The trap is sneaky because the wrong factor looks right. You know 1 m = 100 cm, so when you convert m² it feels natural to multiply by 100. But that's the length factor, and area needs it squared.
What Goes Wrong
Say you're converting 5 m² to cm². The tempting move is 5 × 100 = 500 — grab the "100" you remember from lengths and go. But a square metre isn't 100 square centimetres, it's 10 000 (you proved that in rung 1). So the real answer is 5 × 10 000 = 50 000 cm². The trap answer is a hundred times too small. The toy puts both side by side so you can watch the gap blow out.
Why the Squared Factor Is the Right One
Because area is two-dimensional. When the side grows by a factor of 100 (a metre becoming 100 cm), it grows that much across and that much down. Two directions, two factors, multiplied together: 100 × 100. A single ×100 only stretches one direction — that's a line, not a square.
The Check That Saves You
After any area conversion, ask: "did I square the factor?" Or sanity-check the size. Going to a smaller unit should give a much bigger number than you might expect — if 5 m² came out as just 500 cm², that's suspiciously small for thousands of little squares. Trust the doubt and re-check the factor.