A mathematical model is a deliberate simplification of a situation. It selects quantities, relationships and assumptions so that Mathematics can describe, predict or compare something useful. A good model is not simply one that produces a number. It is one whose assumptions are visible, whose variables have clear meanings, and whose conclusions are interpreted within the model’s limits.
This Secondary 2 Mathematics Learning Guide develops modelling as a full cycle: understand the situation, choose variables, state assumptions, build a mathematical representation, solve, validate against the original problem, interpret the answer, and revise the model when evidence or conditions change.
Secondary Mathematics Hub: S1–S4 Capability Map · Secondary 2 Learning Guide, Batch 7, Guide 2. Companion guides cover functions and multiple representations, units, rate and conversion chains, and problem posing and changing conditions.
Course boundary. Mathematical modelling is a cross-topic process. This guide uses Secondary 2 ideas such as ratio, rate, percentage, equations, graphs, geometry and statistics. The models are teaching examples, not engineering, financial, medical or policy advice. Their value lies in making assumptions and mathematical structure explicit.
Navigate: the modelling cycle · variables · assumptions · building a model · validation · model limits · revision · practice and answers · teaching and transfer.
1. Modelling is a cycle, not a one-way calculation
A useful modelling cycle is:
- understand the situation;
- identify quantities and relationships;
- state assumptions;
- choose a representation;
- perform the Mathematics;
- interpret the result;
- validate against the situation;
- revise if necessary.
The process can loop. A mathematically correct solution may reveal that an assumption was unrealistic. The model is then improved rather than defended simply because the algebra worked.
Worked example 1: simple taxi-like cost model
Suppose a fictional transport model charges a fixed 4 dollars plus 2.50 dollars per kilometre. Let d be distance in kilometres and C be total cost. The model is C = 4 + 2.5d.
For d = 8, C = 24 dollars. But the result is valid only under the assumptions encoded by the simplified model: one fixed charge, one constant per-kilometre rate, no waiting charge, no surcharge and no rounding rule.
2. Variables should represent quantities, not merely unknown letters
Let x be the width of a rectangle in centimetres is stronger than writing let x be unknown. The variable definition includes quantity, role and unit. This matters when several equations or stages appear.
Worked example 2: modelling a perimeter condition
A rectangle has length 4 cm more than its width and perimeter 44 cm. Let width be w cm. Then length is w + 4. The perimeter model is 2w + 2(w + 4) = 44.
Solving gives 4w + 8 = 44, so w = 9 and length = 13. Validation: 2(9 + 13) = 44.
The model is effective because each symbol remains attached to a physical quantity throughout the solution.
The best variable choice often reduces complexity
If two consecutive integers are involved, let the smaller be n and the larger n + 1. If one quantity is 30% more than another, define one base quantity and express the other as 1.3 times it. Good variables make the relationship visible before solving begins.
3. Assumptions are part of the Mathematics
A model cannot include every feature of reality. It chooses what to hold constant, what to ignore and what to approximate. These assumptions decide where the result is useful.
Worked example 3: constant speed
A cyclist travels 36 km in 2 hours. A simple constant-speed model gives speed = 18 km/h and predicts 54 km in 3 hours.
The prediction assumes the same average speed continues. Real travel may include traffic, slopes, stops, fatigue or changing conditions. The calculation is correct within the simplified model; the model may still be an approximation to reality.
Worked example 4: direct proportion assumption
Four identical machines produce 600 units in 5 hours. Under an assumption that output is directly proportional to both machine count and time, 6 machines in 8 hours produce 600 × (6/4) × (8/5) = 1440 units.
This assumes identical machines, constant rate, no setup delay, enough materials and no interference between machines. Without those assumptions, the proportional model may fail.
Assumptions should be neither hidden nor excessive
Do not invent assumptions that do not affect the model. State the ones needed for the mathematical relationship. “Assume constant speed” matters in a distance-time calculation. “Assume the bicycle is blue” does not.
4. Build the model from relationships, not keywords
A modelling problem may require an equation, graph, table, ratio, diagram or probability structure. The representation should expose the relationship that matters most.
Worked example 5: break-even model
A fictional school-club project has fixed setup cost 120 dollars and variable cost 4 dollars per item. Items are sold at 10 dollars each. Let n be items sold.
Cost model: C = 120 + 4n. Revenue model: R = 10n. Break-even occurs when R = C: 10n = 120 + 4n, so 6n = 120 and n = 20.
Validation: at n = 20, both cost and revenue are 200 dollars. The model assumes all items made are sold, price and variable cost remain constant, and there are no additional charges.
Worked example 6: percentage model
A quantity grows by 8% per period under a repeated-growth model. If the initial quantity is 500, one period gives 500 × 1.08 = 540. Two periods give 500 × 1.08² = 583.2.
The repeated multiplier encodes an assumption that the 8% growth applies to the updated quantity each period. A model using simple addition of 40 each period would describe a different process.
Worked example 7: geometric model
A rectangular floor is 8 m by 5 m. A uniform 1 m-wide border is placed inside the edge, leaving a central rectangle 6 m by 3 m. Total floor area = 40 m²; centre area = 18 m²; border area = 22 m².
The model assumes the border width is measured perpendicular to every edge and is uniform. Subtracting only 1 m from each dimension would be wrong because the border appears on both sides.
5. Validation asks whether the result is consistent with the original situation
Validation can include substitution, unit checking, estimation, comparison with data, checking boundaries or using a second representation.
Worked example 8: validate by substitution
Two ticket types total 40 tickets and 310 dollars. Adult tickets cost 10 dollars and student tickets 5 dollars. Solving a + s = 40 and 10a + 5s = 310 gives a = 22, s = 18.
Validation: 22 + 18 = 40 and 220 + 90 = 310. Both original conditions are satisfied.
Worked example 9: validate by scale
A 12% discount on about 200 dollars should reduce the amount by roughly 24 dollars. If a calculator returns a final price above 200, the result conflicts with the model’s direction before any detailed recalculation.
Validation need not be exact to be useful. A rough expectation can catch a structural error.
Data can validate or challenge a model
Suppose a linear model predicts outputs 10, 15, 20, 25 for inputs 1, 2, 3, 4, but observed values are 10, 15, 21, 31. The first values fit reasonably; later values increasingly diverge. The model may be useful over a limited range but poor outside it.
Do not force the data to fit the original model because the equation is convenient. A model is answerable to the situation it is meant to represent.
6. Model limits define where the conclusion should stop
Every model has a domain of reasonable use. A linear relationship may work over a short interval and fail over a long one. A direct-proportion model may ignore capacity limits. A probability model may assume equally likely outcomes that are not equally likely in practice.
Worked example 10: extrapolation risk
A plant-growth model uses h = 12 + 2t, where h is height in centimetres and t is weeks, based on observations over the first four weeks. Predicting h = 112 cm at t = 50 is algebraically possible but may be biologically unrealistic.
The issue is extrapolation far beyond the evidence used to build the model.
Worked example 11: capacity limit
A model says one printer makes 30 pages per minute, so ten printers make 300 pages per minute. This assumes each printer has independent access to power, files, paper and output handling. If one shared system can process only 180 pages per minute, the simple linear model overestimates total throughput.
A capacity constraint changes the model, not merely the final number.
7. Revision improves the model when evidence changes
Model revision might change a parameter, replace a relationship, split the domain into cases or add a constraint.
Worked example 12: piecewise-style revision
A fictional delivery model first uses C = 5 + 2d for all distances d. New information says distances beyond 10 km incur a fixed additional 6-dollar surcharge. The original rule is no longer sufficient for the whole domain.
One revised description is: C = 5 + 2d for d ≤ 10, and C = 11 + 2d for d > 10. The model now reflects the changed condition explicitly.
Worked example 13: revise an assumption, not only a coefficient
Suppose a water-use model assumes usage directly proportional to number of people. Observations show a substantial fixed baseline even when occupancy is low. A better model may include fixed usage plus a per-person term: W = a + bn.
Adding the fixed term changes the relationship family from direct proportion to a general linear model.
8. Model communication should separate result from assumption
A strong modelling conclusion might say: “Under the assumptions of constant speed and no stops, the predicted travel time is 2.5 hours.” This is more precise than “The journey takes 2.5 hours.”
The qualifier does not weaken the answer. It makes the mathematical claim match the model that produced it.
9. Common modelling errors
- Variable not defined: quantity ownership unclear.
- Every number forced into the model: relevance not filtered.
- Assumption hidden: scope of conclusion unclear.
- Correct algebra interpreted without units: quantity meaning lost.
- Contextually impossible root accepted: model constraints ignored.
- Prediction made far outside observed range: extrapolation risk ignored.
- Model defended after evidence disagrees: validation loop missing.
- Result stated as reality rather than conditional prediction: model limit not communicated.
10. Practice: build, solve, validate, limit
Questions 1–6. 1. A fictional service has fixed fee 8 dollars plus 3 dollars per unit. Define variables and write a cost model. 2. Find cost for 12 units. 3. State two assumptions. 4. Reverse the model to find units for cost 68 dollars. 5. Validate by substitution. 6. Explain why the model is not directly proportional.
Questions 7–12. 7. Five identical machines make 400 units in 4 hours. Under direct proportionality, predict output of 8 machines in 6 hours. 8. State three assumptions. 9. Give one reason the real output could be lower. 10. A rectangle has width x and length x + 5, area 84. Form a model. 11. Solve it. 12. State which root is meaningful and why.
Questions 13–18. 13. A linear model fitted over t = 0 to 5 predicts y = 10 + 4t. Find y at t = 4 and t = 50. 14. Which prediction is more defensible from the information given? 15. Explain extrapolation. 16. Observed data at large t fall far below the model. What should happen next? 17. Write one sentence reporting a model result with its assumption. 18. Give one example where a capacity limit breaks direct proportion.
Explained answers 1–6
1. Let n be units and C total cost: C = 8 + 3n. 2. 44 dollars. 3. Examples: fixed fee stays 8; per-unit rate stays 3. 4. 68 = 8 + 3n gives n = 20. 5. 8 + 3(20) = 68. 6. Non-zero fixed term means C/n is not constant and graph does not pass through origin.
Explained answers 7–12
7. 400 × (8/5) × (6/4) = 960 units. 8. Identical machines, constant rate, enough materials, no bottleneck are examples. 9. Maintenance, setup time or shared-resource limits. 10. x(x + 5) = 84.
11. x² + 5x − 84 = 0 = (x + 12)(x − 7), so x = −12 or 7. 12. Width must be positive, so x = 7 and length = 12.
Explained answers 13–18
13. y(4) = 26; y(50) = 210. 14. t = 4 is within the fitted range and therefore more defensible. 15. Extrapolation predicts beyond the range used to establish the model. 16. Reassess assumptions and revise or restrict the model.
17. Example: “Assuming constant speed and no stops, predicted travel time is 2 hours.” 18. Example: adding more workers does not increase output proportionally if one machine limits total production.
11. Teaching sequence: solve less quickly, inspect more carefully
Give a short real-world-style problem and initially forbid calculation. Ask only for variables, assumptions and a representation. Once the model is agreed, solve it. Then ask the learner to identify one condition under which the model would fail.
Next change one assumption and ask whether the same equation survives. This reveals whether the student understands the relationship or merely remembers the formula.
Questions parents and tutors can ask
What are the quantities? Which are inputs and outputs? What are you assuming stays constant? Why did you choose this representation? Does the answer satisfy the original conditions? What would make the model fail? Is your conclusion a fact about reality or a prediction under assumptions?
12. The transfer test: same mathematics, different model purpose
The formula y = 4x + 20 can represent cost, distance from a starting position, a sequence, or a simplified production model. The Mathematics alone does not determine which interpretation is correct. Variable definitions and assumptions supply the meaning.
A strong modeller can use the same algebraic structure in a new context while rebuilding the assumptions from the situation rather than copying them from the previous story.
Define the quantities. State the assumptions. Build the relationship. Solve. Validate. Interpret within the model’s domain. Revise when the evidence or conditions change.
Continue to Dimensional Reasoning, Units, Scale, Rate and Conversion Chains · Return to the Secondary Mathematics Hub.
