Signs of summer coming
This morning, as kidlings and self sat around the dining room table pondering questions such as, “An artist wants to completely cover [sic: split infinitive, ack!] a rectangle with identically sized squares which do not overlap and do not extend beyond the edges of the rectangle. If the rectangle is 60 1/2 cm long and 47 2/3 cm wide, what is the minimum number of squares required? (a) 429 (b) 858 (c) 1573 (d) 1716 (e) 5148,” a speeding sports car (pale brown-beige, not a late model but something a bit older) plowed into a speed limit sign right across the street from our house. The driver, a woman, mowed down not only the sign (which crumpled nearly in half and which was ripped from its concrete mooring), but also grazed a nearby tree. Her car stopped on the grassy strip between sidewalk and road. She got out her car, examined its right front fender (which was the one that made contact with the stationary objects lining the road), and then got back into her car and sped off. It wasn’t even noon yet, but I seriously wondered whether she was drunk. Either that, or she was suffering from dementia.
What if it was all a misunderstanding, and her collision was due to her heroic attempt to avoid hitting the squirrel that hurried across the road in some territorial kamikaze gesture? What if she was inebriated after all? Would the hypothetical squirrel excuse her lousy driving? Would it be worse for her to have been drunk at 11 am rather than 11 pm? Is she any worse than the taxi drivers, who always speed, sober bastards with professional pride, and who seem to think that they are above the law?
I live on a corner. Across the other street (the quiet one, not the thoroughfare) live farmers. I’m kidding. Sort of. But it’s almost true. They highlight my maintenance inadequacies with their year-round yard work. While my yard is in a constant state of genteel disrepair, with clouds of “what if” plans embellishing its general dilapidation, theirs are marshalled into order. (“What if our ‘historic’ 1938 hedge, planted to please Their Royal Highnesses on their 1939 trip up Victoria’s avenue to Government House, were renovated and pruned and hedged and fertilised and brought back into a state of grace, instead of looking like a set of bad teeth, a study in green and brown?”) I could swear I saw my neighbour in late fall, with scissors the size of juvenile anchovies, snipping at his ornamental Japanese maples.
It’s true, however, that they use only hand-tools, along with the occasional electric-powered garden tool, but never any gas-powered gadgets. This is a blessing, since there are others (especially the custodians of the apartments across the busy street) where gas-powered yard maintenance is all the rage. And it does bring out all the rage in the rest of us. An anchovie-sized pruner might make a person laugh, but gas-powered noise-makers make you crazy enough to play in traffic, which can set off all sorts of accidents.
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Comment by Stu Savory — February 20, 2005 #
Correct!
Figures you wouldn’t be able to resist a math problem, Stu!
For those of you who are like me (i.e., unable to leap to spectacular
conclusions of numerical perfection), here’s one way to get at the
solution (and sorry, I can’t write fractions properly, so I’ll write m
/ n instead, and x means times):
The area of the rectangle is 121/2 x 143/3 = (11×11)/2 x (11×13)/3.
From here, we want to try to massage the equation so that it expresses
an integer times a square of some number (it has to be a square since
the little squares would be written as such, remember?, and an integer
because there are so-and-so many of these squares).
So, the above equation is rearranged to say: (11×11)/2 x (11×13)/3 =
(11×13)/6 x 11squared. I.e., you have multiplied the denominators (2,
3) to get 6, and the numerators have become 11×13 (still as a
numerator, over denominator 6) and 11squared. You haven’t changed the
value by writing 11squared (think of it as a numerator with a
denominator of 1), since, written as such [(11×11)/1], we’re still all
ok on the values we had in place before we started rearranging things.
The next part is where I lost it, and had to ask my mathematician
husband for help in following the solution (oh yeah, the problem[s]
come with a sheet of solutions …you didn’t think I can do this by
myself, did you?). You now multiply the expression (11×13)/6 x
11squared by 6 in order to make the denominator (6) a square. Thus you
get: (6x11x13)/6squared x 11squared. At this point, you can further
rearrange the numerical furniture and put the 6squared denominator in
place (so to speak, I know that’s not mathematically correctly
expressed) in place of the denominator 1 under the 11squared, while the
denominator 1 under the 11squared (a denominator we don’t bother
writing out, I’m just mentioning it to avoid confusion) under the
6x11x13. What you end up with is: 6x11x13 x (11squared(/(6squared), or
(11/6)squared. 6x11x13 is 858, so you now have your integer times a
square, namely (11/6)squared. Easy, eh? Well, wish it were, but this is
the kind of thing I can understand when I work through it, but give me
the next problem, and I’m again staring at it, thinking, “Dude…!”
For these examples and many more, visit The Centre for Education in Mathematics and Computing
(U. of Waterloo), grade 9 Pascal test, scroll down and open a couple of
pdfs with past exams. Quite humbling for some of us, let me tell you!
Comment by Yule Heibel — February 20, 2005 #