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. They play an F blues scale -- F-Ab-Bb-B-C-Eb-F.

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.

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