More than an answer
A correct answer is good. A correct answer with a clear reason is much better. To justify a solution means to show why it’s right, so that someone else can follow and be convinced.
What a justification includes
| Part | Example |
|---|---|
| The claim | "x = 4" |
| Evidence | the steps, a diagram, a table |
| Reasoning | why each step is allowed: "I subtracted 3 from both sides to keep the equation balanced." |
| Verification | "Checking: 2(4) + 3 = 11 ✓" |
Ways to justify
1. Show and explain each step
Solve 5x − 7 = 18.
- Add 7 to both sides (inverse of subtracting 7): 5x = 25
- Divide both sides by 5: x = 5
2. Verify by substitution
5(5) − 7 = 18 ✓
3. Use a different method
A table, a graph or guess-and-check gives the same answer.
4. Use a diagram or model
A bar model or area model shows the relationship visually.
5. Cite a property or theorem
"The angles are equal because they are alternate angles between parallel lines."
Writing a clear explanation
- Use complete sentences for key steps.
- Use correct vocabulary: sum, product, coefficient, perpendicular.
- Use "because", "so" and "therefore" to connect ideas.
- Label units and diagrams.
Weak: "The answer is 12 because I did the math."
Strong: "The rectangle’s area is 48 cm² and its width is 4 cm. Since area = length × width, the length is 48 ÷ 4 = 12 cm. Check: 12 × 4 = 48 ✓."
Justifying "always", "sometimes", "never"
- Always: requires a general argument (often algebra).
- Never: requires a general argument.
- Sometimes: one example where it’s true and one where it’s false.
Example: "A square number is even." Sometimes: 4 is even, 9 is odd.
Justifying with estimation
"Is $4.89 × 19 about $93?" Justify: $5 × 20 = $100, and both were rounded up, so the answer should be a bit less than $100. $93 is reasonable.
Worked examples
Example 1: Justify that 7/8 > 5/6. Common denominator 24: 21/24 > 20/24. ✓
Example 2: Justify that a triangle with sides 5, 12, 13 is right-angled. 5² + 12² = 169 = 13², so by the converse of the Pythagorean theorem it is.
Example 3: Is "the product of two odd numbers is odd" always true? (2a + 1)(2b + 1) = 4ab + 2a + 2b + 1 = 2(2ab + a + b) + 1, which is odd. Always.
Common mistakes
- Restating the answer instead of explaining.
- Skipping steps a reader needs.
- Using one example to "prove" something is always true.
Try it yourself
- Justify that 0.6 = 3/5.
- Is the sum of three odd numbers always odd? Justify.
- Justify that x = −2 solves 3x + 10 = 4.
- "Dividing makes a number smaller." Always, sometimes or never?
Answers: 1) 0.6 = 6/10, and dividing top and bottom by 2 gives 3/5. 2) Always: odd + odd = even, and even + odd = odd. 3) 3(−2) + 10 = 4 ✓ 4) Sometimes (8 ÷ 2 = 4, but 8 ÷ 0.5 = 16).
Quick recap
- Justify with evidence, reasoning and verification.
- Explain clearly using maths vocabulary.
- Verify with substitution, estimation or another method.