use and interpret algebraic notation, including:
- \(ab\) in place of \(a×b\)
- \(3y\) in place of \(y+y+y\) and \(3×y\)
- \(a^2\) in place of \(a×a\), \(a^3\) in place of \(a×a×a\), \(a^2b\) in place of \(a×a×b\)
- \(\dfrac{a}{b}\) in place of \(a÷b\)
- coefficients written as fractions rather than as decimals
- brackets

substitute numerical values into formulae and expressions, including scientific formulae

understand and use the concepts and vocabulary of expressions, equations, formulae, identities, inequalities, terms and factors

simplify and manipulate algebraic expressions (including those involving surds and algebraic fractions) by:
- collecting like terms
- multiplying a single term over a bracket
- taking out common factors
- expanding products of two or more binomials
- factorising quadratic expressions of the form \(x^2+bx+c\), including the difference of two squares
- factorising quadratic expressions of the form \(ax^2+bx+c\)
- simplifying expressions involving sums, products and powers, including the laws of indices

understand and use standard mathematical formulae;

rearrange formulae to change the subject

know the difference between an equation and an identity;

argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments and proofs

where appropriate, interpret simple expressions as functions with inputs and outputs;

interpret the reverse process as the ‘inverse function’;

interpret the succession of two functions as a ‘composite function’ (the use of formal function notation is expected)

work with coordinates in all four quadrants

plot graphs of equations that correspond to straight-line graphs in the coordinate plane;

use the form \(y=mx+c\) to identify parallel lines and perpendicular lines;

find the equation of the line through two given points, or through one point with a given gradient

identify and interpret gradients and intercepts of linear functions graphically and algebraically

identify and interpret roots, intercepts and turning points of quadratic functions graphically;

deduce roots algebraically and turning points by completing the square

recognise, sketch and interpret graphs of linear functions and quadratic functions, simple cubic functions, the reciprocal function \(y=\dfrac{1}{x}\) with \(x≠0\), exponential functions \(y=k^x\) for positive values of \(k\), and the trigonometric functions (with arguments in degrees) \(y=\sin{x}\), \(y=\cos{x}\) and \(y=\tan{x}\) for angles of any size

sketch translations and reflections of a given function

plot and interpret graphs (including reciprocal graphs and exponential graphs), and graphs of non-standard functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration

calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts (this does not include calculus)

recognise and use the equation of a circle with centre at the origin;

find the equation of a tangent to a circle at a given point

solve linear equations in one unknown algebraically (including those with the unknown on both sides of the equation);

find approximate solutions using a graph

solve quadratic equations (including those that require rearrangement) algebraically by factorising, by completing the square and by using the quadratic formula;

find approximate solutions using a graph

solve two simultaneous equations in two variables (linear/linear or linear/quadratic) algebraically;

find approximate solutions using a graph

find approximate solutions to equations numerically using iteration

translate simple situations or procedures into algebraic expressions or formulae;

derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution

solve linear inequalities in one or two variable(s), and quadratic inequalities in one variable;

represent the solution set on a number line, using set notation and on a graph

generate terms of a sequence from either a term-to-term or a position-to-term rule

recognise and use sequences of triangular, square and cube numbers and simple arithmetic progressions, Fibonacci-type sequences, quadratic sequences, and simple geometric progressions (\(r^n\) where \(n\) is an integer and \(r\) is a rational number \(> 0\) or a surd) and other sequences

deduce expressions to calculate the nth term of linear and quadratic sequences