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

Notes: it is expected that answers will be given in their simplest form without an explicit instruction to do so.

substitute numerical values into formulae and expressions, including scientific formulae

Notes: unfamiliar formulae will be given in the question.

See the Appendix for a full list of the prescribed formulae. See also A5

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

to include identities

Notes: this will be implicitly and explicitly assessed.

simplify and manipulate algebraic expressions by:
- collecting like terms
- multiplying a single term over a bracket
- taking out common factors
- simplifying expressions involving sums, products and powers, including the laws of indices

simplify and manipulate algebraic expressions (including those involving surds) by:
- expanding products of two binomials
- factorising quadratic expressions of the form \(x^2+bx+c\), including the difference of two squares

simplify and manipulate algebraic expressions (including those involving surds and algebraic fractions) by:
- expanding products of two or more binomials
- factorising quadratic expressions of the form \(ax^2+bx+c\)

understand and use standard mathematical formulae

rearrange formulae to change the subject

Notes: including use of formulae from other subjects in words and using symbols.

See the Appendix for a full list of the prescribed formulae. See also A2

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

to include 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’

Notes: understanding and use of \(f(x)\), \(fg(x)\) and \(f^{−1}(x)\) notation is expected at Higher tier.

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

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

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

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

deduce turning points by completing the square

Notes: including the symmetrical property of a quadratic. See also A18

recognise, sketch and interpret graphs of linear functions and quadratic functions

including simple cubic functions and the reciprocal function \(y=\dfrac{1}{x}\) with \(x≠0\)

including 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

Notes: see also G21

sketch translations and reflections of a given function

plot and interpret 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

including reciprocal graphs

including exponential graphs

Notes: including problems requiring a graphical solution. See also A15

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

Notes: see also A14, R14 and R15

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

find approximate solutions using a graph

including those with the unknown on both sides of the equation

Notes: including use of brackets.

solve quadratic equations algebraically by factorising

including those that require rearrangement

including completing the square and by using the quadratic formula

find approximate solutions using a graph

Notes: see also A11

solve two simultaneous equations in two variables (linear/linear) algebraically

find approximate solutions using a graph

including linear/quadratic

find approximate solutions to equations numerically using iteration

Notes: including the use of suffix notation in recursive formulae.

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

Notes: including the solution of geometrical problems and problems set in context.

solve linear inequalities in one variable

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

represent the solution set on a number line

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

Notes: students should know the conventions of an open circle on a number line for a strict inequality and a closed circle for an included boundary. See also N1

In graphical work the convention of a dashed line for strict inequalities and a solid line for an included inequality will be required.

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

Notes: including from patterns and diagrams.

recognise and use sequences of triangular, square and cube numbers and simple arithmetic progressions

including Fibonacci-type sequences, quadratic sequences, and simple geometric progressions (\(r^n\) where \(n\) is an integer and \(r\) is a rational number \(> 0\))

including other sequences

including where \(r\) is a surd

Notes: other recursive sequences will be defined in the question.

deduce expressions to calculate the nth term of linear sequences

including quadratic sequences