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
Understand and use set language, notation and Venn diagrams to describe sets. Venn diagrams are limited to two sets. Definition of sets:
A = {x : x is a natural number}
B = {a, b, c…}
C = {x : a \(\leq\) x \(\leq\) b}
Notation:
Number of elements in set A [n(A)]
Complement of set A [A']
Universal set [\(\xi\)]
Union of A and B [A \(\cup\) B]
Intersection of A and B [A ∩ B]
1.3 Powers and Roots
Calculate with the following:
squares
square roots
cubes
cube roots
other powers and roots of numbers
Includes recall of squares and their corresponding roots from 1 to 15, and recall of cubes and their corresponding roots of 1, 2, 3, 4, 5 and 10.
1.4 Fractions, Decimals and Percentages
Use the language and notation of the following in appropriate contexts:
proper fractions
improper fractions
mixed numbers
decimals
percentages
Candidates are expected to be able to write fractions in their simplest form.
Candidates are not expected to use recurring decimal notation.
Recognise equivalence and convert between these forms.
1.5 Ordering
Order quantities by magnitude and demonstrate 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.
Includes:
negative numbers
improper fractions
mixed numbers
practical situations, e.g. temperature changes
1.7 Indices I
Understand and use indices (positive, zero and negative integers).
Understand and use the rules of indices.
1.8 Standard Form
Use the standard form A x 10n where n is a positive or negative integer and 1 \(\leq\) A < 10.
Convert numbers into and out of standard form.
Calculate with values in standard form [core candidates are expected to calculate with standard form only on Paper 3].
1.9 Estimation
Round values to a specified degree of accuracy [includes decimal places and significant figures].
Make estimates for calculations involving numbers, quantities and measurements.
Round answers to a reasonable degree of accuracy in the context of a given problem.
1.10 Limits of Accuracy
Give upper and lower bounds for data rounded to a specified accuracy.
1.11 Ratio and Proportion
Understand and use ratio and proportion to:
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
Calculate a given percentage of a quantity.
Express one quantity as a percentage of another.
Calculate percentage increase or decrease.
Calculate with simple and compound interest.
Percentage calculations may include:
[deposit, discount, profit and loss (as an amount or a percentage), earnings, percentages over 100%]
1.14 Using a Calculator
Use a calculator efficiently
Enter values appropriately on a calculator
Interpret the calculator display appropriately
1.15 Time
Calculate with time: seconds (s), minutes (min), hours (h), days, weeks, months, years, including the relationship between units [1 year = 365 days].
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
Calculate with money.
Convert from one currency to another.
2 Algebra and Graphs
2.1 Introduction to Algebra
Know that letters can be used to represent generalised numbers.
Substitute numbers into expressions and formulas.
2.2 Algebraic Manipulation
Simplify expressions by collecting like terms.
Expand products of algebraic expressions.
Factorise by extracting common factors.
2.3 Indices II
Understand and use indices (positive, zero and negative).
Understand and use the rules of indices [knowledge of logarithms is not required].
2.4 Equations
Construct simple expressions, equations and formulas.
Solve linear equations in one unknown.
Solve simultaneous linear equations in two unknowns.
Change the subject of simple formulas.
2.5 Inequalities
Represent and interpret inequalities, including on a number line.
When representing and interpreting inequalities on a number line:
open circles should be used to represent strict inequalities (<, >)
closed circles should be used to represent inclusive inequalities (\(\leq\), \(\geq\))
2.6 Sequences
Continue a given number sequence or pattern.
Recognise patterns in sequences, including the term-to-term rule, and relationships between different sequences.
Find and use the nth term of the following sequences:
linear
simple quadratic
simple cubic
2.7 Graphs in Practical Situations
Use and interpret graphs in practical situations including travel graphs and conversion graphs [e.g. interpret the gradient of a straight-line graph as a rate of change].
Draw graphs from given data [e.g. draw a distance-time graph to represent a journey].
2.8 Graphs of Functions
Construct tables of values, and draw, recognise and interpret graphs for functions of the following forms: ax + b, ± x2 + ax + b, \(\frac{x}{a}\) (x ≠ 0), where a and b are integer constants.
Solve associated equations graphically, including finding and interpreting roots by graphical methods [e.g. find the intersection of a line and a curve].
2.9 Sketching Curves
Recognise, sketch and interpret graphs of the following functions:
linear
quadratic
Knowledge of symmetry and roots is required. Knowledge of turning points is not required.
3 Coordinate Geometry
3.1 Coordinates
Use and interpret Cartesian coordinates in two dimensions
3.2 Drawing Linear Graphs
Draw straight-line graphs for linear equations.
Equations will be given in the form y = mx + c (e.g. y = -2x + 5), unless a table of values is given.
3.3 Gradient of Linear Graphs
Find the gradient of a straight line (from a grid only)
3.4 Equations of Linear Graphs
Interpret and obtain the equation of a straight-line graph in the form y = mx + c.
Questions may:
use and request lines in the form y = mx + c, x = k
involve finding the equation when the graph is given
ask for the gradient or y-intercept of a graph from an equation
Candidates are expected to give equations of a line in a fully simplified form.
3.5 Parallel Lines
Find the gradient and equation of a straight line parallel to a given line.
4 Geometry
4.1 Geometrical Terms
Use and interpret the following geometrical terms:
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. Use measurements from nets to calculate volumes and surface areas.
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.
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.
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 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°
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
Carry out calculations involving the perimeter and area of a rectangle, triangle, parallelogram and trapezium.
5.3 Circles, Arcs and Sectors
Carry out calculations involving the circumference and area of a circle [answers may be asked for in terms of π].
Carry out calculations involving arc length and sector area as fractions of the circumference and area of a circle, where the sector angle is a factor of 360° [answers may be asked for in terms of π].
5.4 Surface Area and Volume
Carry out calculations and solve problems involving the surface area and volume of a: cuboid, prism, cylinder, sphere, pyramid, cone [answers may be asked for in terms of π].
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 Pythagoras' Theorem
Know and use Pythagoras' theorem
6.2 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 [angles will be given in degrees and answers should be written in degrees, with decimals correct to one decimal place].
Solve problems in two dimensions using Pythagoras' theorem and trigonometry [knowledge of bearings may be required].
7 Transformations and Vectors
7.1 Transformations
Recognise, describe and draw the following transformations:
Reflection of a shape in a vertical or horizontal line.
Rotation of a shape about the origin, vertices or midpoints of edges of the shape, through multiples of 90°.
Enlargement of a shape from a centre by a scale factor [positive and fractional scale factors only].
Translation of a shape by a vector \(\begin{pmatrix}
x \\
y
\end{pmatrix}\).
Questions will not involve combinations of transformations.
8 Probability
8.1 Introduction to Probability
Understand and use the probability scale from 0 to 1.
Probability notation is not required.
Probabilities should be given as a fraction, decimal or percentage. Problems may require using information from tables, graphs or Venn diagrams (limited to two sets).
Calculate the probability of a single event.
Understand that the probability of an event not occurring = 1 - the probability of the event occurring.
8.2 Relative and Expected Frequencies
Understand relative frequency as an estimate of probability.
Calculate expected frequencies.
Includes understanding what is meant by fair, bias and random.
8.3 Probability of Combined Events
Calculate the probability of combined events using, where appropriate [combined events will only be with replacement]:
Sample space diagrams
Venn diagrams [Venn diagrams will be limited to two sets]
Tree diagrams [in tree diagrams, outcomes will be written at the end of the branches and probabilities by the side of the branches]
9 Statistics
9.1 Classifying Statistical Data
Classify and tabulate statistical data e.g. tally tables, two-way tables.
9.2 Interpreting Statistical Data
Read, interpret and draw inferences from tables and statistical diagrams.
Compare sets of data using tables, graphs and statistical measures e.g. compare averages and ranges between two data sets.
Appreciate restrictions on drawing conclusions from given data.
9.3 Averages and Range
Calculate the mean, median, mode and range for individual data and distinguish between the purposes for which these are used [data may be in a list or frequency table, but will not be grouped]
9.4 Statistical Charts and Diagrams
Draw and interpret:
bar charts [includes composite (stacked) and dual (side-by-side) bar charts]
pie charts
pictograms
stem-and-leaf diagrams
simple frequency distributions
9.5 Scatter Diagrams
Draw and interpret scatter diagrams [plotted points should be clearly marked, for example as small crosses (x)]
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
EXTENDED
1 Number
1.1 Types of Number
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
Understand and use set language, notation and Venn diagrams to describe sets and represent relationships between sets. Venn diagrams are limited to two or three sets. Definition of 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
Calculate with the following:
squares
square roots
cubes
cube roots
other powers and roots of numbers
Includes recall of squares and their corresponding roots from 1 to 15, and recall of cubes and their corresponding roots of 1, 2, 3, 4, 5 and 10.
1.4 Fractions, Decimals and Percentages
Use the language and notation of the following in appropriate contexts:
proper fractions
improper fractions
mixed numbers
decimals
percentages
Recognise equivalence and convert between these forms.
Candidates are expected to be able to write fractions in their simplest form.
Recurring decimal notation is required.
Includes converting between recurring decimals and fractions and vice versa.
1.5 Ordering
Order quantities by magnitude and demonstrate 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.
Includes:
negative numbers
improper fractions
mixed numbers
practical situations, e.g. temperature changes
1.7 Indices I
Understand and use indices (positive, zero, negative, and fractional).
Understand and use the rules of indices.
1.8 Standard Form
Use the standard form A x 10n where n is a positive or negative integer and 1 \(\leq\) A < 10.
Convert numbers into and out of standard form.
Calculate with values in standard form.
1.9 Estimation
Round values to a specified degree of accuracy.
Make estimates for calculations involving numbers, quantities and measurements.
Round answers to a reasonable degree of accuracy in the context of a given problem.
Includes decimal places and significant figures.
1.10 Limits of Accuracy
Give upper and lower bounds for data rounded to a specified accuracy.
Find upper and lower bounds of the results of calculations which have used data rounded to a specified accuracy.
1.11 Ratio and Proportion
Understand and use ratio and proportion to:
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
Calculate a given percentage of a quantity.
Express one quantity as a percentage of another.
Calculate percentage increase or decrease.
Calculate with simple and compound interest [problems may include repeated percentage change].
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
Use a calculator efficiently.
Enter values appropriately on a calculator.
Interpret the calculator display appropriately.
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
Calculate with money.
Convert from one currency to another.
1.17 Exponential Growth and Decay
Use exponential growth and decay [e.g. depreciation, population change. Knowledge of e is not required]
1.18 Surds
Understand and use surds, including simplifying expressions.
Rationalise the denominator.
2 Algebra and Graphs
2.1 Introduction to Algebra
Know that letters can be used to represent generalised numbers.
Complete the square for expressions in the form ax2 + bx + c.
2.3 Algebraic Fractions
Manipulate algebraic fractions.
Factorise and simplify rational expressions.
2.4 Indices II
Understand and use indices (positive, zero, negative and fractional).
Understand and use the rules of indices.
Knowledge of logarithms is not required.
2.5 Equations
Construct expressions, equations and formulas.
Solve linear equations in one unknown.
Solve fractional equations with numerical and linear algebraic denominators.
Solve simultaneous linear equations in two unknowns.
Solve simultaneous equations, involving one linear and one non-linear.
Solve quadratic equations by factorisation, completing the square and by use of the quadratic formula.
Change the subject of formulas.
Candidates may be expected to give solutions in surd form.
2.6 Inequalities
Represent and interpret inequalities, including on a number line. When representing and interpreting inequalities on a number line:
open circles should be used to represent strict inequalities (<, >)
closed circles should be used to represent inclusive inequalities (\(\leq\), \(\geq\))
Construct, solve and interpret linear inequalities.
Represent and interpret linear inequalities in two variables graphically.
List inequalities that define a given region.
The following conventions should be used:
broken lines should be used to represent strict inequalities (<, >)
solid lines should be used to represent inclusive inequalities (\(\leq\), \(\geq\))
shading should be used to represent unwanted regions (unless otherwise directed in the question).
Linear programming problems are not included.
2.7 Sequences
Continue a given number sequence or pattern.
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. Subscript notation may be used, e.g. Tn is the nth term of sequence T.
2.8 Proportion
Express direct and inverse proportion in algebraic terms and use this form of expression to find unknown quantities [includes linear, square, square root, cube and cube root proportion. Knowledge of proportional symbol (\(\alpha\)) is required]
2.9 Graphs in Practical Situations
Use and interpret graphs in practical situations including travel graphs and conversion graphs [includes estimation and interpretation of the gradient of a tangent at a point].
Draw graphs from given data.
Apply the idea of rate of change to simple kinematics involving distance-time and speed-time graphs, acceleration and deceleration.
Calculate distance travelled as area under a speed-time graph [areas will involve linear sections of the graph only].
2.10 Graphs of Functions
Construct tables of values, and draw, recognise and interpret graphs for functions of the following forms:
axn (includes sums of no more than three of these)
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.
2.11 Sketching Curves
Recognise, sketch and interpret graphs of the following functions:
linear [ax + by = c]
quadratic [y = ax2 + bx + c]
cubic [y = ax3 + b, y = ax3 + bx2 + cx]
reciprocal [y = \(\frac{a}{x}\) + b]
exponential [y = arx + b]
Where a, b and c are rational numbers and r is a rational, positive number.
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.
2.12 Differentiation
Estimate gradients of curves by drawing tangents.
Use the derivatives of functions of the form axn, where a is a rational constant and n is a positive integer or zero, and simple sums of not more than three of these [\(\frac{dy}{dx}\) notation will be expected].
Apply differentiation to gradients and stationary points (turning points).
Discriminate between maxima and minima by any method. Maximum and minimum points may be identified by:
an accurate sketch
use of the second differential
inspecting the gradient either side of a turning point
Candidates are not expected to identify points of inflection.
2.13 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))
Candidates are not expected to find the domains and ranges of composite functions.
This topic may include mapping diagrams.
3 Coordinate Geometry
3.1 Coordinates
Use and interpret Cartesian coordinates in two dimensions.
3.2 Drawing Linear Graphs
Draw straight-line graphs for linear equations.
3.3 Gradient of Linear Graphs
Find the gradient of a straight line.
Calculate the gradient of a straight line from the coordinates of two points on it.
3.4 Length and Midpoint
Calculate the length of a line segment.
Find the coordinates of the midpoint of a line segment.
3.5 Equations of Linear Graphs
Interpret and obtain the equation of a straight-line graph.
Questions may:
use and request lines in different forms, e.g. ax + by = c, y = mx + c, x = k
involve finding the equation when the graph is given
ask for the gradient or y-intercept of a graph from an equation
Candidates are expected to give equations of a line in a fully simplified form.
3.6 Parallel Lines
Find the gradient and equation of a straight line parallel to a given line.
3.7 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 [includes use of scale factor].
Solve problems and give simple explanations involving similarity [includes showing that two triangles are similar using geometric reasons].
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 I
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
4.8 Circle Theorems II
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
Carry out calculations involving the perimeter and area of a rectangle, triangle, parallelogram and trapezium.
5.3 Circles, Arcs and Sectors
Carry out calculations involving the circumference and area of a circle.
Carry out calculations involving arc length and sector area as fractions of the circumference and area of a circle [answers may be asked for in terms of π, includes minor and major sectors]
5.4 Surface Area and Volume
Carry out calculations and solve problems involving the surface area and volume of a:
[cuboid, prism, cylinder, sphere, pyramid, 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 Pythagoras' Theorem
Know and use Pythagoras' theorem
6.2 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 [knowledge of bearings may be required].
Know that the perpendicular distance from a point to a line is the shortest distance to the line.
Carry out calculations involving angles of elevation and depression.
6.3 Exact Trigonometric Values
Know the exact values of:
sin x and cos x for x = 0°, 30°, 45°, 60° and 90°
tan x for x = 0°, 30°, 45° and 60°
6.4 Trigonometric Functions
Recognise, sketch and interpret the following graphs for 0° \(\leq\) x \(\leq\) 360°
[y = sin x, y = cos x, y = tan x]
Solve trigonometric equations involving sin x, cos x or tan x, for 0° \(\leq\) x \(\leq\) 360°
6.5 Non-Right-Angled Triangles
Use the sine and cosine rules in calculations involving lengths and angles for any triangle [includes problems involving obtuse angles and the ambiguous case]
6.6 Pythagoras' Theorem and Trigonometry in 3D
Carry out calculations and solve problems in three dimensions using Pythagoras' theorem and trigonometry, including calculating the angle between a line and a plane.
7 Transformations 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 [positive, fractional and negative scale factors may be used].
Translation of a shape by a vector \(\begin{pmatrix}
x \\
y
\end{pmatrix}\).
Questions may involve combinations of transformations.
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
Represent vectors by directed line segments.
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
Understand and use the probability scale from 0 to 1.
Understand and use probability notation [P(A) is the probability of A, P(A') is the probability of not A]
Calculate the probability of a single event [probabilities should be given as a fraction, decimal or percentage]
Understand that the 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.
Includes understanding what is meant by fair, bias and random.
8.3 Probability of Combined Events
Calculate the probability of combined events using, where appropriate:
Sample space diagrams
Venn diagrams
Tree diagrams
Combined events could be with or without replacement.
The notation P(A ∩ B) and P(A \(\cup\) B) may be used in the context of Venn diagrams.
8.4 Conditional Probability
Calculate conditional probability using Venn diagrams, tree diagrams and tables.
Knowledge of notation, P(A|B), and formulas relating to conditional probability is not required.
9 Statistics
9.1 Classifying Statistical Data
Classify and tabulate statistical data e.g. tally tables, two-way tables
9.2 Interpreting Statistical Data
Read, interpret and draw inferences from tables and statistical diagrams.
Compare sets of data using tables, graphs and statistical measures e.g. compare averages and measures of spread between two data sets.
Appreciate restrictions on drawing conclusions from given data.
9.3 Averages and Measures of Spread
Calculate the mean, median, mode, quartiles, range and interquartile range for individual data and distinguish between the purposes for which these are used.
Calculate an estimate of the mean for grouped discrete or grouped continuous data.
Identify the modal class from a grouped frequency distribution.
9.4 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
Stem-and-leaf diagrams [Stem-and-leaf diagrams should have ordered data with a key]
9.5 Scatter Diagrams
Draw and interpret scatter diagrams [plotted points should be clearly marked, for example as small crosses (x)].
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
9.6 Cumulative Frequency Diagrams
Draw and interpret cumulative frequency tables and diagrams [plotted points on a cumulative frequency diagram should be clearly marked, for example as small crosses (x), and be joined with a smooth curve].
Estimate and interpret the median, percentiles, quartiles and interquartile range from cumulative frequency diagrams.
9.7 Histograms
Draw and interpret histograms [on histograms, the vertical axis is labelled 'frequency density']
Frequency density = \(\frac{\text{frequency}}{\text{class width}}\)
