Problem-Solving Strategies

Introduction

The FSCE 11+ exam is designed to give you problems you've never seen before. That might sound scary, but here's the good news: you don't need to have seen a problem before to solve it. What you need are strategies — general approaches that work for almost any unfamiliar problem.

Think of problem-solving strategies like tools in a toolkit. A builder doesn't carry a different tool for every single job. Instead, they carry a small set of versatile tools (hammer, screwdriver, saw, drill) that can be used in many different situations. In the same way, the strategies in this lesson can be applied to problems in maths, English, science, and beyond.

The Six Key Strategies

graph TD
    A["PROBLEM"] --> B["Strategy 1: Break It Down"]
    A --> C["Strategy 2: Find What You Know"]
    A --> D["Strategy 3: Look for Patterns"]
    A --> E["Strategy 4: Work Backwards"]
    A --> F["Strategy 5: Try a Simpler Case"]
    A --> G["Strategy 6: Eliminate Impossible Answers"]
    B --> H["SOLUTION"]
    C --> H
    D --> H
    E --> H
    F --> H
    G --> H
    style A fill:#fce4ec
    style H fill:#e8f5e9

Strategy 1: Break It Down

Big problems can feel overwhelming. But almost every big problem is actually several small problems combined. If you break the problem into smaller steps, each step becomes manageable.

How to use it:

1. Read the problem and identify all the different parts.
2. Solve each part separately.
3. Combine your answers.

Strategy 2: Find What You Know

Before focusing on what you don't know, focus on what you DO know. Write down every piece of information the question gives you. Often, the path to the answer becomes clear once you've organised what you know.

How to use it:

1. Underline or list all the facts given in the question.
2. Write down what you're trying to find.
3. Ask: "How can I get from what I know to what I need to find?"

Strategy 3: Look for Patterns

Many problems contain patterns. If you can spot the pattern, you can use it to find the answer without doing every single calculation.

How to use it:

1. Look at the numbers or information you've been given.
2. Ask: "Is there a pattern? Does anything repeat? Is there a rule?"
3. Test your pattern to make sure it works.

Strategy 4: Work Backwards

Sometimes the easiest way to solve a problem is to start from the answer and work backwards to the beginning.

How to use it:

1. Start with the end result.
2. Reverse each step.
3. Check your answer by working forwards.

Strategy 5: Try a Simpler Case

If a problem seems too complex, try a simpler version first. Once you understand the simpler case, you can apply the same approach to the harder problem.

How to use it:

1. Replace the difficult numbers with easier ones.
2. Solve the simpler problem.
3. Use the same method with the original numbers.

Strategy 6: Eliminate Impossible Answers

In multiple-choice questions, you can often eliminate answers that are clearly wrong. This narrows your choices and increases your chances of getting the right answer.

How to use it:

1. Read all the options before answering.
2. Cross out any that are obviously wrong.
3. Focus on the remaining options.

Problem-Solving Flowchart

Use this flowchart when you're stuck on any problem:

graph TD
    A["Read the problem carefully — TWICE"] --> B["What do I KNOW?"]
    B --> C["What do I need to FIND?"]
    C --> D{"Do I see a way to solve it?"}
    D -->|Yes| E["Solve it and CHECK your answer"]
    D -->|No| F{"Can I BREAK IT DOWN into smaller parts?"}
    F -->|Yes| G["Solve each part separately"]
    F -->|No| H{"Can I spot a PATTERN?"}
    H -->|Yes| I["Use the pattern"]
    H -->|No| K{"Can I WORK BACKWARDS from the answer?"}
    K -->|Yes| L["Start from the end"]
    K -->|No| M{"Can I TRY A SIMPLER CASE?"}
    M -->|Yes| N["Simplify, solve, then apply to original"]
    M -->|No| O["ELIMINATE impossible answers"]
    G --> E
    I --> E
    L --> E
    N --> E
    O --> E
    style A fill:#e3f2fd
    style E fill:#e8f5e9

Worked Examples

Worked Example 1: Break It Down

Problem: "At a school fair, 120 tickets were sold. Adult tickets cost £3 and child tickets cost £1.50. The total money collected was £270. How many adult tickets and how many child tickets were sold?"

Breaking it down:

• Step 1: Let's say the number of adult tickets = A. Then child tickets = 120 - A (because total is 120).
• Step 2: Money from adults = A x £3. Money from children = (120 - A) x £1.50.
• Step 3: Total money: 3A + 1.50(120 - A) = 270
• Step 4: 3A + 180 - 1.50A = 270
• Step 5: 1.50A = 90
• Step 6: A = 60

Answer: 60 adult tickets and 60 child tickets. Check: (60 x £3) + (60 x £1.50) = £180 + £90 = £270. Correct!

Worked Example 2: Find What You Know

Problem: "Emma is older than Tom. Tom is older than Sarah. Sarah is older than Jack. Jack is 8 years old. There are exactly 2 years between each person's age. How old is Emma?"

What I know:

• Jack is 8
• 2 years between each person
• Order from youngest to oldest: Jack, Sarah, Tom, Emma

Solution:

• Jack: 8
• Sarah: 8 + 2 = 10
• Tom: 10 + 2 = 12
• Emma: 12 + 2 = 14

Answer: Emma is 14 years old.

Worked Example 3: Look for Patterns

Problem: "What is the 50th number in this sequence: 3, 6, 9, 12, 15, ...?"