# Common Core Standards: Math

### Expressions and Equations 6.EE.C.9

9. Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
Toward the end of May each year, you might notice a bit of a buzz in your classroom. Students are more energized and less focused, homework is abandoned, and the sound of backpacks being zipped up echoes throughout the classroom noticeably earlier than before. It's no secret why this happens; it's because students are eager for the school year to be over and for summer vacation to start.

We know you experience the same kind of anticipation before teaching this standard. After all, it's basically the bedrock on which all of algebra is built. The sooner you teach students about independent and dependent variables, the sooner they'll start learning about lines, functions, and before you know it, they'll bee up to their ears in derivatives and integrals.

But before they do any of that, they've got to start here. And believe us: there's a lot to start with.

Students should begin by understanding that two-variable equations are all about relationships. In fact, the equation tells us exactly how one variable (called the independent variable) affects the other variable (not surprisingly called the dependent variable). Give them concrete examples of the differences between these two types. For instance, your height depends on your age, but the reverse isn't true; you don't magically get older just because you're taller.

After students have the differences between independent and dependent variables down, they should start taking a look at equations of the forms y = px or y = x + p. (Anything more complicated than that, and they might start to hyperventilate.)

Show students how we can convert these equations into tables and graphs of ordered pairs, and that the tables and graphs can tell us everything we need to know about the relationship between the two variables. No need to ask their couples' therapist.

Once students can turn these two-variable equations into tables and graphs on their own ("Look, Ma! No hands!"), stop giving them the equation outright. Instead, force them to come up with it themselves, stressing the fact that we want to express the dependent variable in terms of the independent variable. They should be able to interpret the relationship between the variables in terms of the context, too, because math helps us understand real life, and reality is 100% context.