDL, 15:12:10 represents a major triad on IV, while a full major triad on I is unplayable because of its fifth.

Instead, the best we can do is 20:16:13, which is gC-wE-oG. We know that ocher G is closer to white G# than to white G, and so this triad sounds more augmented than major. But notice that even without the perfect fifth, today's song "Pop Goes the Weasel" might be converted to 20EDL, since the fifth isn't emphasized in that melody.

Officially, white E is the only third (over the root gC) in 20EDL. The interval gC-uD# sounds sort of like a minor third (at 281.4 cents, it's wider than a superminor 3rd), but technically speaking, this is an augmented 2nd, not a minor 3rd. Besides, we already have two EDL's (12EDL, 18EDL) in which a just minor third is playable, so we don't need this so-called "minor 3rd."

The interval 19/16 (that is, kC#-wE) is considered to be a minor third. Its size is 297.5 cents, close to the minor third of 12EDO (the usual scale). The otonal triad 16:19:24 (the "fawn triad") therefore sounds very close to a minor triad. This isn't playable in Mocha since we can't play a perfect fifth above the 19, but it does suggest that the utonal triad 24:19:16 (the "khaki triad") is close to a major scale. This triad is available in 24EDL, and it's is the closest we can get to a major triad over the root of an EDL before we reach 30EDL.

The root note of the fundamental 20EDL scale is green. We know that Kite prefers to think in terms of the root note being white. So we rename all of the intervals by adding the opposite color of green, "yellow," to the name of each interval:

The 20EDL scale:
Degree     Ratio     Interval
20            1/1         white unison
19            20/19     khakiish semitone
18            10/9       yellow 2nd
17            20/17     umberish 2nd
16            5/4         yellow 3rd
15            4/3         white 4th
14            10/7       reddish 4th
13            20/13     ocherish 5th
12            5/3         yellow 6th
11            20/11     amberish 7th
10            2/1         white octave

Notice that in Kite's notation, "red" plus "yellow" is "reddish." Many other colors also add "-ish" to indicate the color yellow.

Here are the roots of all the 20EDL scales available to us in Mocha:

Possible 20EDL root notes in Mocha:
Degree     Note
20            green C
40            green C
60            green F
80            green C
100          deep green Ab
120          green F
140          greenish D
160          green C
180          green Bb
200          deep green Ab
220          amber-green G
240          green F
260          ocher-green Eb

Recall that Degree 260 is Sound 1, the lowest note playable in Mocha. Thus 20EDL is one of the simplest scales containing this low note.

Today is the penultimate day of the semester -- the day I usually devote to exploring the MTBoS, the Math Twitter Blogosphere. For today's post, I wish to focus on Algebra I teachers on the MTBoS, since I'm preparing to teach a summer Algebra I class.

Who is the most famous Algebra I teacher on the MTBoS? That's easy -- it's no contest. The most famous Algebra I blogger is Sarah Carter of Oklahoma:


Even though my favorite type of math is Geometry, I can't help but use many Carter-isms, such as "Slope Dude" and "DIXI-ROYD," whenever I sub in Algebra I classes. (Actually, I doubt that Carter is the originator of "Slope Dude," but "DIXI-ROYD" is 100% Carter.) And many other Algebra I bloggers frequently link to Carter's website.

I must make one correction. The most famous Algebra I blogger was Sarah Carter of Oklahoma. The reason for the correction is Carter's recent announcement that she's no longer an Algebra I teacher:


After six years at Drumright High School, I have turned in my keys and said some tearful good byes. I will be returning to my hometown of Coweta, Oklahoma to teach Algebra 2, Pre-AP Algebra 2, and Pre-Calc at Coweta High School in the fall. I will miss teaching at the school that made me a teacher, but I'm excited to return to the school I graduated from as a teacher.

Oops -- don't let SteveH or other traditionalists see that "Pre-AP" class there. Presumably, Pre-AP Algebra II is for juniors headed for AP Calculus AB as seniors. According to SteveH, the proper AP Calculus track is for eighth graders to take a full Algebra I course, so that they're ready to take Algebra II as sophomores, not juniors.

Anyway, the MTBoS can no longer depend on Carter to post her great ideas for Algebra I. But her new Algebra II classes don't begin until the fall, and so most of the old posts currently available on her blog are still geared towards Algebra I. And besides, current Algebra I teachers aren't necessarily posting in the summer anyway, so we might as well go back to old posts.

And in fact, Carter's most recent post is all about photos of her old classroom -- back when she was still an Algebra I teacher:


Last year, I used the "infinity" poster (from her tenth photo) in my own classroom. I'm not sure what I'll be able to post in this class, since I won't be there until a few days before summer school starts.

The cornerstone of Carter's Algebra I course is the interactive notebook. Here's an old post where she explains what an interactive notebook is and how to set it up:


And here is an Algebra I unit (on "sequences") in the interactive notebooks:


So in many ways, interactive notebooks and foldables are similar (and in fact, Carter uses the label "foldable" for one of the interactive notebooks above).

I've actually seen interactive notebooks used in some of the classes where I sub. For example, today the second period student who learned the four methods of solving quadratic equations showed me his interactive notebook. And a few weeks ago, I was in a science class where one period chose to submit all work in an interactive notebook to be graded, while another period eschewed the interactive notebooks. A problem occurred when a teacher was out for multiple days, and the notebooks were sitting in a pile on a desk to be graded, preventing the students from taking them home to study or do more work in them.

I can see why interactive notebooks can be useful -- students are more likely to take notes in an interactive notebook or on a foldable than in a regular notebook. Earlier, I was considering using foldables for my summer class, since Glencoe encourages them in the text.

That, of course, was before I found out about Edgenuity and the computer-based course. Now it's awkward to have any sort of foldable or interactive notebooks at all when most lessons will be taught on Chromebooks.

With Carter out of the picture, I'm still in the process of finding some good Algebra I blogs. What makes it tricky is that some Algebra I teachers are in middle school. Many of them teach both Algebra I and Common Core Math 8 -- and when I read the blogs, most of the posts turn out to be about Math 8 than Algebra I. Other teachers turn out to be Integrated Math I teachers, or else international teachers (which are also integrated math teachers, of course). But I have yet to find an Algebra I blog as prominent as Carter's has been for the past few years.

Hopefully, I'll find some good Algebra I blogs by the time my summer class begins.

No comments:

Post a Comment