Choosing and evaluating models | Year 10

01 / LEARN

Understand the idea.

Start with the mechanism

Choose a model by the process producing the change. Constant additions suggest linear behaviour; constant percentage changes suggest exponential behaviour; constant nonzero second differences at equal inputs suggest a quadratic candidate.

Define the input, output and time origin before fitting a rule. In $Q = A b^{t}$ , the coefficient $A$ is the value at $t = 0$ , while $b$ is the multiplier for each unit of the chosen time scale.

Test rather than just fit

Test predictions against observations that were not used to choose the model. A residual is observed minus predicted: a positive residual lies above the prediction and a negative residual lies below it.

Interpolation predicts within the observed input range; extrapolation predicts beyond it. A good short-term fit does not justify unlimited growth. Resource limits, changing rates or measurement uncertainty can all affect how far a model should be trusted.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Build a candidate model

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

Worked solution

1
Compare the consecutive ratios.
$\frac{90}{60} = 1.5, \frac{135}{90} = 1.5$
Equal time steps have the same multiplicative change.
2
Use the starting value and factor.
$Q = 60 {(1.5)}^{t}$
The value at $t = 0$ is $60$ , and each unit step multiplies by $1.5$ .
3
Predict at the requested time.
$Q (3) = 60 {(1.5)}^{3} = 202.5$
Three steps apply the same factor three times; this is an extrapolated model prediction.

Answer

$Q = 60 {(1.5)}^{t}$ ; 202.5

EXAMPLE 02

Check a new observation

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

Worked solution

1
Calculate the year-two prediction.
$\hat{N} = 100 {(1.2)}^{2} = 144$
The exponent counts the two annual periods in the supplied model.
2
Subtract predicted from observed.
$residual = 139 - 144 = - 5$
The sign convention is observed value minus model prediction.
3
Interpret the sign.
$139 = 144 - 5$
The observation is $5$ below the prediction; deciding whether that error is acceptable needs context.

Answer

−5

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

−3

Use observed minus predicted, keeping that order: $52 - 55 = - 3$ . The measured value is $3$ below the model prediction.

Keep practising

6 worksheet questions · 13 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

Start with the mechanism

Test rather than just fit

See the method in action.

Build a candidate model

Compare the consecutive ratios.

Use the starting value and factor.

Predict at the requested time.

Check a new observation

Calculate the year-two prediction.

Subtract predicted from observed.

Interpret the sign.

Try it yourself.

Check your understanding

Keep practising