G2 Mathematics K210 calculations can inherit uncertainty from measured, rounded or graph-read inputs. A correct formula does not create more information than the inputs contain. Some operations preserve a small uncertainty; others amplify it.
This one-hundred-and-thirty-fourth Learner’s Guide develops error-propagation control without requiring advanced uncertainty formulae. It extends Vol 0086 Bounding Checks and Vol 0094 Sensitivity Checks.
The propagation protocol
Identify which inputs are exact and which are measured, rounded or estimated. Perturb uncertain inputs within plausible bounds and observe how the output changes. Keep full working precision until the final presentation unless the question specifies otherwise.
1. rounded input to sum
For rounded input to sum, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
2. rounded input to difference
For rounded input to difference, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
3. rounded input to product
For rounded input to product, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
4. rounded input to quotient
For rounded input to quotient, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
5. measurement bound to area
For measurement bound to area, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
6. measurement bound to perimeter
For measurement bound to perimeter, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
7. measurement bound to volume
For measurement bound to volume, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
8. radius uncertainty to circumference
For radius uncertainty to circumference, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
9. radius uncertainty to area
For radius uncertainty to area, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
10. length uncertainty to rectangle area
For length uncertainty to rectangle area, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
11. two measured sides
For two measured sides, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
12. speed from distance/time
For speed from distance/time, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
13. density from mass/volume
For density from mass/volume, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
14. rate from amount/time
For rate from amount/time, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
15. population density
For population density, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
16. unit price from total/quantity
For unit price from total/quantity, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
17. mean from rounded values
For mean from rounded values, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
18. percentage from rounded part/whole
For percentage from rounded part/whole, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
19. ratio from rounded quantities
For ratio from rounded quantities, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
20. currency conversion with rounded rate
For currency conversion with rounded rate, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
21. compound stages with rounded intermediate
For compound stages with rounded intermediate, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
22. repeated multiplication
For repeated multiplication, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
23. repeated division
For repeated division, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
24. early rounding
For early rounding, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
25. late rounding
For late rounding, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
26. calculator full precision
For calculator full precision, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
27. display precision
For display precision, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
28. exact fraction then decimal
For exact fraction then decimal, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
29. π approximation
For π approximation, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
30. scale factor uncertainty
For scale factor uncertainty, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
31. area scale amplification
For area scale amplification, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
32. volume scale amplification
For volume scale amplification, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
33. difference of close numbers
For difference of close numbers, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
34. small denominator
For small denominator, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
35. large denominator
For large denominator, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
36. percentage with small base
For percentage with small base, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
37. relative error
For relative error, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
38. absolute error
For absolute error, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
39. upper bound
For upper bound, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
40. lower bound
For lower bound, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
41. worst-case direction
For worst-case direction, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
42. error cancellation possibility
For error cancellation possibility, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
43. error reinforcement possibility
For error reinforcement possibility, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
44. independent errors conceptually
For independent errors conceptually, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
45. shared systematic error
For shared systematic error, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
46. unit conversion exact factor
For unit conversion exact factor, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
47. counting exact quantity
For counting exact quantity, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
48. measured quantity
For measured quantity, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
49. given exact integer
For given exact integer, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
50. estimated graph reading
For estimated graph reading, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
51. interpolated graph reading
For interpolated graph reading, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
52. extrapolated estimate
For extrapolated estimate, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
53. table rounding
For table rounding, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
54. class interval midpoint estimate
For class interval midpoint estimate, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
55. weighted mean estimate
For weighted mean estimate, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
56. financial cents rounding
For financial cents rounding, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
57. minimum integer after approximate quotient
For minimum integer after approximate quotient, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
58. maximum integer after approximate quotient
For maximum integer after approximate quotient, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
59. capacity safety margin
For capacity safety margin, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
60. threshold near rounded value
For threshold near rounded value, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
61. breakpoint uncertainty
For breakpoint uncertainty, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
62. case selection near boundary
For case selection near boundary, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
63. root near domain edge
For root near domain edge, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
64. probability from estimated frequency
For probability from estimated frequency, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
65. rate comparison close values
For rate comparison close values, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
66. ranking reversal under uncertainty
For ranking reversal under uncertainty, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
67. sign robust despite uncertainty
For sign robust despite uncertainty, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
68. bound robust
For bound robust, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
69. answer interval
For answer interval, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
70. sensitivity to input error
For sensitivity to input error, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
71. high sensitivity
For high sensitivity, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
72. low sensitivity
For low sensitivity, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
73. condition number intuition without formalism
For condition number intuition without formalism, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
74. error propagation versus mistake
For error propagation versus mistake, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
75. uncertainty versus arithmetic error
For uncertainty versus arithmetic error, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
76. systematic bias versus propagated uncertainty
For systematic bias versus propagated uncertainty, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
77. verification with bounds
For verification with bounds, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
78. verification with alternative route
For verification with alternative route, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
79. report sensible precision
For report sensible precision, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
80. do not invent precision
For do not invent precision, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
81. error-propagation final rule
For error-propagation final rule, create a low-input and high-input case consistent with the stated precision. Calculate the corresponding output interval or at least the direction in which uncertainty moves the answer.
For practice, compare early rounding with late rounding. If repeated operations magnify the difference, explain why preserving intermediate precision protects the final result. Then choose a final precision that does not imply information the inputs never contained.
Distinguish propagated uncertainty from a mistake. An interval caused by measurement precision can remain even when every operation is correct. A sign error, wrong formula or wrong unit is not uncertainty; it is an error to repair.
Sensitivity controls propagation
If a small input change produces a large output change, input uncertainty matters more. If the output barely changes, the result is robust. This is why sensitivity checks and bounds are natural tools for examining propagation.
Links
Use the Mathematics Hub, Vol 0122 Breakpoint Reasoning, Vol 0114 Forward-Substitution Verification, the Examination Craft hub and the PSLE Learner’s Guide.
Official-source discipline
For the current 2027 SEC G2 school-candidate framework, use the official SEAB G2 syllabus directory and linked K210 Mathematics syllabus. Error-propagation control here is an eduKateSengkang quantitative-reasoning framework, not an additional SEAB syllabus topic.
Final rule
A calculation cannot be more certain than the information feeding it. Preserve precision during working, test plausible input bounds, and report a final answer whose precision matches the data.
