## Thursday, October 20, 2011

### Play dough math

I've been having fun with play dough lately, making stuff like this with my kids:

I think it would be fun to do in the math classroom with younger students, as questions like this pop up:
You need to make a 30-second video, with 24 frames per second, and each image takes up 4 frames - how many images do you need?
Having the dinosaur not walk like a spaz could be done by taking into account the length of time for each step and the rate the images are being played back. I didn't do this, as you may have noticed.
You could show the application of quadratic functions by having students try to make a projectile look realistic in claymation. Maybe I've spent too long around the play dough.

## Friday, October 14, 2011

### Free math poster for your classroom!

I noticed this little guy on a wall as I was walking along the other day, and thought about the difference math education makes in perceiving the snail.

A snail builds its shell at a rate proportional to its size (see Nature's Numbers), resulting in exponential growth. An understanding of the exponential function is so important in today's world (see examples of its application) that I was inspired to make a poster. Feel free to print it for your classroom, and let me know what you think!

## Wednesday, October 5, 2011

### Flash Math Art

Without an Internet connection in our new apartment, I was inspired to be creative.  Here are a few experiments with Flash, using math to make art.

1. Paths are generated by altering the 2nd-derivative of the x and y components.  It's not as tricky as it sounds.  A common way to simulate gravity (which has a constant acceleration, or second derivative) is to adjust position as follows.
x = x + dx;   //just increasing x by the amount dx in each time step.
dx = dx + g; //just increasing dx (velocity) by a constant amount (acceleration) in each time step.

In the case here, the amount g changes sinusoidally (for no particular reason), making the paths wiggle around and do weird stuff. There's some more stuff in there (I forget why they sometimes just go straight), but remember - but there's no randomness in how the paths are made (they are deterministic). Click to start:

Click to stop the first one so it doesn't slow you down.
2. Similar idea as the first one, but this time not as smooth. The value g changes as the remainder when the time is divided by some value (modulus).

3. This time the x and y derivatives make zig-zag graphs, so the x and y positions are smooth curves (almost like sinusoids). I call these guys Lissajou Worms:

4. The last one got me thinking about Lissajou curves (x and y are sine functions of time), so I started with this:

Click to stop, and again to restart with new conditions. Watch the x position go all the way left, then all the way right, and the y position go all the way up and all the way down. So we have two functions to control position: x(t) = sin(b*t + c) and y(t) = sin(d*t), where b, c, and d are random above.  When one of the periods is a multiple of the other one, you get a static shape like one of these.

We'll come back to this in a minute.

5. What would make these shapes more interesting?  Both x and y have a constant amplitude, so maybe we should mix that up a bit.  I added another component to the x function - now x(t) = f(t) + g(t), where f and g are both sine waves.  Watch for y continuing to go all the way up and down, while x does more complicated stuff:

6. What would it look like if y were the sum of two waves as well?  Well, it looks like craziness.

7. To tone down the craziness, I made sure the periods coincided better.  Here, if
x(t) = f(t) + g(t)
y(t) = j(t) + k(t)

then the period of f is a multiple of j's period (or vice versa), and g and k have the same period.  That way we get a periodic curve of some kind:

Click twice to get a new design. I hope you enjoyed your dose of math art!