Lesson 6.3.3: Solving Systems by Elimination, Continued (Days 131-132)

Today is the seventeenth, and so this is my monthly "A Day in the Life" post for March. It is Friday, often a day for department or other meetings. But there is no such meeting today.

Of course, today is also St. Patrick's Day. I used to have a St. Paddy's Day pencil giveaway, but no such pencils are sold in stores these days. Thus I no longer have one.

8:30 -- First period arrives. This is the first of two Math III classes.

This is the second day of SBAC Prep Boot Camp. According to the pacing guide, today's task is to answer fifteen practice test questions on Illuminate. This platform is mentioned in the song I perform yesterday -- it's always used for district Benchmarks, but non-Benchmark tests can be taken there as well. Indeed, last year I used it for some Calculus and Trig tests, and this year another teacher has set up a practice SBAC there on Illuminate.

The pacing guide also states that the Chapter 8 DeltaMath test corrections are assigned as homework, which is atypical for me. But the other teachers want to make sure that the corrections don't take an extra day away from the SBAC Prep Boot Camp.

The Exit Pass question comes from a SBAC question. Eva wants to buy a bike for $240, but she only has $42 in the bank. If Eva is paid $12 per hour as a babysitter, then how many hours must she work in order to afford the bike? The correct answer is 16.5 hours -- but for the Exit Pass, I have the students round off the answer to 17 hours, because today's date is the seventeenth. After the Exit Pass, the students turn in their notebooks, since Chapter 8 is essentially over.

9:25 -- First period leaves for nutrition.

9:40 -- Second period arrives. This is the first of three Math I classes.

Normally, I give some sort of quiz or assessment every Friday in Math I. But this week, I'm leaning towards counting the poster project as this week's quiz. So instead I try to give a lesson -- unfortunately, there are several problems with today's lesson.

I'm labeling today's post as Lesson 6.3.3, but I'm torn between an extra of 6.3.2 and starting 6.3.3. Here the CPM lesson numbers refers to the three levels of the elimination method as defined by DeltaMath -- Level 1 is for pure addition or subtraction, Level 2 is where one of the equations must be scaled, and Level 3 means that both equations are scaled.

Today is Sixday on the Eleven Calendar:

Resolution #6: We implement all parts of our projects.

Yesterday was the first day of Lesson 6.3.2. As I explained in my last post, I decided to go over the DeltaMath questions on guided elimination last time and save Desmos for today (especially since the Desmos questions lead into Level 3, or Lesson 6.3.3, as well).

Unfortunately, the Desmos lesson doesn't go well. The first system in this activity is written in an unexpected format, C = By + Ax instead of Ax + By = C. It throws the students -- and me -- off, and it takes some time to figure out where we go wrong. And the other systems include fractions -- one as a coefficient, another in the solution. Indeed, the very first Level 3 system has a fractional solution.

I must admit that today's lesson is influenced by several rigidities -- patterns in how I teach the lessons that I uphold just because they're the way I've always done things. One of those rigidities -- the weekly Friday quiz -- is not in force today. But some others are -- including some influenced by the existence of the weekly Friday quiz.

First is the interactive notebook. In order to avoid having too many notebook pages (which would force the kids to buy three or more notebooks to span the year), I make sure that I assign no more than one page on most days (excluding the Warm-Up page) -- and no pages on Fridays (when, once again, I'd normally just give a quiz anyway).

As I explained in my last post, on the last block day I passed out a guided note worksheet containing the lyrics to the rap (with blanks for the kids to fill in). That worksheet took up the entire page, so there was no room to write anything else (like, say, a few system solutions worked out), since I didn't want to have them write on more than one page. Then today is Friday -- the day that there's no page. Thus during the entirety of Lesson 6.3.2, the students write only rap lyrics, with no Level 2 examples. And of course, I couldn't have saved the rap for today because I only perform on block days, not Fridays.

Yesterday's DeltaMath lesson was far more successful than today's Desmos, so perhaps I should have done more DeltaMath instead. The problem is that I set the DeltaMath assignment at the beginning of each week, to be due the following Friday (again because there's usually a quiz). And we already solved all the planned DeltaMath questions yesterday. Some students regularly finish the work the first day it's assigned, so they wouldn't appreciate suddenly having extra questions, especially on the day it's due.

There's also a problem with avoiding arguments. A few students are eating in class again. I tell them that I'll write the eaters' names down and decide on a punishment later, in order to avoid arguments.

Then I pass out the application forms for students hoping to take Math II this summer so that they can advance to Math III next fall (to reach Steve-level classes -- junior Pre-Calc and senior Calculus). One girl gets upset at this -- she calls herself and others the "D" word, meaning not very smart.

I try to tell her that she has a point -- I could have handed out the applications more discreetly. But then she assumes that I'm trying to argue with her over the food that she's eating. And so, although I'm trying to avoid arguing, an argument ensues anyway. That's the problem with arguing -- the students are so used to my arguing, they interpret anything I say as an argument. And that's why it's important to avoid arguments from the very beginning.

It's tricky to find a good Exit Pass question whose answer involves today's date. A possibility is:

5x - 3y = -1

-3x + 2y = 4

which is a Level 3 system with solution (10, 17). But since I don't reach Level 3 questions, I can't give this Exit Pass (though it's a much better question than the one that appears on Desmos).

10:35 -- Second period leaves. Third period is my conference period -- which means that it's time for my weekly tutoring session with three Math III students and my "prep period buddy" teacher.

Yesterday was Part II of the Chapter 8 Test -- the paper portion. Nevertheless, there's still something for these students to prepare for -- and I don't mean the SBAC. Next week is the Math III midterm exam.

Unfortunately, my prep buddy must cover a class this period -- a Spanish class. She's a native Spanish speaker, so she can handle this class -- after all, that's why we set up tutoring sessions in the first place, namely to help my English learners  (though she tells me that she -- just like me -- took French in high school, not Spanish). We decide to hold the tutoring session anyway -- I help out the math students while the other teacher watches the Spanish class and translates for us from time to time.

We work on a few questions from the last homework assignment, including the end behavior of polynomials -- a simpler topic, but one that I only briefly touched upon in class. I want to make sure that the students understand this ahead of the midterm.

11:40 -- Fourth period arrives. This is the second of three Math I classes.

Fourth period goes much more smoothly than second. I do finish both Level 2 questions, but we still don't reach the Level 3 question. Then again, I'm not quite sure whether Level 3 elimination will even be on the upcoming Chapter 6 Test.

Entering today, I'm slightly ahead of my neighbor teachers. They've just barely begun Level 1 of elimination, with perhaps an intro to Level 2. On the other hand, I'm behind my prep buddy, who's completing substitution today (after teaching elimination first). She could give the chapter test early next week, though she tells me that she'll spend most of the week on review.

I hear one of my neighbor teachers tell his class that Level 3 might not be on the test. The issue hasn't quite been decided yet -- but of course, if it's not on the test, it's not as crucial that I teach it. Instead, I should make sure that they're comfortable with Level 2.

I also attempt to hand out the summer Math II letters more discreetly than in second period, but still, one guy wonders why he doesn't get a letter. (Some teachers might take the time to put letters in envelopes and just hand out sealed letters to the proper students.)

12:40 -- Fourth period leaves for lunch. There is a "pie your teacher in the face" contest during lunch, but I'm not one of the victims. This probably should have occurred on Pi Day, but it takes place outdoors and thus was delayed due to rainy weather.

1:25 -- Fifth period arrives. This is the second of two Math III classes.

Fifth period works on the SBAC Practice Test on Illuminate. This is after I show them end behavior questions on the Warm-Up.

By the way, after mentioning how I overused Desmos in Math I, I underused Desmos in Math III. The teacher who created the Math III Desmos lessons also wrote the Chapter 8 Tests, so if I'd assigned Desmos, my students would have been better prepared for the test.

2:20 -- Fifth period leaves and sixth period arrives. This is the third of three Math I classes.

We do barely make it to the Level 3 question in this class -- but once again, it might not matter if there's no Level 3 on the test. And there's only one student recommended for summer Math II, so I easily give her the letter without others noticing.

3:20 -- Sixth period leaves, thus completing my day.

Last year, this was the last week before spring break. This year, spring break will be the last full week in March, so there's still one week left in the "Big March." Enjoy the rest of your St. Paddy's Day!

Chapter 6 Intro (Day 113)

Today is the seventeenth of the month, and so this is my monthly "A Day in the Life" post for February. I wish to start with "A Day in the Life" for yesterday, the sixteenth, since what happened yesterday had a huge impact on my lessons for today.

8:00 -- A meeting for Math I was called yesterday by -- who else -- the TOSA for Math I. He arranged for this meeting to last a half-day -- period subs ended up covering our first two classes. (This is the meeting I alluded to in my last post.)

The purpose of the meeting is to discuss Chapter 6 of the CPM text, on systems of equations. But recall that we're supposed to be emphasizing Stats more this year -- and so the TOSA decided that we should incorporate data analysis into the current chapter on solving systems.

In fact, the chapter is supposed to culminate with another project. But unlike previous projects where I kept leeching off my neighbor teachers for implementation ideas, this time the TOSA asked each of us to come up with our own plan for a project that involves data and systems of equations.

Suddenly put on the spot, I couldn't think of any good ones. Two of the other Math I teachers came up with some sports-related ideas, including one based on LeBron James and his recent pursuit of Kareem Abdul-Jabbar's scoring record (which goes along with the Steph Curry project in Chapter 4). But of course, I couldn't just steal that teacher's idea.

Then the TOSA suggested that I look at a Financial Algebra text. (Our school is considering having a Financial Math class next year, but it's OK for me to use it now.) He showed me the following link:

https://www.ngpf.org/math/financial-algebra/

Unit 3 of this text is called "Saving & Systems of Equations," so he told me to look there. In Lesson 3.5, I found a rather interesting graph comparing income earned vs. net income and expenses. The income levels are divided into bins of unequal size (less than $15K, $15-30K, $30-40K, $40-50K, and so on), and so it's not completely logical. (For example, it shows that someone making $50K has a take-home pay of over $50K, since that value refers to the entire bin $50-70K and not just $50K.)

Still, both take-home pay and expenses can be approximated by a linear equation (in other words, a line of best fit), where x is the income earned and y is take-home pay or expenses. They form a system of equations whose solution indicates where income equals expenses. (From the graph, this appears to be somewhere in the $50-70K bin).

I'm not the only Math I teacher to do a finance-related lesson for this chapter. One of my neighbor teachers will give an activity on paying fines that she found online. Considering that this is written as a three-act lesson, the idea behind this lesson ultimately goes back to Dan Meyer (a name familiar to most MTBoS members, as this is a MTBoS-labeled blogpost).

It will take some work for me to make this into a coherent unit. Still, the TOSA told us to return to this underlying idea several times during the lesson, until we finally end Chapter 6 with the project where the students solve the implied system of equations.

11:45 -- The meeting ended and fifth period arrived (as Thursday's block schedule went 1-2-5-6). This was my lone Math III class of the day.

I sang "Benchmark Tests" to this class and had this class begin working on -- well, their Benchmark Tests from the district. (This explains why I mentioned "Benchmark Tests" in yesterday's post.)

1:15 -- Fifth period left for lunch.

2:05 -- Sixth period arrived. This was my lone Math I class of the day.

I sang "Sequence Tests" to this class and had this class begin working on -- well, their Chapter 5 Test, on which they must test sequences (to see whether they are arithmetic or geometric).

3:30 -- Sixth period left, thus completing my day.

OK, so that's "A Day in the Life" for yesterday. Now let's do today:

7:40 -- Friday (and Monday) mornings before school are for meetings. Today's meeting is a "CT2" meeting, where "CT" stands for "curriculum team" and "2" stands for the secondary class that we teach, as opposed to our primary class.

Logically speaking, since I teach three sections of Math I and two sections of Math III, Math I should be my CT1 and Math III my CT2. But the Math III leader teaches a zero period class, and so he usually holds only CT1 morning meetings but not CT2 meetings.

Thus I've decided that Math III is my CT1 and Math I my CT2, so I can attend both meetings. In other words, this morning I attend yet another meeting to discuss Math I after yesterday's half-day workshop.

For this meeting, the lead Math I teacher and the TOSA decide on a pacing plan for Chapter 6. They tell us that today is the first day of the chapter, and it should be mainly a "hook" to introduce the underlying activity that each of us chooses for the chapter.

For me, a hook is straightforward, since Lesson 3.5 of the Financial Algebra text linked above already has a hook of sorts. I copy the first page of the lesson and plan to hand it out to each Math I class.

This is to be followed by lessons on solving systems (graphically, elimination, as you'd expect), and then ending with the test and the project. Chapter 6 should take us all the way up to spring break (to be followed by the Geometry chapter I'm looking forward to the most, Chapter 7).

8:30 -- The meeting ends and first period arrives. This is the first of two Math III classes.

A few students who were absent yesterday finish their Benchmarks today. (This is one reason that I gave the Math III Benchmarks yesterday -- it gave my students something to do on the sub day without having to lose a lesson.) The rest of the students move on to Section 8.1.3 of the CPM text, which is on finding equations giving the roots.

I mainly show the students some DeltaMath problems. There are three levels of quadratic equations in standard form that the students must find -- at the first level both roots are rational, at the second both roots are irrational but real, and at the third both roots are imaginary.

The Warm-Up question is a review from an earlier lesson. The students must likewise find an equation, but they are given a graph. The equation is cubic, but they get to leave the answer in factored form.

For the Exit Pass, the students must find the quadratic equation with roots 4 + i -- and I spot them most of the equation, x^2 - 8x + _____ = 0. The missing constant term is the product of the roots, which works out to be 17 -- and of course, today's date is the seventeenth.

9:25 -- First period leaves for nutrition.

9:40 -- Second period arrives. This is the first of three Math I classes.

Yesterday -- as often happens when a regular teacher is asked to cover a class during prep period -- the teacher took my class to his own room. (This is another reason why I gave the test yesterday -- if I'd instead given the kids a regular assignment, say with a page for them to glue into their notebooks -- that stack would have been left untouched in my own room.)

Unfortunately, when I look at the DeltaMath grades, it appears that a whopping dozen students decided not to take the test at all today -- they probably just opened DeltaMath and then pretended to work while playing on phones or who knows what else? You might question my wisdom of assigning a high stakes 50-point test on a sub day -- but notice that if the freshmen won't even work on a test worth such a large part of their grade, how much less likely are they to work on a regular assignment (even if it's computer-based, like Desmos)?

Today is the second day of the week on the Eleven Calendar. (In previous posts, I sometimes referred to the second day as "Saturday" and intend to be a replacement for the Jewish Sabbath.)

Resolution #2: We avoid arguing in the classroom.

And you can figure out why I'm mentioning this resolution at this point in my post -- the kids turn my confrontation of yesterday's lazy students into an argument, and they start throwing glue sticks, scissors, and other objects around the room. (The correct purpose of the glue and scissors, of course, is to glue the Chapter 6 title page and hook into their interactive notebooks.)

So I make every effort to respond to this act without arguing. I immediately add six extra questions to the DeltaMath homework assignment -- one for each object that was thrown. And I add that if I ever find out who exactly is throwing the objects, a major disciplinary report will be sent to the office.

10:35 -- Second period leaves and third period conference begins. There's actually something that's been going on recently during my prep period on Fridays (which I haven't mentioned on the blog until now, since I rarely post on Fridays).

In my fifth period Math III class, there are two Spanish-speaking students who are bright in math, but often have trouble understanding me in English. They have a relationship with one of their previous math teachers -- she's a Math I teacher who speaks Spanish, but is rusty with Math III material.

And so, since we have the same conference period, we've decided to hold weekly tutoring meetings in the other teacher's classroom -- both of us teachers, my two students, and another student who has a different Math III teacher but can use the extra tutoring as well. (Yes, this is why I was so eager to meet my "prep period buddy" earlier this week -- I wanted to set up today's tutoring session. She jokes that this has become our "CT3" meeting.)

Unfortunately, today's meeting doesn't go as well as I hoped. Only one of my two students is able to get a note from his third period teacher to attend today, plus the guy from the other Math III class. (My other student, a girl, is out for an entire week.)

And since the last Math III lesson (before Benchmarks) was on graphing polynomials, I was hoping to set up DeltaMath on the Promethean boards in her classroom so they can graph them directly onto the computer (as I was struggling to do so in my classes earlier this week). But the Internet didn't work in her classroom, so we couldn't access DeltaMath.

Since the Wi-Fi issues were local to her classroom, perhaps I should have asked the group to move over to my own room for today's session. Instead, I had the students work on the whiteboard (an impromptu VNPS of sorts) -- I gave the two guys equations such as f (x) = (x + 2)(x - 1)^2 and asked the pair to find the x- and y-intercepts before we drew the rest of the graphs together.

So even though this isn't the best of sessions, at least you know now what I've been doing during my conference period on Fridays (ever since the first semester final exam).

11:40 -- Fourth period arrives. This is the second of three Math I classes.

All fourth period classes have been asked to give a PowerPoint presentation today on the upcoming visit from the WASC committee. For those of you outside of California (or "the West," as WASC stands for "Western Association of Schools and Colleges"), every high school must have its accreditation renewed every three to six years, or else its diplomas won't be recognized by outside organizations, such as colleges. Our school is in fact overdue for its WASC visit due to the pandemic.

In short, the students learn that they must be on their best behavior during the WASC visit (which is two and a half weeks away), and that they must know something if a WASC committee member visits the classroom and asks them a question (such as "What are you learning now?").

Of all my Math I classes, fourth period is the best behaved by far, so they shouldn't be a problem if WASC visits this class. My other Math I classes are a different matter.

I immediately follow up the WASC PowerPoint with a two-minute video on saving money -- the hook for the Chapter 6 project. (I won't link to the video here -- instead, follow the NGPF link above to Lesson 3.5 and you can find the entire lesson, including the video and graphs, right there.)

12:40 -- Fourth period leaves for lunch.

1:25 -- Fifth period arrives. This is the second of two Math III classes.

1:30 -- A lockdown drill begins. Yes, lockdown drills are now a regular occurrence -- and I assume this one is scheduled due to all the recent mass shootings in the news.

1:50 -- The lockdown drill ends. Assuming in advance that the students would be distracted during the drill, I leave the Warm-Up on the board during the entire drill and don't start the main lesson until after it concludes.

2:20 -- Fifth period leaves and sixth period arrives. This is the third of three Math I classes.

Sometimes sixth period behaves just as poorly as second -- in fact, I had to submit a major disciplinary report form earlier this week for someone who threw objects in sixth period (a sock one day, and then an empty water gun and water bottle the next day).

Fortunately, today this class is much better-behaved. We get through the introductory Chapter 6 lesson without too many problems.

3:20 -- Sixth period leaves, thus concluding my day.

Wow -- today just happens to be such a jam-packed day, with things going on almost every period. Still, this is when I usually post my monthly reflection. Monday is Presidents' Day -- the last holiday of the "holiday stretch." Next week is the start of the Big March -- or is it?

On one hand, spring break is one week later than last year (in order to include Cesar Chavez Day). Last year I wrote that four weeks  isn't long enough to be considered a "Big March," but with five weeks this year, it will feel more like a true Big March.

On the other hand, it won't be five weeks without a day off for our students. As I wrote in a previous post, there will be a teacher day when students won't have to attend. For us teachers it will feel like a Big March, but not as much for our students.

On the third hand, I've now declared that the Big March is now a permanent part of the school calendar (along with the Willis Unit and DEVOLSON), and it refers to the stretch between Prez Day and spring break, no matter how many weeks it is or whether there's a teacher day. Therefore, next week really is the start of the Big March.

I often write that the most difficult chapter of the year seems to be taught during the Big March. In many ways, Chapter 6 -- which will span the entire Big March -- will be hard for my Math I kids. Many of them struggled with solving equations in Chapter 1 and graphing lines in Chapter 2 -- and now Chapter 6 will ask the students to do both of those. And now we're adding in Stats from Chapter 4, and many of them had trouble there too.

Meanwhile in Math III, Chapter 8, our Big March chapter, is on polynomials. While polynomials can be difficult for some students, it's not necessarily worse than Chapter 7, where logs, sines, and cosines served to confuse the class.

That being said, will I sing my "Big March" song on Tuesday? That will be revealed in Tuesday's post, my first post of the Big March after the Presidents' Day holiday.

(And yes, I will actually be posting during the Big March this year, since I want to post the songs that I'll be performing during that stretch of the year.)

Future Freshman Showcase (Days 108-109)

This is my "A Day in the Life" post for the special day "Open House." But as I've mentioned before, Open Houses at high schools have morphed into Future Freshman Showcases.

8:30 -- First period arrives. This is the first of two Math III classes.

Chapter 8 begins in earnest as the students learn about polynomials. In this lesson, they are given the graph of a polynomial and must find an equation for it. First they inspect the roots (x-intercepts), which provide the equation up to some constant multiple a, then use the y-intercept to determine a.

On DeltaMath, all of the roots have multiplicity 1 or 2. But the lead Math III teacher wants to include triple roots as well. At the Math III meeting last Monday, he showed us how to copy his DeltaMath assignments in which he wrote his own triple root problems. It also provides us with a new way to present the material to our classes. The 15-question assignment contains seven problems that are common to all students and can be used as classwork, while the remaining questions become HW.

But there is one question that I can't quite reach today -- it's on identifying polynomials with even and odd degrees, number of relative extrema, and so on. Fortunately, this is a simpler topic that can be taught on the minimum day tomorrow. (In other words, on the minimum day, the students will learn about local minimum -- and maximum.)

Today is Fiveday on the Eleven Calendar:

Resolution #5: We sing to help us remember math.

In all classes today I perform the same song that I played yesterday in fourth period -- "Count the Ways" from Square One TV and the Judds.

9:55 -- First period leaves for nutrition.

10:15 -- Second period arrives. This is the first of two Math I classes that meets today.

This is the same lesson that fourth period got yesterday. I ask the students a few questions from Lesson 5.2.2 of the CPM text and have the students answer in their notebooks.

I could have done VNPS today, but this class has fallen behind in the quiz corrections. But as it turns out, we don't really get to do quiz corrections or VNPS in this class, because some students decide to start arguing with me. And the topic of the argument is pencils.

It is my policy not to give pencils away when students ask for one. A few years ago, I used to, and then the students would complain when I tried to give away pencils on holidays. After all, what's so special about something they can get all the time? And of course, making the students bring their own pencils is meant to foster a sense of responsibility in them.

I've struggled over the years with what to do if a student asks for a pencil on a non-holiday. Ever since Resolution #2 (the one about avoiding arguments), I decided just to say "ten cents please" when asked for a pencil -- the student won't want to pay the dime and will look for another way to get a pencil. And even if the kid mutters "I won't pay that much for a cheap pencil" or something of the like, I could just ignore it. (Again, I don't mind if the student gets the last word -- I just want there to be a last word.)

But as usual for this class, it doesn't end the argument. The two students asking for pencils then accuse me of disrespecting them for ignoring them -- in other words, they don't respect any teacher who doesn't first respect them. And they keep talking during what's supposed to be quiz corrections, meaning that the corrections don't fully happen today either (after being delayed from Tuesday due to yet another argument attempt).

These students have a valid point -- respect is a two-way street and must be earned. But from my perspective, it is arguing with the students that's disrespectful, and that the only real way to respect them is to avoid arguments. But as they implied today, nothing short of just handing them a pencil would silence them (and even then, these two would likely just whisper during corrections instead).

The real problem, of course, was that I argued early in the year before writing Resolution #2, and so I have a reputation for argument that the students seek to exploit. If I'd avoided arguments the entire year, the students likely say nothing after I tell them "ten cents please."

The kids claim that they can't afford a ten-cent pencil, yet they can afford a $1000+ phone. I don't say this in class or respond to their claim, since there's no answer that would avoid further argument.

Then again, this is the sort of thing that Malcolm Kirkpatrick (mentioned on the blog during winter break) addresses in his theories. Students who are willing to pay $1000+ for a phone but not ten cents for a pencil are saying that entertainment is that much more important to them than education -- and markets are supposed to respond to what people want.

(This is why his plan allows students to graduate early and start adult work. They'd have no problem spending a dime for a pencil if doing so helps them learn faster, pass the GED, and earn the wage subsidy that's much more than ten cents. And as for teaching them responsibility, he'd claim that they'd learn responsibility more effectively at a job that they wish to keep, rather than in a class that they really don't want to take.)

11:40 -- Second period leaves and fifth period arrives. This is the second of two Math III classes. (Once again, I don't have third period conference on Thursdays.)

Lately, in fifth period I've been using a Promethean board to display the DeltaMath examples. After giving each graph, I randomly choose a student to go up to the board and enter the correct polynomial into the Promethean.

And believe it or not, some students are still making transfers for the second semester. There is a new guy in my fifth period today.

1:15 -- Fifth period leaves for lunch.

2:05 -- Sixth period arrives. This is the second of two Math I classes.

Sixth period is much like second period, except that the quiz corrections were done yesterday. So we do get to answer two questions on VNPS (a table and a graph). This class does struggle the most on the written questions where they must solve a multi-step equation for x and find the slope of a line.

3:30 -- Sixth period leaves.

5:00 -- The Future Freshman Showcase begins.

Here's how the showcase works -- all teachers save for those who coach a sport or sponsor a club must go to their department booth out on the quad. Then we must answer questions from any students and parents who visit our booth -- especially middle school students (as in "future freshmen").

Most eighth graders who visit our booth are interested in the summer honors program. They are either current Math 8 students who wish to take Math I in the summer in order to reach Math II in the fall (the equivalent of Steve level, except that SteveH opposes Integrated Math), or else current Math I students who wish to take Math II in the summer in order to reach Math III in the fall (Bruce level). And so they end up speaking mostly to the teacher who says that she'll teach the summer Math I and II courses.

Recall that a decade ago, I tutored students who were trying to accelerate to Bruce level. The summer Algebra II teacher assigned so much work that one student was unable to keep up -- and then I actively discouraged another kid from attempting the same jump the following summer. Likewise, the Math II teacher says that she'll assign about thirty problems per day in her class, but the teachers primed to teach Math III and Precalculus are likely to assign double, if not triple that.

During the showcase, I decide to sing a few math songs, just as I did for Parent Conferences. I begin with the official song of the day, "Count the Ways," and then proceed with Weird Al's "Patterns" and "Function Rap" (which one of the other Math I teachers wants me to send her). I don't get to sing the song that one of my neighbor teachers says he likes -- "No Zeroes" (the "No Scrubs" parody). All of these songs are vocal, since I leave my guitar in the classroom.

7:00 -- I sing my final song of the day as the showcase ends -- The Fat Boys' "Working Backwards." I intentionally choose this one because the final verse of the song takes place at 7:00.

Day 109 is tomorrow, the minimum day. And Monday is Lincoln's Birthday, and so my next post will be my usual Tuesday post on Day 110.

Statistics Project (Day 91)

Today is the seventeenth of the month, and so this is my monthly "A Day in the Life" post for January.

8:30 -- On block Tuesdays, periods 2, 3, 4, and 5 meet. So today begins with second period. It's the first of two Math I classes meeting today.

I begin with a Warm-Up -- the students copy the statement. "This week our project will contain both individual and group components." Then they are to work on corrections for the first true quiz of the semester that they took last Friday.

Before I describe how my students do on this project, let me explain what the project is and how I ended up getting there. And that takes us back to last Friday morning's math department meeting.

At that meeting, the Math I TOSA (the de facto head of the curriculum team) was present, and he met with each of the Math I teachers during our respective conference periods. At that time, he described how we should run the projects.

First, there is a Desmos component. We have a choice of two Desmos activities -- one is on the correlation between income and SAT score, and the other is on NBA superstar Steph Curry, who plays for the Golden State Warriors (where the "Golden State" is right here in California). I choose the latter, figuring that my kids will find this more interesting. (The SAT one might have been a good choice for my Ethnostats class last year, since demographics were a major theme of that class.)

For the Steph Curry project, the students are asked "What do you notice?" and "What do you wonder?" as they look at the superstar's field goals made and attempted over his entire 15-year career. (These are common questions to ask during projects popular on the MTBoS.) His current season is incomplete, and so only his stats for the season so far (at the time this Desmos was created -- he was 230 for 450, with just over a quarter of the season completed so far).

As the Desmos proceeds, students create a line of best fit for the relationship between shots attempted (x) and shots made (y), based on Curry's career. Then they use this line to extrapolate how well he'll shoot this season, based on this line and how well he's shot in its first few months.

After the Desmos component, the kids move on to group portion. They will answer a question from the CPM text and analyze the correlation between amount of medicine taken and length of a cold. The idea for this question came from one of my neighbor teachers -- he suggested it after school last Friday.  I'm taking my neighbors' input into consideration, rather than just the department chair or the TOSA.

And indeed, because I had so many tough decisions to make (Curry vs. SAT, neighbors vs. TOSA), I arrive at school this morning still undecided about what exactly the project will look like. What ends up making the decision for me was my inability to print files from my computer on campus. Originally I considered starting the Desmos today, and having them glue a helper page into their notebooks (with instructions on how to complete the Desmos in English and Spanish). But once that page doesn't print, I instead photocopy the medicine problem directly from a print copy of the CPM text, and then have the students glue it into their notebooks (even though we won't get to it until later this week).

He also suggests that I leave out the part where the groups do their own research based on an NBA player of their choice. As we've seen with "Linear Art," if we give the students a choice in their project (draw a picture, choose an NBA player), many will instead choose to do nothing.

9:10 -- About halfway through the period, it's time for music break. Today is another short 14EDL song in AA format (verses only), and it describes this project:

HOW MANY MORE?

First Verse:

What do you now notice?

Three-point shots, Steph Curry.

Three-points shots, Steph GOAT is.

Making shots. Don't worry!

Second Verse:

What do you now wonder?

Three-points shots, Steph Curry.

Now he's made five hundred,

How many more? Hurry!

I'll write more about this song after finishing "A Day in the Life."

9:15 -- I begin the project now, after allowing the students time to work on quiz corrections. But as is usual for this period, the project doesn't run smoothly.

First of all, only about half the students who scored below 100% on Friday even bother working on the quiz corrections -- instead, they spend the whole time either on phones or taking about non-math. And then this continues into the main Desmos lesson.

The Desmos lesson contains ten slides. After the experience of the fateful Linear Art project, I know better than to have the kids just work on Desmos on their own -- many of them won't. Instead, we go over the first six slides together, and I choose names at random to answer the questions (such as "What do you notice?") from the slides.

On the fifth slide that we reached, we must "predict" the number of shots Curry made in the 2015-16 season, based on his actual number of shots attempted, 1598. I use the line of best fit that I'd eyeballed earlier in the activity, y = (1/2)x - 17, but the students can use their own equation as well.

By the way, adjustments are still being made to the schedule. One new freshman guy joins the class -- at first he wasn't on my roster, and then he was placed in a different period. But now his schedule has settled with him in my second period class. I send him to the office to get Google Classroom access. He returns, and he's the first to figure out that to extrapolate Curry's first 1/4-season stats to the entire season, we should multiply by four.

9:55 -- Second period leaves for nutrition, but not without incident. One guy tries to leave early, or at least line up at the door during the last few minutes of class. When I tell him that he must return to his seat, he claims that he's about to "throw up," which is why he must be at the door. (Of course, he only seems to have that urge during the last few minutes of class each day.) So I assign him a one-minute detention, which he then tries to avoid serving. I must block the door to prevent him escaping, which then causes a huge traffic jam at the door.

After nutrition is my third period conference. During this time, I remember that today is Fourday on the Eleven Calendar:

Resolution #4: We start our Warm-Ups and the main lesson promptly.

I keep this mind as I prepare for the next class, since this resolution shows me how I can run the project more smoothly.

11:45 -- Fourth period arrives. This is the second of two Math I classes today.

First of all, in order to honor the fourth resolution, I change the Warm-Up. Now I ask the students, "What grade did you earn on last Friday's quiz?" And then I tell them that I won't stamp their Warm-Up sheets unless they either earned 100% on the quiz, or else completed the quiz corrections.

This goes much better. Only two students fail to finish their corrections. Then we move on to starting the project -- even starting it a little before music break. We reach my goal of matching the six slides that second period completed.

Of course, the fourth resolution isn't the only one on my mind. All three new resolutions come into play today -- most obviously the sixth resolution on implementing all components of projects.

The fourth period class also has a new student, a girl. Technically she's not new -- she missed class all of last week for unbloggable reasons. She quickly makes up her missing quiz and assignment and prepares to work on the project.

1:10 -- With only a few minutes left in class, I put up the same Exit Pass as second period: "Using my line of best fit y = (1/2)x - 17, how many shots do you expect Curry to have made in 1598 attempts?"

The answer isn't the date, although the date (17) does appear in the question. (OK, I admit it -- when I eyeball my equation earlier, I intentionally include the date.) While I sometimes find other ways to incorporate the date besides the answer, perhaps in hindsight, today isn't the best day to do so. After all, there are new kids in both second and fourth period, and I want to introduce them to the idea that the answer is usually the date. Also, this Exit Pass might lead the students to believe that they should all just use my equation in their project rather than attempt to find an equation themselves.

Instead, I should have just asked them to write "Mr. Walker's line of best fit is y = (1/2)x - _____," and had them fill in the blank.

1:15 -- Fourth period leaves for lunch.

2:05 -- Fifth period arrives. This is a Math III class.

We begin the Warm-Up question -- condense the expression log a + 3 log c into a single log. Then we move on to the main lesson -- the second part of Lesson 7.1.2, on properties of logs. Today, instead of condensing, the students must now expand log expressions.

After music break, we move on to a short activity -- Concentration. The students are supposed to divide into pairs, cut out the game pieces, and then flip them over. If the two expressions are equal (one condensed, the other expanded), then it counts as a match. Then they must write them as an equation in their notebooks.

I want the students to work with their neighbor as a partner -- indeed, I pass out the gameboards to cut to every other row, so that each pair has exactly one gameboard. But then some students try to partner up with those other than the ones I choose for them, which leads to an argument.

And then that invokes the second resolution, on avoiding arguments. In fact, as soon as I realize that I'm arguing, I lose the will to talk about the activity -- I'm afraid I might start arguing again. A few pairs do successfully complete the activity after all. I don't even put up a valid Exit Pass question and end up just stamping papers for merely writing the date.

By the way, there are certain things I say during today's argument that I only say because it makes me feel better -- not because it's worth saying. It's like a back scratch when your back is itchy. It's hard to scratch my own back physically, but when I need a mental back scratch, I should do so -- scratch my own back in my mind, and don't say anything out loud at all.

And in order to get myself to argue less, I actually give school PBIS tickets to the two students I argue with the most. I'm trying to force myself to stop arguing so much -- and this goal is so important that I will reward students who catch me arguing (but only if I start arguing, not if they argue with me).

How could I have avoided this argument altogether? One way is to tell the students in advance that they must work with their assigned partner sitting next to them, or otherwise it's a detention.

Another, more interesting way is to pass out index cards with matching numbers, and then tell the students that they must work with their match. This way actually fits Concentration, which is after all a game about making matches. I could even put log expressions on the cards, but I must be careful -- if students get confused, they might match up with the wrong people. I should use only trivial expressions such as log(x^1) = 1 log x, log(x^2) = 2 log x, log(x^3) = 3 log x, ..., log(x^n) = n log x with a different n for each card.

This also goes with another MTBoS strategy for pairing and grouping -- VRG, which stands for "visibly random grouping." This idea means that students are more willing to work with others who aren't their friends if it is obvious ("visible") that the choice is random (such as a name generator, or random index cards as in this example). If the pairing instead appears forced, then they will argue (even if it's just the same pairing that we could have obtained randomly).

Unlike second and fourth periods, there are no new students in this class. But one student, a senior girl who had apparently passed the first semester as a junior but needs to repeat the second semester, started to enroll in my class last week -- and she promptly transferred out the next day. When I started teaching that complicated lesson on solving exponential equations, she felt as if she didn't belong here and asked her counselor to move to another class.

Today's lesson is much easier. I suspect she wouldn't have been scared away if last Tuesday had been more like today. Just change "Part 2" of today's log properties lesson to "Part 1," and keep the Concentration game (with VRG to avoid arguments), and I think she'd still be in my class.

So for tomorrow's first period Math III class, I think I'll set up the index cards. Of course, this means that I won't pass out gameboards until after the students have paired.

3:30 -- Fifth period leaves, thus completing my day.

The "holiday stretch" continues. Yesterday was the MLK holiday, and of course we have the usual Presidents' holidays coming up in February.

And my goal to reduce and eliminate arguments continues. Once again, the idea is to say the right things and put enough structures (such as index cards for choosing partners, a la VRG) so that most arguments are avoided.

One fifth period guy is on the baseball team, so he enjoys using sports analogies. It's important to avoid arguments, whether in the classroom or on the baseball field. Of course, arguing with an umpire will get a player ejected. In the classroom though, I am the umpire -- and it does no good for the umpire to start arguing on the field. So neither should I argue in the classroom.

Let's get to the Mocha code for today's song:

https://www.haplessgenius.com/mocha/

10 N=8

20 FOR V=1 TO 8

30 FOR X=1 TO 6

40 READ A,T

50 SOUND 261-N*A,T

60 NEXT X

70 RESTORE

80 NEXT V

90 END

100 DATA 14,6,14,6,11,4,13,12,9,2,12,2

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

Once again, as an AA song, it consists of a single riff. Unlike the last tune, this is a true 14EDL song where the tonic F# really does appear.

Since Degrees 14 and 11 appear in the first bar, it's best to treat 14/11 as a major third, and so we should write Degree 11 as Bb rather than B. So Degree 13 becomes G in order to make 13/11 a minor third. We can now write the repeated line of melody as F#-F#-Bb-G-D-A.

This ought to be written as song in the Superlocrian scale. But with Degree 10 missing, our ear naturally wants to fill in the perfect fifth (C#), making the song sound more like Phyrgian (or Phyrgian Dominant, like a Middle Eastern scale). The resulting riff then becomes F#-G (both chords major).

Also, originally I wanted to sing our 14EDL songs in the F#2-F#3 range (as my main vocal range is Octave 3, the octave below middle C). But with the high tonic at Degree 7 missing, this song was easier for me to sing with Degree 14 as F#3 instead of F#2. The range of the song becomes F#3-D4, which aligns better with my vocal range.

Soon we will reach more complex 14EDL songs, with a verse and chorus (AB). These generated songs are more likely to include the entire 14EDL scale rather than skip some tones.

2023 Challenge (Day 86)

This counts as my "A Day in the Life" post for the special day "first day after winter break."

8:30 -- First period arrives. This is the first of two Math III classes.

I begin this period with a Warm-Up of reflection -- "What was your first semester grade? Are you satisfied with your grade? What will you do second semester to maintain your grade?"

Then it's time for the main task -- Sarah Carter's 2023 Challenge. I won't link to it here, but I assume that you readers are already familiar with Carter's blog. I print out the 100 version, and the top students in this class are able to figure about twenty or so in the allotted time.

As an Exit Pass, this is the rare time that I can use a problem directly from the Rapoport Calendar. On her Mathematics Calendar 2023, Rebecca Rapoport writes:

4! / 4 + (4 - 4 / 4)

This works out to be 4! / 4 + 3 = 24 / 4 + 3 = 9. Therefore the desired answer is nine -- and of course, today's date is the ninth. This question looks just like the classic "For Four 4's" puzzle -- plus a 4 -- that I gave this class during the first week of school (from which the 2023 Challenge is derived.) It uses add, subtract, divide, parentheses, and factorial -- all of which are used in the 2023 Challenge today.

9:25 -- First period leaves for nutrition.

9:40 -- Second period arrives. This is the first of three Math I classes.

This class is noisy, as usual. Only nine students are actually working on the 2023 Challenge. At the end of class, I threaten to assign standards to the kids who aren't doing the assignment.

10:35 -- Second period leaves. Third period is my conference period.

11:40 -- Fourth period arrives. This is the second of three Math I classes.

While fourth period is quieter than second period (as usual), even fewer students are working on the 2023 Challenge -- just five. During the conference period, I'd decided to set up some standards for those students who aren't working. After all, there will be a Chapter 4 Project starting soon -- we can't have another project where hardly anyone's doing the work. We're past the time for threats -- now I have to assign the standards for real.

12:40 -- Fourth period leaves for lunch.

1:25 -- Fifth period arrives. This is the second of two Math III classes.

Today most of fifth period is on task (admittedly a rarity). Only a few students end up having to write down standards.

2:20 -- Sixth period arrives. This is the third of three Math I classes.

Sixth period ends up being just like fourth period -- only five kids are on task. But problems occur when it's time for me to assign standards.

Here's what happens -- first of all, this class often has too many students ask for restroom passes, some almost every day that the class meets. So I've purchased a clipboard to use as a pass, but with a limit -- once a student asks for the twentieth time in the semester (starting today) for a restroom pass, I won't give it to them -- if they leave, they do so at their own risk (of being stopped by security).

Sometimes students ask for passes to the drinking fountain, not the restroom. This is a gray area. My decision is that if they're willing to go without a pass, then they can do so at their own risk. But if they want to go with a pass, then they must wait for someone to bring back the pass. Today a few thirsty kids choose to wait for the pass to return.

But then, near the end of class, the group waiting to go to the fountain decide to leave all at once without a pass -- not coincidentally, it's right when I announce that they must write standards. (In other words, they were really just trying to avoid writing standards.) My policy is that students who are assigned standards must stay for detention or else I must contact the parent (for avoiding the work, the standards, and the detention). Only one student stays, thus forcing me to contact four parents today.

Of these four, at least two are intentional non-learners -- they almost never do any work whatsoever. If I'm going to have this standards policy, then I might end up contacting their parents all the time (which is counterproductive).

This is a tricky one. Today is an all-classes day, so it's understandable that many students are thirsty (as opposed to a block day when sixth is just after lunch, so they could have had water then). I could allow more kids to leave the room for water earlier in the class, so that they can't use it as an excuse to leave at standards time. But those intentional non-learners may just leave to get water a second time anyway.

My other two classes have students who do very little work, but all are special ed students. Sixth is the only class with students who don't leave for special ed pull-outs, instead putting their heads on their desk when it's time to work.

So how to deal with these students is something I must figure out as we approach the Stats project.

3:20 -- Sixth period leaves, thus completing my day.

Even though the holidays are over, the "holiday stretch" continues. For teachers, the holiday stretch runs from Veterans Day to Presidents' Day -- the time of year when most school off-days occur.

Let's get back to the habit of posting how I'm keeping up with my New Year's Resolutions. Today is Sevenday on the Eleven Calendar:

Resolution #7: We earn our grades through hard work and dedication.

Well, as I mentioned above, the Warm-Up was a reflection of the students' first semester grades, so we all think about today's resolution. Hopefully, they will take this rule to heart and plan to maintain or improve their grades in the new semester.

My next post will be tomorrow -- the regular Tuesday post.

Baptism Sunday (Yule Blog Challenge #12)

Table of Contents

1. Introduction

2. Yule Blog Challenge #18: My Goals/One Word Challenge for 2023

3. Previewing Math III Chapter 7

4. Music in the 14EDL Scale

5. Problems in Fifth Period Math III 

6. Rapoport Question of the Day

7. Conclusion

Introduction

Today is Baptism Sunday on the Christian Calendar. It is defined as the Sunday after Epiphany, and is based on an interpretation of the Bible that Christ was baptized around his 30th birthday. Thus Baptism Sunday is the last holiday of the "Christmas season."

Orthodox Christians celebrated Christmas yesterday, and their Epiphany (or Theophany), will be twelve days later, on January 19th. This is also their day to celebrate Baptism -- indeed, it's the Baptism, not the Three Kings, that is more significant in Eastern Churches.

What are my Hispanic students celebrating today? After doing some research, I did read about how some Spanish-speakers observe Octavitas, an eight-day celebration for Epiphany and Baptism. But this is mainly observed in the Caribbean, not Mexico. Thus, unless I find out that one of my students is of Caribbean descent, I shouldn't assume that my mostly Mexican-American students are celebrating any holiday today. In Mexico, the Three Kings are more significant than the Baptism.

Baptism Sunday is the end of the Christmas season, and sometimes (though not always) it marks the end of winter break, including this year. I often post my twelfth Yule Blog post on Baptism Sunday as well.

Yule Blog Challenge #18: My Goals/One Word Challenge for 2023

Last year, Shelli -- the leader of the Yule Blog Challenge -- came up with a "one word challenge" for the new year. In other words, we should select one word to describe the entire year. Last year, she chose "journey" as her one word. But there's no evidence that she's doing a One Word Challenge this year.

Last year, I selected "travail" as my one word. I got the idea from 365 New Words-a-Year Calendar, where "travail" was the word for January 2nd. And last year, making our way through this stage of the pandemic depended on the travails of all of us -- that is, our hard work.

This year, I turn to that same calendar to choose a word for 2023. But some of the words on that calendar aren't appropriate. The January 1st word was "ab initio,"  Latin for "from the beginning," as in from the beginning of the year -- good for New Year's Day, but inappropriate for One Word. And the January 2nd word was "shard," as in "shards of glass." I might choose "shard" if I were planning a trip to England in 2023 to visit that country's tallest building, The Shard (which really is shaped like a shard of glass), but I'm not.

Fortunately, there's one more "hidden" New Year's Day word on the calendar. One thing about the calendar is that it often combines weekends, with Saturday and Sunday sharing a page, even though each day has its own word. The weekend of December 31st-January 1st was tricky because it was split between years. Although the 2023 calendar printed "ab initio" on its own page, the 2022 calendar had words for both the 31st and the 1st. And so New Year's Day had an extra word this year -- and this word is more appropriate.

And so my one word for 2023 is (drumbeat) --

blithesome

The calendar defines "blithesome" as "joyfully exuberant" or "merry." And with all the arguments I've been having with my students lately, I definitely need to be more blithesome in the classroom this year, and so this is a great word to choose for the One Word Challenge in 2023.

Shelli once chose "joy" as her One Word. "Blithesome" means "joyful" -- indeed, "joy" itself can be said as "blithe." But "blithesome" comes from Old English (unlike "joy" with is French). It's been some time since I blogged about Shapelore and Anglish (the thought that Old English words are better than French or Latin words like ab initio). So for those folk who like Anglish words more, let's keep this year's One Word as "blithesome" instead of "joy." ("Shard," by the way, also comes from Old English.)

Last year, I mentioned a Desmos activity that's related to the One Word Challenge:

https://teacher.desmos.com/activitybuilder/custom/5ff4fde1dad0813c40aee6f4?collections=5efb5e06d6b26b71b2e1090d

And I wrote the following:

I haven't decided whether I'll assign One Word to my students this year or not. In my last post, I wrote that I might have more Desmos lessons in my classes, and so I could give this activity. But the one above is all about equations of lines, which is not what any of my students are learning.

Well, this year, equations of lines fit the Math I curriculum, so I could give it this year. After glancing at this Desmos though, I notice that the students are to choose their One Word, and then try graphing an image of the word on Desmos.

In other words, it's just like the infamous Linear Art Project in Math I -- and we know that my students didn't find that project very joyful -- uh, blithesome -- at all. The last thing I want to do is give another version of that project, especially when it will take time away from the Stats project. But hey -- at least you blog readers get to see what such a project actually looks like.

(It's too bad that I couldn't have delayed the Linear Art Project to January. Perhaps this version might have gone more smoothly than the November version, especially if I kept the Desmos lesson open during the entire project. But the Linear Art Project goes with Chapter 2 -- and there's no way that we should still be doing Chapter 2 stuff in second semester.)

Well, I've already written about Math I in my last post, so that's enough about Math I prokects. In today's post, I'll preview for the upcoming chapter for Math III instead.

We just finished Chapter 5 on inverses and logarithms. Chapter 6 of the Math III CPM text is a bit like Chapter 4 of the Math I text -- it focuses on Stats. But unlike Math I, the Math III department has decided to skip over the Stats chapters and proceed directly to Chapter 7.

Previewing Math III Chapter 7

Chapter 7 of the Math III CPM text is called "Logarithms and Triangles." Section 7.1 deals with logs and is a continuation of the content taught in Chapter 5.

We know that Math III follows the pacing guide much more strictly than Math I. So let me write out the plan for this chapter with the exact dates:

January 9th: Desmos Reflection

Like Math I, the first day back is a sort of "ease back into math class" day. The Desmos Reflection asks students to give a New Year's Resolution -- I could have them write One Word about their goals (but they won't have to graph it, like the Desmos link above). And since this should be completed quickly, there should still be time for Sarah Carter's 2023 Challenge as well.

January 10th-12th: Lesson 7.1.1, Using Logarithms to Solve Exponential Equations.

Two block periods are used to introduce exponential equations and the idea of taking logs of both sides to solve them.

January 13th-18th: Lesson 7.1.2, Investigating the Properties of Logarithms.

One Friday and one block period (after MLK Day) are devoted to the properties of logs, including the product, quotient, and power rules. There are worksheets associated with this lesson, but nothing to do with Desmos.

January 19th-20th: Lesson 7.1.3, Writing Equations of Exponential Functions

One block period and one Friday are devoted to exponential functions and their equations. There is a Desmos activity associated with this lesson.

January 23rd-25th: Lesson 7.1.4, Applications of Logarithms

One Monday and one block period are devoted to word problems involving logs. One day is for compound interest and the other day is for a murder mystery (where Newton's Law of Cooling is used to determine when the murder took place).

January 26th-27th: Lesson 7.2.2, Law of Sines

One block period and one Friday are devoted to the "Triangles" mentioned in the chapter title. And this leads me to wonder, how exactly is Trigonometry taught in Integrated Math classes?

Back when it was Algebra I/Geometry/Algebra II, right triangle Trig began in Geometry classes (and indeed, it appears in Chapter 14 of the U of Chicago Geometry text). The fact that the first appearance of Trig in the Math III text is on the Law of Sines suggests that right triangle Trig is taught in Math II (as it's definitely not in our Math I text).

On the other hand, when I taught that one semester of Trig last year, I moved through the text slowly and didn't quite make it to the Law of Sines. (Of course, my students should have at least seen it in their Math III classes the previous year -- but that was the pandemic year, so who knows?)

January 30th-February 1st: Lesson 7.2.3, Law of Cosines

Naturally, if there's a Law of Sines, there needs to be a Law of Cosines. One Monday and one block period are devoted to this lesson. One day should be solving for missing sides and the other day for missing angles.

February 2nd: Lesson 7.2.4: The Ambiguous Case

One block period is devoted to the ambiguous case of the Law of Sines -- that is, SSA. But according to the pacing guide, we won't actually have the students solve for two triangles. Instead, we will give a Desmos lesson and then direct DeltaMath to give SSA questions with only one possible triangle on the chapter test. (This is what the U of Chicago text calls the SsA case -- when the side opposite the given angle is longer than the adjacent side, then there is only one possible triangle.)

February 3rd-6th: Chapter 7 Test

As usual, there is a written part and a DeltaMath part. In the first semester, I've been giving the written part first. But one of my fellow Math III teachers usually gives the DeltaMath part first -- and the Math III leader apparently agrees with her, since the pacing guide now directs us to give the DeltaMath part on Friday and the written part the following Monday.

There are a few things worth mentioning about Chapter 7. We are directed to have DeltaMath give only questions with one possible triangle, not two. But as we've seen before, we can tell DeltaMath to give only certain types of questions on the test, but not the homework. I wish to avoid the situation where students can't do their homework because DeltaMath keeps asking them for two triangles.

Fortunately, there's a way to avoid this. Since the Ambiguous Case is the last lesson before the test, the homework that night should be test review. So the only practice in class will be the Desmos lesson, and then the night's HW is on reviewing everything except SSA for the test.

But before the test, there should be a quiz. As usual, we're directed to give the students a quiz, but there's no day on the pacing guide devoted to a quiz. In the first semester, I usually give quizzes on the second Monday after the chapter begins -- this works out to be January 23rd. The pacing plan mentions a quiz on Illuminate (not the usual DeltaMath) after Lesson 7.1.3, so this fits the 23rd nicely.

On the other hand, a more natural dividing point is between 7.1.4 and 7.2.2, in order to separate the logs from the Trig. This suggests a quiz day on one of the block days that week.

In the end, I'll watch to see whether the quiz is on Illuminate or DeltaMath, and whether or not it contains 7.1.4 material. If the quiz contains 7.1.4, a possible plan is to give it on that Thursday, and then start the Law of Cosines (not Sines) on Friday. The plan devotes one day for missing sides and the other day for missing angles for both Laws. But the only time one ever needs to use the Law of Sines to find a missing angle is the SSA case -- which is already taught as the Ambiguous Case. So we can afford to have one fewer day for the Law of Sines and replace it with the quiz day.

By the way, if February 3rd-6th are the test days, then Chapter 8 should begin the following block day, February 7th or 8th. But we know that in reality, an extra day is needed for test corrections (especially due to the Math III department's "70% rule" where students must first do corrections to raise their grade to 70%, and then they can do the test retake). Now I don't mind using February 7th for test corrections, because that's a special day anyway -- 2/7 is "e Day." (Yes, we just finished celebrating Phi Day, and now I'm already looking ahead to e Day.)

But here's the thing -- recall that e didn't appear in Chapter 5 (which led to a problem with the test as DeltaMath kept asking questions with e and ln). So surely e and ln appear in Chapter 7, where logs are studied in more detail, right?

Wrong -- in the CPM text, e doesn't appear until Chapter 10, on series! In fact, we must also tell DeltaMath not to include continuous compound interest questions (A = Pe^rt), since there's no e yet.

So that means we really shouldn't celebrate e Day. But then again, since a day will be wasted for test corrections anyway, we might as well have an "e Day/Chapter 7 Test make-up/corrections" party. Those who need to work on the test can do so, and those who don't can celebrate instead.

February 7th falls on a Tuesday, when fifth period Math III meets, but not first period. The morning class can celebrate the next day, "e Day Observed/Chapter 7 Test make-up/corrections." After all, remember that e starts out 2.71.828, so both 2/7 and 2/8 are e Days (as is 1/8 -- hey, that's today).

Looking ahead to Pi Day, we notice that Chapter 9 is on trig functions, including radians and pi. But it appears that Chapter 9 is for Honors Math III only -- my class will skip directly to Chapter 10. (In other words, my class will finally learn about e on or just after Pi Day.) After all, recall that in the old days, radians were taught only in Honors Algebra II/Trig, or else not until Pre-Calculus (or a separate Trig class like the one I taught last year.)

So there goes my Pi Day party. Then again, the schedule shows us finishing the Chapter 8 Test (or some sort of midterm) just before Pi Day. So March 14th is the "Pi Day/Chapter 8 Test make-up/correction" party, it appears.

Music in the 14EDL Scale

Now is a good time to start thinking about songs that I might perform in the upcoming weeks. Indeed, whenever I look ahead to the next chapter, I always think about the songs.

There are two songs previously posted to the blog that fit Math III Chapter 7. One is "The Compound Interest Rap," and the other is "Whodunnit" (about a murder mystery). Unfortunately, both of those songs go with Lesson 7.1.4 -- and one of those lessons might land on a Monday (that is, a non-singing day) and the other on a quiz day. I'll be playing close attention the week of January 23rd to see which of these songs might go with these lessons.

The Mega Millions jackpot for Tuesday will be over one billion dollars. Even though I've already performed "One Billion Is Big" when the Powerball jackpot was that high, it might be fun to let this rap be the first performance of the new year.

All remaining songs will most likely be for Math I. I've already done the first Stats song, "Rudolph the Statistician," before winter break. The only other Stats song I have is "Measures of Center," but this one about mean, median, and mode was intended for middle school students. These aren't included in Chapter 4 because they're to be considered below high school level. (It's too bad that I couldn't have performed for Ethnostats classes last year, as some of those songs might have fit this class.)

So that means I'll need some brand new songs for Math I. And there will be a new scale to compose them in -- our main scale for January and February will be 14EDL.

As a reminder, here is the 14EDL scale:

Degree     Note

14             ru F#

13             thu G/G#

12             wa A

11             lu Bb/B

10             gu C

9               wa D

8               wa E

7               ru F#

This scale has seven notes, just like the usual major and minor scales. But while its length is familiar, its actual notes aren't. It's the first EDL scale to include Degree 13, which is about halfway between G and G#, just as Degree 11 is about halfway between Bb and B.

Our familiar seven-note scales are called "modes." If we play all the white notes starting from C, this is the major mode. If we start at D, it's the Dorian mode (which I used as part of my UCLA fight song parody in order to honor quarterback Dorian Thompson-Robinson). If we start at E, it's the Phyrgian mode, and so on.

If we interpret Degrees 13 and 11 as G and B, then we obtain a mode -- the Locrian mode. If we play all the white notes starting from B, we obtain B Locrian. Starting from F# and then playing white notes from G to E is called F# Locrian (related to the G major scale).

But interpreting 13 and 11 as G and B is awkward, since 13/11 sounds more like a minor third. So we might use the 13=G# or 11=Bb interpretations. Here 13=G# produces the Locrian sharp-2 (or #2) scale, since its second note is sharpened. If we use 11=Bb instead, the resulting scale is called Superlocrian.

F# Locrian: F#-G-A-B-C-D-E-F#

F# Locrian #2: F#-G#-A-B-C-D-E-F#

F# Superlocrian: F#-G-A-Bb-C-D-E-F#

Knowledge of the three Locrian scales can help us come up with chords for our 14EDL songs. For example, here's a link to Levi Clay, a guitarist who composes in the Superlocrian scale:

https://www.premierguitar.com/lessons/beyond-blues-how-to-use-the-super-locrian-scale

Clay's examples use B Superlocrian, but we can change it to F# by starting at the fret 2 instead of 7. To Clay, the tonic chord for his B Superlocrian scale is something called "B7alt," which is a B7 chord with the fifth and ninth degrees altered. Converting this to F# Superlocrian, we get these F#7alt chords:

b5b9: F#-Bb-C-E-G (14-11-10-8-13)

#5b9: F#-Bb-D-E-G (14-11-9-8-13)

b5#9: F#-Bb-C-E-A (14-11-10-8-6)

#5#9: F#-Bb-D-E-A (14-11-9-8-6)

Notice that all of these interpret 14/11 as a major third. Indeed, 14EDL is the first EDL scale to have two plausible thirds -- major (14/11) and minor (14/12 = 7/6).

Of course, at first I'll want to figure the new scale out. I'll save these altered 7th chords for later on. And as usual, we'll start out with simple AAA songs (that is, with a verse only) in the new scale before attempting more complex tunes.

Let me run the random song generator on my TI to see what 14EDL tunes look like. My first attempt produces the following melody:

9-7-8-7-11-10-9-9-9 (or D-F#-E-F#-Bb-C-D-D-D)

This is a lousy excuse for a 14EDL song. Degrees 12, 13, 14 are all left out, so this is really just a glorified 12EDL song. If we needed to play one of the F#7alt chords it would likely be F#7#5, but I'm likely just to play a D chord with so many D's in the song.

A second run of the generator produces the following melody:

11-9-13-8-8-9-8-10-9 (or Bb-D-G-E-E-D-E-C-D)

In many ways this is worse -- while Degree 13 is included, there is no tonic (Degree 14 or 7). We might play this melody over a b9 altered chords, probably F#7#5b9, but again it doesn't sound right. The way this is going, I'd do best to start with "One Billion Is Big" until I get 14EDL figured out.

Problems in Fifth Period Math III

I've been having problems avoiding arguments and staying blithesome in some of my Math I classes, but this is to expected since the students are freshmen. But in some ways, the most problematic class of all isn't any of the freshman classes, but the fifth period Math III class.

In this class, a group of students talks through almost every lesson. And this talking didn't stop on the day of the final exam. I'm still doing some last minute sorting out regarding which students deserve zeroes before the first semester grades are due.

(By the way, I normally watch The Simpsons, but tonight there's an episode of Bob's Burgers on instead of The Simpsons. And of course, the whole episode is about a girl who's accused of cheating on a test.)

Anyway, this week I checked the rosters -- every year, students transfer in and out of classes and make schedule changes at the semester. Every class is losing students, and every class except sixth period is gaining students as well.

The class that's losing the most students by far is fifth period -- a whopping six are transferring out. (All other classes are losing just two.) And while there could be many reasons for them to be moving (such as making the number of students in each class equal), one obvious possible reason is that they find my class too noisy -- in other words, because I'm a bad teacher. After all, several of the moving students did sit near the loudest corner of the room.

And even if just these six are transferring out, more likely think the class is too loud. Several of them requested to take the final exam in another classroom. That should never happen. Indeed, this is what prompted me to start handing out zeroes in the first place -- if the class is too noisy during the test, someone needs to get a zero.

I also can't help but notice that of the six leaving students, five are girls. With many of the loud students being guys, I'm wondering whether these girls accuse me of being sexist. The loud guys are trying to take over the class, and these girls can't learn -- and I, a male teacher, have an environment in which the guys are preventing the girls from learning. My biggest fear is that as a male teacher, I could be making subconscious decisions that favor the guys -- and this is one reason why many girls don't do well in math or seek out STEM careers.

Earlier, I wrote that I want to make more connections with my freshmen and avoid being the teacher than none of them want to have the next three years. While this is more important than with the older Math III students, I see right here that being an adequate teacher matters for these students as well. On Aeries, I can see the destinations of the six leaving students. If these girls requested a schedule change, then I can figure out which teachers they are flocking to -- and then that's the sign for me to be more like those popular teachers.

Of the six transferring students, one is going to the continuation school. (This is a bit surprising for a Math III junior who seems to be on grade level in math, but she must have failed some of her other classes that aren't math.) Two of them are seniors -- both are going to the department chair's class.

The three remaining juniors are all moving to the same teacher's class -- as it happens, her open math class is first period, not fifth. This might mean that I'm such a bad teacher that these students are willing to rearrange their schedule just to get rid of me (and the fact that they choose the lone female Math III teacher could be because they think I'm sexist).

By the way, she's also the teacher who gives her students the DeltaMath test before the written test. So that already gives me an easy way to emulate her -- give my students the DeltaMath test first. But there are other habits I need to get from her, or any other veteran popular teacher -- how to keep a group of loud students quiet, and how to be fair to all students regardless of gender.

Because so many students are leaving fifth period, it's also the class gaining the most students. Of the three new students, one is moving from my first period. And I feel sorry for her -- she's leaving my very quiet first period for the very loud fifth period. (But she's a special ed student who regularly takes tests in another room, so at least my noisy guys won't affect her during tests.)

The other two have been demoted from the Math III Honors class due to a low first semester grade -- along with a third girl who is moving into my first period. Of the three demoted students, one got an F, but the other two have D+'s -- that is, they just barely missed staying in Honors (since I assume that C- is the grade they need to stay).

The Honors classes have moved at a slightly faster pace. They finished Chapter 7 at the end of the semester in order to have more time for extra chapters (like Chapter 9, the Trig chapter). I hope that the three demoted girls remember something from Chapter 7 in December, so they can get off to a good start by passing the Chapter 7 Test now.

At any rate, I must make sure that the new girls don't have a sexist teacher this semester. I must treat all students equally without regard to gender.

Rapoport Question of the Day

Today on her Mathematics Calendar 2023, Rebecca Rapoport writes:

cot^2(pi/6) + csc(5pi/6) + 3tan^2(pi/6) + sec(11pi/6)cot(7pi/6)

This is the sort of question that Honors Math III will see in Chapter 9, but my regular Math III students won't see at all. (Indeed, the only trig they'll see this year are sine and cosine -- and these are only the two functions that don't appear in this function.)

The reference angle for all of the angles in this problem is pi/6, so that makes it easy:

= sqrt(3)^2 + 2 + 3/sqrt(3)^2 + 2sqrt(3)/sqrt(3)

= 3 + 2 + 1 +2

= 8

So the desired answer is eight -- and of course, today's date is the eighth.

Conclusion

And so I officially make it to twelve Yule Blog posts for the second straight year. I'm still not the champion, though, since Shelli had thirteen posts (and with a shorter winter break to boot).

Several more posts are coming in the next several days. Tomorrow is my "first day after winter break" post, which is a special "Day in the Life" post. Then Tuesday and Wednesday are my regular block day posts as we continue to focus on my Math I classes and the songs I sing in them.

Epiphany Post (Yule Blog Challenge #11)

Table of Contents

1. Introduction

2. Yule Blog Prompt #16: What Worked (and What Didn't) in 2022

3. An Ethnostats Lesson That Worked in 2022

4. Previewing Math I Chapter 4

5. Conclusion

Introduction

Today is Epiphany, marking the end of the Twelve Days of Christmas on the Christian Calendar. On the mathematical calendar, it is Phi Day, celebrating the number (1 + sqrt(5))/2 -- a value that's also known as the golden ratio.

For many people of Hispanic descent, today is Dia de los Tres Reyes -- Three Kings Day.

Yule Blog Prompt #16: What Worked (and What Didn't) in 2022

Hmmm, let's see -- you already know that the chapter I looked forward to the most was Math I Chapter 3, on transformations. Within that chapter, one lesson I particularly liked was the November 30th Desmos activity, on Tetris Transformations.

Indeed, in comparing that activity to the previous Desmos activity -- Transformation Golf -- there's a reason I like the Tetris one more. The Tetris one is a bit more realistic -- in that game, players really do translate and rotate the tetrominos in order to win. The golf one directs students to translate, rotate, and reflect a "ball" so that it fits in the "hole" -- but the "ball" is really an asymmetrical tetronimo. Had the ball been a real circle, one could win the game via translations only and never reflect the ball.

Notice that the Transformation Golf activity in the U of Chicago Geometry text (Lesson 6-4) is a more realistic example of reflections. No one reflects the ball -- instead, the course is reflected, and then by aiming the ball where the image of the hole is, the ball will bounce off the wall and land in the hole. So this is why, based on how the Desmos activities were set up, I like the Tetris one better.

But Shelli's question here isn't which activity I liked -- it's what worked in 2022. And I must admit that in the end, the Tetris activity didn't work with my students. The main reason it didn't work was timing.

In fourth period, this was one of the lessons I gave during the World Cup, when I knew that many students were paying more attention to the Mexican and Argentinian teams in Qatar. I decided that I'd rather have a Desmos activity than attempt to teach a regular lesson during the matches. And then in sixth period, the soccer games were over -- but then there was a power outage. The blackout lasted into the next day, so that second period didn't get the Desmos lesson on Tetris either.

Once again, if I had followed my neighbor teachers, transformations wouldn't have been taught until the following week, so the World Cup and blackout wouldn't interrupt the lesson. But the lesson was reduced to a packet. The packet was superior to the worksheet that I'd given my own kids, but then it would leave no time for any Desmos activities.

The only way to make these Desmos lessons work better would have been for the Math I department to treat Chapter 3 with more respect. Instead, most of the semester was spent on Chapters 1-2, and transformations were relegated to the last week of the semester.

I know how I'd improve these lessons in future years, and I even know what songs I'd perform those days -- parodies of The Simpsons (for Transformation Golf, referring to the classic episode when Bart goes mini-golfing) and Tetris them songs. But I won't be able to do so unless the department allows more time for Chapter 3 in Math I.

Still, looking at these lessons and seeing what worked and didn't work prepares me for Chapter 4, when I hope more of my lessons will work. Since this is one of my last winter break posts, I want to look ahead to the new semester. Today's post will be all about the second semester of Math I in general, and Chapter 4 in particular.

But before we begin, let's look at one lesson that did work -- from my Ethnostats class. Since Math I Chapter 4 is a Stats chapter, looking at old statistics lessons will help me prepare as well.

An Ethnostats Lesson That Worked in 2022

Believe it or not, one of the best lessons I had at the old magnet school was something that originally didn't work (due to technology), then ultimately worked on a sub day. And it's on interpreting data and correlation, so it's ultimately related to Math I Chapter 4 as well.

I wrote about it on the old Stat blog in February. I will cut-and-paste it here below:

Let me start by describing what is going on in this class, going all the way back to last Wednesday when I received the fateful COVID test. My original plan had been to start the first project of the semester (in lieu of a chapter test). It's another project that goes back to my Ethnostats predecessors -- the students were to read the New York Times interactive article "Money, Race, and Success." They would enter the name of our district and several other districts around the nation, and observe the correlation between percentage of students who are white and number of grade levels they are behind.

But unfortunately, although I was able to access the article on my computer, the students hit a paywall when they tried to read it. So instead, I changed the assignment to answering the review questions at the end of Chapter 12 -- which they'd normally do before a quiz or test. I eventually figured out that if I type in the name of the article in a Google search, then I can see the article. But if I create a link to the New York Times in Google classroom, that's when the paywall appears.

So on Friday, the day of the sub, I finally assigned the project. The students kept track of their data in their Stats Scrapbooks (interactive notebooks). As it turns out, the sub that day has a math background as he's working on his credential. It's always lucky when a math teacher gets a math sub (though there weren't much calculation in the assignment -- it's just recording and interpreting data).

Previewing Math I Chapter 4

Let's look at the pacing guide to see what the upcoming plan for Chapter 4 looks like. First of all, recall that the Math I department chair has shown us his own pacing guide. Both of them devote two weeks to this chapter, although one of them lists the Stats project as an unlabeled week (not Week 1 or 2). I suspect that in reality, it's going to take closer to three weeks to complete the chapter. I will try to stay within a week of the chair's pacing guide, just as I did first semester.

Since the chair places the project between "Week 1" and "Week 2," I'll base the following with "Week 1" the first week, the project the second week, and then "Week 2" the third week. But keep in mind that I'm starting the project when my neighbors begin it, rather than when any pacing guide says to start it.

Week 1 Monday: Nothing.

This pacing plan was originally written last year. But one thing has changed since last year -- recall (from the old Stats blog) that the second semester started on Wednesday, not Monday, due to a COVID surge and a last-minute decision to require COVID testing at a pharmacy. This year, the semester will begin on a Monday as usual, leaving a hole in the schedule.

It appears that teacher-blogger Sarah Carter is planning a 2023 Challenge this year, similar to last year's 2022 Challenge (and similar ones in other years). So I can do this on Monday as well.

Week 1 Block 1: Lesson 4.1.0, Fit Fights

This is a Desmos introducing lines of best fit. I assume the lesson number 4.1.0 means that it's merely an introduction to the material in Section 4.1 of the CPM text.

Week 1 Block 2: Lesson 4.2.2, Correlation and Association

This is a Desmos on the difference between correlation and association. Last year, I taught that correlation is for quantitative data, and association is for qualitative data. Correlation coefficients appear in this lesson.

Week 1 Friday: Section 4.1 Quiz

This is on DeltaMath. Originally I was considering making the first quiz of the new semester be a Hero Quiz, because the students might not have learned enough for a full quiz yet. But this guide suggests that the kids will be ready for a true quiz after all.

Week 2 Block 1 (after MLK holiday): Chapter 4 Project (Part 1)

OK, now let's get to the project. It appears that last year, a Desmos was set up for a project where kids collect data from around the class or school, and that the data should be related to the topics from the first week (lines of best fit, correlation, association). But in a Math I teachers meeting before winter break, the chair and TOSA discussed having a project that's related to sports data instead.

That's why it's imperative that I not start any project until I see other teachers doing so. The last thing I want to do is start a project based on this Desmos, ask the students to go around and survey others to collect data, and then suddenly tell them that they should forget about the surveys and just use the sports data.

By the way, if the project ends up being on sports data, then it will be similar to the Ethnostats project from February that I listed above -- we give the students the data and they interpret any correlations they see in the data. If they must collect data instead, then it becomes similar to the data collection project from earlier in the Ethnostats class. (That project worked too -- the only reason I didn't list earlier is because it was still 2021, so it's not an example of something that worked in 2022.)

Week 2 Block 2: Chapter 4 Project (Part 2)

This might end up being the first day of the project again, depending on other teachers. The new version of the project might have a completely different Desmos set up for it, or it might not have a Desmos component at all. Whatever it contains, I must follow my new Resolution #6 and make sure that I implement all components of the project.

Week 2 Friday: Chapter 4 Project (Part 3)

I notice that there is no Chapter 4 Test -- like Chapter 2, the project is the test. I might end up giving a Hero Quiz this week depending on how the project is going. And the project will likely count as 50 points, and weighted under the Chapter Tests category (25% of the grade).

Week 3 Monday: Lesson 4.1.1, Charge!

This is a continuation of the previous week's lesson on lines of best fit. It is on Desmos, which is often tricky to do on a Monday.

Week 3 Block 1: Lesson 4.1.4, Residuals and LSRL

This is a Desmos on lines of best fit and residuals between the actual and predicted data.

Week 3 Block 2:

Two possible assignments are given here. One is a Polygraph on Desmos, labeled as a "quiz." The other is a worksheet -- apparently, my predecessor was absent that day and left this for the sub. (Maybe he tested positive for COVID in early 2022, just before I did.)

If I do give the Polygraph that day, I likely won't call it a "quiz." There was a similar "quiz" in Chapter 3 that only confused the students.

Week 3 Friday: Section 4.2 Quiz

This is a DeltaMath quiz, so I will give this one. And this ends the chapter -- once again, there is no Chapter 4 Test due to the project.

Looking ahead in the semester, if I stay within a week of the pacing guide, then Chapters 5-6 will be covered by spring break, allowing us to reach Chapter 7 -- the next Geometry chapter -- after the break.

Also, I notice that there are three district Benchmarks in the year. Only one was given in the first semester, so two are given in the second semester. Recall that there was a huge discussion over whether those should count as a midterm grade. With two Benchmarks in the semester, the midterm question will be even trickier.

Conclusion

Let's conclude our Phi Day post with our usual Phi Day song, courtesy Michael Blake: