Numbers provide a way to mark events. This can usually be done in more than one way. We can “slice” reality in more than one way.

The picture above shows a common way to look at a single stock on a stock exchange. Price is shown in yellow, volume is shown in blue. At any moment in time (this chart covers 1/29/1999 through 1/30/2004) you can find the price at which the stock changed hands (the yellow line, with its scale indicated on the left) and you can find the number of shares that changed hands (the blue line, with its scale indicated on the right). For example, somewhere in the spring of 2003, some 3,600,000 shares changed hands on a single day, at prices around $15.

It is not my intent to suggest that price/volume charts tell you a whole lot about stocks or what drives the stock market. Quite the contrary. I simply want to point out that each of its components, price and volume, tell something about what happened that you couldn’t have determined from simply knowing the other. High prices can combine with low volumes or with high volumes; and high volumes can combine with low prices or high prices, with prices going up or with prices going down.

Sometimes, the different ways we have to slice reality are highly correlated. When I take my basket to the checkout counter, there are different numbers I can associate with that basket. I am interested in how much it weighs, for I’ve been carrying the darn thing through the store, and if it gets too heavy I might switch over to a cart. I am interested in how many items are in the basket, since the express checkout lane is only for people with no more than 15 items in their basket. I am interested in how much the items in the basket cost, for that is what I am going to end up having to pay to be allowed to take these things home. These three aspects, weight, number and cost, are correlated, at least in the following way. If I add more to my basket, this will increase the weight, increase the number and increase the cost.

This is a familiar situation. When a boy grows taller, he expects to get heavier, and wear larger shoe sizes. When a lawyer spends more time on your case, you expect his bill to be higher. When the landscapers spend more time on your property, you expect it to be more expensive. When you plant wheat on more acres, you expect to get more of a wheat crop.

Sometimes quantities relate so perfectly, in a way that’s sometimes called a “direct proportion”: Two quantities go up in unison and go down in unison; moreover, when one doubles, the other one doubles too. When one is cut in half, the other one is cut in half too.

The perfect cases may also be trivial: when you measure the length in inches, and I measure the length in feet, our numbers will be in direct proportion. When you measure a distance in miles, and I measure the distance in lightyears, the numbers will be in direct proportion. When you measure the cost in dollars, and I measure the cost in cents, the numbers will be in direct proportion. Sometimes the relationship between quantities is in direct proportion up till a certain point: the price I pay for buying shares of stock is in direct proportion to the number of shares I buy, unless I buy so much that my buying itself drives up the price. Sometimes the price I pay for buying bottles of a certain wine is in direct proportion to the number of bottles I buy, unless I buy enough to buy by the case, in which I expect a discount over the price per bottle.

In an earlier post, I traced one of these perfect direct proportions: One quantity is an amount in dollars, from 0 to $48; in parallel there is a percentage, from o% to 100%. For each amount, there is a corresponding percentage, for each percentage, there is a corresponding amount. When you double the percentage, you double the corresponding amount. Let me steal a picture from that post:

I also want to bring back a picture of the relationship the seventh grade kids reliably seemed to see in this kind of a two-column relationship, a relationship that is actually richer and more encompassing than just noting that when you double the percentage you double the amount and vice versa:

This insight, that when you add two numbers in one column, the corresponding numbers in the other column are added also, is known by a very fancy name: *distributive property*. For most kids, the material about the distributive property in school is one of those gauntlets: kids relate to it as being beaten with sticks, falling, and not being able to get back up. As educators, we fail miserably in conveying our enthusiasm for a(b+c)=ab+ac and related ham-handed expressions of it, and it is the more discouraging since, from my direct observations, kids have such a solid grasp on the idea that adding numbers on the left corresponds to adding numbers on the right. What’s *missing *is that those same kids don’t relate their insight to multiplication.

And yet, what we have here is another way to look at multiplication: multiplication as a relationship between two quantities where the additivity insight is valid (the fancy name to say this: where the distributive property holds).

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