IGCSE MATHS SUBJECT
1.Number, set notation and language
Core
Identify and use natural numbers, integers
(positive, negative and zero), prime numbers, square numbers, common factors
and common multiples, rational and irrational numbers (e.g. π, 2 ), real
numbers; continue a given number sequence; recognise patterns in sequences and
relationships between different sequences, generalise to simple algebraic
statements (including expressions for the nth term) relating to such sequences.
Supplement
Use language, notation and Venn diagrams to
describe sets and represent relationships between sets as follows: Definition
of sets, e.g. A = {x: x is a natural number} B = {(x,y): y = mx + c} C = {x: a
Y x Y b} D = {a, b, c, …} Notation
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 A is a subset of B A ⊆ B A is a proper subset of B A ⊂ B A is not a subset of B A ⊈
B A is not a proper subset of B A ⊄
B Union of A and B A ∪ B Intersection of A and B A∩B
2. Squares and cubes
Core
Calculate squares, square roots,
cubes and cube roots of numbers.
3. Directed numbers
Core
Use directed numbers in practical situations
(e.g. temperature change, flood levels).
4. Vulgar and decimal fractions and percentages
Core
Use the language and notation of
simple vulgar and decimal fractions and percentages in appropriate contexts;
recognise equivalence and convert between these forms.
5. Ordering
Core
Order quantities by magnitude and
demonstrate familiarity with the symbols =, Œ, K, I, [,Y
6. Standard form
Core
Use the standard form A × 10n
where n is a positive or negative integer, and 1 Y A I=10
7. The four rules
Core
Use the four rules for
calculations with whole numbers, decimal fractions and vulgar (and mixed)
fractions, including correct ordering of operations and use of brackets.
8. Estimation
Core
Make estimates of numbers,
quantities and lengths, give approximations to specified numbers of significant
figures and decimal places and round off answers to reasonable accuracy in the
context of a given problem.
9. Limits of accuracy
Core
Give appropriate upper and lower bounds for
data given to a specified accuracy (e.g. measured lengths).
Supplement
Obtain appropriate upper and
lower bounds to solutions of simple problems (e.g. the calculation of the
perimeter or the area of a rectangle) given data to a specified accuracy.
10. Ratio, proportion, rate
Core
Demonstrate an understanding of the elementary
ideas and notation of ratio, direct and inverse proportion and common measures
of rate; divide a quantity in a given ratio; use scales in practical
situations; calculate average speed.
Supplement
Express direct and inverse
variation in algebraic terms and use this form of expression to find unknown
quantities; increase and decrease a quantity by a given ratio.
11. Percentages
Core
Calculate a given percentage of a
quantity; express one quantity as a percentage of another; calculate percentage
increase or decrease.
Supplement
Carry out calculations involving reverse
percentages, e.g. finding the cost price given the selling price and the percentage
profit.
12. Use of an electronic calculator
Core
Use an electronic calculator
efficiently; apply appropriate checks of accuracy.
13. Measures
Core Use current units of mass,
length, area, volume and capacity in practical situations and express quantities
in terms of larger or smaller units.
14. Time
Core
Calculate times in terms of the 24-hour and
12-hour clock; read clocks, dials and timetables.
15. Money
Core
Calculate using money and convert
from one currency to another.
16. Personal and household finance
Core
Use given data to solve problems on personal
and household finance involving earnings, simple interest and compound interest
(knowledge of compound interest formula is not required), discount, profit and
loss; extract data from tables and charts.
17. Graphs in practical situations
Core
Demonstrate familiarity with Cartesian
co-ordinates in two dimensions, interpret and use graphs in practical
situations including travel graphs and conversion graphs, draw graphs from
given data.
Supplement
Apply the idea of rate of change
to easy kinematics involving distance-time and speed-t ime graphs, acceleration
and deceleration; calculate distance travelled as area under a linear
speed-time graph.
18. Graphs of functions
Core
Construct tables of values for functions of
the form ax + b, ±x2 + ax + b, a/x (x Œ 0) where a and b are integral
constants; draw and interpret such graphs; find the gradient of a straight line
graph; solve linear and quadratic equations approximately by graphical methods.
Supplement
Construct tables of values and draw graphs for
functions of the form axn where a is a rational constant and n = –2, –1, 0, 1,
2, 3 and simple sums of not more than three of these and for functions of the
form ax where a is a positive integer; estimate gradients of curves by drawing
tangents; solve associated equations approximately by graphical methods.
19. Straight line graphs
Core
Interpret and obtain the equation of a
straight line graph in the form y = mx + c; determine the equation of a
straight line parallel to a given line.
Supplement
Calculate the gradient of a straight line from
the co-ordinates of two points on it; calculate the length and the co-ordinates
of the midpoint of a straight line segment from the co-ordinates of its end
points.
20. Algebraic representation and formulae
Core
Use letters to express
generalised numbers and express basic arithmetic processes algebraically,
substitute numbers for words and letters in formulae; transform simple
formulae; construct simple expressions and set up simple equations.
Supplement
Construct and transform more
complicated formulae and equations.
21. Algebraic manipulation
Core
Manipulate directed numbers; use brackets and
extract common factors.
Supplement Expand products of
algebraic expressions; factorise where possible expressions of the form ax + bx
+ kay + kby, a2x2 – b2y2; a2 + 2ab + b2; ax2 + bx + c manipulate algebraic
fractions, e.g. 243 −+ xx, ()23532 − − xx, 3543aab×, 10943aa −, 3221 −+ − xx factorise
and simplify expressions such as
26 222 +− − xx xx
22. Functions
Supplement
Use function notation, e.g. f(x) = 3x – 5, f:
x a 3x – 5 to describe simple functions, and the notation f–1(x) to describe
their inverses; form composite functions as defined by gf(x) = g(f(x))
23. Indices
Core
Use and interpret positive, negative
and zero indices.
Supplement
Use and interpret fractional
indices, e.g. solve 32x = 2
24. Solutions of equations and inequalities
Core
Solve simple linear equations in one unknown;
solve simultaneous linear equations in two unknowns.
Supplement
Solve quadratic equations by
factorisation, completing the square or by use of the formula; solve simple
linear inequalities.
25. Linear programming
Supplement
Represent inequalities graphically and use
this representation in the solution of simple linear programming problems (the
conventions of using broken lines for strict inequalities and shading unwanted
regions will be expected).
26. Geometrical terms and relationships
Core
Use and interpret the geometrical
terms: point, line, parallel, bearing, right angle, acute, obtuse and reflex
angles, perpendicular, similarity, congruence; use and interpret vocabulary of
triangles, quadrilaterals, circles, polygons and simple solid figures including
nets.
Supplement
Use the relationships between
areas of similar triangles, with corresponding results for similar figures and
extension to volumes and surface areas of similar solids.
27. Geometrical constructions
Core
Measure lines and angles; construct a triangle
given the three sides using ruler and pair of compasses only; construct other
simple geometrical figures from given data using protractors and set squares as
necessary; construct angle bisectors and perpendicular bisectors using straight
edges and pair of compasses only; read and make scale drawings.
28. Symmetry
Core
Recognise rotational and line
symmetry (including order of rotational symmetry) in two dimensions and
properties of triangles, quadrilaterals and circles directly related to their
symmetries.
Supplement
Recognise symmetry properties of
the prism (including cylinder) and the pyramid (including cone); use the
following symmetry properties of circles: (a)
equal chords are equidistant from the centre (b) the perpendicular bisector of a chord passes
through the centre (c) tangents from an
external point are equal in length.
29. Angle properties
Core
Calculate unknown angles using
the following geometrical properties: (a)
angles at a point (b) angles at a
point on a straight line and intersecting straight lines (c) angles formed within parallel lines (d) angle properties of triangles and
quadrilaterals (e) angle properties of
regular polygons (f) angle in a
semi-circle (g) angle between tangent
and radius of a circle.
Supplement
Use in addition the following
geometrical properties: (a) angle
properties of irregular polygons (b)
angle at the centre of a circle is twice the angle at the circumference
(c) angles in the same segment are equal
(d) angles in opposite segments are
supplementary; cyclic quadrilaterals.
30. Locus
Core
Use the following loci and the
method of intersecting loci for sets of points in two dimensions: (a) which are at a given distance from a given
point (b) which are at a given distance
from a given straight line (c) which are
equidistant from two given points (d)
which are equidistant from two given intersecting straight lines.
31. Mensuration
Core Carry out calculations
involving the perimeter and area of a rectangle and triangle, the circumference
and area of a circle, the area of a parallelogram and a trapezium, the volume
of a cuboid, prism and cylinder and the surface area of a cuboid and a
cylinder.
Supplement Solve problems
involving the arc length and sector area as fractions of the circumference and
area of a circle, the surface area and volume of a sphere, pyramid and cone
(given formulae for the sphere, pyramid and cone).
32. Trigonometry
Core
Interpret and use three-figure bearings
measured clockwise from the North (i.e. 000°–360°); apply Pythagoras’ theorem
and the sine, cosine and tangent ratios for acute angles to the calculation of
a side or of an angle of a right-angled triangle (angles will be quoted in, and
answers required in, degrees and decimals to one decimal place).
Supplement
Solve trigonometrical problems in two
dimensions involving angles of elevation and depression; extend sine and cosine
values to angles between 90° and 180°; solve problems using the sine and cosine
rules for any triangle and the formula area of triangle = 2 1 ab sin C, solve
simple trigonometrical problems in three dimensions including angle between a
line and a plane.
33. Statistics
Core
Collect, classify and tabulate
statistical data; read, interpret and draw simple inferences from tables and
statistical diagrams; construct and use bar charts, pie charts, pictograms,
simple frequency distributions, histograms with equal intervals and scatter
diagrams (including drawing a line of best fit by eye); understand what is
meant by positive, negative and zero correlation; calculate the mean, median
and mode for individual and discrete data and distinguish between the purposes
for which they are used; calculate the range.
Supplement
Construct and read histograms with equal and
unequal intervals (areas proportional to frequencies and vertical axis labelled
'frequency density'); construct and use cumulative frequency diagrams; estimate
and interpret the median, percentiles, quartiles and inter-quartile range;
calculate an estimate of the mean for grouped and continuous data; identify the
modal class from a grouped frequency distribution.
34. Probability
Core
Calculate the probability of a
single event as either a fraction or a decimal (not a ratio); understand and
use the probability scale from 0 to 1; understand that: the probability of an
event occurring = 1 – the probability of the event not occurring; understand
probability in practice, e.g. relative frequency.
Supplement
Calculate the probability of
simple combined events, using possibility diagrams and tree diagrams where
appropriate (in possibility diagrams outcomes will be represented by points on
a grid and in tree diagrams outcomes will be written at the end of branches and
probabilities by the side of the branches).
35. Vectors in two dimensions
Core
Describe a translation by using a
vector represented by
e.g. y x, AB or a; add
and subtract vectors; multiply a vector by a scalar.
Supplement
Calculate the magnitude of a vector y x
as 22 xy + . (Vectors will be printed as
AB or a and their magnitudes denoted by
modulus signs, e.g. AB or a. In their answers to questions candidates are expected to indicate
a in some definite way, e.g. by an arrow or by underlining, thus AB or a) Represent vectors by directed line
segments; use the sum and difference of two vectors to express given vectors in
terms of two coplanar vectors; use position vectors
36. Matrices
Supplement
Display information in the form
of a matrix of any order; calculate the sum and product (where appropriate) of
two matrices; calculate the product of a matrix and a scalar quantity; use the
algebra of 2 × 2 matrices including the zero and identity 2 × 2 matrices;
calculate the determinant and inverse A–1 of a non-singular matrix A
37. Transformations
Core
Reflect simple plane figures in
horizontal or vertical lines; rotate simple plane figures about the origin,
vertices or midpoints of edges of the figures, through multiples of 90°;
construct given translations and enlargements of simple plane figures;
recognise and describe reflections, rotations, translations and enlargements.
Supplement
Use the following transformations
of the plane: reflection (M); rotation (R); translation (T); enlargement (E);
shear (H); stretch (S) and their combinations (if M(a) = b and R(b) = c the
notation RM(a) = c will be used; invariants under these transformations may be
assumed.) Identify and give precise descriptions of transformations connecting
given figures; describe transformations using co-ordinates and matrices
(singular matrices are excluded).
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