Monday, July 15, 2013

My Mind Is Blown!!!

Today is the official start of my second week here at VSA. I started my morning with a 7:00 AM run. This time, there were only 5 of us (including the proctor). Usually we have 12+, but we speculated that the others either forgot or were too tired to get up before 7. After the run, I went to take a shower. Both showers in my local bathroom were full, so I went across the hall to another bathroom. This is the first time I've had this problem, but most people were up by the time we got back, so it makes sense. All the usual items were present at breakfast. 

Today's puzzler was about probability. Imagine that you are a gold miner in 1849. You walk out of a saloon, and a man in a suit approaches you, offering a game of chance. He has a bag with three cards. One is red on both sides, one is red on side and green on the other, and one is green on both sides. He tells you to inspect the cards, and then put them back in the bag. Then you are to draw one and up it on the table, not revealing the other side. You draw a card with a red front. The man bets you that the other side is a red. Should you take the bet? If you want to solve, STOP reading.

At first glance, the odds seem 50-50. We know it is not the GG card, so it is either the RR or the RG. But it's not 50-50. Think about the number of red faces. You could be seeing the front RR, the back of RR, or the front of RG. In 2 of the 3 cases, the con man wins, as red in on the other side. You should not take the bet. Next, Dawson changed it to be 2 RR, 3 RG and 1 GG. You should not take this bet either. There are 4 RR faces you could be looking at and 3 RG faces you could be looking at. Therefore, the con man wins 4 out of 7. Next Dawson asked us to change the amount of the various cards so that the game it fair. (50-50.) In order to do this, the number of RG cards must be double the number of RR cards. Possibilities we came up with included 1 RR, 2 RG & 1 GG; 2 RR, 4 RG, and 1 GG; and 3 RR, 6 RG, and 5 GG. The number of GG cards does not matter, as in every scenario, a red faced card is drawn. I initially though the game was fair (50-50), but later realized it is not fair. Just like with the lottery odds, these exercises teach us to calculate the odds, so you can make an informed decision. Informed decisions are almost always better than uninformed ones. 

After the Puzzler, we had Independent Study for 50 minutes. Remember those surveys we filled out on Friday? Well, Dawson read them. He told us that overall, we like the class, but we had mixed feelings about Independent Study. Some people wanted more of it, other wanted less. I enjoy independent study because it is a very important skill, and I like being able to do things on my own. You cannot learn everything from your teacher/professor. Don't get me wrong, I love Dawson's lectures, but Dawson will not be my teacher for ever, and there are certain things you must do on your own. I finished Section 3 during the 50 minutes, so tomorrow I'll move on to Section 4. After Section 4 there is another quiz.

After Independent Study, Dawson gave a extremely long lecture. (It lasted from 10:30 to 2:30, with an hour break for lunch.) He told us before hand that it would be the longest lecture he does, but that it is extremely important for this week's activities. I enjoyed every minute of the lecture on Number Theory, but the vast majority of the class did not. Number Theory is some intense stuff. We started with the basics including the definition and equations of even and odd numbers. We then moved on to proving what happens when you add and multiply numbers (even and even, even and odd, and odd and odd.) All of these can be generalized with equations, so we proved all of them. Next we proved that if you square any odd integer, the answer is a multiple of 8, plus one. (8k + 1, where k is an integer.) For example, 9 squared is 8 * 10 + 1 or 80 + 1, or 81. Next we talked about prime numbers, and the Sieve of Eratosthenes. The Sieve of Eratosthenes is just a method of finding prime numbers. It is in a grid format. We started with going up to 25, but afterwards went to 200. You find the first prime number (2). You cross circle it, and cross off all the multiples of 2 (4, 6, 8, 10 etc.). Then you go to the next prime number (3) and do the same. You know when you are done, when you have found all the prime numbers less than the square root of the last number of your grid (in these cases 25 or 200). Any uncircled numbers are prime.

Next Dawson told showed us the two most famous Number Theory problems: Goldbach's Conjecture and Fermat's Last Theorem. Goldbach's conjecture was created by a man named Goldbach (big surprise) back in the 1700s. This conjecture is still unproven to this day. It is very possible Goldbach had a proof, but he never published it. As of today, it was been proven up to 4 * 10 to the 18th power, by trial and error. (Well, no errors so far.) The other big problem has been proven (Fermat's Last Theorem.) Fermat wrote this theorem in the 1600s. The theorem says that a to the n + b to the n = c to the n, only when N is 2 or less. (Assuming that a, b, c and n are all integers.) He said he could prove it, but that it would not fit in the newspaper, so he did not publish it. It was finally proven in the 1990s, with math that had not existed in the 1600s. This 1990s proof was 150 pages long. Dawson then showed us a clip from a Simpson's episode that made fun of this theorem. The equation was 1782 to the 12 + 1841 to the 12 = 1922 to the 12. If you punch the left side of the equation in your calculator, you get the same answer as the right side. However, it you subtract 1922 to the 12 from the other side of the, you get -7*10 to the 29, instead of 0. The writers of the Simpsons very cleverly engineered the equation to look equal on a standard 10 digit calculator, but it is not actually true. After this we went to lunch.

Since it was Monday, they had lasagna, macaroni and cheese, green beans and salmon at the dining commons. I always feel we are treated like second class citizens on these days. Yes we get the "better food," but that's only because the prospective students are coming. We are also crammed up on the balcony. I've even heard tour guides apologize to their groups that we are here. Anyway, for the first time, I noticed that they also offered hamburgers and hot dogs (just like every other day) on Mondays and Fridays. Of course, they are pretty well hidden in an alcove near the exit of the regular line.

After just 90 minutes of lecture on Number Theory, I was intrigued and amazed. It's so complex, and lots of it remains unproven. I was ready to learn more.

After lunch, we continued the Number Theory lecture, by talking about congruences. I cannot explain them well, as I cannot use the symbols necessary in the blog. Congruences revolve around modulos. The basic idea is that a is congruent to b mod m. This means that (a - b) / m results in an integer. There are various properties of these equations including: the Reflexive Property, the Symmetric Property, and and the Transitive Property. Each of these help you rewrite and solve congruence equations. Every mod equation also has Equivalence Classes, that are based on every integer from 0 to one less than the mod number. As you might have guessed, these sets contain values that are equivalent in the congruence equation. Next we talked about finding b if you are given a and m.

The last topic we covered was Arithmetic, Multiplicative Functions. This refers to functions that have positive x-values, and are such that f(mn) = f(m)f(n). We experimented with various functions to find that functions can be completely multiplicative, multiplicative for certain values, or never multiplicative. Dawson then showed us three arithmetic multiplicative functions that are "special". These functions are used to calculate the number of integers < n that are relatively prime to n (relatively prime means the GCF is 1), the number of positive divisors of n, and the sum of the positive divisors of n. These functions are the Euler Phi function, the Nu function, and the Sigma function respectively. We went over the proofs for all these functions as well. All of this number theory is very interesting to me. I enjoy math, I always want to know why things are, and want to see the evidence/proof behind it.

After the lecture, Dawson introduced our next project. Instead of researching a mathematician, this time we have to prepare a 45 minute lesson to teach/review Algebra 1/Geometry topics. Dawson loves this assignment, and gives you 2-3 pages typed, single spaced of feedback afterwards. If you do not fill the 45 minutes, Dawson will start throwing problems at you to explain. To make it even more interesting, Dawson tells certain people to undermine your efforts to teach such asking stupid questions, talking throughout the class etc. You are out of the room when he assigns these roles, so you have no idea who or what to expect. Since my parents are elementary school teachers, I know that this will be a very realistic simulation. No matter what class you teach, there is always one person who tries to be the class clown, and/or some who does not care, and wants the teacher to fail.

If Dawson is bored, he also undermines your efforts, but if your lesson is going well, he make helpful suggestions. We have until Friday to finish preparing the lesson. Each day, two to three student lessons will take place. I chose to do my lesson on Mixture Problems, so I teach on Friday. For those who don't know, here's an example of mixture problem: A chemist has a 100 mL solution that is 50% acid. How much water must be added to dilute it down to a 25% solution? I have a pretty good advantage here, as most people have forgotten how to do these. Dawson also gave me an Algebra 1 textbook to base my lesson on. This book happens to be the same textbook I learn Algebra 1 from. I am really excited about this assignment. It combines my presentation and math skills, something I don't get to do often. It will also be fun to try to mess with other students while they try to teach. This assignment will also make us appreciate teachers more.

Tomorrow we are going to do activities involving Number Theory. It's going to be fun!!!

No comments:

Post a Comment