Author: Lensyl Urbano

93 Ways to Prove Pythagoras’ Theorem

Geometric proof of the Pythagorean Theorem by rearrangemention from Wikimedia Commons' user Joaquim Alves Gat. Animaspar.

Elegant in its simplicity but profound in its application, the Pythagorean Theorem is one of the fundamentals of geometry. Mathematician Alexander Bogomolny has dedicated a page to cataloging 93 ways of proving the theorem (he also has, on a separate page, six wrong proofs).

Some of the proofs are simple and elegant. Others are quite elaborate, but the page is a nice place to skim through, and Bogomolny has some neat, interactive applets for demonstrations. The Wikipedia article on the theorem also has some nice animated gifs that are worth a look.

Cut the Knot is also a great website to peruse. Bogomolny is quite distraught about the state of math education, and this is his attempt to do something about it. He lays this out in his manifesto. Included in this remarkable window into the mind of a mathematician are some wonderful anecdotes about free vs. pedantic thinking and a collection of quotes that address the question, “Is math beautiful?”

Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.

– Bertrand Russell (1872-1970), The Study of Mathematics via Cut the Knot.

Adding Positive and Negative Numbers with Dice

We want students to become as involved as possible with their work, but a lot of math is going to be repetitively working similar questions. I believe that giving students any additional degree of control over the questions they’re answering will be helpful to some degree. So, for working with addition of positive and negative integers on a timeline, letting students generate their own problems might add a little interest, and be a little more engaging than just answering the questions in the text. There are a number of ways of doing this, but using two sets of differently colored dice might be fairly easy to put together and appeal to the more tactile-oriented students.

So give each student a set of dice, say six, of two different colors, say red and wooden-colored. Have them roll them then organize them in a line. Your red dice are positive integers and your wooden dice are negative integers.

Set of positive (red) and negative (wood) dice. Dice images assembled from Wikipedia user AlexanderDreyer.

The dice in the image above would produce the expression:

5 + (-1) + 3 + (-4) + (-6) + 1 =

It might make sense to start with two then move up to four and six (or even use odd numbers as long as you include different colors). You could use a more specialized dice with different numbers of sides, but I think the standard six-sided ones would be sufficient for this exercise.

History, Captured in the River Fleet Sewer

Under London, in the River Fleet. Image by suburban.com via Flickr.

History is hard sometimes, when all you have are dates and events to remember. It helps to have context. Montessori schools build a lot of history and social science on the concept of the needs of people. While the need for electronics excites many of my students, another fundamental need is for sanitation.

RJ Evans has a wonderful post, full of excellent photography that will go a long way toward capturing the imagination, which encapsulates the history of London by looking at the evolution of the River Fleet – from a “clear and sparkling” stream in medieval times, to a chartered, elegant, underground sewer system built by excellent, Victorian engineers that still functions today.

Everything is in place, thanks to the ingenuity of the Victorian engineers, to ensure that the Fleet is confined to these tunnels. Yet it was not always like that. If we travel back a few centuries we find a different story altogether – one which is not without its own pathos if such an emotion can be felt for a river.

– Evans, 2011: The Fleet – London’s Underground River in Kuriositas.

Match Stick Rockets

A great, simple, and slightly dangerous way of making rockets. There are a number of variations. I like NASA’s because they have a very nice set of instructions.

How to make a match stick rocket. By Steve Cullivan via NASA.

With a stable launch platform that maintains consistent but changeable launch angles, these could be a great source of simple science experiments that look at the physics of ballistics and the math of parabolas (a nice video camera would be a great help here too) and statistics (matchsticks aren’t exactly precision instruments).

The Spirit of the Law

A You are the Ref strip by Paul Trevillion.

Every week, artist Paul Trevillon poses, in text and cartoon form, some truly idiosyncratic situations that might come up in a soccer match in his You are the Ref strip on the Guardian website. Readers get a week to propose their solutions and then referee Keith Hackett give his official answers.

It’s a fascinating series, the subtext of which is that, while there is a lot of minutiae to remember – the actual diameter of a soccer ball is important for one question – the game official is really out there to enforce the spirit of the laws, enabling fair and fluid play to the best of their ability. This is a useful lesson for adolescents who tend toward being extremely literal, and have to work on their abstract thinking skills, especially when they relate to questions of justice. For this reason, I find that when refereeing their games it’s useful to take the time during the game, and afterward in our post-match discussions, to talk about the more controversial calls.

Cyberwar

Perhaps it’s cultural conditioning, or maybe it’s genetic a predisposition, but adolescent boys to seem to have more of a predilection for war games than their female peers. The games they like tend to be first-person-shooters, like Call of Duty, and, given the trends toward improved video game graphics and remote, kinetic military action, real and simulated life seem to be converging.

Recently however, the Stuxnet virus exposed a much less glamorous picture of the future of cyber warfare. Kim Zetter has an excellent, extensive article in Wired on the computer scientists and engineers who reverse engineered the virus to try to figure out who made it and what it did. Since the virus seems to have been aimed at damaging Iran’s nuclear enrichment plants their work brought them to the edge of the world of international espionage. And they still don’t really know who created this remarkably sophisticated virus, though they suspect the U.S. and Israel.

One of the most interesting take-home messages from Zetter’s article is the amazing degree of international collaboration it took to figure things out. The virus was discovered by someone in Belarus. Researchers from the anti-virus company Symantec’s offices in California, Tokyo and Paris worked together passing information from one office to the next to keep the project going 24 hours a day. They published their findings to share them, and when they ran into stumbling blocks they couldn’t solve they put out calls for help on the internet – and people responded, bringing in expertise from Germany and the Netherlands.

The virus’ secretive creators and the open, diverse collaborators who untangled the virus reflect two conflicting aspects of the future that computer technology and the internet are making possible. And this conflict is showing up more and more in different areas – take Wikileaks for example – so it will be very interesting to see where the future takes us. Of course, we are not simply flotsam on the tides of history. As citizens of the internet, we have been enabled. We have a certain power, and a concomitant responsibility, to shape what we have for the benefit of our fellow citizens and those that come after us.