Tuesday, July 9, 2013

In the Matrix

This morning at 7 AM, Day 2 of my Vanderbilt adventure began. As is the norm, I went to breakfast at 8, and went to class (Special Topic in Math) at 9. Today's puzzler was much more math related. Here's the problem. A man is 75 miles from his destination. He is driving at 75 mph. If he decreases his speed my 1 mph for every mile traveled (such that a 74 miles to go he is going 74 mph, and a 73 miles to go he is at 73 mph), how long, to the nearest second, will it take him to reach his destination? For those of you who want to solve this, STOP reading, as the process and answer are about to be revealed. If you work backwards you find that it will take 1/1 hours to go the last mile, 1/2 hours to go the second-to-last mile, 1/3 hours to go the third to last mile etc. This means you can add up 1/1+1/2+1/3+1/4 all the way to 1/75 to get the total time. 

Next Dawson challenged us to find the answer to the sequence of adding numbers. He told us there was an easier way to add them together than to manually punch them in, but he let us try to figure it out. Eventually, he showed us the shortcut: to use Sigma (∑). Sigma has three parts the starting integer value that goes at the bottom, the ending value which goes at the top, and the equation which goes on the right. Sigma plugs in every whole number value in the range given, and plugs it in the equation. It them adds all the results, to give you the final answer. We plugged in 1 for the minimum value, 75 for the max, and 1/N for the equation. This gave us the amount of time in hours it took to get there. We then had to convert to minutes and seconds. 

Next he had us solve the same problem, but this time with the speed dropping by 3 mph every 3 miles. From the fractions we created (3/75, 3/72, 3/69, etc.), we found that the range for Sigma was 1 to 25. We then did the same problem again, but with speed dropping by half mph every half mile. This resulted in the same Sigma equation with a range of 150. After these experiments, Dawson confirmed out theory that you use the same Sigma equation for any question like this, and the range is just the number of time intervals.

After the Puzzler, we did independent study for an hour. It continued to be review for me, but the last question of section one took me a long time. We had to prove that a 14 in radius wheel going 35 mph does 420 revolutions per minute. I knew I had to change the units from miles per hour all the way to inches per minute, and divide by the circumference. The only problem was I was not getting 420 as my answer. After checking my work for 10 minutes, I asked Dawson for help. He told me it was a calculator issue. When I got down to inches per minute I put in my calculator "36945/28π, thinking that π was on the bottom. Apparently, the calculator reads this as 36945/28 times π. In order to get the π on the bottom, you have to use parenthesis: "36945/(28π)." I was very glad Dawson showed me this. Without this important information, I would not only get problems wrong in this class, but also in high school Pre-Calculus and Calculus classes, all because I did not know how to use my calculator correctly.

After independent study, we learned more about matrices. This time learned about the Elementary Row Elimination Rules, Gaussian Elimination, and how to solve a system of equations with a matrix. After solving systems of equations with a matrix, I can never go back to the old way, especially with there are three or more equations/unknowns. With the matrix, you can multiples of a row to another row, making it much easier than the standard elimination, where you can only combine equations and multiply the equation by any number.

After lunch, we did more independent study, and the 12th student in our class finally showed up. He is a rising junior, and was already taken AP Calculus. Next year he is taking AP Statistics. This truly amazed me. First of all, he was a sophomore in Calculus. Secondly, his high school offers AP Statistics. This means that enough people make it past AP Calculus at his school to make offering AP Statistics worthwhile. (Granted, there my only be one class, but that's still 30-40 people.) He must go to a school full of really intelligent people for that to happen. Anyway, after independent study we played a math game. Dawson divided the class into two teams. On the board, he and Emily would write the same matrix, and each team competed to finish first. The catches were only one person was allowed at the board at a time, each person could only complete one step, and we could not talk to the person at the board. The first three rounds went really well, as we won all of them. Then the tides turned, and the other team won the next two. These were all 3x3 matrices. For the final one, he had us do a 4x4 Matrix. For this one, he amended the rules so that we could all be at the board, and could all talk there. Although we spent 15 minutes on it, neither team got the correct answer. Both sides made small mathematical errors, and got off track multiple times, even though Dawson corrected us. Overall, the game was an excellent learning experience. It demonstrated/strengthened our ability to work together, and helped others get catch on. At the beginning, some people would only go up to the board if someone told them exactly what to do first. By the end, these people were making suggestions about how to solve the 4x4 matrix.

Next up was study hall, so we headed to the computer lab to work on our mathematician presentations. Today's lesson actually kind of foiled my plan. I was going to explain Gaussian Elimination, but now that Dawson taught it, I will have to explain other things Gauss developed like the heptadecagon construction with just a straightedge and compass. It's not that big of a deal: I just have to do a little more research.

Once again, I enjoyed Arête very much. Today we made a Columbus cube. Columbus cubes are regular cubes, but they have a triangular indentation on one of the corners. This allows them to stack on and balance on that corner (See the picture. The cube is on the flat corner, not a side) We also made ninja stars. Both of these projects are called "Modular Origami Projects," because you make many of the same piece, and then assemble them. (The cube is six pieces and the ninja star is eight.) The ninja star also converts into a disc-like object with a hole in the center. (See the picture on the right.)

As usual, we had dinner at six. This time I remembered to take my camera to the dining hall, so I took a picture of my dinner. They always have a wide variety of choices. As you can see from my plate, they have hot dogs, pasta, potatoes, chicken, and salad. Other options include burgers, veggie burgers, grilled chicken, fries, mozzarella sticks and fruit. The salad bar has every salad topping you can think of from shredded carrots to broccoli to cheese to beans to dressings. The variety is so much, that every meal, I get a slightly different salad.

My dinner.
After dinner, we had our first SOFT (Sign-out Free Time) Night. Keli'i, Kim, Loan and I went to CVS to buy various items and get quarters of the laundry machines. I needed an eraser for my math class. Once we headed back, we (as you might have already guessed) did laundry. The laundry room has seven washing machines, and 12 dryers. The dryers are are stacked two high on one wall, and the washers are unstacked on the other wall. In the middle, there is a table that seats eight. Although the laundry room is small, there is plenty of comfortable seating just outside the laundry room. Unfortunately, I forgot to grab my camera, so I do not have a picture. I'll be sure to take one next time though. Since we were all doing laundry, we took turns leaving to go take a shower. The laundry took a long time. Washing was only 27 minutes, by drying took 60 minutes. The other annoying factor was that a load of laundry costs $1.25 per load, ever though the Student Handbook says it's only a $1 per load. It's not that I think it costs too much, it's that they told us it was a dollar.

After doing laundry, I folded my clothes and went to my Proctor Meeting. Today's was very short. Hugh reminded us of the athletic services offered tomorrow morning (the gym, and a 3 mile run). He told us to sign out on his door if we leave in those groups. He handed out a schedule for the evenings this week. For the rest of the week, we have SOFT nights except for Friday when we have a dance. Hugh also told us to write down any song requests we have for the dance. 

Tomorrow we will move on from matrices, but I do not know yet what to. We will also have a logic puzzle as our Puzzler tomorrow. Read my blog tomorrow for more details!

No comments:

Post a Comment