Unit 04 · Lesson 02 · Open-Age

Multiplying and Dividing Rational Expressions

Factoring before multiplication often exposes common factors and reduces work. Division means multiplying by the reciprocal of the divisor, with all original and reciprocal restrictions recorded.

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Learning objectives

Explain the central structure of multiplying and dividing rational expressions using precise mathematical language.
Apply the governing relationship and verify results with an alternate representation or substitution.
State domain, unit, precision, and context restrictions whenever they affect the result.
Analyze a misconception and communicate a correction with evidence.

Essential question: How do excluded values, asymptotes, holes, and rates of change shape rational expressions and models?

Governing relationship

Multiply numerators and denominators; factor before canceling.

Factoring before multiplication often exposes common factors and reduces work. Division means multiplying by the reciprocal of the divisor, with all original and reciprocal restrictions recorded.

Worked examples

Read each example, close it, and reconstruct the reasoning independently. Then verify by substitution, graph, table, or alternate method.

Multiply (x²−1)/(x²−4) · (x−2)/(x+1).Factor and cancel to obtain (x−1)/(x+2), with original restrictions x≠±2,−1.

Divide x/(x−3) by 2x/(x²−9).Multiply by (x²−9)/(2x) to get (x+3)/2, with x≠0,±3.

Guided and independent practice

Multiply two factored rational expressions.
Divide and list restrictions.
Verify numerically at an allowed input.
Create one original example involving multiplying and dividing rational expressions and solve it completely.
Use the Khaemenes Scientific Calculator to verify one result, then document the setup and reasonableness check.
Write a one-paragraph explanation connecting symbolic, graphical, numerical, and contextual representations.

Differentiated pathways

Foundation

Review prerequisite vocabulary, annotate each worked step, complete the first four prompts, and use a table or graph to support symbolic work.

Core

Complete every prompt, explain domain and verification, correct errors, and submit the applied task with clear mathematical communication.

Extended

Generalize the relationship, compare methods, use technology strategically, analyze sensitivity, and create a proof, counterexample, or advanced model.

Application laboratory

Simplify a compound rate or ratio model while preserving excluded values.

Required evidence: defined quantities and units; at least two representations; transparent calculations; verification; one limitation; and a defensible conclusion.

Misconception and correction

Common misconception: Forgetting restrictions created when taking the reciprocal of a divisor.

Write the first invalid step, the governing rule that corrects it, and one counterexample or verification that demonstrates the correction.

Mastery evidence

Accurate symbolic work with preserved equivalence.
Correct domain, interval, unit, and precision statements.
At least one independent verification.
A correction explaining why an incorrect approach fails.
A clear application conclusion that distinguishes mathematical result from recommendation.

Reflection

What structure did you recognize? Which representation was most useful? What assumption or restriction mattered most? What will you do differently on the next problem?