Sermon Stats

In sermon notes in a church bulletin it stated, "The probability Jesus could have fulfilled even eight of these prophesies is 1 in 10 to the 17th power (1 in 100,000,000,000,000,000)". This was a statistic taken from a book, although I don't know the title. I thought there is a math application in there somewhere.

I thought that small a number might be almost incomprehensible to most. Maybe to everyone. It

reminds my of something David Letterman said once regarding buying a lottery ticket. A particular lottery was at a near record amount and lots of people were buying them. He wanted people to consider that if you buy a ticket, your chance of winning is only slightly more than if you don't buy one. Incidentally, I was in the audience for one of his shows during his final month. Hilarious. I am including a picture for no other reason than I love Dave. Back to math.

I considered a couple of ways to tie this probability to other situations. How does this probability compare with chances in rolling a die? In flipping a coin?

Well, the chances of rolling a "6" are one in six. How many consecutive rolls would correspond to the above probability?

1 / 10¹⁷ = 1 / 6^x
10¹⁷ = 6^x
Taking the common log of each side, we get:
17 = x(log6)
x = 21.85

So, at least 21 consecutive rolls coming of 6.

Similarly with flipping the coin. The coin has only two outcomes, so:

1 / 10¹⁷ = 1/2^x
After a few steps we get x = 56.47

56 heads in a row. Unlikely.

If worried about church vs state issues, a teacher could come up with other kinds of problems. The actually probablility of winning a certain lottery, winning the grand prize in the McDonald's Monopoly Game. For example, I just looked up on-line that the probability of getting the Boardwalk piece - 1 in 602,000,000.

Good Luck.

Hidden Figures

This is kind of a review of the movie/book Hidden Figures. Before seeing either one, I was a little concerned that it was going to be preachy. We need to treat African American people / women with respect. Hollywood, you don't need to tell us that. The people that already know that don't need to be told. And the people that don't know that probably aren't going to pay money to see this movie anyway. But it wasn't like that - the movie or the book. It just told their story.

If you want the true story, books are usually a better bet than the movie. The movie seemed quite in line with the book, but there were a few things. The two hour movie obviously had to leave a lot of stuff out that was in the book. There was also a time one of the ladies was at a chalkboard. She was impressing the room with the math she was doing. Some of it was a little unrealistic. Most people can't rattle off sin(23) to ten-thousandths place from memory - stuff like that. But, I'm quibbling. The movie was really good.

I got the book. And to be honest, I did some skimming in parts. I may get around to reading every word at some point, but that will be a ways off. There were a couple of cool things dealing with math applications that I thought was interesting. First of all, the book is Hidden Figures and is written by Margo Lee Shetterly. There. I hope that covers me from violating any copy write issues. Regardless, here we go.

Where do systems of equations take place in real life? Well check this out: "Modeling flight at transonic speeds was a particularly knotty problem, because of the subsonic and supersonic winds that passed over the plane or model simultaneously. Aerodynamic equations describing transonic airflows might contain as many as thirty-five variables. Because each point in the airflow was dependent on the others, an error made in one part of the series would cause an error in all the others. Calculating the pressure distribution over a particular airfoil at a transonic speed could easily take a month to complete for the most experienced of mathematicians." (pages 137, 138)

I once had the father of a student tell me about an older child of his that was in the NASA astronaut program. He said that their astronauts had to have a very good mathematics background in case something happened to the on-board computers. Another part of this book spoke to that issue:  "An astronaut stranded hundreds of thousands of miles from Earth is like a mariner from a previous age, adrift in the most remote part of the ocean. So what do you do when the computers go out? This was precisely the question Katherine [Johnson] and her colleague Al Hamer had asked in the late 1960s, during the most intense preparations for the first Moon landing. And in 1967, Johnson and Hamer coauthored the first of a series of of reports describing a method for using visible stars to navigate a course without a guidance computer and ensure the space vehicle's safe return to earth. This was the method that was available to the stranded astronauts aboard Apollo 13." (page 248)


Speaking of outer space, I give both the book and the movie, four stars.

World Population Growth

Here is an interesting graph. (https://ourworldindata.org/world-population-growth/) It shows the world's population growth up to the present day and then someone's estimates as to what will happen in the next few decades. It probably takes a little looking at for it to make sense. I combines two graphs in one. The horizontal axis shows the passage time in years and the vertical shows growth rates. The graph also shows the total population although these numbers are just recorded on the graph rather than being recorded on the vertical axis. As line graphs go, it's a pretty busy graph.

It seems to me that math teachers could make use of the graph in pretty much any high school mathematics class.

This actually started for me with information I found in the 2017 World Almanac. It showed population estimates going much farther back in time than this graph shows. You could look at that information as a set of ordered pairs with years being represented as x-values and world population (in billions) as y-values. The almanac states that in the year one the population was an estimated 300 million. That gave me an ordered pair of (1,0.3). Proceeding in this manner gave me ordered pairs of (1,0.3), (1250,0.4), (1500,0.5), (1804,1), (1927,2), (1960, 3), (1974,4), (1987,5), (1999,6), (2011,7).

Just using the raw data, an Algebra I class might simply write ordered pairs, or without seeing the above graph, choosing appropriately labeled axes to graph the data.

Higher math classes could look at finding an equation to model the data. I had more trouble than I thought I would. I guess that is because, as the graph shows, the rate has varied over time just in the last couple centuries, let alone millennia. Leaving out the first few ordered pairs and adjusting the data such as changing (1804,1) to (0,1) and so on, I was able to find an equation that had a correlation of r = .9647. Students could maybe experiment with similar things to get a best fitting curve.

Calculus students would be able to examine the blue population growth curve and discuss how it ties into first and second derivatives. It is interesting that the person making the future projections seems to think our current point in time seems to correspond to an inflection point. Students could discuss what that really means and mathematically and socially.

Best Music Video Ever

Best video ever. Check it out at okgo.net. It features my now favorite band, OK Go. It's not their only

awesome video either. Did I tell you to check it out? Do so.

The band was approached by a Russian airline about using a plane to do some flights for a video that would involve weightlessness. I know. A little hard to believe. But I read about it in an issue of Smithsonian Magazine. If they aren't a credible source, who would be?

The weightless part of the flight doesn't last long and to do anything substantial, it probably takes several runs at it. You might be familiar with the concept. you can fly a plane or rocket in a parabolic path to achieve weightlessness. It's how the Tom Hanks film, Apollo 13, had its weightless moments shot.

The article (December, 2016, pages 52,53) explained the process.

1. Steady horizontal flight.
2. Hypergravity (1.5 to 1.8 g's) for 20 to 25 seconds at a 47 degree ascent.
3. Microgravity (0 g's) for 20 to 25 seconds.
4. Hypergravity (1.5 to 1.8 g's) for 20 to 25 seconds at a 47 degree descent.
5. Steady horizontal flight.

The weightlessness takes place during that 20 to 25 second sweet spot at the top of the curve. As the authour Jeff MacGregor stated "Then came the math. The song is 3 minutes and 20 seconds long, give or take. Weightlessness during parabolic flight occurs in roughly 25-second increments. That's at the top of each parabola. And for every parabola, it takes five minutes of flight to reset for the next one. To get a single continuous weightless take lasting 3:20 would require eight parabolas - more than 45 minutes of actual flying time. 

This is perhaps not the best ever math application. However, have used it in movies and videos and it has even been used for weddings. Maybe it is the best ever math application.

Golden Gate Bridge

I'm going to be out for a couple of weeks, so I thought I should squeeze in one more post before I take off.

Last week I ran a half marathon. I know, foolish. But the main reason I did it is because it was in San Francisco and the course crossed the Golden Gate Bridge twice. You can see the bridge from a distance in downtown San Francisco. You can also see it driving across in a car, but it goes by pretty quick. Neither of those are the same experience as crossing it on foot. The cable droops down almost all the way to the road. It was fun to get right up close to it. Its just about head level at its lowest point.

An interesting thing I noticed is that the roadway is curved. You definitely run a bit uphill then down. The highest point, though, is not in the middle. I'm sure engineers had a reason for that, but that would be beyond me.

You can get many statistics regarding the bridge on-line. I was going to cut and paste them here, but they are easy to find. A teacher could fashion math applications for anything from arithmetic to calculus. A cool one is to find the equation of a parabola approximating the cable.

Since the cable is about head-height, using the roadway as the x-axis, we can take the center of the cable to be the point (0,6). The distance between the two towers is 4,200 feet. The height of the towers above the roadway is 500 feet. So, two other points on the cable could then be (2,100, 500) and (-2,100, 500). Using a system of equations with those three points could give you an equation of a parabola.

Also, the distance from the mean high water mark to the road is 220 feet. So a parabola could be found with the Pacific Ocean represented by the x-axis.

Or it could be found using meters rather than feet.

You could probably spend a week just studying the bridge. And it would make for a cool field trip.

Baseball Distances

I stumbled onto an interesting website. It had baseball statistics and had some stats I didn't know they even kept track of.

It gives numbers on average flights of batted balls for each player. It's interesting to look at as a math application. I tried it out using formulas and didn't get the quite the same answer. However, the trajectory equations don't account for the air resistance encountered. And, of course, I might have just done the math wrong. More on this after I let you know how far off I was.

The categories were "Average Launch Speed", "Average Distance", "Average Velocity", "Average Launch Angle", and "Average Height". For example Evan Longoria (no relation to the actress) had:

• Average Launch Speed: 92.17 miles per hour
• Average Distance: 248.2 feet
• Average Launch Angle: 14.39 degrees
• Average Height: 46.06 feet

I assume Launch Speed could be found with a radar gun. Launch Angle perhaps by camera, although it seems like that would depend on where the camera is in relation to the camera. Ideally, the camera would be pointed perpendicularly to the ball's trajectory, I would think, but that wouldn't always be the case.

So, I wondered if I could compute what they had for Average Distance. I'm guessing that is how far the ball before it hits the ground. But what if Evan hits a line drive and is caught? It went a certain distance, but would have gone farther without the fielder there? Anyway, here we go.

First I figured I need to get its average speed into feet per second to match with the other categories.

92.17 miles per hour = 286,657.6 feet per hour = 135.183 feet per second

I then used the formula:  y(t) = h + (vsinA)t -16t^2.

I'll assume an height of the ball when making contact with the bat to be 5.5 feet. I want to see how long it takes to hit the ground (y(t) = 0).

0 = 5.5 + 135.183sin(14.39)t - 16t^2

Using the quadratic formula, this game me two answers, the positive one being 2.25 seconds. 

Then I used this to find how far it went with x(t) = v(cosA)t = 135.183(cos(14.39))2.25 = 294.62 feet

According to that website (http://m.mlb.com/player/446334/evan-longoria) the distance is only 248.2 feet. 

I was ready to call this a big old fail. But, perhaps not. Like I mentioned before, I'm not sure how they figure balls that are caught before they land or balls that bounce off the outfield fence. And are those distances found by observation of where the ball seems to land? Air resistance slows down the ball quite a bit. They say that the Colorado Rockies in mile-high Denver is the easiest place to hit home runs because of its thin atmosphere. The math equations assume a vacuum, so the formula would give a greater distance. 

So, maybe my math is all right. Regardless, it's a nice math application.

 

Fractal Video

I saw an interesting video on fractals. It was produced by NOVA called Fractals: Hunting the Hidden Dimension. Being for the general public, it, of course, didn't get too hard core with the mathematics, but it didn't completely back away from it each.

Because of that, a few of the applications left one with some questions. Such a case was when someone in the video said, "Fractals are important in code breaking." Then they leave it there because to try to explain it would cause most people's heads to explode. In a lot of cases, to have to skip over the math is somewhat unsatisfying, but probably pretty much unavoidable.

Interesting to me was how Benoit Mandlebrot first got involved with applications of fractals. As computers were just starting to communicate, there were problems. Computer data was being sent over telephone lines. However, it often wasn't getting through as intended.

Benoit B. Mandelbrot, then an employee of IBM, noticed a certain pattern of interference over, say, a ten minute span. He then noticed that same pattern would appear if he looked at maybe a five minute span, then a two and a half minute span, etc.

The video called it, "self-similarity". The fact that this self-similarity was taking place, told him this situation could be modeled with fractals.

It is a good video. It is from 2011, so not too out of date. Students no doubt will be chagrined at how excited the math nerds in the video get over these fractals. Even with that - a good video.

Space Equation

I try to focus on high school math applications in this blog. Therefore, this picture probably doesn't quite fit. However, it is such a cool picture that I wanted to put it in. I saw it, or at least a portion of it in the September, 2016 issue of Reader's Digest. I went looking for the picture and found an expanded view of it on-line at http://rarehistoricalphotos.com/nasa-scientists-board-calculations-1961/. The Reader's Digest piece said that it was taken on October 10, 1957. These are equations related to satellite orbits. The picture was taken six days after the launch of Sputnik, putting the USSR up 1-0 in the space race. That seemed to get things going in the United States. NASA was created the next month and two months later, the U.S. had launched its own satellite.

Initially, I thought the photographer did some kind of time-lapse photography and these were all the same guy. Although, one person did the writing - astronomer Samuel Herrick - these are all different scientists. Its been my observation that everyone from the 1950s looked more or less the same. I think that is the reason for my confusion.

At the above website, I got some more information about the photo. The point was made that there are no calculations here - just equations that they might use. That makes sense being at the start of the space race and smack dab in the middle of the cold war. So no top level information was being given away in this photo.

Usually I think its a poor idea to present applications that are over the heads of students, but I think an exception could be made here. There is virtually no calculus here and concepts in trigonometry, "e", etc would be recognizable to many high school students.

The article ended with this:  For a complex equation that deals with time-steps and feeds back on itself, the prominent scientists of NASA would have “math parties”!!! [exclamation points, mine]. Everyone would master one part of the equation. Then the first guy would do his part and hand it off to the next guy and so on. Eventually the final guy would go back to the first person and give him the new inputs for 1ms [microsecond?] further in time. After a few hours you could have a nice neat graph of everything over a 1-2 second period. That is how the first nuclear reactors, nuclear bombs and a lot of aerospace calculations were done.

Marathon Percentages

I came across this in Runner's World magazine (Page 88, May, 2016). It seemed like a good application for a math class. It had the percentage of women in the field of the Boston Marathon. It seemed like it was something close to a linear relationship. Here is the data:

Year          # Women          # Men          % of Women

1966          1                      415               0.3%
1972          8                      1,210            0.7%
1980          237                  3,428            6.5%
1990          1,434               6,516            18.0%
2000          5,469               10,199          34.9%
2010          9,560               13,161          42.1%
2015          12,018             14,580          45.2%

I tried it out on an on-line calculator. First it is years vs. percentage of women. I used the point (0,0.3), (6,0.7), (14,6.5), etc.

The best fit was a linear equation of y = 1.0232x-3.8943. It had a correlation of 0.9866, so pretty good. I graphed it and perhaps some kind of logistic growth model perhaps some kind of logistics growth model might be a little better. The rate of increase of the percentages is starting to fall off a little.

I wasn't going to do this, but for fun I graphed years vs. number of women. It looked fairly parabolic. After plugging them in - (0,1,), (6,8), etc. I got the best fit to be:

y = 0.3888x^2.5833 with a correlation of 0.9751.

So, some interesting applications could be looked at from Algebra I through Advanced Math.

Who Wants to Be a Millionaire II

Last week's application dealt with a mathematical situation involving the winnings board on the game show Who Wants to Be a Millionaire? The question was whether there is a best fitting curve that fits the numbers fairly well.

We just kind of left it right there. Again, I apologize for doing this in Euros or what ever those numbers are. I guess that just gives another opportunity to do the project in dollars.

I used a website http://www.had2know.com/academics/regression-calculator-statistics-best-fit.html. It allowed you to input ordered pairs and then would give you four answers: Linear, Exponential, Power, and Logarithmic. The are other websites that have even more options.A teacher could get a lot of mileage out of this. Students could take guesses at which of the four would be best and which would be worst, and what the winning equation might look like. Enough suspense. The equations the website came up with are as follows:

Linear: y = 42877.1492x-209477.1429
                    Correlation:  0.6949

Exponential:  y = 41.5145(1.9468^x)
                    Correlation:  0.9992                  

Power:   y = 11.9805(x^3.5337)
                     Correlation:  0.9267                  

Logarithmic: y = -202534+180689.7131lnx
                      Correlation:  0.512

The winner? Judging by the correlations, the exponential equation, although the power equation wasn't bad. The others, fairly bad.

Students could graph the original 15 ordered pairs along with these four equations for some conformation.
                   

Who Wants to Be a Millionaire?

I thought about the Who Wants to Be a Millionaire money board as a good math application. I've thought how much pressure it would be to be a contestant. Once you get up to the big money, each step is a big deal. It's tempting to go on, even though you could lose everything.

The relationship clearly isn't linear. If it was, it would make the decision making a lot easier. But what is going on with these numbers? I thought it would be interesting to try to fit an equation to this set of numbers.

It might be good to first have students take a shot at it without any help. Most would probably see that it isn't linear. Most in fact would see a two and an exponent would be involved somehow. There could be a pretty even split, though, between 2^x and x^2.

It might help students to graph the values 0 (1,100), (2,200)...

Clearly, some kind of doubling is going on. It goes from 100 to 200. It takes a bit of a detour at 300, then 500, 1,000, 2,000, 4,000, 8,000, 16,000, 32,000, 64,000. Then a bump at 125,000, but then rights itself to the doubling pattern for the last few.

An astute student might think that the relationship is something like 2^x. However, when it doesn't double, it come up a little short of doubling. So, the base value might be a little less than 2.

By the way, you might have noticed that this is not in dollars. It must be the European version and those are pounds or Euros or something. I don't remember what that symbol means. I meant to do the dollar version. I didn't even know there was another. The dollar version has similar, but not identical numbers. This same project could be done with those numbers as well.

Anyway, most students are familiar with the game, so it might be a fun exercise for them. Next week I'll share some final results.

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."