Pythagorean Theorem and Volume

Lesson 6

Math

Unit 7

8th Grade

Lesson 6 of 16

Objective


Understand the Pythagorean Theorem as a relationship between the side lengths in a right triangle.

Common Core Standards


Core Standards

  • 8.G.B.6 — Explain a proof of the Pythagorean Theorem and its converse.

Criteria for Success


  1. Understand the relationship between the legs and the hypotenuse of right triangles, named the Pythagorean Theorem$${a^2+b^2=c^2}$$.
  2. Use the Pythagorean Theorem to verify the relationship between the legs and hypotenuse of right triangles.
  3. Understand that the hypotenuse of a right triangle is the longest side of the triangle located across from the right angle.

Tips for Teachers


In this lesson, students are introduced to the famous relationship that exists between the side lengths of right triangles. In the next lesson, students will investigate informal proofs of the Pythagorean Theorem, followed by lessons where students will apply the theorem in various problems.

Lesson Materials

  • Graph Paper (1 sheet per student)
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Anchor Problems


Problem 1

Several right triangles are shown below (not drawn to scale). In a right triangle, the two side lengths that form the right angle are called legs, and the side length opposite the right angle is called the hypotenuse. 

Use the triangles to investigate the question: What relationship do you see between the measures of the legs and the measure of the hypotenuse?

Guiding Questions

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Problem 2

Use the Pythagorean Theorem to show the relationship between the sides of the right triangles shown below.

Guiding Questions

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Problem 3

A right triangle has side lengths $$6$$ units, $$4$$ units, and $${\sqrt{20}}$$ units. 

Which side length represents the hypotenuse?

Guiding Questions

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Problem Set

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Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

Target Task


Problem 1

Lee tried to use the Pythagorean Theorem on the triangle shown below and found that the relationship did not hold true. 

 


Explain why the Pythagorean Theorem did not show a true relationship in Lee’s triangle. 

Problem 2

Which three measures could be the side lengths of a right triangle? Explain or show your reasoning.

$$5$$                $$2$$                $$7$$               $$\sqrt{24}$$

Student Response

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Additional Practice


The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Include problems where students use given side lengths or identify side lengths from a group for a right triangle and verify the relationship using the Pythagorean Theorem.
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Lesson 5

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Lesson 7

Lesson Map

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Topic A: Irrational Numbers and Square Roots

Topic B: Understanding and Applying the Pythagorean Theorem

Topic C: Volume and Cube Roots

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