Key Stage 4 maths · Algebra

GCSE higher algebra: fractions, factorising, proof and functions

This topic gathers the algebra aimed at pupils working at the top of GCSE maths, usually those entered for the higher tier. They simplify and combine algebraic fractions, factorise quadratics where x² has a number in front, build short algebraic proofs, work with inverse and composite functions, find perpendicular lines from gradients and complete the square to locate the turning point of a quadratic graph.

Algebraic fractions, proof and functions, explained

An algebraic fraction is a fraction with letters in the top, the bottom or both, such as (x + 3) ÷ x. It follows the same rules as a numerical fraction. To simplify, divide the top and bottom by a common factor. To add or subtract, find a common denominator: 1/x + 1/(x + 1) becomes (x + 1) ÷ [x(x + 1)] plus x ÷ [x(x + 1)], which is (2x + 1) ÷ [x(x + 1)]. Simplifying usually means factorising first, which is why the two skills are taught together.

In the standard GCSE content pupils factorise quadratics such as x² + 5x + 6, where x² has no number in front. The harder version, ax² + bx + c with a not equal to 1, is higher-tier content. A common method is to multiply a by c, find two numbers that make that product and add to b, split the middle term using them, and factorise in pairs. Some teachers prefer a grid or an organised trial of brackets. Any reliable method is fine, and every answer can be checked by expanding the brackets.

Algebraic proof uses letters to show that something is true for every number, not just the ones tried. Even numbers are written 2n, odd numbers 2n + 1, and consecutive whole numbers n, n + 1, n + 2. To prove that the sum of three consecutive numbers is a multiple of 3, write n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1). A good proof finishes with a sentence saying what has been shown. In “show that” questions the result is given, so every line of working has to be visible.

A function is a rule that turns an input into an output, such as f(x) = 2x + 3. A composite function applies one function after another: fg(x) means do g first, then f. The inverse function f⁻¹ reverses the rule. To find it, write y = 2x + 3, make x the subject to get x = (y − 3) ÷ 2, then swap the letters, giving f⁻¹(x) = (x − 3) ÷ 2. Note that f⁻¹(x) does not mean 1 ÷ f(x).

Two straight lines are perpendicular when their gradients multiply to make −1. A line with gradient 2 is perpendicular to one with gradient −½, so the second gradient is the negative reciprocal of the first. Completing the square rewrites x² + 6x + 1 as (x + 3)² − 8. That form shows the graph’s lowest point, its turning point, is at (−3, −8), because the squared bracket is never negative and is zero when x = −3. The x-coordinate of the turning point is the opposite sign to the number in the bracket.

Words your child will hear

Algebraic fraction: A fraction with an algebraic expression on the top, bottom or both, such as (x + 3) ÷ x.
Factorise: Write an expression as a product of brackets, such as 2x² + 7x + 3 = (x + 3)(2x + 1).
Proof: A chain of algebra showing a statement is true for every number of the kind described.
Composite function: Two functions applied in turn. In fg(x), g acts first and then f.
Inverse function: The function that undoes another, so that f⁻¹(f(x)) = x.
Negative reciprocal: Flip the fraction and change the sign: the negative reciprocal of 2 is −½.
Completing the square: Rewriting a quadratic as a squared bracket plus or minus a number, to read off the turning point.

What the National Curriculum says

In Key Stage 4, pupils should be taught to:

Simplify and manipulate algebraic expressions involving algebraic fractions
Factorise quadratic expressions of the form ax² + bx + c
Use algebra to support and construct proofs
Interpret the reverse process of a function as the ‘inverse function’; interpret the succession of 2 functions as a ‘composite function’
Use the form y = mx + c to identify perpendicular lines
Deduce turning points of quadratic functions by completing the square

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

Factorise 2x² + 7x + 3, then use your answer to simplify (2x² + 7x + 3) ÷ (2x² + x).

Multiply the number in front of x² by the number term: 2 × 3 = 6. Look for two numbers that multiply to 6 and add to 7. They are 6 and 1.
Split the middle term using those numbers, then factorise in pairs. Both pairs leave the same bracket, (x + 3).
```
2x² + 6x + x + 3
= 2x(x + 3) + 1(x + 3)
= (x + 3)(2x + 1)
```
Check by expanding: 2x² + x + 6x + 3 = 2x² + 7x + 3. It matches.
Factorise the bottom of the fraction too: 2x² + x = x(2x + 1).
The fraction is now (x + 3)(2x + 1) over x(2x + 1). The bracket (2x + 1) is a common factor of the whole top and the whole bottom, so it cancels, leaving (x + 3) over x.

2x² + 7x + 3 = (x + 3)(2x + 1), and the fraction simplifies to (x + 3) ÷ x (for x not equal to 0 or −½).

Common mistakes and how to help

Cancelling terms instead of factors, so (x + 3) ÷ x becomes 3

In arithmetic, cancelling is quick and satisfying, and the x on top and bottom looks like it should go. But the x on top is only part of an addition.

What helps: Only things that are multiplied can be cancelled. Test with a number: if x = 2, (2 + 3) ÷ 2 = 2.5, not 3.

Doing composite functions in the wrong order, or reading f⁻¹(x) as 1 ÷ f(x)

Pupils read fg(x) left to right and apply f first. The −1 looks like a power, so it is taken as a reciprocal.

What helps: Read fg(x) as “f of g of x”, with g next to the x. For f(x) = 2x + 3, f⁻¹(5) = 1 because f(1) = 5, whereas 1 ÷ f(5) = 1 ÷ 13.

Treating a few examples as a proof

Trying 2 + 3 and 4 + 7 and seeing odd totals feels convincing, and in science, repeated results are often evidence enough.

What helps: Explain that examples only show that no counter-example has turned up yet. Use letters (2n, 2n + 1) so one line of algebra covers every case, and end with a sentence of explanation.

Getting a perpendicular gradient by changing only the sign, or only flipping

The rule has two parts, and remembering one of them feels like remembering the rule. So a line perpendicular to y = 3x + 1 is given gradient −3 or ⅓ instead of −⅓.

What helps: Both gradients should multiply to −1. Check: 3 × (−⅓) = −1. On squared paper, a line going 2 across and 1 up turns through a right angle to go 1 back and 2 up, gradient −2.

How to help at home

Check by expanding. Whenever your teenager factorises, ask them to expand the brackets to see if they get back to the start. It takes ten seconds and removes most careless errors from homework.
Prove a number trick. Think of a number, double it, add 6, halve it, subtract the number you started with. The answer is always 3. Ask them to prove it with n: 2n, then 2n + 6, then n + 3, then 3.
Function machines in reverse. Say “my rule is times 2 then add 3” and give inputs. Then give outputs and ask what went in. This is the idea behind inverse functions, and talking it through often clears up the notation.
Perpendicular lines on squared paper. Draw a line that goes 3 across and 2 up. Rotate the paper a quarter turn and see what the line now does. Count the steps and compare the two gradients.
Teach it back. Ask your teenager to explain one proof or one factorising method as if you were a Year 9 pupil. Explaining shows quickly what is secure and what is only half-remembered.

Builds on

Key Stage 3: Algebraic notation and formulae
Key Stage 3: Solving linear equations
Key Stage 3: Graphs of linear and quadratic functions
Key Stage 3: Sequences and the nth term