Estimation to check answers to calculations

Activities to try

1. 1

   Estimate Detective: flag the unreasonable answer

   Why it works: A quick independent estimate shows roughly what size the answer should be, so an answer sitting far from the estimate is visibly wrong (487 × 23 cannot be 1121 when the estimate is ~10000). The estimate's job is to be roughly right, fast — not to match exactly.

   1. Face a calculation with a proposed answer, e.g. 487 × 23 with a candidate answer shown — some candidates are correct, others sabotaged by a misplaced decimal point or a wrong operation.
   2. Round each number to a convenient, friendly value (e.g. 487 → 500, 23 → 20) and compute the easy estimate (500 × 20 = 10000).
   3. Place the estimate and the proposed answer as points on the same number line, then judge whether the proposed answer is 'reasonable' or 'unreasonable' by how far it sits from the estimate.
   4. State the general direction rule — rounding BOTH factors up over-estimates (so the true answer is a bit less than the estimate) and rounding BOTH down under-estimates (so the true answer is a bit more) — then notice that the 500 × 20 example is MIXED (487 rounds up, 23 rounds down), so its direction is NOT cleanly predetermined; for a mixed case rely on the estimate's MAGNITUDE to catch gross errors rather than declaring a precise over- or under-estimate.
   5. Rule each proposed answer reasonable or not WITHOUT computing the exact product first.
2. 2

   How accurate does it need to be?

   Why it works: Appropriate accuracy is a contextual judgement, not a fixed rule: the purpose of the answer sets how precise it must be, and sometimes a fast ballpark beats an exact figure.

   1. Meet real contexts — buying enough paint, sharing money fairly, counting a crowd, working out how many buses are needed — and decide whether an exact answer or an estimate is appropriate.
   2. Choose the sensible precision for each: money to 2 decimal places, the number of buses or tins of paint needed rounded UP so there is enough, a fair 'about' for a crowd size.
   3. Explain the consequence of choosing wrongly, e.g. rounding the number of buses down leaves people without a seat; giving money to 3 decimal places is spurious precision.
   4. Name a situation where a rough estimate is actually more useful than an exact answer (when a ballpark is enough, or when speed matters more than precision).
3. 3

   Does that seem about right? — reasoning aloud

   Why it works: Estimation is a stance toward reasonableness and its limits: it can rule out place-value-sized errors but cannot verify the last digits, so it complements — not replaces — an exact or inverse check.

   1. Be presented with suspicious proposed answers and, before any reveal, say whether each seems about right and why.
   2. Answer probing questions: 'If your estimate was 10,000 and your worked answer was 1,150, which do you trust — and what do you do next?'
   3. Explain whether an estimate can ever PROVE an exact answer is 100% correct, and why not.
   4. Describe how you would round 487 × 23 to check it, and justify why you rounded that way.