Welcome to Modelling Real-World Situations!

In Paper 3 (Problem Analysis and Solution), you aren't just solving a single math problem. You are taking a complex "real-world" scenario—like running a theme park, organizing a delivery route, or managing a shop's inventory—and turning it into a mathematical or logical model.

In this chapter, we will learn how to build a basic model and then refine it (make it more detailed) as the question adds new rules or changes. Don't worry if this seems tricky at first; Paper 3 is designed to lead you through these stages one step at a time!

1. What is a "Model"?

A model is a simplified version of reality. Just like a map isn't the actual ground but helps you navigate, a mathematical model uses numbers and rules to represent a situation so we can make predictions or find solutions.

The Building Blocks of a Model:
- Information: The data you are given (e.g., "The bus holds 40 people").
- Variables: Things that can change (e.g., "The number of tickets sold").
- Constraints: Limits you must follow (e.g., "The bus cannot leave before 09:00").

Analogy: Think of a model like a recipe. The ingredients are your data, and the instructions are your logical rules. If you change the number of guests (a variable), you must refine the recipe (the model) to make more food!

2. Step-by-Step: Developing and Applying a Model

In Paper 3, questions often ask you to develop a model from a description. Follow these steps:

Step A: Identify Relevant Information

Scenarios in Paper 3 are long. Your first job is to ignore the "fluff" and find the numbers and rules that actually matter for the calculation.

Step B: Set Up the Calculation

Organize the calculations needed to solve the problem. For example, if you are calculating total cost, your simple model might be:
\( \text{Total Cost} = (\text{Price per Item} \times \text{Quantity}) + \text{Shipping Fee} \)

Step C: Identify Cases that Satisfy Criteria

Sometimes the model isn't just a formula; it's a set of conditions. You might need to check which options (e.g., different delivery vans) meet the requirements of your model (e.g., can carry at least \( 500\text{kg} \) and cost less than \( \$200 \)).

3. Refining the Solution in Stages

A key feature of Paper 3 is that the problem develops. A "refined" solution is just a "better" or "more realistic" solution that accounts for new information.

How models are refined in exam questions:
- Introducing a Change: "What if the price of fuel increases by \( 10\% \)?" You must adapt your method to include this new cost.
- Adding a Constraint: "The driver must now take a 30-minute break every 4 hours." This refines your schedule model.
- Complex Deductions: Moving from a simple "yes/no" answer to finding the optimal solution (the best possible outcome, like the lowest cost or the shortest time).

Quick Tip: When the question says "The rules have now changed...", don't throw away your earlier work! Usually, you just need to add one extra step or adjust one number in your previous calculation.

4. Finding the Optimal Solution

An optimal solution is the "best" answer given the constraints. This usually means:
- Maximizing something (like profit or number of people helped).
- Minimizing something (like waste, time, or cost).

To find the optimal solution, you often need to search for solutions. This might involve testing the "extreme" cases. For example, if you want to minimize cost, test the cheapest options first and see if they satisfy all the rules of your model.

5. Communicating Your Reasoning

From 2028 onwards, the syllabus places a heavy focus on communicating reasoning. Even for exams in 2027, showing your work is essential for "method marks" if your final answer is wrong.

How to justify your solution:
1. Label your numbers: Don't just write "\( 10 \times 5 = 50 \)". Write "\( 10\text{ toys} \times \$5 = \$50 \text{ income} \)".
2. Explain your steps: Use short phrases like "Total cost including tax" or "Time after adding the break".
3. State Assumptions: If the problem doesn't give a specific value and you have to make a logical guess to proceed, state it clearly (e.g., "Assuming the van travels at a constant speed").

Key Takeaway Box

1. Understand: Pick out the key numbers and rules.
2. Model: Create a simple calculation or list of conditions.
3. Refine: Adjust the model when the question adds new "real-world" complications.
4. Optimize: Find the highest or lowest possible result.
5. Communicate: Label your units and show every step of your logic!

6. Common Mistakes to Avoid

  • Ignoring Units: If the model uses minutes but the answer asks for hours, always convert. (e.g., \( 90 \text{ minutes} = 1.5 \text{ hours} \)).
  • Missing a Constraint: Read the "small print." Does the rule say "more than 5" or "5 or more"? This changes your model!
  • Forgetting the "Refinement": When you move to the next part of a Paper 3 question, check if the previous rules still apply or if they have been replaced.
  • Not Showing Working: In Paper 3, a "Yes" or "No" without a calculation to back it up will often receive zero marks, even if the "Yes" is correct.

Did you know? In Paper 3, the questions usually get more difficult as you go through the paper, but they also get more rewarding! The 15-mark questions (in the 2028 format) allow you to pick up many marks for "correct steps" even if you don't reach the very final optimal solution.

Cross-reference: Once you have mastered modelling, you will use these skills to Combine problem solving skills across an extended scenario in the next stage of your Paper 3 preparation.