Wednesday, September 28, 2022

Lesson 1.3.2: Zero and Negative Exponents (Days 35-37)

U-N-I-T RATE! RATE! RATE!

Please don't be sad,
To multiply powers, just add.
And it is a fact,
To divide powers, just subtract.
Zero powers are fun,
'Cause the answer's always one.
Don't be negative, don't frown,
To get rid of them, move down.
U! N! I! T!
U-N-I-T! Rate! Rate! Rate!

This is a song that I first posted to this blog six years ago. It's a parody of the fight song of UCLA -- my alma mater. It has an artefact title -- originally, I wrote this song at the old charter middle school, when my eighth graders were learning the laws of exponents at the same time my kids in Grades 6-7 were learning about unit rates. (Nowadays I no longer attempt to incorporate Math I and Math III material as verses of the same song.)

So perhaps I should change the title and hook to refer to something about exponents. But it's just so to drop U-N-I-T, which starts with the same letter as U-C-L-A and so links the parody to its parent song -- thus far, I can't think of a suitable replacement line.

For parodies, I don't post any melody, since it's similar to that of its parent song. On the guitar, I usually play the song in the key of D major. (I wish to sneak in an extra line, playing a D scale without any sharps or flats, D-E-F-G-A-B-C-D. This is called the Dorian mode, which I want to play in order to honor the fifth-year quarterback for UCLA, Dorian Thompson-Robinson. But I need to figure out what lyrics, if any, to sing over that scale.)

Since I'm not posting any Mocha code today, there are other things I wish to mention in this post. First of all, last weekend was the first big library book sale since before the pandemic. I found no recreational math books (and we no longer do side-along reading on the blog anyway). But I did obtain three textbooks that are indirectly related to the classes that I'm currently teaching:

  • The U of Chicago Algebra I text (second edition, teachers edition, part one)
  • The IMP Integrated Math I text
  • A college-level Intermediate Algebra text
I don't wish to tie up this post with long discussions of these textbooks (although this blog was first created to discuss the U of Chicago text). Instead, I'll briefly mention these texts in comparison to the CPM text from which I'm currently teaching.

Meanwhile, today's not an official "Day in the Life" day. But it's such a strange day that I can't help but describe it via "A Day in the Life":

8:30 -- First period begins. This is a Math III class. But a sub (actually another teacher) arrives to cover my class, because I have a special ed meeting. (A few years ago, I was the sub who surprised regular teachers who've forgotten their meetings, and now I know how it feels to be on the other side.)

The meeting is being held via Google Meet, so I depart for the classroom of another math teacher who has first period conference. But a few minutes before it's set to begin, there's a power outage. The blackout lasts several minutes, and it takes a little longer for Wi-Fi to be restored. So the big meeting ultimately turns to a short discussion.

9:55 -- First period ends for nutrition. On Wednesdays, the next block is third period conference. But now I'm scheduled for another meeting, this time with the TOSA who's assisting us with Math I. He meets with me and another Math I teacher who also has third period prep.

I discuss the scores from last week's group test. There are a few strong students, but also many students who are still struggling. I also inform the TOSA that I've been struggling to get my students participate during group work -- too many spend the time discussing non-math entertainment, then claim "I don't know" when I ask them a question.

The TOSA, in turn, talks about the path ahead. I was considering giving the exponents quiz this Friday and the individual part of the Chapter 1 Test next week. But he suggests letting last week's group test be the only Chapter 1 assessment, so we can dive directly into Chapter 2 (on linear equations and slope) next week. According to the original pacing guide, we should be starting Chapter 2 this week. But all Math I teachers have fallen behind because many of our students are struggling with equations (1.1) and functions, domain/range (1.2).

Let me now compare our CPM text to the U of Chicago text that I purchased over the weekend. This text appears to start slowly. Chapter 1 is a review of Pre-Algebra, then Chapters 2-5 are all about solving one-step equations -- one chapter for each operation (add, subtract, multiply, divide), as well as one-step inequalities. Not until Chapter 6 do we get to solve two-step equations. (Second edition has these same chapters but in a different order -- and Part 1 of the teachers edition ends with Chapter 6.)

But after seeing my actual students struggle in the classroom, I'm now starting to appreciate the U of Chicago approach of introducing equations slowly. If our school were following the U of Chicago text, we might be in Chapter 3 by now -- but we'd still be working on Pre-Algebra and solving one-step equations with addition and subtraction (multiplication and addition in the newer edition). There would be no functions, domain, range, exponents, or slope for the students to worry about.

Of course, this is an Algebra I text, not Integrated Math I. In the past, I spent several posts discussing a possible integrated course based on the U of Chicago text. While my proposal was needlessly complex, perhaps we couldn't do much worse than naively covering the first half of the Algebra I text and then the first half of the Geometry text (Chapters 1-7), with the second halves saved for Math II. (The only awkward part is that slope would be delayed to Math II, with many might find to be especially late.)

The IMP book is a very progressive (that is, anti-traditionalist) text. Indeed, its units are titled "Patterns," "The Game of Pig," "The Overland Trail," "The Pit and the Pendulum," and "Shadows." So it's difficult to imagine what my current Math I course would look like if we were using the IMP text.

Let's return to today's meeting. The other teacher reveals that she hasn't given the group test yet -- this week she'll give an extended version of the test with an extra question on exponents.

She also gives me some pointers about keeping students engaged in groups. She regularly has her groups do work on one of two Promethean boards, or at least on smaller whiteboards (VNPS). They are never simply answering dry questions from the text, since, as I found out earlier, there is less chance of students being distracted or doing no work.

11:45 -- Fourth period arrives. This is a Math I class.

Before today's meetings, I'd already planned to continue group work on exponents. But I want to incorporate some of what the TOSA and my fellow Math I teacher has shown me. I still need to learn more about how to use my own Promethean board, but I can implement VNPS right away -- indeed, I'd already used VNPS for quiz reviews before learning more about CPM group roles, and so it's rather straightforward to get right back to using them now.

And so far, it seems to be successful -- there's much more engagement during VNPS than when the students are answering book problems. This is something I should think about for future groupwork.

1:15 -- Fourth period leaves for lunch.

2:05 -- Sixth period arrives. This is a Math I class.

Due to the block schedule, sixth period is always a day behind fourth period on Wednesdays. So this means that sixth period got yesterday's period 4 lesson -- no VNPS for them until tomorrow. And so I had to deal with the frustration of students not working during group time -- indeed, I had to assign one group a short detention because they kept saying "I don't know" instead of discussing the problem (just as I had to do with fourth period yesterday).

Given a choice between using consequences like detention to keep the kids on task and using structures (like VNPS and my Math I colleagues' other suggestions), the latter is almost always preferable. So I'm expecting a much smoother ride in sixth period tomorrow.

3:30 -- Sixth period leaves, thus ending my teaching day.

At this point, one girl from my first period arrives. She's done well in her previous classes up to Math II, but she's having a little trouble in Math III. It doesn't help that I had to leave an entire lesson on completing the square/vertex form for the sub due to the semi-expected meeting.

And so I tutor her on completing the square and a few previous lessons. I believe that she is now starting to understand the material. The trick is to make sure that she, and her classmates, understand the material the first time without need for extra tutoring sessions.

Speaking of Math III, the recently acquired text for Intermediate Algebra (that is, Algebra II, which is nearly the same as Math III) starts out with lots of Algebra I review. Completing the square and vertex form don't appear until Chapter 8 of the text (along with x = -b/2a, the formula I mentioned in the song from yesterday).

4:25 -- The tutoring session ends, and so I finally get to go home.

I'll make my final decision regarding Math I Chapter 2 next week, after meetings on Monday. That's right -- Monday is the student free day. It's still the Vortex or DEVOLSON, since teachers won't get a day off until Veteran's Day. My next post will be on Tuesday, which will be Day 38.

Tuesday, September 27, 2022

Lesson 1.3.1: Simplifying Exponential Expressions, Continued (Day 34)

MISTAKE, RIGHT WAY

First Verse:
Mistake? h cubed times h fourth.
Mistake? h to the twelfth.
Right way! Add little numbers.
Right way! h to the seventh.

Chorus:
Laws of exponents for you.
Laws of exponents for me.
Laws of exponents stay true.
Laws of exponents, be free!

Second Verse:
Mistake? x squared to the fourth.
Mistake? x to the sixth.
Right way! Times little numbers.
Right way! x to the eighth.
(to Chorus)

This is the sixth song that I'm writing in 10EDL, our main scale for September and October. The next few 10EDL songs will be written in ABAB format, with verses and a chorus.

After the group test last week, we're back on track with the exponents lesson in Math I. The lyrics come directly from the CPM text, an exercise where the students must identify common errors and fix them. Thus the song mentions both the "mistake" and the "right way."

Here's a link to the Mocha code for the song:


10 N=8
20 FOR V=1 TO 2
30 FOR X=1 TO 39
40 READ A,T
50 SOUND 261-N*A,T
60 NEXT X
70 RESTORE
80 NEXT V
90 END
100 DATA 10,8,6,8,6,4,9,4,6,4,8,4
110 DATA 10,8,6,8,6,4,9,4,6,4,8,4
120 DATA 10,8,6,8,6,4,9,4,6,4,8,4
130 DATA 10,8,6,8,6,4,9,4,6,4,8,4
140 DATA 5,4,9,4,6,4,6,2,6,2,7,8,9,8
150 DATA 5,4,9,4,6,4,6,2,6,2,7,8,9,8
160 DATA 5,4,9,4,6,4,6,2,6,2,7,8,9,8
170 DATA 5,4,9,4,6,4,6,2,6,2,7,8,9,8

Don't forget to click Sound before you RUN the program.

This tune contains perhaps our most complex chorus so far, with seven notes that must be repeated, hence it's the most tedious to type. The notes in the chorus are high C-D-A-A-A-F#-D, with the first five notes corresponding to the syllables in "Law of exponents."

The guitar riffs I play with this song are C-Am in the verses and C-D(7) in the chorus. At least one other chording of this song is possible -- perhaps something like Am-Dm in the verse and Am-D7 in the chorus, but then we'd lose the tonic chord C.

Meanwhile, in Math III we return to graphing quadratic formulas. This means that I perform "Quadratic Weasel," with the new bridge on parabola properties. One problem is that my song focuses on the standard form of the quadratic (like the Quadratic Formula itself), while the lesson itself deals more with the vertex form of the quadratic. This reflects a transition that started with the advent of Common Core -- back when I took Algebra I and II as a young student, the vertex form wasn't taught, and we needed x = -b/(2a) to find the x-coordinate of the vertex. But now the focus is more on transformations, including translations, from the parent graph, and so the vertex form y = a(x - h)^2 + k is seen more.

So perhaps I need a new song that focuses on the vertex form of a parabola. But most of my current songs are on Math I material, not Math III, and so I likely won't get around to it.

Wednesday, September 21, 2022

Chapter 1 Review, Continued (Days 30-33)

TRANSFORMATION

First Verse:
Parent graph? That's square root.
Orientation? Up to you.
Shift left four, then up one.
Stretch factor? Stretch by two.

Chorus:
Reflection.
Translation.
Compression.
Transformation.

Second Verse:
Parent graph? Hyperbola.
Orientation? Let's get down.
Shift right three, then down two.
Stretch factor? None to be found.
(to Chorus)

This is the fifth song that I'm writing in 10EDL, our main scale for September and October. The next few 10EDL songs will be written in ABAB format, with verses and a chorus.

Progress in the Math I text has stalled, as we're stuck in the section on functions, domain, and range until we get past the group test. So I decided to write a song for Math III instead. This chapter is on parent graphs and transformations. The verses refer to f (x) = 2sqrt(x + 4) + 1 and g(x) = -1/(x - 3) - 2.

Here's a link to the Mocha code for the song:


10 N=8
20 FOR V=1 TO 2
30 FOR X=1 TO 36
40 READ A,T
50 SOUND 261-N*A,T
60 NEXT X
70 RESTORE
80 NEXT V
90 END
100 DATA 7,4,5,4,7,8,10,6,8,6,5,4
110 DATA 7,4,5,4,7,8,10,6,8,6,5,4
120 DATA 7,4,5,4,7,8,10,6,8,6,5,4
130 DATA 7,4,5,4,7,8,10,6,8,6,5,4
140 DATA 8,16,7,8,6,8
150 DATA 8,16,7,8,6,8
160 DATA 8,16,7,8,6,8
170 DATA 8,16,7,8,6,8

Don't forget to click Sound before you RUN the program.

The riff for this song is simple, D(7)-C, switching to C-D(7) for the chorus. Once again, the chorus consists of a few long notes so that there's less code to type. Even though the first two notes generated by the TI are F#-C, in practice I sing it today as F#-A, since F#-C sounds too much like the chorus from yesterday's song -- again an artefact of 10EDL having so few notes.

Indeed, the TI just happened to avoid Degree 9 (the note D), The no-nines version of 10EDL sounds a little like 8EDL with an extra note at Degree 10 -- and indeed, we might wish to treat it has a true 8EDL song with Degree 8 (E) as the tonic, perhaps with an E (or Em) chord in the chorus. It's possible to make the song end on whichever tonic we prefer (E or C) by singing the last syllable of the title ("Transformation") on the chosen tonic.

In fact, I suspect that if I'd chosen 12EDL instead of 10EDL, the TI might have produced a no-elevens 12EDL song, which might sound better than this no-nines 10EDL song. Oh well.

The lyrics refer to the functions we're using in a matching activity. Even though the verses refer to a vertical stretch (a > 1), the word "compression" (0 < a < 1) sounds better in the song.

Tuesday, September 20, 2022

Chapter 1 Review, Continued (Day 29)

DIXI-ROYD

First Verse:
Domain, what's that? It's what you start with.
Input function? It's what you put in.
x-values, plug it in for x.
Independent, choose 'em all again.

Chorus:
DIXI! ROYD! (repeat four times)

Second Verse:
Range, what's that? It's what you end with.
Output function? It's what you get out.
y-values, f of x is y.
Dependent, that's what it's about.
(to Chorus)

The mnemonics in this song come from Sarah Carter (DIXI-ROYD).

This is the fourth song that I'm writing in 10EDL, our main scale for September and October. The next few 10EDL songs will be written in ABAB format, with verses and a chorus.

At this point, I've found a new strategy for adding guitar chords to my 10EDL songs. First, I use the TI calculator to create two measures/bars in the 10EDL scale. Then I add a sequence of chords (often referred to as a riff) that sounds good with the two bars. Then this riff is repeated throughout the verse.

Then I have the TI create another two-bar sequence for the chorus. As it turns out, the TI chose a pair of whole notes for the chorus -- Degree 7 (F#) and Degree 5 (high C). So far, I've been using a D (or D7) chord for the F# note, followed by our tonic C major chord. Notice that in the standard scale, F#-C is a dissonant tritone, though in true 10EDL this becomes a 7/5 interval -- a better-sounding tritone. The riff for the verses, meanwhile, is Am-D7.

Here's a link to the Mocha code for the song:


10 N=8
20 FOR V=1 TO 2
30 FOR X=1 TO 44
40 READ A,T
50 SOUND 261-N*A,T
60 NEXT X
70 RESTORE
80 NEXT V
90 END
100 DATA 6,6,7,6,10,2,10,2,7,2,8,2,8,4,9,4,10,4
110 DATA 6,6,7,6,10,2,10,2,7,2,8,2,8,4,9,4,10,4
120 DATA 6,6,7,6,10,2,10,2,7,2,8,2,8,4,9,4,10,4
130 DATA 6,6,7,6,10,2,10,2,7,2,8,2,8,4,9,4,10,4
140 DATA 7,16,5,16
150 DATA 7,16,5,16
160 DATA 7,16,5,16
170 DATA 7,16,5,16

Don't forget to click Sound before you RUN the program.

The tricky part was coding these riffs in Mocha. While a single riff can easily be coded using a simple FOR loop with RESTORE, having different riffs for the verse and chorus is trickier -- the only feasible way is to type in each line four times (100-130 for the verse, 140-170 chorus). While the verse contains nine notes in its melody, the chorus contains just two whole notes, so it's not as bad to type in. Later on, more complex refrains will require us to type more (or find some clever way to enter the code).

Saturday, September 17, 2022

Monthly "Day in the Life" Post for September (Day 28)

This is my monthly "Day in the Life" post for September, since today is the seventeenth. It is a Saturday -- one of two months this school year when the 17th falls on the weekend. (Here Day 28 refers to this upcoming Monday, since my next post after today isn't until Tuesday.) And there are plenty of things to discuss and reflect upon today, despite it being the weekend.

First of all, we've definitely entered DEVOLSON -- the "Dark Evil Vortex of Late September, October, and November" as defined at the following link:

https://www.weareteachers.com/5-ways-of-coping-with-devolson-the-dark-evil-vortex-of-late-september-october-and-november/

(Kelly Treleaven is the creator of the term.) At my school, there are no more days off for teachers until Veteran's Day, although there will be a student-free day for professional development in October. So while outside of school, this time of year is associated with the NFL, playoff baseball, pumpkin spice, and Halloween, for teachers this time of year is a vortex.

Meanwhile, I'm thinking ahead to progress reports. Last year on the other blog, I mentioned how this is one of a growing number of districts where the semester is divided into thirds, with progress reports given one-third and two-thirds of the way through the semester. Theoretically, the progress report days should be on Days 30 and 60, but in reality they are slightly later in the 30's and 60's. (Last year, my magnet only had progress reports in the 60's, then labeled it by the misnomer "end of the first quarter.")

I still like to use the name "hexter" for one-third of a semester -- the name can refer to either "six weeks" or "one-sixth of a school year." (The name was originally coined to mean "half of a trimester.")

In theory, there are supposed to be "curriculum team" (CT) teacher meetings for both of my classes, Integrated Math I and III. But  I've only ever attended CT meetings for Math I, my primary CT. There was supposed to be a meeting for the secondary CT ("CT2") last month, but it was cancelled due to early year confusion with the calendar. Then yesterday's CT2 meeting was also cancelled because Math I decided to have a second CT1 meeting instead.

I guess I don't mind. I have three Math I classes and only two Math III classes, so Math I is my priority (and I've even made it my focus here on the blog, with the "Math I" label). And there's not really that much to discuss in the Math III meetings anyway. There is a strict pacing guide to follow, and the teachers regular share quizzes and tests with each other. I will give the Chapter 2 Quiz (transformations of functions) in class on Monday.

We'll come very close to finishing Chapter 2 in time for the first hexter progress report. Like all chapter tests, the Chapter 2 Test will be divided into two parts -- paper and DeltaMath. The paper test will be given before the grading window closes, but not the DeltaMath part. The only problem here is in deciding when to do the Chapter 2 notebook check -- I'd like to collect them after the test is completed so the students can use them to study, but I want to include them in the progress report. The solution is to collect them after part 1 of the test -- the assumption is that any studying for the DeltaMath part of the test is done by doing DeltaMath problems, not referring to the notebook.

So that takes us to Math I. Our meetings are led by a "teacher on special assignment" -- and I should be very grateful for his presence. After all, it was the creation of this TOSA position (and this former classroom teacher's promotion to that position) that ultimately opened up a teaching spot for me -- had it not been for the TOSA, I'd be stuck working at a charter middle school right now. He tells us that he's working with us because Math I (like the old traditional Algebra I before it) is a gateway class -- so many students fail it, yet success here is critical to the student's path in math and other STEM classes.

While Math I has a pacing guide, it isn't followed as strictly as the Math III pacing guide. Still, I wanted to follow the pacing guide for several reasons. First, the Math I guide places the Chapter 1 Test a week before the Math III Chapter 2 Test. This will allow me to collect the Math I notebooks a week before the Math III notebooks and get them all graded before progress reports, as opposed to having to grade I and III on the same weekend just days before grades are due. (Last year with my tiny classes, I found it convenient to give tests in all classes at the same time -- but now with full-sized classes, it's important to avoid having to grade 170-180 students' work in the same weekend.)

Also, the Math I guide has quizzes nearly every week (unless there's a test or another assessment during that week). I want to have a strict "no food" policy in class, but this leads to arguments with students who find it suffocating to have a "no food" policy. So my compromise is to hand out Red Vines candy on days after quizzes to high-scoring students -- on Mondays, since the quizzes are on Fridays -- and then allow food in the classroom on those days. Still, one student this week arrived to my fourth period class with a Chick-Fil-A bag (on a day other than Monday) and had just taken it for granted that I'd allow him to eat in class -- to such students, 90+% of all teachers let them eat as much as they want in class and a small minority of teachers ever forbid food.in class, even though almost no K-12 teacher in the 20th century ever allowed students to eat in class, except at parties once or twice a year.

(An aside -- a few years ago, I subbed at a continuation school, hosted at an adult school. Each room had a sign, posted by the adult school, that food was forbidden except for one party per semester. Yet students were eating in that classroom all day -- and if I recall, one guy told me I was the first teacher who ever tried to tell him that he couldn't eat in class!)

And third, this week I began Lesson 1.3.x, on the laws of exponents. I was looking forward to singing one of my old songs -- the UCLA fight song parody, specifically the verse on exponent laws (as opposed to the verse on unit rates), and I was planning to sing it on Wednesday and Thursday.

OK, so far I've mentioned three reasons behind my pacing -- to hurry up and give the Chapter 1 Test just to avoid being swamped with work to grade before progress reports, to hurry up and give quizzes every week just to avoid arguing with kids over food in the classroom, and to hurry up and reach my favorite songs. What's missing here is what should be the most important factor regarding pacing -- how much the students are learning.

There are three ways to pace any math class. The fast way is to make sure that all chapters are covered by the end of the year, even if some lessons are covered briefly or not at all. The slow way is to stop and avoid proceeding unless the students show understanding -- after all, it's better for the students to learn a little something than a whole lot of nothing. There's also a middle path -- cover the lessons and repeat the basic lessons if the students aren't getting them, but move forward over certain lessons as well. It's a compromise between breadth and depth of understanding.

Originally, I thought that the pacing guide represented the slow path -- at least at first. After all, there are eleven chapters in the CPM text, but only three are covered in the first semester. So we're slowing down to make sure that the students understand Chapters 1-3 of the text. The tests for Chapters 1, 2, and 3 are given just before the end of the respective hexters, which is convenient for progress reports.

But so far, the pacing guide appears to represent the fast path. Of all the Math I teachers, one or two others, like me, began Lesson 1.3.x this week. Two teachers, including both of my next door neighbors, reached Lesson 1.2.x, on functions. And one or two teachers are still in Lesson 1.1.x on equations -- more time was spent there on reviewing negative numbers and fractions.

The official pacing guide had a "Mock Chapter 1 Test," ahead of the real Chapter 1 assessment. During the first meeting on Monday, the TOSA suggested making it into a group test instead -- and counting it as an actual part of the grade. The test follows the middle path, covering up to 1.2.x. Then yesterday, I was told that I still can't give the test yet, because I hadn't explicitly taught group norms or roles yet (as given in the CPM text -- the roles of "task manager," "resource manager," "facilitator," and "recorder").

And so yesterday, I divided the Math I classes into groups and had them choose their roles. For the most part I grouped them based on their current seats, but I had to change many seats around so that each group contained three or four students (with frequently absent students placed into groups of four, so that there will be at least three kids in each group who regularly attend class). Then I tried to have them work on a question from Lesson 1.2.1 in groups -- only to have many of them have trouble accessing the CPM online text. (Like many science classes, now our math class has only an online text.)

Oh well, at least the students have their groups and roles. Their test won't be on Monday, so they will have at least one more day to show me they can work together in groups before the team test. But this now means that the pacing guide is now in shatters.

It's clear that the students need more time to proceed through the chapter. The students are struggling in all three lessons -- equations, functions, and for those teachers who reached it, exponents. (That they'd have trouble with equations and exponents is understandable, but I'm surprised that they're having trouble identifying what is a function, or what is its domain or range -- tasks that require no addition, subtraction, multiplication, or division.)

But now I'm wondering what I should do about pacing -- and specifically, the three private reasons I stated above for wanting to follow the originally guide. First, I have no idea when Chapter 1 will be completed -- at this rate, it might even be after progress reports. But I don't necessarily have to wait for the end of the chapter to collect the notebooks -- for example, if the team test is next week, then I can just collect the notebooks after that test, which will be still be before I collect Math III notebooks or do progress reports.

Second, as far as weekly quizzes and food days are concerned, recall that before the year started, I considered having Hero Quizzes that covered review material, or even basic material. Once I saw those weekly quizzes on the pacing guide, I dropped my Hero Quiz idea. But now that we're going more slowly than the guide, there's room for Hero Quizzes once again. Thus whenever we reach the end of the week and the students aren't ready for the next full quiz, then declare it to be a Hero Quiz. Then all students can earn a Red Vine and have a food day the following Monday.

That leaves us with the songs. I was fortunate to find Alex Milanese's song on functions, but now I'm wondering what to sing about next week. Indeed, the Milanese song includes almost everything the kids are learning about functions, and now I'm stumped as to sing. While of course I can repeat songs as needed, I still like to have one new song per block period.

Moving forward with the pacing, there's one key time by which I'm hoping we're back on pace -- the week of the district benchmarks. The first assessment is just before the end of the second hexter, so there's likely to be Chapter 2 material on that test. And Chapter 2 is on linear functions and slope -- perhaps one of the most difficult chapters in Math I. It's not the sort of chapter I'd want to rush the week before the district assessment just because we spent so much extra time in Chapter 1. I'm hoping that our TOSA will tell us what to do about the district benchmarks in our next CT meeting.

In this post, I want to discuss my class demographics, in hopes that by now the classes are stable. There were still some class changes this week due to students switching schools -- or did they switch? Two guys dropped my class to transfer to other districts at the start of this week -- and by the end of the week, both had returned to my class! (This happened once last year as well -- a student moved from our magnet to the main high school, only to return a week later.) And I might be on the other side of this phenomenon as well -- a new guy appeared in my fifth period, only to leave (for his old school?) a few days later.

I've decided that the focus period for the blog will be fourth period Math I, as it's my largest class. My roster has 37 students -- which is tough, as there are only 35 desks in my classroom. But one student is a special ed student who attended my class only once -- on Day 2 -- yet remains on my roster.

As far as racial demographics are concerned, the vast majority of the students are Hispanic/Latino, resembling the surrounding city. According to Aeries, one student in this class is white, and there appear to be two mixed-race students (one white/Hispanic, the other listed as "other Asian/Hispanic"). I do have a few full Asian students and black students in other classes, but not in fourth period.

(Speaking of ethnicity, one of my neighbor teachers, in addition to Math I, teaches Ethnostats -- the class I taught last year. She asked me for some materials to help her teach this class. This week, I told her about last year's https://www.lathisms.org/ website for Hispanic Heritage Month. She also informs me that with no district benchmarks in this class, she might go even more slowly than in Math I.)

Since there are four group roles according to CPM, yesterday I divided the 36 regularly-attending kids into nine groups of four. The one tricky thing involves special ed students -- three students are often pulled out by a paraprofessional, so it's convenient to group them together (so, in case they are pulled out on group test day, they can work on the test during the pull-out). But then this group needs a fourth member, and so there's one guy who will be left out of group if the others are pulled out. There's also the 36th student who must sit at the side table rather than a desk (and this isn't the same as the guy who's grouped with the special ed students, so he must move during group assignments).

Of course, I'll make further changes to both the seating chart and the groups if something changes (such as one student leaving, the "ghost" student appearing for the first time since Day 2, and so on).

There's one thing that I'm still worried about -- am I making good connections with my students? Last year, I wrote that I often had trouble connecting with my seniors, many of whom had established prior relationships with the more experienced teachers. This year, now that I have freshmen, I want to try to connect with my students more. If I remain at this school, these are the kids who might take future classes with me as upperclassmen. The question is, will I be the sort of teacher whose classes they'll want to take and enjoy, or will they seek to avoid my classes in the future?

The one thing that I definitely want to reduce and/or eliminate is arguing. If I'm the teacher who argues a lot, then no one will want to take my classes in the future.

But unfortunately, arguments have crept up lately, especially in my two toughest classes -- second period Math I and fifth period Math III. Second period's argument was over the aforementioned food rules, while the fifth period debate was over two students going to the restroom at the same time -- I only let one out at a time, but neither realized the other was out because my restroom log was full. (As for fourth period, the behavior is class is right in the middle -- they're better-behaved than second and fifth periods, but not as good as first or sixth periods.)

What can I do to reduce or eliminate my arguments in these two classes? Well, in the case of fifth period, I can use a weekly larger restroom log. In previous posts, I mentioned that I allow my students nine passes per semester, or about one every two weeks. So if all my students use up all of their passes, then we can expect about half of my kids to use a pass per week -- about 88 students. Thus I will print out a weekly restroom log with room for 88 students. While this might not eliminate all arguments, it should at least reduce arguments over not knowing whether there's one student out or two.

The food argument is a bit trickier -- but more important, as these involved the freshman classes. Here I want to separate with "forbidding food" with "arguing over food." Once again, most of my teachers I had back as a young student in the 20th century forbade food -- but they were able to ban food without arguing over it. This is what I must do -- find a way to convince a student to put their food away and not bring it to the next class (unless it's Monday).

On the other hand, if a freshman decides that the mere banning of food (as opposed to arguing over it) is suffocating, and chooses to avoid my classes in the future in favor of teachers who allow food, then there's not much I can do short of allowing food in my class too.

Of course, many students enjoy the songs I sing in class -- but music isn't enough to sustain a class if I'm spending the rest of the time arguing. Reducing and eliminating arguments are the only way to make the class more enjoyable, survive the vortex of DEVOLSON, and set up the students for success this year and beyond. I knew going into this year (a typical year) that it would be more difficult than last year (an unusual year at the tiny magnet school), so I must rise to deal with these challenges.

Wednesday, September 14, 2022

Chapter 1 Mock Exam (Days 25-27)

https://sites.google.com/site/thefunctionsong/lyrics

The Function Song - by Alex Milanese

F-U-N-C-T-I-O-N spells function, get it?
Everybody’s wondering what we’re talking about,
well you put something in and you get something out, it’s a function, oh yeah.
Well you can’t spell “function” without “fun”, f(x) = 2x+1.
That’s a function, I mean, that’s just one example—there’s a bunch of them.
Hey girl, what’s your name?
My name is D, I live in the domain.
What you looking at me like that for? Do you think it’s strange
that I only go to one place in the range?
Singing inputs and outputs, I never get a rest.
Can someone hook me up with a vertical line test?
‘Cause I got to know if you’re a function baby.
Show me what you’re working with.
If a vertical line passes through more than one of your points,
I know I’ve got to get up on out of this joint, ‘cause you’re not a function baby.
I hate to be the one to tell you, but if you can’t pass a vertical line test then
you’re definitely not a function. Got it? Alright good, here we go.
Now before you make any sort of computation,
make sure you’ve got these things written in function notation.
And that f(x) part does not mean multiply. It means the value of f
at that given value x. So just keep that part straight, ok?
Now some of you are thinking that this is hysterical,
but I always run around with an independent variable.
And it can be whatever you want it to be because it’s independent.
That’s kind of the point of being independent, right? Alright.
Well if you still need help I’ll be your flight attendant.
The variable you get out is called dependent, so it’s a dependent variable.
Because it depends on the input. I don’t know who came up with that,
But that’s pure genius, so will use it. Alright.
Evaluate a function, I’m as serious as cancer.
Just put in the input and out comes the answer.
That’s straight forward; I understand. That’s why I like math so much.
I love math, so I write songs about it. Alright, anyway, what’s next?
So you want to graph it? Some of you don’t care.
How we going to do it? Just make an ordered pair.
Put in that input, get an output, turn it into an ordered pair,
and plot those points, and you’re good to go.
Seems pretty straight forward right? I know, I agree.
Eventually there’s more that you got to know,
but as for me and my crew we like to take it slow,
so for right now you are good to go.
Don’t forget to holla atcha boy – word! Ok…


Yes, I will still play songs that I find on YouTube from time to time. This time, I decided that I wanted another song about functions, so this song by teacher Alex Milanese (from a decade ago) fits the bill.

Milanese plays this song using the B minor and E minor chords -- but as you can see, he's clearly using a capo on the guitar, so he's really playing Em and Am. Since I don't have a capo on my guitar, I play it today using the Em and Am chords. This also fits my vocal range better, as the melody now emphasizes the notes E and G that are closer to the center of my range.

Tuesday, September 13, 2022

Chapter 1 Review (Day 24)

NOTATION SCIENTIFIC

Verse 1:
When you're multiplying,
In notation scientific,
Little numbers get added. Easy!
Multiply bigs. Easy! (to Pre-Chorus)

Verse 2:
When you are dividing,
In notation scientific,
Little numbers subtracted. Easy!
Then divide bigs. Easy! (to Pre-Chorus)

Pre-Chorus:
Hey! In decimal,
Anywhere is point.
In scientific, just one,
Digit, then place the point.

Chorus:
Point left, little plus!
Point right, little minus!
Notation scientific.

This is one of the songs that I wrote over the summer.

Originally, this song was written in 18EDL, in ABCABCD format (with a bridge). Once again, this was back when I thought I'd be teaching in a Math 8 class now, using a textbook in which scientific notation is taught near the end of the year (when I'd be using more complex scales and formats).

So I kept most of the lyrics, except for the bridge and third verse (since these verses are about adding numbers in scientific notation, which has been dropped). Also, I'd written the lyrics down but not the 18EDL melody (which was posted on the blog, but my Internet stopped working last night, so the entire tune was inaccessible). So I started a new 10EDL tune from scratch, tweaking it to fit the lyrics. The last line ("That's terrific!") was dropped as it didn't fit the new melody.

Here is a link to the Mocha code for the song:


10 N=8
20 FOR V=1 TO 2
30 FOR X=1 TO 72
40 READ A,T
50 SOUND 261-N*A,T
60 NEXT X
70 RESTORE
80 NEXT V
90 END
100 DATA 7,2,6,2,7,4,8,4,8,2
110 DATA 6,2,8,6,10,6,9,2,6,2
120 DATA 5,8,10,8,6,8,10,8,5,6
130 DATA 6,6,5,2,8,2,10,8,7,8
140 DATA 9,6,7,6,8,4,9,4,10,4
150 DATA 10,4,5,4,9,6,8,6,8,4
160 DATA 8,6,8,6,5,4,5,6,5,6
170 DATA 7,4,9,16,5,8,9,8,9,2
180 DATA 10,2,9,6,6,6,7,4,10,4
190 DATA 5,8,9,8,8,2,7,2,8,2
200 DATA 9,2,9,8,7,4,9,2,7,2
210 DATA 7,2,8,2,7,6,6,6,9,8
220 DATA 7,4,9,2,7,2,10,6,7,6
230 DATA 6,2,8,2,7,16,8,16,5,16
240 DATA 9,4,10,12

Don't forget to click Sound before you RUN the program.

Adding the guitar chords for this 10EDL song is somewhat tricky. The main verse sounds good with D7 and C, perhaps with D7-G7-C as the last three chords. Then when we get to the pre-chorus, we might start with some different chords, E7-Am, before ending with D7-G7 as we head for the chorus. The chords in the chorus resemble that in the verse.

The last few notes are written as whole notes but they go with the last line of the chorus -- no F#, tation E, scien high C, tif D, ic low C. (The original reason for "notation scientific" rather than "scientific notation" was that the former way fit the lyrics and the tune, not to mention the rhyme with terrific that has since been dropped.)

Wednesday, September 7, 2022

Domain and Range Review (Days 20-23)

DOMAIN AND RANGE

First Verse: (Draw a function like (x) = x + 4 over the given domain.)
DIXI, what's our domain?
1 less than x less than 5.
And ROYD, what is our range now?
5 less than y less than 9.

Second Verse: (Draw a function like f (x) = 6x - x^2 over the given domain.)
DIXI, what's our domain?
1 less than x less than 5.
And ROYD, what is our range now?
5 less than y less than equal to 9.

Third Verse: (Draw a function like f (x) = 2^x.)
DIXI, what's our domain?
All real numbers it contains.
And ROYD, what is our range now?
y greater than 0, for a change.

The mnemonics in this song come from Sarah Carter (DIXI-ROYD).

This is the second song that I'm writing in 10EDL, our main scale for September and October. The first few 10EDL songs will be written in AAA format, with verses only.

Here's a link to the Mocha code for the song:


10 N=8
20 FOR V=1 TO 3
30 FOR X=1 TO 25
40 READ A,T
50 SOUND 261-N*A,T
60 NEXT X
70 RESTORE
80 NEXT V
90 END
100 DATA 7,2,7,2,9,6,8,6,10,8
110 DATA 8,8,9,4,9,2,8,2,9,4
120 DATA 8,4,7,12,6,4,8,2,6,2
130 DATA 5,4,6,4,9,4,5,8,5,8
140 DATA 5,4,9,4,5,4,6,4,5,16

As usual, click on Sound before you RUN the program.

For this song, I simply sing through each verse as the music plays, with no repeated lines. The chords are a bit tricky to play, though, now that Degree 7 (F#) appears more prominently throughout the song, including the first two notes. The first three notes are F#-F#-D so our D7 chord from yesterday's song fits somewhat. But I might find better ways to chord our 10EDL songs (or just cheat and switch to G to make it a true pentatonic melody).

Also, another thing about having a simple scale like 10EDL is the limited selection -- the last five notes already sound like the first four notes of yesterday's song! As I perform it today, I instead insert Degree 4 (a high E) instead of the Degree 9 (a low D) listed in the Mocha code above -- and this is despite my saying earlier that I wouldn't sing that high.

So once again, this is a work in progress, and I might find ways to improve this song.

Tuesday, September 6, 2022

Lesson 1.3.1: Simplifying Exponential Expressions (Day 19)

EXPONENTIAL EXPRESSION

First Verse:
When John's saying 1, Mia's saying 1.
When John's saying 1 times 3, Mia's saying 3 to first.
When John's saying 1 times 3 times 3, Mia's saying 3 to second.
Rule: Multiply powers, add little numbers.

Second Verse:
When John's saying 5 to fourth, Lauren's saying 5 to fourth.
When John's saying 5 to fourth o'er five, Lauren's saying 5 to third.
When John's saying 5 to fourth o'er five o'er five, Lauren's saying 5 to second.
Rule: Divide powers, subtract little numbers.

Third Verse:
When John's saying 2 squared, Lachlan's saying 2 to second.
When John's saying 2 squared squared, Lachlan's saying 2 to fourth.
When John's saying 2 squared cubed, Lachlan's saying 2 to sixth.
Rule: Power of powers, multiply little numbers.

Fourth Verse:
When John's saying 2 squared times 3, Kelly's saying 2 squared times 3.
When John's saying 2 squared times 3 all squared, Kelly's saying 2 fourth 3 squared.
When John's saying 2 squared times 3 all cubed, Kelly's saying 2 sixth 3 cubed.
Rule: Power of product, distribute little numbers.

The names in this song come from the Desmos activity on which this is based. In particular, John and Lachlan are the names of the activity creators. (I don't know where the girls' names come from.)

This is the first song that I'm writing in 10EDL, our main scale for September and October. The first few 10EDL songs will be written in AAAA format, with verses only.

Here's a link to the Mocha code for the song:


10 N=8
20 FOR V=1 TO 4
30 FOR X=1 TO 25
40 READ A,T
50 SOUND 261-N*A,T
60 NEXT X
70 RESTORE
80 NEXT V
90 END
100 DATA 5,4,9,4,5,4,6,4,9,2
110 DATA 8,2,9,8,9,2,7,2,5,2
120 DATA 9,2,7,6,9,6,10,8,9,4
130 DATA 9,4,9,16,10,12,8,2,8,2
140 DATA 8,4,6,4,8,4,8,4,10,16

Don't forget to click Sound before you RUN the program.

This is also the first song for which I play guitar chords. The 10EDL scale goes C-D-E-F#-A-C, but I was wondering whether we should change F# to G to make a traditional pentatonic scale. Well, for this song I retain the F# note. The F#'s are surrounding by D's, suggesting the D(7) chord in addition to G(7) and C chords. For the eight bars, the chord pattern is C-G7-D7-G7-G7-C-G7-C.

The tricky part is making the lyrics fit the tune here. I consider making each of the first three lines fit the first four bars, and then the last line of each stanza fitting the last four bars. The problem is that there are both short lines ("When John's saying 1, Mia's saying 1") and long lines (such as all the "2 squared times 3" lines), and no single line of music fits both easily.

Thus there is room for improvement in this song. I might consider going back and changing some of the lines in the verses in order to make both mathematical and musical sense.