July 2011 – Montessori Muddle

Interactive Model showing the Kinetic Energy of a Gas

I really like this little video because it’s relatively dense with information but its visual cues complement each other quite nicely; the interactive model it comes from is great for demonstrations, but even better for inquiry-based learning. The model and video both show the motion of gas molecules in a confined box.

In the video, the gas starts off at a constant temperature. Temperature is a measure of how fast the particles are moving, but you can see the molecules bouncing around at different rates because the temperature depends on the average velocity (via Kinetic Energy), not the individual rates of motion. And if you look carefully, you notice the color of the particles depends on how fast they’re moving. A few seconds into the video, the gas begins to cool, and you can see the particles slow down and gradually the average color changes from blue (fast) to red and then some even fade out entirely.

In the interactive, VPython model I’ve put in a slider bar so you can control the temperature and observe the changes yourself. The model is nicely set up for introducing students to a few physics concepts and to the scientific method itself via inquiry-based learning: you can sit them down in front of the program, tell them it’s gas molecules in a box, have them observe carefully, record what they see, and then explain their observations. From there you can branch off into a lot of different places depending on the students’ interests.

Temperature (T) – a measure of the average kinetic energy (KE_average) of the substance. In fact, it’s proportional to the kinetic energy, giving a nice linear equation in case you want to tie it into algebra:
$! T = c {KE}_{average}$
where c is a constant.

Of course, you have to know what kinetic energy is to use this equation.
$! KE = \frac{1}{2} m v^2$
Which is a simple parabolic curve with m being the mass and v the velocity of the object.

The color changes in the model are a bit more metaphoric, but they come from Wein’s Displacement Law, which relates the temperature of an object, like a star, to the color of light it emits (different colors of light are just different wavelengths of light).

$! T = \frac{b}{l}$

where b is another constant and l is the wavelength of light. This is one of the ways astronomers can figure out the temperature of different types of stars.

Notes

The original VPython model, from Chabay and Sherwood’s (2002) physics text, Matter and Interactions, comes as a demo when you install their 3D modeling program VPython.

I’ve posted about this model before, but I though it was worth another try now that I have the video up on YouTube.

93 Ways to Prove Pythagoras’ Theorem

Pythagoras-2a-300 — 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.

math-pos-neg-int-dice — 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.

Thinking Like a Cartoonist

The Guardian website has some very interesting glimpses into the mind of the artist, with a series of videos where Paul Trevillion describes his thought process as he draws different players for his You are the Ref strip.

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.

match_rocket_howto — 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

You-are-the-Ref-Pele-004 — 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.