Thursday, March 22, 2018

Lesson 13-4: Indirect Proof (Day 134)

Lesson 13-4 of the U of Chicago text is called "Indirect Proof." In the modern Third Edition of the text, indirect proof appears in Lesson 11-3.

Today I subbed in an instrumental music class. It's at the same middle school as the recent Digital Film class -- and in fact, the regular teacher is out for the exact same reason. Her class is on an out-of-state field trip -- this time to the Disney World Music Festival.

I mentioned Florida in recent posts to a recent hot topic -- of course, I'm speaking of the Year-Round Daylight Saving Time bill. Orlando, where Disney World is located, would be on Year-Round Eastern Daylight Time, equivalent to Atlantic Standard Time. (On the Eleven Clock, all of Florida would be united in the Eastern Time Zone.)

Last week, I did a "Day in the Life" for the first day of the Digital Film class even though it's not math, since I wanted to focus on classroom management on a multi-day subbing assignment. But this time, there's a student teacher who takes charge of all classes. Normally, he teaches only half a day at this school and spends the rest of the day at a high school in the same district. But once he found out that a music specialist sub wasn't available, he rearranged his schedule to cover all classes today.

Thus there's no reason for a "Day in the Life" today, as classroom management isn't an issue. But I do wish to write a little about the music classes for today -- Day 126 in the new district.

There are three eighth grade classes and two seventh grade classes. Within each grade, there is one band class (typical band instruments) and a strings class (mostly violins and cellos, of course, with a few violas and one guy playing the bass in each class). That's only four classes -- well, the fifth class is jazz band. These eighth graders play the same instruments as regular band, including saxes (three types), trombones, and trumpets. (There would have been a drummer, but he's in Florida.)

In previous posts, I mention the musical notes Bb-A-Bb in connection with band classes. Let me remind you what these notes are -- a common "Warm-Up" for band is to play the notes Bb-A-Bb, then Bb-Ab-Bb, then Bb-G-Bb, and so on. I brought it up in connection with the traditionalist debate, where traditionalists lament that musicians know how important repetitive practice (Bb-A-Bb) is to being proficient musicians, yet progressive reformers oppose repetitive practice in math.

I actually never subbed for band on a Bb-A-Bb day before -- I only know about it because I would sometimes overhear Bb-A-Bb while covering another class next door. Now that I'm actually in a band class today, you might be wondering, do I hear Bb-A-Bb today?

Well, the answer is sort of. I do hear repeated notes as a "Warm-Up," but it's not Bb-A-Bb. Instead, the "Warm-Up" begins with F, not Bb. So the eighth graders play F-E-F, F-Eb-F, F-D-F, and so on all the way down to F-F-F (an octave down and up), all in half notes. The seventh graders have a slightly easier "Warm-Up" -- F-E-F-Eb-F-D-F, all the way down to the octave, in whole notes (or longer).

All classes have breathing exercises before F-E-F -- even the string players. And after F-E-F, all classes move on to scales. The seventh grade string players begin with C major and proceed via the circle of fifths until they reach B major. On most band instruments, it's easier to play flat scales than sharp scales, so they play the other side of the circle of the fifths -- from F major to Gb major. The eighth grade classes only play a single scale, but it's one of the more difficult scales -- B major for strings and Db major for band. The jazz class plays a completely different scale -- the "blues scale" commonly used in their genre. I believe they play a G blues scale -- G-Bb-C-C#-D-F-G.

Again, traditionalists wonder why math classes can't promote mastery of arithmetic and p-sets -- the mathematical equivalent of breathing exercises, F-E-F, and scales. In the past, I point out that students are willing to practice for something that's easy, fun, or high-status. So F-E-F and scales are neither easy nor fun, but they lead to the high status of being a musician. On the other hand, being a mathematician isn't high status for many teens. They view mastering arithmetic and completing their p-sets as "not worth it," whereas F-E-F and scales are "worth it."

And so here are the songs the students play after their scales -- both grade level strings work on a song called "Ancient Ritual." Eighth grade band plays "The Great Locomotive Chase," while seventh grade band is beginning a brand new song today. Titled "Best of the Beatles," it's a medley of a trio of Fab Four songs -- "Ticket to Ride," "Hey Jude," and "Get Back." The jazz students don't play a new song, but instead practice jazz standards from their textbook, including "Birdland."

I'll have more to say about music after my second and final day in this class tomorrow.

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

But Section 13-4 is the big one. This section is on indirect proof. I've delayed indirect proofs long enough -- now is the time for me to cover them. Actually, indirect proofs aren't emphasized in the Common Core Standards, but they were in the old California State Standards, where they were known as "proofs by contradiction."

What, exactly, is an indirect proof or proof by contradiction, anyway? The classic example in geometry is to prove that a triangle has at most one right angle. How do we know that a triangle can't have more than one right angle? It's because if a triangle were to have two right angles, the third angle would have to have 0 degrees -- since the angles of a triangle add up to 180 degrees -- and we can't have a zero angle in a triangle. Therefore a triangle has at most one right angle.

And voila -- that was an indirect proof! Notice what we did here -- we assumed that a triangle could have two angles -- the opposite (negation) of what we wanted to prove. Then we saw that this assumption would lead to a contradiction -- a triangle containing a zero angle. Therefore the original assumption must be false, and so the statement that we wanted to prove must be true. QED

Indirect proofs are often difficult for students to understand. One way I have my students think about it is to imagine that they are having a dream. Normally, when one is dreaming, one can't tell that they are having a dream, unless something impossible happens, such as a pig flying in the background, or the dreamer is suddenly a young child again. I recently had a dream where I was suddenly younger again, and I was flying off the ground! Naturally, as soon as those impossible events happened, I knew that I was in a dream.

And so a proof by contradiction works the same way. We begin by assuming that there is a triangle with two right angles, and then we see our flying pig -- a triangle with a zero angle. And as soon as we see that flying pig, we know that we were only dreaming that there was a triangle with two right angles, because there's no such thing! And so all triangles really have at most one right angle. So an indirect proof is really just a dream.

We saw how an indirect proof was needed when we were trying to prove that there exists a circle through any three noncollinear points A, B, and C. The proof that such a circle exists requires an indirect proof to show that the perpendicular bisectors m of AB and n of BC actually intersect. The indirect proof goes as follows: assume that they don't intersect -- that is, that they are parallel. Then because, m is perpendicular to AB and parallel to n, by our version of the Fifth Postulate, AB must be perpendicular to n. Then, now that n is perpendicular to both AB and BC, by the Two Perpendiculars Theorem, AB and BC are parallel. But B is on both lines, so we must have, by our definition of parallel, that a line is parallel to itself -- that is, AB and BC are on the same line. But this contradicts the assumption that AB, and C are noncollinear. Therefore the perpendicular bisectors m and n aren't parallel -- so that they actually exist.

Returning to 2018, let me post my worksheets. I begin with the second side of the worksheet that I posted yesterday. Then, since this lesson naturally leads itself to activity, I also include some old logic problems that I did post last year.

By the way, it's back-to-back scientists featured in the Google Doodle. Today's Doodle features Japanese geochemist Katsuko Saruhashi. Her specialty is measuring pollution in water, in particular radioactive pollution.

Wednesday, March 21, 2018

Lesson 13-3: Ruling Out Possibilities (Day 133)

Lesson 13-3 of the U of Chicago text is called "Ruling Out Possibilities." In the modern Third Edition of the text, ruling out possibilities appears in Lesson 11-1.

This is what I wrote about Lesson 13-3 last year:

Here are a few things that I want to point out. First of all, some texts refer to the Law of Ruling Out Possibilities in Section 13-3 by another Latin name, modus tollens. Here is a link to the Metamath reference to modus tollens.

As we can observe in the proof at the above link, modus tollens is essentially modus ponens (The Law of Detachment) applied to the contrapositive (Law of the Contrapositive, or contraposition.)

Section 13-3 is another section that lends itself to an activity, since many of its questions are actually logic problems, like the ones that often appear in puzzle books.

I don't have much else to say today. After so many lengthy posts about circle clocks and circle constants, the Queen of the MTBoS and the King of Traditionalism, and so much subbing, today is a much-needed short post.

(OK, I can at least mention today's Google Doodle since it honors a scientist -- Guillermo Haro, the first Mexican member of the Royal Astronomical Society.)

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:

Monday, March 19, 2018

Lesson 13-1: The Logic of Making Conclusions (Day 131)

Chapter 13 of the U of Chicago text is called "Logic and Indirect Reasoning." It is a strange chapter, for two reasons.

First, in past years on the blog, I didn't like the fact that some lessons on logic that appear early in most texts don't appear until Chapter 13 of the U of Chicago text. For example, "Reasoning and Proof" is only Chapter 2 of the Glencoe text. And so I broke up Chapter 13 and scattered its lessons among other chapters.

And second, it turns out that the modern Third Edition of the U of Chicago text does the same thing! I see that Chapters 13 and 14 of the new version correspond to Chapters 14 and 15, respectively, of the old version. The content of the old Chapter 13 is now included in other chapters.

On this blog I continue to follow my old Second Edition of the text. Therefore, we will continue to cover the logic of Chapter 13 as a separate chapter.

Before we start Chapter 13 though, the Queen of the MTBoS has spoken. That's right -- Fawn Nguyen posted on Saturday, St. Patrick's Day. And her St. Paddy's Day post deals with an issue that comes up too often these days: cell phones and technology.

I’m sipping hot sake while waiting for my food. I scan the restaurant, about half full already for an early Friday evening. Two kids are on their smartphones at the table with their parents. They don’t even look up as the waiter arrives to take their orders; I guess the parents already know what to order for them. At the next table, I see a young child sitting in his high chair and watching a video on a propped up smartphone. Nearly every kid in the restaurant is doing something on his/her phone. Never mind the adults.
This scene is all too familiar, too common — so common that it would be “odd” if we didn’t see this. And we’ve been seeing it for some time now.
I’m happy and grateful that technology is here to stay. But I hope we seek opportunities to connect more humanly.

There are several things going on in this post. First of all, Nguyen is a middle school teacher, so of course she's in a room surrounded by kids on phones all day. The problem, of course, is that she's describing a restaurant, not a classroom.

It's easy to relate the restaurant to the classroom here. The students want to use phones in the class, but it's not because the lesson is particularly boring. It's because phones are the only things that entertains these youngsters, as evidenced by their use at the restaurant. In other words, if the kids had their way, they'd spend almost 100% of their waking hours using the phones.

And, most important, they believe that anyone who gets between them and their 100% waking hour phone use is "mean" or "unfair." Of course, this means that the teachers who tell them that they can't use phones in class are "mean." Nguyen's story takes place late Friday afternoon -- and to the kids, it's the start of 60+ hours of no teachers telling them to put their phones away. That's an entire weekend with no teachers between them and their 100% waking hour phone use.

We expect middle school students to be interested in 100% waking hour phone use. It's a shame that at least elementary school teachers can't be protected from children who desire 100% phone use, as Nguyen's statement tells us here:

At the next table, I see a young child sitting in his high chair and watching a video on a propped up smartphone.

I've heard of two-year-olds using phones before -- and sometimes I wonder how this is even possible, since they wouldn't even know what buttons to press, or how to spell words to text. Well, I guess this answers the question -- the parents play videos for them, so the toddlers only watch them.

The problem is that not only do the kids view teachers who take their phones away as "mean," but so their parents who won't buy them the phones. And presumably, any parent who gets anything less than an unlimited plan is also getting in the way between the children and their 100% phone use goal.

I don't write about family that much on the blog. But I have said in old posts that I have no children (for example, in old Daylight Saving Time posts where I mention that I prefer Year-Round DST since I have no kids, whereas parents tend to prefer Year-Round Standard Time). So the question I ask myself is, what would I do about phones if I had children of my own?

In our society there are some items which have a minimum age requirement. For example, we can't legally purchase alcohol or tobacco until age 21 -- the latter a California law. (One issue mentioned in last week's walkout is whether guns should be added to the list.) Of course, youngsters often become interested in alcohol or tobacco well before the age of 21 -- but it's not usually as early as six. Most six-year-olds are interested in drinking chocolate milk and soda pop, not alcohol.

Likewise, I'd want my children to be "too young even to be interested in using cell phones" -- as long as possible. For starters, this means I wouldn't play them videos such as the one Nguyen's toddler watched in the restaurant. That two-year-old didn't beg his parents to buy him a phone -- the parents just showed them the phone. If I were a parent, I wouldn't do that.

Sooner or later though, the inevitable happens. My children would see another student at school with a phone, and then the begging begins. So the next goal is for my children to know that I won't buy them a phone without them thinking I'm mean -- as long as possible.

Here's what I'd want them to know -- the older generations criticize and make fun of younger generations who are addicted to cell phones. Fawn Nguyen is either a late Boomer or an early Gen X'er, and in this very post she's criticizing young people. Avoiding cell phones and finding other ways to entertain yourself, then, is a way of making older people like you.

My own generation -- the late X'er/early Millennial cusp -- is caught in the middle. I remember that as a young child, I often played cheap handheld games, such as baseball and Yahtzee. Most often, these games were kept in the car, and so I played them on long car rides. It never occurred to me that I should spend 100% of my waking hours playing these games, or that anyone who stopped me from playing them 24-7 was mean. I often went days without playing these handheld games -- it never occurred to me that I should bring them to school, much less play them during class. But of course, nowadays parents would entertain their kids on long car rides with phones, not handheld games -- and it's the phones that kids want to use during 100% of their waking hours.

I'd like to tell my children that the kids who use phones 100% of waking hours are future dropouts, and that students who earn A's and B's don't own phones. But this is probably false -- even future valedictorians most likely own cell phones.

But I can make a clearer relationship between math and phones. This is the idea behind the phrase I made my own students say -- "Without math, cell phones wouldn't exist." After all, anyone who wants to work for Google or Apple should have a STEM degree, which requires being good at math.

So here's the idea -- I set some appropriate age for my children to have their first phone -- let's say seventh grade. But this doesn't mean that I buy them their phone as soon as they've completed the sixth grade -- it means I get the phone as soon as they complete sixth grade math. If they are below grade level, then they don't get the phone until they earn a passing grade in the final trimester of Math 6 (or a higher class). If they're above grade level, they can get the phone earlier. If I held myself to this standard, I'd get my phone in second grade -- the year I independently studied Pre-Algebra.

What I really want is to show my children that all the STEM disciplines are relevant to cell phone technology, not just math. So once they get their phones, I'd like to implement the second part of my academic incentive -- the grades the students receive in all STEM classes that appear on the report card (math, science, maybe computers) determine how much money I spend on the phone plan. So if the STEM grades are all A's, then I get an unlimited plan. If the STEM grades are B's then I get some sort of limited plan, all the way down to no plan for D's and F's.

The problem, of course, is that the phone company probably wouldn't like it if I kept on switching between different plans every month. Even if I simplified this a little -- the child gets an unlimited plan and I pay the phone bill if all the STEM grades are C or better, and I skip a month if one of the STEM grades is a D or F -- my credit rating might suffer whenever I skip a payment.

Well, at least I should be able to implement the first part of the plan -- purchase the phone as soon as the student completes sixth grade math -- without any troubles with the phone company.

Returning to Nguyen's post, her main topic is whether schools should embrace technology and find ways to incorporate it into the lessons. She quotes her own tweet:

At BTSA mentor training, 1 of the prompts was "How do u incorporate tech into a lesson?" My knee-jerk response, "You don't." It's back to that tech for tech’s sake that irks me. It's like asking, "How do u add aspirin into your diet?" @ddmeyer

In many ways, the technology debate mirrors the traditionalist/progressive debate. Progressive reformers tell us that students who have a 100% phone use mindset might complete an assignment if it's online, whereas they won't even answer Question #1 on a p-set in a printed textbook. Indeed, last year at my old school, the history teacher painstakingly scanned every page of the text and posted it online for exactly this reason.

But notice that Nguyen is actually on the traditionalist side of the technology debate. She doesn't believe that technology should be incorporated into a lesson for the sake of including it.

Her mention of aspirin is a reference to the former King of the MTBoS, Dan Meyer. Back during the summer of 2015, Meyer wrote several posts about how certain math topics, such as factoring in Algebra I, are like aspirin without a headache. For example, we see the following post:

But here Meyer was writing about particular math topics from Algebra I and beyond -- and not something more general like technology. Traditionalists might oppose forcing technology in the curriculum, but they have no problem with forcing factoring into the curriculum.

Today, during the last day of subbing for the digital film class, of course the students used Chromebooks as the elective class is all about technology. But I had two encounters with cell phones during class. The first occurred during silent reading time. In fact, I haven't mentioned it yet on the blog, but this middle school has SSR as part of the regular school schedule. (Last Wednesday -- the one day I wrote a "Day in the Life" at this school -- there was no SSR as both homeroom and SSR were dropped to accommodate the walkout.)

I believe that one of the most difficult things for a sub to enforce is SSR. When a student decides not to do a written or online assignment, the regular teacher will eventually discover it when it's time to check the work. But if the student starts whispering or avoids reading, the teacher may never find this out -- and the students know this. At this school, SSR is always after lunch -- and with the rotation, it means that SSR is attached to a different class each day. Fortunately, today SSR is attached to fourth period -- the best-behaved class.

Anyway, here's the connection to technology -- some students don't have a book to read, so insist on reading something on their phone. I must apologize to Fawn Nguyen, who writes:

(I still need a real book to read from, however, like this one that just came in the mail because the Internet said I should read it.)

As it turns out, one girl has her phone, but there are no books uploaded on the phone either. The TA for the class lends her a book to read.

The other phone encounter is in fifth period -- the worst behaved class of the day. And here's a common problem with students who bring phones to school -- they often lose them. One student thinks he left the phone in another class -- and so during today's entire class, he acts out. He speaks loudly and out of turn and continually leaves his seat.

Back in my January 6th post, I wrote about one girl -- the special scholar -- who also loses her phone (in September). It's difficult to know what a teacher should do in this case. The student desperately wants the phone, knowing how angry the parents will be if they discover its missing. But the teacher doesn't want to waste class time on something the students shouldn't have anyway. I consider calling the office (cf. the January 6th post) but I ultimately don't. (Some schools have it built directly into the rules -- no class time can be spent searching for phones, no matter how expensive they are.)

Let's end our discussion of Nguyen's post with the commenter Pamela Baker, who specifically mentions Geometry in her post:

Pam Baker:
Completely agree with you. I teach Geometry and find it completely fascinating that most geometric discoveries can be done without “technology”. I have tried to do a lot of explorations with paper folding, cutting and coloring. It amazes me that many of my 9th & 10th graders have not had the opportunity to “play” with paper. 

Last year, some of the STEM projects indeed allowed the students to "play" with paper. But other parts of the Illinois State curriculum required a computer (e.g., the online homework). And in hindsight, I realize that my school saved money by not purchasing printed science texts. I should have followed the history teacher and tried to have the students in all grades learn science online using the online Illinois State text (not Study Island). It's not until I read Nguyen's post and subbed in certain classes did I realize that classes can be completely dependent on online texts. (Saving money is arguably a valid reason to eschew printed books in favor of online texts.)

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

The other is that I've been meaning to move the first two lessons of Chapter 13 -- namely 13-1 on the Logic of Making Conclusions and 13-2 on Negations -- up to Chapter 2. Dropping Lessons 2-3 leaves a hole right in the middle of Chapter 2, and conveniently, 13-1 and 13-2 fit here. Indeed, 13-1 on Making Conclusions makes perfect sense right after Lesson 2-2 on If-then Statements.

[2018 update: Not only does the modern Third Edition of the text include the old Lesson 13-1 in another chapter, namely Chapter 11, but just as I did in past year, the new edition also combines Lessons 13-1 and 13-2 in its new lesson, which is Lesson 11-2.]

This chapter focuses on mathematical logic, which ultimately helps the students write proofs. I mentioned earlier that the Law of Detachment is often known by its Latin name, modus ponens. In fact, I pointed out that on the Metamath website -- a website full of mathematical proofs -- modus ponens is one of the most used justifications:

Notice that I only mention the Metamath website for general information. This website is definitely not suitable for use in a high school math classroom. At Metamath, even a simple proof like that of 2+2=4 is very complex:

 In fact, believe it or not the proof was once even more complicated because it tried to use pure set theory to prove that 2+2=4, and then later on more axioms (postulates) were added to make the proofs easier -- similar to the postulates for real numbers mentioned in Lesson 1-7. To repeat, the basic idea is that one makes a proof simpler by adding more axioms/postulates.

This is when students often ask, "Why do we have to learn proofs?" Of course, they ask because proofs are perhaps the most difficult part of a geometry course. The answer is that even though mathematical proofs may not be important per se -- but proofs are. Many fields, from law to medicine, depend on proving things. We don't want to guess that a certain person is guilty or that taking a certain medicine is effective -- we want to prove it. For centuries, the dominant way to learn how logical arguments work was to read Euclid. Let's learn about how Honest Abe learned about logical arguments from Euclid:

Unfortunately, the above link is a political and religious website. Well, I suppose it's impossible to avoid politics when discussing Lincoln, but the webpage is also a Catholic site.


Friday, March 16, 2018

Lesson 12-10: The Side-Splitting Theorem (Day 130)

Lesson 12-10 of the U of Chicago text is called "The Side-Splitting Theorem." This is what I wrote last year about today's lesson:

The U of Chicago version of the theorem is:

Side-Splitting Theorem:
If a line is parallel to a side of a triangle and intersects the other two sides in distinct points, it "splits" these sides into proportional segments.

And here's Dr. Wu's version of the theorem:

Theorem 24. Let triangle OPQ be given, and let P' be a point on the ray OP not equal toO. Suppose a line parallel to PQ and passing through P' intersects OQ at Q'. ThenOP'/OP = OQ'/OQ = P'Q'/PQ.

Notice that while the U of Chicago theorem only states that the two sides are split proportionally, Wu's version states that all three corresponding sides of both sides are proportional.

Moreover, the two proofs are very different. The U of Chicago proof appears to be a straightforward application of the Corresponding Angles Parallel Consequence and AA Similarity. But on this blog, we have yet to give the AA Similarity Theorem. So how can Wu prove his theorem?

We've seen several examples during the first semester -- a theorem may be proved in a traditionalist text using SSS, SAS, or ASA Congruence, but these three in Common Core Geometry are theorems whose proofs go back to reflections, rotations, and translations. Instead, here we skip the middle man and prove the original high-level theorem directly from the transformations. We saw this both with the Isosceles Triangle Theorem (proved from reflections in the U of Chicago) and the Parallelogram Consequences (proved from rotations in Wu).

So we shouldn't be surprised that Wu proves his version of the Side-Splitting Theorem using transformations as well. Naturally, Wu uses dilations. In fact, the names that Wu gives the points gives the game away -- O will be the center of the dilation, and P and Q are the preimage points, while P' and Q' are the images.

Here is Wu's proof: He considers the case where point P' lies on OP -- that is, the ratio OP'/OP, which he labels r, is less than one. This is mainly because this case is the easiest to draw, but the proof works even if r is greater than unity. Let's write what follows as a two-column proof:

Given: P' on OPQ' on OQPQ | | P'Q'r = OP'/OP
Prove: OP'/OP = OQ'/OQ = P'Q'/PQ

Statements                                           Reasons
1. P' on OPQ' on OQPQ | | P'Q'      1. Given
2. OP' = r * OP                                    2. Multiplication Property of Equality
3. Exists Q0 such that OQ0 = r * OQ  3. Point-Line/Ruler Postulate
4. For D dilation with scale factor r,    4. Definition of dilation
    D(Q) = Q0, D(P) = P'
5. P'Q0 | | PQP'Q0 = r * PQ               5. Fundamental Theorem of Similarity
6. Lines P'Q0 and P'Q' are identical      6. Uniqueness of Parallels Theorem (Playfair)
7. Points Q0 and Q' are identical          7. Line Intersection Theorem
8. OQ' = r * OQOP' = r * OP,            8. Substitution (Q' for Q0)
    P'Q' = r * PQ
9. OP'/OP = OQ'/OQ = P'Q'/PQ = r    9. Division Property of Equality

Now the U of Chicago text also provides a converse to its Side-Splitting Theorem:

Side-Splitting Converse:
If a line intersects rays OP and OQ in distinct points X and Y so that OX/XP = OY/YQ, then XY | | PQ.

The Side-Splitting Converse isn't used that often, but it can be used to prove yet another possible construction for parallel lines:

To draw a line through P parallel to line l:
1. Let XY be any two points on line l.
2. Draw line XP.
3. Use compass to locate O on line XP such that OX = XP.
4. Draw line OY.
5. Use compass to locate Q on line OY such that OY = YQ.
6. Draw line PQ, the line through P parallel to line l.

This works because OX = XP and OY = YQ, so OX/XP = OY/YQ = 1.

The U of Chicago uses SAS Similarity to prove the Side-Splitting Converse, but Wu doesn't prove any sort of converse to his Theorem 24 at all. Notice that many of our previous theorems for which we used transformations to skip the middle-man, yet the proofs of their converses revert to the traditionalist proof -- once again, the Parallelogram Consequences.

Another difference between U of Chicago and Wu is that the former focuses on the two segments into which the side of the larger triangle is split, while Wu looks at the entire sides of the larger and smaller triangles. This is often tricky for students solving similarity problems!

Now it's time to give the Chapter 12 Test. Again, it's awkward to combine Lesson 12-10 with the test, but that's the way it goes. I can argue that by doing so, we're actually following the modern Third Edition of the text. Yesterday's lesson on SAS~ and AA~ is the final lesson of Chapter 12 in the newer version (Lesson 12-7). In the Third Edition of the text, the Side-Splitting Theorem is the first lesson of the next chapter, Lesson 13-1.

Test Answers:

10. b.

11. Yes, by AA Similarity. (The angles of a triangle add up to 180 degrees.)

12. Yes, by SAS Similarity. (The two sides of length 4 don't correspond to each other.)

13. Hint: Use Corresponding Angles Consequence and AA Similarity.

14. Hint: Use Reflexive Angles Property and AA Similarity.

15. 3000 ft., if you choose to include this question. It's based on today's Lesson 12-10.

16. 9 in. (No, not 4 in. 6 in. is the shorter dimension, not the longer.)

17. 2.6 m, to the nearest tenth. (No, not 1.5 m. 2 m is the height, not the length.)

18. 10 m. (No, not 40 m. 20 m is the height, not the length.)

19. $3.60. (No, not $2.50. $3 is for five pounds, not six.)

20. 32 in. (No, not 24.5 in. 28 in. is the width, not the diagonal. I had to change this question because HD TV's didn't exist when the U of Chicago text was written. My own TV is a 32 in. model!)

This is a traditionalist post. Let's look at Barry Garelick's latest post:

For those who are wondering what future math teachers learn in ed school, here is a concise summary:
Traditional mathematical teaching has never worked and has failed thousands of students.
The standard ed school catechism is that traditional math teaching is based on rote memorization with no understanding, and no connection between concepts.  Another is that the conceptual underpinning of math procedures are not explained. According to ed school teaching, procedures are presented as a “bag of tricks” (such as “keep, change, flip” for dividing fractions). The evidence presented is simply that many adults do not remember how to solve certain problems. This stands as proof that the traditional methods are not effective–if they were, they would “stay with us.”
Of course, Garelick disagrees with this "catechism." So far, this post has drawn five comments. Wow, SteveH must be losing his edge -- he wrote only one comment to this post. Before we look at SteveH's comment, let's look at the previous commenter. This commenter tells a Common Core horror story, and there's some confusion as to what he's describing. So he clarifies himself in a subsequent comment:

I think I have jumped over the actual test item.
The aim was to achieve the solution to an arithmetical 2 digit multiplication by the “area strategy”, and the student was expected to follow the strategy exactly, with a sequence of steps.
Marks: 1 to succeed totally, 0 for inadequate explanation,

Here is the relevant Common Core Standard:

Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

We've seen enough of these stories to know what's happening here. The fourth graders are taught some "area strategy" in order to multiply a pair of two-digit numbers. Let's say they are asked to multiply 98 by 39. I suspect the "area strategy" means that they're supposed to draw a rectangle whose dimensions are 98 and 39. The length is to be divided into 90 and 8, while the width is to divided into 30 and 9. These produces four rectangles whose area they find separately, and finally they add up the areas:

(98)(39) = (90 + 8)(30 + 9)
               = (90)(30) + (90)(9) + (8)(30) + (8)(9)
               = 2700 + 810 + 240 + 72
               = 3822

The author tells us the rubric here. He strongly implies that if students use any method other than the area strategy -- like the standard algorithm, of course -- then they receive a score of 0, even if the final answer is correct. And naturally, the standard algorithm isn't taught until fifth grade:

Fluently multiply multi-digit whole numbers using the standard algorithm.

We already know what the traditionalists would prefer -- just teach the standard algorithm in to fourth grade and skip the "area strategy" altogether. I'm sympathetic to this idea, but I still believe that some students learn multiplication more easily using the lattice method (the "rectangular arrays" mentioned in the standard?) than using the standard algorithm.

Then again, one solution that should suit both traditionalists and reformers alike is simply to ask for the product of 98 and 39, and award one point if the answer is correct, regardless of what algorithm is used to find the solution. Then fourth grade teachers might teach the lattice method and fifth grade teachers the standard algorithm, and students can choose which method they prefer without penalty (including fourth graders who know the standard algorithm).

OK, let's see what SteveH has to say in the comment thread:

That’s just plain bad teaching and you offer no description or indication of anything better. The traditional AP Calculus (or IB) track in high school is the only success path for proper STEM careers. Other “integrated” high school math options have lost the battle and are not what colleges are looking for.

Here we go again -- SteveH claims that colleges aren't looking for Integrated Math students. We know this is false, because colleges admit thousands of international students, all of whom (except for the Vietnamese) took Integrated Math in high school.

For example, MIT famously sent out its admission letters this week. Of course, data for this year's class isn't available yet, but here's a link to data for the current freshman class:

According to the link, 11% of the incoming class is international. Let's give SteveH the benefit of the doubt and assume that 1% of the class is from Vietnam (likely an overestimate). So this leaves 10% of the class having taken Integrated Math. And it goes without saying that many if not most MIT students are STEM majors. So even though Integrated Math isn't "what colleges are looking for," there surely are a lot of MIT admits who took Integrated Math. (Again, MIT admits typically have taken Calculus, but I'm talking about the classes they took in the years before Calculus.)

My son will be graduating from college in May and one of his degrees is in abstract math. He went through a proper AP Calculus track in high school and he now sits next to the best math students from around the world.

The best students from around the world have taken Integrated Math in their home countries.

They offer no understanding advantage (everything they do is proof-based) and they all went through “traditional” math programs. You can’t just throw out a pejorative complaint without backup. If you offer a process or pedagogy that works better than the traditional math curriculum track, I might be your biggest supporter. I won’t hold my breath.

If we're using results (international math scores, admissions to MIT) to measure, then Integrated Math clearly "works better than the traditional math curriculum track."

Returning to the original Garelick post, I notice that he writes about tracking. As usual, it's impossible to write about tracking without writing about race (but I'm protected by the "traditionalists" label):

Traditional math failed to adequately address the realities of educating a large, diverse, and rapidly changing population during decades of technological innovation and social upheaval.
This argument relies on the tracking argument, when many minority students (principally African Americans) were placed into lower level math classes in high school through courses such as business math. It goes something like this: “Most students did not go on in math beyond algebra, if that, and there were more than enough jobs that didn’t even require a high school diploma.  Few went to college.  Now most students must take advanced math, so opting out is not an option for them like it was for so many in the past.”
First, in light of the tracking of students which prevailed in the past, the traditional method could be said to have failed thousands of students because those students who were sorted into general and vocational tracks weren’t given the chance to take the higher level math classes in the first place — the instructional method had nothing to do with it.  Also, I don’t know that most students must take advanced math in order to enter the job market. And I don’t think that everyone needs to take Algebra 2 in order to be viable in the job market.
OK, notice that Garelick concedes that Algebra II isn't necessary for everyone -- as opposed to SteveH, for whom it seems to be AP Calculus or bust.

Indeed, Garelick proceeds to write about traditionalism not in Algebra, but in Pre-Algebra:

Secondly, while students only had to take two years of math to graduate, and algebra was not a requirement as it is now, many of today’s students entering high school are very weak with fractions, math facts and general problem solving techniques. Many are counting on their fingers to add and rely on calculators for the simplest of multiplication or division problems.  In the days of tracking and weaker graduation requirements, more students entering high school than now had mastery of math facts and procedures including fractions, decimals and percents.

I think about the three eighth graders I see in the morning on this, my third day of subbing for the digital film teacher. Since I already introduced race in this post, I might as well point out that the three students are white, black, and Asian. I help the black student with her Common Core 8 homework on the Pythagorean Theorem. She's confused when she reaches the questions that ask her to find a leg rather than the hypotenuse. Yet she does very well in performing all the arithmetic by hand, only needing a calculator to find the square roots. This includes the one question with decimals. This is a very traditional worksheet -- the Pythagorean Theorem is a topic that's impossible to teach any other way than traditionally, using the formula.

The white and Asian students are in Algebra I. Their assignment is on finding angle measures. One worksheet is more like a puzzle -- there's a single diagram with dozens of angles to find, using linear pairs, vertical angles, and parallel lines cut by a transversal. Both of them add and subtract the angle measures without a calculator. When I check their homework, I notice that their answers are slightly different -- some of them are off by three degrees. Unfortunately, I'm stumped as well -- the puzzle format makes it difficult to tell where the errors are made, and in fact, I wonder whether one of the angles given by the teacher is off by three degrees.

When the two of them come in later at lunch to work on their video assignment, I ask them what their math teacher said about the homework. They inform me that she showed them a worksheet with the correct answers, so apparently they made a mistake somewhere that I can't find. These puzzle problems are tricky -- an early student mistake ruins the entire worksheet. But I know that some traditionalists are fond of this type of worksheet -- this is the deeper thinking that they want to see, as opposed to "explain what method you used to get your answers."

All three students have strong arithmetic skills. I can't be sure about their ability to handle fractions, since there are no fractions on either worksheet.

OK, that's all I have to say about traditionalists. Here are today's worksheets -- one for Lesson 12-10 and the other for the Chapter 12 Test.