Friday, April 7, 2017

Lesson 13-4: Indirect Proof (Day 134)

This is what Theoni Pappas writes on page 97 of her Magic of Mathematics -- a table of contents for Chapter 4, "The Magic of Numbers":

-- Discovering the Magic of Numbers
-- Quaternions & the Games Numbers Play
-- Cantor & Infinite Cardinal Numbers
-- Number Fantasies
-- What About Perfect Squares?
-- The Parable of Pi
-- The Prime Fascination
-- Cantor and the Uncountable Reals
-- Euclid's Proof of Unending Primes
-- Number Magic
-- Playing the Numbers

But it will be a while before I actually post about any of these topics. That's because the blog calendar follows the LAUSD, where spring break is the week before Easter. In addition, my charter school has an additional day off for students on Easter Monday. This means that after today, the students don't return to school until Tuesday, April 18th.

I dropped out of the "Day in the Life" challenge, but it's interesting to note that my monthly posting date, the 18th, would have been the first day after spring break. Also, notice that Tina Cardone, the leader of that challenge, declared the day before Thanksgiving break and the days before and after winter break to be special posting days, but not the days before or after spring break.

The existence of Cesar Chavez Day on March 31st and the fact that spring break is the week before Easter conspire to make the Big March less severe in LAUSD than in other districts. Next year, Easter falls on April 1st. This means that Cesar Chavez Day will be observed on April 2nd, thereby extending spring break to Easter Monday not only at my charter school but in the entire LAUSD.

Oh, and before I leave the topic of calendars, let me add one more thing about President's Day. I wrote earlier that in some states, such as New York, President's Day extends into a weeklong break -- President's Day vacation or February break. But I didn't think that any school here in California would observe the February break.

Well, I think I finally found a California school with a February break. Here's a link to the poster Right on the Left Coast -- "left coast" means California, "right" means right wing or political conservatism (but let's not get into politics today -- save it for next week):

Darren, the author of this blog, is a Northern California high school math teacher. He never actually states that his school was closed for a week. But notice that he apparently took a trip to Iceland during the exact week that New York schools were closed. The conclusion is obvious -- either his California school has a February break, or he decided to leave his class with a sub while he took a vacation across the Atlantic.

By the way, New York schools don't simply take either the week before Easter or the week after the holiday as spring break. Instead, New York schools close for the Jewish Passover (in addition to the High Holidays, which are also observed by LAUSD). This year, Passover begins on Tuesday, April 11th, and it lasts until the 18th. Naturally, no one wants a one-day week on Monday the 10th, so that date is also included in spring break. Therefore today is also the last day for New York students, and they'll return to school one day later than my charter students. It also means that New York's spring break is one day longer than the district's winter break -- which, as you recall, didn't begin until Christmas Eve and lasted until the day after New Year's.

OK, that's enough about the calendar -- let's finally get into today's lesson. Lesson 13-4 of the U of Chicago text is on indirect proof. Just as with 13-3 yesterday, 13-4 is another lesson for which I created a worksheet two years ago, yet I never actually covered it last year.

Indeed, last year I tantalizingly mentioned indirect proofs several times throughout the year -- especially when referring to proofs that are more difficult than U of Chicago proofs. Yet at no point during the 2015-16 school year did I ever title a post "Lesson 13-4: Indirect Proof" and declare it to be that day's lesson -- or even combine it with another lesson.

This is what I wrote two years ago about indirect proof:

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 2017, 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. For those of you who are expecting an Easter activity on the last day before spring break, sorry, but these logic problems fit today better.

My next school day post will be on Tuesday, April 18th. As usual, I will make one or two posts during the spring break.

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