Eighth Grade · Math · week 13

Linear vs. nonlinear

Distinguish linear functions (constant rate of change) from nonlinear functions using tables and equations.

A relationship is linear only if the rate of change is constant -- equal steps in the input always give equal steps in the output. Test any table by computing the difference between EVERY consecutive pair of outputs; if that difference ever changes, the relationship is nonlinear.

Equations with an exponent on x (like x^2), a root, or x in a denominator are common signs of nonlinear behavior -- but when in doubt, build a quick table and check the differences.

📋 For the teacher
Teach it (10–15 min):
  • The gold-standard test works on any table: compute consecutive differences in the output; constant means linear, changing means nonlinear.
  • Spot likely-nonlinear equations by sight once they have seen a few -- an exponent, a root, or x in a denominator are red flags, but a quick table always settles it.
  • Point out that a doubling pattern (like Table 2 in the example) is a specific, very common kind of nonlinear pattern called exponential -- worth naming, even before it is studied formally.
  • Nonlinear does not mean "wrong" -- lots of real things (compound interest, area, falling objects) genuinely grow in a nonlinear way, which is exactly why this test matters.
Talk about it:
  • Why do you need at least three outputs (two differences) to be sure a table is linear, not just two?
  • What does it mean about a graph if the equation has an x^2 in it?
  • Can you think of something in real life that grows in a nonlinear way?

Mastery looks like: Given a table or equation, they test for a constant rate of change (or recognize an exponent/root on x) and correctly classify it as linear or nonlinear.

Common stumbles: Checking only one pair of differences and stopping; assuming any equation with numbers and a variable must be linear.

Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
← Back to todayAll coursesYear planner