by Dave Maier
Some students first run into trouble when they get to fractions, usually in sixth grade or so. Now we are writing the same number in rather different ways (1/2 = 2/4 = 0.5, and so on), and we can’t really count on our fingers either. All of a sudden there are a whole bunch of numbers between 2 and 3. In fact, as it turns out, there are an infinite number of such numbers. Infinity was okay when it was the biggest number of all, all the way on the end (or ends) of the number line and thus safely out of the way, but now we’re using it to count things, and those things are themselves not only the things we count with, but the numbers between what we seem now to be calling the “counting” numbers. (It even turns out – although they don’t make a big deal of this in sixth grade, thank goodness – that there are more numbers between 2 and 3 than there are “counting” numbers on the whole number line, even though both numbers are infinite. Yikes!)
Again, though, in arithmetic at least we’re just talking about numbers. Every problem has a single right answer, even if we now get to write that answer in different ways. But then, all of a sudden, straight up ahead: algebra.
Now, besides numbers, we have letters: x’s and y’s and stuff. x, they tell you, is the unknown. At first this seems not that different from arithmetic: after all, before we do the arithmetic problem, the answer is just that: unknown. Why not call it x? When there’s only x, there’s just the one solution, most of the time anyway. For example,
(1) x – 7 = 14
– an algebra problem – doesn’t seem that far away from
(2) 7 + 14 = _______
– an arithmetic problem. Each simply requires that we add 7 and 14 to get 21; the only difference is that we write the answer to (1) as “x = 21”, where the answer to (2) – the thing that goes in the blank – is just “21”.
But what about when there are both x’s and y’s? Like this:
(3) 3x + 5y – 13 = 5x + 4y – 12
Using the algebra rules they taught me, which are pretty much the same as the rules of arithmetic (you can add the same number to both sides, for example), I can simplify this to “-2x + y = 1”, but now it looks like I’m stuck. Not only do I not know what x is; I don’t know what y is either. Help!
Now of course at the same time as they’re introducing x’s and y’s, they’re also telling us about lines, and demanding that we figure out their slopes and y-intercepts, and this is often enough for our eighth-graders to make the connection between the algebra of x’s and y’s and the geometry of the x-y plane. But I have found that there is often a lingering conceptual unease about variables, even in students who did well in 8th grade and are now struggling with pre-calc, an unease which should probably have been addressed at the time. So let’s press on.
The concept of a variable isn’t really, or simply, the weird math thing it looks like. We already know what variables are, because we use them in English as well, (presumably) perfectly well.
First, compare (2) to
(4) ________ was the third president of the USA.
and (1) to
(5) Someone was the third president of the USA.
In each case the “answer,” if that’s how we want to look at (5), is “Thomas Jefferson”. Of course, sometimes English predicates have more than one thing they apply to:
(6)/(7) ________ / Someone is on the current roster of the Boston Red Sox.
But I digress.
So far, so good. Our problem with math, though, concerns the two-dimensional case: x’s and y’s. (4) and (5) involve one-place, or monadic, predicates. Let’s try a two-place predicate. It takes two people for anyone to be a sister:
(8) ______ is a sister of ______.
Anyone who can use the word “sister” properly understands this perfectly. It may look funny to write
(9) Someone is the sister of someone else.
but it’s just the natural analogue of (5) for binary predicates. If it has an “answer,” that answer would, as with (6)/(7), be a set: here, a set of ordered pairs of persons such that the first is the sister of the second, e.g.
Not to get too technical [*cough* referential vs. substitutional interpretations of variables *cough*], but if we think of things like (8)/(9) not as problems to be solved but simply as definitions of sets of ordered pairs, this makes (9) look less funny. After all, I don’t need to list all, or even any, pairs of people such that the first is the sister of the second, in order to tell you what sisters are (= what “sister” means), although of course (8) or (9) isn't any help all by itself. This might also – more to the point – help us with (3).
Returning to math, then, let’s say we take (3) this way, less as a problem to be solved – although of course it does need to be simplified, and thus might indeed appear on a math test – than as defining a set of ordered pairs of numbers. Of course there are an infinite number of such sets, most of which are not at all interesting. The x-y plane, a.k.a the Cartesian coordinate system, allows us to visualize the more interesting patterns associated with such sets and use them for things like physics and the other sciences.
It shouldn’t be too surprising that if our equation isn’t very complicated – only x’s or x-squareds and y’s or y-squareds, plus constants (= no cubes or higher, no fractions [e.g. 1/x] or fractional exponents) – the shape our set of ordered pairs forms on the x-y plane is simple too: so simple, in fact, that we may even have a geometrical name for it, with its own geometrical definition.
In fact this is pretty obvious in the case of (3), which we can also write as
(3a) y = 2x + 1
which we may now recognize as a line written in its “slope-intercept” form, so called because we may simply read off from it its slope (2) and y-intercept (the y coordinate of the point at which the line intersects the y axis, in this case (0,1)). We can be asked to generate or make sense of other forms of that same line, such as
(3b) y – 3 = 2(x –1)
– which is in “point-slope” form, allowing us to read off not the y-intercept but instead another particular point on the line, here (1, 3). Or, more likely in math class, we use it when we are given the slope of our line and a point on it like (1, 3), and asked to get to (3a) from there.
We can also be asked to find the “solution set” of a “system of simultaneous equations” both of which look like (3a). This turns out (most of the time) to be a single value of x paired with a single value of y. When we understand the connection to geometry, this is not at all surprising, since of course we knew already that two (non-identical, non-parallel) lines intersect in a single point (x,y).
Lines actually get boring pretty fast, but in algebra II, which most students hit in tenth grade, we get things of the form
(10) A(x-squared) + Bx + C(y-squared) + Dy + E = 0
for various values (usually integers, thank goodness) for A, B, C, D, and E. (Later on there’s an xy term as well, but that’s nasty so let’s skip that.) This could be a number of things, but if we’ve absorbed today’s lesson we know that even this is probably just a fairly simple pattern of points on the x-y plane. And indeed everything we can get from something like this is (I think) what we call a “conic section”: a curve we get as a cross section when we cut a three-dimensional cone (or, in one case, two cones meeting at their pointy ends) with a plane (examples of conic sections are circles, ellipses, parabolas, and hyperbolas). Take this one, where A = C = 1 (and actually all we need is the A = C part, because we can always just divide the whole business by A to get them to be 1).
(11) x-squared + 6x + y-squared – 4y – 12 = 0
Using algebra rules we will learn for this very purpose, we can write this as
(12) (x + 3)-squared + (y – 2)-squared = 25
So which pattern do these points make on the x-y plane? Well, consider first
(13) x-squared + y-squared = 25
Now for any point (x, y) on the plane, we can draw a vertical line (of length y) up or down to the x-axis, and then a line from the origin (0, 0) to meet that line at (x, 0), and then a third line from the origin to (x,y). Now we have a right triangle which has one leg with length x and another with length y, and so using the Pythagorean theorem
(14) x-squared + y-squared = r-squared
we get that r-squared = 25 and r = 5; and now we can see why I called that length “r”, as it is not simply the hypotenuse of our triangle but the radius of a circle, one made up of all the points on the plane which are at distance 5 from the origin. So (13) picks out a circle centered at (0, 0) with radius 5. And this makes perfect sense, as the geometric definition of a circle is the set of points all of which are the same distance (= r) from a given point – and of course when we draw a circle with our compass, that's exactly the set of points we're picking out.
(15) (x – 1.2)-squared + (y + 0.5)-squared = 1
Now all our anxiety should be dispelled, yes? At least until trigonometry anyway!