Singapore School Mathematics Operating Manual · Chapter 8
A calculation can be numerically correct and still describe an impossible object. A bus count cannot be 4.3. A measured length can. A class can contain 31 pupils but not 31.4 pupils. A probability can lie anywhere between zero and one. A graph showing water height through time may be continuous even though the number of containers used is discrete.
This chapter develops the distinction between discrete and continuous mathematical models. Discrete quantities take separated admissible values, often integers. Continuous quantities can vary through an interval. Many school problems combine both types, so the learner must know which operations belong to which layer.
The central habit is simple: after obtaining a numerical result, return to the object being represented and ask what values that object is allowed to take.
Counts and measurements · Rounding decisions · Graphs · Probability and expectation · Practice · Worked answers
1. A count is not a measurement
Suppose 95 pupils are divided equally among 4 rooms as far as possible. The quotient is 23.75. That number does not mean a room can contain 23.75 pupils. It describes an average allocation if fractional people were allowed, which the actual model does not permit.
A feasible distribution is 24, 24, 24 and 23 pupils. The total remains 95. The count values are integers because pupils are indivisible units in this model.
Now compare a 95-metre rope cut into four equal lengths. Each length is 23.75 metres. Here the decimal is physically meaningful because length is being modelled as continuous.
The same arithmetic operation produces the same quotient. The interpretation changes because the objects belong to different domains.
2. Discrete does not always mean “whole number from zero upward”
Integers are a common discrete set, but discrete models can use other separated values. Shoe sizes, test grades, ticket categories, dice outcomes and positions on a board are all discrete even when their labels are not simple counts.
A lift might stop only at numbered floors. Time itself can be modelled continuously while recorded appointment slots are restricted to every 15 minutes. Money can be modelled to the nearest cent in an accounting exercise even though an abstract mathematical amount could be continuous.
The key property is not “looks like an integer”. It is that the model permits separated states rather than every value in between.
Always read the object and its rules before assigning a domain.
3. Continuous does not mean infinitely precise in real life
Mathematical models often treat length, mass, time and temperature as continuous variables. That does not mean a real instrument measures them with infinite precision.
A ruler may report a length to the nearest millimetre. The underlying mathematical model can still be continuous while the observation is rounded. The uncertainty then belongs to the measurement record, not to the continuity of the modeled quantity.
This distinction matters. A student should not conclude that because a measurement is recorded as 12.3 cm, the object can only have values with one decimal place. The true value may lie anywhere within the relevant rounding interval under the stated measurement model.
The Rounded Data chapter develops that interval reasoning separately.
4. Whole-number decisions require context-sensitive rounding
Suppose 112 pupils need transport and each bus holds at most 26 pupils. The quotient 112/26 is approximately 4.31.
Four buses hold 104 pupils, which is insufficient. Five buses hold 130. Therefore five buses are required.
This is not the rule “always round up”. If the question asks how many complete groups of 26 can be formed from 112 pupils, the answer is four complete groups with eight pupils left over.
The same quotient supports two different integer decisions. The requested object determines whether the fractional remainder forces another container or is simply left over.
5. Normal rounding can be mathematically wrong for a capacity decision
Consider 41 passengers and vehicles holding 40 each. The quotient 41/40 = 1.025. Ordinary rounding to the nearest whole number gives 1. But one vehicle cannot carry all 41 passengers.
The minimum-container decision requires the smallest integer at least as large as the quotient. In higher Mathematics, this operation is the ceiling function.
By contrast, if 41 items are packed into as many complete boxes of 40 as possible, there is one complete box and one leftover item. That answer corresponds to the integer part or floor-like decision.
Rounding rules are therefore consequences of the model, not shortcuts to apply before understanding the target.
6. Integer restrictions can turn an interval into a finite list
Solve 2x + 3 < 12. Over the real numbers, x < 4.5.
If x is a positive integer, the admissible values are 1, 2, 3 and 4. If x is a non-negative integer, include 0. If x is any integer, the set also contains all negative integers.
The inequality defines a continuous boundary. The domain selects which discrete points inside that region are actually allowed.
This is why writing only x < 4.5 can be incomplete when the question asks for integer values, and writing only 1, 2, 3, 4 can be incomplete when the domain is real.
7. A discrete graph is not automatically joined by a continuous curve
Suppose n is the number of books purchased and cost is exactly $8 per book. The relation C = 8n is algebraically linear.
If books can only be purchased as whole copies, the practical graph consists of points at n = 0, 1, 2, 3, … . The line y = 8x contains intermediate points such as x = 2.5 that do not represent an allowed purchase count.
Drawing a continuous line can still help reveal the underlying proportional relationship, but the interpretation must remember the discrete domain.
Conversely, if x is distance travelled and fuel use is modelled continuously, a connected graph may be appropriate. The graph style should follow the quantity, not just the formula.
8. Step graphs model discrete decisions driven by continuous inputs
A taxi-like invented model charges one booking unit for any positive trip up to 5 km, two units for trips above 5 km up to 10 km, and so on. Distance is continuous; the number of charge units is discrete.
The output graph is a step function. Small changes in distance within one interval do not change the charge unit count. Crossing a threshold causes a jump.
This mixed structure appears in packaging, capacity, staffing, storage and queue models. A continuous input can feed a discrete decision.
The reverse can also occur: a discrete count can feed a continuous outcome. The number of identical tiles is discrete, while the total covered length may be modelled as a continuous physical measurement subject to tolerances.
9. Histograms and bar charts encode different kinds of variables
A bar chart commonly represents categories or discrete values, with separated bars emphasising distinct categories.
A histogram represents continuous numerical intervals, with adjacent bars reflecting connected class intervals. When widths differ, frequency density rather than raw frequency may be required for area to represent frequency correctly.
The distinction is conceptual before it is visual. A learner should ask whether the horizontal axis lists categories or partitions a numerical continuum.
Simply copying the appearance of a familiar chart can hide the wrong data model.
10. Mean values can be non-admissible individual values
Three classes contain 28, 31 and 35 pupils. Their mean size is 94/3 = 31⅓ pupils.
No actual class contains one-third of a pupil. The mean is a continuous summary value calculated from discrete observations. It describes the centre of the data, not necessarily an observed admissible count.
This is not a contradiction. Summary statistics live in a different mathematical role from individual observations.
A learner should therefore avoid rejecting a non-integer mean simply because the original data are integers.
11. Expected value can also be non-admissible as a single outcome
For a fair six-sided die, the expected value is (1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5.
No single roll can produce 3.5. Expected value is a weighted average over possible outcomes, not a prediction that the next outcome must be an allowed result nearest to 3.5.
Over many repeated trials, the average outcome may approach the expected value under the model. Individual outcomes remain discrete.
This distinction is important whenever a calculated average, expectation or rate is mistaken for a literal object.
12. Probability itself is continuous even when outcomes are discrete
A die has six discrete outcomes, but probabilities assigned to events are real numbers between zero and one.
The event “roll an even number” contains three discrete outcomes and has probability 3/6 = 1/2. The probability value is not itself one of the die outcomes.
A binomial random variable takes discrete count values, while a normal model uses a continuous random variable. Both can describe uncertainty, but their probability calculations have different structures.
The learner should distinguish the domain of the random outcome from the domain of the probability used to describe that outcome.
13. Discrete optimisation may not occur at the continuous optimum
Suppose a continuous model suggests an objective is maximised at x = 7.4, but the actual decision variable x must be an integer.
The value 7.4 is not admissible. The learner should compare nearby allowed integers, commonly x = 7 and x = 8, and evaluate the objective under the original constraints.
Rounding 7.4 automatically to 7 may fail if the function is asymmetric or if constraints affect the neighbouring choices differently.
A continuous relaxation can guide the search, but the final discrete model owns the decision.
14. Geometry mixes discrete structure with continuous magnitude
A polygon has a discrete number of sides. Its side lengths and angles are usually modelled continuously.
A triangle has exactly three sides as a combinatorial object, but one side may be 4.2 cm and an angle may be 37.6°. The count and the measurements coexist inside one model.
Likewise, a graph can have a discrete number of vertices while edge weights are continuous distances or costs.
Do not classify an entire problem as discrete or continuous too quickly. Different variables within the same problem can belong to different domains.
15. Time can be continuous even when a timetable is discrete
A bus journey may last 17.4 minutes in a continuous time model. But a timetable might schedule departures only at 10-minute intervals.
If a passenger arrives at 8:03 and buses depart at 8:00, 8:10, 8:20 and so on, the next scheduled departure is 8:10. The waiting decision is tied to a discrete schedule even though clock time itself is continuous.
A learner who subtracts 8:03 from a hypothetical continuously available departure time is solving a different model.
School word problems often hide this distinction in ordinary language: continuous physical processes interacting with discrete operational rules.
16. Sampling converts a continuous population question into discrete evidence
A population proportion can be modelled as a real number between zero and one. A sample of n people produces a discrete count of successes from 0 to n.
The sample proportion k/n therefore takes only n + 1 possible values for fixed n, even though the underlying population parameter is continuous.
This distinction helps explain why an observed sample proportion is not identical to the unknown population probability. One is a statistic generated from a finite discrete sample; the other is a model parameter.
This is an extension beyond many school stages, but the principle matches the rest of the chapter: identify the mathematical object before interpreting the number.
17. Integer programming begins when the decision itself is indivisible
Suppose an invented school needs to place identical tables, each seating 6 pupils, for 95 pupils. The continuous ratio is 95/6 ≈ 15.83.
If partial tables are meaningless, at least 16 tables are required. That gives 96 seats, leaving one unused seat.
If the same numerical problem represented 95 litres divided equally among containers with no capacity restriction, 15.83 litres per container could be perfectly meaningful.
Integer constraints are not cosmetic restrictions placed on a continuous answer. They can change the feasible set and therefore the optimum itself.
18. A practical domain audit
Before finalising a result, identify four things: what does the variable represent; what values are admissible; does the calculation produce an admissible value; and if not, what model-specific decision converts it into one?
For counts, ask whether zero and negative values are allowed. For measurements, ask about units and precision. For probabilities, check the interval from zero to one. For scheduled choices, list the permitted times or categories. For integer optimisation, inspect nearby admissible values.
This audit turns vague “common sense” into explicit mathematical control.
19. Independent practice
1. 73 pupils are placed into groups of at most 8. Find the minimum number of groups.
2. How many complete groups of 8 can be formed from 73 pupils, and how many pupils remain?
3. A 73-metre cable is cut into 8 equal pieces. Find each length.
4. List the positive integer solutions of x < 6.4.
5. A shop model sells pens only as whole items at $2.50 each. Describe the practical graph of cost against quantity.
6. Four class sizes are 29, 30, 31 and 34. Find the mean and explain why it need not be an admissible class size.
7. Find the expected value of a fair die and explain why the value is not a possible single roll.
8. A continuous optimisation gives x = 12.6, but x must be an integer. What should be checked next?
9. A car park charges by complete or partial hour: $3 for each started hour. Write the charge for stays of 0.2 h, 1 h, 1.1 h and 2.9 h.
10. Explain why the statement “the average family has 2.4 children” does not mean that an actual family can contain 2.4 children.
20. Worked answers
1. 73/8 = 9 remainder 1, so nine groups are insufficient for all pupils. Ten groups are required.
2. Nine complete groups can be formed, with one pupil remaining. This is a different target from the minimum number of groups needed to include everyone.
3. 73/8 = 9.125 metres per piece. Length is continuous in this idealised model, so the decimal is admissible.
4. 1, 2, 3, 4, 5 and 6.
5. The algebraic relation is C = 2.5n, but practical values occur only at whole-number n ≥ 0. The graph is a set of discrete points on the line rather than every point of the continuous line.
6. Mean = (29 + 30 + 31 + 34)/4 = 124/4 = 31. Here the mean happens to be an admissible count. In general, means of integer counts can be non-integers because the mean is a summary, not necessarily an observed member of the data set.
7. Expected value = 3.5. Individual die outcomes are 1 through 6 only. The expectation is a weighted average over repeated-trial behaviour.
8. Evaluate the objective at nearby admissible integers, especially 12 and 13, and check all constraints. Do not round mechanically without testing the actual discrete objective.
9. Under the stated model: 0.2 h costs $3; 1 h costs $3; 1.1 h costs $6; 2.9 h costs $9. The output changes in steps when a new hour begins.
10. Family size is a discrete count. The average 2.4 summarises a collection of families; it is not claimed to be the size of a particular family.
21. Continue through the operating manual
Use Boundary Cases, Endpoints and Extreme Values when a discrete decision changes at a threshold. Use Case Splitting and Piecewise Reasoning for step functions and branch-based models. Use Reversible Steps and Extraneous Solutions when domain restrictions affect transformations.
Return to the BTT Mathematics Hub for Batch 02.