Singapore School Mathematics Operating Manual · Chapter 10
Mathematics allows enormous freedom to change form. We regroup numbers, rearrange equations, redraw diagrams, scale shapes, substitute variables, transform graphs and rewrite expressions. Yet every valid transformation has something it must preserve.
That preserved quantity or property is an invariant. Sometimes it is obvious: adding the same amount to both sides of an equation preserves equality. Sometimes it is hidden: rotating a rigid shape changes its orientation but not its side lengths or angles. Sometimes it becomes a powerful solving tool: if a total is conserved, the missing part can be recovered from what remains.
This chapter treats invariants as an operating principle across school Mathematics. The word itself need not appear in every syllabus or examination question. The underlying habit is more important: when the surface changes, ask what is not allowed to change.
Equality and balance · Conserved totals · Parity and remainder · Geometry invariants · Ratio and scale · Practice · Worked answers
1. Equality is an invariant under balanced operations
Consider 4x + 7 = 31. Subtracting 7 from both sides gives 4x = 24. Dividing both sides by 4 gives x = 6.
The expressions change, but equality is preserved. Each operation is applied symmetrically to both sides.
This is the deeper reason behind “whatever you do to one side, do to the other”. The rule is not a classroom ritual. It is an invariant requirement: the two sides must continue to represent the same quantity.
A student who subtracts 7 from only one side has not merely broken a procedure. The new equation no longer represents the same relationship.
2. Place-value regrouping preserves quantity
In 403 − 178, a learner may regroup 403 as 3 hundreds, 9 tens and 13 ones.
The written form changes, but the total quantity remains 403. One hundred has been exchanged for ten tens; one ten has been exchanged for ten ones.
Regrouping is therefore an invariant-preserving transformation. It changes representation while preserving value.
This Primary idea grows directly into algebra, where expressions are expanded, factorised or rearranged while their value remains unchanged for all allowed inputs.
3. Conserved totals solve missing-part problems
A class has 36 pupils. If 14 are absent, then 22 are present because absent + present = total.
The total is conserved across the partition. The categories may change, but every pupil belongs to exactly one of the two groups under the model.
This same structure appears in money spent and money remaining, probability of an event and its complement, angle parts summing to a whole, and frequencies divided across categories.
A useful solving move is to identify the conserved whole before chasing individual pieces.
4. Conservation can expose impossible data
If a survey of 80 pupils reports 47 choosing option A and 39 choosing option B, with each pupil required to choose exactly one option, the counts sum to 86. The data cannot all be correct under the stated model.
The total check is cheap and powerful.
Similarly, percentages representing a complete partition should total 100%, subject to possible rounding. Probabilities of mutually exclusive exhaustive outcomes should total 1.
Conservation checks often detect errors without repeating the original calculation.
5. The sum of probabilities is a conservation law
For a fair six-sided die, the six single-outcome probabilities each equal 1/6 and sum to 1.
If a probability table assigns values 0.2, 0.3 and 0.6 to three exhaustive outcomes, the total is 1.1, which is impossible.
When one probability is missing, the total of 1 can recover it. If two known probabilities are 0.25 and 0.40, the remaining exhaustive probability is 0.35.
This is the complement principle in another form: P(not A) = 1 − P(A).
6. Parity can remain invariant under some moves
Adding an even number to an integer preserves its parity. If n is even, n + 4 is even. If n is odd, n + 4 is odd.
Adding an odd number flips parity. Multiplying by an even number forces the product to be even.
These rules allow us to track parity without knowing exact values.
For example, if a counter starts on an even-numbered square and each move changes the position by an even number, it can never reach an odd-numbered square. Exact movement may be complicated, but parity remains invariant.
7. Remainders can provide stronger invariants than parity
Parity tracks remainder modulo 2. We can generalise to other moduli.
Suppose an integer is repeatedly increased by 3. Its remainder when divided by 3 does not change. Starting from a number congruent to 1 modulo 3 means every later number in the process remains congruent to 1 modulo 3.
If the target is divisible by 3, such a process can never reach it.
This kind of invariant can solve puzzles and impossibility questions much faster than simulating every move.
8. Digit-sum checks are modular invariants
A base-10 integer has the same remainder modulo 9 as the sum of its digits because 10 is congruent to 1 modulo 9.
Thus 572 has digit sum 14, then 1 + 4 = 5, so 572 leaves remainder 5 modulo 9.
This provides a quick arithmetic check. If 572 × 4 is claimed to be 2280, compare remainders: 572 has remainder 5, times 4 gives remainder 20, hence 2 modulo 9. But 2280 has digit sum 12, remainder 3. The claim must be wrong.
The check can detect an error, though passing it does not prove the calculation correct.
9. Rigid transformations preserve geometric structure
Translations, rotations and reflections preserve distances and angle measures. A shape may move or change orientation without changing its size.
Thus congruence can be understood through invariants: corresponding side lengths and angle measures survive a rigid motion.
Reflection reverses orientation while preserving lengths and angles. Rotation changes direction while preserving distance from the centre of rotation.
These preserved properties explain why geometric transformations are useful for proving equality of lengths or angles.
10. Similarity preserves shape but not size
In similar figures, corresponding angles are preserved and corresponding lengths scale by a common factor.
Length itself is not invariant, but ratios of corresponding lengths are controlled by the scale factor.
If a triangle is enlarged by factor 3, every corresponding length triples, areas multiply by 9, and volumes of similar solids multiply by 27.
The invariant is proportional structure, not absolute measurement.
11. Ratios can remain invariant under common scaling
If red:blue = 2:5, then multiplying both parts by the same positive factor preserves the ratio.
4:10 and 6:15 represent the same proportion.
This explains why simplifying a ratio is valid. Dividing both terms by a common factor changes the numbers while preserving the multiplicative relationship.
Adding the same number to both terms does not generally preserve the ratio. 2:5 is not equal to 3:6.
12. Unit conversion preserves the physical quantity
2.4 metres and 240 centimetres represent the same length.
The numeral changes because the unit changes, but the physical quantity is invariant.
This is why valid unit conversion can be represented as multiplication by a conversion factor equal to one, such as 100 cm / 1 m.
The existing Units, Scale and Measurement guide develops the mechanics. The invariant viewpoint explains what a correct conversion must preserve.
13. Area decomposition preserves total area
A composite shape can be cut into smaller pieces, rearranged, and recombined. If no area is lost or overlapped, total area is preserved.
This makes decomposition powerful. An L-shape can be treated as a large rectangle minus a missing rectangle, or as the sum of two smaller rectangles.
Different decompositions should give the same total area. If they do not, one route has miscounted or mismeasured a region.
Two methods become independent checks through the invariant total.
14. Algebraic identities preserve value for every allowed input
(x + 2)² = x² + 4x + 4 for every real x.
Expansion changes the expression’s form but not its value.
Likewise x² − 9 = (x − 3)(x + 3). Factorisation reveals different structure while preserving the function value.
Recognising identities as invariant-preserving rewrites helps students understand why algebraic manipulation is legitimate rather than merely symbolic.
15. Completing the square preserves a quadratic while revealing a new invariant description
x² − 6x + 13 can be rewritten as (x − 3)² + 4.
The function is unchanged. The second form reveals that its minimum value is 4 and its axis of symmetry is x = 3.
Changing representation can expose properties that were hidden without changing the object itself.
This is one of the most important uses of invariance in algebra: transform the representation while preserving the mathematical thing being represented.
16. Gradient is invariant along a straight line
For a non-vertical straight line, any two distinct points on the line produce the same gradient.
If A(1, 3), B(3, 7) and C(5, 11) lie on one line, gradient AB = 4/2 = 2 and gradient BC = 4/2 = 2.
This constant rate of change is the defining linear structure.
A table claimed to represent a straight-line relationship can therefore be checked by comparing gradients or first differences where x-steps are equal.
17. The mean preserves total when the count is fixed
For n values with mean m, total = nm.
If five scores have mean 12, their total is fixed at 60 even though the individual scores may vary.
This invariant total can recover a missing observation. If four scores sum to 49, the fifth must be 11.
The data set is not uniquely determined by the mean, but its sum is.
18. Exchange operations can reveal invariants in word problems
Suppose a puzzle repeatedly replaces two 1-dollar coins with one 2-dollar coin. The number of coins changes, but total monetary value remains invariant.
If the process begins with $20, no valid sequence of such exchanges can end with $19 or $21.
This kind of reasoning separates state variables that change from conserved quantities that do not.
Many Primary and Secondary puzzles become simpler once the learner asks which total survives each legal move.
19. Invariants can prove impossibility
Suppose a token begins on square 1 and every move changes its square number by a multiple of 4. Every reachable square must remain congruent to 1 modulo 4.
Square 10 is congruent to 2 modulo 4, so it cannot be reached under the stated rules.
No search through possible move sequences is needed.
Impossibility proofs often become short when a preserved property separates the start from the target.
20. A check should use a different invariant where possible
After solving an equation, substitute back to check equality.
After calculating a probability table, check the total is 1.
After converting units, check the physical magnitude remains plausible.
After rearranging a geometric shape, check area is preserved.
After scaling similar figures, check corresponding length ratios remain constant.
A good check is powerful because it watches a different invariant from the one used in the main route.
21. Not every quantity should be conserved
In a percentage increase, the original value is not preserved. In an enlargement, lengths change. In differentiation, constants disappear. In probability without replacement, branch probabilities change.
The invariant question is therefore not “what stays the same?” in a vague sense. It is “what does this particular transformation guarantee will stay the same?”
Using the wrong invariant can be as misleading as ignoring invariants entirely.
22. A transformation audit
Whenever a mathematical object changes representation, ask:
What is being changed? What must remain the same? What is allowed to change? What new structure becomes visible? What restriction must still be carried?
For factorisation, value is preserved while product structure becomes visible. For unit conversion, physical quantity is preserved while numerical representation changes. For similarity, angles and proportional shape are preserved while length scale changes. For balanced equations, the solution set is preserved.
This audit turns transformations into controlled reasoning.
23. Independent practice
1. Solve 5x − 8 = 27 and state the invariant preserved by each algebraic step.
2. A class has 42 pupils. 17 are absent. Find the number present using a conserved total.
3. A probability table has outcomes A, B and C with probabilities 0.24, 0.51 and p. Find p.
4. Explain why adding 6 to any integer preserves parity.
5. A token starts on a number congruent to 2 modulo 5 and every move adds 10. Can it ever land on a multiple of 5?
6. A shape is rotated 90°. Which geometric properties are preserved?
7. Two similar triangles have length scale factor 4. State the area scale factor.
8. Convert 3.2 m to cm and identify what is invariant.
9. Five values have mean 14. Four sum to 59. Find the fifth.
10. Show that 247 × 6 = 1482 passes a modulo-9 digit-sum check.
11. Explain why x² − 4x + 7 and (x − 2)² + 3 represent the same quadratic.
12. A rectangle is cut into two smaller rectangles. Explain why the sum of their areas equals the original area if the cut creates neither overlap nor loss.
24. Worked answers
1. Add 8 to both sides: 5x = 35. Divide both sides by 5: x = 7. Equality and the solution set are preserved.
2. Present + absent = 42, so present = 42 − 17 = 25.
3. Exhaustive probabilities sum to 1, so p = 1 − 0.24 − 0.51 = 0.25.
4. Six is even. Adding an even number does not change whether an integer is even or odd.
5. No. Adding 10 changes the number by 0 modulo 5, so the remainder 2 modulo 5 is invariant. A multiple of 5 has remainder 0.
6. Distances, side lengths, angle measures, area and shape are preserved. Orientation relative to the page changes.
7. Area factor = 4² = 16.
8. 3.2 m = 320 cm. The physical length is invariant while the unit and numeral change.
9. Total = 5 × 14 = 70. Missing value = 70 − 59 = 11.
10. 247 has digit sum 13, remainder 4 modulo 9. Multiplying by 6 gives remainder 24, hence 6 modulo 9. The claimed product 1482 has digit sum 15, also remainder 6. The calculation passes this check, though the check alone does not prove correctness.
11. Expanding (x − 2)² + 3 gives x² − 4x + 4 + 3 = x² − 4x + 7. The transformation preserves value for every real x.
12. The two smaller rectangles partition the original region. If every point of the original belongs to exactly one piece, their areas add to the total area.
25. Continue through the operating manual
Use Given, Deduced, Assumed and Conjectured Information to track the authority of facts. Continue to Backward Reasoning and Target Decomposition when an invariant suggests what intermediate result is needed, and Parameter Sensitivity and Stability when the key question is what changes rather than what stays fixed.
Return to the BTT Mathematics Hub for Batch 03.