How to Study for NSC Mathematics Paper 1: The Algebra & Functions Playbook

NSC Mathematics Paper 1 is the paper most Grade 12 learners fear, but it is also the most predictable of the two mathematics papers. Every November the Department of Basic Education publishes a paper that draws from six clearly defined topics: algebra and equations, number patterns and sequences, functions and inverse functions, financial mathematics, differential calculus, and probability. Once you understand how those sections repeat year after year, Paper 1 becomes a game you can actually train for.

Understand the structure before you touch a single sum

Paper 1 is 150 marks and three hours long. The marks are distributed roughly as follows: algebra and equations (25), number patterns (25), functions (35), finance (15), calculus (35) and probability (15). If you can consistently score 60% on the first three sections you are already sitting on a comfortable pass. That is where your revision effort should go first.

Print out the last five official DBE papers from our Subjects page and highlight, on each one, where the marks live. You will very quickly notice that the same skills are tested every year: solving quadratic equations by three methods, simultaneous equations with one linear and one non-linear equation, exponential equations, and inequalities. These carry roughly 25 marks and take fifteen minutes if you drill them.

Section 1: Algebra & equations (do this first every day)

Warm up every study session with five short algebra questions. Rotate through: (1) factorising and solving a quadratic; (2) solving a quadratic with the formula and giving the answer to two decimal places; (3) solving simultaneous equations; (4) solving an exponential or surd equation; (5) a nature of roots discussion. Set a five-minute timer per question. Speed matters — the paper allows about one minute per mark, so 25 algebra marks must be finished in 25 minutes.

Common traps: forgetting to check for extraneous solutions when squaring both sides, dropping the negative solution of a quadratic without a reason, and confusing "real and equal" with "real and rational" when discussing the discriminant.

Section 2: Number patterns

Two families of patterns dominate: arithmetic and geometric sequences, and quadratic patterns. Memorise the four formulas — $T_n$ and $S_n$ for both arithmetic and geometric sequences — and practise deriving them, because IEB and enrichment questions sometimes test the proof. The one pattern skill that catches learners out is finding the value of $r$ that makes a geometric series converge and then applying $S_\infty = \dfrac{a}{1-r}$.

Work through the November 2022, 2023 and 2024 pattern questions back-to-back. You will notice the examiner alternates between "prove the sum" and "find the position of the term". Once you have seen both, you have seen the whole topic.

Section 3: Functions and inverse functions

This is the section that separates 70% learners from 80% learners. You must be able to sketch, from memory, the six CAPS graphs: straight line, parabola, hyperbola, exponential, logarithmic and cubic. For each one write down, on a single flashcard, the shape, the effect of $a$, $p$ and $q$, the asymptotes, and how the inverse is constructed.

Nine times out of ten the "sketch" question gives you a graph and asks for the equation, then asks you to find the length of a vertical line between two graphs, the maximum turning point of a difference, or the values of $x$ for which one graph is above the other. Train the two-step routine: first read off intercepts and asymptotes to build the equation, then answer the sub-questions from the equations, not the sketch.

Section 4: Financial mathematics

Only 15 marks, but the easiest 15 in the paper. Learn the four formulas — simple interest, compound interest, future value and present value annuities — and how to identify which one the examiner wants. Draw a timeline for every question. A timeline stops you from mixing up "how much have I saved" (future value) with "how much did I have to borrow" (present value).

Section 5: Differential calculus

This section is a marathon of small techniques: first principles, rules of differentiation, cubic sketches, tangents, and optimisation. Two things transform your calculus mark. First, know first principles by heart, including the "$\lim_{h\to0}$" notation — every year it is worth five marks and it is exactly the same proof. Second, when you sketch a cubic, always find the $x$-intercepts by factorising (usually one factor comes from inspection, then long division), the $y$-intercept, and the stationary points from $f'(x) = 0$. Label the coordinates of every important point on the sketch.

Optimisation questions look intimidating because they mix geometry with calculus. Always write down two equations first: the constraint (given in the question, usually about area or perimeter) and the thing you are trying to maximise. Substitute the constraint into the thing you want to maximise so you have a single-variable function, then differentiate and set to zero.

Section 6: Probability

Fifteen marks that many learners waste. Learn the difference between mutually exclusive, independent, and complementary events. Memorise the product rule for independent events and the addition rule for mutually exclusive events, and know how to draw a Venn diagram from a word problem. The counting question — arrangements and choices — is almost always solved with the multiplication counting principle. Read the question twice to check whether repetition is allowed and whether order matters.

How to actually revise in the last four weeks

Spend one hour a day on Paper 1 for the four weeks before the exam. Divide the hour like this: fifteen minutes on algebra drills, thirty minutes on one full section from a past paper, and fifteen minutes marking your own work using the official memo. Never mark your own work in a different colour and never skip the memo — the marks come from the exact wording the examiner accepts.

On the day of the paper, do the sections in the order that feels safest. Most learners should start with algebra, patterns and finance, then functions, calculus and probability last. Keep an eye on the clock: if a section is running long, move on and come back. Marks left on the table because you spent forty minutes on a fifteen-mark question are gone.

Consistent, focused practice with the past papers, plus honest marking against the memo, is what separates a 60% from an 80% on this paper. Now open our past papers library, pick November 2024, and start.