Overview (Leaving Cert Applied Maths): Revision Notes
Overview
Mathematical modelling is a powerful approach that transforms complex, messy real-world situations into clear mathematical problems that can be solved systematically. This process bridges the gap between everyday challenges and mathematical solutions, making it an essential tool across many fields from business to science.
Mathematical modelling essentially acts as a translator between the complexity of real-world problems and the precision of mathematical solutions, allowing us to apply logical, systematic approaches to messy, complicated situations.
What is mathematical modelling?
Mathematical modelling is the process of using mathematical concepts, techniques, and tools to represent and solve real-world problems. It involves creating simplified mathematical versions of complex situations so we can:
- Test different solutions before implementing them
- Make informed decisions based on mathematical analysis
- Predict future outcomes
- Optimise processes and systems
The key strength of mathematical modelling lies in its systematic approach, which ensures problems are tackled in a logical and structured manner.
The problem-solving cycle
Mathematical modelling follows a cyclical process with five main stages. This cycle is iterative, meaning you often need to go back and refine your work if the initial solution isn't adequate.
The modelling process is iterative by design - don't expect to get the perfect solution on your first attempt. Most real-world problems require several cycles of refinement to achieve satisfactory results.
Stage 1: Identifying the problem
This initial stage involves clearly understanding what needs to be solved. The key activities include:
- Recognising a real-world situation that needs solving
- Clearly defining what you want to achieve
- Understanding the context and constraints
Worked Example: Problem Identification
A car rental company wants to maximise their profits and needs to determine the best pricing strategy.
Problem identification:
- Situation: Car rental pricing
- Goal: Maximise company profits
- Context: Competitive market with price-sensitive customers
Stage 2: Formulating the problem
During formulation, you need to transform the real-world problem into a form that can be mathematically analysed. This involves several critical steps:
- Research the background: Gather relevant context and data about the situation
- Identify variables: Determine which quantities affect the situation (such as time, distance, speed, or cost)
- Determine relevant information: Focus on what's important and ignore unnecessary details
- Break the problem down: Separate complex issues into smaller, manageable parts
- Make assumptions: Simplify the situation without losing essential accuracy
Worked Example: Virus Modelling Formulation
Problem: Model the spread of a virus through a population
Formulation process:
- Variables: Infection rate, recovery rate, population size, time
- Key assumption: Recovery provides permanent immunity
- Simplification: Assume random mixing of the population
- Background research: Study similar epidemiological models and historical data
Stage 3: Translating into mathematics
This stage involves abstraction - stripping away unnecessary details to capture the essence of the system. You'll need to transform your formulated problem into mathematical language:
- Build the model: Express relationships using equations, graphs, or diagrams
- Link assumptions with mathematics: Turn real-world statements into mathematical symbols and expressions
- Create equations: Develop mathematical expressions that represent the relationships in your problem
Worked Example: Falling Ball Translation
Real-world assumption: Air resistance is negligible for a dropped ball
Mathematical translation: Distance equation:
Where:
- = distance fallen
- = initial velocity
- = time
- = gravitational acceleration (9.8 m/s²)
Stage 4: Computing solutions
Once you have your mathematical model, you need to apply appropriate mathematical techniques to find solutions. This stage focuses on solving and analysing:
- Apply mathematical techniques: Use algebra, calculus, or numerical methods to solve your equations
- Use computational tools: Employ software or calculators for complex calculations
- Analyse the model: Test how sensitive your results are to changes in input values
Worked Example: Demand and Profit Optimisation
Given model: Demand function where is price
Profit calculation: Profit = Price × Demand =
Finding maximum profit:
Solution:
Result: Maximum profit occurs at a price of €25
Stage 5: Evaluating solutions
The final stage involves checking and improving your work to ensure it provides meaningful real-world solutions:
- Check accuracy: Does your solution make sense in the real world?
- Refine assumptions: If results seem unrealistic, improve your model by adjusting assumptions
- Predict and improve: Use your model to forecast outcomes and suggest improvements
- Communicate clearly: Present your solutions in written reports that others can understand
Evaluation is often the most critical stage because it determines whether your mathematical solution actually solves the real-world problem. A mathematically correct answer that doesn't make practical sense indicates the need to revisit earlier stages.
Worked Example: Weather Model Evaluation
Initial model: Basic temperature prediction using only historical averages
Evaluation findings: Predictions were inaccurate during seasonal transitions
Refinements made:
- Added wind speed variables
- Included humidity factors
- Incorporated pressure system data
Result: Significantly improved prediction accuracy
Real-world applications
Mathematical modelling is used extensively across numerous fields, demonstrating its versatility and practical value:
- Business: Optimising profits, supply chain management, market analysis
- Medicine: Predicting disease spread, treatment effectiveness
- Weather forecasting: Climate prediction and weather warnings
- Engineering: Structural design, system optimisation
- Economics: Market behaviour, risk assessment
The breadth of applications shows that mathematical modelling is not just an academic exercise - it's a practical toolkit used daily by professionals to solve real problems and make informed decisions across virtually every industry.
Exam tips
Critical Exam Strategies:
- Always show each stage of the modelling cycle clearly in your answers
- Justify your assumptions - explain why they're reasonable for the given context
- Check that your final answer makes sense in the original real-world situation
- Practice identifying which variables are most important in different scenarios
- Remember that modelling is iterative - be prepared to explain how you might refine your approach
- Show all mathematical working clearly, especially in the computation stage
Key Points to Remember:
- Mathematical modelling transforms real-world chaos into solvable mathematical problems
- The five-stage cycle (identify, formulate, translate, compute, evaluate) provides a systematic approach to problem-solving
- Making reasonable assumptions is crucial - they simplify problems without losing essential accuracy
- The process is iterative - expect to refine and improve your models based on results
- Successful modelling requires both mathematical skills and real-world understanding to create practical solutions that actually work in practice