Unit Plan 25 (Grade 8 Math): Volume of Cylinders, Cones, and Spheres
8th graders explore and apply volume formulas for cylinders, cones, and spheres through hands-on modeling and real-world problems. They justify why formulas work, solve composite and missing-dimension tasks, and check units, conversions, and reasonableness.
Focus: Derive and apply formulas to solve real-world and mathematical volume problems.
Grade Level: 8
Subject Area: Mathematics (Geometry • Measurement & Modeling)
Total Unit Duration: 5 sessions (one week), 45–60 minutes per session
I. Introduction
This week students connect geometry, measurement, and modeling by working with the volumes of cylinders, cones, and spheres. They build intuitive understanding for the formulas, then apply them to real contexts: containers, tanks, scoops, packaging, and composite figures. Emphasis is on units, reasonableness, and structure (base area × height; one-third relationship for cones; sphere relationships with cylinders).
Essential Questions
- Why does a cylinder’s volume equal base area times height, and why is a cone with the same base and height exactly one-third of that?
- How can I use cross-sections or comparisons to make sense of the sphere volume formula?
- How do unit conversions, precision, and scaling (linear scale k → volume scales by k^3) affect real-world volume problems?
II. Objectives and Standards
Learning Objectives — Students will be able to…
- State and use volume formulas:
- Cylinder: V = π·r^2·h
- Cone: V = (1/3)·π·r^2·h
- Sphere: V = (4/3)·π·r^3
- Give informal rationales: cylinder as base area × height; cone as one-third of matching cylinder (same base and height); sphere related to cylinder/cone through cross-sections or comparison.
- Solve real-world and mathematical problems, including composite solids, missing dimensions, and unit conversions (for example, cm^3 ↔ mL; in^3 ↔ fl oz approximation if provided).
- Evaluate reasonableness and communicate solutions with labeled diagrams, correct units, and clear steps.
Standards Alignment — CCSS Grade 8
- 8.G.9: Know the formulas for the volumes of cylinders, cones, and spheres and use them to solve real-world and mathematical problems.
Success Criteria (student-friendly)
- I can pick the right formula, substitute correct dimensions, and report volume with cubic units.
- I can explain (in a few sentences) why the cone is one-third of a cylinder and how the sphere formula relates to a cylinder.
- I can solve multi-step problems (composites, missing radius/height) and check if the answer makes sense.
- I consistently include units and appropriate rounding or exact π form when requested.