Unit Plan 20 (Grade 5 Math): Understanding Volume—Cubes to Formulas
5th graders explore volume conceptually by packing unit cubes into rectangular prisms, counting layers, and deriving the formulas V = l × w × h and V = B × h. Students connect visual models to formulas, label cubic units correctly, and explain why the volume formulas work in real-world contexts.
Focus: Develop volume as packing with unit cubes; build from counting layers to deriving and using the formulas V = l * w * h and V = B * h for rectangular prisms.
Grade Level: 5
Subject Area: Mathematics (Measurement & Data)
Total Unit Duration: 5 sessions (one week), 45–60 minutes per session
I. Introduction
Students build a concrete-to-abstract understanding of volume. They start with unit cubes, pack boxes, and count layers. From there, they generalize to base area (B) times height (h) and the dimension product l * w * h. Emphasis is on units (cubic units), clear representations, and explanations of why the formulas work.
Essential Questions
- What does volume measure, and how is it different from area?
- How does counting unit cubes lead to B * h and l * w * h?
- How do I choose and justify a strategy (counting, layers, or formula) for a given prism?
- How do units help me verify that my answer makes sense?
II. Objectives and Standards
Learning Objectives — Students will be able to:
- Define volume as the number of unit cubes that fill a 3D space with no gaps or overlaps.
- Build rectangular prisms with layers and show that volume = (cubes on the base) * (number of layers).
- Derive and use V = B * h (B = area of base) and V = l * w * h for whole-number side lengths.
- Solve real-world and mathematical problems involving rectangular prisms, labeling answers in cubic units.
- Explain reasoning with models, equations, units, and reasonableness checks.
Standards Alignment — CCSS Grade 5
- 5.MD.3: Recognize volume as an attribute of solid figures; understand it as counting unit cubes.
- 5.MD.4: Measure volume by counting unit cubes (e.g., cubic centimeters, cubic inches).
- 5.MD.5a: Relate volume to multiplication; find volume of a rectangular prism with whole-number edge lengths using V = l * w * h and V = B * h.
- Mathematical Practices emphasized: MP.1 (persevere), MP.2 (quantitative reasoning), MP.3 (justify), MP.4 (model), MP.5 (tools), MP.6 (precision), MP.7 (structure).
Success Criteria — Student Language
- I can explain volume using unit cubes and layers.
- I can find B (area of the base) and use V = B * h.
- I can use V = l * w * h and show it matches B * h.
- I can label answers in cubic units and check if they are reasonable.