What 'Prove' Really Demands in an NSC Maths Exam

Here is the rule, stated as plainly as possible: to prove something, you must start from what is given or already known, and work forward, step by step, until you arrive at the statement you were asked to prove. You are not allowed to start from the statement itself and work backward to something true. You are not allowed to write the two sides of an equation next to each other and manipulate both at once until they meet in the middle. Either of those approaches assumes the very thing you were supposed to demonstrate, and a marker reading it will award zero for the sub-question, even when every individual algebraic step is correct.

This is the single biggest difference between 'prove' and almost every other instruction word in the CAPS Mathematics vocabulary. Most instructions, determine, calculate, solve, judge you on the destination. Prove judges you on the direction of travel.

Why a Correct Answer Still Earns Zero

Imagine you're asked to prove that (x² − y²) / (x + y) equals x − y, where x is not equal to negative y. A student under time pressure often writes the left-hand expression on one line, the right-hand expression on the next, and starts adjusting both until they look the same. It feels efficient. It is also circular: you have assumed the two sides are already equal and simply rearranged both of them until that assumption is visually confirmed, which proves nothing.

The correct approach starts from one side only, almost always the more complicated-looking one, and works it down until it exactly matches the other side, which is never touched. Here, you'd factorise the numerator on the left as a difference of squares, (x + y)(x − y), cancel the shared (x + y) factor with the denominator, and arrive at x − y. That final line matches the right-hand side exactly, and because you only ever worked from the left, the logic holds. Same algebra, same final line, but only one of these two approaches is a proof.

'Prove' and 'Show That' Are the Same Instruction

Students sometimes treat 'show that' as a softer, less formal version of 'prove', something you can get away with skipping steps on. CAPS makes no such distinction. Both words demand identical rigour: a one-directional chain of logical steps from what's given to what's required, with every step justified. If anything, 'show that' appears more often in questions where the result is later used in a 'hence' sub-question, which makes getting the direction right even more important, because a broken proof also breaks whatever is built on top of it.

In Euclidean Geometry, Every Step Needs a Reason

Proof questions look slightly different in Euclidean geometry, but the underlying discipline is identical, and the stakes for skipping it are higher. A geometry proof isn't just a sequence of correct conclusions, each conclusion must be paired with the specific theorem or fact that justifies it. Writing 'angle ABC = angle ADC' without stating why, for example 'angles subtended by the same arc' or 'angles in the same segment', earns no marks even if the angle relationship you've stated is true.

A real past paper example: the November 2023 Paper 2 exam, question 8.3, asked candidates to prove that AB is a tangent to a circle at point B. Getting the correct geometric relationship isn't enough here. The proof has to move through a specific chain, usually involving the tan-chord theorem or a converse circle theorem, with each line naming both the conclusion and the reason. Markers are checking the reasoning trail, not just whether you landed on 'AB is a tangent' as a final statement.

Algebraic Proofs: 'Real Roots' and Other Common Types

Outside geometry, the most common 'prove' or 'show that' questions in Grade 10 to 12 Mathematics fall into a small number of recurring types:

• Proving an identity or simplification is correct, starting from one side of an equation and manipulating it algebraically until it matches the other side, exactly as in the (x² − y²) / (x + y) example above.
• Proving the roots of a quadratic equation are real, rational, equal, or non-real, by calculating the discriminant (b² − 4ac) and reasoning about its sign, rather than solving for the actual roots. Solving the equation fully, when the question asked you to prove something about the nature of the roots, answers a different question and misses the marks tied to the discriminant argument.
• Proving a trigonometric identity, working from the more complex side, converting everything to sine and cosine where useful, and never touching the other side of the equation until the final line matches it.
• Proving a general statement holds 'for all values of k' or similar, which usually means showing an inequality or discriminant condition is true regardless of the specific value substituted, not just testing one convenient number.

A real example of the second type: the November 2022 Paper 1 exam, question 3.4, asked candidates to show that the roots of a given equation are real for all values of k. The correct method calculates the discriminant in terms of k, then argues why that expression can never be negative, often by showing it's a sum of a positive constant and a squared term. Substituting a few values of k and showing the roots come out real each time is not a proof, it's a demonstration for specific cases, and it earns little to no credit against a 'for all values' instruction.

A Simple Check Before You Start Writing

Before you write a single line of a proof, ask yourself two questions. First: which side, or which starting point, am I working from, and am I going to leave the other side completely untouched until the final line? Second, for geometry specifically: for every conclusion I write, do I have a named reason ready to go next to it? If you can't answer both of those clearly, you're not ready to start writing, and starting anyway is exactly how a correct piece of Mathematics turns into a proof that earns nothing.

Half a Proof Is Not Half the Marks

Proof questions don't distribute marks the way calculation questions do. In a 'calculate' question, a wrong final answer with mostly correct method still picks up several method marks along the way. In a 'prove' question, the logical structure is often the whole point, so a proof that assumes its own conclusion, no matter how much correct algebra surrounds it, tends to score far lower than the amount of correct Mathematics in it would suggest. The fix isn't more algebra practice. It's the specific discipline of starting from one place, moving in one direction, and stating a reason for every step you take.