A mathematical model is a controlled simplification of a situation. It keeps the relationships needed for the question and leaves out details that do not matter to the calculation. The difficult part is not always solving the equation. It is deciding what the equation should represent, which information belongs inside it and where the model stops being trustworthy.
This forty-sixth Secondary 4 Mathematics Learning Guide develops AO2 model building as a complete loop: select, represent, solve, interpret, validate and refine. It belongs to the Secondary Mathematics Hub and S1–S4 Capability Map.
The current SEC G3 Mathematics syllabus expects students to identify relevant mathematics, translate information, connect topics, formulate problems mathematically, select relevant information and interpret results in context. Real-world problems may combine several topics. Official reference: SEAB 2027 SEC G3 Mathematics K310 syllabus.
A model has a boundary
Every model answers a particular question under particular assumptions. A taxi-cost equation may describe price as distance changes while ignoring traffic. A floor-plan model preserves scale while ignoring wall thickness unless thickness is relevant. A compound-interest model assumes a stated rate and compounding rule continue over the specified periods.
A good model is not reality copied perfectly. It is reality simplified for a purpose.
The model-building loop
- Define the question. What exactly must be found, compared or decided?
- Select relevant information. Which facts affect the required quantity?
- Define variables and units. What does each symbol mean?
- State assumptions. What simplifications are being made?
- Build relationships. Equations, inequalities, diagrams, ratios, graphs or tables.
- Apply constraints. Physical, numerical and contextual limits.
- Solve. Use appropriate techniques.
- Interpret. Translate the mathematical result back into the problem.
- Validate. Check units, magnitude, conditions and model behaviour.
- Refine if necessary. Change the model if the result exposes a bad assumption.
Relevant information is question-dependent
The same scenario can contain information that matters for one question and not another.
Worked Example 1 | Carpet cost
A hall is 20 m long, 15 m wide and 5 m high. It contains 180 chairs. Carpet costs $42 per square metre. Find the cost of carpeting the floor, ignoring wastage.
Relevant: length, width, cost per square metre.
Irrelevant to this question: ceiling height and chair count.
Area=20×15=300 m².
Cost=300×42=$12,600.
If the question were about air-conditioning volume, the height would suddenly become relevant. Relevance belongs to the question, not permanently to the datum.
Variables should have one precise meaning
“Let x be the answer” is weak modelling. Better: “Let x be the number of adult tickets sold.” The variable definition helps determine domain, units and interpretation.
Worked Example 2 | Ticket model
A theatre sells adult tickets at $18 and student tickets at $12. A total of 240 tickets are sold for $3600. Let a be adult tickets and s be student tickets.
a+s=240
18a+12s=3600
The variables are counts, so a and s must be non-negative integers.
Substitute s=240−a:
18a+12(240−a)=3600.
6a=720, so a=120 and s=120.
The algebraic solution also satisfies the integer and non-negative constraints.
Assumptions are invisible inputs
A model may rely on assumptions even when no symbol represents them. For example, constant speed assumes speed does not vary during the modelled interval. A unit-price comparison assumes products are comparable enough for price per unit to be meaningful.
Worked Example 3 | Travel-time model
A car travels 180 km at an average speed of 60 km/h. Estimate travel time.
t=180/60=3 h.
The model assumes the stated average speed represents the whole 180 km interval. If a separate 30-minute stop must be included in total elapsed time, the model must add it.
Constraints reduce the solution space
Equations may produce values that are mathematically valid but contextually impossible. Counts cannot be fractional when whole objects are required. Lengths are normally positive. A denominator cannot be zero. A budget cannot exceed a stated cap. A point may need to remain inside a physical boundary.
Worked Example 4 | Packaging constraint
A box can hold at most 25 kg. Packaging weighs 1.8 kg and each product weighs 2.3 kg. Let n be the number of products.
1.8+2.3n≤25.
2.3n≤23.2, so n≤10.086…
Because n counts whole products, maximum n=10. Rounding to 10 is not ordinary numerical rounding; it is a constraint decision.
A model can combine topics
Real-world questions may move through percentage, geometry and rates in one chain. The learner should not ask “Which chapter is this?” but “Which relationship is needed at this stage?”
Worked Example 5 | Floor plan, area and percentage allowance
A rectangular room measures 5.4 cm by 4.2 cm on a 1:100 floor plan. Flooring is ordered with an 8% allowance for cutting. Find the ordered area.
Actual dimensions=5.4 m by 4.2 m.
Floor area=22.68 m².
Ordered area=22.68×1.08=24.4944 m².
If the supplier sells only whole square metres or fixed packs, another practical constraint would determine the purchase quantity.
Validation begins with units
Units can expose a broken model. If speed in km/h is multiplied by time in minutes without conversion, the numerical result has no coherent distance meaning.
Worked Example 6 | Unit validation
A runner moves at 12 km/h for 30 minutes. Find distance.
30 minutes=0.5 h.
Distance=12×0.5=6 km.
The unit chain km/h × h = km confirms the structure.
Validate magnitude before trusting precision
An answer of 6200 km for a short school bus journey should trigger suspicion even if the calculator arithmetic was entered correctly. Estimation gives the result a scale before exact computation gives it precision.
Worked Example 7 | Reasonableness check
A rectangular garden is approximately 18 m by 12 m. Before exact multiplication, estimate area as about 20×10=200 m². An exact result near 216 m² is plausible. An answer 2160 m² likely contains a place-value or unit error.
Validate with boundary cases
A useful model should behave sensibly at simple inputs. In C=8+2.5d, putting d=0 gives C=8. If the problem says an $8 fixed charge exists even with zero distance, that boundary is sensible.
If the model instead predicted a negative cost at zero usage without explanation, it would need investigation.
Worked Example 8 | Validate a linear cost model
A delivery fee is modelled by C=6+1.8d.
- d=0 → C=6, matching a fixed $6 charge.
- Increasing d by 1 increases C by $1.80, matching the stated unit rate.
- For d≥0, cost remains positive.
These checks do not prove the real company uses the model in all circumstances. They show the model is internally consistent with its stated assumptions.
Model limits should be stated when they matter
A model built from a short interval may not remain valid far outside it. A straight-line trend in a small data range does not guarantee indefinite linear growth. A price model may change after a threshold. A physical object may not behave as an ideal geometric solid.
Worked Example 9 | Linear extrapolation limit
A plant height is modelled as h=12+2.5t centimetres over the first 8 weeks. The model predicts 262 cm at t=100 weeks.
The calculation is correct under the equation, but the model may be biologically unreasonable far beyond the observed period. Mathematical correctness and model validity are different questions.
Piecewise models: one rule may not cover the whole situation
Many real prices, fares and tariffs use thresholds. A single straight line may fail because the rate changes after a boundary.
Worked Example 10 | Tiered charge
A hypothetical service charges $0.50 per unit for the first 100 units and $0.30 per additional unit. Find the cost for 160 units.
First 100: $50.
Next 60: $18.
Total=$68.
Using 160×0.30 would incorrectly apply the second-tier rate to the whole quantity.
Models can be compared, not merely solved
Suppose Plan A has A=12+2x and Plan B has B=4+2.8x. The equations let us ask when each is cheaper and where they break even.
Worked Example 11 | Break-even model
Set 12+2x=4+2.8x.
8=0.8x, so x=10.
At x=10, both cost $32.
For small x, Plan B has the lower fixed cost. For sufficiently large x, Plan A’s lower variable rate dominates.
Probability models need a defined sample space
Probability calculations assume a sample space and event structure. “Without replacement” changes later probabilities. “Independent” means one event does not alter the probability of the other under the stated model.
Worked Example 12 | Changing sample space
A bag contains 5 red and 3 blue counters. Two are drawn without replacement.
P(two red)=5/8×4/7=5/14.
The denominator changes from 8 to 7 because the model explicitly removes the first counter.
Model validation by a second route
If a triangle side is found by trigonometry, Pythagoras may sometimes provide an independent check in a right-triangle case. If a break-even point is found algebraically, substituting into both cost equations verifies equality.
Independent checks are powerful because they do not merely repeat the same arithmetic.
The difference between assumption and fact
| Statement | Role |
|---|---|
| “The diagram marks AB∥CD.” | Given fact/constraint |
| “Assume constant speed during the interval.” | Modelling assumption |
| “n counts buses, so n is a non-negative integer.” | Domain constraint |
| “The cost model is used only for 0≤x≤100.” | Validity range |
| “The answer 14.2 buses is rounded to 15.” | Contextual interpretation |
Common failure modes
| Failure | Why it fails | Repair |
|---|---|---|
| Uses every number in the question | Relevance not analysed | Link each datum to the required quantity |
| Defines x vaguely | Domain and units become unclear | Name the quantity precisely |
| Accepts negative time or fractional people | Context constraints ignored | Filter mathematical roots through meaning |
| Uses one rate across a tiered tariff | Model boundary ignored | Build piecewise stages |
| Trusts calculator precision | Model plausibility unchecked | Estimate magnitude and inspect units |
| Extrapolates indefinitely | Validity range ignored | Ask whether assumptions survive outside observed data |
| States an assumption as if it were given | Evidence and modelling blurred | Label assumptions explicitly |
Independent practice
- A hall is 30 m by 18 m by 7 m and floor tiles cost $28/m². Which dimensions are relevant to tiling the floor?
- A lift can carry 600 kg. It already carries 140 kg and each box weighs 32 kg. Form an inequality and find the maximum number of boxes.
- A plan uses scale 1:250. A path measures 8 cm. Find actual distance in metres.
- Plan A costs 5+3x and Plan B costs 17+2x. Find the break-even x and cost.
- A model y=4+2t is intended for 0≤t≤12. Explain why t=100 may be mathematically evaluable but outside the model’s stated validity.
- A bag contains 4 red and 6 blue counters. Find P(two red without replacement).
Explained answers
1. Length 30 m and width 18 m. Height is irrelevant to floor area.
2. 140+32n≤600. Thus 32n≤460 and n≤14.375. Maximum whole number=14 boxes.
3. 8×250=2000 cm=20 m.
4. 5+3x=17+2x gives x=12. Cost=41.
5. The equation can produce a number at t=100, but the model was only asserted for 0≤t≤12, so the assumptions supporting the relationship may not hold at 100.
6. 4/10×3/9=2/15.
The examination habit: write the model before operating it
Before the first calculation, identify four things in the margin or working:
- Variable: what does x mean?
- Relationship: what equation, ratio or geometric constraint connects the quantities?
- Domain: what values are actually allowed?
- Check: what would make the result obviously impossible?
This short control layer prevents many long solutions from drifting away from the situation they were meant to model.
Final thought
Model building is where Mathematics becomes answerable to the world. The equations must not only be solvable; they must represent the right quantities, respect the right constraints and produce conclusions that survive a return to context.
Select what matters, state what you assume, constrain what is possible, solve the structure, then ask whether the world agrees.
Continue with AO3 Mathematical Argument or return to the Secondary Mathematics Hub.
