Find a pure maths tutor for your course and goals

Pure maths tuition might focus on a difficult exam topic, the step into Further Maths, or a proof in a university module. For exam preparation, give the full qualification name and board. The examples below distinguish named courses; use your own specification if you follow a different board or UK qualification.

For a learner studying differentiation, consider y = x³ − 3x. Finding where the derivative is zero is only part of a stationary-point question. The derivative gives the gradient; at a stationary point the gradient is zero.

Differentiating gives y′ = 3x² − 3. Here y′ denotes the first derivative. Setting it to zero gives x² = 1, so x = −1 or x = 1. Substitute into the original expression to find the coordinates: (−1, 2) and (1, −2).

The second derivative is y″ = 6x. At x = −1 it is negative, so the gradient is decreasing through zero: (−1, 2) is a local maximum. At x = 1 it is positive, so the gradient is increasing through zero: (1, −2) is a local minimum. A stationary point is not automatically a maximum or minimum; classification needs a reason.

An independent follow-up is y = x³ − 12x. The answers are a local maximum at (−2, 16) and a local minimum at (2, −16). Reproduce the differentiation, substitution and classification before checking those answers. These are constructed teaching examples of the differentiation skills in AQA A-level Mathematics 7357.

To prove that two consecutive integers always have an odd sum, checking 2 + 3 = 5 and 8 + 9 = 17 only shows two cases. A proof must cover any pair.

Let n be any integer. The next integer is n + 1, so their sum is n + (n + 1) = 2n + 1. Twice an integer is even; adding one makes it odd. This establishes the claim for every integer n, including zero and negative integers.

This introductory example shows what a stronger written response contains: a defined variable, justified steps and a conclusion that answers the question. AQA A-level Mathematics 7357 includes this kind of logical reasoning from assumptions. In tuition, a useful next step is to explain each line aloud before writing a fresh proof without the earlier prompts.

Latimer’s service is online-first and one-to-one. Microsoft Teams is the default lesson platform; you and the tutor can agree another. Before the first teaching session, agree how you will show handwritten calculations, graphs or proofs.

A clear photo of the whole attempt lets a tutor trace the reasoning before the final line. During a lesson, a camera view of paper or a shared writing surface may suit you; discuss what you can use comfortably. Check that small details such as minus signs, powers and fraction bars remain readable. For longer proofs, number the lines so you can refer to the exact step that needs explanation.