Numbers and their operations

• primes and prime factorisation

• finding highest common factor (HCF) and lowest common multiple (LCM), squares, cubes, square roots and cube roots by prime factorisation

• negative numbers, integers, rational numbers, real numbers and their four operations

• calculations with calculator

• representation and ordering of numbers on the number line

• use of <, >, ⩽, ⩾

• approximation and estimation (including rounding off numbers to a required number of decimal places or significant figures and estimating the results of computation)

• use of standard form A × 10n, where n is an integer, and 1 ⩽ A < 10

• positive, negative, zero and fractional indices

• laws of indices

Ratio and proportion

• comparison between two or more quantities by ratio

• relationship between ratio and fraction

• dividing a quantity in a given ratio

• ratios involving rational numbers

• equivalent ratios

• writing a ratio in its simplest form

• map scales (distance and area)

• direct and inverse proportion

Percentage

• expressing percentage as a fraction or decimal

• expressing one quantity as a percentage of another

• comparing two quantities by percentage

• percentages greater than 100%

• increasing/decreasing a quantity by a given percentage

• finding percentage increase/decrease

• reverse percentages

Rate and speed

• relationships between distance, time and speed

• writing speed in different units (e.g. km/h, m/min, m/s and cm/s)

• calculation of speed, distance or time given the other two quantities

• average rate and average speed

• conversion of units (e.g. km/h to m/s)

• using letters to represent numbers

• interpreting notations:
ab as a × b

• Cartesian coordinates in two dimensions

• graph of a set of ordered pairs as a representation of a relationship between two variables

 • linear functions (y = ax + b) and quadratic functions (y = ax2 + bx + c)

• graphs of linear functions

• the gradient of a linear graph as the ratio of the vertical change to the horizontal change (positive and negative gradients)

• graphs of quadratic functions and their properties: ∗ positive or negative coefficient of x2 ∗ maximum and minimum points ∗ symmetry

• graphs of power functions of the form y = axn, where n = −2, −1, 0, 1, 2, 3, and simple sums of not more than three of these

• graphs of exponential functions y = kax , where a is a positive integer

• estimation of the gradient of a curve by drawing a tangent

• solving linear equations in one variable

• solving simple fractional equations that can be reduced to linear equations such as:

–          x/3 + x-2/4 = 3

–          3/2-3 = 6

• solving simultaneous linear equations in two variables by
∗ substitution and elimination methods
∗ graphical method

• solving quadratic equations in one variable by
∗ factorisation
∗ use of formula
∗ completing the square for y = x2 + px + q ∗ graphical method

• solving fractional equations that can be reduced to quadratic equations

• formulating equations to solve problems

• solving simple inequalities in the form ax ⩽ b, ax ⩾ b, ax < b and ax > b where a and b are integers

•right, acute, obtuse and reflex angles

• vertically opposite angles, angles on a straight line and angles at a point

• angles formed by two parallel lines and a transversal: corresponding angles, alternate angles, interior angles

• properties of triangles, special quadrilaterals and regular polygons (pentagon, hexagon, octagon and decagon) including symmetry properties

• classifying special quadrilaterals on the basis of their properties

• angle sum of interior and exterior angles of any convex polygon

• construction of simple geometrical figures from given data using compasses, ruler, set squares and protractor, where appropriate

• congruent and similar figures

• properties of similar triangles and polygons: ∗ corresponding angles are equal ∗ corresponding sides are proportional

• enlargement and reduction of a plane figure

• scale drawings

• properties and construction of perpendicular bisectors of line segments and angle bisectors

• solving simple problems involving similarity and congruence

•congruent and similar figures

• properties of similar triangles and polygons:

∗ corresponding angles are equal

∗ corresponding sides are proportional

• enlargement and reduction of a plane figure

• scale drawings

• properties and construction of perpendicular bisectors of line segments and angle bisectors

• solving simple problems involving similarity and congruence

• use of Pythagoras’ theorem

• determining whether a triangle is right-angled given the lengths of three sides • use of trigonometric ratios (sine, cosine and tangent) of acute angles to calculate unknown sides and angles in right-angled triangles • extending sine and cosine to obtuse angles

• use of the formula 2 1 absinC for the area of a triangle

• use of sine rule and cosine rule for any triangle

• problems in two and three dimensions including those involving angles of elevation and depression and bearings

• area of parallelogram and trapezium

• problems involving perimeter and area of composite plane figures

 • volume and surface area of cube, cuboid, prism, cylinder, pyramid, cone and sphere

• conversion between cm2 and m2, and between cm3 and m3

• problems involving volume and surface area of composite solids

• arc length as fraction of the circumference and sector area as fraction of the area of a circle

• area of a segment

• use of radian measure of angle (including conversion between radians and degrees)

• problems involving the arc length, sector area of a circle and area of a segment

• problems involving perimeter and area of composite plane figures

• volume and surface area of cube, cuboid, prism, cylinder, pyramid, cone and sphere

• conversion between cm2 and m2, and between cm3 and m3

• problems involving volume and surface area of composite solids

• arc length as fraction of the circumference and sector area as fraction of the area of a circle

• area of a segment

• use of radian measure of angle (including conversion between radians and degrees)

• problems involving the arc length, sector area of a circle and area of a segment

• analysis and interpretation of: ∗ tables ∗ bar graphs ∗ pictograms ∗ line graphs ∗ pie charts ∗ dot diagrams ∗ histograms with equal class intervals ∗ stem-and-leaf diagrams ∗ cumulative frequency diagrams ∗ box-and-whisker plots

• purposes and uses, advantages and disadvantages of the different forms of statistical representations

• drawing simple inference from statistical diagrams

• explaining why a given statistical diagram leads to misinterpretation of data

• mean, mode and median as measures of central tendency for a set of data

• purposes and use of mean, mode and median

• calculation of the mean for grouped data

• quartiles and percentiles

• range, interquartile range and standard deviation as measures of spread for a set of data

• calculation of the standard deviation for a set of data (grouped and ungrouped)

• using the mean and standard deviation to compare two sets of data

• probability as a measure of chance

• probability of single events (including listing all the possible outcomes in a simple chance situation to calculate the probability)

• probability of simple combined events (including using possibility diagrams and tree diagrams, where appropriate)

• addition and multiplication of probabilities (mutually exclusive events and independent events)