I’ve only just begun to work on some of these, but I’m having a good time so far.

The first one that caught my eye was this:

Can you find a fraction so that the difference between the fraction and its reciprocal is exactly equal to 1?

That is

The problem gave the example that

D’oh – off by

Then the question asks – Can you find another fraction that gets closer than this?

I approached this from a couple of different ways – first I took the original equation and created a common denominator to get or and .

I didn’t pursue this past that point, but did come back to it later.

Next, I broke out the spreadsheet and set it up to take all the numbers from 1-30 and create fractions and their reciprocals from these and subtract them.

Looking at all that, I noticed a few places where the difference between 1 and the was smaller than .

This happened for and .  Then it was time for class.

The numbers in the fractions that were getting close to 1 had caught my eye yesterday and this morning when I came in, I sat down with it again and saw that they were all consecutive Fibonacci numbers.  So, I made a new spreadsheet with Fibonacci numbers and the fractions and reciprocals and noticed that the difference was approaching 1.

At some point yesterday afternoon I went to Wolfram Alpha and typed in just to see what I would get and it provided a relationship between the two variables that comes from treating one of the variables as a constant so that

In other words a and b cannot both be integers!  And of course, some of you may have picked up on this sooner than I did – the number that produces the exact value of 1 is phi – The Golden Ratio.

.

FUN STUFF!

Here’s a link to what I did get done.

These tests are being used to reward/punish students, teachers, schools and communities.

John Ewing, the former Executive Director of the American Mathematical Society has a great article in the recent Notices of the American Mathematical Society about the use of these tests to make decisions that effect the lives of the students, teachers schools and communities.

But using tests to evaluate teachers, schools, or programs has many problems. (For a readable and comprehensive account, see [Koretz 2008].) Here are four of the most important problems, taken from a much longer list…

4. Inflation. Test scores can be increased without increasing student learning. This assertion has been convincingly demonstrated, but it is widely ignored by many in the education establishment [Koretz 2008, chap. 10]. In fact, the assertion should not be surprising. Every teacher knows that providing strategies for test-taking can improve student performance and that narrowing the curriculum to conform precisely to the test (“teaching to the test”) can have an even greater effect. The evidence shows that these effects can be substantial: One can dramatically increase test scores while at the same time actually decreasing student learning. “Test scores” are not the same as “student achievement”.

This last problem plays a larger role as the stakes increase. This is often referred to as Campbell’s Law: “The more any quantitative social indicator is used for social decision-making, the more subject it will be to corruption pressures and the more apt it will be to distort and corrupt the social processes it is intended to measure” [Campbell 1976]. In its simplest form, this can mean that high-stakes tests are likely to induce some people (students, teachers, or administrators) to cheat…and they do [Gabriel 2010]. But the more common consequence of Campbell’s Law is a distortion of the education experience, ignoring things that are not tested (for example, student engagement and attitude) and concentrating on precisely those things that are.

The project of corporate style management as a solution to the country’s educational problems has been beset by fraud just as corporate fraud brought our economy to the brink of collapse two years ago.

The blog Seattle Education 2010 has collected the various stories of fraudulent test results from the schools that were run more like businesses in Atlanta, Houston, New York City, Washington D.C., and Charleston, S.C.

In Atlanta, the investigation has been turned over to law enforcement (Georgia Bureau of Investigation).  USA Today has a good article detailing its own investigation into test scores in six states (Arizona, Florida, Ohio, California, Colorado and Michigan) and Washington, D.C.

I’ll end with another quote from Diane Ravitch’s article:

When parents complained that their children’s high scores didn’t make sense, since they were still struggling to do basic math, they were ignored.

This is the real issue – the focus on the tests is a classic example of mistaking the map for the territory.  It’s not the test score that’s important.  It’s what the test score is supposed to reflect – a student’s abilities.  The problem is that when it becomes obvious that the scores don’t reflect the students’ abilities, nobody pays attention, because everyone is being paid based on the test – not the learning.

The posts for that course are here.

The most recent post was about the French mathematicians of the 17th century – Viète, Mersenne, Fermat, Descartes and Pascal.

French Mathematics of the 17th century

Francois Viète (1540-1603)

Francois Viète was the son of a lawyer in 16th century France.  He is credited with devising a scheme* in which unknown quantities in algebra would be represented by letters that are vowels and constant quantities would be represented by letters that are consonants.  At the time, the Arabic algebra that had been transferred to Europe over the previous 500 years was based on prose writing – everything was described in words.  After Viète’s initial use of letters for unknowns and constants, René Descartes later began to use letters near the end of the alphabet for unknowns (x, y, z) and letters from the beginning of the alphabet for constants (a, b, c).  This practice continues today.

In 1593, the Dutch ambassador to France said to French King Henry IV that a well-known Dutch mathematician had posed a problem that was beyond the capabilities of ANY French mathematician.  Henry IV passed the problem along to Viète and Viète was able to solve it.

Viète began a correspondence with Roomen, the Dutch mathematician who had posed the problem originally and became one of the first internationally recognized French mathematicians.  He worked mainly in trigonometry, astronomy and the theory of equations.

*This link is a paper written by a college student at Rutgers University in New Jersey.  Papers on other subjects by other students in the same course can be found here.

Marin Mersenne (1588-1648)

Marin Mersenne was a French monk best known for his research into prime numbers.  He also did important research into the musical behavior of a vibrating string, showing that the frequency of the vibration was related to the length, tension, cross section and density of the material.

Mersenne primes are prime numbers of the form , where p is a prime number itself.  For example

which is prime

and so on.

Mersenne was also interested in the work that Copernicus had done on the movement of the heavenly bodies and despite the fact that, as a monk, he was closely tied to the Catholic church, he promoted the heliocentric theory in the 1600′s.

Mersenne was also known as a friend, collaborator and correspondent of many of his contemporaries.  Fermat, Pascal, Descartes, Huygens, Galileo, and Torricelli all corresponded with Mersenne and the exchange of ideas among these scientists promoted the understanding of music, weather and the solar system.

René Descartes (1596-1650)

René Descartes is probably best known for two things.  One is the conclusion “I think therefore I am” (Cogito ergo sum in Latin and Je pense donc je suis in French) and the other is the geometric coordinate system generally known as the Cartesian plane.

Descartes joined the army of Prince Maurice of Nassau in 1619 and was in Bavaria (southern Germany) and Bohemia (Czech Republic) during the beginning of the Thirty Years War.

The importance of the Cartesian Plane is difficult for us to understand today because it is a concept that we are taught at a young age.  Locating objects on a grid by their horizontal and vertical coordinates is so deeply embedded in our culture that it is difficult to imagine a time when it did not exist.

Before Descartes’ grid system took hold, there was Geometry:

and there was Algebra:

(Click on photo for larger view)

…and they were separate fields of endeavor.  The idea that a geometric shape like a parabola could be described by an algebraic formula that expressed the relationship between the curve’s horizontal and vertical components really is a ground-breaking advance.  It is so ground-breaking that once it happened, people began to forget that it hadn’t always been that way.

Once this new method for describing curves was developed, the question of finding the area under a curve was addressed.  This is the general problem of Integral Calculus.  Descartes (among others) saw that, given a polynomial curve , the area under the curve could be found by applying the formula

These were the rudimentary beginnings of the development of the Calculus that would be devised by Isaac Newton and Gottfried Leibniz in the ensuing years.

Fermat (1601-1665)

Pierre Fermat is also mostly remembered for two important ideas – Fermat’s Last Theorem and Fermat’s Little Theorem.  Fermat’s Last Theorem is a simple elegant statement – that Pythagorean Triples are the only whole number triples possible in an equation of the form .

Pythagorean Triples are interesting groups of numbers that satisfy the Pythagorean relationship .  Triples such as {3,4,5} {6,8,10} {8,15,17} {7, 24, 25} can be found that satisfy the equation.  But – Fermat’s Last Theorem says that if the in the original equation is any number higher than two, then there are no whole number solutions.

It’s true – but very difficult to prove.  Mathematicians tried for 350 years or so to prove this theorem before it was finally accomplished by Andrew Wiles in 1995.

By the way, you can generate Pythagorean Triples using the following formulas:

Pick two numbers and , with

Fermat’s Little Theorem is a useful and interesting piece of number theory that says that any prime number divides evenly into the number , where is any number that doesn’t share any factors with .

Blaise Pascal (1623-1662)

Blaise Pascal was the son of Etienne Pascal, who was a lawyer and amateur mathematician.  Etienne Pascal knew Marin Mersenne and often visited him at his Paris monastery, and when Blaise was a teenager he sometimes accompanied his father on these visits.

Pascal’s first published paper was a work on the conic sections.  He also did research on the composition of the atmosphere and noticed that the atmospheric pressure decreased as the elevation increased.  This led him to believe that beyond the atmosphere there existed a vacuum in which there was no atmospheric pressure.

René Descartes visited Pascal in 1647 and they argued about the existence of a vacuum beyond the atmosphere.  Descartes felt that this was impossible and criticized Pascal, saying that he must have a vacuum in his head.

Pascal is known for the structure of Pascal’s Triangle, which is a series of relationships that had previously been discovered by mathematicians in China and Persia.

Here is Pascal’s version:

Here is the Chinese version:

Here is a version that we often see in textbooks:

Each successive level is created by adding the two numbers above it, so in the 6th row {1,5,10,10,5,1} the 10 is created by adding the 4 and the 6 from the row above it.  These number patterns are actually quite useful in a wide variety of situations.

In raising a binomial to a power like , the coefficients of each term are the same as the numbers from the 6th row:

These numbers are also related to Discrete Mathematics and Combinatorics which describes how many ways there are to choose something from a series of possibilities.

There was a lot of great mathematics happening in Italy, England, Holland and Germany during the 17th century, but this collection of French mathematicians spanning nearly 100 years produced a tremendous amount of very important mathematical ideas.

The English, Germans and Swiss would make great contributions to mathematics in the 18th century with Newton, Leibniz, the Bernoullis, Euler and others, while the French would still contribute with the works of Laplace, Lagrange and Legendre.

As part of these talks, I’ve had to make explicit the difference between the Cartesian Plane and the Argand Diagram.  I’ve been doing this by showing the mapping of the points from the x-real number line to the y-real number line.  George Abdo and Paul Godfrey have a nice website that shows this process.

OK, then if we cross the x and y number lines to create the Cartesian Plane, we’ll see the picture of the graph that we’re used to.

Then, we have to consider the complex mapping.  In this situation, each x value is two-dimensional and is mapped to a two-dimensional y coordinate, like so:

(Click the picture for a better view, or check out the link to Prof. Abdo’s website)

But to show these together, as the Cartesian Plane does for real-valued functions would require 4 dimensions, which creates difficulty for human beings who normally have enough trouble with 3 dimensions.  What people have done instead is to take the x values from the complex plane and color them based on their corresponding y values.

Lawrence Crone at American University has some nice pictures showing this effect.

Andrew Bennett at Kansas State University has a nifty online complex graphing calculator on his website.  You can type in a formula and the software will show you a representation of the mapping.  I prefer the top view to see the roots, but the side view is interesting as well.

The graphing utility window is limited to complex x values a+bi in which a and b are both between 2 and -2, but this can be adjusted by right clicking on the window.

Something that shows up nicely on the complex grapher at the previous link are roots of unity – typing in z³-1 will show the cube roots of 1, both real and complex…

I’ve written about the complex or Gaussian primes in a previous post, but I realized that I had never put up a picture of the complex primes, which is what motivated me in the first place.

From Wikipedia:

Also, from the Dutch company Sanny de Zoete, here is a picture of linens made in the pattern of the complex primes:

What we see here is that, although China and Japan do have substantial holdings of U.S. government debt, they are far from being majority stakeholders.

At its heart, his comparison of Ireland and Nevada highlights the issues very clearly,

Climate, scenery and history aside, the nation of Ireland and the state of Nevada have much in common. Both are small economies of a few million people highly dependent on selling goods and services to their neighbors. (Nevada’s neighbors are other U.S. states, Ireland’s other European nations, but the economic implications are much the same.) Both were boom economies for most of the past decade. Both had huge housing bubbles, which burst painfully. Both are now suffering roughly 14 percent unemployment. And both are members of larger currency unions: Ireland is part of the euro zone, Nevada part of the dollar zone, otherwise known as the United States of America.

But Nevada’s situation is much less desperate than Ireland’s.

First of all, the fiscal side of the crisis is less serious in Nevada. It’s true that budgets in both Ireland and Nevada have been hit extremely hard by the slump. But much of the spending Nevada residents depend on comes from federal, not state, programs. In particular, retirees who moved to Nevada for the sunshine don’t have to worry that the state’s reduced tax take will endanger their Social Security checks or their Medicare coverage. In Ireland, by contrast, both pensions and health spending are on the cutting block.

Also, Nevada, unlike Ireland, doesn’t have to worry about the cost of bank bailouts, not because the state has avoided large loan losses but because those losses, for the most part, aren’t Nevada’s problem. Thus Nevada accounts for a disproportionate share of the losses incurred by Fannie Mae and Freddie Mac, the government-sponsored mortgage companies — losses that, like Social Security and Medicare payments, will be covered by Washington, not Carson City.

And there’s one more advantage to being a U.S. state: it’s likely that Nevada’s unemployment problem will be greatly alleviated over the next few years by out-migration, so that even if the lost jobs don’t come back, there will be fewer workers chasing the jobs that remain. Ireland will, to some extent, avail itself of the same safety valve, as Irish citizens leave in search of work elsewhere and workers who came to Ireland during the boom years depart. But Americans are extremely mobile; if historical patterns are any guide, emigration will bring Nevada’s unemployment rate back in line with the U.S. average within a few years, even if job growth in Nevada continues to lag behind growth in the nation as a whole.

Over all, then, even as both Ireland and Nevada have been especially hard-luck cases within their respective currency zones, Nevada’s medium-term prospects look much better.

What does this have to do with the case for or against the euro? Well, when the single European currency was first proposed, an obvious question was whether it would work as well as the dollar does here in America. And the answer, clearly, was no — for exactly the reasons the Ireland-Nevada comparison illustrates. Europe isn’t fiscally integrated: German taxpayers don’t automatically pick up part of the tab for Greek pensions or Irish bank bailouts. And while Europeans have the legal right to move freely in search of jobs, in practice imperfect cultural integration — above all, the lack of a common language — makes workers less geographically mobile than their American counterparts.

The issue here is that the difficulty of creating a monetary union without a corresponding political fiscal union creates huge problems for the member states and their citizens.

Although it doesn’t introduce new concepts, students always have some difficulty with the format -

which is matching.

Also, it is the type of matching where you can’t use the process of elimination as some questions have more than one answer, and some equations satisfy more than one question.

FRIDAY FUN

Match each description below with the appropriate letter(s).

Some statements may satisfy more than one of the equations.

Find the equation of a line:

1) Through the point (1, −1)
2) Through the points (5, 2) and (−1, − 4)
3) With a slope of − 3
4) With a slope of 0.5 through the point (5, 6)
5) Through the point (−3, − 6)

A) 3y + x = −21
B) y = x − 3
C) y + 3x = 2
D) 2y = x + 7
E) y = 2x − 3

AND

Find the equation of a line:

1) Through the points (4,−4) and (12,2)
2) Through the point (8,−1)
3) With a slope 1/3 through the point (5,−7)
4) Through the points (1,4) and (−3,1)
5) With a slope of − 2 through the point (5, 5)
6) With a slope of 3/4

A) 4y − 3x = 13
B) y = 0.75x − 7
C) y+7= 1/3(x-5)
D) 2x + y =15