2. Algebra

Notation, vocabulary and manipulation

A1 use and interpret algebraic manipulation, including:
• ab in place of a × b
• 3y in place of y + y + y and 3 × y
• a2 in place of a × a,

${a}^{3}$

in place of a × a × a,

${a}^{2}b$

in place of a × a × b

$\frac{a}{b}$

in place of a ÷ b
● coefficients written as fractions rather than as decimals
● brackets

A2 substitute numerical values into formulae and expressions, including scientific formulae
A3 understand and use the concepts and vocabulary of expressions, equations, formulae, identities, inequalities, terms and factors.

A4 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

${x}^{2}+bx+c$

● simplifying expressions involving sums, products and powers, including the laws of indices.

A5 understand and use standard mathematical formulae; rearrange formulae to change the subject
A6 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
A7 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).

Graphs

A8 work with coordinates in all four quadrants
A9 plot graphs of equations that correspond to straight-line graphs in the coordinate plane; use the form y = mx + c to identify parallel and perpendicular lines; find the equation of the line through two given points or through one point with a given gradient
A10 identify and interpret gradients and intercepts of linear functions graphically and algebraically
A11 identify and interpret roots, intercepts, turning points of quadratic functions graphically; deduce roots algebraically and turning points by completing the square
A12 recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function

$y=\frac{1}{x}$

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
A13 sketch translations and reflections of a given function.

A14 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
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 (this does not include calculus)

A16 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

Solving equations and inequalities

A17 solve linear equations in one unknown algebraically (including those with the unknown on both sides of the equation); find approximate solutions using a graph
A18 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
A19 solve two simultaneous equations in two variables (linear/linear or linear/quadratic) algebraically; find approximate solutions using a graph
A20 find approximate solutions to equations numerically using iteration
A21 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.

A22 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.

Sequences

A23 generate terms of a sequence from either a term-to-term or a position-to term rule
A24 recognise and use sequences of triangular, square and cube numbers, 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
A25 deduce expressions to calculate the nth term of linear and quadratic sequences.