Thinking Skills: Critical Thinking and Problem Solving

3.11 Choosing and using models 119

The current structure of income tax collection

in Bolandia is that the first $2000 of annual

earnings are tax-free (this is called the tax

threshold), then 20¢ of tax is charged on

every dollar earned over this (this could also

be described as a 20% tax rate).

The most obvious and familiar use of the word
‘model’ is that of a replica of an object, for
example a car, at a smaller scale. In this book
the word is used in a wider sense. Models can
be pictures, graphs, descriptions, equations,
word formulae or computer programs, which
are used to represent objects or processes.
These are sometimes called ‘mathematical
models’; they help us to understand how
things work and give simplified
representations that can enable us to do ‘what
if?’ type calculations.
Architects, for example, use a wide range of
models. They may build a scale model of a
building to let the client see it and to give a
better impression of how it will look. Their
drawings are also models of the structure of
the building. In modern practice, these
drawings are made on a computer, which will
contain a three-dimensional model of the
building in digital form. This may be used to
estimate material costs and carry out
structural calculations as well as producing a
three-dimensional ‘walk-through’ picture on
the screen.
This chapter deals with the recognition and
use of appropriate models. A simple example
of a model is a word formula used to calculate
cost. The amount of a quarterly electricity bill
can be described as ‘A standing charge of $35
plus 10¢ per unit of electricity used’. This may
equally be shown algebraically as:

c = 35 + 0.1u

where c is the amount to pay (in dollars) and
u is the number of units used.
A more complicated example of a model
would be the type that governments set up to
simulate their economies. These usually

3.11 Choosing and using models

consist of large numbers of equations and

associated data, and are implemented on

computers. They can predict (with varying

success) things such as what will happen to

the inflation rate if interest rates are raised.

Such models are gross simplifications because

there are too many variables contributing to

the condition of a national economy and all

factors can never be included.

Scientists also use models, for example in

predicting population growth. Such a model,

for example, to predict fish stocks in fishing

areas, can be invaluable as it may be used to

control quotas on fish catches to ensure that

fishing does not reduce stocks to

unsustainable levels.

In both of these cases, the model has been

produced as a result of a problem-solving

exercise. The actual development of a model to

represent a process is beyond the multiple-

choice questions in the lower-level thinking

skills examinations and will be dealt with in

Chapter 5.2. Multiple-choice questions on

choosing and using models test some of the

basic skills involved in modelling and the

extraction of data from mathematical models.

In the following activity you are asked to

use different models to compare calculations.

This example is close to a real-life situation.

Activity