2 Equations, formulae and identities
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2.1 Use of symbols
2.1.ASymbols in equations, expressions and formulae
understand that symbols may be used to represent numbers in equations or variables in expressions and formulae
2.1.BGeneralised arithmetic rules in algebra
Understand that algebraic expressions follow the generalised rules of arithmetic.
2.1.CInteger index notation
use index notation for positive and negative integer powers, including zero
2.1.DIndex laws
use index laws in simple cases: xᵐ × xⁿ = xᵐ⁺ⁿ, xᵐ ÷ xⁿ = xᵐ⁻ⁿ and (xᵐ)ⁿ = xᵐⁿ
2.1.HAIndex notation involving fractional, negative and zero
use index notation involving fractional, negative and zero powers
2.2 Algebraic manipulation
2.2.AEvaluate expressions by substituting numerical values
evaluate expressions by substituting numerical values for letters
2.2.BCollect like terms
collect like terms
2.2.CExpanding a single term over a bracket
multiply a single term over a bracket
2.2.DTaking out common factors
take out common factors
2.2.EProducts of two linear expressions
expand the product of two simple linear expressions
2.2.FFactorising simple quadratic expressions
understand the concept of a quadratic expression and factorise expressions limited to x² + bx + c
2.2.HAProducts of multiple linear expressions
expand the product of two or more linear expressions
2.2.HBFactorising quadratic expressions
understand the concept of a quadratic expression and factorise quadratic expressions
2.2.HCAlgebraic fractions
manipulate algebraic fractions where the numerator and/or denominator can be numeric, linear or quadratic
2.2.HDCompleting the square
complete the square for a given quadratic expression
2.2.HEAlgebra to support and construct proofs
use algebra to support and construct proofs
2.3 Expressions and formulae
2.3.AUnknowns and variables
understand that a letter may represent an unknown number or a variable
2.3.BCorrect notational conventions for algebraic
use correct notational conventions for algebraic expressions and formulae
2.3.CSubstitution in expressions and formulae
substitute positive and negative integers, decimals and fractions for words and letters in expressions and formulae Evaluate 2x – 3y when x = 4 and y = −5
2.3.DFormulae from mathematics and other real-life contexts
use formulae from mathematics and other real-life contexts expressed initially in words or diagrammatic form and convert to letters and symbols
2.3.EDerive a formula or expression
derive a formula or expression
2.3.FChanging the subject of a formula
change the subject of a formula where the subject appears once
2.3.HAAdvanced changes of subject
manipulate formulae or equations to change the subject, including cases where the subject appears twice or a power of the subject occurs
2.4 Linear equations
2.4.ASolving linear equations
solve linear equations with integer or fractional coefficients in one unknown, where the unknown appears on either or both sides
2.4.BForming linear equations
set up simple linear equations from given data The three angles of a triangle are a°, (a + 10)°, (a + 20)°. Find the value of a
2.5 Proportion
2.5.HA
set up direct or inverse proportion problems and relate algebraic solutions to graphs, using y ∝ x, y ∝ 1/x, y ∝ x², y ∝ 1/x², y ∝ x³, y ∝ 1/x³, y ∝ √x and y ∝ 1/√x
2.6 Simultaneous linear equations
2.6.AExact solutions of simultaneous linear equations
Calculate exact solutions of two simultaneous linear equations in two unknowns.
2.6.HAHigher-tier simultaneous linear equations
Calculate exact solutions of higher-tier simultaneous linear equations in two unknowns.
2.6.HBGraphical interpretation of simultaneous equations
interpret the equations as lines and the common solution as the point of intersection
2.7 Quadratic equations
2.7.AFactorising simple quadratic equations
solve quadratic equations by factorisation, limited to x² + bx + c = 0
2.7.HAFactorising quadratic equations
solve quadratic equations by factorisation
2.7.HBQuadratic formula and completing the square
solve quadratic equations by using the quadratic formula or completing the square
2.7.HCForming quadratic equations
form and solve quadratic equations from data given in a context
2.7.HDLinear and quadratic simultaneous equations
solve simultaneous equations in two unknowns, one linear and one quadratic
2.8 Inequalities
2.8.AInequality symbols and compound inequalities
understand and use the symbols >, <, ≥ and ≤, including double-ended inequalities
2.8.BOpen and closed intervals
understand and use the convention for open and closed intervals on a number line
2.8.CSolving linear inequalities
solve simple linear inequalities in one variable and represent the solution set on a number line
2.8.DLinear inequalities on Cartesian graphs
Represent simple linear inequalities on rectangular Cartesian graphs.
2.8.ERegions defined by linear inequalities
identify regions on rectangular Cartesian graphs defined by simple linear inequalities Conventions for the inclusion of boundaries are not required
2.8.HASolving quadratic inequalities
solve quadratic inequalities in one unknown and represent the solution set on a number line
2.8.HBHarder regions defined by linear inequalities
Identify harder regions on Cartesian graphs defined by linear inequalities.