Tuesday, March 20, 2018

Lesson 13-2: Negations (Day 132)

Lesson 13-2 of the U of Chicago text is called "Negations." In the modern Third Edition of the text, negations appear as part of Lesson 11-2 (which is the old Lessons 13-1 and 13-2 combined).

Today on her Mathematics Calendar 2018, Theoni Pappas writes:

Find AB^2.

(We're given Triangle ABC with right angle at B, altitude to the hypotenuse BD, AD = 2, DC = 8.)

This is clearly a case of the Right Triangle Altitude Theorem, Lesson 14-2:

Right Triangle Altitude Theorem:
In a right triangle,
b. each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to the leg.

So this tells us that AB is the geometric mean of the hypotenuse AC = 10 (= 2 + 8) and AD = 2. As we're asking for AB^2 rather than AB, we don't even need to use square roots:

AB^2 = AC * AD
AB^2 = 10 * 2
AB^2 = 20.

So the desired value is 20 -- and of course, today's date is the twentieth. It's the spring equinox -- and by definition, the moment of the equinox (9:15 AM Pacific Time) is Nowruz, Persian New Year.

By the way, our students haven't seen Chapter 14 yet. But our students might be able to figure it out using the similar triangles of Chapter 12 -- though Triangle ABC ~ ADB is a bit tricky.

Today I subbed in a middle school English class. So there is no "Day in the Life" today, but I do want to follow yesterday's Fawn Nguyen post with another issue that comes up today -- restroom passes.

The teacher I'm covering today has a strict restroom pass policy -- one pass per trimester. (It is Day 124 in the new district, hence it's near the start of the third trimester.) The rule is that the kid must produce a student planner, which the teacher signs. Presumably, the teacher uses the planner to determine whether the student's single pass has been used up that trimester.

Let's approximate how many passes this teacher gives in a typical day. She has six classes (five English and a yearbook elective) with about 30 students in each class, so 180 students. Each student gets one pass per trimester, so 180 passes per trimester. Finally, a trimester is about 60 days, so we estimate that she gives out about three passes per day. Hence I would have considered it good if only three students ask to go to the restroom today.

So how many students ask to go the restroom today? No, it wasn't three -- try ten. As a comparison, if the regular teacher gave out three passes per day, every student would have used up his or her pass by the time the SBAC is given. And with those daily two-hour blocks for testing, the students will wish that they still have their restroom passes!

Here's the breakdown by period -- one student each in fourth and fifth periods ask for a pass. And two students in each of the other classes ask to go. And for comparison, the ten students who ask to go today is more than the number who go during the four days of the digital film class combined -- which is not what I'd expect in a class with such a restrictive restroom rule.

It's obvious that some students ask because they assume that I, as a sub, don't even know about the regular teacher's restroom rule. The two students in first period -- the rotation actually starts with first today -- find out that I read the pink lesson plan where she lists rules when I ask for their planner. In second period, one of the two students who ask doesn't have his planner, and so he doesn't go. The other student tells me that the pass page has fallen out so that I should sign elsewhere in his planner. I wouldn't be surprised if he's hidden the signature page intentionally so that he can ask the regular teacher to sign for a second pass later on.

In third period, I begin to get upset with the fifth student -- two beyond my goal of three -- asks for a restroom pass. I start warning him about using up his pass so early in the trimester and not having a pass for later on. This warning is enough to convince the sixth student to change his mind about the restroom -- as does the lone student who asks when fourth period begins.

In fifth period, a student asks to go just as the class is about to start. This would aggravate me even if the teacher didn't have a one-pass-per-trimester rule -- because he's asking right after lunch. If he had gone to the restroom ten minutes earlier, he wouldn't have needed to ask a teacher -- and his one pass for the trimester would still be intact.

Of course, we all know what's going on here. The student would rather miss class time than lunch time with his friends. To him, sacrificing his pass is worth it if it means he doesn't have to lose a precious second with his friends.

When I was a young student, I was the opposite. I'd rather spend my lunch in the restroom -- sometimes going twice during break -- rather than miss even one second of class. My eighth grade English class was always after lunch, and I didn't miss a second of class time. Yes, English wasn't my favorite subject, and yes I sometimes found it boring, and yes I sometimes thought it was irrelevant to my future career -- but I didn't want to miss a single second. I didn't always care about spending time with friends.

Given a choice between being popular with my fellow teenagers and being popular with adults, I chose the latter -- and it wasn't even close. Also, I wasn't trying to be a "teacher's pet." The name "teacher's pet" implies that teachers judge students by feelings rather than by merit. I wanted teachers to like me for meritocratic reasons. I tried my best in English class, even if I didn't always get the highest grades. For example, I earned a "C" grade in the third quarter -- the quarter when our class read The Diary of Anne Frank (the play this class is reading today). I'd much rather be reading the story then asking my teacher for a pass.

And if I'd been a student in a class with a one-pass-per-trimester policy, I definitely would have saved the third trimester pass for SBAC two-hour testing block season, not blow it well before testing -- especially if I'm an eighth grader who should remember two-hour testing from seventh grade.

Here's what I do when the fifth period student asks for a pass today -- I tell him that since lunch begins at 12:45, he should have gone to the restroom at 12:46, or 12:47, or 12:48, or 12:49, and so on for all forty minutes of lunch, until 1:25. Yes, I name every single minute out loud. Some of the students start to laugh -- but it's effective, since the student changes his mind about going.

The reason I name all the minutes is because of a new classroom management idea I thought of. If I ever have my own class again and a student asks to go to the restroom right after break, I'll let the student go, but then the student owes me standards -- one line for each minute of the break during which he or she should have gone:

At 12:46, I could've gone to the restroom, but I didn't.
At 12:47, I could've gone to the restroom, but I didn't.
At 12:48, I could've gone to the restroom, but I didn't.
At 12:49, I could've gone to the restroom, but I didn't.
At 1:25, I could've gone to the restroom, but I didn't.

Again, this is management that is based on how students actually think. Unlike my young self, many students value friendship time over class time and are willing to miss class for friendship time, so no argument about favoring adults' opinions over their friends' (or what I used to do, tell them about how I went to the restroom thousands of times over three years of school without missing single a second of class) can be effective. Now it's, I'll go to the restroom during lunch so that I won't have to write 40 lines in class.

Recall what Fawn Nguyen writes about restroom passes, in her big classroom management post:


A few years ago I was sitting in the lodge at CMC-North in Asilomar when someone recognized me from my Ignite talk and came over to chit chat. He was lamenting the frequency in which his students were asking to use the restroom pass. I asked some related questions to learn more before I realized that he wasn’t lacking classroom management skills as much as just lacking a good lesson.

This explains why I had eight restroom passes during five periods of English when I didn't have as many passes in sixteen periods of digital film -- film is more interesting than English. Students rarely ask for passes if they find the lesson interesting or fun -- which unfortunately, Anne Frank isn't to many students.

That takes us to sixth period Yearbook class. Yearbook, like Digital Film, is an elective, so perhaps the stream of kids of students asking for passes would end here.

Well, there are a few other issues going on here. That same pink lesson plan also mentions that the students must be in their assigned seats, and that (going back to yesterday's post) cell phones may be used only for listening to music, not texting or taking photos. (In particular, the phones must be either face down on the desk or in a pocket.) But the students complain that their assignment is to get together in groups (thus violating the seating chart) and upload the photos they took on phones for the yearbook assignment.

I concede that maybe the teacher doesn't have Yearbook in mind when she lists these rules. But there's one rule that she explicitly applies to Yearbook class -- "zero level of talking." Apparently, there are four levels of acceptable noise, and of course "zero level" is silence. I can't make those words disappear from the pink lesson plan, and so I'm obligated to enforce it. The students continue to object, since "zero level" would interfere with the group projects to which they are assigned.

And this is when a student asks for a restroom pass. She has her planner ready for me to sign, and it's not the period right after lunch. The only problem I have is that she's the ninth student to ask for a pass today when my goal is three passes or fewer. Again -- students tend to ask for passes during boring classes, and a class with "zero level of talking" is boring.

So this is when the argument begins. I tell her that she's the ninth student asking for a pass when there should have been only three -- and I suspect that six students have asked me, the sub, for a pass when they wouldn't have asked the regular teacher. She counters, "How can you 'calculate' when someone needs to go to the restroom?"

In the end, I finally sign the planner, but I continue to argue along the way. The argument ends when security, in the form of an academic coach, finds out what's going on here. The students tell her their side of the story and I tell her mine. In the end, she decides to make it "level one of talking" and allows one more girl -- the tenth student -- to use the restroom.

I admit that I don't really want to enforce the "zero level" rule today anyway. Students who have completed their projects are allowed to use the class as a study hall. Two students are working on math -- an eighth grader is beginning rotations, while a seventh grader is finding area. The older girl doesn't draw her 180 degree rotation correctly and the younger girl calculates the perimeter instead of the area -- and yet I can't help either one! For not only would helping them make it harder to enforce the "zero level" rule, I'd be violating it -- I'd want to ask the students questions about their homework and they'd answer, which then wouldn't be "zero level" of talking.

It's notable that the pink lesson plan is actually the same sub plan she uses everyday -- the actual lesson for the day (at least for English) is posted in the Google Classroom. And so the Yearbook lesson plan doesn't change -- perhaps some days they're working on individual assignments with "zero level," and today they're working on a group assignment.

I keep thinking that the Yearbook class would have been the best class I subbed for in a while had it not been for having to enforce the pink rules -- or if the teacher had specified different rules for the Yearbook class (such as "The students are seated in their assigned groups. It's OK for them to talk quietly among their group -- Level 1. It's OK for them to upload photos from their phones"). Perhaps at least one of the two girls wouldn't have been bored enough to want a restroom pass. And I could have helped one or both of the other girls with their math assignments.

It's ironic that I'd have trouble keeping students away from restroom passes on a day when the students are reading Anne Frank. After all, the young Jewish girl was only allowed to use the "WC," or water closet (bathroom), at night. Our students think it's oppressive when we make them wait until snack or lunch to use the restroom, but that's nothing compared to what Anne Frank had to endure.

But this doesn't mean that my strategy for keeping the kids away from the restroom should have been to compare them to Anne Frank (or to the guard at the Tomb of the Unknowns, or to myself and my thousands of restroom trips at school, as I have in previous posts). As a regular teacher, I should have had the students write standards as mentioned above. And as a sub -- well, the students, I assume, already have some penalty if they make a second restroom visit in a trimester, and that wasn't enough to stop the demand for passes today.

So instead, I could have an incentive for the students, to be implemented from now on whenever the regular teacher mentions a strict restroom policy in the lesson plan. At the start of the day, I tell the students that according to the teacher, she usually gives out only three passes a day (which is technically true, albeit indirectly inferred from the calculation). So our goal is for there to be no more than three total passes during the entire day.

Now within each period, there's a reward if the class goes the whole period without a pass. So what would be a suitable reward for the entire class?

Well, last year I sang songs in class as part of music break. So here's the idea -- as the students enter, I sing (without instruments, since I don't carry them around when subbing) the first few lines of one of the more popular songs that I played last year. I know that these are math songs, but I suspect that kids in any class will enjoy some of them. Then I tell them that I'll sing the rest of the song during the last few minutes of class if all of them can avoid using a restroom pass the entire period.

Of course, suppose one student takes the pass at the start of class. This means that they fail to earn my reward -- so they think they might as well all go to the restroom that period. Well, here's the sneaky part of the reward -- as long as the student returns before singing time, I'll keep the reward alive but just sing half of the song. The goal is now to avoid a second student asking for the pass the rest of the period.

Now suddenly, the students have a real reason to skip using the restroom pass -- one that should be more effective. Just maybe, the Yearbook girls who insisted on using the restroom would have been the second and third students to go, not the ninth and tenth. And the earlier classes might have earned a song at the end of their Anne Frank reading. This is definitely something I want to consider, especially as we head into SBAC two-hour testing block season.

This is what I wrote last year about today's lesson:

For example, the statement:

All unicorns are white.

is actually true -- after all, we have never seen a unicorn that isn't white (precisely because there exists no unicorns at all, much less ones that aren't white). Another way of thinking about this is that there are zero unicorns in this world, and all zero of them are white! In if-then form this statement becomes:

If an animal is a unicorn, then it is white.

The hypothesis is false (since there are no unicorns), so the entire conditional is true. This statement has no counterexamples (unicorns that aren't white), and conditionals without counterexamples are normally called true.

The book then derives, from the statement 1=2, the statement 131=177. There is a famous example of a derivation of a false conclusion from a false hypothesis, often attributed to the British mathematician Bertrand Russell, about a hundred years ago. From the statement 1=2, Russell proved that he was the Pope:

The Pope and I are two, therefore the Pope and I are one.

that is, he used the the Substitution Property of Equality from the hypothesis 1=2.

In today's lesson, the U of Chicago text introduces the symbol not-p for the negation of p. In other texts, the notation ~p is used, but I have no reason to deviate from the U of Chicago here.

Before leaving this site, let me point out that this [Metamath] site gives yet a third way of writing the "not" symbol used in negations:


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