Variable as a Batch of Numbers

There are a couple of interesting lines from the Common Core State Standards for Mathematics (CCSS-M), referencing the meaning of a variable in an equation, which have been on my mind lately. The first is from 6.EE.B.5 and the second from 6.EE.B.6. I have emphasized in red the bits that I think are significant to this post:

Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true?

Understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

One of the reasons these are interesting (to me) is that, almost universally as far as I can tell, curricula in 6th grade mathematics (CCSS-M-aligned) limit themselves to equations like \(\mathtt{5+p=21}\) and \(\mathtt{2x=24}\), which only have one solution. So, it’s not possible to talk about the values (plural) that make an equation true; nor is it possible to talk about a variable as representing any number in a specified set when all of our examples will essentially resolve to just one possibility.

How do curricula then cover the two standards above? Well, it’s possible to still hit these two standards when you interpret multiple solutions as something that belong with inequalities. Inequalities are part of the “or” statement at the end of 6.EE.B.5 and can be seen as part of “the purpose at hand” in 6.EE.B.6. This, it seems, is the interpretation that most curricula for 6th grade (again, as far as I can tell) have settled on.

Stipulation and Functions

A reason this may be problematic is that it introduces a stipulation (or continues one, rather)—one which, as far as I can tell, is not effectively stretched out in Grades 7 or 8. That stipulation is this: a variable in an equation represents a single number. We dig this one-solution trench deeper and deeper for two to three years until one day we show them this. In this object, a function, the \(\mathtt{x}\) most certainly does not represent a single value.

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But, crucially, \(\mathtt{x}\) doesn’t have to represent a single value even back in 6th grade. That is, when solving an equation in middle school, the variable may wind up to be one number, but we don’t HAVE to make students think that it always will. An equation—even a simple 6th-grade equation—can have no solutions, one solution, or all kinds of different solutions. For example, \(\mathtt{x = x + 2}\) has no real solutions; \(\mathtt{6x = 2(3x)}\) has an infinite number of solutions; whereas the tricky \(\mathtt{x = 2x}\) or \(\mathtt{x = 2x + 2}\) each have one solution apiece. (The latter is a 7th-grade equation, though.)

Once Is Not Enough

The point that an unknown in an equation does not automatically represent one value could be made a little better if solving quadratics or absolute value equations typically preceded an introduction to functions. But even if the content were moved around to fit those topics before functions, the trench is dug mighty deep in middle school. Further, the 8th-grade standard that references different numbers of solutions as we did above, 8.EE.C.7a, is too late, and is often interpreted by curricula as comparing two linear expressions (e.g., \(\mathtt{y = x}\) vs. \(\mathtt{y = x + 2}\); parallel so no solutions), thus keeping the one-solution stipulation ironically intact.

Frequent reminders starting when variables are introduced through the introduction of functions would serve students better, I think, especially when they tackle concepts such as domain and range. The notion that an unknown can represent 0, 1, or multiple values could also help to make linear algebra a bit more approachable when it is introduced.

variable


Check out John Redden’s and Paul Gonzalez-Becerra’s Open Graphing Calculator, which I used in this post.