G2 Mathematics K210 checking becomes faster when learners know whether an output should move consistently upward or downward as one input increases. This is a practical use of monotonicity: direction of change across a stated range.
This one-hundred-and-twenty-sixth Learner’s Guide develops monotonicity checks. It extends Vol 0082 Direction-of-Change Checks and Vol 0094 Sensitivity Checks.
The monotonicity test
Hold relevant quantities fixed, increase one input and predict whether the output must increase, decrease or stay unchanged over the range being considered. Then compare the exact calculation with that directional expectation.
1. unit price to total cost
For unit price to total cost, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
2. quantity to total cost
For quantity to total cost, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
3. discount rate to sale price
For discount rate to sale price, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
4. markup rate to selling price
For markup rate to selling price, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
5. positive interest rate to amount
For positive interest rate to amount, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
6. time to simple interest
For time to simple interest, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
7. growth rate to future value
For growth rate to future value, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
8. depreciation rate to future value
For depreciation rate to future value, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
9. numerator to positive fraction
For numerator to positive fraction, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
10. denominator to positive fraction
For denominator to positive fraction, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
11. distance to required speed
For distance to required speed, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
12. time to required speed
For time to required speed, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
13. speed to travel time
For speed to travel time, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
14. radius to circumference
For radius to circumference, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
15. radius to circle area
For radius to circle area, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
16. side to square area
For side to square area, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
17. length to rectangle area
For length to rectangle area, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
18. length to perimeter
For length to perimeter, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
19. scale factor to image length
For scale factor to image length, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
20. scale factor to area factor
For scale factor to area factor, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
21. scale factor to volume factor
For scale factor to volume factor, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
22. high data value to mean
For high data value to mean, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
23. added above-mean value to mean
For added above-mean value to mean, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
24. added below-mean value to mean
For added below-mean value to mean, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
25. maximum to range
For maximum to range, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
26. minimum to range
For minimum to range, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
27. favourable outcomes to probability
For favourable outcomes to probability, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
28. total outcomes to probability with favourable fixed
For total outcomes to probability with favourable fixed, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
29. rise to positive gradient
For rise to positive gradient, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
30. run to positive gradient
For run to positive gradient, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
31. slope coefficient to line steepness
For slope coefficient to line steepness, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
32. intercept to vertical position
For intercept to vertical position, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
33. population to density
For population to density, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
34. area to density
For area to density, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
35. usage to utilisation
For usage to utilisation, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
36. capacity to utilisation
For capacity to utilisation, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
37. spending to percentage budget used
For spending to percentage budget used, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
38. budget to percentage used
For budget to percentage used, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
39. revenue to profit
For revenue to profit, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
40. cost to profit
For cost to profit, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
41. distance to journey time at fixed speed
For distance to journey time at fixed speed, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
42. work rate to completion time
For work rate to completion time, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
43. flow rate to fill time
For flow rate to fill time, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
44. data rate to transfer time
For data rate to transfer time, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
45. cash to runway
For cash to runway, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
46. burn rate to runway
For burn rate to runway, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
47. efficiency to fuel needed
For efficiency to fuel needed, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
48. consumption rate to fuel needed
For consumption rate to fuel needed, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
49. direct-proportion x to y
For direct-proportion x to y, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
50. inverse-proportion x to y
For inverse-proportion x to y, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
51. positive constant to direct y
For positive constant to direct y, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
52. positive constant to inverse y
For positive constant to inverse y, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
53. arithmetic common difference to later term
For arithmetic common difference to later term, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
54. geometric ratio above one to later term
For geometric ratio above one to later term, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
55. ratio between zero and one to decay term
For ratio between zero and one to decay term, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
56. load to extension in stated linear region
For load to extension in stated linear region, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
57. temperature to rate in limited stated trend
For temperature to rate in limited stated trend, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
58. concentration to rate in limited stated trend
For concentration to rate in limited stated trend, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
59. graph increasing interval
For graph increasing interval, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
60. graph decreasing interval
For graph decreasing interval, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
61. graph constant interval
For graph constant interval, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
62. turning point
For turning point, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
63. threshold then increase
For threshold then increase, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
64. threshold then plateau
For threshold then plateau, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
65. piecewise increasing
For piecewise increasing, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
66. piecewise decreasing
For piecewise decreasing, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
67. non-monotonic relation
For non-monotonic relation, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
68. U-shaped relation
For U-shaped relation, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
69. inverted-U relation
For inverted-U relation, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
70. same output two inputs
For same output two inputs, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
71. monotonicity and inverse
For monotonicity and inverse, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
72. monotonicity and bounds
For monotonicity and bounds, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
73. monotonicity and sensitivity
For monotonicity and sensitivity, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
74. monotonicity and graph
For monotonicity and graph, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
75. monotonicity and table
For monotonicity and table, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
76. monotonicity and MCQ
For monotonicity and MCQ, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
77. monotonicity and estimate
For monotonicity and estimate, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
78. monotonicity and context
For monotonicity and context, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
79. monotonicity and integer ceiling
For monotonicity and integer ceiling, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
80. monotonicity and floor
For monotonicity and floor, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
81. monotonicity and rounding
For monotonicity and rounding, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
82. monotonicity final rule
For monotonicity final rule, construct two nearby input values and compare their outputs. State whether the relation is increasing, decreasing, constant or not monotonic over the tested interval.
For practice, create one wrong calculation whose output moves in the impossible direction. Use monotonicity to detect the error before redoing arithmetic. Then identify the likely source: reciprocal, sign, denominator, multiplier, unit or branch.
Do not extend monotonic behaviour beyond the range that justifies it. A relation can increase before a turning point, flatten at saturation or change rule at a breakpoint. State the interval whenever direction can change.
Monotonicity is stronger than a vague plausibility check
“This looks about right” has no mathematical direction. “Increasing the denominator with a fixed positive numerator must reduce the fraction” is a structural prediction. That prediction can reject an answer immediately.
Links
Use the Mathematics Hub, Vol 0122 Breakpoint Reasoning, Vol 0086 Bounding Checks, 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. Monotonicity checking here is an eduKateSengkang reasoning framework, not an additional SEAB syllabus topic.
Final rule
Before trusting the exact number, ask which way the answer must move when the input increases. Use monotonicity inside a justified range, and treat a wrong-direction result as a reason to inspect the model before submission.
