YR6 pupils should be taught to describe positions on the full coordinate grid (all four quadrants)

KS3 pupils should be taught to model situations or procedures by translating them into algebraic expressions or formulae and by using graphs

KS3 pupils should be taught to work with coordinates in all four quadrants

KS3 pupils should be taught to recognise, sketch and produce graphs of linear and quadratic functions of one variable with appropriate scaling, using equations in x and y and the Cartesian plane

KS3 pupils should be taught to interpret mathematical relationships both algebraically and graphically

KS3 pupils should be taught to reduce a given linear equation in two variables to the standard form y = mx + c; calculate and interpret gradients and intercepts of graphs of such linear equations numerically, graphically and algebraically

KS3 pupils should be taught to use linear and quadratic graphs to estimate values of y for given values of x and vice versa and to find approximate solutions of simultaneous linear equations

KS3 pupils should be taught to find approximate solutions to contextual problems from given graphs of a variety of functions, including piece-wise linear, exponential and reciprocal graphs

KS4 pupils should be taught to interpret and construct tables and line graphs for time series data

KS4 pupils should be taught to use the form y = mx + c to identify parallel {and perpendicular} lines; find the equation of the line through 2 given points, or through 1 point with a given gradient

KS4 pupils should be taught to interpret the gradient of a straight line graph as a rate of change; recognise and interpret graphs that illustrate direct and inverse proportion

KS4 pupils should be taught to identify and interpret roots, intercepts and turning points of quadratic functions graphically; deduce roots algebraically {and turning points by completing the square}

KS4 pupils should be taught to {interpret the gradient at a point on a curve as the instantaneous rate of change; apply the concepts of instantaneous and average rate of change (gradients of tangents and chords) in numerical, algebraic and graphical contexts}

KS4 pupils should be taught to recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function y =  with x not equal to 0, {the exponential function 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}

KS4 pupils should be taught to sketch translations and reflections of the graph of a given function

KS4 pupils should be taught to 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

KS4 pupils should be taught to {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}

KS4 pupils should be taught to {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}

KS4 pupils should be taught to 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

KS4 pupils should be taught to solve 2 simultaneous equations in 2 variables (linear/linear {or linear/quadratic}) algebraically; find approximate solutions using a graph

KS5 pupils should be taught to understand and use the equation of a straight line, including the forms y – y1 = m(x – x1) and ax + by + c = 0; Gradient conditions for two straight lines to be parallel or perpendicular. Be able to use straight line models in a variety of contexts

KS5 pupils should be taught to locate roots of f (x) = 0 by considering changes of sign of f(x) in an interval of x on which f(x) is sufficiently well behaved. Understand how change of sign methods can fail

KS5 pupils should be taught to understand and use the coordinate geometry of the circle including using the equation of a circle in the form (x – a)² + (y – b)² = r² Completing the square to find the centre and radius of a circle; use of the following properties: the angle in a semicircle is a right angle, the perpendicular from the centre to a chord bisects the chord, the radius of a circle at a given point on its circumference is perpendicular to the tangent to the circle at that point

KS5 pupils should be taught to work with quadratic functions and their graphs. The discriminant of a quadratic function, including the conditions for real and repeated roots. Completing the square. Solution of quadratic equations including solving quadratic equations in a function of the unknown.

KS5 pupils should be taught to understand and use the parametric equations of curves and conversion between Cartesian and parametric forms

KS5 pupils should be taught to use parametric equations in modelling in a variety of contexts

KS5 pupils should be taught to understand and use graphs of functions; sketch curves defined by simple equations including polynomials. The modulus of a linear function. Reciprocal graphs including their vertical and horizontal asymptotes. Interpret algebraic solution of equations graphically; use intersection points of graphs to solve equations. Understand and use proportional relationships and their graphs

KS5 pupils should be taught to understand the effect of simple transformations on the graph of y = f(x), including sketching associated graphs: y = af(x), y = f(x) + a, y = f(x + a), y = f(ax) and combinations of these transformations