Estimation Duels

A quantity goes on the board — the slope of a tangent nobody is allowed to differentiate yet, the angle between two arrows, how fast a puddle’s area grows the moment its radius passes a metre — 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

1. Commit in writing first. A number without a reason scores nothing.
2. Defend: what did you compare it to, and which way did rounding push you?
3. Bracket as a class, then calculate the reveal.

One duel: the slope of $y=x^2$ at $x=3$

• “More than 5 — because the secant from $x=2$ to $x=3$ has slope $\frac{9-4}{1}=5$, and the curve is steepening, so the tangent at 3 must beat every secant that arrives from the left.”
• “Less than 7 — because the secant from $x=3$ to $x=4$ has slope $\frac{16-9}{1}=7$, and by the same steepening the tangent must lose to every secant that leaves to the right. So it is cornered between 5 and 7 before anyone computes anything.”
• “Split the difference with the secant that straddles the point: from $x=2$ to $x=4$, slope $\frac{16-4}{2}=6$. I said 6.”
• The reveal: exactly 6. For a parabola, the straddling secant is not an estimate at all — it lands on the tangent dead centre. Why that happens for this curve, and not for every curve, is a question worth carrying into The Derivative.

One variation

Vector duels: the angle between $⟨3,4⟩$ and $⟨5,0⟩$. Less than $90^{\circ }$, because both arrows lean the same general way; more than $45^{\circ }$, because the first arrow rises faster than it runs. Cornered between $45^{\circ }$ and $90^{\circ }$ before any formula appears — the reveal is about $53^{\circ }$, and The Dot Product is the machine that turns the bracket into an exact number.

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. Tangent slopes need it most: a secant on each side of the point corners the tangent between them, which is the entire strategy of The Limit wearing estimation clothes.