Singapore School Mathematics Operating Manual · Chapter 24
Many Mathematics questions can be answered without calculating everything.
If one positive number is larger than another, squaring preserves the order. If a quantity is inversely proportional to another, increasing one decreases the other. If a function is strictly increasing, comparing inputs is enough to compare outputs. If a probability is bounded above by 0.2, there is no need to compute an exact decimal to know it is less than one half.
This chapter develops monotonicity, ordering, comparison and bound-based decision-making. These ideas reduce unnecessary calculation and provide powerful independent checks.
Ordering · Monotonicity · Order-preserving transformations · Bounds · Decision without full calculation · Practice · Worked answers
1. Comparison is a mathematical operation in its own right
When asked which value is larger, the target is an ordering—not necessarily two exact decimal expansions.
For example, compare 7/12 and 5/8.
Cross-multiply positive denominators: 7×8 = 56 and 5×12 = 60, so 7/12 < 5/8.
No decimal conversion is required.
2. A common benchmark can simplify comparison
49/100 is less than 1/2.
51/100 is greater than 1/2.
Therefore 49/100 < 51/100 immediately.
Benchmarks such as 0, 1/2, 1 and powers of ten reduce cognitive load.
3. Difference comparison is often easier than evaluating both quantities
To compare A and B, inspect A − B.
If A − B > 0, then A > B.
If A − B = 0, they are equal.
If A − B < 0, then A < B.
This converts comparison into sign analysis.
4. Ratio comparison is useful for positive quantities
If A and B are positive, compare A/B with 1.
A/B > 1 means A > B.
This is especially useful when A and B are products or scaled quantities.
5. An increasing function preserves input order
If f is strictly increasing and a < b, then f(a) < f(b).
For f(x) = 3x + 2, larger x always gives larger y.
Therefore if x₁ < x₂, no separate output calculation is needed to know f(x₁) < f(x₂).
6. A decreasing function reverses input order
If f is strictly decreasing and a < b, then f(a) > f(b).
For positive x, f(x) = 1/x is decreasing.
Thus 1/5 < 1/3 because 5 > 3.
The reciprocal reverses order on positive numbers.
7. Monotonicity can replace repeated substitution
Suppose y = 2x + 7 and x lies between 3 and 5.
Because the function is increasing, y lies between 13 and 17.
Only endpoint evaluation is needed.
8. Monotonicity can depend on domain
f(x) = x² is decreasing on x < 0 and increasing on x > 0.
It is not globally increasing over all real numbers.
Any ordering conclusion must respect the interval where monotonicity has been established.
9. Derivatives provide a monotonicity test
Where f′(x) > 0, f is increasing locally over that interval.
Where f′(x) < 0, f is decreasing.
For f(x) = x² − 4x, f′(x) = 2x − 4.
Thus f decreases for x < 2 and increases for x > 2.
10. Monotonicity supports uniqueness
If f is strictly increasing, the equation f(x) = c can have at most one solution.
This is because two distinct inputs cannot produce the same output.
If one solution is found, uniqueness follows.
11. Adding the same number preserves order
If a < b, then a + c < b + c for every real c.
This allows inequalities to be translated left or right without changing direction.
12. Multiplying by a positive number preserves order
If a < b and c > 0, then ac < bc.
This is why multiplying an inequality by 3 leaves the inequality sign unchanged.
13. Multiplying by a negative number reverses order
If a < b and c < 0, then ac > bc.
For example, 2 < 5 but −2 > −5.
The sign reversal is not a memorised exception; it reflects the orientation reversal of the number line.
14. Squaring preserves order only under conditions
For non-negative numbers, a < b implies a² < b².
But over all reals, −5 < −2 while 25 > 4.
Squaring is not globally increasing on ℝ.
Order-preserving rules always need domain awareness.
15. Square root preserves order on non-negative inputs
If 0 ≤ a < b, then √a < √b.
Therefore comparing √17 and √20 is immediate because 17 < 20.
16. Logarithms preserve or reverse order depending on base
For base b > 1, log_b x is increasing.
For 0 < b < 1, it is decreasing.
This is an extension beyond some school levels, but it illustrates how parameter conditions control ordering.
17. Exponential functions behave similarly
For b > 1, b^x is increasing.
For 0 < b < 1, b^x is decreasing.
Thus 2^5 > 2^3 without direct calculation.
18. Bounds can settle a comparison before exact evaluation
If A < 10 and B > 12, then A < B.
No exact values are needed.
This is especially powerful when A and B are complicated expressions but easy to bound.
19. Upper and lower bounds support guaranteed decisions
If a measured quantity is known to lie between 4.8 and 5.0, and a threshold is 6, the quantity is guaranteed below the threshold.
If the interval were 5.8 to 6.2, the decision would be uncertain.
The complete interval, not the central estimate, controls the guarantee.
20. Bounding products requires sign awareness
If 2 < x < 3 and 4 < y < 5 with both positive, then 8 < xy < 15.
With negative factors, endpoint pairing becomes more subtle.
Bounds should respect sign and monotonicity of the operation.
21. Triangle inequality is a comparison bound
For a non-degenerate triangle with sides a, b and c, each side is less than the sum of the other two.
This bound rules out impossible configurations without calculating angles.
22. Probability bounds can settle qualitative questions
If P(A) ≤ 0.1, then A is less likely than an event with probability 0.4.
If P(A) ≥ 0.8, it is more likely than not.
Exact probability may be unnecessary if the task is comparative.
23. Decision questions often require only which side of a threshold the result lies
Suppose a requirement is x ≤ 100.
If you can prove x < 90, there is no need to compute x exactly.
If you can only show 80 < x < 120, more information is needed.
Decision sufficiency is weaker than exact evaluation but can be enough for the task.
24. Ranking can be easier than scoring
Suppose three positive values are proportional to 2, 5 and 7.
Their ranking is immediate even if the common multiplier is unknown.
A shared positive scale factor preserves order.
25. Common transformations can simplify ranking
To compare positive fractions with a common numerator, the smaller denominator gives the larger fraction.
Thus 5/7 > 5/9.
No decimal calculation is needed.
26. Common denominators simplify fraction ordering
Compare 3/8 and 5/12.
Common denominator 24 gives 9/24 and 10/24, so 3/8 < 5/12.
27. Standard form simplifies order-of-magnitude ranking
Compare 4.2×10⁶ and 9.8×10⁵.
The first is larger because 10⁶ scale exceeds 10⁵ scale.
Coefficient comparison is only needed when powers of ten match.
28. Monotonicity can identify parameter thresholds
Suppose C(n) = 20 + 5n and we need C(n) ≤ 100.
Because C increases with n, once the threshold is crossed it remains crossed.
Solving 20 + 5n ≤ 100 gives n ≤ 16.
The monotonic structure makes the feasible set an interval.
29. Binary-search thinking uses monotonicity
If a yes/no condition changes only once as an ordered input increases, test the middle.
If the middle already fails, search lower. If it passes, search higher.
This is the principle behind binary search and many threshold-finding strategies.
It is enrichment, but the school-level idea is valuable: monotone decisions let us discard large regions safely.
30. Optimisation can use monotonicity to avoid calculus
If an objective is strictly increasing over the entire feasible interval, its maximum occurs at the upper endpoint and minimum at the lower endpoint.
No stationary-point calculation is needed.
The structure already decides the location of the optimum.
31. Comparison can reveal impossible calculator outputs
If x is known to be between 2 and 3, then x² must lie between 4 and 9.
A calculator output of 14 for x² is impossible.
Bounds can therefore verify numerical work.
32. Ordering helps with roots
Since 3² = 9 and 4² = 16, √10 lies between 3 and 4.
Since 3.1² = 9.61 and 3.2² = 10.24, √10 lies between 3.1 and 3.2.
Repeated comparison narrows an approximation without needing a root key.
33. Sign charts are ordering tools
For (x − 2)(x + 1), critical points −1 and 2 divide the number line into intervals.
Testing signs shows the product is positive for x < −1 and x > 2, and negative for −1 < x < 2.
The sign chart converts algebra into ordered interval reasoning.
34. Absolute value is distance ordering
|x| compares distance from zero.
|x| < 3 means x lies within 3 units of zero, so −3 < x < 3.
|x| > 3 means x lies farther than 3 units from zero, giving two outer intervals.
35. Correlation does not create a simple monotone rule for individual cases
A positive correlation means larger x values tend to be associated with larger y values in the data.
It does not mean every pair respects strict ordering.
Statistical trend and deterministic monotonicity are different types of claim.
36. A graph can make monotonicity visible but algebra should justify exact claims
A sketch may suggest increasing or decreasing behaviour.
For formal interval conclusions, use known function structure, derivative sign or other valid reasoning.
Visual evidence is a guide; exact claims need exact support.
37. A practical ordering audit
Ask:
Do I need exact values or only a comparison? Is there a common benchmark? Can I compare a difference or ratio? Is the relevant function increasing or decreasing on this domain? Which transformations preserve order? Can bounds settle the threshold? Would endpoint evaluation be enough?
38. Independent practice
1. Compare 7/10 and 11/15 without decimal conversion.
2. If 2 < x < 5, bound 3x + 1.
3. If 4 < a < 9, bound √a.
4. Explain why 1/7 < 1/5.
5. Solve −3x < 12.
6. State where f(x) = x² − 6x is increasing and decreasing.
7. A quantity is guaranteed below 8.5. Is it automatically below 9?
8. Compare 3.1×10⁷ and 8.9×10⁶.
9. If a strictly increasing function satisfies f(2) = 7, can f(5) be 6?
10. Find all n such that 4n + 3 ≤ 35.
11. Explain why x² is not globally increasing over all real x.
12. Give one example where a bound is enough and an exact value is unnecessary.
39. Worked answers
1. Cross-multiply: 7×15 = 105 and 11×10 = 110, so 7/10 < 11/15.
2. Multiply by positive 3: 6 < 3x < 15. Add 1: 7 < 3x + 1 < 16.
3. Square root is increasing on non-negative inputs, so 2 < √a < 3.
4. Reciprocal is decreasing on positive numbers; larger denominator gives smaller reciprocal.
5. Divide by −3 and reverse the inequality: x > −4.
6. f′(x) = 2x − 6. Decreasing for x < 3 and increasing for x > 3.
7. Yes. If x < 8.5, then x < 9.
8. 3.1×10⁷ is larger.
9. No. Since 5 > 2 and f is strictly increasing, f(5) must exceed 7.
10. 4n ≤ 32, so n ≤ 8.
11. It decreases on negative x approaching zero and increases on positive x.
12. If a measurement is proven below 5 and the requirement is only “below 10?”, the bound already settles the decision.
40. Batch 06 as one control layer
Assumptions and Model Validity asks whether the mathematical relationship belongs to the situation.
Scale, Order of Magnitude and Estimation asks whether the size of the result is plausible.
Equivalence and Canonical Forms asks which representation makes the object easiest to compare or use.
This chapter asks whether ordering alone can settle the mathematical decision before full evaluation.
Return to the BTT Mathematics Hub for Batch 06.