The Algebraic Alphabet: Unknowns, Variables, and Parameters
"Constants aren't. Variables won't."
In the transition from arithmetic to algebraic reasoning students begin to use letters (or other symbols) in addition to numbers. As early as the second grade, students learn to solve addition problems with the unknown in any position (Grade 2, Operations and Algebraic Thinking, #1). Later they use letters in many different ways. Beginning in sixth grade, students are expected to use variables in expressions and equations (6.EE.2 and 6.EE.6). At this point, letters become identified with multiple terms: variable, parameter, and unknown. Students must learn to make correct assumptions about what the letters mean. One way to help them is to emphasize that the meaning of the letter will depend on how you use it in the context where you find it.
Here’s an analogy, using a piece of wood. Once the wood is cut into a useful piece, you might call it a board. If you are going to lay that piece on a set of brackets, you might call it a shelf. On the outside of a house you might call that piece siding, and if you are a pirate, you might call it a plank!
Key idea: In mathematics and science, how a letter is used determines its meaning.
We'll explore some ideas and examples to help clarify the meaning of terms that people may mistakenly use interchangeably. You may find the thinking behind these distinctions helpful in your work with students, even if you don't present them in the detail you'll find here.
In early introductions to algebraic ideas, we begin to use letters as unknowns. For example, consider the equation 4 + m = 6. Here only one value of m will satisfy the equation. If the equation were m² – 4 = 0, there are two values of the unknown that satisfy the equation, 2 and –2. When the letter in an equation has one or more specific values that we can find, we call that letter the unknown. Although some people may call m a variable, it is not a variable because it does not vary.
Key idea: We solve an equation for the unknown(s).
As algebraic tasks get more sophisticated, we introduce letters as variables. Consider the equation y = 3x, which defines a relationship between two variables, x and y. As x “varies,” so does y. Values that make the equation true come in pairs, so that if a student picks a value for one of the variables (say x = 2) , it determines the other (y = 6). We usually think of one of the variables as depending on the other. By convention, x is the independent variable and goes on the horizontal axis of a graph; and y is the dependent variable, on the vertical axis. We call these (x, y) pairs ordered pairs or points. We can display a set of points that make the equation true in a table or coordinate graph (7.RP.2c). For example, we can graph y = 3x as a line with slope 3. Variables can also show up in expressions, such as 5x + 2y. Unlike an equation, an expression does not imply a relationship between x and y. Instead, the whole expression will take on a particular value for given values of x and y.
Key idea: A variable is a letter that can take on a set of values and varies within a particular expression, equation, or situation.
Note: In statistics, we often use the terms predictor and response instead of independent and dependent, respectively. Some teachers find these terms more vivid and understandable.
While parameters can change value, they are different from variables because within an expression, equation, or situation they take on a particular value. In our example of the proportional relationship y = 3x, we could have written a more general form y = mx. This general equation still relates the variables x and y, but now contains m, a parameter called the “constant of proportionality.” Here’s an example that illustrates the phrase “constant of proportionality.”
Question: The number of meals (y) eaten in the cafeteria is directly proportional to the number of people (x) that attend the school. Choose an appropriate constant of proportionality and justify why it makes sense. Then use your constant to write an equation relating the number of people in the school to the number of meals eaten.
Answer: Say that you figure that about one third of the students bring their lunch and the rest eat in the cafeteria. Then you could write the equation as y = (2/3)x, where y is the dependent variable and x is the independent variable (the number of meals depends on the number of people). The number 2/3 is called the constant of proportionality because y and x are proportional to each other. If you rewrite the equation as y/x = 2/3, you can see that the constant of proportionality m is actually the ratio of y to x, which is constant: in this model the ratio is always 2/3.
With proportional relationships we have a constant of proportionality, but the more general name for m is parameter. The term parameter acknowledges the fact that the model has a quantity m that can vary, but will take on a single particular value for a given situation. In contrast, the variables x and y will still take on a range of values. For example, you might have some data that you think follows a proportional relationship. The goal could be to find the value of the parameter m that makes the model y = mx best fit the data.
Key idea: A parameter is a special kind of variable. It can take on a range of values, one of which will be appropriate for a given model.
Lines: two variables and two parameters
One standard form for a line, y = mx + b, has four letters. Now x and y are variables while m and b are parameters. The best way to make the distinction is to consider a specific line. Once we are thinking about a particular line, its equation has only one value for the parameter m and only one value for the parameter b. Thus m and b are parameters that become fixed as constants for a particular line. Change m or b and you have a different line. In contrast, a single line with a fixed m and b is made from an infinite number of points. The points are x-y pairs that have a special relationship determined by the equation y = mx + b and the parameters m and b. The relationship dictates that when you choose an x, then you get a specific value of y. Thus x and y are variables because within a single line, they take on many values.
Sometimes within a single problem, letters will shift roles. For example, when studying lines, a common task is to find the values of the parameters m and b given some points on the line. For example: a line runs through the points (2, 3) and (–1, 5). Write the equation of the line.
There are many ways to solve this problem, but if you are fond of y = mx + b, you might first find m using
and then plug that m and one point into y = mx + b to find b. Finally you would rewrite the equation using the parameters m and b you just found, and leave x and y as variables. When you are finding m and b, they are temporarily the unknowns, even though they are parameters of the line.
In another example, the role of a letter shifts from variable to unknown. Given the equation of a line, such as y = –3x + 5, you could ask, “what is the value of x when y = –2?” Now x is temporarily an unknown, even though it is a variable in the original problem. These shifts, and the ambiguity that they imply, demonstrate why it is especially important to emphasize that when we encounter a letter in mathematics or science, we should think about how the letter is used.
What's in a name?
As a final example of shifting roles, let’s consider the letter m, which we have used as a parameter representing the slope of a line. In mathematical modeling or science courses, m might serve as the abbreviation for meter or m might represent the mass of an object. When students study geometry, we sometimes use the letter m to mean measure. “What is m∠B?” means “What is the measure of angle B?” So while a few letters have common uses, such as g for the gravitational constant or π for the ratio between a circle’s circumference and diameter, almost no letter means the same thing in every context. For example, some statisticians use π for a "population proportion," and g is also the abbreviation for gram.
Take home message: In the same way that wood takes on different names depending on how we use it, letters take on different meanings in mathematics and science depending on the context in which they appear and how they are used in that context. We can help students recognize the meaning of letters in mathematics and the language we use to identify them by emphasizing how those letters are used.
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