A storage box without a lid is shaped like a right rectangular prism with length 22text{ cm}, width 13text{ cm},...

1. TRANSLATE the problem information

• Given information:
  • Rectangular box without lid (open top)
  • Length = 22 cm, Width = 13 cm, Height = 14 cm
  • Paint only exterior surfaces, excluding the open top
• What this tells us: We need the areas of the bottom and the four vertical sides

2. INFER which faces to include

• Since the box has no lid, we exclude the top face
• Include: bottom base + 4 vertical sides (2 long sides + 2 short sides)
• This gives us 5 faces total to paint

3. SIMPLIFY by calculating each face area

• Bottom base area: length × width = \(22 \times 13 = 286\) cm²
• Two long sides (length × height): \(2 \times (22 \times 14) = 2 \times 308 = 616\) cm²
• Two short sides (width × height): \(2 \times (13 \times 14) = 2 \times 182 = 364\) cm²

4. SIMPLIFY to find total surface area

• Add all face areas: \(286 + 616 + 364 = 1266\) cm²

Answer: 1266

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "without a lid" and either include the top face in their calculation or get confused about which faces to paint.

They might calculate the full surface area (including top):

\(2(22\times13) + 2(22\times14) + 2(13\times14) = 572 + 616 + 364 = 1552\) cm², leading to an incorrect answer of 1552.

Second Most Common Error:

Poor INFER reasoning: Students correctly identify that the top is excluded but miscalculate by only including 4 sides and forgetting the bottom base.

They calculate:

\(2(22\times14) + 2(13\times14) = 616 + 364 = 980\) cm², missing the base area of 286 cm² and getting 980 instead of 1266.

The Bottom Line:

This problem tests whether students can visualize a 3D shape and systematically account for all relevant faces. The key insight is recognizing that "without a lid" means the top face is open, but all other faces (including the bottom) need painting.