September 03, 2003

One Hundred Interesting Mathematical Calculations, Puzzles, and Amusements: Number 19: The Distributive Law, or The Get-Out-of-the-Way Problem

Number 19: The Distributive Law, or the Get-Out-of-the-Way Problem

"So do you understand why the distributive law of multiplication over addition is important?"

"Yes. It is important because it provides math teachers with yet another way to torment Thirteen-Year-Olds."

"It's a free country--or it's supposed to be."

"The right answer is that the distributive law allows you to rewrite equations--to replace one equation with another that means the same thing--and that the main point of algebra is to keep rewriting and rewriting equations, expanding, factoring, and cancelling, until you get to a form where the answer is obvious."

"Sure."

"No. I mean it. Take... will you believe me if I say that if you fire a cannonball straight up with an initial upward velocity of 640 feet per second, then its height above the earth measured in feet t seconds after liftoff is given by the equation:

h = 640t - 16t2 ?"

"If you say so."

"Well, now, let's factor this equation. Do the two terms on the right hand side share anything?"

"They share a sixteen."

"So we can rewrite the equation as:

h = (16)(40t) - (16)(t2),

right?"

"I guess so."

"And do they share anything else? do the 40t and the (16)(t2) share anything else?"

"I guess each is a multiple of t..."

"So we can pull out another factor of t, and rewrite the equation as:

h = (16t)(40) - (16t)(t) = (16t)(40 - t) ?"

"Yep. But you need to get to the point."

"Well, just look at:

h = (16t)(40 - t)

and ask yourself, 'When will the left hand side be zero? It will be zero when the right hand side is zero, and that is...?"

"Well, the right hand side is two things multiplied together, and so it will be zero only when one or the other is zero..."

"Exactly. And the first thing on the right hand side is zero when...?"

"When t = 0..."

"And that's the moment that the cannonball is fired upward. So that makes sense. And the second thing on the right hand side is zero when...?"

"When t = 40"

"So that in the form:

h = 640t - 16t2

the equation doesn't tell you much. But in the form:

h = (16t)(40 - t)

it tells you that, after firing the cannon straight up, you have forty seconds to get out of the way--or else!"

"Hmmm..."

"But the two equations are the same--the distributive law guarantees it."

"So you're saying that people who don't know the distributive law regularly get klonked on the head by their own cannonballs?"

"That's not the point..."

Posted by DeLong at September 3, 2003 08:22 PM | TrackBack

Are you telling me they don't teach the metric system in the US? Still?

Posted by: Eccles on September 4, 2003 12:44 AM

Eccles, they teach both and then do everything in English measures. They do it that way because English measures are less confusing to retired people. :)

Posted by: Stan on September 4, 2003 05:53 AM

Ahhh, glib answers from youngsters about laws of mathematics. My favorite turned up on an undergrad statistics for business folks exam. The question ran something like, "You have a friend who has trouble explaining the Law of Large Numbers -- help your friend out and provide a concise account of that law." One student answered, "I would not have as a friend someone who cared about the Law of Large Numbers."

I gave a few points just for honesty/chutzpah

Posted by: David on September 4, 2003 06:02 AM

Stan writes:
> Eccles, they teach both and then do everything in English
> measures. They do it that way because English measures
> are less confusing to retired people. :)

Then I say: let retired people learn to use Google!

Seriously, bring up everybody's favorite search engine and type:

55 miles per hour in km per hour

or

37 celsius in fahrenheit

or

640 ft/s in m/s

or

450 horsepower in watts

or even

70 mph in furlongs per fortnight

Yeah, the Google calculator even does a pretty good job with real arithmetic, although I haven't found a way to make it do anything involving a variable, there doesn't seem to be any way to do integrals, it is ignorant of linear algebra, and creates no graphs.

But it would instantly have solved a problem I posed the other day in my cog psych class:

If you could make 1 move per second doing the Tower of Hanoi Puzzle, how long would it take you to solve the 64 disk puzzle? (Remember: the number of moves is 2^n -1
where n is the number of disks.)

The fun thing about that problem is that undergrads here are always off by at least a factor of 1 million. Even after
I tell them that:

2^10 ~= 1000

and

3.15 x 10^7 seconds in a year.

It ain't just the metric system that our students don't know...

Posted by: Jonathan King on September 4, 2003 07:21 AM

Those of us who had math teachers from the Cloud Cuckoo Land school can appreciate how lucid and motivating this explanation is!

Posted by: John on September 4, 2003 07:24 AM

Is this calculation right..? Surely, at t=40 seconds, the cannonball would be high in the air, about to begin its descent. At t=80 seconds, you'd best be walking away..

Posted by: Dan Ryan on September 4, 2003 05:00 PM

Metric units are often inconvenient. Try woodworking in metric. You will quickly find yourself using fractions of a centimeter. The inch is better than the centimeter. For one thing it’s about the width of your thumb and divides up nicely into quarters, eighths and sixteenths. And of course a foot is close to, well the size of your foot. A yard is about the distance from your nose to the tip of your forefinger. In short English units are right sized for human dimensions. A degree F is finer than a degree C. We usually deal in tenths of a mile so there is really no advantage over kilometers and meters over miles and feet or yards. About the only advantage for the US to convert to the metric system is to bring us into conformity with most of the rest of the world. It is annoying to have to keep two sets of some tools. For example I have both metric and English socket and Allen wrenches, somewhat of an annoyance. Too bad Napoleon conquered Europe.

Posted by: A. Zarkov on September 4, 2003 10:52 PM

If you are good with the distributive and communitive laws you might solve this one. Write the inverse of a complex matrix solely in terms of real matrices and the inverses of real matrices. You do it like you do the reciprocal of a complex number, 1/z = 1/(x+ i y). But since matrix multiplication is generally not communitive, you run into a problem. So you need a little trick. I give this to a graduate math class I taught, but none of them got it.

Posted by: A. Zarkov on September 4, 2003 11:03 PM

"But since matrix multiplication is generally not communitive"

You mean "commutative", right?

Posted by: Abiola Lapite on September 5, 2003 03:26 AM

Whoops. Yes, commutative.

Posted by: A. Zarkov on September 5, 2003 08:48 AM

Dan,

the calculation is correct. In another sense, the maximum distance the cannonball travels is when 640 - 32t = 0 or at 20 seconds. So, it spends the first 20 seconds going up and the next 20 coming back.

the relevant formula is: s = ut + 0.5 g t^2

Posted by: Suresh Krishnamoorthy on September 5, 2003 02:08 PM

A. Zarkov writes: Metric units are often inconvenient. Try woodworking in metric. You will quickly find yourself using fractions of a centimeter.

Those are called millimetres. They're useful.

The inch is better than the centimeter. For one thing it’s about the width of your thumb and divides up nicely into quarters, eighths and sixteenths.

Those of us who had metric educations wonder what's so nice about sixteenths. I will grant you that the one-syllable "inch" is easier to talk about than the mouthful, "centimetre".

And of course a foot is close to, well the size of your foot.

No doubt that's useful to cobblers who make shoes for people with huge feet.

And a metre is about the same.

In short English units are right sized for human dimensions. A degree F is finer than a degree C.

How often do you care about the difference between 77 F and 78 F?

We usually deal in tenths of a mile so there is really no advantage over kilometers and meters over miles and feet or yards.

I wonder how many Americans can actually tell me how many yards there are in a quarter mile.

About the only advantage for the US to convert to the metric system is to bring us into conformity with most of the rest of the world.

Yup, that's the one. To provide a standard system that everyone could accept is what the metric system was created for, and is still the most powerful argument for using it.

Posted by: colin roald on September 21, 2003 10:46 PM

I agree with A. Zarkov to a degree. Miles or kilometers are both fine once you have a sense of how long one is. But the point of F or C is that you get a feel for how hot or cold it is within 10s. That is, "It is in the 80s means hot" "In the 90s" is really hot while "in the 70s" is warm. Try that with "It's in the teens" with centigrade. You can always use a number like "17 degrees outside," but it sounds too clinical. The world can go metric - except the temperature. That should be the deal at the next WTO meeting.

Posted by: remo williams on October 13, 2003 11:40 AM

Here in Québec, the metric system is the official system, but in every day life, we use both. But not both at the same time. Each as its use. For example, we measure the temperature of the water in the pool in Fahrenheit but the outside temperature in °Celsius. So if you tell me : "the water is at 20°C" I don’t really know if its warm or not and if you tell me : "its 70F outside, i don’t know if I should put a jacket on...

Best of both world or worst of both world...?

Perhaps we could all use Kelvins ? (Hey! Put a sweater, its only 283 Kelvins outside!)

(and of course, in Québec, French is the official language so don’t be too harsh, I don’t write in English very often...)

Posted by: É on December 16, 2003 10:39 AM