Math Occupations

I copied this from the University of Northern British Columbia website. I'm claiming ignorance as to copy write laws, but I'm giving them credit, so i think I'm fine. This seems like a good list. I edited this down just a bit. The full page is at http://www.unbc.ca/math-statistics/careers-mathematics

Careers with a Mathematics Degree

Aside from being useful to becoming a High School teacher, a degree in Mathematics is quite useful in the following job areas (where people with a Mathematics degree have been hired).

1. It prepares you for an MBA (Masters of Business Administration) degree and become, for example, an Accounting Manager or a Certified Public Accountant.
2. Can work as a computer scientist at the National Institute of Standards and Technology.
3. Can work as a mathematician in places like Rockwell International Corporation.
4. With a degree in mathematics you can go to Law School.
5. Can work as a project manager in a company like Hewlett-Packard.
6. You can go on to an engineering school and become an industrial engineer or work in companies like Westinghouse Wireless Solutions.
7. Informations Systems Consultant.
8. Manager at a place like Advanced Research Computing Services.
9. Can work as an actuary for a life insurance company.
10. Senior software engineer (e.g. at Harris Scientific Calculations).
11. With a MSc in Mathematics one can work as an aerospace mathematician in space centers like the NASA Goddard Space Flight Center.
12. Can be a mathematics editor for a publisher (e.g. Simon & Schuster, Inc.).
13. A computer systems specialist for a chemical company.
14. A professional relations representative for a health care company.
15. Educational markets manager for companies like Texas Instruments.
16. Operations research analyst (e.g. for FedEx or any courier company).
17. Director for inventory control.
18. Can work in companies like Exxon Production as a research specialist.
19. A statistician at a health research laboratory.
20. Can work at a Boeing company as instructor and/or consultant on quality control.
21. An environmental mathematician for an engineering company.
22. With a double major with economics you can become an economist for an oceanographic company.
23. Some national laboratories hire mathematicians to analyze problems numerically (Numerical analysis).
24. A statistician in a census bureau.
25. Telephone companies are known to hire mathematicians, even ones with only a BSc degree.
26. Financial analyst.

Rating Baseball Players

I stumbled onto something called Elo Rater. It is a way of rating former or current baseball players if they were to face off in a head to head match up. It was developed by Arpad Elo. He was born in what was at that time Austria-Hungary in 1903 and passed away in 1992. Arpad was a physics professor at Marquette University and was an avid chess player. He developed his rating system originally to rank chess players.

Frankly, I don't completely understand every bit of this. There are original point values for the players. I'm not sure how those are determined. And I don't know who decides how it is determined that one player goes against another. It looks like maybe people can go to the website and pick a couple players and they play each other. Since he was a college professor, I'm going to assume he knew what he was doing. Also his ratings seem pretty accurate. His top five hitters of all time are:

1. Babe Ruth
2. Stan Musial
3. Ty Cobb
4. Lou Gehrig
5. Mike Schmidt

You can check out his full lists at http://www.baseball-reference.com/friv/elo.cgi

Here is an example used on the site.

RA is the rating for Player A and RB is the rating for Player B. Working out the probability that Player B wins where RA =  2450 and RB = 2500:

P(B wins) = 1 / (1 + 10^((RA - RB) / 400)) 

= 1 / (1 + 10^((-50) / 400)) 

= 1 / (1 + 10^(-0.125)) = 

= 0.571

To analyse how these come out means looking at fraction exponents and negative exponents. You can find the details of the process at http://www.baseball-reference.com/about/elo.shtml

Hawking's Imaginaries

Stephen Hawking wrote the bestselling book, "A Brief history of Time. In it he spoke of how imaginary numbers are used in relativity theory. It's not totally satisfying as a high school mathematics application. To do it justice mathematically would probably make it incomprehensible. And to tone it down is to miss the application.

Since there aren't a lot of imaginary number applications to share in a high school class, I thought I might quote from the book. At least students can see that there is a reason for their existence.

"We don't yet have a complete and consistent theory that combines quantum mechanics and gravity. However, we are fairly certain of some features that such a unified theory should have. One is that it should incorporate [physicist Richard] Feynman's proposal to formulate quantum theory in terms of a sum over histories. In this approach, a particle does not have just a single history, as it would in a classical theory. Instead, it is supposed to follow every possible path in space-time, and with each of these histories there are associated a couple of numbers, one representing the size of a wave and the other representing its position in the cycle (its phase). The probability that the particle, say passes through some particular point is found by adding up the waves associated with every possible history that passes through that point. When one actually tries to perform these sums, however, one runs into severe technical problems. The only way around these is the following peculiar prescription: one must add the waves for particle histories that are not in the"real" time that you and I experience but take place in what is called imaginary time... (For those that don't know there is an interlude of a brief and undoubtedly insufficient explanation of what imaginary numbers are) ... To avoid Feynman's sum over histories, one must use imaginary time. That is to say, for the purposes of the calculation one must measure time using imaginary numbers, rather than real one. This has an interesting effect on space-time: the distinction between time and space disappears completely. A space-time in which events have imaginary values of the time coordinate is said to be Euclidean...In Euclidean space-time there is no difference between the time direction and directions in space. On the other hand, in real space-time, in which events are labeled by ordinary, real values of the time coordinate, it is easy to tell the difference - the time direction at all points lies within the light cone, and space directions lie outside. In any case, as far as every day quantum mechanics is concerned, we may regard our use of imaginary time and Euclidean space-time as merely a mathematical device (or trick) to calculate answers about real space-time."

Area of Colorado II

A blog from a couple weeks ago has bothered me. If we can learn from our mistakes, then I have an opportunity here.

I thought I would try finding the area of Colorado. It is a rectangle. That was mistake #1. It is bordered by lines of longitude of 102W and 109W. That is a difference of 7 degrees. but distances between lines of longitude don't stay the same. The distance between them narrows as you move from the equator going toward the poles where they all meet. So Colorado is really more of a trapezoid. And there we have mistake #2. It really isn't a trapezoid since it is on the surface of a sphere. So there are non-Euclidean aspects to deal with. I won't be dealing with that as I'm having enough problems with this. I'll stick with Euclid.

In mistakes #3 and counting, I found that it either can't be done, or I can't figure out how to get those horizontal boundary distances. I resorted to a website where you can plug in latitudes and longitudes and it will compute the distance using something called the haversine formula.

Long story short, doing so told me the south boundary is 386.2 miles and the north boundary is 363.9 miles. A couple weeks ago I had correctly computed the height as lines of latitude are parallel and thus stay the same distance apart. That value was 276.4 miles. Using the formula for area of a trapezoid, I got 103,056 square miles. The internet says it is 103,718 square miles. I'm calling that a win.

I'm not sure how they actually figure the areas of states. Maybe its just an estimate. Online I found at least three different values for the area of the state on sites that seemed to be fairly credible. So, my guess is as good as theirs.

Amount of Daylight

We come to the time of the year when there is the minimum amount of daylight. That is depressing, but it's balanced by the fact that it is the holiday season. A trig equation could be written that would show the amount of daylight for each day of the year. It can be done without a whole lot of information.

Sounds like fun. Here we go.

We need to come up with A, k, and c for the equation y = Asin(kx+c).

Some basic information is that there are 365 days in a year. The least amount of daylight is on the first day of Winter - around December 21. The most is around June 21. It will be even amounts on the equinox dates - March 21 and September 21. 

With this info, we can examine amplitude, period, and phase shift. 

Amplitude - Let's assume we get three extra hours on the first day of summer and three fewer hours on the first day of winter. So A = 3. That was easy.

Period - Since Period = (2pi)/k and the daylight cycle is 365 days, so 365 = (2pi)/k. Therefore k = (2pi)/365 = 0.0172

Phase Shift - It would have been convenient if the spring equinox fell on January first. There would have been no phase shift. Instead, it falls about 80 days later. Phase shift = -c/k, So 80 = -c/0.0172. We get c = -1.376.

Our equation can be written y = 3sin(0.0172x-1.376). The x stands for the number day of the year, such as for January 12, x = 12. For February 1, x = 32. The y stands for the amount of extra or less sunlight. I tried it out for a few values and it seems to work. On December 21 we have three fewer hours. On January 1, we have 2.94 fewer hours - a slight improvement. 

We could change the formula so it could stand for the full amount of daylight by tacking on a +12 to the end:  y = 3sin(0.0172x-1.376)+12

Chance of Being Undefeated

They said that now, at the midpoint of the NFL season, there are still three undefeated teams. That is the most there have ever been at this point in the season. What is the chance we have at least one of those teams staying undefeated for the whole season?

There are seven games to go for each team. First we would want to know what the chance of winning a single game would be. I'm going to go with an 80% chance. That sounds about right. They're undefeated at this point, so obviously pretty good, yet they wouldn't be invincible.

The chance of any one team of winning their next seven games is (0.8)^7 = 0.21. Now what is the probability of New England, or Cincinnati, or Carolina remaining unbeaten? The easiest way to deal with a three team "or" problem is to look at it negatively. There is a 1-0.21 = 0.790 chance of a team having at least one loss the rest of the way. The chance all three are defeated is 0.493. So, there must be a 1-0.493 = 0.507 of that not being the case. So chances are at least one team will be undefeated - a 50.7% chance.

It could be done the straight forward way, but its a little more cumbersome.

0.21+0.21+0.21-(0.21)(0.21)-(0.21)(0.21)-(0.21)(0.21)(0.21)+(0.21)(0.21)(0.21) = 0.507 = 50.7%

For full disclosure, since I thought about it yesterday, Cincinnati lost on Monday Night Football. I was just giving them the win. I probably shouldn't have done that.

Now we're down to two teams. There is still a (0.8)^7 = 0.21 chance of any one team winning seven in a row. The chance that either Carolina or New England does it is:

(0.21)+(0.21)-(0.21)(0.21) = 0.347 = 34.7%

If you take exception to my 80% single win assumption, it's easy to substitute another value. This makes a bigger difference in the final probabilities than I would have thought. For example, if you figure the chance of a team of this caliber wins a single game is 85%, the chance one of the remains unbeaten is now 53.9%. If 90%, the chance of having an unbeaten is all the way up to 73.8% .

Area of Colorado

I thought it might be interesting to try to compute the area of Colorado by using arc lengths - just to see if it comes out right. It doesn't. At least not how I did it. Maybe its of use to someone that can learn from my mistake(s).

One troubling thing is that I looked on the internet and got three different values for the area of the state. I would think in the age of GPS we would have that figured out to a pretty precise amount. The three amounts were separated by 94 square miles. At least that gives me some wiggle room.

I figured I could find both the length and width of the state would be with:

                                          (arc length)/360(2pi(radius of the earth))

The radius of the earth is 3,959 miles. The arc length I figured would be the differences in the latitudes or the longitudes. The height came out great. I got 276.4, and the web says 276. The width wasn't even close: my 483.7 to their 387. Mine was too big. Arc length would include the curvature of the earth. I thought maybe I could use Law of Cosines to get a closer figure.

                                C^2 = (3959)^2+(3959)^2-2(3959)(3959)Cos(7)

It was interesting that I got 483.4 as compared to an arclength of 483.7. What was not interesting was the web says the length of Colorado is 387.

What I learned:

 1.  I noticed their published lengths and widths didn't multiply to get their published area. It was 104,091 to 106,812 square miles. So, they must not use length x width to get the area. That makes sense, because that would only work on rectangular states of which there are not many.

2. Why my method didn't work. I knew this but didn't think about it - lines of longitude are not the same distance apart. They are a certain distance apart at the equator and shrink to nothing at the poles. I guess that is why my north-south distance came out accurately but my east-west was way off.

So, I don't know how they find the area of a state or any region for that matter.
There's a good project for you or your students.

The Law of Large Numbers and MVP's

As you do repeated trials, the mean average of those trials will approach the theoretical mean. Or if you are looking at a sample, as your sample grows, its mean average will get closer to the mean of the entire population.

The law of large numbers is a good title for this. Many have the idea that the law of probability would imply that a .250 hitter that goes 0 for 3 is now "due" for a hit. Flipping a coin 10 times means you'll get 5 heads and 5 tails. For most of us, our own life experience would show statements like these to be incorrect. It would not be weird for a coin being flipped 10 times to have 3 heads and 7 tails. However, we would think something was up if we flipped it 1,000 times and got 300 heads and 700 tails.

Like many math teachers, I would have classes do some coin flipping experiments. Always a fun day. For years I would write down the results and keep a running total. I don't know where that is now. I wish I had kept that. I was up to something like 20,000 flips. It wasn't 50-50, but pretty close. Maybe something like 49.7% to 50.3%.

I'm reminded of this in something I read in "The Signal and the Noise: Why So Many Predictions Fail - But Some Don't" by Nate Silver. It's an interesting book. One example: At what is a major league baseball player at his peak? You can make a pretty good case for it being 27 years of age. To make his case, he looked at 50 MVP award winners. Granted, 50 is possibly not to be considered a "large number", but it's what was used in this case.

"A baseball player...peaks at age twenty-seven. Of the fifty MVP winners between 1985 and 2009, 60 percent were between the ages of twenty-five and twenty-nine, and 20 percent were aged twenty-seven exactly."

This doesn't prove anything for sure, but then again, surveys never do. I would think we would have a better idea as we can look at additional MVP's in coming decades, giving more applicability to the law of large numbers. Also, the results might have been more convincing if they didn't include Barry Bonds steroid-assisted MVP awards in his late thirties.

Torus Formulas

I've had a week to reflect on the torus. Most of my reflections have been in the past few minutes leading up to my current act of typing, but I like to think my subconscious has been mulling it over.

Remember that last week we learned that the plural of torus is tori. True, even though my spell checker has put a squiggly red line under it.

I did think of another application. Remember a torus is a donut shape. Inner tubes of tires are also tori. So the square inch amount of rubber in the tire can be found with the surface area formula and the air in the tire would be found with the volume formula.

Also, there are apparently applications I would never have come up with.

In our model of cosmometry, the torus is the fundamental form of balanced energy flow found in sustainable systems at all scales. It is the primary component that enables a seamless fractal embedding of energy flow from micro-atomic to macro-galactic wherein each individual entity has its unique identity while also being connected with all else. 

This is from the website http://cosmometry.net/the-torus---dynamic-flow-process. I do not know what this all means. I'm not even sure what all the individual words mean, but it certainly sounds very important.

Last time I listed the formulas for the torus. I found there are other ways to find volume and surface area. Instead of using the variables in the way we used last week, these formulas use r, the distance from the center of the torus to the inner edge and R the distance from the center to the outer edge. 

V =  1/4(pi)^2(r+R)(R-r)^2

S.A. = (pi)^2(R^2-r^2)

(Again, I apologize for my inability to write exponents any other way.)

Beside finding volumes and surface areas of donuts, inner tubes, and various balanced energy flows (?) there is another nice application here. To find the ratio of surface area to volume of any three-dimensional shape is an important concept. To do so with the above formulas is especially cool as it simplifies down a lot.

Donut / Torus

I thought donuts would be an interesting topic. They are actually the mathematical shape called a torus. I first heard of this sitting in an undergraduate math class. Our professor told us we might try to find out about the torus before the next class. I actually looked it up. I was the only one in the class to do it and was able to talk about it the next time we met. I'm sure I got labeled as a nerd at that point. I wouldn't mind that, but when a room full of mathematicians think you're a nerd, that is probably an especially bad sign.

It turns out there is a lot I didn't know about this topic. Like the plural of torus. (It's tori, with a long i sound.) Is it donut or doughnut? (The consensus by those that decide these things seems to be "doughnut" although they seem to put up with "donut". "Donut really didn't come into regular use until Dunkin' Donuts started up in the 1950's.) I thought maybe this had a tie-in to the car, but no. It is spelled "Taurus" and I'm guessing has to do with the zodiac sign.

Imagine two circles linked as a chain. If one makes a full lap following the path of that first circle, we have a torus. Let's say the moving circle is radius r and the stationary circle has radius R.

The surface area is S = 4(pi^2)Rr. (Sorry, I do not know how to make my blog write the pi symbol or how to do exponents.) The derivation of this formula is more easily seen if written S = (2(pi)r)(2(pi)R). It is the circumference of the moving circle taking the a path along the circumference of the big circle.

The volume is V = 2(pi^2)(r^2)R. While this is the simplified version, again it is easier to see where it comes from by writing it differently: V = ((pi)(r^2))(2(pi)r). It is the area of the moving circle again taking the a path along the circumference of the big circle.

What can we use these formulas for? Not important, but there are a lot of them - donuts. Important, but none actually exist - the space station shown in the movie 2001. In the picture notice that it seems more of a rectangle than a circle on the outer edge. I think that is still a torus. The torus definition from different sources I found say "a closed curve", "a closed curve, especially a circle", or simply "a circle".

Enough for now. We'll look into this topic more next time.

The Sophomore Jinx

The sophomore jinx, or sophomore slump, takes place when the second round is not as good as the first. The second album, the second season, doesn't seem to be quite as good. They got everyone's hopes up after a great debut. What's with that?

Does it really happen, anyway? Maybe its just hit or miss. The Grammy Award for Best New Artist in 1964 was a group called the Beatles. Good call. But the year before, the Best New Artist was Ward Swingle. First let's look at the case for there being such a thing as a sophomore jinx. With a quick look at the internet one can find plenty of examples that seem to support this idea.

Album sales by some pretty well known names:

Terence Trent D'arby - Album Number One - 12 million, Album Number Two - 2 million.
Spin Doctors - Album Number One - 5 million, Album Number Two - 1 million.
Christopher Cross -  Album Number One - 5 million, Album Number Two - 500,000 thousand.
Hootie and the Blowfish - Album Number One - 16 million, Album Number Two - 3 million.

You get the idea. Aaron Gleeman in an article titled The Sophomore Slump looked at all of the Rookie of the Year award winners, comparing their first and second seasons by using a baseball statistic called win shares. He found that 73 of the winners got worse in season two, while only 37 improved. Four stayed the same.

Rick Sutcliffe was the National League Rookie of the Year in 1979. Overall, he had a fine career, winning 179 games. His first year he won 17 games and lost 10. He gave up about three and a half runs a game. Next year he won 3 and lost 9 and gave up about five and a half runs a game.

This so-called sophomore jinx, can be explained at least in part statistically with the concept of the regression to the mean. The dictionary says, "In statistics, regression toward (or to) the mean is the phenomenon that if a variable is extreme on its first measurement, it will tend to be closer to the average on its second measurement.

If we flip a coin 100 times and get 63 heads, would we do better next time? Yes, maybe. But probably not. But if we got 40 heads on a first try, chances are, next time we'll see an increase. In either case we go back toward or regress toward the mean.

The examples we have seen have something in common. All of these first years were very good. They got our attention. People are wondering what they'll do for a follow up. All of these burst upon the scene with a great debut. Perhaps they were far above what their usual production would be. That can happen, but chances are in any given effort, we will do what our historical average would suggest.

Are there cases where the sophomore slump doesn't happen? Consider the baseball player that has a first year that is a bit below what he is capable of. Likely, he will improve the next year. The public really didn't notice his first year because it was nothing spectacular. We we're all paying attention to the Rookie of the Year winners. 

Batting Average

Algebra I students start out solving one-step equations. A good application is finding a baseball player's batting average. The batting average is the ratio of hits to official at-bats. An at-bat that is not "official" would refer to getting on base by means other than your batting ability. Being walked or being hit by a pitch is not counted. Batting average is expressed as a decimal rounded to thousandths place. A person getting one hit in four at-bats is hitting 0.250, pronounced "two fifty". A person going two for three is batting 0.667, pronounced "six sixty-seven". Do not get mathematically correct and pronounce this "six hundred sixty-seven thousandths". Expect blank stares or ridicule if you do. Incidentally, when baseball people are talking about Ted Williams being the last person to hit four hundred, they mean the last person to have his hit to at-bat ratio being greater than or equal to 0.400.

Since baseball's regular season just ended a couple days ago, let's take some batting averages from the last season.

Miguel Cabrerra won the American League batting title by having 145 hits in 429 at-bats. Students could use the formula BA = H/A to get his batting average (.338). Or, given his batting average was .338 and he was at bat 429 times, how many hits did he get? Or, Cabrerra was had a batting average of .338 with 145 hits. How many times was he up to bat? This also leads to an opportunity to talk about round off error as the last question could be answered by saying he was up to bat 428.99 times.

The National league batting title was won by Dee Gordon, having a .333 average by having 205 hits in 615 at-bats. He edged out Bryce Harper who was 172 for 521 for a .330 average. Actually, going into the final day of the season, they were tied. Each had a .331 batting average. OK, not exactly tied. Harper had an average of .33075 and Gordon was at .33061. On the final day of the season, Gordon was 3 for 4 and Harper was 1 for 4, giving Gordon the title.

Leaving out some information and being creative, a person could probably come up with a number of algebra problems from that scenario.

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.

Y2K

I recently bought a magazine put out by Popular Science entitled, Mistakes and Hoaxes: 100 Things Science got Wrong. One of the hundred was concerning the Y2K event. It was predicted that on January 1, 2000 computers would go nuts. Apparently computer programmers hadn't accounted for the fact that computers might take the date of 1/1/00 to be January 1, 1900. At least that was the assumption. The disaster that was predicted didn't take place. I don't know if we should label that a mistake or not. I don't recall if anyone said it would happen or that it just could happen. And maybe the work that was done in the months leading up to that date did, in fact, stave off the disaster.

Anyway, according to the same article, we dodged a bullet in 2000, but there is still another computer issue pending. To quote the article, "Many computers still operate on a 32-bit system, referring to the way a computer processor handles information. These systems use a binary code to track time as a running tally of elapsed seconds, beginning on January 1, 1970. at 12:00:00. But a 32-bit system can only handle a value up to 2,147,483,647, which is exactly how many seconds will have elapsed between January 1, 1970 and January 19, 2038. Luckily, programmers have already started updating computers to a larger 64-bit system, hopefully staving off a massive computer shutdown for 292 billion years."

This could lead to a number of great little questions for students to tackle. Such as:

1. Where did they get the number 2,147,483,647 from?
2. Are there really that many seconds between the 1970 and 2038 dates?
3. Would a 64-bit system really translate to 292 billion years?

OK, get to work.

I myself am going to take a dramatic pause here and reexamine this next week to find some answers.

Math Videos

Many math teachers lament the dearth of good math videos. How many Friday afternoons have you thought how great it would be to be a history teacher and just pop in one of thousands of video they have? From some one who has shown "Donald Duck in Mathemagicland" maybe a millions times, the value of a good math video cannot be overstated. I was somewhat surprised to see how many good math related things there are on the PBS website. I literally stumbled on to (via http://www.stumbleupon.com/) a video entitled "The Great Math Mystery". It can be found at http://www.pbs.org/wgbh/nova/physics/great-math-mystery.html. I think it's good. It perhaps tries to do a little too much, but it is a quality video. It shows many good applications in mathematics. It doesn't get into the nitty-gritty of the math, making it showable to pretty much any level of student.  And at 53 minutes, you could probably milk it for two days worth.

Along with other videos, there are also a number of well-written articles. For example one called "Describing Nature with Math" is at http://www.pbs.org/wgbh/nova/physics/describing-nature-math.html. There are quite a few more you can find in their "physics & math" section.

Home Field Advantage

While a lot of people would say that the law of averages would say that flipping a coin means there will be five heads and five tails. Anyone who has flipped coins know this isn't necessarily true. However, things do even out in the long run. In ten flips of the coin, we would not be surprised to get seven heads and three tails. However, we would be quite surpised to flip a thousand times and get seven hundred heads and three hundred tails.

Sometimes the probablilities will sometimes change because circumstances change. We expect coins to fall 50-50 regardless what year we do the flipping. The average life span though has changed through the years.

How about home field advantage for a major league baseball team. Things have certainly changed there. A hundred years ago there were no night games, all the players were white, and spitballs were legal. Fenway Park and Wrigley Field were brand new stadiums. It wouldn't be surprising if home field advantage, if there really is such a thing, didn't change some through the years. Check out the home field winning percentage by decade. (From www.baseballprospectus.com/)

1900-1909     53.3%
1910-1919     54.0%
1920-1929     54.3%
1930-1939     55.3%
1940-1949     54.4%
1950-1959     53.9%
1960-1969     54.0%
1970-1979     53.8%
1980-1989     54.1%
1990-1999     53.5%
2000-2009     54.2%

Maybe even more striking (pun!) is to look at figures rounded to the nearest percentage.

1900-1909     53%
1910-1919     54%
1920-1929     54%
1930-1939     55%
1940-1949     54%
1950-1959     54%
1960-1969     54%
1970-1979     54%
1980-1989     54%
1990-1999     54%
2000-2009     54%

A good intro to this might be to have students share if they think if there is such a thing as home field advantage, what it might be, and would it have changed in the past century.

Coke Rewards

I've collected the codes attached to Coca Cola products for awhile now. I'm about ready to give that up for several reasons. For one, I probably don't need to give up hours recording those codes to finally have enough to get items such as a free t-shirt advertising their company. Anyway, those codes seem longer than they need to be. Here is one I recently used - 5KBMNOMN6FWPFW. There doesn't seem to be a reason for it. I figure the possible combinations are astronomical. There doesn't seem to be anything in particular that would limit the possibilities. For example, I thought maybe the first entry might always be a digit. No, sometimes a digit and sometimes a letter. They do state that the letter O (oh) and the number 0 (zero) are registered the same. This is also true entering the letter I and the number 1.

I had given this some, but not a lot of thought previously. But I read on their website's FAQ section the following:

Why did My Coke Rewards change from 12 digit codes to 14 digit codes? 

Due to the popularity of our program, we’ve made the transition from 12 digit codes to 14 digit codes, to ensure we have a steady supply of codes for you, our loyal members. Please note that 12 digit codes are ineligible effective 8/1/2014.

Really? Twelve are not enough? How many is that anyway? Without knowing of any other limiting factors I assume there are 34 possibilities for the first part of the code. (The 26 letters of the alphabet, the 10 numeric digits, and throwing out the two repeats.) So how many for the 12 digit codes?

                      34x34x34x34x34x34x34x34x34x34x34x34 = 2.386x10^18

I understand there are about 8 billion people on the Earth currently. Dividing those two numbers we see there are enough codes that each person on Earth could have 2,983,000,000 of them. Seems like enough. But apparently there was a need to go to a fourteen digit code. (Since Coke uses the term "digit" to designate both numbers and letters, I will too.)

Students can brush up on their scientific notation a little more by finding the amount of 14 digit codes.
   
                           2.386x10^18 x 34 x 34  = 2.758x10^21

I looked on-line and found that scientists have a rough estimate of grains of sand of all the beaches of the world. There very roughly, 7.5x10^18 grains of sand in the world. Hopefully 14 digits are going to be enough for the Coke folks.


   
   

Mileage

I took a fairly long trip and I thought that made for a good and not extremely difficult application problem. It is a two-parter, with a little algebra, but not to hard.

I drove roughly 600 miles. I was on a freeway the entire time, so was getting a healthy 39 miles per gallon. The price of gas seemed was about $3.60 per gallon. How much did my trip cost me?

How much it cost me depends on how many gallons I used. So, how many gallons did I use? Since miles per gallon = miles/gallons, we have 39 = 600/x. It turns out that x = 15.39 gallons.

Again we can use a fairly obvious equation: cost per gallon = cost/gallons. Since the cost for each gallon is $3.60, we have 3.60 = x/15.39. This gives us x = $55.40.

Tycho Brahe - World's Coolest Scientist

I was reading the book Grapes of Math, written by Alex Bellos. In it he mentioned a couple of things about the scientist Tycho Brahe. Alex mentioned that he had a silver nose and that had a pet deer who fell to his death after drinking too much beer. What? I knew just a little bit about Mr. Brahe, but how did this information get past me?

Students often see mathematicians and scientists as being fairly plastic and dull. A few stories about Tycho could probably change their minds. He was not plastic. Part metal, but not plastic. Here are some top points about Tycho Brahe.

1. He was a nobleman and created something of a scandal by having children with a lowly peasant girl.

2. The silver nose thing seems to be quite true. Several sources back this up. As a twenty year old, he got into a mathematically induced fight with another student. It seems it was either over some mathematical point or over who was the better mathematician of the two. It did not end well for Tycho who lost a good part of his nose in a sword duel. It was stylishly covered with a metal nose piece.

3. There has been some debate whether the nose piece was gold, silver, brass, or copper. Tycho's body was exhumed in 2010. (For those into such things, the exhuming can be viewed on the internet.) One of history's mysteries solved - it was brass.

4. He did not agree with Copernicus. Copernicus died three years before Brahe was born. The Copernican theory was out there, but Tycho did not fully buy into it. Brahe was somewhere between Earth and Sun centered. He believed the Moon and Sun orbited the Earth and the other known planets orbited the Sun.

5. He built and made observations from the finest observatory in Europe. His observations were by eye as the telescope had not been invented yet.

6. He hired Johannes Kepler as his assistant. After Brahe's death Kepler went on to find the great important three laws of planetary motion, based on Brahe's observations.

7. The deer story seems to be true. He owned a tame elk as a pet. Tycho loaned him to a friend. One everning he got into some beer and became drunk. Later that evening, the elk fell to his death down a flight of stairs. (Feel free to insert any desired number of exclamation points.)

8. His death. There has been a theory that he was murdered by mercury poisoning, perhaps by a jealous Kepler. The exhumation in 2010 showed that he did not have elevated levels of Mercury. Johannes is off the hook.

9. It is probable that he drank too much at a banquet and felt it would be rude to dismiss himself. He died of a broken bladder.

Winning a car

I got my mail today. It turns out I might have won a car. First I have to scratch something off to see if I'm in the running. I scratched and it turns out I'm already a winner!!!! Looks like luck is on my side already!!! Now I know I've won at least something. There are only four choices. Three of them are great, and one is just a two dollar bill. So I'm thinking a three out of four chance of winning a big prize.

I checked out the fine print included in the add. It doesn't seem as good as it did a minute ago:

2015 Jeep Patriot or $25,000 - Odds of Winning - 1:25,000

$100 - Odds of Winning - 1:25,000

$2 - Odds of Winning - 24,998:25,000

So first of all, I didn't have a three out of four chance of winning something good as you had a chance to win the $25,000 or the car, but not both. So I have a two out of three chance of winning something good.

Secondly, it turns out I don't have a two out of three chance. I have a two out of 25,000 chance.

Good can come out of this, as there is much students can learn from this.

For one, it could be pointed out that despite what the ad says, these are probabilities and not odds, but why quibble?

Next, students could spot all the mind games that are being played. Getting you hooked by having you scratch off something. Feeling lucky when you are successful (which undoubtedly everyone is). Having what looks like three pretty good prizes, but is only really two.

Students can learn about expected value. The expected value could be found as $25,000 x 0.00004 + $100 x 0.00004 + $2 x 0.99992 = $3.00384

On the plus side, there is no way you can lose, but chances are you don't win very much. In most gambling, a person has to put up some money to play the game. That is the upside for the person/company/casino running the game. In those cases, the house typically sets up the game so the player's expected value is a negative number. What is the company's incentive in this case? They are obviously hoping to win their money on the back side of the game. They're hoping to get extra traffic to their showroom to sell a few more cars.

Gimmicky advertisements can be a great learning experience.

Deflategate math

There are a number laws relating to gasses that show a proportional relationship. One of these is Charles’ Law. Jacques Charles was the uncommon combination scientist/balloonist. He co-developed and rode in the first hydrogen balloon in 1783. Manned balloon flight was it its golden age and Charles played an important part.

Charles’ Law states that with pressure remaining constant, the volume of a gas are in direct variation. The experts say that tire pressure should be measured before driving as the act of driving will cause the air in the tire to heat up and thus expand.

You might be aware that a basketball left in the cold doesn’t bounce very well. That is because the decreased temperature means a reduced volume of air in the ball. If it is brought inside or run under hot water it will regain its bounciness.

A similar law, Gay-Lussac’s Law, states a directly proportional relationship between temperature (measure in degrees Kelvin) and pressure.

Some examples:

- A basketball is left outside at 8 degrees Celsius. Its volume was 444 cubic inches. Brought inside, it warmed up to 32 degrees. What is its volume now?

First of all, we need to be in Kelvins, so K = C+273, means the ball’s temperature went from 281 to 305 degrees Kelvin. Now setting up our proportion we see,

                                             444 : 281 = x : 305   →     x = 483 cubic inches

- NFL Rule 2.1 states that “The ball shall be made up of an inflated (12 1/2 to 13 1/2 pounds) urethane bladder…”. It is claimed that the Patriots football team has under-inflated the ball to try to gain an advantage. A league report claims that at least one of the balls was inflated to only 10.5 pounds per square inch. Suppose the Patriots then claim that according to Gay-Lussac’s Law the football would naturally lose pressure as it was brought from the warmth of the locker room to the field. If the locker room was 75° F (297.04° K) and the field temperature was 40° F (277.5944° K), might the Patriot’s claim that the football was actually within the guidelines?  

Our proportion comparing pressure and temperature could be written as:

                                            10.5 :  277.59 = x : 297.04 

We get an answer of 11.2 psi, which is still under the 12.5 minimum.

100,000 Times

I saw a commercial in which someone said, "My heart beats one hundred thousand times a day." I thought that sounded a little high. So in a week and a half it would beat a million times. That seemed high as well. We should check that out.

Sometimes students just aren't sure how to get started. One thing that can help to look at the labels. For example ,if someone wants to find out miles per hour, the label mi/hr would imply one would divide miles by hours. If someone had traveled at 23 miles per hour for 1.3 hours, how far would they go? The labels work out if the miles per hour quantity is multiplied by the number of hours. The labels would be (mi/hr) x (hr/1). The hours cancel leaving us with miles, just like we want.

Suppose a student wants to see if the 100,000 value is plausible, but has no idea how to begin. One could start with just looking at the labels and see if that works out. The Mayo Clinic says normal is somewhere between 60 and 100 beats per minutes. That is a pretty broad range. I have heard 72 in the past, so let's go with that. We want to see a final answer that is in beats per day. So the student is starting with beats per minute and we want to end up with beats per day. How do we get there? Hopefully they can see that 60 beats per minute will bridge that gap.

                    72 beats/min x 60 min/hr x 24 hours/day
                         = 72 beats/min x 60 min/hr x 24 hours/day
                         = 103,680 beats per day

Nicely done.

What number of beats per minute would correspond to 100,000 beats per day? As in the last problem, we can examine labels. This is not the only way, and not even necessarily the best way, but for some students it might make the most sense.

So we start with 100,000 beats per day and want to end up with so many beats per minute. We have to find some linking labels to get from one to the other. We have to fill in the blanks for:  100,000 beats/day x ?????? = x beats/min

                   100,000 beats/day x ?????? = B beats/min
                   100,000 beats/day x day/24hr x hr/60 min = B beats/min          
                   100,000 beats/day x day/24hr x hr/60 min = B beats/min
                            B = 69.444... beats per minute

Expected Value for Robbing a Bank

"When to Rob a Bank" is a book written by Steven Levitt and Stephen Dubner. It is a collection of stories from their blog. They are two economists who previously wrote the bestselling "Freakonomics".

An article in this latest book gives some statistics on bank robbery in the United States. It's not as lucrative a business as I thought. Bank robbers get away with it 65% of the time. Chances are, the average bank robber gets away with it, but there is a pretty solid chance he doesn't. Another drawback is that they don't get nearly as much money as I thought. The average haul is only $4,120. That is quite a bit of money I suppose, but it isn't going to make you rich. You would have to rob a bank a month to get yourself to a middle class income. And a lot more than that to get rich.

When I read this, I wondered what the expected value would be? Expected value is the average value you expect to gain in an experiment with a large number of trials. In this case, you have a 65% chance of making $4,120. But how do you put a value on getting caught? You would be going to jail, I'm guessing for roughly 5 to 10 years. How much would you pay to have your freedom instead? In other words - How much would you pay for a get out of jail free card? I'm guessing conservatively that has to be worth at least $10,000 to you. 

                    Expected Value = 0.65(4,120) + 0.35(-10,000) = -$822 

We've established mathematically that crime doesn't pay.

What if we didn't just rob one bank. Let's try robbing two. Basically three things could happen.

1. You successfully rob both banks. Probability = (0.65)(0.65) = 42.25%. Payoff = $8,240.

2. You rob one and then get caught trying to rob the second. Probability = (0.65)(0.35) = 22.75%. Payoff = -$10,000. We're assuming they won't let you keep the money from the first bank and you still are going to jail for 5 to 10.

3. Probability you are caught the first time. Probability = 0.35. Payoff = -$1,000.

                    Expected Value = 0.425(8,240)+0.2275(-10,000)+0.35(-10,000) = -$2,293.60

Crime doesn't pay.

Equal Temperatures

Students know that the same warmth registers differently on the Celsius and the Fahrenheit scales. They might be surprised to know that there is a point at which they are the same. That temperature could be found by using a system of equations. Celsius and Fahrenheit are related with the equation F=(9/5)C+32. Since we want to find out when the two scales are the same, we also need the equation F=C.

Initially, students might make up a table of values and notice that when it is warmer, temperature readings are farther apart -

                                 100 degrees Celsius = 212 degrees Fahrenheit
                                   40 degrees Celsius = 104 degrees Fahrenheit
                                     0 degrees Celsius = 32 degrees Fahrenheit

The differences are getting closer together - 112 degrees apart, then 64, then 32.

If we change the variables to x's and y's, the equations would perhaps look more familiar to students:

                                                 y=(9/5)x+32   and   y=1x+0

In this form students should see these are both linear equations, but have different slopes. While they might not know where it is yet, clearly those lines must have an intersection someplace. So, being straight lines, there must be one and only one temperature that is the same for each scale.

To get that temperature, the substitution method is probably the easiest way to go.

                                                               x=(9/5)x+32
                                                                5x=9x+160
                                                                  -4x=160
                                                                      x=-40

And so if x is -40, y must also be -40.

Being linear equations and already in slope-intercept form, algebra students should be able to easily graph them. Doing so, they would find an intersection point (or at least fairly close) of (-40, -40).

"Large" Soft Drinks

I was in a local fast food restaurant, who will be left nameless. Anyway, after getting my Mcdrink I thought it seemed a little small. It was called a "large" after all. So I was thinking it should be at least a 32 ounce or 40 ounce drink. Being something of a connoisseur of these things, I felt this was well off.

It reminded me of a time when I verified to a class the measurement of a supposed 32 ounce cup. It was like an episode of Mythbusters. It turned out to indeed be the as advertised 32 ounce container.

I began my investigation by Googling "volume of cup sizes" which led me to websites that were not even close to helpful. Another approach was necessary. The plastic cups found in convenience stores or fast food restaurants are close to, but not quite, cylinders. They slope somewhat so that the top of the cup is a little wider than the bottom. Mathematically, this could be called a truncated cone or a frustrum. To find the volume we multiply the area of the base by the height. The area of the base can be found by using the average of the radii of the two bases.

Getting as close as I could on the measurements, I found radii of 4.85 cm and 3.15 cm and a height of 16.3 cm. We then have an average radius of 4.0 and a total volume of (Pi)(4)(4)(16.3) = 819.3 cubic centimeters. One cubic centimeters is equal to 0.033814 fluid ounces, so we can multiply to find a volume of 27.7 ounces of fluid.

While I know the term "large" is relative, I was a little mcdisappointed nonetheless.

John Nash

As my first post, let's look at the life of the late John Nash. He and his wife Alicia were tragically killed in a traffic accident in May of 2015 - just about a week ago from when I'm writing this. For students, he might be the mathematician they are most aware of, being the focus of the best-selling book and Academy Award winning movie, A Beautiful Mind.

He was born in Bluefield, West Virginia in 1928. He showed early promise, even skipping a year in school. He graduated from the Carnegie Institute of Technology and then obtained his Doctorate from Princeton at the age of 22. He then worked of the Rand Corporation and taught at M.I.T.

He majored in mathematics, specifically focusing on game theory. Is it in a country's best interest to go to war? Should my team pass or run the ball? What should my investment strategy be? Should we drop our prices and attempt to undercut our competition? Should I buy more land or put motels on the land I have? Game theory, and especially the concept of the Nash Equilibrium, have applications in many different areas. It is for game theory that Nash would later win his Nobel Prize.

Interestingly, his and Albert Einstein's paths crossed at Princeton. Nash presented some of his thoughts on relativity to Einstein. The meeting wasn't totally satisfactory. Einstein let Nash know he needed to go learn some more physics.

His whole life, John Nash had always been what kindly could be called eccentric. His behavior became increasingly bizarre resulting in a diagnosis of schizophrenia. The National Institute of mental Heath's website states that, "People with the disorder may hear voices other people don't hear. They may believe other people are reading their minds, controlling their thoughts, or plotting to harm them. This can terrify people with the illness and make them withdrawn or extremely agitated."

Image courtesy Wikipedia.org

John Nash certainly seemed to fit that description. Thankfully, in the 1980's, he began showing improvement. It turns out this improvement was not as miraculous as some have assumed. The literature shows that ten years after its onset, approximately 25% of schizophrenics are "much improved". After thirty years, this increases to 35%. The chances of recovery seem improved with having a home, job, and hope. Thanks to Alicia and the support of colleagues John had these.

In his later years Nash was granted a number of honors including his Nobel Prize in 1994. He continued to work a Princeton University for the rest of his life.