Why a 4% error on each side becomes 12% on the area, and the bound notation examiners insist on.
Ask whether rounding two measurements to the nearest unit is a big deal.
They will say no, and for a single measurement they are right: 4.17% at worst. Then reveal the area, which can be anywhere from 74.75 to 93.75 against a nominal 84.
Multiply quantities and the percentage errors roughly add. That is the one sentence to leave on the board, and it is the whole reason this sub-topic exists rather than being a rounding exercise.
4.05 ≤ x < 4.15. Closed below, open above, because 4.15 would round up to 4.2. Students write both as ≤ and it is marked wrong.
And the pairing flips. Greatest area is upper times upper, but greatest difference is upper minus lower, and greatest quotient is upper over lower. Make them think about which bound makes the answer big rather than copying a pattern.
| 1. Lower bound of 350 to 2 sf | The unit is 10, so 345. |
| 2. Percentage error of 3.14 for π | 0.0507%. |
| 3. Greatest area | 12.5 × 7.5 = 93.75. |
1 markBounds with the correct inequality signs.
1 markThe right pairing of bounds for the quantity asked for.
1 markDividing by the exact value in a percentage error, and rounding only at the end.
| They give | What it means |
|---|---|
| 349.5 (Q1) | Treated 350 as exact to the nearest unit. To 2 sf the rounding unit is 10. |
| 355 (Q1) | Gave the upper bound. |
| 0.0405 (Q2) | That is the error for 22/7, which is the better approximation. |
| 0.00159 (Q2) | Has the fraction, has not multiplied by 100. |
| 84 (Q3) | Gave the nominal area rather than the greatest possible. |
| 84.5 (Q3) | Added half a unit to the area instead of to each side first. |
"Why not just round at the end?" You should, and that is the point. Rounding mid-calculation is a second source of error on top of the measurement error.
"Is 22/7 better than 3.14?" Yes, 0.0402% against 0.0507%, which surprises people. Worth doing because it shows a percentage error is a real comparison and not a formality.
"How many significant figures should I give?" Match the data. Three is the default in these courses unless the question says otherwise, and an answer to seven figures from data given to two is its own error.
| What is happening | |
|---|---|
| 1 | Is rounding a big deal? Reveal the area range. |
| 2 | Bounds, with the inequality signs laboured. |
| 3 | Percentage error, including the pi comparison. |
| 4 | Compound quantities: area, then a difference, then a quotient. |
| 5 | Choosing a sensible accuracy for an answer. |
Do not write both bounds with ≤. They will copy it and lose the mark.
Do not say percentage errors "add" as if it were exact. They roughly add for products; the exact answer comes from the bounds.
Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.
The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.
The same skill inside a real situation, where the first job is working out what is being asked.
Reasoning, working backwards, or spotting an error. These are where the top grades are decided.
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