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The Pythagorean Theorem

Eighth graders explore the famous relationship in every right triangle, where the squares of the two legs add up to the square of the longest side, and use it to find missing lengths.

Grade 8The Pythagorean Theorem50 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…

  • ✓Identify the legs & hypotenuse.
  • ✓State a² + b² = c².
  • ✓Find a missing side.
  • ✓Know when to use it.
Essential Question

There is a hidden rule connecting the three sides of every right triangle, powerful enough to measure distances you cannot walk in a straight line. What is it?

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Lesson Phases
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Vocabulary Terms
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Standards Aligned
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Interactive Task
Angle Explorer · Interactive

Explore the Theorem

Project this and open each card to explore the parts of the Pythagorean theorem, from the legs and hypotenuse to a² + b² = c².

📐 Explore the Pythagorean theoremTry it

The Lesson · Gradual Release (I Do · We Do · You Do)

50 minutes, five moves

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

1

Hook — The Slanted Distance

5 min

How far is it straight across a corner, when you only know the two sides?

👩‍🏫 Teacher Moves

  • Show a right triangle.
  • Ask about the long side.
  • Set the goal.

🎒 Student Actions

  • Think.
  • Guess.
  • Get ready.
2

Use — Explore the Theorem

12 min

Students explore the parts of the theorem.

👩‍🏫 Teacher Moves

  • Send students to Explore the Theorem.
  • Open each card.
  • Note the formula.

🎒 Student Actions

  • Explore.
  • Read.
  • Note it.
3

Modify — Find a Side

13 min

Students solve for a missing length.

👩‍🏫 Teacher Moves

  • Label the legs and hypotenuse.
  • Plug into a² + b² = c².
  • Solve for the unknown.

🎒 Student Actions

  • Label.
  • Plug in.
  • Solve.
4

Create — Build a Problem

15 min

Students write a real-world problem.

👩‍🏫 Teacher Moves

  • Pick a real situation.
  • Draw the right triangle.
  • Solve it with the theorem.

🎒 Student Actions

  • Pick.
  • Draw.
  • Solve.
5

Reflect — Only Right Triangles

5 min

Students reflect on when it applies.

👩‍🏫 Teacher Moves

  • Which side is c?
  • Why only right triangles?
  • Complete the exit ticket.

🎒 Student Actions

  • Say it.
  • Say it.
  • Complete the exit ticket.
Standards Alignment

Built to the standards you report on

Aligned to the Common Core State Standards for Mathematics.

CCSS
8.G.B.6

Explain a proof of the Pythagorean theorem and its converse.

CCSS
8.G.B.7

Apply the Pythagorean theorem to determine unknown side lengths in right triangles.

CCSS
8.G.B.8

Apply the Pythagorean theorem to find the distance between two points in a coordinate system.

CCSS
MP.7

Look for and make use of structure.

Differentiation

One lesson, every learner

Multilingual Learners

ELL / EMERGING READERS
  • Right-triangle cards labeled a, b, c.
  • Sentence frame: “The ___ is the longest side.”
  • Identify one part at a time.

Support & Access

IEP / 504
  • Start by labeling legs vs hypotenuse.
  • Use grid squares to show a² and b².
  • Work the 3-4-5 triangle first.

Stretch & Extend

GIFTED / EARLY FINISHERS
  • Solve for a leg, not just the hypotenuse.
  • Apply the converse to test triangles.
  • Use it to find distance on a coordinate grid.
Materials

What to gather

  • 📽️Projector / board
  • 📐Right-triangle cutouts
  • 💻Explore the Theorem
  • 🧮Calculators
  • ✏️Pencils
  • 🎫Exit-ticket slips
Vocabulary

Key terms — hover for a quick definition

Pythagorean theoremthe rule that a² + b² = c² in a right triangleright trianglea triangle with one 90-degree anglelegone of the two shorter sides that form the right anglehypotenusethe longest side of a right triangle, opposite the right angleright anglea 90-degree angle, like the corner of a squaresquare (a number)the result of multiplying a number by itselfsquare roota value that, multiplied by itself, gives the numberconversethe reverse of a statement, used to test for right triangles
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
In a right triangle, which side is the hypotenuse?
The longest side, which is always opposite the right angle (labeled c).
QUESTION 2
Write the Pythagorean theorem.
a² + b² = c², where a and b are the legs and c is the hypotenuse.
QUESTION 3
A right triangle has legs of 6 and 8. How long is the hypotenuse?
10, because 6² + 8² = 36 + 64 = 100, and the square root of 100 is 10.

Create: build a problem.

Have students invent a real-world situation (like a ladder against a wall), draw the right triangle, and use a² + b² = c² to solve it. A printable Pythagorean-theorem worksheet is in the Mathematics 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.

🃏 The Pythagorean TheoremFlip
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Practice · Quiz

Check your understanding

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

📝 The Pythagorean TheoremQuiz
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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.

🖨️ The Pythagorean TheoremPrint
Name: ________________________
Date: ____________

Part A · Write the word that matches each meaning

Word bank: converse, hypotenuse, leg, Pythagorean theorem, right angle, right triangle, square (a number), square root
  1. the longest side of a right triangle, opposite the right angle
  2. a 90-degree angle, like the corner of a square
  3. a value that, multiplied by itself, gives the number
  4. the reverse of a statement, used to test for right triangles
  5. one of the two shorter sides that form the right angle
  6. the result of multiplying a number by itself
  7. the rule that a² + b² = c² in a right triangle
  8. a triangle with one 90-degree angle

Part B · Show what you learned

  1. In a right triangle, which side is the hypotenuse?
  2. Write the Pythagorean theorem.
  3. A right triangle has legs of 6 and 8. How long is the hypotenuse?
Answer key — Part A: 1) hypotenuse · 2) right angle · 3) square root · 4) converse · 5) leg · 6) square (a number) · 7) Pythagorean theorem · 8) right triangle
Part B: 1) The longest side, which is always opposite the right angle (labeled c). 2) a² + b² = c², where a and b are the legs and c is the hypotenuse. 3) 10, because 6² + 8² = 36 + 64 = 100, and the square root of 100 is 10.