Tuesday, April 4, 2017

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

This is what Theoni Pappas writes on page 94 of her Magic of Mathematics:

"Artists over the centuries have used different mediums to create their works -- watercolor, oils, acrylic, chalk, etc. There are some artists who feel the computer is an artificial means that lacks freedom of spontaneous expression."

And yet, as we already know, computer art is becoming increasingly important in this digital world. I point out that on this page, Pappas shows us how even Leonardo da Vinci, if he had lived in the present day, could have used computers to create a sketch of a horse.

I was thinking earlier today about motivating students to learn math. If we ask what students who are uninterested in math are actually interested in, a common answer may be art -- of course I myself have observed students drawing during class. If I were teaching math to a class full of artsy students, it would be great to show them how math connects to art. And Pappas is the one whose book demonstrates how math is important in art.

The Illinois State text also believes that art is important in math. Part of the idea behind our learning centers is that one station should be used for arts projects using a die cut machine. Many of the die cut shapes we have are digits and other math symbols. This is tricky in middle school, since the text only directly provides arts projects for fifth grade and below. There are still arts projects for middle school, but they are more difficult to find. Illinois State fully supports the idea that students can use art to learn concepts in math better -- and so STEM should be replaced with "STEAM," where the extra A is for art.

In the U of Chicago text, we now begin Chapter 13, on Logic and Indirect Reasoning. Lesson 13-1 of the U of Chicago text is "The Logic of Making Conclusions."

For Chapter 12, it was easy to include worksheets from last year. But for Chapter 13, this is tricky as I made so many modifications to this chapter over the first two years of the blog. Recall that the first year, I combined Lessons 13-1 and 13-2. Then last year, I moved these lessons up to the start of the year with the justification that in most other Geometry texts, logic appears in Chapter 1 or 2, not 13.

This year I'm fed up of moving lessons and chapters around. And so I want to teach Chapter 13 in its original order in the U of Chicago text -- that is, after Chapter 12. Yes, compared to most other texts, this is late to teach logic, but that's the way it goes. But it also means that it will be difficult to post last year's worksheets (that were based on jumping and skipping chapters and lessons) now that I want to respect that U of Chicago order.

I decided that I will only post the first part of last year's Lesson 13-1/13-2 worksheet. I feel that I can get away with only posting a single side in light of all the extra worksheets that I posted yesterday.

No, this isn't to say that logic isn't important. On the contrary, it's significant as Geometry is the only high school class in which students learn about proofs and logic. It's getting the short shrift only because this is a time when I don't feel like creating new worksheets.

This is what I wrote last year about today's lesson (including the reasons for jumping to it last year):

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.

So here's the plan -- I will post both my 2-3 and my 13-1/13-2 from last year. Recall that I'm not leaving Chapter 13 intact this year (just as Chapter 3 is already broken up). Chapter 13 is best divided up, with Lessons 13-1 and 13-2 included here with the logic of Chapter 2, Lesson 13-5 included with the circles of Chapter 15, and Lesson 13-7 included with the inequalities that are spread throughout the U of Chicago text (including Lessons 1-9 and 7-8).

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.

2017 update: I often point out that the Metamath website used to post April Fool's Day proofs. Since it's still the first week in April, let me say that sadly, there was no April Fool's Day proof this year -- and there hasn't been such a joke proof since 2014.

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