Eleventh Grade · Math · week 4

Piecewise-defined and absolute value functions

Evaluate piecewise-defined functions correctly and rewrite absolute value expressions as an equivalent piecewise rule.

Not every function follows one rule everywhere. A piecewise function gives DIFFERENT rules for different parts of the domain -- like a phone plan that charges one rate for the first few gigabytes and a different rate after that. The only new skill is bookkeeping: before you evaluate, you must first decide which piece's condition the input actually satisfies.

Absolute value is secretly a piecewise function in disguise. |x| means "the distance from 0," which is x itself when x is already positive or 0, and -x (which flips a negative number positive) when x is negative. Writing |x - a| as a piecewise rule is a skill that resurfaces when we solve absolute value equations and inequalities, so get comfortable with it now.

📋 For the teacher
Teach it (10–15 min):
  • Make them say the condition out loud before writing a single number: "is x less than 0, or is it not?" -- the condition check is the whole skill this week.
  • Use a real-life piecewise story (a rideshare fare, a tax bracket, a phone data plan) to show WHY a function would ever need more than one rule.
  • For absolute value, connect it back to "distance from zero" concretely: |-7| and |7| are both 7 because both points are 7 steps from 0 on the number line.
  • Watch the boundary point closely (like x=0 or x=4 in the examples) -- ask "which piece owns this exact point?" every time one shows up.
Talk about it:
  • Why must you check the condition before evaluating a piecewise function, instead of just plugging into whichever piece looks easiest?
  • How is |x| itself an example of a piecewise function?
  • What happens at the boundary point between two pieces -- could a function accidentally give two different outputs there?

Mastery looks like: They correctly identify which piece applies for a given input, evaluate accordingly, and can rewrite a simple absolute value expression as a piecewise rule.

Common stumbles: Using the wrong piece because the boundary condition (<= versus <) was misread; forgetting the negative sign on the second piece when rewriting absolute value.

See it visually

Interactive, spoken-aloud tools that show this idea in motion.

FractionsSlice bars and circles into equal parts and compare them.try →
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.
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