The commonest objection to revising an IGCSE topic over the summer is not laziness. It is that nobody has said what the topic is for.

“Practise surds” is a chore. “Practise surds because Pure Mathematics 1 expects exact answers and you will be rationalising denominators inside calculus questions in week five” is a reason.

Here is the map, taken from what Cambridge 9709 actually lists rather than from general advice.

Index laws unlock differentiation

The clearest pair in the whole syllabus.

Cambridge 9709 Pure Mathematics 1, section 1.7, requires candidates to use the derivative of xⁿ for any rational n. That is the core rule of the entire differentiation topic.

The catch is that A Level questions rarely present terms as clean powers. They give you a root, or a fraction with x in the denominator, or both. Differentiating means first rewriting that as a power — and rewriting is index laws.

So a student whose index laws are shaky does not have a differentiation problem. They have an index laws problem that only becomes visible during differentiation, several weeks after anyone would have thought to check.

Surds unlock exact answers

At IGCSE, a decimal was usually acceptable. At A Level, exact form is frequently the required answer, and the working carries marks in its own right.

Cambridge 9709 names simple manipulation of surds directly in its prior-knowledge statement, alongside the shapes of graphs of the form y = kxⁿ. Those two items are the only specific skills the syllabus calls out beyond “IGCSE 0580 Extended or O Level 4024/4029”.

Surds then appear everywhere downstream: exact trigonometric values, coordinate geometry distances, results of integration. And because 9709 gives no marks for unsupported answers from a calculator, reaching for a decimal is not a shortcut you can take.

Completing the square unlocks circles

This is the connection students find most surprising.

Coordinate geometry in 9709 uses the circle in centre-radius form, (x − a)² + (y − b)² = r², and asks for problems involving lines and circles: a tangent perpendicular to a radius, the angle in a semicircle, symmetry.

Exam questions do not hand you the tidy form. They give the expanded equation, and getting from one to the other is completing the square, twice, in two variables at once.

It also does its original job: completing the square gives you the vertex of a quadratic, which is how you find a maximum or minimum without calculus and how you read off the shape of a curve at a glance.

One IGCSE technique, two separate A Level topics.

Rearranging formulae unlocks functions

Section 1.2 of Pure Mathematics 1 is Functions, and the work there is finding inverses, composing functions and identifying domains and ranges.

Finding an inverse is rearranging, done under a different name. Write y = f(x), swap the variables, make y the subject. The last step is the one people find hard, and it is exactly the skill they practised at IGCSE as changing the subject of a formula.

Students who found rearranging fiddly at IGCSE often conclude that functions are conceptually difficult. Usually the concept is fine and the algebra underneath it is what is slowing them down.

The unit circle unlocks trigonometric equations

Section 1.5 is Trigonometry, and two requirements sit at its centre.

First, understanding the definition of a radian and the relationship between radians and degrees. Radians are not a stylistic preference; arc length and sector area formulae depend on them, and so does calculus with trigonometric functions later.

Second, solving trigonometric equations lying in a specified interval. Not just finding an angle, but finding every angle in a given range that satisfies the equation.

That second one is where the unit circle earns its place. Knowing that sine is positive in two quadrants, and where the symmetry sends the second solution, is what turns one calculator answer into the complete set the mark scheme wants.

Simultaneous equations unlock intersections

One more pair, because it is the one students least expect to matter.

Simultaneous equations at IGCSE were an algebra exercise: two equations, two unknowns, solve. At A Level they become the standard method for finding where two graphs meet.

That turns up constantly in coordinate geometry. Where does this line cut this circle? Solve the pair simultaneously, which produces a quadratic, and the number of solutions tells you whether the line is a secant, a tangent, or misses entirely.

The discriminant then does the final job. A repeated root means the line touches at exactly one point, which is how tangency gets proved algebraically rather than by drawing.

So three IGCSE techniques — simultaneous equations, solving quadratics and the discriminant — combine into one A Level method. None of them is difficult on its own, and the combination only works if all three are fluent at once.

Why the map is worth having

Two practical uses, beyond satisfying curiosity.

It tells you what to prioritise. The prior-knowledge list is short, but not everything on it carries the same weight downstream. Index laws feed differentiation, which is a whole examined topic. That earns more of your August than a skill appearing once.

It converts revision into something with a point. The reason unsupervised summer study collapses is rarely difficulty. It is that “revise algebra” has no visible destination. “Get index laws automatic so differentiation is about the calculus” does.

Why the order is not arbitrary

Reading the map top to bottom shows something a topic list alone does not: the prerequisites cluster, and they all sit upstream.

Index laws, surds, completing the square, rearranging and simultaneous equations are all algebra. Differentiation, coordinate geometry, functions and trigonometry are the A Level topics they feed. Almost every arrow in the map points from the same place to somewhere else.

That is why “get algebraic fluency automatic” is better advice than a list of nine separate jobs. The individual techniques are not the point; the point is that A Level asks you to use several of them inside a single question about something else entirely.

It also explains the sequence any sensible preparation follows. There is no value in practising trigonometric equations while surds are still slow, because the exact answers those questions want are written in surd form. Work upstream first and the downstream topics get easier without being touched.

Doing this for your own subjects

The same exercise works outside maths, and takes about twenty minutes per subject.

Open your syllabus on the Cambridge website. Read the Previous study or Prior knowledge section, then skim the subject content list for the first year. For each assumed skill, find where it shows up in the content: Physics 9702 prints a mathematical requirements list and says teaching it belongs inside the Physics course; Economics 9708 assumes a mathematics course, and the data response papers are where that assumption gets used.

Write the pairs down. That is your map, and it is more useful than any general revision plan because it is specific to what you are actually taking.

Where to start

Tutopiya’s Pre-A Level Mathematics is organised along these lines. Algebraic Fluency comes first and breaks into nine sub-topics — Index Laws, Surds and Rationalising, Expanding and Factorising, Solving Quadratics, Completing the Square, Algebraic Fractions, Rearranging Formulae, Simultaneous Equations and Speed and Accuracy — before the course moves on to Functions and Graphs, Trigonometry and Radians, and Coordinate Geometry.

Read that ordering against the map above and it is the same list in the same order: the prerequisites first, then the topics they unlock.

For the wider picture, five things A Level assumes you can already do covers the cross-subject version, and what to prepare before A Levels separates the gaps a summer can close from the ones it cannot.

Frequently asked questions

Which IGCSE maths topics matter most for A Level? +

Index laws, surds, completing the square, rearranging formulae and the unit circle. Each is a direct prerequisite for a named item in Cambridge 9709: the derivative of x to the power n, exact answers, the equation of a circle, function work, and solving trigonometric equations in a specified interval.

Why do I need index laws at A Level? +

Because Pure Mathematics 1 requires you to use the derivative of x to the power n for any rational n. Differentiating a term written as a root or a reciprocal means rewriting it as a power first. If that rewriting is slow, the calculus behind it never gets a fair hearing.

What does completing the square unlock at A Level? +

Two things. It gives the vertex of a quadratic, and it converts the expanded equation of a circle into the centre-radius form that Cambridge 9709 uses. Coordinate geometry problems involving lines and circles depend on being able to make that conversion quickly.

Why does A Level maths use radians instead of degrees? +

Because arc length, sector area and calculus with trigonometric functions all require them. Cambridge 9709 asks you to understand the definition of a radian and the relationship between radians and degrees, then to solve trigonometric equations lying in a specified interval using either measure.

Ready to Excel in Your Studies?

Get personalised help from Tutopiya's expert tutors. Whether it's IGCSE, IB, A-Levels, or any other curriculum — we match you with the perfect tutor and your first session is free.

Book Your Free Trial