The Difference Of Two Triangles


[ Trapezoid Week continues! ]

Yesterday I set out a diagram, showing off one example of a trapezoid, with which I mean to show one way to get the formula for a trapezoid’s area. The approach being used here is to find two triangles so that the difference in area between the two is the area of the trapezoid. This can often be a convenient way of finding the area of something: find simple shapes to work with so that the area we want is the sum or the difference of these easy areas. Later on I mean to do this area as the sum of simple shapes.

For now, though, I have the trapezoid set up so its area will be the difference of two triangle areas. The area of a triangle is a simple enough formula: it’s one-half the length of the base times the height. We’ll see much of that formula.

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Eliminating Your Footprints


When last we discussed divisibility rules, particularly, rules for just adding up the digits in a number to tell what it might divide by, we had worked out rules for testing divisibility by eight. In that, we take the sum of four times the hundreds digit, plus two times the tens digit, plus the units digit, and if that sum is divisible by eight, then so was the original number. This hasn’t got the slick, smooth memorability of the rules for three and nine — just add all the numbers up — or the simplicity of checking for divisibility by ten, five, or two — just look at the last digit — but it’s not a complicated rule either.

Still, we came at it through an experimental method, fiddling around with possible rules until we found one which seemed to work. It seemed to work, and since we found out there are only a thousand possible cases to consider we can check that it works in every one of those cases. That’s tiresome to do, but functions, and it’s a legitimate way of forming mathematical rules. Quite a number of proofs amount to dividing a problem into several different cases and show that whatever we mean to prove is so in each ase.

Let’s see what we can do to tidy up the proof, though, and see if we can make it work without having to test out so many cases. We can, or I’d have been foolish to start this essay rather than another; along the way, though, we can remove the traces that show the experimenting that lead to the technique. We can put forth the cleaned-up reasoning and look all the more clever because it isn’t so obvious how we got there. This is another common property of proofs; the most attractive or elegant method of presenting them can leave the reader wondering how it was ever imagined.

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A Quick Impersonation Of Base Nine


I now resume the thread of spotting multiples of numbers easily. Thanks to the way positional notation lets us write out numbers as some multiple of our base, which is so nearly always ten it takes some effort to show where it’s not, it’s easy to spot whether a number is a multiple of that base, or some factor of the base, just by looking at the last digit. And if we’re interested in factors of some whole power of the base, of the ten squared which is a hundred, or the ten cubed which is a thousand, or so, we can find all we want to know just by looking at the last two or last three or last or-so digits.

Sadly, three and nine don’t go into ten, and never go into any power of ten either. Six and seven won’t either, although that exhausts the numbers below ten which don’t go into any power of ten. Of course, we also have the unpleasant point that eleven won’t go into a hundred or thousand or ten-thousand or more, and so won’t many other numbers we’d like.

If we didn’t have to use base ten, if we could use base nine, then we could get the benefits of instantly recognizing multiples of three or nine that we get for multiples of five or ten. If the digits of a number are some strand R finished off with an a, then the number written as Ra means the number gotten by multiplying nine by R and adding to that a. The whole strand will be divisible by nine whenever a is, which is to say when a is zero; and the whole strand will be divisible by three when a is, that is, when a is zero, three, or six.

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How To Recognize Multiples Of Ten From Quite A Long Way Away


I got so caught up last week talking about the different possible bases that I forgot to the interesting thing I had wanted to talk about those bases. I suppose that will happen as long as I write to passion rather than plan. It gives me something to speak about today, at least.

Here is one thing implied by having a consistent base for all these numbers in which position is relevant: a one in each column represents the base-number of units of whatever the next column over represents. That is, in base ten, a one in the tens column represents ten units of one; a one in the thousands column represents ten units of one hundred. I mention this obvious point because it is so familiar and simple as to pass into invisibility. (It also extends past the decimal point; a one in the hundredths column is equivalent to ten units of a thousandth. But I want to talk about divisibility, in the whole numbers, and so leave fractions for some later time.)

This is tidy, in a way that we don’t see in variable bases. It will give us one tool for neat little divisibility rules. That tool appears just by writing things in the appropriate way, which is the best sort of tool. It saves on time trying to prove it works.

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In Defense Of FOIL


I do sometimes read online forums of educators, particularly math educators, since it’s fun to have somewhere to talk shop, and the topics of conversation are constant enough you don’t have to spend much time getting the flavor of a particular group before participating. If you suppose the students are lazy, the administrators meddling, the community unsupportive, and the public irrationally terrified of mathematics you’ve covered most forum threads. I had no luck holding forth my view on one particular topic, though, so I’ll try fighting again here where I can easily squelch the opposition.

The argument, a subset of students-are-lazy (as they don’t wish to understand mathematics), was about a mnemonic technique called FOIL. It’s a tool to help people multiply binomials. Binomials are the sum (or difference) of two quantities, for example, (a + 2) or (b + 5). Here a and b are numbers whose value I don’t care about; I don’t care about the 2 or 5 either, but by picking specific values I avoid having too much abstraction in my paragraph. The product of (a + 2) with (b + 5) is the sum of all the pairs made by multiplying one term in the first binomial by one term in the second. There are four such pairs: a times b, and a times 5, and 2 times b, and 2 times 5. And therefore the product (a + 2) * (b + 5) will be a*b + a*5 + 2*b + 2*5. That would usually be cleaned up by writing 5*a instead of a*5, and by writing 10 instead of 2*5, so the sum would become a*b + 5*a + 2*b + 10.

FOIL is a way of making sure one has covered all the pairs. The letters stand for First, Outer, Inner, Last, and they mean: take the product of the First terms in each binomial, a and b; and those of the Outer terms, a and 5; and those of the Inner terms, 2 and b; and those of the Last terms, 2 and 5.

Here is my distinguished colleague’s objection to FOIL: Nobody needs it. This is true.

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