Eleventh Grade · Math · week 21

Adding and subtracting rational expressions

Add and subtract rational expressions with unlike denominators using a common denominator.

Just like numeric fractions, rational expressions need a COMMON denominator before you can add or subtract their numerators. The least common denominator is built from all the distinct factors involved (after factoring each denominator), each raised to the highest power it appears with.

Once the denominators match, add or subtract the numerators and keep the shared denominator underneath -- then simplify if anything cancels. Subtraction needs extra care: distribute the negative sign across the ENTIRE second numerator, not just its first term, or the whole computation goes wrong.

📋 For the teacher
Teach it (10–15 min):
  • Practice building an LCD as its own separate skill first, with no addition attached -- just identify the LCD for a few denominator pairs.
  • Watch the subtraction sign trap closely: have them put the second numerator in parentheses and distribute a literal minus sign across it, every time.
  • Show the numeric-fraction parallel side by side (like 1/2 + 1/3) so the unlike-denominators rule feels like something they already know, just with letters.
  • After combining, always check whether the resulting numerator and denominator share a factor that simplifies further.
Talk about it:
  • How do you find the least common denominator of two rational expressions?
  • Why must you distribute the negative sign across an entire numerator when subtracting?
  • What domain restrictions apply to a sum or difference of rational expressions?

Mastery looks like: They find a correct common denominator, combine numerators accurately (including sign distribution for subtraction), and state domain restrictions.

Common stumbles: Adding denominators together instead of finding a common one; forgetting to distribute a negative sign across every term of the second numerator.

See it visually

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

SubtractionTake away and count back, step by step.try →AdditionSee groups combine — with counters, blocks, and the number line.try →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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