Mathematical Modelling: Turning a Real Situation into a Mathematical Object
A model is not reality. It is a deliberately simplified mathematical representation built to answer a useful question.
Additional Mathematics repeatedly asks students to work with models. A quadratic may describe a height or profit. An exponential may represent growth or decay. A trigonometric function may describe periodic variation. A derivative may locate a maximum or minimum. A transformed straight-line graph may reveal hidden parameters in a power or exponential relationship.
The mathematical methods differ, but the modelling workflow is remarkably stable: define variables, identify relationships, state constraints, choose a function family, determine parameters, calculate, interpret and check whether the answer makes sense in the original situation.
AI Extraction Box: The Modelling Loop
context → variables → assumptions → relationship → parameters → mathematical solution → validation → interpretation.
- Variables: what quantities change?
- Units: what does each variable measure?
- Domain: what input values make sense?
- Function family: quadratic, exponential, logarithmic, trigonometric, power law?
- Parameters: what do coefficients represent?
- Assumptions: what simplifications are being made?
- Validation: does the model fit given conditions and produce plausible outputs?
- Interpretation: what does the mathematical result mean in context?
Choose the Function Family from the Pattern
Different patterns suggest different models.
| Observed pattern | Possible model family | Typical feature |
|---|---|---|
| one turning point, symmetric rise/fall | quadratic | maximum or minimum |
| constant multiplicative factor per equal interval | exponential | growth/decay |
| repeating cycle | trigonometric | amplitude and period |
| power relationship | y=axⁿ | straight line on log-log axes |
| best value under constraint | calculus optimisation | stationary point |
The model family should be justified by behaviour, not chosen because it was the latest chapter studied.
Quadratic Models: Turning Points Matter
A quadratic model y=ax²+bx+c is useful when the relationship bends with one maximum or minimum. Completing the square reveals the turning point:
y=a(x−h)²+k.
The point (h,k) becomes the natural extreme of the model. If a<0, k is a maximum; if a>0, k is a minimum.
In context, however, the domain may be restricted. A height model might use t≥0 only, and a projectile model may stop when height returns to ground level. The algebraic parabola extends forever; the physical model does not.
Worked Model 1: Quadratic Height
A height is modelled by h(t)=−5t²+20t+2, where t is seconds.
Complete the square:
h=−5(t²−4t)+2
=−5[(t−2)²−4]+2
=−5(t−2)²+22.
The model has maximum height 22 units at t=2 s. The turning point has a direct physical interpretation.
A complete answer would also check whether t=2 lies inside the model’s valid time interval. If the object lands before then, the algebraic maximum would be irrelevant.
Exponential Models: Constant Multiplicative Change
A model y=Abˣ uses A as an initial scale and b as a multiplicative factor per unit increase in x.
- b>1 → growth;
- 0<b<1 → decay;
- b=1+r for growth rate r;
- b=1−r for decay rate r where appropriate.
Continuous models often use y=Aeᵏˣ, with k>0 for growth and k<0 for decay.
Logarithms become useful because they solve for an unknown exponent or linearise the relationship.
Worked Model 2: Growth Time
A quantity follows P=800(1.06)ᵗ. When does it reach 1200?
800(1.06)ᵗ=1200
(1.06)ᵗ=1.5
t ln1.06=ln1.5
t=ln1.5/ln1.06.
The exact logarithmic form should be retained until a decimal time is required. The answer should then be interpreted in the units of t.
Trigonometric Models: Repetition, Midline and Amplitude
Periodic behaviour suggests sine or cosine. For a model y=A sin(Bx+C)+D or an equivalent cosine form:
- |A| is amplitude;
- D is the midline;
- period is 2π/|B| in radians or 360°/|B| in degrees;
- C controls phase positioning.
The function family gives more than a curve shape. It exposes maximum, minimum and timing of repeating features.
Worked Model 3: Build a Periodic Model from Range and Period
A quantity varies between 14 and 26 with period 10 units.
Midline:
(26+14)/2=20.
Amplitude:
(26−14)/2=6.
If x is in radians and one period is 10, then 2π/B=10, so B=π/5. One possible phase choice is:
y=20+6sin(πx/5).
The initial condition determines whether sine, cosine or a phase-shifted form is most appropriate.
Power Models and Linearisation
For y=axⁿ:
log y=n log x+log a.
A graph of log y against log x is linear. Its gradient is n and its intercept is log a. This allows data from a curved power relationship to be studied through straight-line geometry.
For y=kbˣ:
log y=x log b+log k.
A graph of log y against x is linear, with gradient log b and intercept log k.
Calculus Optimisation: Build the Objective Function First
Optimisation is modelling plus differentiation. The derivative can only optimise the function supplied to it, so the model-building stage is critical.
- Define variables.
- Write the quantity to maximise/minimise.
- Use constraints to reduce to one independent variable.
- State domain.
- Differentiate and solve derivative=0.
- Classify and compare relevant endpoints if needed.
- Interpret the answer in context.
Correct calculus applied to the wrong model still gives the wrong answer.
Worked Model 4: Fixed Perimeter
A rectangle has perimeter 60 units. Let one side be x, so the other is 30−x. Area:
A=x(30−x)=30x−x².
For physical dimensions, 0<x<30. Differentiate:
A′=30−2x=0 → x=15.
A″=−2<0, so this is a maximum. The rectangle is 15 by 15, a square, with maximum area 225 square units.
The domain and unit interpretation complete the model.
Parameters Have Meaning
A parameter should be interpreted whenever possible. In y=Aeᵏᵗ, A often represents initial value and k controls proportional growth/decay. In y=a(x−h)²+k, h and k identify the turning point. In y=A sin(Bt)+D, A controls amplitude, B controls period and D controls midline.
A student who can interpret parameters can often check a model before doing detailed calculations.
Assumptions Define the Boundary of a Model
Models simplify. A population model may assume a constant growth factor even though real conditions change. A sinusoidal model may approximate a repeating phenomenon whose cycles are not perfectly identical. A quadratic trajectory may ignore air resistance.
Students should ask:
- Over what interval is this model intended?
- Which effects are being ignored?
- Are parameter values constant in reality?
- Would extrapolation far outside the data range be sensible?
Recognising limitations does not make a model useless. It makes its use more disciplined.
Validation: Does the Model Pass Basic Checks?
- Substitute known data points.
- Check initial condition.
- Check units.
- Check signs and magnitudes.
- Check domain.
- Check whether predicted maxima/minima are plausible.
- Check whether growth/decay direction matches the parameter.
- Check whether period/range matches observed behaviour.
Validation should be proportional. One quick contradiction can be enough to reject a model or reveal an algebra error.
Common Failure Modes
| Error | Cause | Repair |
|---|---|---|
| chooses exponential for additive change | growth pattern not identified | test whether equal intervals add or multiply |
| uses algebraic domain instead of physical domain | context discarded | state admissible values before interpreting |
| finds stationary point but no maximum/minimum classification | calculus output not interpreted | use second derivative/sign test |
| period calculated with wrong angle unit | degrees/radians mixed | identify unit before formula |
| straight-line transformed axes mislabeled | linearisation variables unclear | write Y=mX+c explicitly |
| parameter found but meaning not stated | model treated as pure algebra | return coefficient to context |
| extrapolates far beyond reasonable interval | model limitation ignored | state intended domain and assumptions |
A 50-Minute Modelling Session
- 8 minutes: classify six contexts by likely function family.
- 10 minutes: interpret parameters in quadratic, exponential and trig models.
- 8 minutes: solve one exponential time question.
- 8 minutes: build one periodic model from range and period.
- 8 minutes: linearise a power/exponential relationship and identify gradient/intercept meanings.
- 8 minutes: solve one constrained optimisation model and audit domain/units.
What Mastery Looks Like
- The learner selects model families from behaviour rather than chapter cues.
- The learner defines variables, units and domain clearly.
- The learner interprets parameters instead of treating them as anonymous constants.
- The learner uses logarithms and transformed graphs to recover model constants.
- The learner builds optimisation objectives before differentiating.
- The learner validates models against known data and contextual constraints.
- The learner can state where a model is useful and where its assumptions may fail.
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