How to Test Whether a Mathematical Model Makes Sense — Secondary Mathematics

Testing a model is a form of mathematical quality control. The student asks whether the output is sensible, whether the relationship fits known information, whether the units are consistent, and whether the model behaves reasonably at important values.

Test 1: check the units

Suppose a model claims distance d=v+t. If v is measured in km/h and t in hours, the right side attempts to add unlike dimensions. The model is structurally invalid.

By contrast, d=vt gives km/h×h=km. Dimensional consistency does not prove the model is correct, but inconsistency can reject it quickly.

Test 2: use known values

A fictional service charges $12 fixed plus $5 per use. Proposed model A is C=12+5n. At n=0, it gives C=12, matching the fixed charge. At n=1, it gives 17.

Proposed model B is C=12n+5. At n=0, it gives 5, immediately contradicting the stated starting cost. Boundary values are powerful tests.

Test 3: check the sign

If a model predicts a negative number of people, negative physical length or negative elapsed time in a context where those quantities cannot be negative, either the domain has been exceeded or the model is inappropriate.

Negative numbers are not automatically wrong. A temperature difference, profit/loss balance or coordinate can be negative. Context determines the allowed sign.

Test 4: inspect the intercept

For y=mx+c, c is the value when x=0. Ask whether that starting value makes sense. A model for total cost may reasonably have a positive fixed charge. A model for distance travelled from a starting point may be expected to begin at zero if distance is measured from the start.

Test 5: inspect the gradient or rate

If a graph models water draining from a tank, a positive gradient would mean volume increases over time, which may contradict the described situation unless inflow is also present. The sign and magnitude of the gradient should match the story.

Test 6: use extreme or boundary cases

If a formula models ticket cost, test zero tickets. If it models area from a length, test a very small positive length. If it models a tank, test the capacity boundary. If it models percentage growth, ask what happens after many periods.

Extreme cases can expose hidden assumptions that ordinary values do not reveal.

Test 7: compare prediction with data

Suppose a model predicts y=3x+2. Observed data are (1,5), (2,8), (3,11), (4,14). The model fits these points exactly.

If observed values were 5,8,12,17, the errors would grow. That pattern suggests the model may be missing curvature or another changing-rate effect.

Residual thinking without formal statistics

A residual is observed value minus predicted value. Students can compare these differences informally. Small mixed positive and negative residuals may indicate a reasonable simple fit; systematic residual patterns may indicate model bias.

Test 8: check whether the relationship type is plausible

A fixed percentage growth process should usually be multiplicative, not additive. A fixed unit price with no fee should usually be proportional. A rectangle’s area depends on a product of lengths, not their sum.

Choosing the wrong family of relationships can produce reasonable answers near one point and poor behaviour elsewhere.

Worked model test 1: constant rate

A tank contains 50 L and fills at 10 L/min. Model V=50+10t. At t=0, V=50. At t=5, V=100. Units are consistent. If tank capacity is 120 L, the model should only be used until t=7 unless overflow is included.

Worked model test 2: percentage growth

A student models 5% yearly growth from 1000 as A=1000+50n. This gives correct first-year growth but assumes a fixed increase of 50 every year. If the intended model compounds 5% on the current amount, the correct relationship is A=1000(1.05)^n.

Testing n=2 distinguishes them: linear model gives 1100; compound model gives 1102.50.

Worked model test 3: geometry

A student models the area of a circle as A=2πr. Units are length, not area, so dimensional reasoning rejects the formula immediately. A=πr² has area units.

Worked model test 4: line fit

Observed points are approximately (0,10), (1,14), (2,18), (3,22). Model y=10+4x is plausible because it matches the starting value and constant increase. A quadratic model could also be made to fit selected points, but it would add complexity without evidence that curvature is needed.

Simple versus complicated models

A more complicated model is not automatically better. If a straight line answers the question accurately enough, adding many parameters may make interpretation harder and prediction less stable.

Use the simplest model that captures the important structure for the intended purpose.

Model validation is purpose-dependent

A rough estimate may be enough for planning. A safety-critical engineering calculation needs much tighter validation. In Secondary Mathematics, students should learn that “good enough” depends on what the model is being used for.

Common checking failures

Checking only arithmetic. Correct calculation does not validate assumptions.

Testing only one convenient value. A wrong model may accidentally fit one point.

Ignoring domain. Extrapolating far beyond the observed range can be dangerous.

Ignoring units. Dimensional errors often expose structural mistakes immediately.

Assuming a graph that looks close must be correct. Visual fit depends on scale and precision.

A model-testing routine

1. Check units.
2. Check the value at zero or another natural boundary.
3. Check signs and domain restrictions.
4. Check rate, intercept and graph shape.
5. Compare with known data.
6. Test an extreme case.
7. Interpret whether errors are acceptable for the intended purpose.
8. Revise the model if the failure is systematic.

Diagnostic checkpoints

• Can the student use units to reject an impossible formula?
• Can they test a natural boundary value?
• Can they distinguish interpolation from extrapolation informally?
• Can they compare prediction with data?
• Can they explain why model quality depends on purpose?

Practice prompts

1. Model C=5+3n for a fixed-charge service. What does C(0)=5 mean? The fixed charge.

2. Model d=v+t. Reject or accept? Reject dimensionally if d is distance, v speed and t time.

3. Linear model y=2x+1 predicts 11 at x=5; observation is 11.4. Residual=0.4.

4. A growth model predicts negative population after many periods. What should be questioned? Domain and model form.

The larger mathematical habit

Testing a model trains students to challenge mathematics responsibly. The question is not only “Can I solve it?” but “Does this representation deserve to be trusted here?”

Build from What Is a Mathematical Model? and From a Real Situation to Variables, Assumptions and Equations, then continue to When a Model Fails. Return to the Secondary Mathematics Master Index.