What does the Algebra content in OCR GCSE Maths cover?

Algebra covers manipulating expressions (expanding, factorising and rearranging formulae), solving linear, quadratic and simultaneous equations, generating and finding the nth term of sequences, straight line graphs with the equation y equals mx plus c, and solving and representing inequalities alongside recognising non-linear graphs. It is the second foundation area, after Number, and threads through every other topic.

Which algebra topics are Higher tier only in OCR maths?

Higher tier adds the quadratic formula and completing the square, solving one linear with one quadratic equation simultaneously, quadratic inequalities, the nth term of a quadratic sequence, perpendicular lines, simplifying algebraic fractions and algebraic proof. Foundation candidates still meet linear equations, basic factorising, linear sequences and straight line graphs, but not these harder techniques.

How do you find the nth term of a sequence?

For a linear sequence, the nth term is the common difference times n, adjusted by a constant to match the first term, so a sequence rising by three each time starts with three n. For a quadratic sequence, the coefficient of n squared is half the constant second difference, and you then subtract the n squared part to find a linear remainder. Recognising square, cube, triangular and Fibonacci sequences also helps.

What are the three ways to solve a quadratic equation?

Factorising is fastest when the quadratic factorises, setting each bracket to zero. The quadratic formula, given on the OCR Higher formulae sheet, works for any quadratic and is the standard method on the calculator paper. Completing the square, a Higher-tier method, both solves the equation and reveals the turning point of the curve. The discriminant tells you how many real roots there are.

How do you know if two lines are parallel or perpendicular?

Compare their gradients once each equation is in the form y equals mx plus c. Parallel lines have equal gradients. Perpendicular lines have gradients that multiply to give negative one, so each gradient is the negative reciprocal of the other. If a line has gradient two, a perpendicular line has gradient negative one half.

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OCR GCSE Mathematics Algebra: a complete overview of manipulation, equations, sequences, graphs and inequalities

A deep-dive OCR GCSE Mathematics guide to the Algebra content. Covers algebraic manipulation, solving linear, quadratic and simultaneous equations, sequences, straight line graphs, inequalities and other graphs, with the methods and exam patterns OCR repeats across Foundation and Higher tier.

Generated by Claude Opus 4.816 min readJ560 AUpdated 2026-06-02

Reviewed by: AI editorial process; not yet individually human-reviewed

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What the Algebra content demands
Algebraic manipulation
Solving linear equations
Quadratic equations
Simultaneous equations
Sequences
Straight line graphs
Inequalities and other graphs
Check your knowledge

What the Algebra content demands

Algebra is the language the rest of the course is written in. OCR uses it everywhere, from geometry proofs to compound-interest formulae, so weak algebra leaks marks across the whole paper. The content runs from manipulating expressions through solving every kind of equation to graphs and inequalities, with the hardest techniques reserved for Higher tier. Because OCR puts the non-calculator paper in the middle of the three, algebraic fluency by hand is essential.

This guide walks through the seven areas of the Algebra content and ties together the matching dot-point pages, each of which has its own practice questions.

Algebraic manipulation

The toolkit is simplifying (collecting like terms, the index laws), expanding (single and double brackets, using FOIL), factorising (common factors, quadratics, and the difference of two squares $a^2 - b^2 = (a+b)(a-b)$ ), and changing the subject of a formula by inverse operations. At Higher tier you also simplify algebraic fractions by factorising and cancelling common brackets.

Solving linear equations

Solve by keeping the equation balanced: do the same to both sides, undoing operations in reverse. Expand brackets first, clear fractions by multiplying through, and when the unknown is on both sides, collect the unknowns on one side and the numbers on the other. Forming an equation from a worded or geometric situation, such as angles summing to $180^\circ$ , is a frequent source of marks.

Quadratic equations

Three methods: factorising (set each bracket to zero), the quadratic formula $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ for any quadratic, and completing the square at Higher tier, which also gives the turning point. The discriminant $b^2 - 4ac$ tells you whether there are two, one or no real roots. The formula is on the OCR Higher formulae sheet, but you must know when to use it.

Simultaneous equations

Two equations, two unknowns. Solve by elimination (match coefficients, then add or subtract) or substitution (rearrange one and substitute into the other). The solution is where the graphs intersect. At Higher tier, one linear and one quadratic are solved by substitution, giving up to two solution pairs.

Sequences

A linear sequence has a constant first difference and nth term $dn + c$ . A quadratic sequence has a constant second difference, and its $n^2$ coefficient is half that difference. Recognise square, cube, triangular, Fibonacci and geometric sequences on sight. The nth term lets you find any term and test whether a value belongs.

Straight line graphs

The equation $y = mx + c$ packages the gradient $m$ and $y$ -intercept $c$ . Gradient is the change in $y$ over the change in $x$ . Parallel lines share a gradient; perpendicular gradients multiply to $-1$ . To find a line's equation, find the gradient then use a point to find $c$ , rearranging into $y = mx + c$ where needed.

Inequalities and other graphs

Solve inequalities like equations, but reverse the sign when multiplying or dividing by a negative. Show solutions on a number line with open or closed circles. Quadratic inequalities (Higher) use the parabola's sign. Recognise the shapes of quadratic, cubic, reciprocal and exponential graphs.

Check your knowledge

A mix of manipulation, equation and graph questions covering the Algebra content. Attempt them under timed conditions, then check against the solutions.

Expand and simplify $(x + 6)(x - 2)$ . (2 marks)
Factorise $x^2 - 49$ . (1 mark)
Solve $5x - 3 = 2x + 12$ . (3 marks)
Solve $x^2 + x - 12 = 0$ by factorising. (3 marks)
Find the nth term of $4, 9, 14, 19$ . (2 marks)
A line passes through $(0, -1)$ with gradient $3$ . Write its equation. (1 mark)
Solve $3x + 1 > 13$ . (2 marks)
State the gradient of a line perpendicular to $y = 2x + 5$ . (1 mark)

Solutions

Step 1: Q1 - expanding double brackets

Use FOIL (First, Outer, Inner, Last) to multiply every term in the first bracket by every term in the second, then collect like terms:

x^2 - 2x + 6x - 12 = x^2 + 4x - 12

Step 2: Q2 - difference of two squares

Recognise $x^2 - 49$ as $x^2 - 7^2$ , which factors using the identity $a^2 - b^2 = (a + b)(a - b)$ :

(x + 7)(x - 7)

Step 3: Q3 - solving a linear equation

Collect the unknown on one side by subtracting $2x$ from both sides, then undo the constant term:

Subtract $2x$ : $3x - 3 = 12$ . Add $3$ : $3x = 15$ , so $x = 5$ .

Step 4: Q4 - solving a quadratic by factorising

Find two numbers that multiply to $-12$ and add to $+1$ : those are $4$ and $-3$ . Set each bracket equal to zero:

(x + 4)(x - 3) = 0 \implies x = -4 \text{ or } x = 3

Step 5: Q5 - nth term of a linear sequence

The common difference gives the coefficient of $n$ . Adjust the constant so the first term matches: if $5n + c = 4$ when $n = 1$ , then $c = -1$ :

5n - 1

Step 6: Q6 - equation of a straight line

The gradient is $3$ and the $y$ -intercept is $-1$ (the line passes through $(0, -1)$ ), so substitute directly into $y = mx + c$ :

y = 3x - 1

Step 7: Q7 - solving a linear inequality

Solve just like an equation, subtracting $1$ from both sides then dividing by $3$ . There is no sign change because we divide by a positive number:

Subtract $1$ : $3x > 12$ , so $x > 4$ .

Step 8: Q8 - perpendicular gradient

Perpendicular gradients multiply to $-1$ . The gradient of the given line is $2$ , so the perpendicular gradient is its negative reciprocal:

-\frac{1}{2}

Sources & how we know this

OCR GCSE (9-1) Mathematics (J560) specification — OCR (2015)