Unit Plan 22 (Grade 7 Math): Volume—Prisms and Pyramids
7th graders explore volume through right prisms and right square pyramids, deriving V = B·h and V = ⅓B·h. They compare packing and stacking, reason with cubic units, convert measurements, and solve real-world modeling problems.
Focus: Solve real-world and mathematical volume problems for right prisms and right square pyramids; compare packing/stacking.
Grade Level: 7
Subject Area: Mathematics (Geometry • Measurement & Modeling)
Total Unit Duration: 5 sessions (one week), 45–60 minutes per session
I. Introduction
Students build volume understanding from unit cubes to formulas. For right prisms, they establish V = B * h (B = area of base). For right square pyramids, they discover and use V = (1/3) * B * h, comparing packing/stacking and simple fill experiments (three congruent pyramids fill one prism with same base and height). Emphasis: cubic units, base identification, height measured perpendicular to the base, and contextual reasoning (tanks, packaging, soil, sand).
Essential Questions
- How does stacking equal layers lead to V = B * h for prisms?
- Why is a right square pyramid’s volume one-third of a matching prism’s volume?
- How do units, conversions, and rounding affect real-world volume calculations?
II. Objectives and Standards
Learning Objectives — Students will be able to…
- Identify the base of a right prism or right square pyramid and the perpendicular height.
- Compute B for common bases (rectangles, triangles, squares) and use V = B * h for prisms.
- Use V = (1/3) * B * h for right square pyramids and explain the 1/3 factor informally (packing/filling comparison).
- Solve multi-step application problems with unit conversions (cubic units) and reasonable rounding.
- Communicate clear unit-based conclusions that answer the question asked.
Standards Alignment — CCSS Grade 7
- 7.G.6: Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. (Extended here to right square pyramids through modeling and comparison.)
- Mathematical Practices emphasized: MP.1 (make sense), MP.3 (justify), MP.4 (model), MP.5 (use tools), MP.6 (precision), MP.7 (structure).
Success Criteria — Student Language
- I can find B (area of the base) and h (height) and use the correct volume formula.
- I can explain, in words or a sketch, why a matching pyramid has one-third the prism’s volume.
- I can keep track of cubic units and convert when needed.
- I can check that my answer makes sense with an estimate or bound.
- I can write a clear final statement that answers the question.