Algebra
Choose your practice ↓Algebra is the largest content area on a Higher paper at 30%, and 20% of Foundation. It starts with notation and substitution, moves through collecting like terms, expanding brackets, taking out common factors and factorising quadratics, and builds into solving equations: linear equations with the unknown on both sides, quadratic equations, and simultaneous equations. Alongside the manipulation sits the graphical half of the subject — coordinates in all four quadrants, gradients and intercepts, y = mx + c and the lines parallel and perpendicular to a given line, quadratic curves and their roots and turning points, and real-life graphs including distance-time and velocity-time. Sequences close the area, from generating terms to deducing the nth term. Higher extends almost every strand: factorising ax² + bx + c, completing the square, the quadratic formula, algebraic fractions, linear/quadratic simultaneous equations, quadratic inequalities, inverse and composite functions, graph transformations, iteration, and the equation of a circle with its tangent.
📚 What you learn here
- Algebraic notation, substitution into formulae, and the vocabulary of expressions, equations, formulae, identities and inequalities
- Collecting like terms, expanding products of two or more brackets, and taking out common factors
- Factorising x² + bx + c and the difference of two squares; factorising ax² + bx + c (Higher)
- Rearranging formulae to change the subject
- Solving linear equations, including with the unknown on both sides
- Solving quadratic equations by factorising, by completing the square and by the quadratic formula (the last two are Higher)
- Simultaneous equations, linear/linear and (Higher) linear/quadratic
- Linear inequalities on a number line; quadratic inequalities and set notation (Higher)
- Straight-line graphs, gradients and intercepts, parallel and (Higher) perpendicular lines
- Quadratic, cubic, reciprocal, exponential and trigonometric graphs, and (Higher) translating and reflecting them
- Sequences: term-to-term and position-to-term rules, the nth term of a linear sequence, and (Higher) quadratic sequences
Frequently asked questions
What is the difference between expanding and factorising?
Expanding removes brackets by multiplying out: 3(x + 4) becomes 3x + 12. Factorising is the reverse — it puts brackets back by taking out what the terms share: 3x + 12 becomes 3(x + 4). If a question says 'fully factorise', check whether a number as well as a letter can come out.
When do I use the quadratic formula rather than factorising?
Factorise first if the quadratic factorises with integers — it is quicker and less error-prone. Use the formula (or completing the square) when it does not, or when the question asks for a solution to a given number of decimal places or in surd form. Both are Higher only; Foundation solves quadratics by factorising.
How do I find the equation of a line through two points?
Find the gradient m as the change in y divided by the change in x between the two points, then substitute one point into y = mx + c to find c. Getting the gradient upside down is the usual slip — divide the rise by the run, not the run by the rise.
Are simultaneous equations on Foundation?
Linear/linear simultaneous equations are on both tiers. The linear/quadratic case, where one equation is a curve, is Higher only.
Is the nth term of a quadratic sequence on Foundation?
No. Foundation deduces the nth term of a linear sequence; quadratic sequences are the Higher part of that statement.
✍️ Written by the MathsUK teamChecked against the National Curriculum and GCSE specifications