6.EE.C.9 6th Grade Expressions & Equations

Independent vs. Dependent Variables

Use variables to represent two quantities that change in relationship to one another, distinguishing the dependent and independent variables.

How to explain it

At this standard, students identify independent and dependent variables in real-world contexts, write equations expressing the dependent variable in terms of the independent variable, complete tables of values, and analyze relationships using graphs. Note: Problem 15 (R-D=ii) asks students to write an equation AND verify an ordered pair satisfies it — multi-representation synthesis bridging this standard with coordinate-plane work from #22.

The anchor students hold onto: x = independent (input) · y = dependent (output)

Variable relationships extend to inequalities (6.EE.B.8) and later drive proportional reasoning (7.RP.A.2) and linear function graphing (8.EE.B) in grades 7 and 8.

Worked examples

Example 1 Identify Variables

Notebooks cost 3 dollars each.

Step 1Independent: number of notebooks (n) — you choose how many

Step 2Dependent: total cost (c) — depends on number of notebooks

Step 3Equation: c = 3 × n

Answerx = n (notebooks); y = c (cost); c = 3n

Example 2 Table to Equation

Table: (1,8), (2,16), (3,24).

Step 1Rate: 8 ÷ 1 = 16 ÷ 2 = 24 ÷ 3 = 8 dollars per hour

Step 2Independent: hours (x); Dependent: pay (y)

Step 3Equation: y = 8 × x

Answery = 8x

Common mistakes

What students write Writing x = [rate] × y instead of y = [rate] × x — treating the dependent variable as the input.

The fix Ask which quantity changes freely — that is always x. The dependent variable (y) goes alone on the left side: y = [rate] × x.

Try this Student work: Independent variable: total distance (d) Dependent variable: hours driven (h) Equation: h = 8 × d (WRONG) A bicycle travels 8 miles per hour. The student above analyzed the situation. Identify the error and write the correct equation.

What students write Calling the output variable "independent" because it sounds more important or self-contained.

The fix Independent means it changes on its own — not that it is more important. Hours, distance, or items chosen freely by the user are the independent variables.

Teacher tip

Head off the two predictable errors before they happen. First: Ask which quantity changes freely — that is always x. The dependent variable (y) goes alone on the left side: y = [rate] × x. Second: Independent means it changes on its own — not that it is more important. Hours, distance, or items chosen freely by the user are the independent variables.