Estimation, rounding and simple error intervals · Foundation
| English | Español |
|---|---|
| significant figure/sɪɡˈnɪfɪkənt ˈfɪɡə/ | cifra significativa |
Would the estimate catch a typo?
- A calculator reports 49.8×19.7÷10.2. Could 962 be a reasonable answer, or does a quick estimate expose an input error?
- This lesson studies significant figure 有效数字: A digit counted from the first nonzero digit when reporting numerical precision.
Choose the mathematical structure
- For a rough check use easy nearby numbers: 50×20÷10=100. Decimal places count digits after the point; significant figures start at the first nonzero digit. Round only the final result. Nearest-unit rounding has an interval extending half a unit either way; positive truncation keeps values from the stated value up to the next unit.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines significant figure?
A digit counted from the first nonzero digit when reporting numerical precision.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
The calculation is about 96.18, consistent with the estimate 100. The number 0.004786 rounds to 0.0048 at 2 significant figures, but to 0.005 at 3 decimal places. If length L rounds to 8.0 cm at 1 decimal place, 7.95≤L<8.05. If a positive value is truncated to 8.0 at 1 decimal place, 8.0≤L<8.1 instead. Reported precision must fit the question. Seventeen items packed six per box need three whole boxes, since two boxes hold only twelve items. Rounding 17/6 to two boxes would fail the physical requirement. For a nearest-0.1 reading of 8.0, each possible value differs from the report by at most 0.05; a claim of 8.08 lies outside the interval. Carry full calculator precision until a final money, length or accuracy requirement is applied.
Estimation, rounding and simple error intervals
For a rough check use easy nearby numbers: 50×20÷10=100
Compare the model with the worked case and explain one change.
Find the rough estimate 50×20÷10.
50×20/10=100.
Test a tempting shortcut
- Zeros before the first nonzero digit do not count as significant figures. Rounding and truncation give different intervals. Do not turn an approximate check into an exact answer, or round every intermediate result.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Rounding a positive value and truncating it to the same decimal place always give the same error interval. This claim is false. Explain which definition or assumption it violates.
Round 0.004786 to 2 significant figures.
Start at digit 4; the next digit after 7 is 8, so round to 0.0048.
Rounding a positive value and truncating it to the same decimal place always give the same error interval.
Zeros before the first nonzero digit do not count as significant figures. Rounding and truncation give different intervals. Do not turn an approximate check into an exact answer, or round every intermediate result.
Interpret a new situation
- AQA N14–N16 require accuracy interpretation at both tiers. Foundation uses simple intervals and limits of accuracy. Higher combines upper and lower bounds for calculated quantities in the separate bounds lesson.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the upper endpoint of the nearest-0.1 interval for 8.0.
Half of 0.1 is 0.05; 8.0+0.05=8.05.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Foundation · 3.1. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A digit counted from the first nonzero digit when reporting numerical precision. Choose the relationship, show the method, check its assumptions and interpret the result.