Welcome to the Art of Justification!
Have you ever solved a tricky puzzle and had someone ask, "How do you know that’s right?" In the Cambridge Thinking Skills (9694) syllabus, being able to find the answer is only half the battle. The other half is justifying it. Justifying means proving, through a logical "paper trail," that your solution is the only one that works or the best one possible.
This skill is especially important for Paper 3 (Problem Analysis and Solution). Because the problems in Paper 3 are more complex and "real-world" than those in Paper 1, examiners want to see that you haven't just guessed—you've reasoned your way to the finish line.
Note: This chapter focuses on "Justifying" your solution. To learn how to describe the steps you took, see the chapter on Explain reasoning. To learn about stating the "hidden" facts you relied on, see State assumptions made within reasoning.
1. What does "Justify" actually mean?
According to the official syllabus command words, to Justify means to support a case with evidence or argument. In a problem-solving context, this usually involves two things:
- Showing it fits: Proving your answer meets all the rules (constraints) of the problem.
- Showing it’s the best: If the question asks for an "optimal" solution (like the cheapest or fastest way), you must prove that no better version exists.
Quick Review: In Paper 3, if a question asks for a "Yes" or "No" answer, you might get zero marks if you don't provide the justification, even if your "Yes" or "No" is correct!
2. The Building Blocks of a Logical Justification
To justify a solution effectively, you need to use Logical Reasoning. Think of this as building a bridge from the information given (the scenario) to your final answer.
A. Checking Constraints
Most problems have "rules." A justification often involves listing these rules and showing how your answer follows them. For example, if a delivery schedule says a driver cannot work more than \( 8 \) hours and must have a \( 30 \)-minute break, your justification would point out: "The total driving time is \( 7.5 \) hours and a break is scheduled at noon, so the solution satisfies all safety rules."
B. The Process of Elimination
Sometimes the best way to justify an answer is to show why other answers are wrong. This is very common when you are asked to "Find the only possible time that..." You justify your choice by showing that every other time slot results in a conflict.
C. Proving Optimality
In the 2028 syllabus, you are specifically tested on finding optimal solutions. To justify that a solution is "optimal" (e.g., the minimum cost):
- State your solution and its value (e.g., \( \$50 \)).
- Explain why any attempt to make it lower (e.g., \( \$45 \)) would break one of the rules of the problem.
Did you know?
An optimal solution is not just "a good answer"—it is the absolute best possible answer under the given circumstances (the maximum, the minimum, the fastest, etc.).
3. Step-by-Step: How to Write a Justification
Don't worry if this seems tricky at first. Follow these three steps to structure your thoughts:
Step 1: State the Conclusion
Start by clearly stating your answer (e.g., "The minimum number of buses required is \( 4 \).")
Step 2: Provide the Evidence (The "Positive" Case)
Show the math or logic that makes your answer work.
"With \( 4 \) buses, we have a total capacity of \( 4 \times 50 = 200 \) seats, which accommodates the \( 195 \) students."
Step 3: Provide the Logical Argument (The "Negative" Case)
Show why one step "better" fails.
"If we used only \( 3 \) buses, the capacity would be \( 3 \times 50 = 150 \), which is less than the \( 195 \) students required. Therefore, \( 4 \) is the minimum."
4. Working with Complex Models
In Paper 3, you will often deal with complex models. These are scenarios with many moving parts (like a tax system, a sports league, or a factory production line). When justifying a solution within a model:
- Identify logical relationships: Use phrases like "Because \( X \) depends on \( Y \)..." or "Since the price increases when \( Z \) happens..."
- Label your values: The syllabus explicitly asks you to label your calculations. Instead of just writing \( 10 \times 5 = 50 \), write "\( 10 \text{ tickets} \times \$5 \text{ per ticket} = \$50 \text{ total income} \)." This makes your reasoning much easier for the examiner to follow.
5. Common Mistakes to Avoid
The "Because it looks right" trap: Never justify an answer by saying it "seems" correct or is "the most likely." Use the numbers and rules provided in the text.
Circular Reasoning: Avoid saying "The answer is \( 10 \) because when you calculate it, you get \( 10 \)." Instead, show the specific constraint that forces the answer to be \( 10 \).
Forgetting Units: If the problem involves money, time, or weight, always include the units (e.g., \( \text{kg}, \text{ minutes}, \$ \)) in your justification. The 2028 syllabus places extra emphasis on this for "Communicating Reasoning."
6. Summary Key Takeaways
Key Takeaway 1: Justification is about proving why your solution is valid and/or optimal using the rules provided in the scenario.
Key Takeaway 2: In Paper 3, showing your working and providing an explanation is often required to earn full marks, especially for "Yes/No" questions.
Key Takeaway 3: A strong justification often shows that your answer meets all constraints and that a "better" answer would break those rules.
Key Takeaway 4: Always label your numbers with words (e.g., "\( 5 \text{ hours of labor} \)") to make your logical reasoning clear to the examiner.