CCSS.MATH.CONTENT.HSA.CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
CCSS.MATH.CONTENT.HSA.CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
A variable is a letter that represents an unknown number. Sound familiar? It should. This is how many textbooks define a variable. But we are going to give several reasons why variables deserve a better definition—one that will support more robust student understanding.
To start, not all letters used in mathematics represent an unknown number. Think about i and e. Each of these is a letter and they’re used in math, but they are not unknown, in fact, they are rather well known.
So would it be better to define a variable as a letter that sometimes represents an unknown number? Well, not exactly.
We should also note that not all variables are even letters. They can be other symbols too. In fact, many students have seen these sorts of variables since first grade, they just weren’t always letters.
So maybe we should define variables as symbols that sometimes represent an unknown number?
It’s getting better, but we’re still not there yet. You see, variables don’t have to represent an unknown number. They often do, but they don’t have to. Variables can represent many different types of mathematical objects such as expressions or matrices or functions.
Okay, so variables are symbols that sometimes represent an unknown mathematical object?
We’re getting very close now. The last problematic bit is the word “unknown.” Unknown typically means a specific value that we don’t happen to know yet. For instance, X is unknown in the equation X plus 3 equals 5… until you realize that it’s 2, then it’s known.
But unknown can also mean it is unknown because there are many possibilities, as in this case. Here, the variables X and Y represent a whole domain and range of values. In other words, X and Y vary.
Okay, I think we’re ready for the final version of our definition: Variables are symbols that represent unknown or varying mathematical objects.
So now that we have a more robust definition, how do we accommodate learners by using these ideas? The key is to help students understand the different uses of variables. Students tend to think about variables like the typical textbook definition—as letters that represent an unknown quantity. So we need to perturb them into expanding their notion of variables to include not just unknowns but also values that can vary. We should be explicit about the differences. Help them recognize when they’re looking for an unknown but fixed value versus when there are many possible values—and over time, push them to be able to tell you about the difference. If they’re having trouble seeing the difference, go ahead and let them plug in many different values for an equation like X plus 3 equals 5. And when they see different results, have them replace the equal sign with something that matches the situation. This flexibility can help them realize the difference between variables as unknowns and variables as varying values. Another way they can be flexible and creative is to use different symbols besides x and y as variables. The more their experiences vary, the better their understanding will be.