Introduction to Mathematical Modelling
In A-level Mathematics, you aren't just learning how to solve equations; you are learning how to use those equations to describe the real world. This process is called mathematical modelling.
Real life is messy and complicated. If you tried to calculate the path of a falling leaf by accounting for every tiny gust of wind and every twist of the leaf's shape, the math would be impossible! Instead, we create a model—a simplified version of reality that captures the most important features so we can use math to predict what will happen.
The Modelling Cycle
Modelling isn't a one-step process. It is a loop (often called the problem-solving cycle) that looks like this:
1. Set up the model: Take a real-world situation and turn it into mathematical language. This involves choosing variables (like \(t\) for time or \(h\) for height) and making assumptions to simplify the problem.
2. Solve the math: Use the techniques you’ve learned—like differentiation, integration, or trigonometry—to find a solution.
3. Interpret the results: Look at your mathematical answer and see what it means in the real world. If your model for a diver's height gives a result of \(h = -50\) meters, something might be wrong!
4. Evaluate and Refine: Compare your prediction to what actually happens. If the prediction is off, go back to step 1 and change your assumptions to make the model more realistic.
The Power of Assumptions
Assumptions are the "shortcuts" we take to make the math manageable. In the AQA syllabus, you are expected to know why we make these assumptions and how they affect our answers.
Common Assumptions in Mechanics
In Paper 2 (Mechanics), you will often see these terms. They are specific types of assumptions:
"A Particle": We assume an object has mass but no size. This means we can ignore air resistance and rotational movements.
"Light String": We assume the string has no mass. This means the tension is the same all the way along the string.
"Inextensible String": We assume the string does not stretch. This means two connected objects will have the exact same acceleration.
"Smooth Surface": We assume there is no friction (\(\mu = 0\)).
"Constant Acceleration": We assume \(g = 9.8 \text{ ms}^{-2}\) and ignore the fact that gravity changes slightly depending on where you are on Earth.
Common Assumptions in Statistics
In Paper 3 (Statistics), modelling often involves choosing a probability distribution (like the Binomial or Normal distribution) to represent data.
Independence: We often assume that one event doesn't affect the next. For example, in a Binomial model, we assume each "trial" is independent.
Random Sampling: We assume our sample represents the whole population fairly.
Key Takeaway: Assumptions make the math easier to solve, but they make the results slightly less accurate. A good mathematician knows how to balance simplicity with accuracy.
Refining a Model
If a model is not accurate enough, we need to refine it. This usually means "removing" a simplifying assumption and replacing it with something more complex.
Example: Modelling a car's journey.
Initial Model: Assume the car travels at a constant speed.
Problem: The car has to stop at traffic lights and speed up on motorways.
Refinement: Use a piecewise function or variable acceleration (using calculus) to better describe the motion.
Example: Modelling population growth.
Initial Model: Exponential growth (\(P = Ae^{kt}\)).
Problem: In the real world, populations can't grow forever because they run out of food or space.
Refinement: Adjust the model to include a "carrying capacity" or a limit to growth.
Evaluating the "Appropriateness" of a Model
The exam may ask you to "comment on the validity" or "evaluate" a model. When you see this, look for:
1. Sensible limits: Does the model predict something impossible? (e.g., a human growing to 10 meters tall or a probability greater than 1).
2. Domain and Range: A model might work for a short time but fail over a long time. For example, an exponential model for a savings account works well for a few years but might not account for a sudden change in interest rates.
3. Context: Does the math actually fit the story? If you are modelling the number of people on a bus, your model should only give integer (whole number) answers.
Quick Review: How to tackle modelling questions
1. Identify the variables: What is changing? (Time, distance, temperature?)
2. List the assumptions: What has been ignored to make the math work? (Air resistance, friction, thickness of a rope?)
3. Check the units: Ensure everything is in SI units (meters, seconds, kilograms) unless stated otherwise.
4. Critique: Always be ready to suggest one way to make the model better (e.g., "To improve this model, I could include the effect of air resistance").
Summary of Key Terms
Variable: A quantity that can change (e.g., \(x, y, t\)).
Parameter: A constant in a model that can be changed to fit different situations (e.g., the \(k\) in \(y = ke^x\)).
Constraint: A limit on the values a variable can take (e.g., \(t \geq 0\) because time cannot be negative).
Interpretation: Explaining what a mathematical result (like a gradient or an intercept) means in the real-world context.
Don't worry if this seems tricky at first! Modelling is a skill that grows as you learn more Pure, Mechanics, and Statistics. Just remember: a model is a tool, not a perfect replica of reality.