Natural numbers, integers (positive, zero and negative), prime numbers, square numbers, cube numbers, common factors, common multiples, rational and irrational numbers, reciprocals
Express as a product of its prime factors
Finding the highest common factor (HCF) of two numbers
Finding the lowest common multiple (LCM) of two numbers
1.2 Sets
Set language, notation and Venn diagrams [Venn diagrams are limited to two or three sets]
A = {x : x is a natural number}
B = {(x, y): y = mx + c}
C = {x : a \(\leq\) x \(\leq\) b}
D = {a, b, c…}
Number of elements in set A [n(A)]
“… is an element of …” [∈]
“… is not an element of …” [∉]
Complement of set A [A']
The empty set [∅]
Universal set [\(\xi\)]
A is a subset of B [A ⊆ B]
A is not a subset of B [A ⊈ B]
Union of A and B [A \(\cup\) B]
Intersection of A and B [A ∩ B]
1.3 Powers and Roots
Squares, square roots, cubes and cube roots
1.4 Fractions, Decimals and Percentages
Proper and improper fractions, mixed numbers, decimals, percentages, equivalence and convert between these forms
1.5 Ordering
Familiarity with the symbols =, ≠, >, < , \(\geq\) and \(\leq\)
1.6 The Four Operations
Use the four operations for calculations with integers, fractions and decimals, including correct ordering of operations and use of brackets
1.7 Indices
Use indices (positive, zero, negative and fractional). Understand and use the rules of indices.
1.8 Standard Form
Standard form A x 10n where n is a positive or negative integer and 1 \(\leq\) A < 10
1.9 Estimation
Round values to a specified degree of accuracy [includes decimal places and significant figures]
1.10 Limits of Accuracy
Find upper and lower bounds of the results
1.11 Ratio and Proportion
Give ratios in their simplest form
Divide a quantity in a given ratio
Use proportional reasoning and ratios in context
1.12 Rates
Use common measures of rate [e.g. hourly rates of pay, exchange rates between currencies, flow rates, fuel consumption]
Apply other measures of rate [e.g. pressure, density, population density]
Solve problems involving average speed [knowledge of speed/distance/time formula is required]
1.13 Percentages
Express one quantity as a percentage of another
Calculate percentage increase or decrease
Calculate with simple and compound interest
Calculate using reverse percentages [e.g. find the cost price given the selling price and the percentage profit]
Percentage calculations may include [deposit, discount, profit and loss (as an amount or a percentage), earnings, percentages over 100%]
1.14 Using a Calculator
1.15 Time
Calculate with time: seconds (s), minutes (min), hours (h), days, weeks, months, years, including the relationship between units
Calculate times in terms of the 24-hour and 12-hour clock
Read clocks and timetables [includes problems involving time zones, local times and time differences]
1.16 Money
Convert from one currency to another
1.17 Exponential Growth and Decay
Use exponential growth and decay [e.g. depreciation, population change]
1.18 Surds
Understand and use surds, including simplifying expressions. Rationalise the denominator.
2 ALGEBRA AND GRAPHS
2.1 Algebraic Manipulation
Simplify expressions, expand products, factorise and complete the square
2.2 Algebraic Fractions
Factorise and simplify rational expressions
2.3 Equations
Construct expressions, equations and formulas
Solve linear (in one unknown), fractional, simultaneous linear (in two unknowns) and quadratic equations
Change the subject of formulas
2.4 Inequalities
Solve and interpret linear inequalities, including on a number line
Interpret linear inequalities in two variables graphically
List inequalities that define a given region
2.5 Sequences
Recognise patterns in sequences, including the term-to-term rule, and relationships between different sequences [includes linear, quadratic, cubic and exponential sequences and simple combinations of these]
Find and use the nth term of sequences
2.6 Proportion
Direct and inverse proportion [includes linear, square, square root, cube and cube root proportion. Knowledge of proportional symbol (\(\alpha\)) is required].
2.7 Graphs in Practical Situations
Travel graphs and conversion graphs
Distance-time and speed-time graphs, acceleration and deceleration
Calculate distance travelled as area under a speed-time graph
2.8 Graphs of Functions
Construct tables of values, and draw, recognise and interpret graphs for functions of the following forms:
axn, abx + c [where n = -2, -1, -0.5, 0, 0.5, 1, 2, 3; a and c are rational numbers; and b is a positive integer]
Solve associated equations graphically, including finding and interpreting roots by graphical methods
Draw and interpret graphs representing exponential growth and decay problems
Estimate gradients of curves by drawing tangents
2.9 Sketching Curves
Recognise, sketch and interpret graphs of the following functions:
[Linear, quadratic, cubic, reciprocal and exponential]
Knowledge of turning points, roots and symmetry is required
Knowledge of vertical and horizontal asymptotes is required
Finding turning points of quadratics by completing the square is required
2.10 Functions
Understand functions, domain and range, and use function notation
Understand and find inverse functions f-1(x)
Form composite functions as defined by gf(x) = g(f(x))
3 COORDINATE GEOMETRY
3.1 Drawing Linear Graphs
Draw straight-line graphs for linear equations
3.2 Gradient of Linear Graphs
Find the gradient of a straight line from the coordinates of two points on it
3.3 Length and Midpoint
Calculate the length and find the coordinates of the midpoint of a line segment
3.4 Equations of Linear Graphs
Interpret and obtain the equation of a straight-line graph
3.5 Parallel Lines
Find the gradient and equation of a straight line parallel to a given line
3.6 Perpendicular Lines
Find the gradient and equation of a straight line perpendicular to a given line
4 GEOMETRY
4.1 Geometrical Terms
Use and interpret the following geometrical terms:
[centre, radius (plural radii), diameter, circumference, semicircle, chord, tangent, major and minor arc, sector, segment]
4.2 Geometrical Constructions
Measure and draw lines and angles [constructions of perpendicular bisectors and angle bisectors are not required]
Construct a triangle, given the lengths of all sides, using a ruler and pair of compasses only
Draw, use and interpret nets [draw nets of cubes, cuboids, prisms and pyramids]
4.3 Scale Drawings
Draw and interpret scale drawings
Use and interpret three-figure bearings
bearings are measured clockwise from north (000° to 360°)
includes an understanding of the terms north, east, south and west
4.4 Similarity
Calculate lengths of similar shapes
Use the relationships between lengths and areas of similar shapes and lengths, surface areas and volumes of similar solids
Solve problems and give simple explanations involving similarity
4.5 Symmetry
Recognise line symmetry and order of rotational symmetry in two dimensions [includes properties of triangles, quadrilaterals and polygons directly related to their symmetries]
Recognise symmetry properties of prisms, cylinders, pyramids and cones [e.g. identify planes and axes of symmetry]
4.6 Angles
Calculate unknown angles and give simple explanations using the following geometrical properties:
sum of angles at a point = 360°
sum of angles at a point on a straight line = 180°
vertically opposite angles are equal
angle sum of a triangle = 180°
angle sum of a quadrilateral = 360°
Calculate unknown angles and give geometric explanations for angles formed within parallel lines:
corresponding angles are equal
alternate angles are equal
co-interior angles sum to 180° (supplementary)
Know and use angle properties of regular and irregular polygons [includes exterior and interior angles, and angle sum]
4.7 Circle Theorems
Calculate unknown angles and give explanations using the following geometrical properties of circles:
angle in a semicircle = 90°
angle between tangent and radius = 90°
angle at the centre is twice the angle at the circumference
angles in the same segment are equal
opposite angles of a cyclic quadrilateral sum to 180° (supplementary)
alternate segment theorem
Use the following symmetry properties of circles:
equal chords are equidistant from the centre
the perpendicular bisector of a chord passes through the centre
tangents from an external point are equal in length
5 MENSURATION
5.1 Units of Measure
Use metric units of mass, length, area, volume and capacity in practical situations and convert quantities into larger or smaller units.
Units include:
mm, cm, m, km
mm2, cm2, m2, km2
mm3, cm3, m3
ml, l
g, kg
Conversion between units includes:
between different units of area, e.g. cm2 ↔ m2
between units of volume and capacity, e.g. m3 ↔ litres
5.2 Area and Perimeter
Calculate perimeter and area of a rectangle, triangle, parallelogram and trapezium
5.3 Circles, Arcs and Sectors
Calculate circumference and area of a circle, involving arc length and sector area as fractions of the circumference and area of a circle
5.4 Surface Area and Volume
Calculate the surface area and volume of cuboid, prism, cylinder, sphere, pyramid and cone
5.5 Compound Shapes and Parts of Shapes
Carry out calculations and solve problems involving perimeters and areas of:
compound shapes
parts of shapes
Carry out calculations and solve problems involving surface areas and volumes of:
compound solids
parts of solids
6 TRIGONOMETRY
6.1 Right-Angled Triangles
Know and use the sine, cosine and tangent ratios for acute angles in calculations involving sides and angles of a right-angled triangle
Solve problems in two dimensions using Pythagoras' theorem and trigonometry
Perpendicular distance from a point to a line is the shortest distance to the line
Calculate angles of elevation and depression
6.2 Non-Right-Angled Triangles
Use sine rule and cosine rule to calculate lengths and angles for any triangle [includes problems involving obtuse angles and the ambiguous case]
6.3 Pythagoras' Theorem and Trigonometry in 3D
Calculate and solve problems in three dimensions using Pythagoras' theorem and trigonometry, including the angle between a line and a plane
7 TRANSFORMATION AND VECTORS
7.1 Transformations
Recognise, describe and draw the following transformations:
Reflection of a shape in a straight line
Rotation of a shape about a centre through multiples of 90°
Enlargement of a shape from a centre by a scale factor
Translation of a shape by a vector \(\begin{pmatrix}
x \\
y
\end{pmatrix}\)
Questions may involve combinations of transformations
Positive, fractional and negative scale factors may be used
7.2 Vectors in Two Dimensions
Describe a translation using a vector represented by \(\begin{pmatrix}
x \\
y
\end{pmatrix}\), \(\overrightarrow{AB}\) or a
Add and subtract vectors
Multiply a vector by a scalar
7.3 Magnitude of a Vector
Calculate the magnitude of a vector \(\begin{pmatrix}
x \\
y
\end{pmatrix}\) as \(\sqrt{x^2 + y^2}\)
The magnitudes of vectors will be denoted by modulus signs, e.g. |\(\overrightarrow{AB}\)| is the magnitude of \(\overrightarrow{AB}\)
7.4 Vector Geometry
Use position vectors
Use the sum and difference of two or more vectors to express given vectors in terms of two coplanar vectors
Use vectors to reason and to solve geometric problems
Examples include:
show that vectors are parallel
show that 3 points are collinear
solve vector problems involving ratio and similarity
8 PROBABILITY
8.1 Introduction to Probability
Probability scale is from 0 to 1
Probability notation [P(A) is the probability of A, P(A') is the probability of not A]
Probabilities should be given as a fraction, decimal or percentage
Probability of an event not occurring = 1 - the probability of the event occurring [e.g. P(B) = 0.8, find P(B')]
8.2 Relative and Expected Frequencies
Understand relative frequency as an estimate of probability
Calculate expected frequencies
8.3 Probability of Combined Events
Calculate the probability of combined events using, where appropriate:
Sample space diagrams
Venn diagrams
Tree diagrams
9 STATISTICS
9.1 Averages and Measures of Spread
Calculate the mean, median, mode and range for individual data
Calculate an estimate of the mean for grouped discrete or grouped continuous data
Identify the modal class from a grouped frequency distribution
9.2 Statistical Charts and Diagrams
Draw and interpret:
Bar charts [includes composite (stacked) and dual (side-by-side) bar charts]
Pie charts
Pictograms
Simple frequency distributions
9.3 Scatter Diagrams
Understand what is meant by positive, negative and zero correlation
Draw by eye, interpret and use a straight line of best fit
A line of best fit:
should be a single ruled line drawn by inspection
should extend across the full data set
does not need to coincide exactly with any of the points but there should be a roughly even distribution of points either side of the line over its entire length
9.4 Cumulative Frequency Diagrams
Estimate and interpret the median, percentiles, quartiles and interquartile range from cumulative frequency diagrams
9.5 Histograms
Draw and interpret histograms [on histograms, the vertical axis is labelled 'frequency density']
Frequency density = \(\frac{\text{frequency}}{\text{class width}}\)
