Unit Plan 24 (Grade 8 Math): Pythagorean Applications & Distance
8th graders apply the Pythagorean Theorem to solve real-world and coordinate problems—finding unknown sides, diagonals, and distances. They derive the distance formula, use it in 2D and 3D contexts, and justify answers with units and reasoning.
Focus: Apply the theorem to find unknown side lengths and compute distances in the coordinate plane.
Grade Level: 8
Subject Area: Mathematics (Geometry)
Total Unit Duration: 5 sessions (one week), 45–60 minutes per session
I. Introduction
Now that students know why the Pythagorean Theorem is true, they’ll use it to solve problems. This week targets two big moves: (1) find unknown side lengths in right triangles that show up in real situations (ramps, ladders, diagonals of rectangles and rectangular prisms), and (2) derive and use the distance formula on the coordinate plane: straight-line distance between two points from horizontal and vertical changes.
Essential Questions
- How do I recognize the right triangle hiding in a real or coordinate problem?
- How do units, rounding, and reasonableness matter when using a^2 + b^2 = c^2?
- How is the distance formula just the Pythagorean Theorem in coordinate clothing?
II. Objectives and Standards
Learning Objectives — Students will be able to…
- Identify right triangles in 2D and 3D settings and use a^2 + b^2 = c^2 to determine unknown side lengths with units and appropriate rounding.
- Derive and use the distance formula: distance d between points (x1, y1) and (x2, y2) is d = sqrt((x2 - x1)^2 + (y2 - y1)^2).
- Solve practical problems: ramp length from rise and run, ladder reach, diagonal of a rectangle, and space diagonal of a rectangular prism using a two-step Pythagorean approach.
- Judge reasonableness of answers (scale, bounds, domain) and communicate steps clearly.
Standards Alignment — CCSS Grade 8
- 8.G.7: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
- 8.G.8: Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Success Criteria (student-friendly)
- I can set up a^2 + b^2 = c^2 correctly by choosing the hypotenuse and labeling legs.
- I can compute a missing length and report it with units and reasonable rounding.
- I can find coordinate distance using d = sqrt((dx)^2 + (dy)^2) where dx and dy are differences in x and y.
- I can explain my setup and why my answer makes sense.