Some Didactical Aspects in Mathematical Modelling

Abstract

Introduction

Mathematical modelling is the adaption of mathematical knowledge in resolving real every-day problems. This is a complex cognitive process, so must be taught as early as feasible. In learning how to apply mathematics to solve such problems, an effective learning method is to deal with selective case studies. In this way, the necessary skills may be developed effectively. The aim of this article is to provide guidance for mathematical modelling at high school level.

The process of mathematical modelling

1. Identifying the problem.

2. Formulation of a mathematical model.

3. Solution of the corresponding mathematical problem.

4. Evaluation and explanation (commonly called validation of the model).

This division can be executed in form of a ``questions'' and ``to do'' list. (1) Identifying the problem. Initial appraisal and recognition of the problem.

• Understanding the problem.
• Differentiating between input and output quantities.
• What information do we possess?
• What solution is desired?
• List all quantities affecting the problem together with their units and dimensions.
• Denote each quantity by an appropriate notation.
• Collect data (of obvious importance).

• Make reasonable assumptions, approximations, estimations and make documentation.
• When necessary draw graphs and diagrams.
• What results can be anticipated?
• Specify relationships between the various quantities.
• Begin with a rough model and refine progressively.

• Interpret the solution and compare it with reality.
• Investigate contributory factors of the solution (sign, continuity, etc.).
• Check the solution for special quantities.
• Does the model fulfill its purpose?
• Does the model consider all important aspects of the problem?
• Is the mathematical process free from errors?
• Documentation.
• Repeat the evaluation process and make corrections until a satisfactory result is attained.

Note: The second step of our model building is undoubtedly the most difficult, being more abstract in character than the other steps. But it is an invaluable aspect of the modelling process and demands a high degree of creativity.

Teaching mathematical modelling

HEIGHT AND WEIGHT

Problem: derive a mathematical relationship between height and weight of adult persons.

(1) Identifying the problem

There would seem to be some relationship between body height and weight, since we often assume that taller persons are heavier than shorter persons. Let us try to clarify this assumption. With the use of appropriate abbreviations we list all known quantities influencing the problem.

To set up a first model we neglect the influence of age, gender and body density. We collect and record data: height and weight of different persons (e.g. parents, teachers, relatives, friends) are measured and listed in a table. Some data collected from adult women are displayed on the next page.

Thus, with unknown constants a,b we formulate the following:

weight = a+b·height.

(3) Solution of the corresponding mathematical problem.
The parameters a,b may be extracted directly from our diagram but are more accurately assessed by using a computer algebra system. By fitting a curve to the data we obtain (MAPLE) the values -56.919 for a and 0.736 for b respectively (see Figure 1(b)).

(4) Evaluation and explanation.
The mathematical model is evaluated by measuring various persons and then comparing with the formula:

weight = -56.919+0.736·height.

We observe from the curve that measurements deviate substantially, and we must conclude that the results of our modelling process are to be considered only as a first approximation.
For further investigation:

• Has our mathematical model fulfilled its purpose ( to appropriate degree of accuracy)?
• Has our mathematical model taken into account all the important aspects of the problem?
• How can our mathematical model be improved (e.g. some non-linear curve)?
• Can our mathematical model be extended to integrate other factors (e.g. age)?

  
  

LIQUID IN TANKS

(1) Identifying the problem.
The volume of the fluid is proportional to the cross-sectional area of the tank and not the depth of the fluid. The question is why? Let us consider the tank with its elliptical cross-section and semi-axes b > a > 0. The quantities influencing the problem are as follows:

V = area of fluid × length of tank .

This area must now be determined; we accomplish this in the next step.

(3) Solution of the corresponding mathematical problem.
From the diagram we observe that the cross-sectional area of the elliptic sector S_ellipse has to be initially determined. It is

Sellipse = 2 ó õ x0 0  b a æ è Ö   a2-x2   dx ö ø -2x0y0.

We consider the integral separately and substitute according to

x = asin( u) ,a > 0.

Thus we derive for the indefinite integral

2 b a ó õ Ö   a2-x2   dx = 2ba ó õ cos2( u) du,

Integrating by parts, we obtain

2ba ó õ cos2( u) du = 2ba æ ç è 1 2 ( u+sin( u) cos( u) ) ö ÷ ø = ba æ ç è arcsin æ ç è x a ö ÷ ø + 1 2 x Ö   a2-x2   ö ÷ ø .

Hence:     area of fluid = area of ellipse -S_ellipse,  that is:

area of fluid = 2pab- é ê ë ba æ ç è arcsin æ ç è x0 a ö ÷ ø + 1 2 x0 Ö   a2-x2   ö ÷ ø -2x0y0 ù ú û .

(1)

The case h < b is reduced to the case h ³ b by rotating the tank through 180^°.

(4) Evaluation and explanation.
Let us consider the case when a fluid-tank has a spherical form with a radius 1m and the height of the fluid in the tank is 0.5m ; we derive [(p)/2]·1000m³ = 1570.8m³ which proves that our mathematical model is correct for this special case. This check gives us confidence in our calculations. Now formula (1) is evaluated for different values of the parameter a,b and d, giving the following results:

A further possible investigation: Derive a corresponding mathematical model for other shaped fluid-containers.

SUGGESTIONS FOR FURTHER PROJECTS:

Many students are interested in sports. Consider for example the Olympic Games as a rich source for problems to resolve through mathematical modelling. Reflect on the following questions:

• In which year can we anticipate that a woman sprinter will run 100 meters in under 10,0 seconds?
• Construct a mathematical model to simulate a tennis match.
• Is there an successful strategy for penalty-kicks in soccer?
• Does rarefied air influence results in long jump (see Bob Beamon, Mexico 1968)?

HINT: Information on recent records in track and field can be found, for example, at the following website:
www.dlv-sport.de/Ergebnisse/weltrekordentwickling.sthml

References

[1] D Burghes, P Galbraith, N D Price, A Sherlock : Mathematical Modelling. Prentice Hall; London, New York, 1996.

[2] F Giordano, M Weir : A First Course in Mathematical Modelling. Brooks/Cole Publishing Company, Belmont California, 1985.