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Equations, formulae, expressions and identities
Year 7
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Year 11
Extension
Use letter symbols to represent unknown numbers or variables; know the meanings of the words term, expression and equation. Recognise that letter symbols play different roles in equations, formulae and functions; know the meanings of the words formula and function. Distinguish the different roles played by letter symbols in equations, identities, formulae and functions.      
Understand that algebraic operations follow the rules of arithmetic. Understand that algebraic operations, including the use of brackets, follow the rules of arithmetic; use index notation for small positive integer powers. Use index notation for integer powers and simple instances of the index laws. Know and use the index laws in generalised form for multiplication and division of integer powers.
Simplify linear algebraic expressions by collecting like terms; multiply a single term over a bracket (integer coefficients).

Brackets using Excel

Simplify or transform linear expressions by collecting like terms; multiply a single term over a bracket. Simplify or transform algebraic expressions by taking out single-term common factors; add simple algebraic fractions. Square a linear expression; expand the product of two linear expressions of the form x ± n and simplify the corresponding quadratic expression; establish identities such as a2 − b2 = (a + b) (a − b).

Area algebra

 Factorise quadratic expressions, including the difference of two squares, e.g. x2 − 9 = (x + 3) (x − 3) and cancel common factors in rational expressions, e.g.
2(x+1)2/(x+1).

Factoring and graphing quadratic equations Autograph

Factorising quadratics
    Simplify simple algebraic fractions to produce linear expressions; use factorisation to simplify compound algebraic fractions.
Construct and solve simple linear equations with integer coefficients (unknown on one side only) using an appropriate method (e.g. inverse operations). Construct and solve linear equations with integer coefficients (unknown on either or both sides, without and with brackets) using appropriate methods (e.g. inverse operations, transforming both sides in same way).

Solving linear equations by balancing

Construct and solve linear equations with integer coefficients (with and without brackets, negative signs anywhere in the equation, positive or negative solution). Solve linear equations in one unknown with integer and fractional coefficients; solve linear equations that require prior simplification of brackets, including those with negative signs anywhere in the equation. Solve equations involving algebraic fractions with compound expressions as the numerators and/or denominators.
Use graphs and set up equations to solve simple problems involving direct proportion. Use algebraic methods to solve problems involving direct proportion; relate algebraic solutions to graphs of the equations; use ICT as appropriate.
Introductory work on simultaneous equations. Solve a pair of simultaneous linear equations by eliminating one variable; link a graph of an equation or a pair of equations to the algebraic solution; consider cases that have no solution or an infinite number of solutions. Explore 'optimum' methods of solving simultaneous equations in different forms. Solve exactly, by elimination of an unknown, two simultaneous equations in two unknowns, where one is linear in each unknown and the other is linear in one unknown and quadratic in the other or of the form x2 + y2 = r2.
  Solve linear inequalities in one variable; represent the solution set on a number line. Solve linear inequalities in one and two variables; find and represent the solution set.  
Use systematic trial and improvement methods and ICT tools to find approximate solutions to equations such as
x2 + x = 20.              

Trial & improvement with Excel

Trial and Improvement Demonstration Excel

 Solve quadratic equations by factorisation. Solve quadratic equations by factorisation, completing the square and using the quadratic formula, including those in which the coefficient of the quadratic term is greater than 1.
Explore ways of constructing models of real-life situations by drawing graphs and constructing algebraic equations and inequalities.
Use simple formulae from mathematics and other subjects; substitute positive integers into linear expressions and formulae and, in simple cases, derive a formula. Use formulae from mathematics and other subjects; substitute integers into simple formulae, including examples that lead to an equation to solve; substitute positive integers into expressions involving small powers e.g. 3x2 + 4 or 2x3; derive simple formulae. Use formulae from mathematics and other subjects; substitute numbers into expressions and formulae; derive a formula and, in simple cases, change its subject. Derive and use more complex formulae; change the subject of a formula, including cases where a power of the subject appears in the question or solution, e.g. find r given that A = πr2. Derive and use more complex formulae; change the subject of a formula, including cases where the subject occurs twice. Derive relationships between different formulae that produce equal or related results.