Singapore School Mathematics Operating Manual · Chapter 15
A statement can be correct in one place and false somewhere else.
A graph may be increasing near x = 2 but decreasing later. A function may have a local maximum that is not the largest value on its whole domain. A pattern observed across several terms may fail farther out. A line of best fit may describe the middle of a data range reasonably well but become unreliable when extrapolated far beyond the observed values. A numerical approximation may be accurate close to one point and poor elsewhere.
This chapter develops the distinction between local and global conclusions. Local reasoning concerns behaviour near a point, inside a small region or under a restricted set of cases. Global reasoning concerns the entire stated domain, full interval, complete population of cases or whole mathematical object.
The central discipline is to match the reach of the conclusion to the reach of the evidence.
Local versus global · Graphs · Extrema · Data and extrapolation · Approximation · Practice · Worked answers
1. A local statement answers “what happens here?”
Suppose f(x) = x².
Near x = 2, the function is increasing. If x moves from 1.9 to 2.0 to 2.1, the output rises.
But it would be false to conclude from this local observation that x² is increasing over all real numbers. For negative x, the function decreases as x moves toward zero.
The correct global statement is that x² decreases on (−∞, 0] and increases on [0, ∞).
A local observation can be perfectly correct without supporting a whole-domain claim.
2. Domain language controls the reach of a conclusion
“f is increasing at x = 3” is informal language about nearby behaviour.
“f is increasing on 1 < x < 5” is an interval statement.
“f is increasing for all real x” is global over ℝ.
These claims require different evidence.
A student should always ask: over which set of inputs is this statement intended to hold?
3. A theorem can be global even when demonstrated with one diagram
A geometry textbook may draw one triangle to illustrate that its interior angles sum to 180° in Euclidean plane geometry.
The diagram is one local representation. The theorem is a universal statement about every triangle in the stated setting.
The proof, not the picture, provides the global reach.
This explains why measuring three angles from one drawing cannot establish a theorem, even if the measurements happen to sum to 180°.
4. Examples are local evidence unless a general argument connects them
Checking a formula for n = 1, 2, 3 and 4 confirms those four cases.
It does not prove the formula for every positive integer.
A proof by induction, algebraic identity or another general method can supply the missing bridge from finite examples to a global result.
The difference is not the number of examples. It is whether the reasoning covers every admissible case.
5. A graph can change behaviour across its domain
Consider f(x) = x³ − 3x.
The derivative is f′(x) = 3x² − 3 = 3(x − 1)(x + 1).
The derivative is positive for x < −1, negative for −1 < x < 1, and positive for x > 1.
Thus the function increases, then decreases, then increases again.
No single local trend describes the whole graph.
6. A tangent is a local linear description
For a differentiable curve, the tangent at one point captures the curve’s local direction there.
For y = x² at x = 2, the tangent gradient is 4. The tangent line is y − 4 = 4(x − 2), or y = 4x − 4.
Near x = 2, the tangent can approximate the curve well.
Far away, it does not reproduce the quadratic globally. At x = 10, the tangent gives 36 while the curve gives 100.
A local linear approximation should not be promoted into a global model without justification.
7. A derivative value is local information
f′(3) = 5 tells us the instantaneous rate of change at x = 3.
It does not imply the gradient is 5 everywhere.
Only if the derivative is identically 5 over an interval can we conclude the function has constant gradient there.
This distinction separates tangent information from straight-line structure.
8. Local maxima and global maxima are different objects
A local maximum is greater than nearby values.
A global maximum is greater than or equal to every value in the stated domain.
For f(x) = x³ − 3x on −2 ≤ x ≤ 2, x = −1 is a local maximum with value 2.
But the endpoint x = 2 also gives value 2, so the global maximum is 2 attained at two points.
A local classification alone does not settle the global optimisation problem.
9. Closed-interval optimisation needs endpoints
For a differentiable function on a closed interval, global extrema can occur at stationary points or endpoints.
Consider f(x) = x² − 4x + 7 on 0 ≤ x ≤ 5.
The stationary point is x = 2, giving f(2) = 3.
Endpoint values are f(0) = 7 and f(5) = 12.
The global minimum is 3 and the global maximum is 12.
Checking only stationary points would miss the global maximum.
10. A local minimum may not exist as a global minimum on an open domain
Take f(x) = x on the open interval 0 < x < 1.
The function is globally increasing, but it has no minimum value because the lower endpoint 0 is excluded.
Values can approach 0 as closely as desired without attaining it.
The infimum is 0, but there is no minimum.
This is an advanced vocabulary distinction, but the operating-manual lesson is straightforward: endpoint inclusion changes global claims.
11. A repeated root is local contact; an intersection count is global over the pair of graphs
When a line is tangent to a parabola, the two graphs touch at a point with matching local gradient.
Algebraically, substitution often produces a quadratic with a repeated root.
The repeated root describes the local contact point.
The statement “these graphs have exactly one real intersection” is global across the complete pair of graphs and requires ruling out any other intersections.
The discriminant can provide that whole-equation classification.
12. Symmetry can turn a local calculation into a global conclusion—but only when justified
If a graph is known to be symmetric about the y-axis, behaviour on x > 0 may determine corresponding behaviour on x < 0.
For f(x) = x², the identity f(−x) = f(x) establishes even symmetry.
That global structural fact allows one side to inform the other.
Visual symmetry in an approximate sketch is not enough. The symmetry must be given, derived or encoded by the function.
13. Local correctness in algebra does not guarantee global validity of an identity
Suppose two expressions agree at x = 1 and x = 2.
That does not prove they are identical functions.
For example, f(x) = x and g(x) = x + (x − 1)(x − 2). They agree at 1 and 2 but differ elsewhere.
An identity must be established across the entire stated domain, usually through valid algebraic transformation or another general argument.
14. A numerical check is local evidence
Substituting x = 5 into an identity candidate can reveal an error if the two sides disagree.
If they agree, the test only confirms one input.
This makes numerical substitution an excellent falsification tool but a weak proof of universal equality.
One failed test can disprove an identity. Many successful tests still do not replace a proof.
15. Interpolation stays inside the observed range
Suppose data are collected for x-values from 10 to 50.
Using a fitted trend to estimate at x = 30 is interpolation because the target lies inside the observed range.
Estimating at x = 100 is extrapolation.
Extrapolation can be much less reliable because the relationship observed over 10 to 50 may not continue unchanged beyond that region.
The arithmetic formula may still produce a number; the evidence supporting that number has changed.
16. A line of best fit is a global summary over a data cloud, not a law for every point
A line of best fit summarises a trend.
Individual points can lie above or below it.
The line does not claim every observation must satisfy the line equation exactly.
Nor does a strong trend prove causation.
Statistical summaries describe data at a different logical level from exact algebraic identities.
17. A sample conclusion is local to the evidence unless inference justifies a wider population claim
A sample mean describes the observed sample.
Using it to estimate a population mean introduces inferential assumptions and uncertainty.
The sample statistic is not automatically the population parameter.
This is an extension beyond many school stages, but the principle is universal: moving from observed cases to a wider universe requires a justified bridge.
18. A frequency observed in a short run is not a global probability law
Suppose a fair coin is tossed 10 times and produces 7 heads.
The observed relative frequency is 0.7.
This does not change the model probability of heads from 0.5 to 0.7.
Local sample behaviour can deviate from long-run model probability.
As sample size grows, relative frequency may stabilise under appropriate assumptions, but any particular finite run remains evidence rather than the definition of the probability.
19. Approximations have regions of usefulness
For small x measured in radians, sin x is close to x.
Near zero, this can be a powerful approximation.
At x = 0.1, sin x ≈ 0.09983, close to 0.1.
At x = 1, sin 1 ≈ 0.84147, much farther from 1.
The approximation is local. Its quality depends on how far the input moves from the expansion point.
20. Rounding error can be locally harmless and globally important
A small rounding error in one isolated final answer may have little practical effect.
The same rounding performed repeatedly inside a long iterative process can accumulate or propagate.
Likewise, a rounded intermediate value can be harmless if the final conclusion is far from a threshold but decisive if the result lies near the boundary.
Error impact depends on where and how the value is reused.
21. A local repair in teaching may not produce global transfer
A student learns to solve one exact form of simultaneous equation.
That is local success.
To know whether the learning is global enough for examination use, change coefficients, wording, representation and topic context.
If performance survives the changes, the method is transferring.
This is why mixed practice and delayed retrieval matter after a repair.
22. A counterexample can destroy a global claim
Suppose someone claims, “Every quadratic has two real roots.”
The equation x² + 1 = 0 has no real roots.
That one case disproves the universal statement.
Global claims are logically vulnerable because every admissible case must satisfy them.
Existential claims have the opposite structure: one successful case proves existence.
23. A local theorem may require a local condition
Some statements are true only under nearby or interval-specific conditions.
If f′(x) > 0 on an interval, then f is increasing on that interval.
The condition does not automatically extend beyond the interval where it has been established.
When the derivative changes sign elsewhere, the monotonicity conclusion changes too.
24. Piecewise functions are globally assembled from local rules
A piecewise function may use one formula for x < 0 and another for x ≥ 0.
Each formula has local scope.
The full function is the global object created by combining the branches.
A global property such as continuity at x = 0 requires comparing the local behaviours from both sides with the actual value at the boundary.
25. Local constraints can create a global feasible region
Suppose x ≥ 0, y ≥ 0 and x + y ≤ 10.
Each inequality defines a half-plane.
The feasible region is their intersection.
A candidate point may satisfy two constraints and still fail globally because it violates the third.
This connects directly to Constraints, Feasible Sets and Admissible Solutions.
26. Global conclusions often need multiple candidate classes
To find a global maximum on a closed interval, compare stationary points and endpoints.
To find every trigonometric solution in a stated interval, include every branch and period that lies in the domain.
To classify all integer solutions, inspect every relevant remainder or parity class.
A global answer is complete only when no admissible class has been omitted.
27. “Always”, “never”, “for all” and “every” are global warning words
These words signal high logical burden.
“Sometimes”, “can”, “there exists” and “for some” signal weaker existence claims.
When reading a question, underline the word that determines the scope.
Many reasoning errors disappear when the learner notices that the task asks “must” rather than “can”.
28. A global result can contain local structure
The global description of f(x) = |x| is piecewise:
f(x) = −x for x < 0 and f(x) = x for x ≥ 0.
The full statement combines two local rules into one global function.
Mathematical understanding often means knowing both levels: how the object behaves in each region and how the regions fit together.
29. A local observation can guide discovery without carrying proof burden
Sketching a graph may reveal where a maximum seems to occur.
Testing small cases may suggest a number pattern.
Measuring a diagram may suggest a hidden symmetry.
These local observations are valuable because they generate conjectures and methods.
The final conclusion must then be supported at the scope the question requires.
30. A practical scope audit
Before accepting a conclusion, ask:
What domain did the evidence cover? Is my conclusion about one point, an interval, a finite set or every admissible case? Did I check endpoints? Did I include all branches? Am I interpolating or extrapolating? Did I use examples as illustration or as proof?
This audit is particularly useful in graph, optimisation, pattern, proof and data questions.
31. Independent practice
1. For f(x) = x², state where the function is decreasing and where it is increasing.
2. Explain why the tangent to y = x² at x = 1 is not a global replacement for the curve.
3. Find the global maximum and minimum of f(x) = x² − 2x on 0 ≤ x ≤ 4.
4. A formula agrees with another formula at x = 1 and x = 2. Does that prove the formulas are identical?
5. Give a counterexample to “every quadratic has two real roots”.
6. Data are observed for 0 ≤ x ≤ 20. Is a prediction at x = 12 interpolation or extrapolation? What about x = 40?
7. A coin shows heads 8 times in 10 tosses. Does that prove P(heads) = 0.8?
8. Explain why checking only stationary points can miss a global maximum on a closed interval.
9. If f′(x) > 0 for 2 < x < 5, what can be concluded on that interval?
10. Give one local statement and one global statement about y = |x|.
11. Explain why one counterexample is sufficient to disprove a universal claim.
12. A student masters one worksheet format. What changed practice would test whether learning has transferred more globally?
32. Worked answers
1. f decreases for x < 0 and increases for x > 0, with minimum at x = 0. On the closed half-interval language, decreasing on (−∞, 0] and increasing on [0, ∞).
2. The tangent matches the curve’s value and gradient at x = 1, but the quadratic curves away from the line elsewhere. It is a local linear description.
3. Complete the square: f(x) = (x − 1)² − 1. Minimum = −1 at x = 1. Endpoints give f(0) = 0 and f(4) = 8, so global maximum = 8 at x = 4.
4. No. Agreement at finitely many points is local evidence only. Different functions can share those values.
5. x² + 1 = 0 has no real roots.
6. x = 12 is interpolation. x = 40 is extrapolation.
7. No. The observed relative frequency is 0.8 for that sample. It does not by itself determine the underlying probability.
8. A global maximum may occur at an endpoint even when no stationary point is present there.
9. f is increasing on 2 < x < 5.
10. Local: for x > 0 near x = 3, y = |x| has gradient 1. Global: |x| ≥ 0 for every real x.
11. A universal claim says every admissible case has the property. One valid case without the property directly contradicts the claim.
12. Change numbers, wording, representation, delay, topic mixture or the order of information, then require independent method selection without a chapter cue.
33. Continue through the School Mathematics manual
Use Notation, Scope and Quantifiers to read how far a statement reaches, Necessary and Sufficient Conditions to control logical direction, and Constraints, Feasible Sets and Admissible Solutions when a global answer must satisfy several local conditions at once.
Return to the BTT Mathematics Hub for Batch 04.