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9.2
Constructing a Mathematical Model
P
urpose
The role this topic plays in quantitative reasoning
A mathematical model that is appropriate for the situation you wish to analyze may not always exist.
Outside of the classroom, many occupations require modeling in an area that is too specific, unique, or
private to use currently available models.
Developing models from scratch or refining existing models is necessary in these situations.
When refining an existing model it is important that you have first completed the Methodology for
Analyzing and Using a Mathematical Model covered in the previous section. Upon completion of this
analysis you will have an explicit need to expand or simplify the model which guides your refinement
procedure. An example of this situation is a business plan for a small company. While many generic
models are available there will most likely be aspects specific to your business type (retail, shipping,
service, etc.) or location (regulations, taxes, permits, etc) which are not included in the public offering.
This explains why most business students learn the process for developing business plans rather than
deferring to someone else’s.
In the case where no appropriate model is available, then one must be constructed using the methodol-
ogy given in this section. In scientific research, new materials or systems are discovered on a frequent
basis. It is the unknown qualities of these mathematical relationships which provoke so much interest
and yet abate the “copy and paste” routine we encounter in a classroom science problem. While you
may design the model for the sole primary purpose, it may become the basis for a completely new field
of study later on.
The purpose of creating models in this section is to learn the process for abstracting the relationship
at hand. Initially we strive to find the essence of a phenomenon and use the most important quantities
and relationships to explain the behavior. The simple model is then compared with the actual situation
to determine its accuracy. The model is refined by incorporating more variables or a more complex
relationship. This cyclic process of formulate, use, analyze, and adjust repeats until the desired accuracy
is achieved. Whether starting from a well-developed tool or your own ideas each situation likely merits
all the steps of the methodology. You already construct mathematical models in your everyday life
without even noticing it. Through cognizance of the process, learning skills in this area will become
efficient and powerful. With practice, modeling will become part of your natural approach to studying
situations so that you can predict, manage, or understand them. In many lucrative fields, this set of skills
makes you an attractive applicant.
L
earning Goals
What you should learn while completing this activity
1. Identify when no currently available model is ideal for analyzing your situation
2. Create a simple mathematical model from a specific situation by selecting the relevant inputs and
outputs and formulating the relationships into equations or algorithms.
3. Use and analyze the model to determine if it needs to be expanded to accurately represent the
given situation.