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Rational Expressions: Algebraic Fractions

Eleventh graders work with rational expressions — fractions whose numerators and denominators are polynomials — simplifying by factoring and canceling, finding where they’re undefined, and adding with common denominators.

Grade 11Rational Expressions55 minutes1 class periodGradual ReleaseExplicit teaching4 StandardsCommon Core
Start the Lesson
Lesson at a Glance

Everything you need before the bell rings

Learning Objectives

Students will be able to…

  • ✓Simplify rational expressions.
  • ✓Find where they’re undefined.
  • ✓Add with a common denominator.
  • ✓Factor to cancel.
Essential Question

A fraction like (x² − 9)/(x + 3) looks scary — but it follows the same rules as ordinary fractions. How do you simplify, add, and analyze these algebraic fractions?

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Lesson Phases
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Vocabulary Terms
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Standards Aligned
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Interactive Task
Model · Interactive Tool

Work the Expression

Project each item and have students simplify or analyze the rational expression. The tool explains factoring, canceling, and restrictions.

🔤 Simplify & analyze rational expressionsTry it
Simplify or analyze each rational expression.
The Lesson · Gradual Release (I Do · We Do · You Do)

55 minutes, five moves

Tap any phase to open the teacher moves and student actions.

1

Launch — Fractions with Variables

5 min

(x² − 9)/(x + 3) — it’s just a fraction with variables. How do we simplify it?

👩‍🏫 Teacher Moves

  • Show a rational expression.
  • Recall fraction rules.
  • Set the goal: rational expressions.

🎒 Student Actions

  • See the expression.
  • Recall fractions.
  • Predict the idea.
2

I Do — Factor & Cancel

10 min

Teacher models.

👩‍🏫 Teacher Moves

  • Factor numerator and denominator.
  • Cancel common factors.
  • Note where it’s undefined.

🎒 Student Actions

  • Factor it.
  • Cancel factors.
  • Find restrictions.
3

We Do — Work Together

13 min

Class uses Work the Expression.

👩‍🏫 Teacher Moves

  • Simplify each one.
  • Find restrictions.
  • Add with a common denominator.

🎒 Student Actions

  • Simplify them.
  • Find restrictions.
  • Add them.
4

You Do — On Your Own

12 min

Students work expressions.

👩‍🏫 Teacher Moves

  • Simplify rational expressions.
  • Find where they’re undefined.
  • Add and subtract them.

🎒 Student Actions

  • Simplify them.
  • Find restrictions.
  • Add/subtract them.
5

Close — Close

5 min

One expression.

👩‍🏫 Teacher Moves

  • Give a rational expression.
  • Simplify it.
  • Hand out the exit ticket.

🎒 Student Actions

  • Simplify it.
  • Explain it.
  • Complete the exit ticket.
Standards Alignment

Built to the standards you report on

Aligned to the Common Core State Standards for Mathematics (High School Algebra II).

CCSS
A-APR.6

Rewrite rational expressions using inspection or long division.

CCSS
A-APR.7

Add, subtract, multiply, and divide rational expressions.

CCSS
A-REI.2

Solve rational equations and note extraneous solutions.

CCSS
A-SSE.2

Use the structure of an expression to rewrite it.

Differentiation

One lesson, every learner

Multilingual Learners

ELL / EMERGING READERS
  • Compare to numeric fractions.
  • Sentence frame: “It’s undefined when ___ = 0.”
  • Factor step by step.

Support & Access

IEP / 504
  • Start with simplifying.
  • Factor first.
  • Highlight the common factor.

Stretch & Extend

GIFTED / EARLY FINISHERS
  • Solve rational equations.
  • Divide with long division.
  • Find extraneous solutions.
Materials

What to gather

  • 📽️Projector / board
  • 🧮Calculators
  • 💻Work the Expression
  • 📄Factoring charts
  • ✏️Pencils
  • 🎫Exit-ticket slips
Vocabulary

Key terms — hover for a quick definition

rational expressiona ratio of two polynomialsnumeratorthe top of a fractiondenominatorthe bottom of a fractionundefinedwhen the denominator is 0simplifyto reduce to lowest termsfactorto write as a productcommon denominatora shared bottom for addingextraneous solutiona false solution to check for
Evaluate

Exit Ticket

Preview the three formative checks. Tap “Sample answer” to see what mastery looks like — hide them before you print for students.

QUESTION 1
Simplify: (x² − 4)/(x − 2).
x + 2 (factor to (x+2)(x−2), cancel x−2).
QUESTION 2
For what value of x is 1/(x + 5) undefined?
x = −5.
QUESTION 3
What must you have before adding two rational expressions?
A common denominator.

Going deeper? Rational equations.

Have students solve rational equations and check for extraneous solutions that make a denominator zero. A printable rational-expressions sheet is in the Math library.

Study · Flashcards

Study the key terms

Tap a card to flip it, then rate whether you knew it. Built from this lesson’s vocabulary.

🃏 Rational ExpressionsFlip
Card 1
Term
Tap to flip →
Meaning
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Nice work!

Practice · Quiz

Check your understanding

A quick self-check with instant feedback, drawn from this lesson’s key terms.

📝 Rational ExpressionsQuiz
Score: 0
1 / 6
Question 1
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Practice · Worksheet

Printable worksheet

A print-and-go review sheet with a built-in answer key. Tap “Show answer key” to reveal answers, or print the clean version for students.

🖨️ Rational ExpressionsPrint
Name: ________________________
Date: ____________

Part A · Write the word that matches each meaning

Word bank: common denominator, denominator, extraneous solution, factor, numerator, rational expression, simplify, undefined
  1. the top of a fraction
  2. to write as a product
  3. a ratio of two polynomials
  4. when the denominator is 0
  5. to reduce to lowest terms
  6. a shared bottom for adding
  7. the bottom of a fraction
  8. a false solution to check for

Part B · Show what you learned

  1. Simplify: (x² − 4)/(x − 2).
  2. For what value of x is 1/(x + 5) undefined?
  3. What must you have before adding two rational expressions?
Answer key — Part A: 1) numerator · 2) factor · 3) rational expression · 4) undefined · 5) simplify · 6) common denominator · 7) denominator · 8) extraneous solution
Part B: 1) x + 2 (factor to (x+2)(x−2), cancel x−2). 2) x = −5. 3) A common denominator.