Develop a model

Welcome to "Developing a Model"

In Thinking Skills, "Developing a Model" isn't about building a miniature airplane or walking down a fashion runway! Instead, it’s about creating a system of rules or calculations that represents a real-life situation. Once you have a model, you can use it to find answers, predict what might happen, or find the best way to solve a problem.

Think of a model as a simplified map of reality. A real city has millions of details, but a subway map only shows the lines and stations because that’s the information you need to travel. Developing a model is the process of deciding which "stations and lines" matter for your specific problem.

1. What Exactly is a Mathematical Model?

In the context of Paper 1 (Problem Solving), a model is a set of logical or mathematical rules that describes how different pieces of information relate to each other.

Real-world example: If you want to know how much a taxi ride will cost, the "model" might be:

Total Cost = \( \text{Fixed Pick-up Fee} + (\text{Rate per Kilometer} \times \text{Distance}) \)

This simple formula is a model. It takes complex variables (traffic, fuel, time) and simplifies them into a rule you can use to calculate your fare.

2. The Three Building Blocks of a Model

When you are asked to develop a model in an exam, you are usually looking for three things:

• Variables: These are the things that can change, like "number of people," "hours worked," or "distance traveled."
• Relationships: This is how the variables affect each other. Does the cost go up when the distance increases? (That is a direct relationship).
• Constraints: These are the "rules" or limits. For example, "the bus can only hold 40 people" or "you cannot work more than 8 hours."

Note: If you are looking at how to use a model that has already been given to you, check out the chapter on "Applying a Complex Model." This chapter focuses on how you create the rules yourself from the information provided.

3. Step-by-Step: How to Develop a Model

Don’t worry if this seems tricky at first! Follow these steps to turn a paragraph of text into a working model:

Step A: Identify the Goal

What is the question asking for? Are you trying to find the total cost, the minimum time, or the maximum number of items? This "Goal" will be the subject of your formula (the part before the equals sign).

Step B: List the Information

Read the scenario and pull out the numbers. Let's say a gym charges a \$20 sign-up fee and \$5 per visit.
- Fixed cost = \$20
- Variable cost = \$5
- Number of visits = \( v \)

Step C: Connect the Dots

Combine these into a logical structure.
Total Cost \( (C) = 20 + 5v \)

Step D: Test Your Model

Try a simple number to see if it makes sense. If you go to the gym twice, does your model work?
\( 20 + (5 \times 2) = 30 \). Does that seem right? Yes!

4. Working with Constraints (The "Rules")

Sometimes, a model isn't just one simple formula; it has "conditions." In Thinking Skills, these are often "if/then" scenarios.

Example: A mobile phone plan costs \$10 a month for the first 100 minutes, but 20 cents for every minute after that.
To model this, you need two rules:
1. If \( \text{minutes} \le 100 \), then \( \text{Cost} = \$10 \)
2. If \( \text{minutes} > 100 \), then \( \text{Cost} = \$10 + 0.20 \times (\text{minutes} - 100) \)

Quick Tip: Always be careful with the "extra" amount. In the example above, you only pay the 20 cents for the minutes above 100, not for all of them!

5. Common Pitfalls to Avoid

• Mixing Units: This is a classic trap! If one part of the problem is in minutes and another is in hours, you must convert them so they match before putting them into your model.
• Ignoring "Fixed" Costs: Don't forget the "one-off" payments, like a delivery fee or a deposit, that don't change regardless of the quantity.
• Off-by-One Errors: If a fence has posts every 2 meters for 10 meters, how many posts are there? It’s not 5... it’s 6 (one at the very start!). Always visualize the model.

6. Exam Skills: Communicating Your Reasoning

The 2028 syllabus edition places a heavy focus on Communicating Reasoning. When you develop a model in the exam:

• Label everything: Don't just write "\( 10 + 5 \times 2 \)." Write "\( \text{Fixed fee} + (\text{rate} \times \text{time}) \)."
• Show your units: Use \$, kg, m, or hours clearly in your working.
• Explain your assumptions: If the problem doesn't state something clearly, state what you assumed (e.g., "I am assuming the price per liter remains constant").

Quick Review Box

A Model is: A simplified representation of a situation using rules or math.
Key Components: Variables (changing parts), Relationships (how they connect), and Constraints (limits).
Pro Tip: Always test your model with a simple "easy" number to make sure the logic holds up before doing the complex calculation.

Note: Developing a model is often the first step before you "Adapt a method of solution in response to a change" or "Find optimal solutions." Master this, and the rest of the section will become much easier!