Tag Archives: math

Algebraically Challenged

I was helping the girls with their Algebra II homework last night, specifically with a Challenge question that was stumping them. It looked something like this (actual values changed so as to avoid giving away the textbook solution):

CHALLENGE: For any three distinct numbers, a, b, and c, a$b$c is defined as a\$b\$c = \frac{-a-b-c}{c-b-a}. Find -5$-8$10.

Forget, for a moment, that the notation a$b$c is not something you’re going to find in real-world math anywhere that isn’t an academic textbook. And let’s also forget, for a moment, that this isn’t even really an algebra problem, let alone Algebra II. (It might qualify as pre-algebra, but only just barely). There’s no balancing an equation, no solving for a variable, nothing that makes this anything more than simple addition/subtraction and division. Once you clear out all the confusing words around the problem, what you’re left with is a basic substitution problem. The problem statement gives you a$b$c = -5$-8$10, which works out to a = -5, b = -8, and c = 10. From there, it’s just a matter of subbing those numbers into the equation and solving (without overlooking all the double negatives, which is really the only tricky part of the whole equation).

So as it turns out, the “challenge” in this question isn’t the problem itself. It’s in interpreting the author’s cryptic and backwards presentation of the problem, which seems designed to do nothing more than make a basic third-grade math problem extravagantly more confusing. I’m trying to decide if this should be a commentary on the quality of math education our kids are receiving in the US now.

Fractional Canceling Error

I love “this post”:http://scienceblogs.com/builtonfacts/2008/11/testing_123.php from Matt Spring on “Built on Facts”:http://scienceblogs.com/builtonfacts/ the other day. In it he describes a mathematical manipulation on a test that “he think[s] should almost be worth _negative_ points.” In this case it was a student who took the part on the left side, cancelled the _m_ to get the part on the right:

frac{F-mg}{m} rightarrow F-g

(1)

Which is, of course, wrong wrong wrong. There’s a little rule in math that, in order to cancel out a denominator, it must be able to cancel out on both sides of the subtraction sign in the numerator. Here’s what the formula _should_ look like, after applying the rule properly:

frac{F}{m}-frac{mg}{m}

(2)

As you can see, the _m_ simply can’t cancel out from both sides. Just for fun I ran a little check on myself just to make sure _I_ did it right (since I’m a little rusty on my math sometimes). Let’s let _F_ be equal to 26, _m_ to 7, and g to 3. Substitute those into the original equation (1) and solve:

frac{26-7(3)}{7} = frac{26-21}{7} = frac{5}{7}

(3)

Now, substitute those same numbers in for (2) and solve:

frac{26}{7}-frac{7(3)}{7} = frac{26}{7}-frac{21}{7} = frac{5}{7}

(4)

Same answer, so I know I did it correctly.

Math-erwocky

Ok, so in my Real Analysis class, we’re finally getting down into the nitty-gritty of discussing mathematical grammar for proofs. We’ve been discussing statements and their negations, converses, and contrapositives. We began with two statements:

P: I eat it.
Q: I see it.

Now, if we combine the two statements, so that P implies Q, we get:

bq. If I see it, then I eat it.

or

bq. I only eat it if I see it.

If we flip them so that Q implies P, we get:

bq. I eat everything I see.

or

bq. I only see it if I eat it.

Our professor called this his Lewis Carroll example. I loved it.

Real Analysis

Calculus

Image by Akash K

It’s official – I start classes again on Tuesday. But I’m not in the Statistics Ph.D. program yet. Turns out, I’m a little weak on some of my math background at the moment, so upon the recommendation of the chair of the stats graduate application committee, I’m registered as a non-degree student so that I can take the classes I’m missing before officially applying to the graduate school.

So the first class I’m taking is an undergraduate math course called Real Analysis. From the looks of the course description and the prerequisite requirements, there’s a bit of a calculus foundation for the class. It’s been several years since I last studied the subject, so I think I’ll be spending a bit of time this weekend with my old textbook and rehashing some basic calc skills.

It’s been two years since I last took a class and several years longer than that since I took anything math-related (my graduate stats classes aside), so needless to say I’m a little rusty. But I’m excited and eager by the opportunity to get back into classes and especially by the opportunity to get into some of the math classes I never had the time or money for during my undergraduate education. It’ll be nice to exercise my brain academically again.

And I’ve got three days to get prepared. Sounds like fun!

Let It… Snow?

With the weather outside the way it is, one would think it was Christmas vacation this week, rather than Thanksgiving. It is beautiful out there, though, and it was a pleasure to walk in it this morning. Snow is definitely preferable at 28 degrees to rain at 36 degrees.

All this snow, however, reminded me of a little childhood wisdom — no two snowflakes are identical. Being now older, wiser, and a little more well-versed in the world of statistics, it has occurred to me to wonder a time or three over the past couple of years just how this can be. Mind you, I wouldn’t put it past an omnipotent God to actually cause every drop of moisture to crystallize into a historically and relationally unique shape once it drops below that all-important threshold of 32 degrees Farenheit (or 0 degrees Centigrade, for those of you using a different system). But by the same token, it occurred to me to wonder just how much proof was really out there on the topic.

So, I ran a “Google”:http://www.google.com “search”:http://www.google.com/search?hl=en&q=snow+no+two+flakes+identical and “this”:http://www.straightdope.com/classics/a3_392.html is what I came up with:

**How do they know with any degree of certainty that no two snowflakes are alike? When I took statistics I was taught that to draw a valid conclusion one had to take a representative sample of the entire population. But considering the impossibly large number of flakes in a single snowfall, let alone that have ever fallen, how could snowologists have possibly taken a sample large enough to conclude that no two are alike? –Leslie B. Turner, San Pedro, California**

They didn’t, of course. Chances are, in fact, that there are lots of duplicates. What the snowologists really mean is that your chance of finding duplicates is virtually zero. It’s been calculated that in a volume of snow two feet square by ten inches deep there are roughly one million flakes. Multiply that by the millions of square miles that are covered by snow each year (nearly one fourth of the earth’s land surface), and then multiply that by the billions of winters that have occurred since the dawn of time, and it’s obvious we’re talking unimaginable googols of flakes. Some of these are surely repeats.

On the other hand, a single snow crystal contains perhaps 100 million molecules, which can be arranged in a gigajillion different ways. By contrast, the number of flakes that have ever been photographed in the history of snow research amounts to a few tens of thousands. So it seems pretty safe to say nobody’s ever going to get documentary evidence of duplication. Still, it could happen, and what’s more, Leslie, it could happen to you. The way I figure, anybody who could dream up a question like this has got to have a lot of time on his hands. Get out and start looking.

There are a whole lot of other mathematical discussions on that page, but unless you’re something of a math geek like me, you’ll probably just find it mind-numbingly boring.