Choosing and evaluating models | Year 10

01 / LEARN

The idea, step by step

Start with the mechanism

Use a linear model for a roughly constant additive change, an exponential model for a roughly constant multiplicative change, and a quadratic model for a suitable second-difference or geometric relationship.

Check that the proposed rule fits the units and time origin. If $t = 0$ means the starting observation, the exponential coefficient should match that value.

Test rather than just fit

Compare predictions with observations not used to choose the model. A residual is observed value minus predicted value; its size helps reveal a poor fit.

Interpolation stays within the observed input range. Extrapolation goes beyond it and depends more strongly on assumptions. Unlimited growth may fail because resources, capacity or conditions change.

A common trap

A curve through two points proves exponential growth forever.

Instead: Two points can fit many models, and a growth assumption may stop being realistic. Use context, further data and stated limits.

02 / WORKED EXAMPLES

See the thinking

Example 1

Build a candidate model

A quantity is $60, 90, 135$ at $t = 0, 1, 2$ . Find an exponential model and predict at $t = 3$ .

The ratio is $1.5$ at each step.
Use $Q = 60 {(1.5)}^{t}$ .
At $t = 3$ , the prediction is $202.5$ .

Q = 60 {(1.5)}^{t}

; 202.5

Example 2

Check a new observation

The model $N = 100 {(1.2)}^{t}$ predicts year 2. The observed value is 139. Find the residual.

The prediction is $100 {(1.2)}^{2} = 144$ .
Residual $= 139 - 144 = - 5$ .
The observation lies 5 below the prediction; investigate whether the difference is reasonable in context.

−5

Check your understanding

A measured value is 52 and its model prediction is 55. Find the residual.

Show answer

−3

Observed minus predicted is $52 - 55 = - 3$ .

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

A quantity is $12, 18, 24, 30$ at equal steps. Choose linear or exponential and justify.
2 marks
Fit an exponential model to $8, 24, 72$ at $t = 0, 1, 2$ .
2 marks
Using $Q = 8 (3^{t})$ , predict at $t = 4$ .
2 marks
A model predicts $250$ and the observation is $238$ . Find the residual.
2 marks
A model is fitted to data from years 0–5. Is using it at year 3 interpolation or extrapolation? What about year 9?
2 marks
A school club grows by $20 %$ monthly in a model. State one reason this may fail over many years.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A quantity is $12, 18, 24, 30$ at equal steps. Choose linear or exponential and justify.
2 marks
The increase is always $6$ , but the ratios vary.
Linear
Fit an exponential model to $8, 24, 72$ at $t = 0, 1, 2$ .
2 marks
Initial value $8$ and ratio $3$ .
$Q = 8 (3^{t})$
Using $Q = 8 (3^{t})$ , predict at $t = 4$ .
2 marks
$8 \times 3^{4} = 8 \times 81 = 648$ .
648
A model predicts $250$ and the observation is $238$ . Find the residual.
2 marks
$238 - 250 = - 12$ .
−12
A model is fitted to data from years 0–5. Is using it at year 3 interpolation or extrapolation? What about year 9?
2 marks
Year 3 lies inside and year 9 outside the observed interval.
Year 3: interpolation; year 9: extrapolation.
A school club grows by $20 %$ monthly in a model. State one reason this may fail over many years.
3 marks
The model assumes a constant percentage and no capacity limit; membership cannot grow beyond all eligible people indefinitely.
A finite school population limits continued growth.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Integer and fractional index laws; connecting fractional powers with roots; simplifying surds; operations with surds; rationalising denominators; exact and approximate values; exponential tables and graphs; growth/decay factors; comparing linear/quadratic/exponential patterns; exponential equations; logarithms as inverses; logarithm laws; interpreting logarithmic scales; graph transformations and parameter experiments; choosing and evaluating models.

Official curriculum source ↗

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