A storage facility uses identical cubic boxes, each with side length 2 centimeters, to organize small items. These cubic boxes...

1. TRANSLATE the problem information

• Given information:
  • Cubic boxes with side length = \(2\) cm each
  • Rectangular container: length = \(8\) cm, width = \(6\) cm, height = \(4\) cm
  • Boxes fit perfectly without gaps or overlaps
  • Need to find: How many cubic boxes fit in the container

2. INFER the solution approach

• This is fundamentally asking: How many small volumes fit into one large volume?
• Key insight: When objects fit perfectly without gaps, we can use volume division
• Strategy: Find volume of one box, find volume of container, then divide

3. Calculate the volume of one cubic box

• Volume of cube = \((\mathrm{side\ length})^3\)
• Volume of one box = \(2^3 = 8\) cubic centimeters

4. Calculate the volume of the rectangular container

• Volume of rectangular prism = \(\mathrm{length} \times \mathrm{width} \times \mathrm{height}\)
• Container volume = \(8 \times 6 \times 4 = 192\) cubic centimeters

5. SIMPLIFY to find the answer

• Number of boxes = Total container volume ÷ Volume per box
• Number of boxes = \(192 \div 8 = 24\)

6. INFER a verification check (optional but recommended)

• We can verify by checking how many boxes fit along each dimension:
  • Length direction: \(8 \div 2 = 4\) boxes
  • Width direction: \(6 \div 2 = 3\) boxes
  • Height direction: \(4 \div 2 = 2\) boxes
• Total: \(4 \times 3 \times 2 = 24\) ✓

Answer: 24

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may misunderstand what "fit perfectly without gaps" means and attempt complex geometric arrangements instead of recognizing this as a straightforward volume problem.

They might try to visualize specific box placements or worry about orientation, leading to confusion and potentially guessing rather than using the systematic volume approach.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up the volume calculation but make arithmetic errors, particularly in calculating \(2^3 = 8\) or in the division \(192 \div 8 = 24\).

This leads them to get an incorrect final answer despite using the right method.

The Bottom Line:

This problem tests whether students can recognize that "perfect fit" scenarios are fundamentally about volume relationships, not complex geometric puzzles. The key insight is translating the physical situation into a simple mathematical division problem.