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

The Algebra content in CCEA GCSE Mathematics is the second half of the Number and Algebra strand and the engine of the whole course. This guide maps the area, from the basic manipulation every question needs to the Higher-tier quadratics and graphs, and shows the methods CCEA repeats across the modules M1 to M8.

Algebraic manipulation

Manipulation is the toolkit: collecting like terms, substituting values, expanding single and double brackets, factorising (into a single bracket, and into two brackets for a quadratic at Higher tier), simplifying algebraic fractions, and changing the subject of a formula. The expanding rule is that each term in one bracket multiplies each in the other; factorising reverses it. Changing the subject undoes the operations on the wanted letter in reverse order. Every later topic depends on this being fluent.

Linear equations and inequalities

A linear equation is solved by balancing: expand brackets, clear fractions, collect the unknown on one side, then undo the remaining operations. Forming an equation from a worded or geometric context carries the reasoning marks. Inequalities use the same steps, with one extra rule: multiplying or dividing by a negative reverses the inequality sign. Solutions are shown on a number line with a filled circle when the endpoint is included and an open circle when it is not.

Simultaneous equations

Two linear equations are solved by elimination (match a coefficient, then add or subtract) or substitution (make a variable the subject, then substitute). At Higher tier a linear equation is solved with a quadratic by substituting the linear into the quadratic and solving the resulting quadratic, giving up to two coordinate pairs. Graphically, the solution is where the two graphs cross.

Sequences

A term-to-term rule continues a sequence; the nth term jumps straight to any position. A linear sequence's nth term begins with the common difference times $n$, adjusted by a constant. At Higher tier a quadratic sequence has a constant second difference, the $n^2$ coefficient is half that second difference, and the remaining part is found by subtraction. Knowing the square, cube, triangular and Fibonacci sequences speeds up many questions.

Straight line graphs

Every straight line is $y = mx + c$, with $m$ the gradient and $c$ the $y$-intercept. The gradient is the change in $y$ over the change in $x$ between two points. At Higher tier, parallel lines share a gradient, and perpendicular gradients multiply to $-1$ (negative reciprocals). To find a line's equation, get the gradient, then substitute a point to find the intercept.

Quadratic equations (Higher)

A quadratic $ax^2 + bx + c = 0$ is solved by factorising into two brackets and setting each to zero, by the quadratic formula $x = \tfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ when it does not factorise, or by completing the square. The discriminant $b^2 - 4ac$ tells you the number of real roots. A quadratic usually has two roots, so always rearrange to equal zero first and give both.

Graphs and functions

Beyond lines, you must recognise and interpret parabolas (quadratics), S-shaped cubics, reciprocal curves with asymptotes, and exponential growth curves, plus the circle $x^2 + y^2 = r^2$ centred at the origin. Roots are where a curve meets the $x$-axis, the turning point is its minimum or maximum, and equations can be solved by reading where curves cross. Real-life graphs such as distance-time use the gradient as a rate of change.

How CCEA examines Algebra

Algebra runs through every module and both the calculator and non-calculator work, and it carries a large share of the AO2 and AO3 reasoning marks, especially when an equation must be formed from a context. Show full, line-by-line working so method marks are secure, keep signs under control, and always give the full set of solutions. Use the dot points below for specification-level detail and worked CCEA-style questions, then test yourself with the Algebra quiz.