Wednesday, December 31, 2008

A Trick With Perfect Squares, or: Why I Love Math

I love perfect squares. Anyone who has conversed with me on the topic will be able to tell you as much, if they paid careful attention.

To reiterate: I love perfect squares.

When I was in 6th grade, we had to memorize all of the perfect squares of integers up to 25, if memory serves (and by extension, I guess, we memorized the perfect squares of all integers between -25 and 25, inclusive, but that's kinda cheating).

We were tested on this, even.

Like most of the 6th graders in the IB program, I was terrified of bad grades and did what any GPA-fearing student would do: I memorized by rote, and wrote them down in a list, like so:

Int -- Sq
1 -- 1
2 -- 4
3 -- 9
4 -- 16
5 -- 25

and so on.

And that's when I noticed a sweet pattern. Take two consecutive integers and their respective squares:

3 -- 9
4 -- 16

The absolute value of the difference between the squares (16 - 9 = 7) is the same as the absolute value of the sum of the integers from which the squares were formed (3 + 4 = 7).

This is true for ANY two consecutive integers (and their respective squares) - if you don't believe me, give it a try on your own.

The thing that's bugged me for the last 11 years or so is why this happens... I couldn't explain it.

UNTIL NOW, that is.

If we call the smaller of the two consecutive integers M, then the larger of the two is (M + 1). To set up our equation mathematically, we end up with

(M + 1)² - M² = M + (M + 1)
(the sum of the integers equals the difference between the squares of those integers).

If my memory of high school algebra serves me right, we can use our good ol' FOIL method to expand and simplify the left hand side of the equation, like so:

(M + 1)² - M² = M + (M + 1)
(M + 1)² - M² = 2M + 1
(M² + 2M + 1) - M² = 2M + 1

2M + 1 = 2M + 1

So we end up with an identity function, basically (I think that's what this is called), when we solve algebraically.

I figured this one out while driving a few months ago, then as soon as I got home I scribbled it all down on a sheet of note-paper to make sure it made sense. Then I showed my freshmen. They were not quite as excited as I am.

I forgave them for this.

* * * * * * * *

Another way to do this is to look at it geometrically. Imagine, if you will, a square with sides of length M (where M is an integer). It might look like this:

Now, we're going to pick the right side and the bottom side, and extend each of them a distance of one (or, to put it another way, we're slapping an (M x 1) rectangle onto the right and bottom sides, each with an area of, you guessed it, M):

Doing this gives us an M² + M + M square units shape.

In order to make it a complete square, we gotta fill in that last gap, which, coincidentally, happens to be a (1 x 1) square.

So we end up with M² + 2m + 1 as the area of our slightly larger square, which happens to have sides of length (M + 1).

Which is pretty friggin' awesome.

* * * * * * * *

What I'd like to do now is figure out if this difference of squares = sum of square roots holds true for non-integers (right now, I'm mostly wondering about decimal numbers (non-irrational / repeating) and fractions... my suspicion is that it does not hold true for decimal numbers (as 1/10th squared is 1/100th, which is a tough number to get when you're adding/subtracting tenths).

But there might be other cool properties that I haven't figured out yet. One can only hope that this is the case.

Thursday, December 25, 2008

This is Getting Ridiculous

I have a Facebook account. It's true.

I also have an e-mail account.

Both accounts are used multiple times per day (not the Facebox account, as I can't access it at school, but I check it at home after school... still multiple times, but on a slightly smaller scale of multiple). I don't feel too terribly bad about this.

Not one bit.

I also use the GMail Notifier (both on my mac and on my PC), which is nice, because it pops up a notification when I get an e-mail (meaning that I don't have to keep my browser open if I want to know whether or not I'm receiving e-mails)... hence the name.

The Gmail notifier even sounds a tone when I'm typing lesson plans feverishly in my word processor (Open Office, yo!), or, more likely, blasting zombies in Left 4 Dead.

This is useful. It lets me know when I have e-mails. Even better, it pops up a little notification thingy letting me know if I have e-mails from real people, or that contain useful information (like where to go if I want cheap prescription drugs (read: penis enhancement) and/or pornography).

But it's getting out of hand. Before I started that last paragraph (the one with the word "penis" in it), my GMail notifier popped up, telling me that I had an e-mail. The e-mail, of course, was from the Facebook people, telling me that someone wrote on my wall. This is not an uncommon event.

So, I click on the little notification, which opens up my inbox. Then I open up the offending e-mail and click on the link that takes me to my Facebook page, at which point I can read the thing on my wall, and, should I feel inclined, respond to it (lately, it's mostly a series of "Your face" jokes... I might be at least partially responsible for this).

So, I get notifications... that tell me that I'm getting notifications... that someone communicated with/to me.

The next logical step, of course: set up my notifiers so that they send notification to my phone, which then forwards it to a courier service, who then prints out a hard copy and delivers it by carrier pigeon, but calls me first to tell me that the pigeons have been sent.

This is all entirely ridiculous.

Of course, I'm not about to change it.

Tuesday, December 23, 2008

TF2 on Ze Macintosh!

There are very few of you who will actually care about this, but I got Team Fortress 2 running on my shiny new Macbook today. Granted, it runs in 800x600, with none of the fancy effects (the medic ubercharge, for example, just applies a blue shading to the models, rather than the shiny metallic thing that goes in in DX 8 or 9, and water is just a flat blue splotch, rather than anything with translucency and/or texture), but it runs.

This whole process was substantially easier than I thought it might be.

First thing I did:
I got a Macbook. Had I not done that, I would not be writing this blog post, and you certainly would not be reading it. It has the aluminum unibody, and it also has "Pretentious Hipster" written all over it.

In invisible ink.

The second thing I did:
Downloaded "Crossfire Games" from (the trial, at least - still making up my mind on whether I want to give it an outright purchase... it's only $40, but I already *have* a PC on which I can play games). The install was painless, and it had a pre-set link to download Steam from the internet (as well as an option to install "The Orange Box" from the CD or DVD or whatever it is that it comes on... I bought it from the internet back when it came out, and as such, I have no disc).

From there, I did the usual Steam thing, and installed TF2.

Easy as pie.

Delicious, team-based combat pie.

Like I said, the graphics aren't too hot, but as stunning as TF2's art direction is (really - they did a fantastic job), I don't play it so that I can gawk at the graphics (well, I do, but that's what my PC is for).

Now, TF2 is a monster battery suck, which limits the options a bit (I'm okay with this...if I'm not somewhere with a convenient plug-in, I probably shouldn't be TF2-ing), but that too is easy enough to deal with (apparently we can plug in our laptops now... who knew?).

I'm still not fully sold on CrossOver Games, as I'm not sure if it's worth $40 to play games that I've already purchased, and which I can already play (on my desktop). The portability is nice (I spend most weekends either in Colorado Springs or in Lincoln), but I bought my laptop mostly for productivity purposes (I like to pretend that I'm a writer, and apparently I do this "teaching" thing as a profession, plus my work with the Nebraska Writers Collective), rather than for gaming purposes.

But if it can run Left 4 Dead (which, by the way, has my vote for "Game of the Year"), and if I can get Fallout 1 and 2 running on it (I bought them from, but haven't yet spent more than 10 minutes on them, which is simply a damn shame), and perhaps a good Doom client (some of my fondest memories of childhood are of running a serial cable between the mouse ports of my computer and my dad's computer, back in the days before any type of reasonable networking, and playing co-op through "Knee Deep in the Dead"... I still get the E1M1 theme stuck in my head from time to time, which is really pretty awesome), then it's like that I'll have accomplished 2 things: 1) written a really, really long sentence, and 2) set myself up to never get anything done ever again.

Now if only I can get rid of this nasty little habit I have of not being able to sleep if I don't feel that I've accomplished enough on a given day, I'll be absolutely golden.

For now, though, I'm off to rage silently against the holiday season some more. And to rock as a Heavy in "Dustbowl".

Wednesday, December 3, 2008

Where to begin...

In an attempted diss, one of my freshmen said this to another: "Maybe if you weren't so busy pwning all the time, you wouldn't be so lame!"

(Cue Inigo Montoya: "That word, you keep using it. I do not think it means what you think it means.")


My 7th hour has been fascinated by witches these last few weeks, more or less since we finished reading The Crucible. After we finished that, we jumped tracks completely and moved into argumentative writing... for every issue, the solution, proposed by at least one or two of my 7th-hour-ers, has been to "Get rid of the witches, who cause (said issue) by ..."

Not exactly airtight logic, but it's funny enough that I can deal.

8th hour just tells me how they want the day to be done. They do not care about witches.


I've been teaching grammar to the freshmen (as it aligns with our curriculum), which, in its very literal sense, is simply a naming of the functions of words, from which a general pattern of how language works presumably emerges (these patterns already exist in the brain, as developmentally, kids use all of the major syntactic patterns by the time they're 5... but the vocabulary gets more complex, and the syntactic patterns start being nested within each other, to some degree), which in turn is supposed to make them into better writers.

I've got some issue, philosophically, with the assumptions beneath the teaching of grammar, but I have to work from within the pre-existing framework for now, as it's "the way we do things", in order to show at least a minimum competence and to then justify my departure from the established framework in years to come.

So I teach grammar.

This week, we've been doing the complements (direct object, indirect object, objective complement, and the 2 subject complements: Predicate Nominative and Predicate Adjective).

Teaching grammar is like pulling teeth, at times, in that the only way for me to do it efficiently is if I numb them first (or knock them unconscious... but I think I get in trouble for that one), so I've been pulling from the "Bag of Tricks", as it were, that teachers I've had have used.

In this case, I set up a series of powerpoint slides, each with a single sentence, and we turned it into a quasi-game show. Kids went up to the board, one by one, and it was a race-type deal, with a point for getting it right, and another for being the first (2 teams of 8 or so) to answer. The deal, then, was that their teammates had to be silent for the first 15 seconds that the question was up (after that, they could help).

Then we kept track of points.

The kids got really excited (not because of grammar, but because of the competitive nature), which turned into them getting really loud, but for once, almost all of them were paying attention, and they were listening to each other.

They learn better when they teach each other than they do when we, as teachers, try to teach them. Call it the increased feelings of self-efficacy that result from seeing a peer perform a task. Call it cooperative learning. But they question each other, they test their hypotheses, and they're more receptive to making mistakes (as it can be figured out).

Since I'm teaching things that are more or less issues of fact (rather than opinion), with the purpose being that students can take a body of knowledge, re-create it, and do something that looks like applying it, I'm more or less an arbiter of right and wrong... I try to recognize that, for a particular problem, there are several possible answers, but usually just one "best answer", but with the curriculum goals and state standards being mostly geared around the transmission of content (and the resultant assessment being whether or not students can re-produce that content), I simply don't know how to move away from being an arbiter of right and wrong and towards someone who guides self-driven learning and authentic problem solving, as there's no room for that in the curriculum.

Which means that moves like the trivia game are purely manipulative in nature, that is, they disguise the transmission of content (and the resultant "drill and kill") by turning it into something else (and tapping into the desire to be competitive).

And I'm on pretty shaky ground, ethically, for pullin' the ol' Game-Show-Switcharoo. But I'm rationalizing it with 1) "It means they're listening to each other and working cooperatively", and 2) If I'm gonna get yelled at for having a loud classroom, I'd rather it be because they're getting excited about the material at hand (even if only tangentially to the thing they're actually excited about) than about something else, and 3) As a teacher in the traditional vein (which hasn't really changed in definition, function, or methodology since, oh, 1890 or so), my job is almost purely one of convincing kids to sit down, shut up, and unscrew their skull-caps so that I can dump knowledge in, that is to say, the measurement of my value and skill is roughly akin to how well I get kids to jump through the hoops I've set out. Something like this makes me look good.

Though I'm not sure that it's actual teaching, and it still seems to be ethically shaky. But it'll have to suffice until I can find a better way to do it.

(And Margaret Spellings, when I do, I am going to take your office. You can empty out your desk, but you'd better believe that I will be sitting in your comfy chair).


I'm not writing as much as I should be. There's a poem about Kansas, a letter, and likely a few more that should be in the works, but aren't being worked on.

Plus, y'know, the diary of a first year teacher, the novel, and doing more research on democratic schooling (I've been reading Freire, but that's not nearly enough).

Drop a line and tell me how lame I am, or something. I probably miss you.