Wise Words of Mr Spooner

Part 1

Divergent Series

"It's like someone with cancer in remission. It looks like they're going to be all right and then bang! - they go off in the other direction! If you've got a divergent series, you may as well just pack your cards in."

Rates of Increase in Functions

"Well, have you ever seen `The Sound of Music'? The Von Trapp family had moved to America during the Second World War, because they'd been evacuated from their big old family house. They didn't want to go back to that house because Hitler had been staying there while conducting a campaign. And one day Hitler was looking out through the house window and heard a soldier whistling a Russian folk song. Those were prohibited, of course, but he didn't know who had been whistling this catchy tune. So he brought the whole platoon out and ordered them to be executed. He didn't just kill one soldier, he killed twelve soldiers, just because of that one Russian ditty. So when the x value of the function goes up by 1, the y value can go up by a lot more than 1."

Domain

"Calculating domain is one of the biggest problems facing Canada today. What is going to be the domain of the dominion of Canada?"

Differentiable Functions

"You see, what we want is a nice, smooth, continuous curve. You know at Frosh Week, when all the skinheads have nice, smooth heads, and everyone wants to run their fingers over them? Well, this is a smooth curve just like that."

Characteristics of Equations

"They're now talking about forms of life existing within computers, and programs which can reproduce and have certain characteristics. There are a lot of arguments about this - by many definitions this is life. I mean, we can only exist on Earth. We can't exist on Jupiter because the gravity would kill us! So what's wrong with living in a silicon environment? These things take on a sort of life of their own. Just like this equation, it takes on a life of its own. It has its own distinct characteristics - it's a circle!"

Discontinuity

"Usually we have continuous functions on a nice smooth curve, but sometimes we get points which aren't on the curve. For example, employees at a certain company might have salaries on this graph according to their seniority. But one particularly good-looking secretary, is getting much more that they should based on the time they've been working there. So the graph is discontinuous at that point. Notice I didn't specify the gender of the secretary - so we have a good- looking male secretary working for a female boss."

Ellipses

"You remember ellipses from Grade 12? They're kind of shaped like eggs, except that eggs are bigger at one end than the other. I wonder what shape the egg would be if it was layed in outer space? You should ask Mr. Monk about that. Heh-heh, he thinks I'm crazy anyways."

Major and Minor Axes

"We have here the major and minor axes of an ellipse. Some of you might be taking music: it's like the major and minor scales in music - the major is supposedly more prominent than the minor one. Although they do often use the minor one in reality to add interest. American blues, for example, is almost all minor."

Rate of Increase

"They have to look at the rate of increase for different diseases people are getting. Maybe we should put more money into cancer research, but there's only so many bucks. There's only so many bucks. Here's some uh... Canadian Tire money and, um, a looney."

Inverse Relationships

"You see it all the time. As one thing goes up, the other goes lower. The more she's interested in him or he's interested in her, the less he's interested in her or she's interested in him. There ain't no justice."

Multiplying Out Derivatives

"This is an example I like to use... In Spain, when they have the bullfights, who always wins? The matador - because they stick spears into the bull beforehand and let blood out. Those crazy Spaniards! So it's like when you're doing the problem and you charge at it with your head lowered. And where are you looking when your head's lowered, Amod?"
"Uh... at the floor?"
"Right! So this guy's here waving a bright red flag and you're snorting and puffing and you zoom right by it! So the bull has to come around for another charge. Those crazy Spaniards!"

Problem Solving

"You need a plan to solve a problem. I know doctors who just say `cut him up and we'll find out what the problem is.' Don't go charging in like that bull we talked about yesterday."

Area of a Trapezoid

"Now this is the area of a trapezoid, which you may remember from Grade 9, was back in the days when we were all young and foolish, the teacher included. Now, that's not an easy formula to use. You can go and charge it like the matador we talked about before, or you can do it a simpler way."

Volume of a Trapezoid

"The sewers, for example, are kind of trapezoidal. And every so often there's a big rain and the storm sewers overflow. And what happens to Richardson Beach and the other beaches then? They get yucky, yucky, yucky and you get ear or eye infections! Why is this? Because the city is bigger now and those storm sewers are 80 years old. And if there's a storm in Kingston, then up in Trenton there is a storm. So if you're interested in the environment, the volume of a trapezoidal prism is a good problem."

Calculus Questions Involving Planes

"Boy, some of these questions make me think that maybe we should just go back to a primitive state and live outside and hunt antelope. Why do we drive ourselves insane with this kind of stuff? They've looked at tribe in South America and it turns out that they just sit around and talk a lot. Sit around smoking pot - hee hee - and whenever they get hungry they go out and get some bananas."

Number of Variables

"We have a duality here, like the Chinese yin and yang. In every society we have the duality of good and evil. In a court you're either guilty or you're not guilty. There's no grey area in there - you can't be just a little bit guilty. Now in Britain they have a parliamentary system with two parties, and it works a lot better than Italy where they have proportional representation so they've had something like thirty governments since World War II. You see, the human brain seems to work best with only two options. So you have the two variables, x and y. And once you get more variables you start screwing up. Like this goddamn Constitutional mess we're in - too many variables! Not even the president of Alcan can figure it out! Now I'll tell you this because my father used to work there. The company wasn't running very well because the president had 11 people reporting to him. So they got some efficiency people in who said `Mr. President, you've got too many people reporting to you.' Now he's only got 4 or 5 vice-presidents. I tell you this so that you won't run into trouble on a test - if you've got too many variables you can't do the problem."

Functions Relating Variables

"I was talking to my friend Ron who has some sheep and cattle. See, he can leave the cows out all winter, because they're so big they produce enough heat. But if you leave the sheep out they'll get frozen, because they're small - kind of like us. If you put 25 head of cattle in the barn it'll will be nice and warm because heat is a function of body size. That's why really fat people don't like to go jogging. Ever think of that? Have you ever seen a guy who's two or three hundred pounds out jogging?"

Fixed and Variable Values

"You see, it's very important to distinguish the fixed and variable values in these problems. The variable things can change, but the fixed things star the same. It's sort of like with gymnasts - it turns out that in order to do all those flips and twirls and so on it helps to be really short. It has something to do with the physics involved; tall people just can't do them as well. That's why you don't see many tall people - say someone who's six-foot-one, to do gymnastics. So the Romanians started giving all of their Olympic gymnasts growth hormone inhibitors. You'd see all these short athletes and the a week later, boom! - they've grown a foot."

Finding a Job

"It's really hard to find a good job these days. It's kind of hard for me to picture - in my day people wanted me... I know it's hard to believe..."

Rates of Growth

"People often buy pigs for pets nowadays. You have these Vietnamese pot-bellied pigs, which don't grow very big. Now, these sellers like to say that what they have is a baby midget pig, not a baby Vietnamese pot-bellied pig. So people buy the pigs and they turn out to be some other kind of pig and grow up to be huge. I knew a family that had this happen and it was awful! It terrorized the house and whenever they sat down to have dinner it would charge the table! Now this, of course, is rate of growth for biology. It's like the rate of growth of your basic cube, y=x3. The pig is kind of like a cube; that's what reminded me of the pig in the first place. Now the pig doesn't care about this, it just keeps on eating and being happy. It doesn't have to be a pig, it can be a hippo, or whatever. It could even be a human. If we landed on Jupiter, our mass would be so big that our arms and bodies would get all bloated and thick and we'd get short, just like a pig or a cube."

Optimization Problems

"Remember when we were young? Remember how the Easter Bunny used to come and carry the eggs? So, remember how you made Easter baskets? Or were you deprived? They took the sheet of paper and folded it up like so, and then stuck it together with that horrible old flour paste (which was good to stick on people's seats.) Then you've got a nice basket to take home to mommy! Now, how does this work? You can get a certain number of eggs in each basket. Now some poor chump might have taken too big a piece out of the side of the basket, and might have gotten the wrong size of basket and wouldn't have been able to carry as many eggs. So if you're a greedy little bastard, you want as many eggs as you can get. You have to optimize the number of eggs in the basket! Anybody know the nursery rhyme? `A tisket, a tasket, a green and yellow basket.'"
[Spends the next several minutes happily constructing a basket with foolscap and a stapler.]

Equilateral Triangles

"You see these things over and over. If you go out and flip a truck over and look at the bottom of it, you'll see equilateral triangles."

Diagrams

"I don't know how many of you have been to the anatomy building over at Queen's. There they chop people up, lengthwise, in order to see what's inside. If you're looking at your body, you can't see what's going on in there, if you've pulled a muscle or have some similar sports injury. But if you look inside poor old chopped-up Alfie there, you can see things like the tiny little nerve endings which carry information to the brain. Now, the nerve coming from your ear is pretty small, about like this. [Holds up point of pen cap.] The one for your sense of touch is about the same size. You may remember in the some of those old movies where burglars filed down their fingertips so they were all tingly and could feel the pins drop in those big combination locks. But the biggest nerve is the one coming in from the eyes. It's about as thick as your finger. Humans are very visual creatures - they have one of the best-developed senses of vision (as long as my new contact lenses stay in.) Our sense of sight is more important that our other senses like hearing or smell. Hunting season started Monday: if a hunter is chasing a deer, he goes downwind so that the deer can't smell him. Same with lions and tigers; they don't want that antelope to smell them. So when you're doing a problem, draw a diagram so you can visualize it!"

Asymptotes

"Now here, this part of the graph is squeezed in-between these points. It's like a watermelon seed. You squeeze a watermelon seed and what happens? It squirts out at your friend, because it has nowhere else to go. What? You haven't done that, Janet? Well, it's also like a canoe. You test the water and everything seems all right and normal, and then you step into the canoe the wrong what and what happens? It tips over."

Minimum/Maximum Points and Roots

"Here you might expect one point to be a maximum point, but it's really a minimum. This is like tying your shoelaces backwards, from top to bottom. Mark, you can start a new fad in Kingston, you can become world-famous, get a job at MuchMusic, lipsyncing, with your groovy new shoes and a wig. My wife has one you might want to borrow.

Making up Math Problems

"It doesn't take a lot of differences to make slightly different problems. There are eighty-eight keys on a piano, but they make trillions of different songs. People come up with new songs every day."

Derivatives Relating to Time

"Everything relates to time... you know, that thing up there on the wall ticking away, the clock, gives the time. So you relate the derivative to time."

Homework in Other Classes

"Oh, watch the chemistry! It's a beast that devours time. I know, Mr. Miedema's assignments go on, and on, and on, and on..."

Inflection Points

"Here's a good example, since tomorrow is November 11. Now, you may know this if you've taken history, which is a very good idea, by the way. It's interesting to note that some countries in Europe, for example, have made history compulsory, because how will you know what your country is if you stopped taking history in Grade 11? Anyways, in the Spanish Civil War, where Norman Bethune was active - if you haven't seen the movie, you should, because Norman Bethune, who later went on to China, was a genuine Canadian hero/saint - the Germans were attacking . They practised their techniques in Ethiopia using airplanes on the poor Ethiopians on horseback. Now, pretend you are a German pilot in 1939, flying in a dive bomber over Poland at two thousand feet. You're cruising along in this crate at 340 miles an hour, which is nothing to be sneezed at. When you see the target, you point your nose down and dive at all the defenceless people, the fleeing Budapests. When he dives he starts going at speeds up to 550 miles per hour with his nose pointing at the target down here. But right after he sends all those people to Kingdom Come, he'd better pull back, unless he wants to become a Kamikaze. The point at which he pulls back is the inflection point, because the slope stops increasing and starts decreasing! You know, it's a good thing we have Remembrance Day, because people often forget this stuff.

Degree Measures

"You see, a circle has 360 degrees. This is how it's been ever since they came up with this in Babylonian times, four thousand years ago. Their system worked better than ours, because they had a system based on the number twelve, which you can divide by three and four."
Student: "But you can't count to twelve on your fingers."
"Yes, they were smarter than us, they didn't have to... well, actually, the genetics were different and produced hands with six fingers on them which then evolved over the years into... but anyways..."

Graphing Calculators

"You know how they make those things, don't you? They take someone from the University of Tokyo and shrink him down to about this big. He runs around inside there, and with an equation like this, he's running pretty hard."

Characteristics of Functions

Every function has a curve. Every function has a personality. So if you know the function has a particular curve, you know just how it will twitch. Your friends have personalities too; you might know just what will make them twitch -- these curves are just like that. Your friends have their highs and lows; so do these curves. I think that there is some big equation that defines everything about the whole world, and we're just little equations within this big equation. So if we could only find out the equation, we could figure out what's wrong with the world. If someone found out my personal equation, the could find out exactly what I'll be doing a year or six months from now. Kinda' scary. Of course, that basically take the fun out of it. [pause] And then one day, we all go up to the Great Equation in the Sky."

Curve Sketching

"Back in the days when dating was a lot more fun that it is now, I was in charge of setting up blind dates for the guys at R.M.C. You could see all the girls from Hotel Dieu looking over the cadets coming in. They'd check them out; since the recruits were from all over Canada, they wouldn't know anyone to take to the big dance coming up. And if they got a date who they didn't like the looks of, they could arrange to be conveniently absent - maybe they'd have a stomach ache or the flu. So it's just like this: you look over to curve to see what it might be like."

Minimum-Maximum Points

"OK, since the derivative of this equation cannot equal zero, there can't possibly be any min-max points! It's like if I'm a lawyer and I can show that my client loves little children and birds and animals and plants and cats, then I can try and convince the jury that my client could not have committed this brutal murder. If I can convince the jury of this, my client goes free! It's one of a lawyer's little tricks."

Fractions

"When we work with fractions, we divide, and conquer."

Graph of y = tan x

"See this graph? It looks like a person's heart pattern. If you watch `General Hospital,' you might see this, when they're not busy chasing each other around the halls and pursuing amorous relations, you can look behind where the nurse and the intern are making out and see something that looks a bit like this. Maybe we should bring a TV in here someday and watch the soaps."

Trigonometric Functions

"You can't cancel the thirty in (cos 30)/30. It's like the cosine is the adjective and the thirty is the noun! You can't just cancel it out! If I run to the window and say, "Oh look, there's a pretty!", it just doesn't make sense. You have to have the noun."

"You see, here you've got the sine of 2x. You can't take just factor out the 2. Remember adjectives and nouns? The 2 and the x are married in heaven and they'll stay together forever."

Checking for Extraneous Roots

"You need to have a system of checks in place. When they fly planes, they have an elaborate system of checking mechanisms to make sure nothing goes wrong. A lot of people say `Oh, there's nothing wrong with this plane,' and think they can just take it up into the sky anyways. And usually there's nothing wrong with that - you can go on all your life that way, but there's a one in a million chance that it will, and if it does... well, they don't even have parachutes on the planes anymore. So you have to check your answers. I once had an injury in the cartilage of my left knee. So I went to this sports doctor in Ottawa who is the best in the area. He does hundreds of knees. I went in and we joked around a bit and he put me under. When I came to I had a bandage on my right knee as well! Now, I was still heavily doped and wasn't sure of what was going on, but sure enough they'd made a mistake! Because which side of you is this on? [Points to left leg.] Right! It's on your right! [laughs] Sue? Did I sue? Well, I thought about it, but you know, he went to Queens and I went to Queens so... [more laughter] It's like the horrible case of this kid who was suffering from diabetes. He and one good leg and one bad leg and guess which one they amputated..."
Student: "Oh, that's not even funny!"
"No, hee-hee, it's not. Heh."

Simplifying Equations

"Sometimes when I'm doing this I feel like a hairdresser. You try this, and you try that, and the next thing the lady or gentleman knows, they're bald!"

Careless Mistakes

"You've got to watch these things, they get on you and breed like germs. Have you ever looked at a germ through a microscope? How about a flea? Have you ever seen a flea under a microscope? Very unpleasant. Have you ever looked at a bloodsucker through a microscope?"
Student: "I'll be right back..."
"The parasites, you know, they cause a lot of human misery. In China, there's this parasite in snails that causes blindness. When they go out barefoot in the rice fields the larvae crawl onto them. They get into the bloodstream and eats out the back of the eyes. It's very tragic. And in Africa, there are certain streams that you just don't go swimming in on a hot day, and it's not because of the crocodiles. We're very lucky here.
"And up in Northern Ontario, people are very poor - we're all quite well off. There's terrible poverty all over. Inflation! It all started when the Roman emperors cut all of the gold coins in half. Inflation started in 200 BC and it can't be stopped. When I went to Greece I got two million drachmas! It used to be that you could buy a jet for that much. I've still got it too, except for the half a million I spent on a model plane. This [pulls out a two dollar bill] isn't really worth anything!"
Student: "I'll take it!"
"Well, I'll keep it a bit longer. But those guys in Watertown are just being nice when they give me something for it, there's no guarantee that it's worth anything."

Visualizing Problems

"Read the problem three, four, or however many times it takes until you can visualize it. There have been experiments and reports on psychology that have shown that humans are very visual creatures. Our sense of hearing isn't so great - there are many animals with better senses of hearing, but we have a very developed sense of sight because we rely so heavily on what we see. That's why the swimsuit issue of Sports Illustrated is such a big seller!"

Diagrams

"Draw the diagram first! When you see the ads on TV for Sports Illustrated, they always talk about which issue? The swimsuit issue!!"
Student: "So do you."

Matrices

"Has anybody here used matrices? You know how they rotate? That was discovered by an Irishman, and he downed at least two hundred bottles of good Irish whiskey before he figured it out."

Phases in Trig Equations

"They had this big power blackout which extended on this huge power grid. The power went out in Toronto, in Montréal, in New York, in Boston, in Washington. The blackout extended as far as Illinois, as far as Maine... all over the place. And they couldn't get it back on for a while because they had to have the sine waves in the right phases. You have to have them in exactly the right phase, not like this [pointing to example on board]. So the blackout lasted three days! And nine months later, guess what happened? Every maternity ward was overflowing because there was a huge baby boom! For a while after that I could ask my class when they were born, and there'd always be several hands going up saying they were born nine months after the blackout. So why did they have the blackout? Well, they had photos from Niagara Falls of UFOs, near the power lines, draining the power out of them."

Test Strategies

"See, even if you only get this much of the problem down, you'll still get two out of six or two out of eight marks instead of zero. And even if a hockey team only goes 50-50 in its road games, they can still win in the league if they win 70 or 80% of their home games. Actually, it's very interesting how they do better in their own territory. It's a basic animal concept -- marking territory. A big bear might go around and shove his big old paw into a tree and say `Look what a big, mean monster I am!! Raargh!' The little grizzly sees this and says [flinging hands in air] `Oh my Lord!' and runs away. It's the same psychology when the Leafs go to Montreal. It could be a Roman coliseum; it's very atavistic. The fans will go mad; they say `Kill 'em! Kill 'em!' People could get violent. When I went to Vermont, some people were killed by some bears which aren't normally in the area, but they were forced south and very protective of their food and cubs. They always protect their territory. Now where did I start all this?"

Overpopulation

"Does anybody know about the Reverend Malthus? He said that there are three ways to keep the population down: disease, famine, and war. You play around with any of those and you're sure to run into trouble. Disease, famine, and war. It's a very pessimistic view; very depressing. Leonard Cohen's latest release is like that: very bleak. I think it's going to do fairly well though, but it's pretty depressing."

Exam Tips

"On the exam, you should try everything. If you get an answer that's right out of there, write a note: `Mr Miedema, I know that an ounce of sulphuric acid can't weight three pounds,' and at least he'll know you're on top of the problem."

"If you don't get the whole question, you can at least get the marks for showing the first step. If you don't get the turkey, you can at least get the stuffing. OK? Of course, some people like the stuffing even better than the turkey. OK, it's a silly analogy, but anyways."

"If you go to sleep now, and wake up two weeks from now, you're not going to do as well as you should."

Area Under a Curve

"Well, the hunting season is over now.. the deer are probably laughing out there. Now it's interesting how firearms have developed over the years. You watch Miami Vice and the drug groups have clips where they can just pop it in and fire off twenty or twenty-seven shots. So it makes it pretty easy to kill a person - even if you miss half the time, you can still make quite a mess out of someone. Has anybody ever thought about how they figure out the stopping power of a bullet? I mean, I use a different bullet if I'm out hunting moose or hunting deer or hunting rabbit. Or hunting squirrels - in the southern United States the squirrel is quite a delicacy. But you've got to be a good shot, because if you don't hit it in the head, you've got no dinner. Matter of the fact, that's how the Americans won the American Revolution. The British would all line up an cry "Charge!!" but the Americans would just sit behind trees and pick them off - doing! doing! So the British would through up their arms and go "Mommy, I want to go home!" Stopping power - haven't you done this with Mr. Knapp? Mass and velocity. A slow and heavy bullet like a .45 can sometimes do more damage. There's a story about that. Anne-Marie, just hang in there. You see, when the Americans went and "liberated" the Philippines, which were owned by Spain, the standard side arm was a .32. So you could fire two shots into a Filipino and he'd still come along and chop your arm off with his machete. So they brought in the .45's which if you got shot in the shoulder with one, by the time it comes out your other side there's not much left of your shoulder. You can find the mass of the bullet using calculus - it's a parabola rotated around the x-axis. Mass and velocity gives stopping power. The trajectory is a parabola too - 'cause with the old .303 you had to really compensate for the trajectory and windage. But now they're a little faster - it makes it easier. Now it's easier for a dumb man to kill a smart man. Or woman."

Mr. Spooner's Commentary on Various Problems

"This is like at a rock concert where the sound comes out in concentric circles from the stage, and a seagull above the stadium can hear it just as well as you can in your $30 dollar seats."

"You should try this one out tonight! Go out at about seven o'clock and find a lamp post. Find your shadow and watch it as you walk away. Then run back and do it again! This is a Peter Pan and Wendy problem, remember? Peter Pan loses his shadow and Wendy has to sew it back on again."

"I don't know if anyone knows about those old radio shows. They used to have this show that would begin with a ghoulish, sickly kind of laugh like this: Hoahahuhuh... [a sound to which no justice can be done on paper]. `Who knows what evil lurks in the hearts of men? Hohohahuh, the Shadow knows! So, you see, we have a shadow in this problem."

"This problem reminds me of something that happened to a friend of my fathers... it's kind of grisly: He was putting the storm windows on his house and he managed to do himself in. You know how when you've got a long ladder and a short house there's this temptation to make the ladder too horizontal. So the guy is walking up this ladder with this bloody window in his hands and, you know, maybe it's a little muddy, or maybe there's a little ice and the ladder starts to slip... So this guy, maybe he's a mathematician and he's counting the bricks on his way down. So now we can find the rate of his descent, since we're given x2 + y2 = 42, we know 2x(dx/dt) + 2y(dy/dt) = 0, cancel the 2s..."

"Now, this is an important problem because if you're walking in a sewer and you get your foot stuck and there's a flash flood, you want to know how fast it'll fill up and you're gonna drown."

"OK, let's do problem `U'. I know you people are just dying to do question number `U.' You've been waiting all weekend just to get here and take up question `U.' 12/20, dear diary..."

"This problem has a lot of really good and important applications, especially with science and nature. For example, if we know the rate that the Earth is warming up, we can.... well, we don't really know that... hmmm.... I've got an article on that somewhere..."

"How many variables do we have here? Too many! It's like taking your mother-in-law along on your honeymoon. There are just too many... well, we'll leave it at that."

"With this problem, there's more than one solution. Now your calculator, your calculator will only give you one answer. But what do you expect - your calculator only cost you thirty-five bucks."

Solving Trig Identities

"You have to be loose and relaxed to solve this problem. You always have to land on your feet. Be like a cat -- you throw a cat out the window, and it still lands on its feet. Not that I'd do that to our dear old cat, of course. They should have put cats up on that pet contest as well as the dogs. In my family we've had a lot of pets - mice, ducks, chipmunks. You know, ducks make great pets. They really do. You can even train them, you know. They're not like chipmunks, who will be nasty and bite you. You see, you have to be open-minded."

[later] "Our neighbours used to really wonder, you know, with all these ducks in our back yard. You see, at night they're afraid of foxes so they'd congregate around our basement door quacking away wanting to get in, because they're afraid of the dark! But ducks are aquatic animals, so they don't use the bathroom or a toilet. They'll just be paddling along and uh, away they go. So we'd have to put newspapers all over the basement floor and it would get a little stinky. But that's the only problem with them - they make very good pets. You should get a duck for a pet, you really should. But anyways..."

In-Vitro Fertilization

"Have you ever heard of in-vitro fertilization? You, know where you have the little petri dish with the even littler cells in it and you grow your baby. It's not a test tube. Anyway, there are all sorts of controversies over this. Oh, that reminds me of my grade 4 music teacher. I wasn't very good one day so she made me learn a piece out of the opera Don Giovanni. Now this damn thing wasn't in English, but I learned it anyways and I read the libretto where Don Giovanni sells his soul to the devil. So if you really want a baby, why not go for in-vitro? Who cares about the consequences? After all, Don Giovanni didn't. Oh, but he was tortured by the devil..."

Uses of Curve Sketching

"You'll find so much of this graphing in your first-year economics text - marginal utility, supply curves, elastic demand. You know about inelastic demand? This is why they keep raising the price of cigarettes. They think they're a luxury item, but for some people, they're a necessity. They're addicted, so the demand is inelastic. They can just keep cranking the price up and they'll still buy it. Of course, the Mafia's now muscling in on that area. There's a lot of bucks to be made in smuggling. Those guys in Ottawa that thought they were so smart forgot about that! So, you know, a bar in Ottawa can just be broken into, and you can steal thousands of dollars worth of cigarettes - and people look at that a lot more kindly than taking a gun and shooting someone."

Multiple Solutions

"With these periodic functions, you can get more than one answer - write them all down. There isn't always just one right answer. I don't know how much you know about nuclear power - in France, most of the power is supplied by nuclear. Environmentalists here are complaining because forty or fifty percent of our power is nuclear! I heard an interesting discussion on nuclear power on the radio - it can be less dangerous and less polluting than oil or natural gas! Now, I don't know, but these environmentalists seem to think that there's only one answer! Bullroar! We've only got enough natural gas to last us out another forty or fifty years. There's more than one solution. A lot of people think that beavers should be preserved -but these people are just locked into one answer. The beavers are turning northern Ontario into a mosquito land! A couple of beavers ruined my cottage!! I could have dynamited the little so-and-so's! Who cares if they're the national animal?! Anyways, it's important to realize that there's more than one answer... if you get two or three, write 'em all down!"

More Problem-Solving Tips

"Sometimes it's a good idea to break a hard problem like this into two smaller problems. Maybe that's the problem with this referendum! People are voting Yes or No for the right or wrong reasons - maybe it was something yesterday or maybe it was two months ago but there's too many parts to the problem! [deep breath:] But anyways..."

"See, when you've got a problem like this, you want to solve one part at a time. It's like if you're in a fight with two people, you don't want to take them on at the same time because they're sure to getcha. If you're clever, you'll tell one guy you'll take him on one day, and the other person that you'll fight a different day. This [pointing at equations on board] is how the Duke of Wellington won the Battle of Waterloo."

"If you've got something like this in your equation, you can't just have this little magician in there who waves his magic wand and - pzzzht! - it's gone. You have to go through and solve it and simplify it. You have to follow the laws of math; you can't just change things as you want. If you try to do that in affairs of heart, or affairs of politics, they'll take you out to KPH [Kingston Pyschiatric Hospital] and lock you up! You've got to obey the fundamental principles of mathematics."

"The first thing you do with these problems is make the diagram! If you've got it in your head, that's OK, but it's best to write it out. If you're trying to decide between Boyfriend A or Boyfriend B, make a list; write out all the merits of Boyfriend A in one column, and all the things about Boyfriend B in another. It's the same with jobs, make a list and see which one's more attractive."

"You have to put your soul into the problem, not just your head. For several years the math department was trying to get people to live mathematics, not just to say `Give me the answer and go away.'"

"Look this stuff over tonight, you've got to learn it as much as you can. This is just like wrestling; you've got to pin this stuff down. If you can pin the guy down, maybe you know a certain move or hold, then you get points - marks. This is college-level wrestling, not that `Pro' stuff they show on TV."

Things that make you go hmmm...

"People at 35 are considered old at 28."

"We've gotten far enough in math now that you can have an opinion. In history and English you've been allowed to have an opinion since Grade 11."

"These two really smart guys wrote a book about this. It took them two years and then they went insane."

"This is a question connecting `rates'. So if you want to boogie with me, [singing and gesticulating wildly with arms:] when this thing changes, this thing changes."

"If you haven't read the book Nineteen Eighty-Four, you should. Do you get that one in your English courses? No? Well, read it anyways - you'll probably get more out of it that if you read it for a course. Studying something can sometimes take all of the fun out of it. Maybe that's why they don't teach much sex in high school."

"It doesn't matter what numbers you stick in here. Instead of a 2-ton bomb you've got a 10-ton bomb - it'll just be a bigger crater!"

"No, you won't have to know these formulas for the exam, but you should learn them anyways. I can't imagine someone not going into more mathematics. Everything is numbers now. Words are numbers. Music is numbers. You turn on a CD, you're not listening to music, you're listening to numbers."

"I didn't recall that I excused the class." [said approximately two and a half minutes after everybody has walked out]

"Is that clear Anne-Marie?" [looks at empty seat and continues] "Good. Now, when you take the derivative..."

"Where's my textbook? Oh, here it is!" [takes a textbook off desk]
Student: "Hey! That's mine!"
"Heh-heh... never trust a teacher!"

"Don't worry, I wouldn't put a question like this on a test."
Students: [in chorus] "On purpose."
[pause] "That's called student abuse..."

"Everybody has problems with problems."

"Just try anything! All I'm doing here is fooling around. [pause] ...with mathematics."

"All's fair in love, war, and mathematics!"

Brett Allen (allen@cs.washington.edu)