A quantity goes on the board — a stack of doublings, a balance left to grow, the angle of the sun over the parking lot — and two duellists commit to estimates before anyone computes. Each defends a reason. Then the class brackets: surely too low, surely too high, squeezed until cornered.
How we play
- Commit in writing first. A number without a reason scores nothing.
- Defend: what did you compare it to, and which way did rounding push you?
- Bracket as a class, then measure or calculate the reveal.
One duel: $100 invested at 8%, compounded yearly, for 50 years
- “More than 8-a-year interest would pay, and compounding only ever beats it.”
- “Less than $10 000 — at 8% the balance doubles roughly every nine years, and fifty years is only about five and a half doublings.”
- “Five doublings of 3200; six is 4500.”
- The reveal: about $4700. The doubling argument had it cornered.
One variation
Doubling duels: how thick is a sheet of paper folded in half twenty times? Commit before you scoff — the estimates in the room usually span a factor of a thousand. Your eyes get their verdict during Folding Paper to the Moon, once The Exponential Function can compute what intuition refuses to.
Bracketing is a life skill
An answer you cannot bracket is an answer you cannot check. Naming “too low” and “too high” first is Checking Your Own Work done in advance — before the mistake instead of after it. Exponential answers need it most, because intuition is linear and they are not.