Introduction: What is a "Simple Model"?

In Thinking Skills, a model isn't a plastic airplane or a fashion runway! In the context of Problem Solving, a model is a set of rules, formulas, or instructions that describe how a real-world situation works. When we "apply a simple model," we take specific information and run it through these rules to find an answer.

Think of a model like a recipe. The "model" is the recipe itself (the rules), and the "data" is your ingredients. If you follow the recipe correctly, you get the right result every time. This skill is vital for Paper 1: Problem Solving.

1. Identifying the Components of a Model

Most simple models in your exam will involve everyday scenarios like taxi fares, mobile phone plans, or reward point systems. To apply a model successfully, you need to identify two things:

A. The Fixed Elements: These are things that do not change. For example, a "booking fee" for a concert ticket that stays the same no matter how many tickets you buy.

B. The Variable Elements: These change based on the situation. For example, the total cost of the tickets depends on how many you buy.

Quick Note: This chapter focuses on applying a model that is already given to you. For "Developing" or "Creating" your own model, see the related chapters in the "Consider wider problems" section.

2. Common Types of Simple Models

Don't worry if the math seems daunting at first. The syllabus limits these models to basic operations. You will usually encounter:

  • Linear Models: These usually look like: \( \text{Total} = \text{Fixed Fee} + (\text{Rate} \times \text{Units}) \).
  • Threshold Models: Rules that change once a certain limit is hit (e.g., "The first \( 10 \) items cost \( \$5 \) each, but every item after that costs \( \$3 \)").
  • Selection Models: Choosing the best option from a set of rules (e.g., "Which gym membership is cheaper if I visit \( 12 \) times a month?").
Example Scenario: The Delivery Service

The Model: A delivery company charges a basic fee of \( \$10 \). Additionally, they charge \( \$2 \) for every kilogram (\( kg \)) the package weighs. However, if the package weighs more than \( 20kg \), the total price is discounted by \( 10\% \).

Let's apply this model to a \( 25kg \) package:

1. Calculate the basic cost: \( \$10 + (25 \times \$2) = \$10 + \$50 = \$60 \).
2. Identify if the threshold is met: Yes, \( 25kg \) is more than \( 20kg \).
3. Apply the discount: \( \$60 - (10\% \text{ of } \$60) = \$60 - \$6 = \$54 \).
4. Final Answer: \( \$54 \).

3. Step-by-Step Guide to Applying a Model

When you face a problem in Paper 1, follow these steps to avoid mistakes:

Step 1: Read the "House Rules"
Identify exactly what the rules are. Look for keywords like "fixed," "per hour," "each," or "maximum."

Step 2: Organize the Inputs
Write down the specific numbers you have been given for the current problem. (See the chapter on Organising the calculation(s) needed to solve a problem for more on this).

Step 3: Perform Operations in Order
Follow the logic of the model. Usually, you calculate the totals first, then apply any special rules like taxes, discounts, or bonuses.

Step 4: Check Constraints
Does your answer make sense? If a model says the "maximum charge is \( \$100 \)" and your calculation gave you \( \$120 \), your final answer must be \( \$100 \).

Key Takeaway: Always look for the limits or special conditions in the text. They are often the "trick" in the question.

4. Communicating Your Reasoning (New for 2028 Syllabus)

The 2028 syllabus edition places a heavy emphasis on 4.1 Explain reasoning and 4.2 Justify the solution. You can no longer just write down a final number. To get full marks, you should:

  • Label your values: Don't just write "\( 50 \)". Write "\( \text{Weight} = 50kg \)".
  • Label your calculations: Instead of just "\( 50 \times 2 = 100 \)", write "\( \text{Variable cost} = 50kg \times \$2 = \$100 \)".
  • Use units: Always include \( \$ \), \( kg \), \( m/s \), or whatever unit the problem uses.

Did you know? In Paper 1, you can often get "working marks" even if your final answer is wrong, but only if the examiner can understand what your labels mean!

5. Common Pitfalls to Avoid

1. Ignoring the "Fixed" part: Students often calculate the variable rate (e.g., price per km) but forget to add the initial "booking fee" or "standing charge."

2. Misapplying Percentages: Ensure you know if a percentage is an increase (add it to \( 100\% \)) or a discount (subtract it from \( 100\% \)).

3. Confusing Units: If the model gives a rate in minutes but the data is in hours, you must convert them so they match before calculating. For example, \( 1.5 \text{ hours} = 90 \text{ minutes} \).

Quick Review Box

Model: A set of rules or a formula.
Inputs: The specific data for the problem.
Key Task: Follow the rules accurately using basic arithmetic (\( +, -, \times, \div \)), ratios, and percentages.
Pro Tip: Label every step of your working with words and units to ensure you get "communication" marks.

Need more help? If the model feels too complicated, check out the chapter on "Make simple deductions" to see how to break down information first!