edemis International College & mont21school
< Montessori community >
GCSE MATHS higher
Advanced Level Pearson Edexcel International GCSE in Mathematics (Specification A) (4MA1)
1. Numbers and the number system
1.1 Integers
A understand and use integers (positive, negative and zero)
B understand place value
C use directed numbers in practical situations
D order integers
E use the four rules of addition, subtraction, multiplication and division
F use brackets and the hierarchy of operations
G use the terms ‘odd’, ‘even’, ’prime numbers’,
‘factors’ and ‘multiples’
H identify prime factors, common factors and common multiples

1.2 Fractions
A understand and use equivalent fractions, simplifying a fraction by cancelling common factors
B understand and use mixed numbers and vulgar fractions
C identify common denominators
D order fractions and calculate a given fraction of a given quantity
E express a given number as a fraction of another number
F use common denominators to add and subtract fractions and mixed numbers
G convert a fraction to a decimal or a percentage
H understand and use unit fractions as multiplicative inverses
I multiply and divide fractions and mixed numbers

1.3 Decimals
A use decimal notation
B understand place value
C order decimals
D convert a decimal to a fraction or a percentage
E recognise that a terminating decimal is a fraction
F convert recurring decimals into fractions

1.4 Powers and roots
A identify square numbers and cube numbers
B calculate squares, square roots, cubes and cube roots
C use index notation and index laws for multiplication and division of positive and negative integer powers including zero
D express integers as a product of powers of prime factors
E find highest common factors (HCF) and lowest common multiples (LCM)
F understand the meaning of surds
G manipulate surds, including rationalising a denominator
H use index laws to simplify and evaluate numerical expressions involving integer, fractional and negative powers

1.5 Set language and notation
A understand the definition of a set
B use the set notation ∪,∩ and ∈ and ∉
C understand the concept of the universal set and the empty set and the symbols for these sets
D understand and use the complement of a set
E use Venn diagrams to represent sets
F understand sets defined in algebraic terms, and understand and use subsets
G use Venn diagrams to represent sets and the number of elements in sets
H use the notation n(A) for the number of elements in the set A
I use sets in practical situations

1.6 Percentages
A understand that 'percentage' means 'number of parts per 100'
B express a given number as a percentage of another number
C express a percentage as a fraction and as a decimal
D understand the multiplicative nature of percentages as operators
E solve simple percentage problems, including percentage increase and decrease
F use reverse percentages
G use repeated percentage change

1.7 Ratio and proportion
A use ratio notation, including reduction to its simplest form and its various links to fraction notation
B divide a quantity in a given ratio or ratios
C use the process of proportionality to evaluate unknown quantities
D calculate an unknown quantity from quantities that vary in direct proportion
E solve word problems about ratio and proportion

1.8 Degree of accuracy
A round integers to a given power of 10
B round to a given number of significant figures or decimal places
C identify upper and lower bounds where values are given to a degree of accuracy
D use estimation to evaluate approximations to numerical calculations
E solve problems using upper and lower bounds where values are given to a degree of accuracy

1.9 Standard form
A calculate with and interpret numbers in the form a * 10 where n is an integer and 1 < a < 10

B solve problems involving standard form

1.10 Applying number
A use and apply number in everyday personal, domestic or community life
B carry out calculations using standard units of mass, length, area, volume and capacity
C understand and carry out calculations using time, and carry out calculations using money, including converting between currencies

1.11 Electronic calculators
A use a scientific electronic calculator to determine numerical results

2. Equations, formulae and identities
2.1 Use of symbols
A understand that symbols may be used to represent numbers in equations or variables in expressions and formulae
B understand that algebraic expressions follow the generalised rules of arithmetic
C use index notation for positive and negative integer powers (including zero)
D use index laws in simple cases
E use index notation involving fractional, negative and zero powers

2.2 Algebraic manipulation
A evaluate expressions by substituting numerical values for letters
B collect like terms
C multiply a single term over a bracket
D take out common factors
E expand the product of two simple linear expressions
F understand the concept of a quadratic expression and be able to factorise such expressions (limited to x2 + bx + c)
G expand the product of two or more linear expressions
H understand the concept of a quadratic expression and be able to factorise such expressions
I manipulate algebraic fractions where the numerator and/or the denominator can be numeric, linear or quadratic
J complete the square for a given quadratic expression
K use algebra to support and construct proofs

2.3 Expressions and formulae
A understand that a letter may represent an unknown number or a variable
B use correct notational conventions for algebraic expressions and formulae
C substitute positive and negative integers, decimals and fractions for words and letters in expressions and formulae
D use formulae from mathematics and other real-life contexts expressed initially in words or diagrammatic form and convert to letters and symbols
E derive a formula or expression
F change the subject of a formula where the subject appears once
G understand the process of manipulating formulae or equations to change the subject, to include cases where the subject may appear twice or a power of the subject occurs

2.4 Linear equations
A solve linear equations, with integer or fractional coefficients, in one unknown in which the unknown appears on either side or both sides of the equation
B set up simple linear equations from given data

2.5 Proportion
A set up problems involving direct or inverse proportion and relate algebraic solutions to graphical representation of the equations

2.6 Simultaneous linear equations
A calculate the exact solution of two simultaneous equations in two unknowns
B calculate the exact solution of two simultaneous equations in two unknowns
C interpret the equations as lines and the common solution as the point of intersection

A solve quadratic equations by factorisation (limited to x2 + bx + c = 0)
B solve quadratic equations by factorisation
C solve quadratic equations by using the quadratic formula or completing the square
D form and solve quadratic equations from data given in a context
E solve simultaneous equations in two unknowns, one equation being linear and the other being quadratic

2.8 Inequalities
A understand and use the symbols >,<, and <
B understand and use the convention for open and closed intervals on a number line
C solve simple linear inequalities in one variable and represent the solution set on a number line
D represent simple linear inequalities on rectangular Cartesian graphs
E identify regions on rectangular Cartesian graphs defined by simple linear inequalities
F solve quadratic inequalities in one unknown and represent the solution set on a number line
G identify harder examples of regions defined by linear inequalities
3. Sequences, functions and graphs
3.1 Sequences
generate terms of a sequence using term-to-term and position-to-term definitions of the sequence
find subsequent terms of an integer sequence and the rule for generating it
use linear expressions to describe the /th term of arithmetic sequences
D understand and use common difference (d) and first term (a) in an arithmetic sequence
E know and use nth term = a + (n - 1)d
F find the sum of the first n terms of an arithmetic series (Sn)

3.2 Function notation
A understand the concept that a function is a mapping between elements of two sets
B use function notations of the form f(x) = ... and f : x ^ ...
C understand the terms 'domain' and 'range' and which values may need to be excluded from a domain
D understand and find the composite function fg and the inverse function f^-1

3.3 Graphs
A interpret information presented in a range of linear and non-linear graphs
B understand and use conventions for rectangular Cartesian coordinates
C plot points (x, y) in any of the four quadrants or locate points with given coordinates
D determine the coordinates of points identified by geometrical information
E determine the coordinates of the midpoint of a line segment, given the coordinates of the two end points
F draw and interpret straight line conversion graphs
G find the gradient of a straight line
H recognise that equations of the form y = mx + c are straight line graphs with gradient m and intercept on the y-axis at the point (0, c)
I recognise, generate points and plot graphs of linear and quadratic functions
J recognise, plot and draw graphs with equation:
y = Ax3 + Bx2+ Cx + D in which:
(i)  the constants are integers and some could be zero
(ii)the letters x and y can be replaced with any other two letters or:
in which:
(i)  the constants are numerical and at least three of them are zero
(ii)the letters x and y can be replaced with any other two letters or:

y = sin x, y = cos x, y = tan x for angles of any size (in degrees)
K apply to the graph ofy = f(x) the transformations
y = f(x) + a, y = f(ax), y = f(x + a),
y = af(x) for linear, quadratic, sine and cosine functions
L interpret and analyse transformations of functions and write the functions algebraically
M find the gradients of non-linear graphs
N find the intersection points of two graphs, one linear (y1) and one non-linear (y2), and and recognise that the solutions correspond to the solutions of (y 2 - yd = o
O calculate the gradient of a straight line given the coordinates of two points
P find the equation of a straight line parallel to a given line; find the equation of a straight line perpendicular to a given line

3.4 Calculus
A understand the concept of a variable rate of change
B differentiate integer powers of x
C determine gradients, rates of change, stationary points, turning points (maxima and minima) by differentiation and relate these to graphs
D distinguish between maxima and minima by considering the general shape of the graph only

E apply calculus to linear kinematics and to other simple practical problems
4. Geometry
4.1 Angles, lines and triangles
A distinguish between acute, obtuse, reflex and right angles
B use angle properties of intersecting lines, parallel lines and angles on a straight line
C understand the exterior angle of a triangle property and the angle sum of a triangle property
D understand the terms 'isosceles', 'equilateral' and 'right-angled triangles' and the angle properties of these triangles

4.2 Polygons
A recognise and give the names of polygons
B understand and use the term 'quadrilateral' and the angle sum property of quadrilaterals
C understand and use the properties of the parallelogram, rectangle, square, rhombus, trapezium and kite
D understand the term 'regular polygon' and calculate interior and exterior angles of regular polygons
E understand and use the angle sum of polygons
F understand congruence as meaning the same shape and size
G understand that two or more polygons with the same shape and size are said to be congruent to each other

4.3 Symmetry
A identify any lines of symmetry and the order of rotational symmetry of a given two-dimensional figure

4.4 Measures
A interpret scales on a range of measuring instruments
B calculate time intervals in terms of the 24-hour and the 12-hour clock
C make sensible estimates of a range of measures
D understand angle measure including three-figure bearings
E measure an angle to the nearest degree
F understand and use the relationship between average speed, distance and time
G use compound measure such as speed, density and pressure

4.5 Construction
A measure and draw lines to the nearest millimetre
B construct triangles and other two-dimensional shapes using a combination of a ruler, a protractor and compasses
C solve problems using scale drawings
D use straight edge and compasses to:
(i)  construct the perpendicular bisector of a line segment
(ii) construct the bisector of an angle
E understand and use the internal and external intersecting chord properties
F recognise the term 'cyclic quadrilateral’
G understand and use angle properties of the circle including:
(i)  angle subtended by an arc at the centre of a circle is twice the angle subtended at any point on the remaining part of the circumference
(ii) angle subtended at the circumference by a diameter is a right angle
(iii) angles in the same segment are equal
(iv)the sum of the opposite angles of a cyclic quadrilateral is 180°
(v) the alternate segment theorem

4.6 Circle properties
A recognise the terms 'centre', 'radius', 'chord', 'diameter', 'circumference', 'tangent', 'arc', 'sector' and 'segment' of a circle
B understand chord and tangent properties of circles

4.7 Geometrical reasoning
A give informal reasons, where required, when arriving at numerical solutions to geometrical problems
B provide reasons, using standard geometrical statements, to support numerical values for angles obtained in any geometrical context involving lines, polygons and circles

4.8 Trigonometry and Pythagoras’ theorem
A know, understand and use Pythagoras' theorem in two dimensions
B know, understand and use sine, cosine and tangent of acute angles to determine lengths and angles of a right-angled triangle
C apply trigonometrical methods to solve problems in two dimensions
D understand and use sine, cosine and tangent of obtuse angles
E understand and use angles of elevation and depression
F understand and use the sine and cosine rules for any triangle
G use Pythagoras' theorem in three dimensions
H understand and use the formula 1 ab sin C for the area of a triangle
I apply trigonometrical methods to solve problems in three dimensions, including finding the angle between a line and a plane

4.9 Mensuration of 2D shapes
A convert measurements within the metric system to include linear and area units
B find the perimeter of shapes made from triangles and rectangles
C find the area of simple shapes using the formulae for the areas of triangles and rectangles
D find the area of parallelograms and trapezia
E find circumferences and areas of circles using relevant formulae; find perimeters and areas of semicircles
F find perimeters and areas of sectors of circles

4.10 3D shapes and volume
A recognise and give the names of solids
B understand the terms 'face', 'edge' and 'vertex' in the context of 3D solids
C find the surface area of simple shapes using the area formulae for triangles and rectangles
D find the surface area of a cylinder
E find the volume of prisms, including cuboids and cylinders, using an appropriate formula
F convert between units of volume within the metric system
G find the surface area and volume of a sphere and a right circular cone using relevant formulae

4.11 Similarity
A understand and use the geometrical properties that similar figures have corresponding lengths in the same ratio but corresponding angles remain unchanged
B use and interpret maps and scale drawings
C understand that areas of similar figures are in the ratio of the square of corresponding sides
D understand that volumes of similar figures are in the ratio of the cube of corresponding sides
E use areas and volumes of similar figures in solving problems

5. Vectors and transformation geometry
5.1 Vectors
A understand that a vector has both magnitude and direction
B understand and use vector notation including column vectors
C multiply vectors by scalar quantities
E calculate the modulus (magnitude) of a vector
F find the resultant of two or more vectors
G apply vector methods for simple geometrical proofs

5.2 Transformation geometry
A understand that rotations are specified by a centre and an angle
B rotate a shape about a point through a given angle
C recognise that an anti-clockwise rotation is a positive angle of rotation and a clockwise rotation is a negative angle of rotation
D understand that reflections are specified by a mirror line
E construct a mirror line given an object and reflect a shape given a mirror line
F understand that translations are specified by a distance and direction
G translate a shape
H understand and use column vectors in translations
I understand that rotations, reflections and translations preserve length and angle so that a transformed shape under any of these transformations remains congruent to the original shape
J understand that enlargements are specified by a centre and a scale factor
K understand that enlargements preserve angles and not lengths
L enlarge a shape given the scale factor
M identify and give complete descriptions of transformations
6. Statistics and probability
6.1 Graphical representation of data
A use different methods of presenting data
B use appropriate methods of tabulation to enable the construction of statistical diagrams
C interpret statistical diagrams
D construct and interpret histograms
E construct cumulative frequency diagrams from tabulated data
F use cumulative frequency diagrams

6.2 Statistical measures
A understand the concept of average
B calculate the mean, median, mode and range for a discrete data set
C calculate an estimate for the mean for grouped data
D identify the modal class for grouped data
E estimate the median from a cumulative frequency diagram
F understand the concept of a measure of spread
G find the interquartile range from a discrete data set
H estimate the interquartile range from a cumulative frequency diagram

6.3 Probability
A understand the language of probability
B understand and use the probability scale
C understand and use estimates or measures of probability from theoretical models
D find probabilities from a Venn diagram
E understand the concepts of a sample space and an event, and how the probability of an event happening can be determined from the sample space
F list all the outcomes for single events and for two successive events in a systematic way
G estimate probabilities from previously collected data
H calculate the probability of the complement of an event happening
I use the addition rule of probability for mutually exclusive events
J understand and use the term 'expected frequency’
K draw and use tree diagrams
L determine the probability that two or more independent events will occur
M use simple conditional probability when combining events
N apply probability to simple problems
 Component/paper code: 4MA1/1H and 4MA1/2HEach paper is 50% of the total International GCSEAvailability: January and JuneEach paper is assessed through a 2-hour examination set and marked by Pearson.The total number of marks for each paper is 100.Questions will assume knowledge from the Foundation Tier subject content.Each paper will assess the full range of targeted grades at Higher Tier (9-4).Each paper will have approximately 40% of the marks distributed evenly over grades 4 and 5 and approximately 60% of the marks distributed evenly over grades 6, 7, 8 and 9.There will be approximately 40% of questions targeted at grades 5 and 4, across papers 2F and 2H, to aid standardisation and comparability of award between tiers.A Higher Tier formulae sheet (Appendix 5) will be included in the written examinations.A calculator may be used in the examinations. 