Practice 1
A prism is 3 unit cubes long, 2 wide and 4 high. Find its volume in cubic units.
Answer and teaching response
24 · Six cubes per layer multiplied by four layers gives twenty-four.
Count a solid by complete layers of unit cubes
Connect unit-cube packing to the area of one layer and the number of identical layers in a rectangular prism.
Get the student and teacher materialsPrepare the group
Learning goal: A learner can justify a three-factor volume calculation and explain why counting only visible cubes is incomplete.
You will need: 24 equal linking cubes, or squared paper for drawing layer plans. Treat each cube as one cubic unit.
Start with this question
A rectangular prism is 4 cubes long, 3 cubes wide and 2 cubes high. How many unit cubes fill it without gaps?
24 unit cubes, so its volume is 24 cubic units. Each complete layer contains 4 × 3 = 12 cubes, and two layers contain 12 × 2.
- Build or draw one complete base layer. Four cubes along the length and three along the width make twelve cube positions.
- Place one cube in every position. The layer must be full; gaps would mean that a rectangular-prism formula no longer describes the actual packing.
- Stack a second identical layer with no overlap or gap. The height of two cubes counts two complete layers of twelve.
- Record 4 × 3 × 2 = 24 cubic units. Distinguish the base's area in square units from the solid's volume in cubic units.
Run the 15-minute workshop
- 3 minutes. Build one layer and have learners label the two counts that describe its rows.
- 4 minutes. Stack or draw identical layers, then explain what the height counts.
- 5 minutes. Complete the practice with a labelled layer sketch alongside each multiplication.
- 3 minutes. Use the independent check and ask learners to explain how their method includes hidden cubes.
A useful error to discuss
A learner might say: Count only the cubes visible on the outside of a drawing.
A solid can contain cubes hidden behind or underneath others. Counting full layers accounts for those positions as well as the visible ones.
Practise, then check independently
Use the first three tasks for guided practice. Try the final task without help. Record a drawing or equation as well as the answer.
Practice 2
A filled prism has 15 cubes in each layer and 3 complete layers. Find its volume in cubic units.
Answer and teaching response
45 · Three identical groups of fifteen cubes fill the prism.
Practice 3
A prism contains 36 cubes in layers of 9 cubes each. How many layers are there?
Answer and teaching response
4 · Four complete layers of nine account for all thirty-six cubes.
Independent check
A filled rectangular prism is 5 unit cubes long, 2 wide and 3 high. Find its volume and explain each factor.
Answer and teaching response
30 · Five times two gives ten cubes per layer; three layers give thirty.
Transfer the idea
If a 4-by-3-by-2 prism is turned so it stands 4 cubes high on a 3-by-2 base, does its volume change?
Discuss a possible response
No cubes are added or removed. The layer description changes from 12 × 2 to 6 × 4, but both count 24 cubic units.
Choose the next teaching step
Look for:
- Describes a complete layer and the number of layers.
- Names cubic units and includes hidden cube positions.
If the idea is not secure: Separate the solid into physical layers and count one layer before stacking them.
If the learner is ready: Remove one unit cube from the packed prism. Explain why the actual volume is now 23 cubic units even though the outer bounding dimensions are unchanged.
Materials for your group
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