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.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Some quantities change together. The longer a car drives at a steady speed, the farther it goes; the more hours a babysitter works, the more she earns. In these relationships one quantity, the independent variable, is chosen or changes freely, and the other, the dependent variable, depends on it. Distance depends on time, and pay depends on hours worked.
Sixth graders write an equation for the dependent variable in terms of the independent one, such as d = 65t for a car traveling at 65 miles per hour, and connect that equation to a table of values and a graph. Each representation shows the same relationship from a different angle: the equation gives a rule, the table lists matching pairs (1 hour, 65 miles; 2 hours, 130 miles), and the graph plots those pairs, with the independent variable on the horizontal axis. Students analyze how a change in one variable affects the other: every extra hour adds another 65 miles. This is the first formal look at functions, which become a central topic in eighth grade.
Students may say time depends on distance in a driving story. Ask which quantity you control or choose: that one is independent.
By convention the independent variable goes on the horizontal axis. Graphing distance across the bottom and time up the side makes the graph hard to read.
For 'earns $12 per hour' students sometimes write h = 12p. Testing with a table row (2 hours, $24) shows the correct form is p = 12h.
Maya earns $12 per hour babysitting. Write an equation for her pay p after h hours, make a table for 1 to 4 hours and describe the pattern.
Answer: p = 12h. Pay rises by $12 for each hour, so 4 hours earns $48.
Start from tables students build themselves from a story, then ask them to describe the pattern in words before writing the equation. Linking each table row to a point on the graph keeps the three representations connected.
Assessment items commonly give one representation and ask for another, or ask students to name the dependent variable. Constant-speed and earnings contexts are the most frequent.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: C) The height of the plant
The height depends on how many weeks have passed, so height is the dependent variable.
Answer: 195 (also accepted: 195 miles)
d = 65 × 3 = 195 miles.
Answer: A) c = 7h
Each cost is 7 times the hours: 7 × 1 = 7, 7 × 2 = 14 and so on. So c = 7h.
Answer: 7
Solve 9n = 63 by dividing by 9: n = 7 tickets.
Answer: C) t, time
The independent variable, time t, goes on the horizontal axis by convention.
A full lesson with slides, activities and an exit ticket on dependent and independent variables, pitched to grade 6 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 6.EE.C.9 with an answer key, ready in about a minute.
Make a worksheet →Turn dependent and independent variables into a quiz students answer online that marks itself, with a class summary for you.
Build a test →Ask which quantity depends on the other. The independent variable is the one that is chosen or changes freely, such as time; the dependent variable responds to it, such as distance.
Yes, informally. Students describe how one quantity determines another using equations, tables and graphs. Functions are named and studied formally in grade 8.