Singapore School Mathematics Operating Manual · Chapter 31
Two quantities can be mathematically related but difficult to compare because they live on different scales.
A score out of 20 and a score out of 80 cannot be compared fairly by raw marks alone. A graph measured in metres and another in centimetres may describe the same shape with different numerals. A quantity may be easier to interpret relative to a baseline of 1, 100, a maximum value, a mean or an initial condition.
This chapter develops normalisation and rescaling: controlled changes of representation that make comparison, pattern recognition and modelling easier without silently changing the underlying relationship.
1. Rescaling changes the numerical representation
2.5 m and 250 cm represent the same length.
The scale changes by factor 100; the physical quantity does not.
Unit conversion is the simplest rescaling example.
2. Normalisation uses a reference value
If a quantity Q is compared with reference R, the ratio Q/R is dimensionless when Q and R have the same units.
If Q=75 and R=100, the normalised value is 0.75.
This can also be written as 75%.
3. Percentages are normalised comparisons
A test score 18/24 becomes 75%.
A score 60/80 also becomes 75%.
Raw marks differ, but normalisation reveals equal proportional performance.
4. The denominator defines the reference frame
20 as a percentage of 80 is 25%.
20 as a percentage of 100 is 20%.
The numerator did not change; the reference did.
Normalised values are meaningless without knowing what they were normalised against.
5. Ratios remove common scale
6:15 simplifies to 2:5.
The common factor is removed while relative structure remains.
Ratio simplification is a form of scale normalisation.
6. Similarity uses normalised shape information
Two similar triangles may have side sets 3,4,5 and 6,8,10.
Absolute lengths differ, but corresponding ratios match.
Normalising by one side makes the common shape structure visible.
7. A unit vector normalises magnitude
Vector (3,4) has magnitude 5.
Dividing by 5 gives unit vector (3/5,4/5).
The direction is preserved while magnitude becomes 1.
This is useful when direction matters independently of scale.
8. Indexing converts a baseline to a standard reference
Suppose a price starts at $50 and later becomes $60.
If the starting point is indexed to 100, the later index is 120.
The index communicates a 20% increase while removing the original currency scale.
9. Growth factors are normalised multipliers
A 5% increase corresponds to factor 1.05.
A 12% decrease corresponds to factor 0.88.
Multipliers allow changes from different starting values to be compared on one common reference of 1.
10. Scale factors separate size from shape
In an enlargement, every corresponding length is multiplied by k.
Normalising each length by a chosen reference side reveals the same dimensionless shape ratios before and after enlargement.
11. Graph axes are rescaling devices
A graph can display x in thousands and y in millions.
The plotted shape may be clear even though axis labels compress large magnitudes.
Reading a graph correctly requires restoring the axis scale before reporting raw values.
12. Broken axes can alter visual impression
A truncated vertical axis may make small differences look dramatic.
The underlying numerical difference does not change, but visual normalisation changes perception.
Always inspect axis origin, increments and units.
13. Standard form normalises magnitude representation
4500000 = 4.5×10⁶.
The coefficient is normalised to a conventional interval while the power of ten records scale.
This makes comparisons across large magnitudes easier.
14. Scientific notation separates significant structure from zeros
3.2×10⁸ and 3.2×10⁻⁸ share coefficient 3.2 but differ radically in scale.
The representation makes that separation explicit.
15. Per-unit quantities normalise by exposure
Cost per item = total cost/item count.
Speed = distance/time.
Density = mass/volume.
Rates allow fairer comparison when totals are based on different amounts of exposure.
16. Raw totals can mislead when denominators differ
Class A has 18 correct answers out of 20. Class B has 70 out of 80.
Raw correct counts favour B, but proportions are 90% and 87.5% respectively.
The comparison target determines whether total or normalised performance is relevant.
17. Per-capita thinking is normalisation by population
Two groups may have different totals simply because one group is larger.
Dividing by population can compare average rate per person.
This does not make total burden irrelevant; it answers a different question.
18. Dimensionless quantities support cross-unit comparison
If two lengths are divided, the units cancel.
A length ratio can therefore be compared whether the original measurements were both in cm, both in m or consistently converted.
Dimensionless structure often reveals the deeper mathematical relationship.
19. Probability is already normalised
Total probability of the complete sample space is 1.
Each event probability is therefore a dimensionless share of the whole.
This common reference makes probabilities comparable across different sample spaces.
20. Frequency can be normalised into relative frequency
If an event occurs 30 times in 50 trials, relative frequency is 0.6.
If another experiment records 600 occurrences in 1000 trials, its relative frequency is also 0.6.
Raw counts differ by factor 20; normalised frequency reveals equal proportion.
21. Standardisation can centre and scale data
A z-score subtracts the mean and divides by standard deviation.
The transformed value reports how many standard deviations an observation lies from the mean.
This enables comparison across distributions with different units or spreads.
This is enrichment beyond some school stages, but the principle extends ordinary normalisation.
22. Rescaling can improve numerical convenience
If every value in a data set is around 1,000,000, working in millions can simplify a graph or table.
Multiplying results back by the scale factor restores original units where required.
23. Rescaling can reveal hidden linearity
A graph may become easier to interpret after transforming units or plotting normalised variables.
However, changing variables can also change slope meaning.
The transformed graph must be interpreted in its own scale before returning to context.
24. A normalised score does not preserve every property
Converting raw marks to percentages preserves order when every denominator is positive and the same comparison meaning applies.
But it may remove information about test length or total number of questions.
Normalisation deliberately discards some scale information.
25. The choice of reference can change interpretation
Growth relative to last year and growth relative to a ten-year average are different normalisations.
Neither is automatically correct without knowing the question.
26. Baseline selection should not be manipulated invisibly
A dramatic percentage can result from choosing a very small baseline.
Responsible mathematical communication names the reference value explicitly.
27. Normalisation is useful in modelling because it separates structure from units
When equations are expressed in dimensionless variables, systems at different physical scales can sometimes be compared more directly.
This is a higher-level extension of school ratio reasoning.
28. Normalisation can make thresholds portable
A threshold stated as 80% can be applied to tests with different total marks after normalisation.
A raw threshold of 40 marks cannot be transported between tests out of 50 and 100 without adjustment.
29. Rescaling should preserve inequalities only under controlled transformations
Multiplying every value by a positive constant preserves order.
Multiplying by a negative constant reverses order.
Normalisation is therefore not automatically order-neutral.
30. Zero reference values create problems
Percentage change from zero is not defined by the ordinary formula change/original×100%.
Division by the baseline fails.
Normalisation needs a valid non-zero reference when it uses division.
31. Near-zero baselines can create huge percentages
A change from 0.01 to 0.02 is a 100% increase despite an absolute change of only 0.01.
Relative and absolute scales tell different stories.
This connects normalisation to sensitivity and conditioning.
32. Recomposition must restore physical meaning
If calculations are performed with normalised variables, final answers may need conversion back to original units.
A dimensionless result may be appropriate as a ratio, but a requested length, cost or time must regain its unit.
33. A practical normalisation audit
Ask:
What quantity is the reference? Why is that reference appropriate? What information does normalisation preserve? What does it discard? Are units cancelled correctly? Does order remain preserved? Do I need to convert back at the end? Could a near-zero baseline distort interpretation?
34. Independent practice
1. Convert 18/24 to a percentage.
2. Compare 45/60 and 72/96 after normalisation.
3. A vector is (6,8). Find its unit vector.
4. A price rises from 50 to 65. If 50 is indexed to 100, find the new index.
5. Convert 0.4 to percentage and ratio out of 100.
6. Why is cost per item often fairer than total cost when quantities purchased differ?
7. What is wrong with calculating percentage increase from 0 using the standard formula?
8. Why can a truncated graph axis exaggerate visual differences?
9. Give one example of a dimensionless ratio.
10. Why must a normalised result sometimes be converted back to original units?
35. Worked answers
1. 18/24=0.75=75%.
2. Both equal 75%.
3. Magnitude=10, so unit vector=(0.6,0.8).
4. 65/50×100=130.
5. 40% and 40:100, simplifying to 2:5.
6. It divides out the quantity purchased and compares unit rate rather than scale.
7. The original value is the denominator, so division by zero is undefined.
8. It visually stretches the displayed variation while leaving the actual values unchanged.
9. Length/length, such as 3 cm / 5 cm = 0.6.
10. The question may ask for a physical quantity such as metres or dollars rather than a dimensionless comparison.
36. Continue through Batch 08
Use Decomposition and Recomposition when modules need common interfaces, Dependency Order to determine when rescaling should occur, and Sampling, Resolution and Aliasing when normalised summaries may hide detail.
Return to the BTT Mathematics Hub for Batch 08.