Teacher Guide: Pythagorean Theorem

Learning Objective 5-10 min lesson

Students will: Apply the Pythagorean Theorem (a² + b² = c²) to find unknown side lengths in right triangles, calculate distances between points on a coordinate plane, and solve real-world problems involving right triangles in two and three dimensions.

Why this matters for FAST: The Pythagorean Theorem is one of the most tested concepts in Grade 8. Students must identify the hypotenuse, set up equations correctly, and apply the theorem in various contexts including coordinate geometry and 3D scenarios.

Materials Needed

Whiteboard/projector
Student Concept Worksheet
Graph paper
Calculators
Rulers
Right triangle manipulatives

Common Misconceptions to Address

Misconception #1: The hypotenuse can be any side

Students substitute values incorrectly, putting the hypotenuse as one of the legs (a or b) instead of c.

How to Address:

"The hypotenuse is ALWAYS the longest side and ALWAYS opposite the right angle. It's always 'c' in our formula. The legs are the two sides that FORM the right angle. Label your triangle FIRST before substituting!"

Misconception #2: Adding a + b instead of a² + b²

Students forget to square the values: 3 + 4 = 7 instead of 3² + 4² = 9 + 16 = 25.

How to Address:

"Remember: it's a-SQUARED plus b-SQUARED equals c-SQUARED. Every single term gets squared! Write out each step: 3² = 9, 4² = 16, so 9 + 16 = 25, and c² = 25 means c = 5."

Misconception #3: Forgetting to take the square root

Students find c² = 25 and write c = 25 instead of c = 5.

How to Address:

"When you find c² = 25, you're not done! You found c-SQUARED, not c. Take the square root of both sides to find c. Since we're measuring length, we only use the positive root."

Lesson Steps

1

Activate Prior Knowledge (1 min)

Review right triangles: "Which angle is 90 degrees? What makes a right triangle special?" Review squares and square roots: "What is 5²? What is the square root of 36?"

2

Introduce the Theorem (2 min)

SAY THIS:

"In any right triangle, if you square the lengths of the two shorter sides (legs) and add them together, you get the square of the longest side (hypotenuse). This is a² + b² = c², where c is ALWAYS the hypotenuse."

The Pythagorean Theorem: a² + b² = c²

a and b = legs (form the right angle)

c = hypotenuse (longest side, opposite the right angle)

Common Pythagorean Triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25

3

Finding the Hypotenuse (2 min)

Example: Find c when a = 6 and b = 8

Step 1: a² + b² = c²

Step 2: 6² + 8² = c²

Step 3: 36 + 64 = c²

Step 4: 100 = c²

Step 5: c = 10

4

Finding a Leg (2 min)

SAY THIS:

"When finding a leg, we rearrange the formula. If we know c and b, then a² = c² - b². We SUBTRACT because the legs are always shorter than the hypotenuse."

Example: Find a when b = 5 and c = 13

a² + 5² = 13²

a² + 25 = 169

a² = 169 - 25 = 144

a = 12

5

Distance Formula Connection (2 min)

Show how the distance formula d = √[(x₂-x₁)² + (y₂-y₁)²] comes from the Pythagorean Theorem. Draw a right triangle on a coordinate plane connecting two points.

The horizontal distance (x₂-x₁) is one leg
The vertical distance (y₂-y₁) is the other leg
The distance between points is the hypotenuse

Check for Understanding

Quick Exit Ticket (Ask the whole class):

"A right triangle has legs of 9 and 12. What is the hypotenuse?"

Correct answer: 15. Check: 9² + 12² = 81 + 144 = 225 = 15². This is a multiple of the 3-4-5 triple (multiplied by 3).

IXL Skills to Assign After This Lesson

Recommended IXL Practice:

Pythagorean theorem: find the hypotenuse Pythagorean theorem: find a missing leg Pythagorean theorem: word problems Distance between two points Converse of the Pythagorean theorem

Differentiation & Extension

For struggling students: Focus on identifying the hypotenuse first (longest side, opposite right angle). Use Pythagorean triples to build confidence before introducing non-perfect squares.

For advanced students: Introduce 3D applications (space diagonals of rectangular prisms) and the converse theorem (proving if a triangle is right).

For home: Send Parent Activity sheet. Families can measure room diagonals and verify with the theorem.