If you've not yet discovered the Wolfram Demonstrations Project site you're in for a great treat. The site has tons of interactive Math applications that you can run with the free Mathematica Player. In other words, you don't need to own Mathematica to run the demonstration apps.
I find many of the apps to be very interesting. One I particularly like is "Enumerating Pythagorean Triangles." It shows a nice relationship between Phythagorean Triples (positive integers a, b, c such that a^2 + b^2 = c^2) and the unit circle.
Below is an image from Wikipedia of a calculating device invented in 1891, Genaille's rods. Each column in the image is a rod that can be placed wherever it's needed in a calculation.
The image shows how to easily calculate 52749x4.
Can you figure out how this arithmetic tool works?
Making a set of these rods would be a great enrichment project. Here is a template (in different sizes) that you or your children or students can print and mount onto cardstock to make your own set. You'll want to make a number of each of the rods so that you can multiply by numbers where one or more digits occurs more than once.
I found these "Math Myths" from his site to be quite interesting:
- You can't take 7 from 3.
- When you multiply, the answer is bigger.
- You have to add from right to left.
- When you subtract the result is smaller.
- Fractions are small numbers.
- There's only one way to do something.
- When you add the result is bigger.
- When you divide the result is smaller.
- I can't do it unless someone tells me how to do it.
- Math is hard and only a few people can do it.
- You have to know everything about whole numbers before you can do fractions.
- You have to know algebra before you can learn about calculus.
How many of these "facts" do we take for granted, even though they're not true?
Here's an interesting exploration, illustration courtesy of Mathematica:
There's a nice animated illustration at Mathematica.
A particularly interesting case of a roulette is a regular n-gon rolling on a "road" composed of a sequence of truncated catenaries, as illustrated above. This motion is smooth in the sense that the geometric centroid follows a straight line, although in the case of the rolling equilateral triangle, a physical model would be impossible to construct because the vertices of the triangles would get "stuck" in the ruts (Wagon 2000).
Mental Floss is one of my very favorite magazines. Here's a simple (if you see it) puzzle from their web-site:
The answer is here.
Here's a very geek way to impress your geek friends. asciiTeX!
Like the name says, asciiTeX takes input similar to that for LaTex and renders mathematical equations in plain ASCII. Wow!
Check it out at Sourceforge.net.
If you're a student, or teacher, or homeschooling parent and you've not heard about the Khan Academy you owe it to yourself to check it out. Salman Khan quit his career in the financial world to teach the world via videos. If you look on Khan's Youtube Channel you'll see videos on biology, linear algebra, chemistry, calculus, statistics, physics, differential equations, algebra, arithmetic, GMAT problem solving, pre-algebra, trigonometry, probability, geometry, precalculus, Singapore Math, and more.
Khan has also made videos of all eight Math SAT practice tests in the College Board's Official SAT Study Guide. Wow!
Here's a great promotional video of Khan's work:
Hat tip to Murray at squareCircleZ.
At TED2010, mathematics legend Benoit Mandelbrot develops a theme he first discussed at TED in 1984 -- the extreme complexity of roughness, and the way that fractal math can find order within patterns that seem unknowably complicated.
Here's some biographical information on Mandelbrot:
Studying complex dynamics in the 1970s, Benoit Mandelbrot had a key insight about a particular set of mathematical objects: that these self-similar structures with infinitely repeating complexities were not just curiosities, as they'd been considered since the turn of the century, but were in fact a key to explaining non-smooth objects and complex data sets -- which make up, let's face it, quite a lot of the world. Mandelbrot coined the term "fractal" to describe these objects, and set about sharing his insight with the world.
The Mandelbrot set (expressed as z² + c) was named in Mandelbrot's honor by Adrien Douady and John H. Hubbard. Its boundary can be magnified infinitely and yet remain magnificently complicated, and its elegant shape made it a poster child for the popular understanding of fractals. Led by Mandelbrot's enthusiastic work, fractal math has brought new insight to the study of pretty much everything, from the behavior of stocks to the distribution of stars in the universe.
And, here's the 17 minute presentation:
There's been a lot of press lately about how Russian mathematician Grigory Perelman has turned down the $1 million prize offered by the Clay Mathematics Institute for solving one of the world's hardest problems, the Poincaré Conjecture. But, these news stories typically provide little information about the problem or the man who conquered it.
In 2006, The New Yorker ran a lengthy piece, Manifold Destiny, providing plenty of background. It's a lengthy but very compelling read despite its age.
Many properties of primes are very difficult to determine and prove. Here's an exploration that's within reach of many of us:
What is interesting about the difference of the squares of most any two primes? In other words, what is interesting about p12-p22 for most primes p1 and p2? When does this property hold? Prove your assertion.
I got the idea for this puzzle/exploration from Standup Maths and adapted it to make it harder!