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Radical Functions & Equations

Eleventh graders work with radical functions and equations — solving by isolating and squaring, watching for extraneous solutions that squaring can introduce, and finding the domain of a square-root function.

Grade 11Radicals55 minutes1 class periodGradual ReleaseExplicit teaching4 StandardsCommon Core
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Lesson at a Glance

Everything you need before the bell rings

Learning Objectives

Students will be able to…

  • ✓Solve radical equations.
  • ✓Check for extraneous solutions.
  • ✓Find the domain of a radical function.
  • ✓Isolate the radical first.
Essential Question

To solve √x = 5, you square both sides. But squaring can create fake answers! How do you solve radical equations safely — and why must you always check?

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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

Solve the Radical

Project each item and have students solve or analyze the radical equation. The tool explains squaring, extraneous solutions, and domain.

√ Solve & analyze radical equationsTry it
Solve or analyze each radical equation.
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 — Squaring to Solve

5 min

To solve √x = 5, we square both sides — but is that always safe?

👩‍🏫 Teacher Moves

  • Show a radical equation.
  • Ask how to undo the root.
  • Set the goal: radical equations.

🎒 Student Actions

  • See the equation.
  • Think “square it.”
  • Predict a catch.
2

I Do — Isolate, Square, Check

10 min

Teacher models.

👩‍🏫 Teacher Moves

  • Isolate the radical.
  • Square both sides.
  • Check for extraneous solutions.

🎒 Student Actions

  • Isolate it.
  • Square it.
  • Check answers.
3

We Do — Solve Together

13 min

Class uses Solve the Radical.

👩‍🏫 Teacher Moves

  • Solve each equation.
  • Check the answers.
  • Find the domain.

🎒 Student Actions

  • Solve them.
  • Check them.
  • Find domain.
4

You Do — On Your Own

12 min

Students solve.

👩‍🏫 Teacher Moves

  • Solve radical equations.
  • Check for extraneous solutions.
  • State the domain.

🎒 Student Actions

  • Solve them.
  • Check them.
  • State domain.
5

Close — Close

5 min

One equation.

👩‍🏫 Teacher Moves

  • Give a radical equation.
  • Solve and check it.
  • Hand out the exit ticket.

🎒 Student Actions

  • Solve it.
  • Check 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-REI.2

Solve simple radical equations and give examples of extraneous solutions.

CCSS
F-IF.7b

Graph square-root and other radical functions.

CCSS
A-CED.1

Create equations to solve problems.

CCSS
F-IF.5

Relate the domain of a function to its graph.

Differentiation

One lesson, every learner

Multilingual Learners

ELL / EMERGING READERS
  • Step-by-step solving cards.
  • Sentence frame: “I must check because ___.”
  • Highlight the radical.

Support & Access

IEP / 504
  • Start with √x = number.
  • Square both sides together.
  • Always check the answer.

Stretch & Extend

GIFTED / EARLY FINISHERS
  • Solve with two radicals.
  • Graph a radical function.
  • Find cube-root domains.
Materials

What to gather

  • 📽️Projector / board
  • 🧮Calculators
  • 💻Solve the Radical
  • 📈Graph paper
  • ✏️Pencils
  • 🎫Exit-ticket slips
Vocabulary

Key terms — hover for a quick definition

radicala root, like √radical equationan equation with a variable under a rootextraneous solutiona false solution introduced by squaringdomainall allowed inputssquare rootthe inverse of squaringisolateto get the radical aloneindexwhich root (2 for square, 3 for cube)radicandwhat’s under the root
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
Solve: √x = 4.
x = 16.
QUESTION 2
Why must you check solutions to a radical equation?
Because squaring both sides can introduce extraneous (false) solutions.
QUESTION 3
What is the domain of f(x) = √x?
x ≥ 0 (no square roots of negatives in the real numbers).

Going deeper? Two radicals.

Have students solve equations with a radical on each side, isolating and squaring carefully, then checking. A printable radical-equations 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.

🃏 Radical FunctionsFlip
Card 1
Term
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Meaning
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Practice · Quiz

Check your understanding

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

📝 Radical FunctionsQuiz
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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.

🖨️ Radical FunctionsPrint
Name: ________________________
Date: ____________

Part A · Write the word that matches each meaning

Word bank: domain, extraneous solution, index, isolate, radical, radical equation, radicand, square root
  1. a root, like √
  2. what’s under the root
  3. to get the radical alone
  4. an equation with a variable under a root
  5. all allowed inputs
  6. which root (2 for square, 3 for cube)
  7. a false solution introduced by squaring
  8. the inverse of squaring

Part B · Show what you learned

  1. Solve: √x = 4.
  2. Why must you check solutions to a radical equation?
  3. What is the domain of f(x) = √x?
Answer key — Part A: 1) radical · 2) radicand · 3) isolate · 4) radical equation · 5) domain · 6) index · 7) extraneous solution · 8) square root
Part B: 1) x = 16. 2) Because squaring both sides can introduce extraneous (false) solutions. 3) x ≥ 0 (no square roots of negatives in the real numbers).