Thursday, February 12, 2015

Review for Chapter 13 Test (Day 111)

Let's begin reviewing for the Chapter 13 test. As usual, here I will discuss the rationale for including the questions that I included.

The last unit had two major sources -- U of Chicago's Chapter 13 (mostly on logic) and Glencoe's Chapter 5 (on inequalities in triangles). The connection between the two is that the indirect proofs of U of Chicago's Section 13-4 are used to prove some of the inequalities in triangles.

My test ended up including most of the logic from Chapter 13, and consequently not as much of the inequalities from Chapter 5. The first question is based on yesterday's lesson. I have separated it into an (a) and (b) part. The (a) part simply asks for the exterior angle of a regular hexagon, while the (b) part asks the student to write a Logo program to draw a regular hexagon using that value of the exterior angle. This way, if the teacher doesn't want to do Logo, the (b) part can be skipped, and only the (a) part will be required.

Question 2 is a simple question on converse, inverse, and contrapositive.

Questions 3-6 ask the students to make conclusions based on logic. I was going to ask the students to name the logical rules that they used (Law of Detachment, etc.), but I decided against it.

Question 7 was inspired by something that I once saw in another text (not Glencoe), for a different student I was tutoring in geometry at least a year ago. I believe that I alluded to this question back when I was in Chapter 2. The students must identify the next three terms of the sequence. One of them is mathematical and straightforward. The second one was a fun one for my student to figure out the answer -- it's the first names of the presidents in order:

George (Washington), John (Adams), Thomas (Jefferson), James (Madison), ...?

and so the next three answers will be the next three presidents:

James (Monroe), John (Quincy Adams), Andrew (Jackson)

I've seen this question modified so that it gives the order of the presidents that appear on money:

George (Washington), Thomas (Jefferson), Abe (Lincoln), Alexander (Hamilton), ...?

with the answer:

Andrew (Jackson), Ulysses (Grant), Benjamin (Franklin)

And yet another variation gives the first ladies:

Martha (Washington), Abigail (Adams), Martha (Jefferson), Dolley (Madison), ...?

with the answer:

Elizabeth (Monroe), Louisa (Adams), Rachel (Jackson)

Once again, with President's Day coming up, I couldn't resist including this question. Don't worry -- the example on the actual test will probably be numeric, since it's unfair to expect the students to know the order of the presidents on a math test.

Question 8 is a logic problem. This one comes directly from the SPUR section of U of Chicago. No, I won't include any of Fireball's so-called "easy" logic problems, as these would not be appropriate for a math test.

Question 9 is on tangents to circles. By the way, even though I had to squeeze in Section 13-5 right in between the indirect proof and inequalities lessons, there is a benefit to including this lesson. I'm expecting that by the time we finally reach Chapter 15 of U of Chicago (on circles), we'll be rushing in order to finish it before the PARCC and SBAC exams. I'm not sure how much of Chapter 15 might appear on the PARCC or SBAC, but at least one topic that's likely to appear -- the fact that tangents to circles are perpendicular to their corresponding radii -- has already appeared right now.

Question 10 is an indirect proof. I would've included the U of Chicago indirect proofs, except that I got tired of the "prove that the square root of 9800 isn't 99" questions. I did notice that one of the questions in the U of Chicago was "prove that the square root of 2 isn't 577/408." In some ways, this sort of question can be said to lead up to one of the most famous indirect proofs -- namely that the square root of 2 is irrational. Here is a link to a common indirect proof that sqrt(2) is irrational:

Neither the U of Chicago nor Glencoe gives the proof outright. But both hint at it -- I just mentioned the U of Chicago's square root proofs. The Glencoe text asks the students to prove that if the square of a number is even, then it is divisible by four. As we can see at the above link, this fact is directly mentioned in the irrationality proof.

I remember once reading the proof of the irrationality of sqrt(2) in my textbook back when I was an Algebra I student. Until then, I had always heard that sqrt(2) was irrational, but I never realized that it was something that could be proved. So I was fascinated by the proof. Naturally, the text only included this as an extra page between the main sections, so it was something that the teacher skipped and most students probably ignored.

The irrationality of sqrt(2) has an interesting history. It goes back to Pythagoras -- he was one of the first mathematicians to use sqrt(2), since his famous Theorem could be used to show that the diagonal of a square has length sqrt(2). The website Cut the Knot, which has many proofs of the Pythagorean Theorem, also contains many proofs of the irrationality of sqrt(2):

Now there is a famous story regarding sqrt(2) and Pythagoras. At the following link, we see that Pythagoras was the leader of a secret society, or Brotherhood:

Now Pythagoras and his followers believed that only natural numbers were truly numbers. Not even fractions were considered to be numbers, but simply the ratios of numbers -- numberhood itself was reserved only for the natural numbers. In some ways, this attitude resembles that of algebra students today -- when the solution of an equation is a fraction, they often don't consider it to be a real answer, even though modern mathematics considers fractions to be numbers. (The phrases real number and imaginary number reflect a similar attitude about 2000 years after Pythagoras -- that some numbers aren't really numbers.) So of course, the idea that there were "numbers" that weren't the ratio of natural numbers at all was just unthinkable.

Pythagoras and his followers must have spent years searching for the correct fraction whose square is 2, but to no avail. Finally, one of his followers, Hippasus, discovered the reason that they were having such bad luck finding the correct fraction -- because there is no such fraction! And, as the story goes, Pythagoras was so distraught, afraid that the secret that sqrt(2) was irrational would be revealed, that he ordered to have poor Hippasus drowned at sea!

But as I said, nowadays students simply complain when they have a fractional, or worse irrational, answer to a problem. No one has to drown any more just because of irrational numbers.

All of this, while interesting, has nothing to do with my test review, since I decided not to put any indirect proofs about square roots on the test. Instead, I decided to write a more geometric indirect proof, based on the Glencoe text. Indeed, my plan is to include the actual problem from the Glencoe text -- you know, the one I mentioned yesterday where Glencoe made an error -- and have the students indirectly prove Glencoe's error!

Question 10 on my test review, therefore, is actually the final step of that proof, since that's the step where the contradiction occurs. They are given a triangle with sides of length 3 and 8, and two angles each 40 degrees (one of which is opposite the side of length 3). The students are to use the Converse of the Isosceles Triangle Theorem to show that the missing side must also be of length 3, and then the Triangle Inequality to show that 3 + 3 must be greater than 8, a contradiction.

When I wrote this problem, I had trouble deciding how difficult I wanted my indirect proof to be. For example, I considered giving 100 as the measure of the angle opposite the side of length 8, and give only one 40-degree angle instead. Then the students would have to use the Triangle Angle-Sum Theorem to find the missing angle as 40 degrees before applying the Isosceles Converse.

Or, to go even further, we can derive a contradiction without making the angle isosceles at all. For example, we could make the angle opposite the 8 side to be, say, 90 degrees instead of 100. Then the missing angle would be 50 instead of 40. If the triangle is drawn so that 50 degrees is opposite the 3 side, then by the Unequal Angles Theorem, the missing side would be less than 3, so the sum of the two legs would still be less than the longest side.

But this might confuse the students even more -- especially if the 90-degree angle is marked with a box (to indicate right angle) rather than "90." A right triangle might lead a student to use the Pythagorean Theorem to find the missing leg. Although this still eventually leads to contradiction -- the missing side would be sqrt(55), which isn't less than 3 -- that irrational side length might still cause some students to drown.

Besides, the question on the actual test is itself somewhat difficult -- after all, if it could confuse the Glencoe authors, it will confuse some students. Having a review question that leads to a discussion of the Pythagorean Theorem and then a test question with no right triangles and completely different theorems (Isoceles Triangle and its converse, Triangle Inequality) required will only frustrate the students who are taking the test.

And so I wrote my Question 10 on the review so that it will actually help the students prepare for the corresponding question on the test. I balance out this tough question with some easier questions about logic (converse, inverse, etc.). Hopefully the test won't be too hard for the students.

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