1984 or A Brave New World

The future according to Orwell vs. Huxley. Image from world-shaker.tumblr.com.

World-shaker tries to draw the modern parallels to 1984 and Brave New World in graphic form.

Orwell’s (1948) distopian view of the future in 1984, warned against the government developing the ability to exert constant, repressive monitoring of everyone, controlling the means of communication and, perhaps more importantly, the use of language. Huxley’s (1932) Brave New World, on the other hand, saw a mass media using your apparent predilection for trivialities to distract you from the important things. These two books are staples of secondary school literature, and it’s easy to see modern parallels; “kinetic military action” is currently my favorite Orwellian term.

Unfortunately, drawing modern parallels to historic literature is fraught with difficulty because it’s so easy: the human brain is predisposed to seeing patterns. World-shaker’s attempt is interesting, but flawed. One of his commenter points out that he compares the entertainment website TMZ to Time.com’s news site, which only gets half as many visitors. However, the New York Times’ site gets three times as many visits as TMZ so perhaps he’s fudging the statistics a little to show the trend toward frivolous media.

There are other examples, but the graphic makes does provide a basis for an interesting conversation. The most interesting aspect is that it shows the U.S.A trending more toward Huxley, while repressive Middle-Eastern regimes seem to be trying to make Orwell’s vision more of a reality.

Equations of a Parabola: Standard to Vertex Form and Back Again

Highlighting the Vertex Form of the equation for a parabola.

The equation for a parabola is usually written as:

Standard form:
! y = ax^2 + bx + c

where a, b and c are constants. This is the form displayed in both the VPython Parabola and Excel parabola programs. However, to make the movement of the curve easier, the VPython program also uses the vertex form of the equation internally:

Vertex Form:
! y = a(x-h)^2 + k

where the point (h, k) is the location of the vertex of the parabola. In the example above, h = 1 and k = 2.

To translate between the two forms of the equation, you have to rewrite them. Start by expanding the vertex form:

y = a(x – h)2 + k

becomes:

y = a(x – h)(x – h) + k

multiplied out to get:

y = a(x2 – 2hx + h2) + k

now distribute the a:

y = ax2 – 2ahx + ah2 + k

finally, group all the coefficients:

y = (a)x2 – (2ah)x + (ah2 + k)

This equation has the same form as y = ax2 + bx + c if:

Vertex to Standard Form:

a = a
b = -2ah
c = ah2+k

And we can rearrange these equations to go the other way, to find the vertex form from the standard form:

Standard to Vertex Form:

! a = a
! h = \frac{\displaystyle -b}{\displaystyle 2a}
! k = c - ah^2 = c - \frac{\displaystyle b^2}{\displaystyle 4a}

Summary

In sum, you can write the standard equation for a parabola as:

Standard form:

And you can write the equation for the same parabola in vertex form as:

Vertex form:

UPDATES

UPDATE 1: This app will automatically convert from standard to vertex form (or back again).

UPDATE 2: Automatically generate and embed graphs using this parabolic grapher app.

Parabola Program

Animation showing the widening and shrinking of the parabola.

So I put together this interactive parabola program using VPython (code here) for students encountering these curves in Algebra.

Simple Excel program to graph a parabola.

It’s a more interactive version of the Excel parabola program in that you can move the curve by dragging on some control points, rather than just having to enter the coefficients of the equation. The program is still in development, but it is at a useful stage right now, so I thought I’d make it available for anyone who wanted to try it.

The program is fairly straightforward to use. You can move the curve (translate it) up and down, and expand or tighten the area within the parabola.

The program also displays the equation of the curve in standard form:
! y = ax^2 + bx + c

.

What the buttons do.

Next Steps

I’m also working making the standard equation editable by clicking on it and typing, and am considering showing the x-axis intercepts, which will give algebra students a nice, visual way to of checking their factoring.

References

Coffman, J., 2011 (accessed). Translating Parabolas. http://www.jcoffman.com/Algebra2/ch5_3.htm

Math Warehouse, 2011 (accessed). Equation of a Parabola
Standard Form and Vertex Form Equations, http://www.mathwarehouse.com/geometry/parabola/standard-and-vertex-form.php#

WolframAlpha.com, 2011 (accessed). http://www.wolframalpha.com/input/?i=a^2+x^4%2Bx^2-r^2%3D0

“Cheat Sheets”

A selection of "cheat sheets".

I let my students bring in one page of handwritten notes, a “cheat sheet” if you will, into their last Physics exam. I’d expected to see some very tiny writing, but some of the notes needed scientific-grade magnification equipment to be read. Seen from a distance, the dense writing did have a certain aesthetic appeal.

Of course the primary reason for letting students bring in the cheat sheets into the exam was to get them to practice taking notes. At one extreme, the students who already take good notes benefit from having to condense them. At the other extreme, the students who don’t take notes at all get a strong incentive to practice. The very act of preparing cheat sheets is a good way to study for exams.

And it worked. As they hand in their papers I usually ask them how the test went, and, this time, I also asked a few student if they found their page of notes useful. One student in particular responded, Well I didn’t need to use it after making it.

Cheat sheets laid out according to note-taking style. Two extremes of note taking styles are highlighted. Equations and diagrams to the left, and text-only to the right.

It was also very interesting to see the different styles of note taking: the strategic use of color; densely packed text; equations; diagrams; columnar organization. What all this means, I’m not sure. I’m particularly interested in how their note taking style relates to students’ preferred learning style.

Indeed, it would be interesting to see if the note taking style co-relates in any way with students’ performance on the test. One could hypothesize that, since we know that students learn better when they encounter material from multiple perspectives, then students whose notes have the greatest mix of styles — diagrams, equations, text etc. — should have learned more (and perhaps perform better on the test).

It’s a pretty simple and crude hypothesis, since there are likely many other factors that affect test performance, but it would still be interesting to look at.

Slingshot Physics

Slingshots came up the other day in physics when we were talking about tension in strings when they’re held at an angle. The larger the angle the greater the tension in the string, which is why it’s harder to do pullups on an overhead bar when your hands are spread apart.

The concept of elasticity also came up. It is the elasticity of the rubber band, its ability to return to its original shape, that provides the potential energy when you pull it back.

Smarter Every Day has a video up that glances at the physics of slingshots.

One of the neater things the video shows is one experiment where they were aiming for a pumpkin but missed. The shot went too low, knocking the piece of wood the pumpkin was sitting on, and practically all the momentum of the shot was transferred to the wood: the shot looses all its velocity while the wood takes off. Once its support is gone, the pumpkin just drops vertically — there’s no horizontal motion — making this also a good demonstration of inertia.

Anomalous Motion: Optical Illusions

Image via boingboing.net (Pescovitz, 2011).

While I do like to use animated gifs, these apparent animations are actually still images that, because of the arrangement of colors and shapes, your brain interprets as moving. Akiyoshi Kitaoka has an extensive gallery. It comes with the warning though: ‘works of “anomalous motion illusion”, … might make sensitive observers dizzy or sick.’ (Kitaoka, 2011).

Rotating Snakes, by Akiyoshi Kitaoka. Click the image for a full sized version where the rotation is accentuated.

The History of the Periodic Table

Fitted to a cylinder, the elements on this periodic table would form a spiral. Image via Wikipedia.

Spurred by Philip Stewart‘s comment that, “The first ever image of the periodic system was a helix, wound round a cylinder by a Frenchman, Chancourtois, in 1862,” I was looking up de Chancourtois and came across David Black’s Periodic Table Videos. They put things into a useful historical context as they explore how the patterns of periodicity were discovered, in fits and starts, until Mendeleev came up with his version, which is pretty much the basis of the one we know today.

The cylindrical version is pretty neat. I think I’ll suggest it as a possible small project if any of my students is looking for one. You can, however, find another interesting 3d periodic table (the Alexander Arrangement) online.