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Variables and Numbers or "When is a letter, not a letter?"

I recently bumped into a nicely written blog item that a fellow tutor wrote to help his students understand what variables are about. It got me thinking about the topic and what follows below is my take on what variables are and how they should be used in algebra (and beyond.)

Let me first start with the notion that variables are only "place holders" for numbers. I'd suggest not only are variable place holders for numbers, they are actually *numbers in their own right* (admittedly numbers with unspecified values, but numbers none the less.)

To help you make sense of what I’m saying, consider the scenario below…

Most algebra and some pre-algebra students as well, will readily tell you that you can change “ tt ” into “ t² ” (because t · t = t²). If that's really so, I then say, I should be able to write “Mary had a lit²le lamb.” If you’re like most of my students, you’ve probably got a look of consternation / amusement / puzzlement on your face at this moment because you do NOT agree that that’s okay… in other words, “lit²le” is not the same thing as “little.”

Give a little thought to what you think the problem is here… (see you back in a moment)

(Imagine you're hearing "Jeopardy" music here while your thinking...)

Hi! Welcome back.

Don’t know what you came up with to explain what the problem is, but I’ll suggest the reason it looks so wrong is that the rule that lets us replace a doubled-letter by the same letter with a raised two is a rule that’s suitable only for numbers and not for letters. So, given that “ t · t = t² ” is perfectly fine in the world of mathematics, it must be because those “letters” are in fact *numbers*.

Now, the reason things got so confusing about these variables is that back in the 3rd to 5th grade where your teachers spent 4 – 6 years helping you see letters and think about the sounds they “make”, your math teacher said one day, “let’s start using letters as placeholders for numbers.” Now, it's unfortunate that math teachers use the word "placeholder" as that word implies that the letters aren’t really numbers… after all, something that’s used as a placeholder usually *isn't* the actual object. If it were the same thing as what it's placeholding for, there'd be no need for a placeholder, right?

It’s no wonder you got confused when your math teacher came along and asked you to put these letters, that represent sounds, into equations that are supposed to be using numbers. And, I’ll bet they didn’t even have the courtesy to tell you what they were really doing (don’t be too hard on them though, because most math teachers were never told either… they just figured out how to make it work.) It’s as if someone out-of-the-blue decided that the word “crumple” will now mean “dog” and then expected you to understand that “The crumple needs to go outside before he pees on the carpet.” is a rather urgent bit of information.

So, where does that leave us? We have symbols such as (3, -5, ½, pi , etc) that are *numbers* and we have these letters, these “placeholder” numbers, that are in reality, numbers. As a matter of fact, the number pi is an excellent example of what I've been talking about... looks like a letter (because it started out as a letter), but it’s really a number that’s approximately equal to 3.1415927… .

This confusion about variables is made even worse by the fact that nearly everyone uses the words “variable” and “constant” as nouns when they are in reality *adjectives*. What do I mean by *that*?

Consider the equation 2 x + b = 20. If we were to categorize the symbols in this equation, most people would quickly say that we have two “variables” (x & b) and two “numbers” (2 & 20) an equal sign and an addition sign and they’d be rewarded for thinking that way when their teacher gave them full-credit for the question.

I assert that there are in reality *four* numbers in the equation (b, x, 2, & 20), and that the numbers b and x are *variable* numbers, whereas the 2 and the 20 are *constant* numbers.

How do I know that the x and b are numbers? Because I’m doing multiplication and addition on them on the left side of the equation and because an equal sign has been placed between them and the universally accepted number on the right side. If x and b were not numbers, I’d never be able to equate them to other numbers nor could I do math on them.

So, to wrap things up, in algebra, there are two basic kinds of numbers, constant numbers like 1, 4, and pi, (constants because their values say constant... 3 is 3 is 3 and not 2.4) and variable numbers that look exactly like letters, except we really know that they’re numbers because they’re showing up in our equations.

Why would we even want to do such a thing? Well, the reason is that it makes it easier to remember formulas. Some students get real comfortable with the idea of always using x & y as the variable (numbers) in their equations. The only problem with this is that if we only use x & y for our variables, the equation for the area of a rectangle A = w · h and the distance equation d = r · t end up looking exactly the same… i.e., A = x · y and d = x · y. by letting different letters stand for different numbers it’s easier to remember both the equations instead of having to figure out whether x · y is supposed to mean distance, area or even something else.

Hope this helps.

- John H.