Explore · unit cube counting
A block has 2 layers. Each layer is 5 cubes long and 3 cubes wide. How many unit cubes fill it?
Singapore Math 5A · Independent learning
Cubic Units
Learning target: Explain cubic measurement as packing equal-size unit cubes; distinguish volume from surface area.
Begin with the exact prerequisite checkpoint below. Afterward, explain why your strategy works, not merely which procedure to use.
Opening task
Prerequisite check (before instruction)
Attempt without help. A weak prerequisite signals review of 4A or 4B material, not a failed 5A lesson.
Count cubes in 3 equal layers of 4 cubes each.
Explain why a cubic centimeter is not a square centimeter.
Lesson-specific prerequisite checkpoint
Do this without a worked example. If you cannot explain it, review the earlier skill before continuing.
Count 5 layers with 6 unit cubes in each.
Understand the idea
Explain cubic measurement as packing equal-size unit cubes; distinguish volume from surface area.
Volume counts equal-size three-dimensional unit cubes, including cubes hidden from view. Count one complete length×width layer, then multiply by the number of equal layers. Face squares or visible corner cubes are not the full volume.
Vocabulary focus: unit cube, cuboid, cubic units, capacity. Make a small labeled model, work through the arithmetic, and explain the result in words.
Watch for: Counting only visible faces or square units.
Model to study: Countable unit-cube layers in an isometric grid.
Worked examples
Worked examples 1 · unit cube counting
A block has 2 layers. Each layer is 4 cubes long and 3 cubes wide. How many unit cubes fill it?
Solution: 24 unit cubes
Why: One layer has 4×3=12 cubes. 2 layers have 12×2=24 unit cubes, so volume is 24 cubic units.
Worked examples 2 · unit cube counting
A block has 4 layers. Each layer is 2 cubes long and 3 cubes wide. How many unit cubes fill it?
Solution: 24 unit cubes
Why: One layer has 2×3=6 cubes. 4 layers have 6×4=24 unit cubes, so volume is 24 cubic units.
Worked examples 3 · unit cube counting
A block has 3 layers. Each layer is 4 cubes long and 4 cubes wide. How many unit cubes fill it?
Solution: 48 unit cubes
Why: One layer has 4×4=16 cubes. 3 layers have 16×3=48 unit cubes, so volume is 48 cubic units.
Check understanding
Check understanding 1 · unit cube counting
A block has 4 layers. Each layer is 3 cubes long and 4 cubes wide. How many unit cubes fill it?
Check understanding 2 · unit cube counting
A block has 4 layers. Each layer is 5 cubes long and 2 cubes wide. How many unit cubes fill it?
Guided practice
Guided practice 1 · unit cube counting
A block has 2 layers. Each layer is 2 cubes long and 2 cubes wide. How many unit cubes fill it?
Guided practice 2 · unit cube counting
A block has 3 layers. Each layer is 5 cubes long and 4 cubes wide. How many unit cubes fill it?
Guided practice 3 · unit cube counting
A block has 3 layers. Each layer is 5 cubes long and 3 cubes wide. How many unit cubes fill it?
Independent practice
Independent practice 1 · unit cube counting
A block has 3 layers. Each layer is 3 cubes long and 3 cubes wide. How many unit cubes fill it?
Independent practice 2 · unit cube counting
A block has 2 layers. Each layer is 2 cubes long and 4 cubes wide. How many unit cubes fill it?
Independent practice 3 · unit cube counting
A block has 3 layers. Each layer is 2 cubes long and 4 cubes wide. How many unit cubes fill it?
Independent practice 4 · unit cube counting
A block has 2 layers. Each layer is 4 cubes long and 4 cubes wide. How many unit cubes fill it?
Independent practice 5 · unit cube counting
A block has 2 layers. Each layer is 3 cubes long and 3 cubes wide. How many unit cubes fill it?
Challenge and error analysis
Challenge and error analysis 1 · worked-error diagnosis
Error-analysis notebook: Learner notebook (LESSON): “I can count one face 2×3=6 and call it all 2 layers of cubes.” Identify exactly which written step or assumption fails, correct it, and independently solve this new task: A block has 2 layers. Each layer is 2 cubes long and 3 cubes wide. How many unit cubes fill it?
Challenge and error analysis 2 · reasoning transfer
Reasoning and transfer: Solve, then explain a different way to check your answer (inverse operation, estimate, model, or units as appropriate). A block has 4 layers. Each layer is 2 cubes long and 4 cubes wide. How many unit cubes fill it?
Targeted mastery: model, calculate, explain
These additional original examples address specific 5A mastery requirements. Compare the visual and the reasoning before trying the independent questions.
Targeted 1 · targeted mastery: layers
A block is 3 cubes long, 2 cubes wide, 3 layers high. Count all cubes including hidden cubes.
Solution: 18 unit cubes
Why: 3×2=6 cubes per complete layer. Three complete layers yield 6×3=18 unit cubes. Hidden cubes still count.
Targeted 2 · targeted mastery: layers
A 4-by-3 base is stacked into 2 complete layers. How many cubes are hidden or visible altogether?
Solution: 24 unit cubes
Why: Each layer has 4×3=12 cubes; 2 layers are 24 cubes. Counting only the drawn visible faces is invalid.
Targeted 3 · targeted mastery: layers
Three full layers each have 2 rows of 5 unit cubes. Find total volume.
Targeted 4 · targeted mastery: layers
A cube packing array has 2 rows, 4 columns, and 4 layers. How many unit cubes?
Exit check
Exit check 1 · unit cube counting
A block has 4 layers. Each layer is 5 cubes long and 3 cubes wide. How many unit cubes fill it?
Exit check 2 · unit cube counting
A block has 3 layers. Each layer is 5 cubes long and 2 cubes wide. How many unit cubes fill it?
Target for independent mastery: all exit-check problems correct plus a justified explanation. A tutor may reteach the concept if calculations are correct without meaningful reasoning.