Defining Variables

Often students are confused with variables: those x's, y's, z's and other letters that begin to replace numbers beginning in algebra and continuing on into geometry, algebra II, Trigonometry, Precalculus, and Calculus. While there are several aspects to variables, one of the best ways to start is to understand a variable as a placeholder for a number. Take the equation: 7 + 3 = 10. Inserting x into this equation for the number 3 produces the "same" equation 7 + x = 10. Yet this equation also shows how a variable is "defined" in that x must equal 3 for the equation to work. More often variables are defined, however, not by a specific number, but with an idea that covers either a to-be-determined range of numbers or an as yet unknown number.

Try this example. Suppose I go to the store planning to put a cheese and cracker tray together for an upcoming football game. I have 24oz. of cheese and 28oz. of crackers and I expect each person to eat 4 oz. of cheese and they need 2oz of crackers for every 1oz. of cheese, so how many people can I serve cheese to, and how many more crackers do I need for all my cheese? Before scrolling down, think about how you would solve this in real life and then think about how you would work this problem for class.

I would begin to define y as the number of people and z as the additional amount of crackers in ounces that I need to buy, i.e. the two questions being asked that I do not know. Then I would link those variables to what I do know. I know that for every person y, they will eat 4oz of cheese, and that should equal the total amount of cheese (24oz), i.e. 4y = 24. I also know that for each of the 24oz's of cheese, I need 2oz of crackers. I also know that the total amount of crackers for that cheese is the 28oz I have plus the z crackers I need to buy. Thus, 24 x 2 = 28 + z. Solving these equations, I find I have enough cheese for y = 6 people, and that I need z = 20 oz. of crackers.

While your teachers may try to complicate things with unneeded information, additional variables or other mathematical operators, most problems involve identifying what you know (the amounts of cheese and crackers you have and the rates at which those items are needed), defining as variables what you need to find out, and preparing equations that relate what you know to the variables you don't know.


John, nice explanation, especially the placeholder analogy. If you don't mind, I'll use it with my math tutee's.
Hello, John M. Enjoyed your post about variable confusion. As a credentialed HS math teacher, I’ve given significant thought to the issue you raised and would like to suggest the issue is deeper than it seems at first. Let me start first with notion that variables are only "place holders" for numbers. I would 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.) While not a rigorous proof, I use the following scenario to help my students understand what I’m claiming when I say the “letters” in math are actually numbers… Most algebra and beyond students, and some preälgebra students as well, will readily tell you that you can change “t t” into t^2 (because t * t = t^2). If that's really so, I then say, I should be able to write “Mary had a lit^2le lamb.” The looks of consternation / amusement / puzzlement are usually quick and profound… they do NOT agree that that’s okay. At this point, I give them a chance to tell me why not. If they can’t verbalize their heartburn with my suggestion, I 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 = t^2” if perfectly fine in the world of mathematics, it must be because those “letters” are in fact *numbers*. The genesis of this confusion happens about the 3rd to 5th grade where teachers after having spent 4 – 6 years helping students link letter-symbols to various sounds / phonemes, they all of a sudden say, “let’s start using letters as placeholders for numbers.” Now, it's an unfortunate that these teachers use the word "placeholder" as that word implies that the letters aren’t really numbers… after all, an object that is 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 students become confused when math teachers come along and proceed to put and entirely different set of meaning on top of the letters that the students have worked so hard to master without so much as a “may I borrow your letter-sound system symbols”. It’s as if someone out of the blue decided that the word “crumple” will now mean “dog” and then expect you to understand that “The crumple(sic) needs to go outside before he pees on the carpet.” is a rather urgent bit of information. The only difference is that at least someone would explicitly tell them that they’d changed the meaning of the word. So, where does that leave us? We have symbols such as (3, -5, ½, p, etc) that are *numbers* and we have these letters, these “placeholder” numbers, that are in reality, numbers. As a matter of fact, the number p is an explicit example of what I've been talking about... looks like a letter, but is really a number. The problem is further compounded 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 statement? 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) and they would be reënforced in that way of thinking by receiving full credit on 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 the mathematical operations of multiplication and addition on them on the left side of the equation and because an equal sign has been placed between them those “numbers” on the left side 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 mathematical operations on them. I could say more about alternate ways to frame the concept of variable numbers, constant numbers and the other category of numbers parameters, but I’ve probably gone too long. Hope you find this helpful. Best wishes in your efforts to assist confused students. Sincerely, John H
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