Monday, December 18

The difference between a “variable” and an unknown constant

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A lot of very knowledgeable, intelligent and well-meaning mathematics teachers all over the united states misuse the term “variable” when discussing particular problems. To illustrate this, I will compare two equations. The first case will be “2x = 16,” while the second case will be “2x = y + 12.” Many viewers will want to call x a variable in both cases. But we shall explore why “x” is a variable in one case, but is something else in the other case.

First Case

The equation “2x = 16” falls under the category of a “one-step” equation. It is explored in 7th grade Pre-Algebra and is mastered by the end of Algebra 1 (this may vary from district to district because some districts use the integrated math curriculum). It is called a one-step equation because by performing one operation, we can find the value for “x.” In this particular case, we will divide both sides of the equation “2x = 16” by both sides, giving us a solution of x = 8.

Second Case

The equation “2x = y + 12” has both an “x” and a “y” present. What does x equal in this equation? There are infinitely many correct answers to this question. We would be completely correct to call x a “variable” in this case because we can assign any value that comes to mind, so long as there is a y value to match. This is in stark contrast to the first case, where x = 8.

So what’;s the point here? Symbols to represent values in Algebra are often loosely referred to as variables, when this is not always the case. In the first case of “2x = 16,” many very well-meaning, intelligent and knowledgeable mathematics educators would call “2” and “16” constants, and would refer to “x” as a variable. But in actuality, x is just as much of a constant as 2 and 16.

There is no valid reason to call x a variable in the first case. The only meaningful thing that x in the first case has in common with x in the second case is that they both contain an “x.” In our first case, x equals 8, while x has infinitely many values in the second case.

Why is this so important?

Algebra is the branch of mathematics that allows us to model trends that we are able to see. In order to effectively understand mathematical modeling, we MUST be able to differentiate between when we are dealing with a variable, and when we are dealing with a constant, and we must understand that replacing a constant with a symbol does not make it cease to be a constant.

Consider the following word problems:

Problem One

If a 2 kilogram ball was thrown and struck an object with 16 Newtons of force, then what was the acceleration of the ball if Force = Mass x Acceleration?

Problem Two

If a vendor spent 12 dollars on a box of gourmet candy bars and sold each candy bar at a price of $2 per candy bar, then

  1. How many bars must the vendor sell to break even?

  2. How many bars must the vendor sell to make twice the amount of money spent.

  3. How many bars must the vendor sell to make 50?

You might notice that the first and second word problems match up with our aforementioned first and second cases respectively. The first word problem has only one answer. Clearly, the question was about one specific moment in time and the ball could not have had more than one rate of acceleration in the same moment. The second word problem, however, describes an ongoing situation, where there is an input and an output. When we modify “2x = y + 12” so that it becomes “2x – 12 = y,” then you can see that x can represent the number of bars sold, where y is the amount of money made.

In examining these problems it can be seen why it is important to be able to differentiate between a variable and a constant when dealing with mathematical models. Problems that deal with solving for unknown constants pertain to static situations. But variables deal with problems that have a situation that may vary. So it is very important that this distinction is correctly made early on in our algebraic education so that our understanding of developing models is not hurt by obfuscation of the two concepts.

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