3.2 Algebra

3.2.1 Notation, vocabulary and manipulation

A1

Basic foundation content	Additional foundation content	Higher content only
use and interpret algebraic notation, including: $a b$ in place of $a \times b$ $3 y$ in place of $y + y + y$ and $3 \times y$ $a^{2}$ in place of $a \times a$ , $a^{3}$ in place of $a \times a \times a$ , $a^{2} b$ in place of $a \times a \times b$ $\frac{a}{b}$ in place of $a \div b$ 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.

A2

Basic foundation content	Additional foundation content	Higher content only
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

A3

Basic foundation content	Additional foundation content	Higher content only
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.

A4

Basic foundation content	Additional foundation content	Higher content only
simplify and manipulate algebraic expressions by:	simplify and manipulate algebraic expressions (including those involving surds) by:	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 simplifying expressions involving sums, products and powers, including the laws of indices
	expanding products of two binomials factorising quadratic expressions of the form $x^{2} + b x + c$ , including the difference of two squares	expanding products of two or more binomials factorising quadratic expressions of the form ${a x}^{2} + b x + c$

A5

Basic foundation content	Additional foundation content	Higher content only
understand and use standard mathematical formulae rearrange formulae to change the subject

Basic foundation content

Additional foundation content

Higher content only

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

A6

Basic foundation content	Additional foundation content	Higher content only
	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

A7

Basic foundation content	Additional foundation content	Higher content only
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’

Basic foundation content

Additional foundation content

Higher content only

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 $f (x)$ , $f g (x)$ and $f^{- 1} (x)$ notation is expected at Higher tier.

3.2.2 Graphs

A8

Basic foundation content	Additional foundation content	Higher content only
work with coordinates in all four quadrants

A9

Basic foundation content	Additional foundation content	Higher content only
plot graphs of equations that correspond to straight-line graphs in the coordinate plane	use the form $y = m x + c$ 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 $y = m x + c$ to identify perpendicular lines

Basic foundation content

Additional foundation content

Higher content only

plot graphs of equations that correspond to straight-line graphs in the coordinate plane

use the form $y = m x + c$ 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 $y = m x + c$ to identify perpendicular lines

A10

Basic foundation content	Additional foundation content	Higher content only
identify and interpret gradients and intercepts of linear functions graphically and algebraically

A11

Basic foundation content	Additional foundation content	Higher content only
	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

A12

Basic foundation content	Additional foundation content	Higher content only
recognise, sketch and interpret graphs of linear functions and quadratic functions	including simple cubic functions and the reciprocal function $y = \frac{1}{x}$ with $x \neq 0$	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

A13

Basic foundation content	Additional foundation content	Higher content only
		sketch translations and reflections of a given function

A14

Basic foundation content	Additional foundation content	Higher content only
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

A15

Basic foundation content	Additional foundation content	Higher content only
		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

A16

Basic foundation content	Additional foundation content	Higher content only
		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

Basic foundation content

Additional foundation content

Higher content only

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

3.2.3 Solving equations and inequalities

A17

Basic foundation content	Additional foundation content	Higher content only
solve linear equations in one unknown algebraically find approximate solutions using a graph	including those with the unknown on both sides of the equation

Basic foundation content

Additional foundation content

Higher content only

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.

A18

Basic foundation content	Additional foundation content	Higher content only
	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

Basic foundation content

Additional foundation content

Higher content only

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

A19

Basic foundation content	Additional foundation content	Higher content only
	solve two simultaneous equations in two variables (linear/linear) algebraically find approximate solutions using a graph	including linear/quadratic

Basic foundation content

Additional foundation content

Higher content only

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

find approximate solutions using a graph

including linear/quadratic

A20

Basic foundation content	Additional foundation content	Higher content only
		find approximate solutions to equations numerically using iteration

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

A21

Basic foundation content	Additional foundation content	Higher content only
	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

Basic foundation content

Additional foundation content

Higher content only

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.

A22

Basic foundation content	Additional foundation content	Higher content only
	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.

3.2.4 Sequences

A23

Basic foundation content	Additional foundation content	Higher content only
generate terms of a sequence from either a term-to-term or a position-to-term rule

Notes: including from patterns and diagrams.

A24

Basic foundation content	Additional foundation content	Higher content only
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

Basic foundation content

Additional foundation content

Higher content only

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.

A25

Basic foundation content	Additional foundation content	Higher content only
deduce expressions to calculate the $n th$ term of linear sequences		including quadratic sequences

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3.1 Number

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3.3 Ratio, proportion and rates of change