Eleventh Grade · Math · week 20

Multiplying and dividing rational expressions

Multiply and divide rational expressions and simplify the result.

Multiplying rational expressions works exactly like multiplying numeric fractions: multiply numerators together, multiply denominators together -- but factor everything FIRST so you can cancel before you multiply, keeping the numbers (and letters) manageable. Dividing is multiplying by the reciprocal of the second expression, the same flip-and-multiply rule from ordinary fractions.

Domain restrictions get trickier here: they come from every original denominator, INCLUDING the one you flip during division (since a flipped denominator becomes a new numerator whose original position still cannot have been zero). Track every denominator that ever appeared before you simplify.

📋 For the teacher
Teach it (10–15 min):
  • Drill "flip the second fraction, then multiply" as a single mechanical first step for every division problem -- it turns division into multiplication instantly.
  • Require factoring before canceling, every time, even when a problem looks simple enough to skip it.
  • For domain restrictions, have them circle every denominator that appears anywhere in the ORIGINAL problem, including inside the fraction that gets flipped.
  • Connect the flip-and-multiply rule back to numeric fractions (like (2/3) / (4/5) = (2/3)(5/4)) so it feels familiar.
Talk about it:
  • Why do you factor before multiplying rational expressions, rather than after?
  • Why does dividing by a fraction become multiplying by its reciprocal?
  • Which denominators count toward the domain restrictions of a division problem?

Mastery looks like: They multiply and divide rational expressions by factoring first, cancel correctly, and state restrictions from every original denominator.

Common stumbles: Multiplying before factoring (missing cancellation opportunities); forgetting that the divisor's own denominator also contributes a domain restriction.

See it visually

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

MultiplicationEqual groups and arrays make products finally click.try →DivisionShare fairly and build equal groups.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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