Preparation for Industrial Careers in the Mathematical Sciences: Improving Market Strategies

The Mathematical Association of America recently published a number of promotional videos showing various mathematics can be used in “the real world.” Here’s the fourth pair of videos describing how mathematics is used in the world of finance. From the YouTube descriptions:

Dr. Jonathan Adler (winner of King of the Nerds Season 3) talks about his career path and about a specific research problem that he has worked on. Using text analytics he was able to help an online company distinguish between its business customers and its private consumers from gift card messages.

Prof. Talithia Williams of Harvey Mudd College explains the statistical techniques that can be used to classify customers of a company using the messages on their gift cards.

Happy Pythagoras Day!

Let me take a one-day break from my current series of posts to wish everyone a Happy Pythagoras Day! Today is 8/17/15 (or 17/8/15 in other parts of the world), and $17^2 = 8^2 + 15^2$ .

Bonus points if you can figure out (without Googling) when the next four Pythagoras Days will be. One of the next four is easy, two others aren’t so hard, but the fourth might take some thought.

Who was kissing in the famous VJ Day picture?

We are approaching the 70th anniversary of VJ Day (August 14, 1945), which marked the end of World War II. And perhaps the iconic photograph of that day is the picture of two anonymous strangers kissing in New York City’s Times Square celebrating the end of the war.

And a question that is still unresolved after 70 years is: Who are they?

The short answer is, Nobody knows for certain. But in a clever bit of geometric and astronomical forensics, physicists at Texas State University (Donald Olson and Russell Doescher) and Iowa State University (Steven D. Kawaler) recently pinpointed the exact time that the photograph was taken: 5:51 pm, or about an hour before President Truman formally announced that the war was over. From the press release:

Overlooked in the right hand background of the photo is the Bond Clothes clock. The minute hand of this clock is clear, but the oblique angle of view and the clock’s unusually short hour hand makes a definitive reading of the time difficult. The clock might show a time near 4:50, 5:50, or 6:50 p.m. A prominent shadow falls across the Loew’s Building just beyond the clock, however, and this shadow could potentially give just as accurate a time reading as the clock.

Every tall building in Manhattan acts as a sundial, its cast shadow moving predictably as the sun traverses the sky. In this case, the Texas State team studied hundreds of photographs and maps from the 1940s to identify the source of the shadow, considering, in turn, the Paramount Building, the Hotel Lincoln and the Times Building. The breakthrough came when a photograph of the Astor Hotel revealed a large sign shaped like an inverted L that advertised the Astor Roof garden.

Calculations showed that only the Astor Roof sign could have cast the shadow, but to be certain, Olson and Doescher built a scale model of the Times Square buildings with a mirror to project the sun’s rays. The location, size and shape of the shadow on the model exactly matched the shadow in Eisenstaedt’s kiss photographs.

So who are the kissers? Again from the press release:

Over the years, dozens of men and women have come forward claiming to be the persons in the photograph. All have different stories, but the one thing they share in common is kissing a stranger in Times Square that fateful day.

“All those people have said they were there and identify themselves in the photograph,” Olson said. “Who’s telling the truth? They all could be telling the truth about kissing someone. They were probably all there, and kisses were common in Times Square on VJ Day.

“I can tell you some things about the picture, and I can rule some people out based on the time of day,” he said. “We can show that some of the accounts are entirely inconsistent with the astronomical evidence”…

“Astronomy alone can’t positively identify the participants, but we can tell you the precise moment of the photograph,” Olson said. “Some of the accounts are inconsistent with the astronomical evidence, and we can rule people out based on the position of the sun. The shadows were the key to unlocking some of the secrets of the iconic VJ Day images–we know when the famous kiss happened, and that gives us some idea of who might or might not have been in the picture.”

From a news report:

“There are probably 50 or 60 sailors who have come forward and say, ‘That’s me! I’m the guy in the photograph.’ Fewer women, maybe five or six women, have said they’re the woman in white. There are articles all over the internet advocating for one [or] the other,” Olson said.

Olson can’t say who is correct, but he can rule out a few.

“What we can do is calculate the precise time, 5:51 p.m., when the photograph was taken. That does appear to rule out some of the widely accepted candidates,” he said.

The full article has been published in the August 2015 issue of Sky and Telescope magazine (sorry, you’ll have to buy a copy in you want to read the article). I also recommend clicking through the photographs in the press release; the captions of the photographs give many details of how the time of 5:51 pm was pinpointed.

Arrangements of Stars on the American Flag

Reasonable star patterns on the American flag correspond to special factorizations; the density of such factorizations is less than the density of values in a multiplication table; Paul Erdös showed this density asymptotically approaches zero by considering the average number of prime factors of an integer. – See more at: http://www.maa.org/programs/maa-awards/writing-awards/lester-r-ford-awards/arrangements-of-stars-on-the-american-flag#sthash.e9PHpilF.dpuf

Read the article here: http://www.maa.org/programs/maa-awards/writing-awards/lester-r-ford-awards/arrangements-of-stars-on-the-american-flag

US vs UK: Mathematical Terminology

Math With Bad Drawings had an amusing essay concerning differences in mathematical nomenclature between American English and British English. I thought that it would be appropriate to share this around the Fourth of July.

From the essay and the comments to the essay:

Math vs. maths
Zee vs. zed
$3.5$ vs. $3\cdot 5$
Trapezoid vs. trapezium
Scientific notation vs. standard form
Exponents vs. indices
Revise and review vs. review and revise
Imperial units vs. metric units
Pythagorean theorem vs. Pythagoras
Slope vs. gradient
Root vs. surd
PEMDAS vs. BIMDAS
Billion vs. trillion (a generation ago)
GCD (greatest common divisor) vs. HCF (highest common factor)

Interesting calculus problems

Source: http://xkcd.com/135/

Engaging students: Completing the square

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Tracy Leeper. Her topic, from Algebra: completing the square.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Muhammad ibn Musa al-Khwarizmi wrote a book called al-jabr in approximately 825 A.D. He was in Babylon and he worked as a scholar at the House of Wisdom. Al-Khwarizmi had already mastered Euclid’s Elements, which is the foundation for Geometry. So in his book he posed the challenge “What must be the square which, when increased by ten of its own roots; amounts to 39?” or in other words: how to solve he turned to geometry and drew a picture to figure out the answer. By doing so, al-Khwarizmi found out how to solve equations by completing the square. He also included instructions on how he solved the problem in words. His book al-jabr become the foundation for our modern day algebra. The Arabic word al-jabr was translated into Latin to give us algebra, and our word for algorithm came from al-Khwarizmi, if you can believe it. Later on, his work was used by other Arab and Renaissance Italian mathematicians to “complete the cube” for solving cubic equations.

How does this topic extend what your students should have learned in previous courses?

In previous courses my students should have already been introduced to prime factorization, the quadratic formula, parabolas, coordinates graphs and other similar topics. Completing the square is another way for students to find the roots of a quadratic equation. The first way taught is by using nice numbers that will factor easily. Then the math progresses to using the quadratic equation for the numbers that don’t factor easily. Completing the square is just another way to solve a quadratic that does not easily factor. Some students prefer to go straight to the quadratic equation, whereas other students will favor completing the square after they learn how to do it. It gives the students another “tool” for their toolbox on how to solve equations, and will enable them to solve equations that previously were unsolvable, such as the quadratic . By giving students a variety of ways to solve a problem, they can pick whichever way they are most comfortable with, which in turn will boost their confidence in their ability to learn math.

How could you as a teacher create an activity or project that involves your topic?

Usually the simplest way to learn something is to see something concrete of what you are trying to do. For completing the square, I can give the students the procedure to follow, but they probably won’t be able to fully understand why it works. In order to help them visualize it, I would use algebra tiles. One long tile is equal to x, since its length is x and its width is 1. The square is equal to since the length and the width are both equal to x. However, when you try to add to the square by a factor of x, you end up having a corner missing. This is the part that is missing from the initial equation. Then the students see that you don’t have a complete square, but by adding the same amount to both parts, we can get a complete square that can then be factored. Like so…

References:

http://bulldog2.redlands.edu/fac/beery/math115/m115_activ_complsq.htm

http://www.youtube.com/watch?v=JXrj5Dtgpss

Engaging students: Graphing parabolas

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Tiffany Wilhoit. Her topic, from Algebra: graphing parabolas.

How did people’s conception of this topic change over time?

The parabola has been around for a long time! Menaechmus (380 BC-320 BC) was likely the first person to have found the parabola. Therefore, the parabola has been around since the ancient Greek times. However, it wasn’t until around a century later that Apollonius gave the parabola its name. Pappus (290-350) is the mathematician who discovered the focus and directrix of the parabola, and their given relation. One of the most famous mathematicians to contribute to the study of parabolas was Galileo. He determined that objects falling due to gravity fall in parabolic pathways, since gravity has a constant acceleration. Later, in the 17^th century, many mathematicians studied properties of the parabola. Gregory and Newton discovered that parabolas cause rays of light to meet at a focus. While Newton opted out of using parabolic mirrors for his first telescope, most modern reflecting telescopes use them. Mathematicians have been studying parabolas for thousands of years, and have discovered many interesting properties of the parabola.

How could you as a teacher create an activity or project that involves your topic?

A fun activity to set up for your students will include several boxes and balls, for a smaller set up, you can use solo cups and ping pong balls. Divide the class into groups, and give each group a set of boxes and balls. First, have the students set up a tower(s) with the boxes. The students will now attempt to knock the boxes down using the balls. The students can map out the parabolic curve showing the path they want to take. By changing the distance from the student throwing the ball and the boxes, the students will be able to see how the curve changes. If students have the tendency to throw the ball straight instead of in the shape of a parabola, have a member of the group stand between the thrower and the boxes. This will force the ball to be thrown over the student’s head, resulting in the parabolic curve. The students can also see what happens to the curve depending on where the student stands between the thrower and boxes. In order for the students to make a positive parabolic curve, have them throw the ball underhanded. This activity will engage the students by getting them involved and active, plus they will have some fun too! (To start off with, you can show the video from part E1, since the students are playing a real life version of Angry Birds!)

How can technology be used to effectively engage students with this topic?

A great video to show students before studying parabolas can be found on YouTube:

The video uses the popular game Angry Birds to introduce parabolic graphs. First, the video shows the bird flying a parabolic path, but the bird misses the pig. The video goes on to explain why the pig can’t be hit. It does a good job of explaining what a parabola is, why the first parabolic curve would not allow the bird to hit the pig, and how to change the curve to line up the path of the bird to the pig. This video would be interesting to the students, because a majority of the class (if not all) will know the game, and most have played the game! The video goes even further by encouraging students to look for parabolas in their lives. It even gives other examples such as arches and basketball. This will get the students thinking about parabolas outside of the classroom. (This video would be perfect to show before the students try their own version of Angry Birds discussed in part A2)

Resources:

Youtube.com/watch?v=bsYLPIXI7VQ

Parabolaonline.tripod.com/history.html

http://www-history.mcs.st-and.ac.uk/Curves/Parabola.html

Preparation for Industrial Careers in the Mathematical Sciences: Finding the Safest Place to Store Nuclear Waste

The Mathematical Association of America recently published a number of promotional videos showing various mathematics can be used in “the real world.” Here’s the first pair of videos describing the process of mathematical modeling. From the YouTube descriptions:

Dr. Genetha Gray talks about her path and about a research problem that she worked on at Sandia National Laboratories. Using quite limited geological data, they had to create a groundwater flow computational model, with parameters to be determined, so that they could study the feasibility and safety of prospective subsurface nuclear waste storage sites.

Prof. Gwen Spencer of Smith College introduces the mathematics behind optimization, calibration, and the quantification of uncertainty in models and in the results that they give.

Engaging students: Factoring quadratic polynomials

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Kristin Ambrose. Her topic, from Algebra: factoring quadratic polynomials.

In previous courses, students would have learned how to solve one-variable linear equations. These kinds of equations would involve variables to the power of one. Quadratic equations extend from this since they add a variable to the equation that is to the power of two. Since students learned how to solve linear equations, they may be curious as to how they can solve quadratic equations that extend from this. Factoring is a way for students to solve these kinds of equations.

Also, in previous courses students would have learned about the ‘factors’ of a number. When talking about numbers, the factors are the numbers you multiply to get another number. For example the positive factors of six are one and six, and two and three. Factoring quadratic polynomials follows this logic, except instead finding the factors of a number, students are finding the factors of an expression. For example, the factors of the expression $x^2+4x+3$ are (x+3) and (x+1). Just like how when we multiply two times three we get six, when we multiply (x+3) times (x+1) we get the expression $x^2+4x+3$ .

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

There is a popular video game called Angry Birds in which the user launches birds to try and knock down structures built by pigs. This game relates to factoring quadratics because if we were to plot the trajectory of the birds being launched on a graph, the result would be a parabola, in other words the graph of a quadratic function. Factoring quadratic polynomials is a way to find the solutions of a quadratic, and the solutions are where the parabola crosses the x-axis. In Angry Birds, we could set our x-axis to be the ground, and our solutions would correspond with where on the ground the bird would land, if nothing were to block its path. If students were given the quadratic equation for the parabola corresponding with the bird’s trajectory, students would be able to factor the equation to solve for where on the ground the bird would land.

What are the contributions of various cultures to this topic?

Indian and Islamic cultures are two major cultures that have contributed to the topic of factoring quadratic polynomials. In Islamic culture, Al-Khwarizmi contributed to this topic by creating a way to solve quadratic equations by reducing the equations to one of six forms, which were then solvable. He described these forms in terms of squares, roots, and numbers, much like the terms we use today when factoring quadratic polynomials. The ‘squares’ related to what would today be our ‘x²’ term, the ‘roots’ related to the ‘x’ term, and the ‘numbers’ to the ‘c’ constant term. One of the forms he described was “squares and roots equal numbers,” in modern terms, “ax² + bx = c.” Today, we factor quadratic polynomials of the form “ax² + bx + c” which is similar to the form Al-Khwarizmi described. (Islamic Mathematics – Al-Khwarizmi)

In Indian culture, Brahmagupta contributed to the concept of factoring quadratics by introducing the idea that a number could be negative. This was significant because it meant a number like 9 could be factored into 3² or (-3)². Since a number could have a negative factor, it followed that quadratic equations could have two possible solutions, since one solution could be negative. Factoring quadratic polynomials like we do today would be impossible without the knowledge that quadratic expressions can have two solutions. (Indian Mathematics – Brahmagupta)

References:

“Islamic Mathematics – Al-Khwarizmi.” The Story of Mathematics. 2010. Web. 17 Sept. 2014. <http://www.storyofmathematics.com/islamic_alkhwarizmi.html>.

“Indian Mathematics – Brahmagupta.” The Story of Mathematics. 2010. Web. 17 Sept. 2014. <http://www.storyofmathematics.com/indian_brahmagupta.html>.