Lesson 13-4: Indirect Proof (Day 134)

Today I subbed in a high school Physics class. This is in my first OC district, and today is the first of a two-day assignment.

I'm not sure whether I want to do "A Day in the Life" today. I often do it for multi-day assignments, but under the hybrid block schedule, I'll only see each student once during these two days. Thus it's more like a pair of one-day assignments than a single two-day.

In the end, I won't do "A Day in the Life" only because it's three straight Physics classes, so I'd only be describing the same class three times. As usual, I will say a little about the class.

The classes are in the unit on mechanical waves -- today, they are learning about sound waves. The students have a Google Slides project for the entire unit, so this time they work on the sound slide. They are supposed to watch some "videos" and then answer questions, but the "video" links are more like interactive webpages. One of them, for example, is to an online piano with visible sound waves, the Chrome Music Lab:

https://musiclab.chromeexperiments.com/Sound-Waves/

The relationship between sound waves and music is right up my alley. After all, the reason we even have EDL scales is that the cutting a string in half causes its frequency to double -- and harmonicity is all about frequencies being in specific ratios.

The song for today is "Count on It" from Square One TV -- in fact, I play it on the piano webpage. Maybe I should have chosen a simpler song such as one of the "Row Row Row" parodies or even "Benchmark Tests" so that the focus would be on the notes/frequencies/waves and not the lyrics. Then again, the message of the song is that "you can count on" seeing math and science applied to other fields -- and one of those fields is sound and music.

I sing the song in the key of A major. I want to sing it in the original key from Square One TV, but I think that's actually in the key of Bb, not A. Still, I emphasize the note A so that I can tie it to the main lesson -- the frequency of Concert A is 440 Hz, and the A an octave higher is 880 Hz. In fact, A and Bb are difficult keys for me to sing this song in -- my best vocal key for this song is probably E. But both A and Bb fit within the span of this online keyboard.

Today is Friday, the first day of the week on the Eleven Calendar:

Resolution #1: We are good at math. We just need to improve at other things.

I only consider this resolution indirectly -- through the "Count on It" song, I remind the students that they are good enough at math to succeed in both music and this science class.

Let's work out the Miller wager for these classes. Since this is Physics, most students are juniors with a few seniors, so I lose $2 for each in-person student:

3rd period (no 1st, since 1st = zero here): 13 in-person, 23 opt out, I lose $3.

5th period: 5 in-person, 19 opt out, I win $9.

7th period: 4 in-person, 24 opt out, I win $16.

So today is very profitable for my Miller wager. Even if Miller insisted on taking an extra $2 for each student who takes the four-day in-person option, that's eight students, so Miller takes back $16. This gives me a profit of $6. I expect to make more money for tomorrow's even periods.

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.

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 A, B, and C are noncollinear. Therefore the perpendicular bisectors m and n aren't parallel -- so that they actually exist.

Let me post the second side of the worksheet that I posted yesterday. This lesson naturally leads itself to activity, but today isn't an activity day. So I omit the extra activity worksheets from past years.