Singapore School Mathematics Operating Manual · Chapter 12
A student solves a problem correctly. Then one number changes.
Does the answer change a little, a lot, or not at all? Does the method still work? Does a whole-number decision jump? Does the graph shift, stretch or reverse? Does the conclusion remain true?
These questions belong to sensitivity: how strongly an output responds when an input, parameter or assumption changes. Stability asks whether a conclusion remains reliable under small changes. What-if analysis deliberately changes one part of a model to learn which relationships actually control the result.
This chapter extends ordinary school problem solving into a deeper form of understanding. Instead of treating every question as a one-time calculation, the learner asks: what changes, what stays the same, and why?
Change one thing · Proportional sensitivity · Thresholds · Graph parameters · Stability · Practice · Worked answers
1. Change one input and hold the rest fixed
Suppose a rectangle has length 8 cm and width 5 cm. Its area is 40 cm².
If only the length changes to 9 cm while width remains 5 cm, the area becomes 45 cm². The 1 cm increase in length produces a 5 cm² increase in area because each extra centimetre of length adds a strip 1 cm by 5 cm.
If instead width changes to 6 cm while length remains 8 cm, area becomes 48 cm².
Changing one input at a time makes the dependency visible. Changing several inputs simultaneously may still be useful, but it becomes harder to identify which change caused which effect.
2. Sensitivity is not the same as importance in every context
A parameter can have a large mathematical effect but be tightly controlled in reality. Another can have a smaller local effect but vary widely.
In school Mathematics, sensitivity usually begins more simply: compare how the formula responds to a controlled change.
For y = 3x + 2, increasing x by 1 always increases y by 3. The sensitivity of y to x is represented by the constant gradient 3.
For y = x², increasing x by 1 does not produce a constant output change. From x = 1 to 2, y increases by 3. From 10 to 11, y increases by 21.
The same input step can have different effects depending on where the system is.
3. Direct proportion has constant relative structure
If y = 4x, doubling x doubles y. Tripling x triples y.
The ratio y/x remains 4 for every non-zero x. This invariant ratio makes proportional sensitivity predictable.
For an invented pricing model at $6 per item, cost C = 6n. Adding one item always adds $6. Increasing quantity by 10% increases cost by 10%, assuming the unit price and all other conditions remain unchanged.
This proportionality breaks when fixed charges, thresholds or discounts are introduced.
4. Percentage change in a product can compound
Area of a rectangle is A = lw.
If both length and width increase by 10%, the new area is 1.1l × 1.1w = 1.21lw.
The area increases by 21%, not 20%.
The extra 1% comes from the interaction of the two increases.
For a cube, increasing every side length by 10% multiplies volume by 1.1³ = 1.331, a 33.1% increase.
This shows how dimensional structure amplifies sensitivity.
5. Inverse relationships respond in the opposite direction
For fixed distance d, travel time t = d/v.
Doubling speed halves time. Increasing speed does not reduce time by the same absolute amount everywhere.
For a 120 km journey, 60 km/h gives 2 hours. Increasing speed to 80 km/h gives 1.5 hours, saving 0.5 hours. Increasing from 80 to 100 km/h gives 1.2 hours, saving only 0.3 hours.
The same 20 km/h speed increase has a smaller time effect at higher starting speeds.
6. Changing the base can reverse a percentage conclusion
A quantity rises from 80 to 100: a 25% increase relative to 80.
Returning from 100 to 80 is a 20% decrease relative to 100.
The absolute change is 20 in both directions, but the percentage sensitivity depends on the reference base.
This is why percentage comparison should always name its denominator.
7. Small continuous changes can cause large discrete jumps
A container holds at most 10 litres. If the required volume changes from 29.9 L to 30.0 L, three containers remain enough. If it changes from 30.0 L to 30.1 L, four containers are needed.
The input changed by only 0.1 L, but the discrete decision changed by one whole container.
This is threshold sensitivity.
Near a capacity boundary, tiny input changes can matter greatly to the final count. Far from the boundary, the same change may have no effect on the discrete output.
8. Rounding can hide threshold sensitivity
If a volume is reported as 30 L to the nearest litre, the true value could lie below or above 30.
A capacity decision based on 10 L containers may therefore be uncertain: some compatible values need three containers, others four.
The companion chapter Rounded Data, Thresholds and Guaranteed Conclusions develops this formally.
Sensitivity adds a useful interpretation: the model is unstable near the threshold because small compatible changes alter the decision.
9. Stable conclusions survive small changes
Suppose a calculated angle is 40° and the decision is simply whether it is acute. A small change to 39° or 41° does not change the classification.
The conclusion “acute” is stable under these perturbations.
If the angle is 89.9°, a small change could cross 90° and change the classification from acute to obtuse. The conclusion is sensitive near the boundary.
Stability therefore depends on distance from the decision boundary, not merely on the number of decimal places in the answer.
10. Sensitivity can reveal whether extra precision matters
If a measurement is far from a threshold, more decimal places may not change the decision.
If a reported length is about 8.2 cm and the only question is whether it is below 10 cm, very fine precision is unnecessary.
If the question asks whether it is below 8.21 cm, precision may become decisive.
The useful question is not “can I calculate more digits?” but “could the unresolved uncertainty change the conclusion?”
11. In y = mx + c, different parameters control different features
For the straight line y = mx + c, m controls gradient and c controls the vertical intercept.
Changing c while holding m fixed shifts the line vertically without changing its gradient.
Changing m while holding c fixed rotates the family of non-vertical lines around the y-intercept point (0, c).
Parameter sensitivity becomes visual: each symbol controls a distinct geometric feature.
12. Quadratic parameters control position and shape
Consider y = a(x − h)² + k.
The parameter h moves the vertex horizontally. The parameter k moves it vertically. The magnitude |a| controls vertical stretch or compression, while the sign of a controls whether the parabola opens upward or downward.
Changing one parameter at a time makes the role visible.
This is a strong way to learn function structure: not as a collection of separate graphs, but as a controlled family.
13. Sensitivity can identify a parameter from observed change
Suppose a linear relationship increases by 12 whenever x increases by 3.
The gradient is 12/3 = 4.
The response pattern identifies the parameter controlling sensitivity.
In a directly proportional model y = kx, observing one non-zero pair gives k = y/x.
Parameters can therefore be interpreted as response controls, not merely letters to solve for.
14. Sequence parameters control long-run behaviour
For the arithmetic sequence uₙ = a + (n − 1)d, changing a shifts every term by the same amount while leaving common difference d unchanged.
Changing d changes the gap between successive terms and its effect grows with n.
If d increases by 1, the first term is unchanged, the second rises by 1, the third by 2, and the nth by n − 1.
A small parameter change can therefore create a large later difference.
15. Exponential growth is highly sensitive to repeated factors
Compare repeated growth factors 1.02 and 1.03 over many periods.
After one period, the difference is small. After 20 periods, the multipliers are approximately 1.486 and 1.806.
A one-percentage-point difference in the per-period growth rate compounds over time.
This is why exponential systems can be much more sensitive to rate changes than linear intuition suggests.
16. Probability sensitivity can be tested by changing one branch
A bag contains 3 red and 2 blue counters. Probability of red on one uniform draw is 3/5.
Add one red counter. The probability becomes 4/6 = 2/3.
Add one blue counter instead. It becomes 3/6 = 1/2.
The total and favourable counts both matter. Sensitivity depends on which part of the sample space changed.
17. Without-replacement probabilities become path-sensitive
If two counters are drawn without replacement, the probability on the second draw depends on the first outcome.
The system’s state changes after the first draw.
With replacement, the composition resets and the second-draw probabilities remain the same as the first.
One small modelling assumption—replacement or no replacement—changes the dependency structure of the entire probability tree.
18. Model assumptions can be stress-tested
Suppose a school problem models cost as C = 5n, where n is the number of items.
Ask what happens if there is a fixed $20 delivery fee. The model becomes C = 20 + 5n.
Ask what happens if a bulk discount begins after 100 items. The model becomes piecewise.
Ask what happens if unit price itself depends on quantity. The simple proportional model no longer applies.
Changing assumptions reveals which conclusions belonged to the original model and which are more robust.
19. A method can be stable even when intermediate numbers change
Suppose a rectangle problem changes its dimensions from 8 by 5 to 9 by 6. The area method remains multiplication.
The numbers change, but the mathematical structure does not.
This is a useful teaching test: change the surface while preserving the relationship.
If the learner can still select and execute the method, the knowledge is transferring.
20. A method change signals structural sensitivity
Now change the rectangle into a triangle while keeping similar numbers.
The area relationship changes. Multiplying two side lengths directly no longer suffices without the appropriate height or included-angle structure.
The learner must notice that the changed condition is structural, not merely numerical.
What-if questions can therefore test recognition as well as arithmetic.
21. Sensitivity helps explain why some mistakes matter more than others
In a long calculation, an early parameter may be reused many times. A small error there can propagate through multiple later steps.
Another quantity may appear only once near the end.
The first variable has greater downstream influence in that particular dependency graph.
This connects to the Linked Parts and Result Handoffs chapter.
22. Relative sensitivity matters when scales differ
A change of 1 unit is large when a quantity is near 2 and small when it is near 10,000.
Relative or percentage change accounts for scale.
Increasing 2 to 3 is a 50% rise. Increasing 10,000 to 10,001 is a 0.01% rise.
Absolute and relative sensitivity answer different questions. Use the one appropriate to the context.
23. Local sensitivity is not always global behaviour
Near x = 0, y = x² changes slowly for small changes in x. Farther away, the same change in x creates a larger output change.
A local observation should not automatically be extended across the entire domain.
Similarly, a graph that looks almost linear over a small interval may be clearly curved over a larger interval.
Sensitivity should be interpreted over the region relevant to the problem.
24. A parameter can change feasibility, not just the answer
Consider x + y = 10 with x and y positive integers. Many solutions exist.
Add the condition x − y = k.
For k = 2, the solution is x = 6, y = 4.
For k = 3, the real-number solution would be x = 6.5, y = 3.5, which is not admissible in the positive-integer model.
Changing one parameter can therefore move a problem from feasible to infeasible.
25. Sensitivity can locate critical values
Suppose a quadratic x² − 6x + k = 0 changes from two real roots to no real roots as k varies.
The discriminant is 36 − 4k.
Two roots occur when k < 9, one repeated root at k = 9, and no real roots for k > 9.
The critical parameter value 9 is the boundary at which qualitative behaviour changes.
This connects sensitivity directly to the Boundary Cases chapter.
26. A what-if table can make relationships visible
For a formula such as A = πr², construct a small table:
r = 1 gives A = π. r = 2 gives 4π. r = 3 gives 9π. r = 4 gives 16π.
The pattern reveals that area responds to the square of radius, not directly to radius.
Doubling radius multiplies area by four.
Tables are useful when the symbolic relationship is understood only weakly. They show the response pattern concretely.
27. Sensitivity can guide checking
If a student changes one input slightly and the answer changes in a wildly unexpected way, investigate.
Sometimes the jump is real because a threshold was crossed. Sometimes it reveals an unstable formula, a division by a very small number, a wrong unit, or a calculation error.
Reasonableness checking should therefore consider not only the answer itself but how the answer behaves under nearby inputs.
A mathematical model that behaves implausibly under tiny harmless changes deserves inspection.
28. Do not confuse sensitivity with uncertainty
Sensitivity asks what the output would do if an input changed.
Uncertainty asks what input values are actually plausible or known.
A model may be highly sensitive to a parameter that is known exactly, or weakly sensitive to one that is highly uncertain.
In school problems, keeping these ideas separate prevents a common mistake: assuming that because an output is sensitive, the input must be uncertain.
29. A practical what-if routine
After solving an important problem, choose one input or assumption and change it deliberately.
Predict the direction of change before recalculating. Then calculate. Compare the result with the prediction. Finally ask whether the method remained valid.
This creates four layers of learning: structural prediction, execution, verification and transfer.
The strongest question is often not “what is the new answer?” but “why did the answer change in that particular way?”
30. Independent practice
1. A rectangle is 12 cm by 5 cm. Increase only the length by 10%. Find the new area and percentage area increase.
2. Increase both dimensions of the same rectangle by 10%. Find the percentage area increase.
3. A fixed 180 km journey is travelled at 60 km/h and then at 90 km/h. Compare the travel times.
4. A bus holds 40 passengers. Compare the required number of buses for 79, 80 and 81 passengers.
5. For y = 2x + c, explain what changes when c increases by 5.
6. For y = a(x − 3)² + 1, explain the effect of changing a from 1 to −2.
7. An arithmetic sequence has a = 4 and d = 3. Find the 20th term. Then increase d to 4 and compare the new 20th term.
8. A bag contains 5 red and 5 blue counters. Find P(red). Then add one red counter and find the new probability.
9. Determine the critical k for which x² − 8x + k = 0 has a repeated root.
10. Explain the difference between saying a conclusion is sensitive to an input and saying the input is uncertain.
11. A continuous optimisation suggests x = 9.4, but x must be an integer. What nearby values should be compared?
12. A measured quantity is far below a threshold. Explain why additional decimal places may not change the decision.
31. Worked answers
1. Original area = 60 cm². New length = 13.2 cm. New area = 13.2 × 5 = 66 cm². Increase = 6 cm², which is 10% of 60. With one dimension fixed, area changes in direct proportion to the other.
2. New dimensions are 13.2 cm and 5.5 cm. Area = 72.6 cm². Increase = 12.6 cm². Percentage increase = 12.6/60 × 100% = 21%.
3. At 60 km/h, time = 180/60 = 3 h. At 90 km/h, time = 2 h. A 50% speed increase reduces time by one-third, not by 50%.
4. 79 passengers require 2 buses. 80 require 2. 81 require 3. The discrete decision jumps immediately after the capacity threshold 80.
5. Increasing c by 5 shifts every y-value upward by 5. The gradient 2 remains unchanged.
6. The vertex remains at (3, 1). The parabola changes from opening upward to opening downward and becomes vertically stretched by factor 2 relative to the a = 1 case.
7. u₂₀ = 4 + 19(3) = 61. With d = 4, u₂₀ = 4 + 19(4) = 80. Increasing d by 1 increases the 20th term by 19.
8. Original P(red) = 5/10 = 1/2. After adding one red counter, probability = 6/11.
9. Discriminant Δ = 64 − 4k. Repeated root requires Δ = 0, so k = 16.
10. Sensitivity describes how strongly the output changes if the input changes. Uncertainty describes how well the input itself is known. They are independent concepts.
11. Compare nearby admissible integers 9 and 10, together with all original constraints. Do not round automatically without evaluating the actual objective.
12. If the full plausible range remains on the same side of the decision boundary, extra precision cannot change the yes/no conclusion. More digits may refine the value without changing the decision.
32. Continue through the School Mathematics operating manual
Use Invariants, Conservation and What Must Stay the Same for the complementary question of what survives change. Use Backward Reasoning and Target Decomposition to identify which inputs actually control a target, and Given, Deduced, Assumed and Conjectured Information to track whether a changed condition is part of the original problem or a deliberate what-if assumption.
Return to the BTT Mathematics Hub for Batch 03.