Introduction: The Art of Proving You’re Right

Have you ever solved a difficult puzzle, only to have someone ask, "How do you know that’s the answer?" In the Thinking Skills (9694) syllabus, arriving at the correct numerical answer is only half the battle. The other half is justifying it.

In your Paper 1 exam, especially under the new 2028 syllabus rules, examiners aren't just looking for a number; they are looking for the logical journey you took to get there. Justifying a solution means proving that your answer is the only one that makes sense based on the information provided. It’s about turning "I think so" into "I can prove it."

What does "Justify" Actually Mean?

According to the official command words, Justify means to support a case with evidence or argument.

When a question asks you to justify a solution, you need to do more than show your math. You need to explain the logic behind your steps. If "Explain reasoning" is telling the story of what you did, "Justify" is the argument for why what you did is correct.

Note: For more on how to simply describe your steps, see the chapter on "Explain reasoning." For the specifics of what to write down, see "Show and label working clearly."

The Core Elements of a Logical Justification

To justify a solution effectively, you should aim to cover these three areas:

  • The Evidence: Which specific facts from the scenario are you using?
  • The Logic: How do those facts connect? (e.g., "If \( A \) is true, then \( B \) must be false.")
  • The Conclusion: Why does this mean your final answer is the only valid one?

1. Identifying Logical Relationships

Often, justifying a solution requires you to show how different pieces of information interact. You might need to point out necessary and sufficient conditions.
Example: "To win the prize, a student must be both over \( 16 \) years old (necessary) and have the highest score (sufficient)." If your solution identifies a winner, your justification must show they meet both criteria.

2. Making Deductions

A deduction is a conclusion reached by logic. In a justification, you should state your deductions clearly.
Example: "There are only \( 7 \) days in a week. If the event cannot happen on a weekend, there are only \( 5 \) possible days left."

3. Ruling Out Alternatives

Sometimes, the best way to justify a solution is to show that no other solution is possible. This is common in "finding the optimal solution" problems.
Example: "Option \( A \) costs \( \$50 \), and Option \( B \) costs \( \$45 \). Since we are looking for the cheapest price, Option \( A \) cannot be the solution."

The "Labeling" Rule (New for 2028)

For students taking exams in 2028 and beyond, the syllabus explicitly states that you should label values and calculations with units and words. This is a vital part of justifying your reasoning. A list of random numbers isn't a justification; a series of labeled steps is.

Bad Example: \( 10 \times 5 = 50 \)

Good Justification: \( 10 \text{ items} \times \$5 \text{ per item} = \$50 \text{ total income} \)

By adding those labels, you are justifying why you multiplied those two numbers: you are calculating the total income.

Step-by-Step: How to Write a Justification

Don't worry if this seems tricky at first! Follow these steps to build a solid justification:

Step 1: State the goal. What was the problem asking for? (e.g., "We need to find the minimum number of trips.")

Step 2: Identify the constraints. Mention the rules you had to follow. (e.g., "The boat can only carry \( 200kg \) at a time.")

Step 3: Show the calculation with labels. (e.g., "Total weight is \( 500kg \). \( 500kg / 200kg = 2.5 \).")

Step 4: Use logic to finalize. (e.g., "Since we cannot take half a trip, we must round up to \( 3 \) trips.")

Common Mistakes to Avoid

The "Answer-Only" Trap: Writing down "The answer is \( 42 \)" without any supporting logic. Even if \( 42 \) is correct, you may lose marks for the "Justify" portion of the question.

Circular Reasoning: Saying something is true just because it’s true. Example: "The price is \( \$20 \) because that is what it costs." Instead, use the data: "The price is \( \$20 \) because the base cost is \( \$15 \) and the tax is \( \$5 \)."

Ignoring Constraints: Forgetting to explain why a simpler-looking answer doesn't work because it breaks one of the rules in the scenario.

Did You Know?

In Paper 1, credit is often awarded for "correct steps towards a solution" even if your final answer is wrong! However, credit can be withheld if you give the right answer but fail to provide the explanation or justification requested. This is why communicating your reasoning is just as important as the math itself.

Quick Review: Keys to Success

  • Use the Facts: Always refer back to the numbers or rules given in the text.
  • Explain the 'Why': Don't just show the math; explain what the math represents.
  • Label Everything: Use units (like \( kg, \$, \text{ or hours} \)) and descriptive words.
  • Be Logical: Show how one step lead to the next using "Therefore," "Because," or "Since."

Key Takeaway: A justification is a logical argument that proves your solution is correct. It connects the data provided in the problem to your final answer using clear, labeled steps and logical deductions.