Straight Lines

[inline]

[script type=”text/javascript”]
var width=500;
var height=500;
var xrange=10;
var yrange=10;

mx = width/(2.0*xrange);
bx = width/2.0;
my = -height/(2.0*yrange);
by = height/2.0;

function draw8587(ctx, polys) {
t=t+dt;
//ctx.fillText (“t=”+t, xp(5), yp(5));
ctx.clearRect(0,0,width,height);

polys[0].drawAxes(ctx);
//drawAxes(ctx);
ctx.lineWidth=3;
//ctx.beginPath();
//ctx.moveTo(x1,y1);
//ctx.lineTo(x2,y2);
//ctx.stroke();
//ctx.closePath();
//y2+=dyp(dy);

// add polynomial timing
// First move (from y=0 to y=1)
if (polys[0].c < 1.0) { polys[0].c = polys[0].c + 2.0*dt*change*move_dir; } // Second move (change from y=1 to y=0.5x+1) else if (polys[0].b < 0.5) { polys[0].b = polys[0].b + dt*change*move_dir; polys[0].order = 1; } // Third move (change from y=0.5x+1 to y=0.5x+4) else if (polys[0].c < 4.0) { polys[0].c = polys[0].c + 2.*dt*change*move_dir; //polys[0].order = 1; } else { //move_dir = move_dir * -1; end_ct = end_ct + dt; if (end_ct >= dt*40) {
polys[0].a = 0;
polys[0].b = 0;
polys[0].c = 0;
polys[0].order = 0;
}
}
ctx.textAlign=”center”;
ctx.textBaseline=”alphabetic”;
ctx.font = “14pt Arial”;
ctx.fillStyle = ‘#f00’;
ctx.fillText (“Equation of a Line”, xp(-5), yp(9));
ctx.fillText (“y = mx + b”, xp(-5), yp(5));
ctx.font = “12pt Arial”;
//ctx.fillText (“y = “+polys[0].b.toPrecision(1)+”x + “+ polys[0].c.toPrecision(2), xp(-5), yp(7));
ctx.fillText (“b = intercept = “+polys[0].c.toPrecision(2), xp(-7), yp(3));

//draw polynomials
ctx.lineWidth=3;
for (var i=0; i

Here’s a prototype for showing how the equation of a straight line changes as you change its slope (m) and intercept (b):
$! y = mx + b$

It starts with an horizontal line along the x-axis, where the slope is zero and the intercept is zero:
$! y = 0.0$

The line moves upward, but stays horizontal, until the intercept is 1.0:
$! y = 1.0$

The slope increases from horizontal (m = 0), gradually, until the slope is 0.5:
$! y = 0.5x + 1.0$

Finally, we move the line upwards again, without changing the slope (m = 0.5), until the intercept is equal to 4.0:
$! y = 0.5x + 4.0$

Now I just need to figure out how to make them interactive.

Dihydrogen Monoxide: Fatal if Inhaled

I just discovered there’s an entire website dedicated to the dangers of dihydrogen monoxide. I particularly like their research page that includes the results from several middle and high school research projects.

Numerical Integration

This is an attempt to illustrate numerical integration by animating an HTML5’s canvas.

We’re trying to find the area between x = 1 and x = 5, beneath the parabola:

$y = -\frac{1}{4} x^2 + x + 4$

By integrating, the area under the curve can be calculated as being 17 ⅔ (see below for the analytical solution). For numerical integration, however, the area under the curve is filled with trapezoids and the total area is calculated from the sum of all the areas. As you increase the number of trapezoids, the approximation becomes more accurate. The reduction in the error can be seen on the graph: with 1 trapezoid there is a large gap between the shaded area and the curve; more trapezoids fill in the gap better and better.

The table below show how the error (defined as the difference between the calculation using trapezoids and the analytic solution) gets smaller with increasing numbers of trapezoids (n).

Number of trapezoids	Area (units²)	Error (difference from 17.66)
1	15.00	2.66
2	17.00	0.66
3	17.37	0.29
4	17.50	0.16
5	17.56	0.10
6	17.59	0.07
7	17.61	0.05
8	17.63	0.03
9	17.63	0.03
10	17.64	0.02

Analytic solution

The area under the curve, between x = 1 and x = 5 can be figured out analytically by integrating between these limits.

$Area = \int_{_1}^{^5} \left(-\frac{x^2}{4} + x + 4 \right) \,dx$

$Area = \left[-\frac{x^3}{12} + \frac{x^2}{2} + 4x \right]_1^5$

$Area = \left[-\frac{(5)^3}{12} + \frac{(5)^2}{2} + 4(5) \right] - \left[-\frac{(1)^3}{12} + \frac{(1)^2}{2} + 4(1) \right]$

$Area = \left[-\frac{125}{12} + \frac{25}{2} + 16 \right] - \left[-\frac{1}{12} + \frac{1}{2} + 4 \right]$

$Area = \left[ 22 \frac{1}{12} \right] - \left[4 \frac{5}{12} \right]$

$Area = 17 \frac{2}{3} = 17.6\bar{6}$

(See also WolframAlpha’s solution).

Update

Demonstration of this numerical integration using four trapezoids in embeddable graphs.

Test HTML5 Canvas

[inline]

[script type=”text/javascript”]
document.write(“hello world!”);

var c2=document.getElementById(“myCanvas2”);
var ctx2=c2.getContext(“2d”);
ctx2.moveTo(10,10);
ctx2.lineTo(150,50);
ctx2.lineTo(10,50);
ctx2.stroke();
[/script]
[/inline]

So I’m experimenting with creating drawings using HTML5. And it works (mostly)!! I have to hardwire in paragraph breaks, but otherwise it works so far.

The key references:

w3schools.com: HTML5 Canvas: for the code for the drawing above.
Volcano’s Inline Javascript plugin for WordPress to make it work on this blog.
The page for Opera developers is also a great HTML5 drawing reference.
Dive into HTML5 is another excellent resource.
Bill Mill’s canvas tutorial.

Should Liberals Homeschool?

Dana Goldstein posts a critique of liberal homeschooling (and, by inference, of private schooling as well).

The crux of her argument is collective (for the greater good); having middle class kids (who often have highly educated parents) benefits the poorer kids in the public schools and thus society at large.

… “peer effects” have a large impact on student achievement. Low-income kids earn higher test scores when they attend school alongside middle-class kids, while the test scores of privileged children are impervious to the influence of less-privileged peers. So when college-educated parents pull their kids out of public schools, whether for private school or homeschooling, they make it harder for less-advantaged children to thrive.

— Goldstein (2012): Liberals, Don’t Homeschool Your Kids in Slate

She backs up her argument with evidence (Schwartz, 2010) that mixed income schools show better outcomes. The fact that school funding and populations come from distinct geographic districts, some of which may be rich while others are poor, is a major part of the problem.

However reasonable the argument is for society at large, it’s going to be a hard argument to make to homeschooling parents who choose that option because of how bad they see the public schools as being. She’s not just asking for some shared sacrifice, but for having parents risk sacrifice their kids’ education.

Until the public schools change in the more progressive directions that these homeschooling parents have a problem with, I don’t think she’ll get much traction. For her idealism I’ll give her the last word, but I suspect that she’s got it backwards.

If progressives want to improve schools, we shouldn’t empty them out. We ought to flood them with our kids, and then debate vociferously what they ought to be doing.

— Goldstein (2012): Liberals, Don’t Homeschool Your Kids in Slate

Tess of the d’Urbervilles: Police Composite

Brian Joseph Davis creates composite sketches of literary characters using the same software used by the police, and the descriptions of the characters in the books.

tess-durbaville — Composite sketch of Tess from Thomas Hardy's novel Tess of the d'Urbervilles. Image by Brian Joseph Davis.

She was a fine and handsome girl—not handsomer than some others, possibly—but her mobile peony mouth and large innocent eyes added eloquence to colour and shape… The pouted-up deep red mouth to which this syllable was native had hardly as yet settled into its definite shape, and her lower lip had a way of thrusting the middle of her top one upward, when they closed together after a word…Phases of her childhood lurked in her aspect still. As she walked along to-day, for all her bouncing handsome womanliness, you could sometimes see her twelfth year in her cheeks, or her ninth sparkling from her eyes…a thick cable of twisted dark hair hanging straight down her back to her waist.

— description by Thomas Hardy, excerpted by Davis (2010): The Composites.

Humans Beings, Super-Predators

Usually, when new, more powerful predators evolve or come in from elsewhere, the local species can often adapt by themselves becoming better defended through a variety of means [larger sizes, thicker shells for example]; but this option seems to be closed when it comes to the evolution of humans as super-predators.

— Geerat Vermeij (2012), quoted in Walker (2012): Super-predatory humans on the BBC website.

Humans, using ingenuity and tools, have become an uniquely, irresistible predator species that the world has never seen before, and to which other species are finding it very difficult to adapt. That’s the premise of a paper by Geerat Vermeij that’s nicely summarized by Matt Walker on the BBC website.

Normally, predators and prey evolve and adapt to each other. Lions are better able to attack and kill smaller buffalo, which means the larger buffalo are more likely to survive, which results, over time, in the average size of the buffalo herds getting larger.

Humans, on the other hand, like to target the larger buffalo, creating a selective pressure the other way. Unfortunately, once the larger specimens are gone, humans will go after the smaller ones, and the intensity of the attacks have often been enough to drive entire species into extinction.

Though humans have been around for a couple hundred thousand years, we still have not seen our full impact on the environment. Which is somewhat interesting to consider.