210. Evaluating Applied Models

Learning Intentions

• Assess whether a model fits the given information.
• Identify assumptions made when modelling change.
• Communicate findings Use mathematical language and evidence.

Pre-requisite Summary

• Know that a mathematical model is a simplified representation of a real situation.
• Know that linear models have constant rates of change and quadratic models have changing rates of change.
• Know how to compare data values with predicted values from a model.
• Know that a model may fit some data better than others.
• Know that assumptions are statements accepted as true when building a model.
• Know that mathematical evidence includes calculations, tables, graphs and comparisons.

Worked Examples

Worked Example 1

A student models the data using the rule .

Actual

a) Use the model to calculate predicted values.

b) Compare the predicted values with the actual values.

c) Decide whether the model fits the data well.

Worked Example 2

A student models the data using the rule .

Actual

a) Use the model to calculate predicted values.

b) Compare the predicted values with the actual values.

c) Decide whether the model fits the data well.

Worked Example 3

A savings model is given by , where is savings in dollars and is weeks.

a) State one assumption made by the model.

b) Explain why the assumption may not always be realistic.

c) Suggest one factor that could affect the accuracy of the model.

Worked Example 4

A ball’s height is modelled by , where is height in metres and is time in seconds.

a) State one assumption made by the model.

b) Explain how air resistance could affect the model.

c) Decide whether the model is likely to work for negative values of .

Worked Example 5

A student claims that the model is appropriate for a phone plan.

The actual costs are shown below.

Months
Actual cost	20$	28$	36$	44$

a) Test the model against the data.

b) Use mathematical evidence to support your conclusion.

c) Explain why the model is reasonable.

Worked Example 6

A student models the area of a square using .

a) Explain why this model may not fit the situation.

b) Suggest a more appropriate type of model.

c) Justify your answer using mathematical language.

Worked Example 7

A business uses the model for profit, where is profit in dollars and is days.

a) State one assumption made by the model.

b) Explain one limitation of the model.

c) Communicate whether the model is useful using mathematical evidence.

Problems

Problem 1

A student models the data using the rule .

Actual

a) Use the model to calculate predicted values.

b) Compare the predicted values with the actual values.

c) Decide whether the model fits the data well.

Problem 2

A student models the data using the rule .

Actual

a) Use the model to calculate predicted values.

b) Compare the predicted values with the actual values.

c) Decide whether the model fits the data well.

Problem 3

A savings model is given by , where is savings in dollars and is weeks.

a) State one assumption made by the model.

b) Explain why the assumption may not always be realistic.

c) Suggest one factor that could affect the accuracy of the model.

Problem 4

A ball’s height is modelled by , where is height in metres and is time in seconds.

a) State one assumption made by the model.

b) Explain how wind resistance could affect the model.

c) Decide whether the model is likely to work for negative values of .

Problem 5

A student claims that the model is appropriate for a subscription service.

The actual costs are shown below.

Months
Actual cost	15$	21$	27$	33$

a) Test the model against the data.

b) Use mathematical evidence to support your conclusion.

c) Explain why the model is reasonable.

Problem 6

A student models the area of a square using .

a) Explain why this model may not fit the situation.

b) Suggest a more appropriate type of model.

c) Justify your answer using mathematical language.

Problem 7

A business uses the model for profit, where is profit in dollars and is days.

a) State one assumption made by the model.

b) Explain one limitation of the model.

c) Communicate whether the model is useful using mathematical evidence.

Exercises

Understanding and Fluency

Exercise 1

For each situation, state whether the model is linear or quadratic.

a)

b)

c)

d)

Exercise 2

Use each model to calculate predicted values.

a) when

b) when

c) when

d) when

Exercise 3

Compare the predicted values with the actual values.

a)

Actual
Predicted

b)

Actual
Predicted

Exercise 4

State whether each model fits the information well.

a) Predicted and actual values are identical.

b) Predicted values differ slightly from actual values.

c) Predicted values are very different from actual values.

Exercise 5

State one assumption for each model.

a) for savings

b) for pay

c) for height

d) for subscription cost

Exercise 6

State one possible limitation for each model.

a) A savings model

b) A ball height model

c) A phone plan model

d) A business profit model

Exercise 7

Explain whether each statement is reasonable.

a) “The model is useful because the predicted values are close to the actual data.”

b) “The model is perfect because it works for one data point.”

c) “The model may become less accurate outside the data range.”

Exercise 8

Use mathematical language to describe each model.

a) “The model has a constant rate of change.”

b) “The model shows increasing growth over time.”

c) “The graph has a turning point.”

d) “The predicted values closely match the observed data.”

Exercise 9

Identify the mathematical evidence used.

a) Comparing predicted and actual values

b) Using a graph to estimate values

c) Using substitution into a formula

d) Calculating first and second differences

Exercise 10

Complete each sentence.

a) A model is a __________ representation of a real situation.

b) An assumption is something accepted as __________ when building a model.

c) A model fits well when predicted values are close to __________ values.

Reasoning

Exercise 11

Explain why a model that perfectly fits a small set of data may still have limitations.

Exercise 12

Explain why assumptions are important when creating mathematical models.

Exercise 13

A student says that a model is useless if one prediction is incorrect.

Explain why this statement is too strong.

Exercise 14

A student claims that is a good model for the area of a square.

Explain why this model confuses area with perimeter.

Exercise 15

A company uses the model for profit.

Explain why the assumption of constant daily profit may not always be realistic.

Exercise 16

Decide whether each statement is true or false. Justify your answer.

a) A useful model must always give exact predictions.

b) Mathematical evidence can include tables, graphs and calculations.

c) A model can still be useful even if it includes simplifying assumptions.

Problem-solving

Exercise 17

A student models the data using .

Actual

a) Calculate predicted values.

b) Compare the predictions with the actual data.

c) Decide whether the model fits well.

d) Communicate your conclusion using mathematical evidence.

Exercise 18

A savings plan is modelled by .

a) State one assumption made by the model.

b) Explain one factor that could affect the accuracy of the model.

c) Explain whether the model is useful over long time periods.

d) Support your answer using mathematical language.

Exercise 19

A ball’s height is modelled by .

a) Explain one assumption made by the model.

b) Explain why negative values of are not meaningful in context.

c) Describe one limitation of the model.

d) Communicate whether the model is still useful.

Exercise 20

Create your own modelling investigation.

Your response must include:

• a real-world situation
• a linear or quadratic model
• at least four data points
• a comparison of predicted and actual values
• one assumption
• one limitation
• a conclusion supported by mathematical evidence

Potential Misunderstandings

• Students may think a model must match every data point exactly to be useful.
• Students may confuse predicted values with actual observed values.
• Students may believe that a model that works for one value must work perfectly for all values.
• Students may not recognise that every model includes assumptions.
• Students may confuse assumptions with proven facts.
• Students may ignore practical limitations such as changing conditions, measurement error or unrealistic input values.
• Students may communicate conclusions without mathematical evidence.
• Students may use informal language instead of mathematical terms such as “rate of change”, “predicted values” or “quadratic growth”.
• Students may make conclusions based on opinion instead of calculations, graphs or comparisons.