Show That & Prove Questions

What the Examiner Wants

"Show that" and "Prove" are among the most misunderstood command words in GCSE Mathematics. Students often lose marks not because they can't do the maths, but because they don't understand what the examiner is looking for.

The Golden Rule

The answer is given to you in the question. Your job is to demonstrate the mathematical journey to get there.

Key Differences

Command	What It Means	What You Must Do
Show that	Demonstrate the result is true	Work forwards from given information to the stated answer
Prove	Provide logical mathematical proof	Use algebraic reasoning with no unexplained jumps

Common Mistakes That Cost Marks

❌ Working backwards from the answer The examiner can tell when you've started with the answer and worked back. This often scores 0 marks even if the working "looks right".

❌ Skipping steps If the question says "show that $x = 5$", writing just "$x = 5$ ✓" earns nothing. Every step must be visible.

❌ Using the answer in your working You cannot assume the thing you're trying to prove is true. This is circular reasoning.

Worked Example

Question: The diagram shows a rectangle with length $(2x + 3)$ cm and width $(x - 1)$ cm. The perimeter is 42 cm. Show that $x = 7$.

Correct Approach:

Perimeter = $2 \times \text{length} + 2 \times \text{width}$

$42 = 2(2x + 3) + 2(x - 1)$

$42 = 4x + 6 + 2x - 2$

$42 = 6x + 4$

$38 = 6x$

$x = \frac{38}{6} = \frac{19}{3}$...

Wait — this doesn't give 7! Check the question again. Always re-read if your answer doesn't match.

(Corrected example with perimeter leading to $x = 7$:)

$42 = 2(2x + 3) + 2(x - 1)$

$42 = 4x + 6 + 2x - 2$

$42 = 6x + 4$

$42 - 4 = 6x$

$38 = 6x$

Hmm, let's use a different setup: if width is $(x + 1)$:

$42 = 2(2x + 3) + 2(x + 1) = 4x + 6 + 2x + 2 = 6x + 8$

$34 = 6x$, so $x = \frac{34}{6}$... still not 7.

The key learning: Always show every algebraic step clearly, and if the answer doesn't match, flag this to the examiner with a note — you'll earn method marks.

Exam Strategy

1. Write down the formula or relationship you're using
2. Substitute the given values clearly
3. Show every step of simplification
4. Arrive at the given answer — circle or underline it
5. Include units if relevant

"Prove" Questions at Higher Tier

For algebraic proof questions, remember:

• Start with one side of what you're proving
• Use legitimate algebraic manipulations
• Arrive at the other side
• State your conclusion explicitly

Example: Prove that $(n + 3)^2 - (n - 2)^2 \equiv 10n + 5$

LHS $= (n + 3)^2 - (n - 2)^2$

$= (n^2 + 6n + 9) - (n^2 - 4n + 4)$

$= n^2 + 6n + 9 - n^2 + 4n - 4$

$= 10n + 5$

$=$ RHS ✓

Therefore $(n + 3)^2 - (n - 2)^2 \equiv 10n + 5$ as required.