Tuesday, August 19, 2014

Section 2-2: "If-then" Statements (Day 9)

Section 2-2 of the U of Chicago text continues the study of logic by focusing on "if-then" statements. I certainly agree with the text when it writes:

"The small word 'if' is among the most important words in the language of logic and reasoning."

There are a few changes that I will make to the text. First of all, the text refers to the two parts of a conditional statement as the antecedent and the consequent -- although it does mention hypothesis and conclusion as acceptable alternatives. I'm going to follow what the majority of texts do and just use the words hypothesis and conclusion. Actually, Dr. Franklin Mason doesn't even use the word hypothesis -- he simply uses the word given -- since after all, the hypothesis of a theorem corresponds to the "given" statement in a two-column proof.

When I teach or tutor students in geometry, one of my favorite examples is "if a pencil is in my right hand, then it is yellow." So I pick up three yellow pencils, and we observe that the conditional is true. But let's suppose that I pick up a blue pencil in addition to the three yellow pencils. Now the conditional is false, since we can find a counterexample -- the blue pencil, since that's a pencil in my right hand yet isn't yellow.

Notice that I decided to replace the word instance with the word example -- so that the connection between examples and counterexamples becomes evident.

The text has to go back to an example from that dreaded algebra again. Of course, it's an important example, since students often forget that 9 has two square roots, 3 and -3. But I decided to include it anyway since it's simple -- it's not as if I'm making students use the quadratic formula or anything like that.

Then the book moves on to a famous mathematical statement: Goldbach's conjecture, named after the German mathematician Christian Goldbach who lived 300 years ago:

If n is an even number greater than 2, then there are always two primes whose sum is n.

At the time the book was written, the conjecture had been verified up to 100 million, but the conjecture had yet to be proved. But what about now -- has anyone proved Goldbach's conjecture yet? As it turns out, the answer is still no -- but now the conjecture has been verified up to four quintillion -- that is, the number 4 followed by 18 zeros.

But there has been work on a similar statement, called Goldbach's weak conjecture:

If n is an odd number greater than 5, then there are always three primes whose sum is n.

This is called weak because if the better-known (or strong) conjecture is true, the weak is automatically true because we can always let the third prime just be 3. Ironically, when Goldbach himself actually stated his conjecture, he stated the weak version of the conjecture. It was a letter from Euler -- you know, the same Euler who solved the bridge problem that we discussed as an Opening Activity -- that convinced Goldbach to state the strong conjecture instead.

Now as it turns out, someone has claimed a proof of Goldbach's weak conjecture -- namely the Peruvian mathematician Harald Helfgott. But Helfgott's proof is still being peer-reviewed -- that is, checked by other mathematicians to find out whether the proof is correct. Perhaps by this time next year, Helfgott's proof will have finally been verified (or dismissed as incorrect).

Dr. M also mentions Goldbach's conjecture, on a worksheet for his Section 2-1. Often students are fascinated when they hear about conjectures that take centuries to prove, such as Goldbach's conjecture or Fermat's Last Theorem. I often use these examples to motivate students to be persistent when trying to come up with proofs in geometry -- if mathematicians didn't give up even after centuries of trying to prove these conjectures, then why should they give up after minutes?

The final example in this section has students rewrite statements into if-then form. I've found that oftentimes, English learners struggle with this part of the lesson. The teacher must point out why, for example, the "something" in the example "all triangles have three sides" must be a figure: "if a figure is a triangle, then it has three sides." So not only must we appease algebra haters when we include algebra in the geometry lesson, but we must also consider English learners when including English in the geometry lesson.

Once again, I decided to include some review questions. Notice that the most of the review questions in this section are from yesterday's lesson, Section 2-1. We skipped Section 1-9 so I threw out the Triangle Inequality question. Once again, that question marked Previous course is an Algebra I question, and so once again, I rewrote it so that the solution is a whole number. Finally, I decided to avoid that inequality question completely.