G2 Mathematics K210 checking can often reject a wrong answer before a full re-solve by asking a directional question: if this input increases, should the output rise, fall, stay fixed or change in a known way? A result that moves opposite to the model is an error signature.
This eighty-second Learner’s Guide develops direction-of-change sanity checks. It complements Vol 0078 Dimensional Consistency and Vol 0055 Error Signatures. Units ask what kind of quantity the answer is. Direction checks ask whether the answer moved the right way.
The direction check
Before or after calculation, hold the other relevant conditions fixed and ask: if this input becomes larger, what should happen to the target? The check is strongest when the relationship is structurally clear—direct, inverse, additive, subtractive, geometric or constrained.
1. positive gradient
Expected direction: as x increases, y increases along a straight line.
Error signal: a negative calculated gradient contradicts the visible direction. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
2. negative gradient
Expected direction: as x increases, y decreases.
Error signal: a positive gradient signals point-order inconsistency or arithmetic error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
3. zero gradient
Expected direction: y remains constant as x changes.
Error signal: a non-zero slope conflicts with the horizontal graph. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
4. direct proportion
Expected direction: increasing x increases y by the same multiplicative relationship.
Error signal: a decrease in y under positive constant of proportionality contradicts the model. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
5. inverse proportion
Expected direction: increasing x decreases y when product remains constant.
Error signal: both variables increasing together contradicts the simple inverse model. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
6. percentage increase
Expected direction: final value should exceed original for a positive increase.
Error signal: a smaller final value signals sign or base misinterpretation. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
7. percentage decrease
Expected direction: final value should be below original.
Error signal: a larger final value conflicts with the stated decrease. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
8. successive increase
Expected direction: each positive percentage increase raises the current base.
Error signal: a stage that falls suggests operation or sign error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
9. successive decrease
Expected direction: each positive percentage decrease lowers the current base.
Error signal: a stage that rises suggests the multiplier was inverted. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
10. discount
Expected direction: sale price should not exceed marked price when discount is positive.
Error signal: a larger sale price reveals direction error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
11. markup
Expected direction: selling price should exceed cost when markup is positive.
Error signal: a lower selling price contradicts positive markup. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
12. simple interest growth
Expected direction: positive interest over positive time raises the accumulated amount.
Error signal: a lower total suggests subtraction or rate-sign error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
13. compound growth
Expected direction: positive growth factor above one increases value over periods.
Error signal: a falling sequence contradicts the multiplier. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
14. depreciation
Expected direction: a positive depreciation rate below 100% lowers value over time.
Error signal: rising value signals use of 1+r instead of 1-r. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
15. exchange conversion
Expected direction: direction depends on quoted rate.
Error signal: sanity-check whether receiving more or fewer currency units is plausible from the rate magnitude. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
16. unit price
Expected direction: buying more identical items raises total cost.
Error signal: a lower total with greater positive quantity contradicts constant positive unit price. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
17. fixed fee plus usage
Expected direction: greater usage should not lower total cost if variable rate is positive.
Error signal: a downward result suggests sign or model error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
18. distance at positive speed
Expected direction: greater positive time increases distance.
Error signal: decreasing distance under the simple model conflicts with d=st. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
19. time for fixed distance
Expected direction: higher speed reduces required time.
Error signal: a higher computed time after speed increase signals reciprocal error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
20. speed for fixed distance
Expected direction: shorter time means higher speed.
Error signal: a lower speed from shorter time contradicts distance/time structure. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
21. area of rectangle
Expected direction: increasing one positive side with the other fixed increases area.
Error signal: a smaller area suggests wrong substitution. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
22. perimeter of rectangle
Expected direction: increasing a side with other dimensions fixed increases perimeter.
Error signal: a decrease is impossible under the simple geometry. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
23. circle radius and circumference
Expected direction: larger positive radius gives larger circumference.
Error signal: a smaller circumference signals formula or input error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
24. circle radius and area
Expected direction: larger positive radius gives larger area, with squared sensitivity.
Error signal: a falling area contradicts the formula. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
25. scale enlargement
Expected direction: scale factor greater than one increases corresponding lengths.
Error signal: smaller image length contradicts enlargement. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
26. scale reduction
Expected direction: scale factor between zero and one reduces lengths.
Error signal: larger image length contradicts reduction. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
27. similarity area
Expected direction: larger length scale factor gives larger area factor squared.
Error signal: area factor moving in the wrong direction signals exponent error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
28. similarity volume
Expected direction: larger length scale factor gives cubic volume growth.
Error signal: using only linear growth understates direction and magnitude. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
29. mean after adding high value
Expected direction: adding a value above the old mean tends to raise the mean.
Error signal: a lower new mean signals total/count error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
30. mean after adding low value
Expected direction: adding a value below old mean tends to lower the mean.
Error signal: a higher new mean is suspicious. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
31. mean after equal shift
Expected direction: adding the same positive constant to every value raises mean by that constant.
Error signal: any other direction contradicts translation. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
32. range after equal shift
Expected direction: adding same constant to every value leaves range unchanged.
Error signal: a changed range signals min/max handling error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
33. probability adding favourable outcomes
Expected direction: with total structure controlled, more favourable share should not reduce probability.
Error signal: a decrease may signal denominator change or counting error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
34. probability restriction
Expected direction: direction depends on which outcomes are removed.
Error signal: state the new universe before predicting change. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
35. frequency increase
Expected direction: greater count raises raw frequency.
Error signal: a lower frequency is a data-copy contradiction. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
36. relative frequency
Expected direction: direction depends on both category count and total.
Error signal: raw count increase does not guarantee proportion increase if denominator grows faster. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
37. ratio common multiplication
Expected direction: multiplying both parts by same positive factor leaves ratio unchanged.
Error signal: a changed simplified ratio signals arithmetic error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
38. ratio adding to numerator part
Expected direction: increasing one part while other fixed increases that part-to-part ratio.
Error signal: a decrease signals orientation error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
39. fraction numerator increase
Expected direction: with positive fixed denominator, increasing numerator increases fraction.
Error signal: a lower quotient is impossible. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
40. fraction denominator increase
Expected direction: with positive fixed numerator, increasing denominator decreases fraction.
Error signal: a higher quotient signals reciprocal or entry error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
41. negative multiplier
Expected direction: multiplying a positive quantity by a negative reverses sign.
Error signal: failure to reverse sign is a directional error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
42. inequality negative multiplication
Expected direction: multiplying or dividing both sides by a negative reverses inequality direction.
Error signal: keeping the sign direction creates a non-equivalent statement. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
43. translation right
Expected direction: x-coordinates increase by the horizontal translation amount.
Error signal: decreasing x contradicts rightward translation. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
44. translation up
Expected direction: y-coordinates increase.
Error signal: decreasing y contradicts upward movement. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
45. reflection
Expected direction: distance to mirror line is preserved while direction changes.
Error signal: size changes indicate wrong transformation. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
46. rotation
Expected direction: distance from centre and size remain fixed.
Error signal: length growth or shrinkage contradicts rigid rotation. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
47. enlargement
Expected direction: distance from centre scales by factor.
Error signal: wrong side of centre or wrong magnitude can be caught from direction. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
48. linear function coefficient
Expected direction: positive coefficient means increasing line.
Error signal: negative visual direction contradicts coefficient sign. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
49. quadratic leading coefficient
Expected direction: positive leading coefficient opens upward; negative opens downward.
Error signal: graph orientation is a quick sign check. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
50. sequence positive common difference
Expected direction: terms increase by constant amount.
Error signal: a decreasing term signals sign or copying error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
51. sequence negative common difference
Expected direction: terms decrease.
Error signal: a rise contradicts arithmetic progression direction. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
52. geometric ratio above one
Expected direction: positive terms grow in magnitude.
Error signal: shrinking terms suggest reciprocal ratio. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
53. geometric ratio between zero and one
Expected direction: positive terms shrink toward zero.
Error signal: growth suggests wrong ratio. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
54. negative geometric ratio
Expected direction: sign alternates while magnitude follows absolute ratio.
Error signal: failure to alternate can reveal lost sign. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
55. coordinate distance
Expected direction: distance cannot be negative.
Error signal: a negative result signals algebraic or square-root error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
56. area
Expected direction: area cannot be negative in ordinary geometry.
Error signal: negative result indicates interpretation or algebra issue. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
57. probability
Expected direction: probability cannot be below zero or above one.
Error signal: range violation is a direction and scale alarm. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
58. count
Expected direction: physical count should not be negative.
Error signal: negative algebraic root may need rejection by context. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
59. capacity remaining
Expected direction: using more capacity reduces remaining capacity.
Error signal: an increase in remaining after added usage signals subtraction direction error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
60. budget remaining
Expected direction: higher spending lowers remaining budget.
Error signal: a larger remainder after extra spending is inconsistent. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
61. profit
Expected direction: higher revenue with fixed cost raises profit.
Error signal: lower profit suggests formula or changed-cost assumption. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
62. loss
Expected direction: higher cost with fixed revenue lowers profit or increases loss.
Error signal: opposite movement signals sign confusion. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
63. population growth
Expected direction: positive net growth raises population.
Error signal: falling value suggests rate sign or migration/death factors omitted. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
64. population decline
Expected direction: negative net change lowers population.
Error signal: rising result contradicts the stated net effect. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
65. temperature difference
Expected direction: direction depends on final minus initial convention.
Error signal: name the subtraction order before interpreting sign. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
66. elevation change
Expected direction: upward movement gives positive change under standard final-minus-initial convention.
Error signal: negative result signals reversed order. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
67. percentage-point increase
Expected direction: higher final percentage gives positive point difference.
Error signal: negative point change contradicts the data order. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
68. index rise
Expected direction: current/base ratio increases when current rises with base fixed.
Error signal: falling index suggests wrong base or inversion. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
69. normalisation
Expected direction: increasing numerator with fixed positive base raises normalised value.
Error signal: opposite direction indicates calculation error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
70. weighted mean shift
Expected direction: increasing a heavily weighted value should pull mean toward it.
Error signal: movement away suggests weighting error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
71. graph interpolation
Expected direction: between increasing endpoints, a reasonable interpolated value should lie between them.
Error signal: outside value suggests wrong scale or extrapolation. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
72. graph extrapolation
Expected direction: direction may continue only if model assumption is accepted.
Error signal: do not treat monotonic continuation as guaranteed outside range. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
73. constraint minimum
Expected direction: raising a minimum threshold cannot make feasibility easier.
Error signal: more solutions after stricter minimum may signal inequality error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
74. constraint maximum
Expected direction: lowering a maximum threshold cannot make feasibility easier.
Error signal: expanded feasible set signals direction error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
75. integer rounding up for minimum
Expected direction: a minimum whole-number requirement may force upward rounding.
Error signal: rounding down violates direction of the constraint. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
76. integer rounding down for maximum
Expected direction: a maximum capacity may force downward choice.
Error signal: rounding up can violate feasibility. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
77. reverse percentage original
Expected direction: for a final value that is 80% of original, original must exceed final.
Error signal: a smaller original signals multiplier inversion. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
78. reverse percentage after increase
Expected direction: if final is 120% of original, original must be below final.
Error signal: a larger original contradicts positive growth. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
79. speed-time graph positive area
Expected direction: positive speed over positive time gives positive distance travelled.
Error signal: negative area in simple context signals sign or axis misread. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
80. cumulative total
Expected direction: cumulative count should not decrease when only additions occur.
Error signal: a fall means subtraction, reset or graph misread. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
81. remaining quantity
Expected direction: remaining equals original minus used.
Error signal: more usage with fixed original cannot increase remainder. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
82. average cost fixed total
Expected direction: increasing item count with fixed total cost lowers average cost per item.
Error signal: higher average signals reciprocal error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
83. average cost fixed unit price
Expected direction: increasing quantity leaves unit price unchanged.
Error signal: changed average suggests a model mismatch. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
84. density fixed mass
Expected direction: increasing volume lowers density.
Error signal: higher density suggests reciprocal or changed-mass assumption. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
85. density fixed volume
Expected direction: increasing mass raises density.
Error signal: lower density signals direction error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
86. people per area fixed people
Expected direction: larger area lowers density.
Error signal: higher density contradicts denominator effect. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
87. area per person fixed area
Expected direction: more people lowers area per person.
Error signal: higher allocation signals reciprocal error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
88. rate fixed numerator
Expected direction: larger denominator lowers the rate.
Error signal: higher rate suggests denominator inversion. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
89. rate fixed denominator
Expected direction: larger numerator raises the rate.
Error signal: lower rate suggests arithmetic error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
90. margin fixed revenue
Expected direction: higher profit raises margin.
Error signal: lower margin contradicts fixed revenue assumption. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
91. markup fixed cost
Expected direction: higher profit raises markup.
Error signal: lower markup signals wrong denominator or changed cost. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
92. survey response fixed invitations
Expected direction: more responses raise response rate.
Error signal: lower rate indicates denominator or copy error. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
93. success rate fixed attempts
Expected direction: more successes raise rate.
Error signal: lower rate contradicts fixed denominator. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
94. error rate fixed attempts
Expected direction: more errors raise error rate.
Error signal: lower rate signals wrong numerator. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
95. comparison baseline
Expected direction: changing the baseline can reverse relative percentages.
Error signal: name the reference before using direction as a check. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
96. absolute versus relative change
Expected direction: larger absolute increase need not mean larger percentage increase.
Error signal: do not apply monotonic intuition across different denominators. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
97. final sanity check
Expected direction: ask whether the answer moved in the direction the model predicts.
Error signal: if not, inspect sign, reciprocal, denominator, conversion or constraint before re-solving everything. For practice, create two nearby input cases and predict the direction before calculating. Then calculate both. If the numbers contradict the prediction, inspect sign, reciprocal structure, denominator, conversion, graph scale or constraint before accepting the result.
The direction check is not a substitute for proof. It is a fast structural filter. When several variables change at once, state what is being held fixed before using monotonic intuition.
When direction is not enough
Some relationships are non-monotonic, piecewise, saturated or dependent on multiple changing variables. In those cases, do not force an increase/decrease rule. Identify the relevant interval or condition first. A direction check is valuable only when the model actually predicts a direction.
Links
Use the Mathematics Hub, Vol 0074 Invariant Hunting, Vol 0072 Verification Asymmetry, the Examination Craft hub and the PSLE Learner’s Guide.
Official-source discipline
For current K210 requirements, use the official SEAB 2027 G2 syllabus directory and linked Mathematics syllabus. If SEAB updates the syllabus, the current official document takes priority.
Final rule
Before trusting a calculation, ask whether the answer moved in the direction the relationship predicts. Wrong direction is often a faster clue than wrong arithmetic. Use it to locate sign, reciprocal, denominator, scale and constraint errors before spending time on a full re-solve.
