Minature Football Field

A little over a year ago I was at the Football Hall of Fame in Canton, Ohio. In a somewhat related note, Canton is also the home and burial place of President William McKinley. The inside of the hall was awesome. But outside had a miniature artificial turf football field!! I think it was 40 yards long. For the most part, my use of my lawn consists of watering it and mowing it. I thought how awesome it would be to turn it into something like that. I figured that could be kind of expensive. I looked online and there was and ad for 10x10 feet of artificial turf for $95. Maybe its doable after all. At the very least, it makes for a cool math problem.

Suppose I want a field of 30 yards. If you are just messing around with folks you don't want to be running 100 yards to score. I don't know if my yard is 30 yards long, but let's say it is. Now, how wide should it be? The real deal is 53 1/3 yards. So mine should be the solution to

100 : 53 1/3 = 30 : x

It actually works out cleaner solving it with fractions. Anyway, x = 16 yards.

For the area, I have 30x16 = 480 square yards. However, I need this in square feet. There are 9 square feet in a yard, so 480x9 = 4,320 square feet. This could get a little expensive.

The cost, y would be found with 100 : 95 = 4,320 : y

So, y = $4,104. Also, I would have to somehow get yard markers, etc painted it. Still, it might be worth it.

Below is a picture of the field in Canton. (Hard to get a good picture from ground level.)

Ryan Braun and Steroids

Ryan Braun is an outfielder for the Milwaukee Brewers. He won the MVP award in 2011. At the end of that season he was accused by Major League Baseball of taking steroids. However, he got off on a technicality. He was accused again in 2013. This one stuck and he was suspended for the rest of the season. That time he admitted it and took his punishment. He has played almost two full seasons since his suspension.

What is interesting in Braun's case is that he is still in the prime of his career. He is 31 years old. Many that have been accused of taking steroids were near the end of their career. If they did come back from a suspension, a decrease in their statistics could be because they are no longer using steroids or just because of father time. A decrease in Braun's statistics would seemingly be due only to him now playing clean.

To compare his statistics pre and post-suspension would be an interesting exercise. It wouldn't be helpful to look at the totals as he played almost seven seasons before the suspension and only two seasons after. However, you could translate those time periods into single 600 at-bat seasons. That is what I did. You can do so by looking at the grand totals and setting up proportions. Also helpful in this exercise is knowing that the definition of batting average is the number of hits divided by the at-bats.

Algebra students would have plenty of opportunity here to review proportions. Let me just skip the messy stuff and go right to the final stats.

Pre-Suspension Statistics:
At-bats 600, Runs 104, Hits 187, Doubles 38, Home Runs 34, RBIs 110, Batting Average .312

Post-Suspension Statistics:
At-bats 600, Runs 90, Hits 166, Doubles 33, Home Runs 26, RBIs 96, Batting Average .277

You could then ask students what they make of these statistics. Some, perhaps with some leading by you, might suggest looking at the percentage decrease. This turns out to be quite interesting. You can easily make the claim that a player is 86 to 87% as effective without using steroids. At least that seems to be the case with Braun in pretty much every area. I've compare post to pre-suspension statistics and changed them into percentages. Check this out:

Runs 87%, Hits 89%, Doubles 87%, Home Runs 76%, RBIs 87%, Batting Average 89%. Its kind of surprising these numbers are all in the same ballpark, so to speak.

Anyway this might make a good review of proportions, takes a topic they've all heard about, and gets students to do some statistical analysis.

Smartest Presidents

I saw an article online which listed the IQ's of each of our presidents. By their own admission, they were doing a bit of guesswork. Since IQ tests weren't developed until about the time of our 26th president, Teddy Roosevelt, there isn't a lot of hard data that can be used. I would think we would take these numbers with a grain of salt. I have some disagreements with a few of these placements and you probably do, too. In spite of that, here we go:

The Top 5 and their estimated IQ scores:

1. John Quincy Adams - 168.8
2. Thomas Jefferson - 153.8
3. John Kennedy - 150.65
4. Bill Clinton - 148.8
5. Woodrow Wilson - 145.1

And the bottom 5:

Andrew Johnson - 125.7
George W. Bush - 124.9
Warren G. Harding - 124.3
James Monroe - 124.1
Ulysses S. Grant - 120.0

We might well ask just how smart these guys really are. Since the mean IQ score is taken to be 100, just like Lake Wobegon, they are all above average. So, President Grant was above average. But was he just a little above or way above?

We could get an idea from looking at how many standard deviations away from the mean he is. Taking the standard deviation for IQ scores to be 16, we see that Grant is 20/16 = 1.25 standard deviations above the mean. Consulting a Z-score table, that puts him in the 89.4 percentile. Pretty good. Even if Ulysses wasn't the sharpest guy ever, it must take a certain amount of intelligence to get elected president twice and to win a war.

What about John Quincy? J.Q. is literally off the chart. Let's go with the runner-up Thomas Jefferson. He is 53.8/16 = 3.36 standard deviations from the mean. This puts him in the upper 99.96 percentile. He's smart. Not John Quincy Adams smart, but smart.

The complete list can be found at http://us-presidents.insidegov.com/stories

I Think, Therefore I Am

Rene Descartes was an interesting and important person. He was a soldier, philosopher, scientist and mathematician.

A big part of mathematical logic is getting the correct set of axioms to begin with. Euclid's issues in doing this led to non-Euclidean geometries being developed a couple thousand years later. It might be interesting to have students in a geometry class develope (and critique) a set of axioms just as Descartes did. He struggled to come up with what anyone absolutely knew about our world. He came up with the fact that he knew his own existence - "I think, therefore I am." That sentence, in isolation, is usually all we hear. I came across his quote in his writings. It is contained in Discourse on the Method of Rightly Conducting the Reason and Seeking for Truth in the Sciences which is often shortened to Discourse on the Method.

What is the context for his famous line? In Part IV Descartes begins with, "I do not know that I ought to tell you of the first meditations there made by me, for they are so metaphysical and so unusual that they may perhaps not be acceptable to everyone." Amen to that.

Halfway through the next paragraph he states, "Thus, because our senses sometimes deceive us, I wished to suppose that nothing is just as they cause us to imagine it to be; and because there are men who deceive themselves in their reasoning and fall into paralogisms, even concerning the simplest matters of geometry, and judging that I was as subject to error as was any other, I rejected as false all the reasons formerly accepted by me as demonstrations. And since all the same thoughts and conceptions which we have while awake may also come to us in sleep, without any of them being at that time true, I resolved to assume that everything that ever entered into my mind was no more true than the illusions of my dreams. But immediately afterwards I noticed that whilst I thus wished to think all things false, it was absolutely essential that the "I" who thought this should be somewhat, and remarking that this truth "I think , therefore I am" was so certain and so assured that all the most extravagant suppositions brought forward by the skeptics were incapable of shaking it, I came to the conclusion that I could receive it without scruple as the first principle of the Philosophy for which I was seeking.

Y2K Answers

This is Part 2 of a two-parter. You can read the previous post to review, but in a nutshell:

A recent article stated computers could go nuts on January 19, 2038. This date is supposedly 2,147,483,647 seconds after January 1, 1970. That is with a 32 bit system. However, a 64 bit system would last a lot longer - 292 billion years.

It's fact checking time:

1. Where did 2,147,483,647 come from? I thought that might be the value of 2^32. It wasn't, but it seemed to be about twice as much as the number. It was in fact exactly 2^31. Frankly, I'm not sure what that worked. I'm no computer whiz, but I guess computers could handle up to, but not including 2^32. Next highest would be 2^31. (?)

2. Where does that many seconds get you? There are approximately 365.25 x 24 x 60 x 60 = 31,557,600 seconds in a year. Dividing those two numbers, I get 68.05. That would be pretty darn close to the time between Jan 1, 1970 and Jan 19, 2038.

3. 292 billion years? Really? Yes, in fact. Using my new-found logic I related previously, I found 2^63. It's big. Divide by the aforementioned 31,557,600 to get 2.92271 x 10^11. This is 292 billion years.