Unit Plan 23 (Grade 8 Math): Pythagorean Theorem—Why It Works
8th graders explore why the Pythagorean Theorem works through visual and similarity-based proofs. They construct right triangles, justify a² + b² = c² with area and triangle reasoning, and prove its converse using clear, labeled diagrams.
Focus: Develop understanding and informal proofs of the Pythagorean Theorem and its converse.
Grade Level: 8
Subject Area: Mathematics (Geometry)
Total Unit Duration: 5 sessions (one week), 45–60 minutes per session
I. Introduction
This week moves beyond “plug and chug” to why the Pythagorean relationship holds. Students build and compare areas on the sides of right triangles, explore a classic rearrangement (dissection) proof, and use similar triangles to justify that for a right triangle with legs a and b and hypotenuse c, a^2 + b^2 = c^2. They then tackle the converse: if a triangle’s side lengths satisfy a^2 + b^2 = c^2, the triangle must be right. Emphasis is on informal arguments with clean diagrams and clear words.
Essential Questions
- What does a^2 + b^2 = c^2 mean geometrically?
- How can area dissections or similarity arguments prove the Pythagorean Theorem?
- Why does the converse (a^2 + b^2 = c^2 implies a right angle) follow?
- How can I communicate a convincing proof using pictures, labels, and sentences?
II. Objectives and Standards
Learning Objectives — Students will be able to…
- Construct right triangles, attach squares on the sides, and describe how leg-square areas relate to the hypotenuse-square area.
- Explain at least one informal proof of the Pythagorean Theorem (rearrangement or similarity).
- Explain an informal proof of the converse: if a^2 + b^2 = c^2, then the angle opposite side c is right.
- Critique and improve peer arguments for clarity, correctness, and diagram precision.
Standards Alignment — CCSS Grade 8
- 8.G.6: Explain a proof of the Pythagorean Theorem and its converse.
Success Criteria (student-friendly)
- I can label a right triangle (legs a, b; hypotenuse c) and state the theorem in words and symbols.
- I can present a clear, step-by-step informal proof that someone else can follow.
- I can justify the converse with a construction or simple congruence argument.
- I can spot and fix gaps in a proof (missing labels, unjustified steps).