Making a mathematical model A model is a representation of a real situation. A real situation will contain a rich variety of detail. A model of it will simplify reality by extracting those features which are considered to be most important. REAL WORLD MODELLING MATHEMATICS

The modelling process Real-world problem Simplifying assumptions Mathematical model (equations etc) Analysis and solve Prediction Interpretation Validation Experiment The pink is the real, physical world which may include experimentation to generate data, where there is a problem. The orange is the interface between the real and the mathematical world. This is where the modelling takes places as the real context is translated into a simplified mathematical problem. This involves making simplifying assumptions, defining constants and variables etc. The problem then becomes a mathematical one which can be solved and/or analysed using mathematical techniques. The solutions should be interpreted fully and any predictions compared with the real situation or tested experimentally. The solution make be a particular one or a generalised results and the interpretation may uncover interesting features of the real context that could lead to further investigation. This a cycle and by altering the initial assumptions or refining the model new solutions will be revealed.

The problem Problem: How far apart should the bumps be placed so that the speed of cars around the estate is kept below 20mph?

Refining a model How reasonable is the answer? Is it an over or an under estimate? How could the model be improved to give a better estimate? What effects would the refinements make – how would you go about calculating these? Justify your thinking.