Mathematics GCE O Level (Secondary 1 – 4 )Syllabus:


Number


  • use natural numbers, integers (positive, negative and zero), prime numbers, common factors and common multiples, rational and irrational numbers, real numbers;

  • continue given number sequences, recognise patterns within and across different sequences and generalise to simple algebraic statements (including expressions for the nth term) relating to such sequences.

  • prime and prime factorization

  • finding Highest Common Factor (HCF) and Lowest Common Multiple (LCM)

  • calculation with the use of calculator


Set language and notation


  • use set language and set 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 =

C =

D = {a, b, c... }


Notation:


Union of A and B: A B

Intersection of A and B: A B

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: Ø

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


Function notation


  • use function notation, e.g. , to describe simple functions, and the notation


and to describe their inverses.


Squares, square roots, cubes and cube roots


  • calculate squares, square roots, cubes and cube roots of numbers.


Decimal, fractions, and percentages


  • use the language and notation of simple vulgar and decimal fractions and percentages in appropriate contexts;

  • recognise equivalence and convert between these forms.


Ordering


  • order quantities by magnitude and demonstrate familiarity with the symbols =,≠ , >, <, ≤, ≥

Standard form

  • use the standard form A×10n where n is a positive or negative integer, and 1 ≤ A < 10

The four operations


  • use the four operations for calculations with whole numbers, decimal fractions and vulgar (and mixed) fractions, including correct ordering of operations and use of brackets.


Estimation


  • 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.


Limits of accuracy


  • give appropriate upper and lower bounds for data given to a specified accuracy (e.g. measured lengths);

  • 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.


Ratio, proportion, rate


  • 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;

  • express direct and inverse variation in algebraic terms and use this form of expression to find unknown quantities.


Percentages


  • calculate a given percentage of a quantity;

  • express one quantity as a percentage of another,

  • percentages greater than 100%

  • calculate percentage increase or decrease;

  • carry out calculations involving reverse percentages, e.g. finding the cost price given the selling price and the percentage profit.

  • problems involving percentage


Measures


  • use current units of mass, length, area, volume and capacity in practical situations and express quantities in terms of larger or smaller units.

Time


  • calculate times in terms of the 12-hour and 24-hour clock;

  • read clocks, dials and timetables.


Money, personal, and household finance



  • solve problems involving money and convert from one currency to another.

  • use given data to solve problems on personal and household finance involving earnings, simple interest, discount, profit and loss, taxation;

  • extract data from tables and charts.


Graphs in practical situations

  • demonstrate familiarity with cartesian coordinates in two dimensions;

  • interpret and use graphs in practical situations including travel graphs and conversion graphs;

  • draw graphs from given data;

  • apply the idea of rate of change to easy kinematics involving distance-time and speed-time graphs, acceleration and retardation;

  • calculate distance travelled as area under a linear speed-time graph.


Graphs of functions


  • construct tables of values and draw graphs for functions of
the form where n = –2, –1, 0, 1, 2, 3, and simple sums of not more than three of these and for functions of the form , where a is a positive integer;

  • interpret graphs of linear, quadratic, reciprocal and exponential functions;

  • find the gradient of a straight line graph;

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

  • graphs of linear equations in two unknowns

  • graphs of quadratic functions and their properties

∗  positive or negative coefficient of x2

∗  maximum and minimum points

∗  symmetry

  • sketching of the graphs of quadratic functions given in the form

∗  y=±(x _p)2+q

∗  y = ±(x _ a)(x _ b)

  • solve equations approximately by graphical methods;

  • estimate gradients of curves by drawing tangents.


Straight line graphs


  • calculate the gradient of a straight line from the coordinates of two points on it;

  • interpret and obtain the equation of a straight line graph in the form

  • calculate the length and the coordinates of the midpoint of a line segment from the coordinates of its end points.


Algebraic representation and formulae


  • use letters to express generalised numbers and express basic arithmetic processes algebraically, substitute numbers for words and letters in formulae;

  • interpreting notations:

  • transform simple and more complicated formulae;

  • translation of simple real-world situations into algebraic expressions

  • construct equations from given situations.


Algebraic manipulation


  • addition and subtraction of linear algebraic expressions

  • simplification of linear algebraic expressions, e.g.

2(3x − 5)+ 4x

  • manipulate directed numbers;

  • use brackets and extract common factors;

  • expand products of algebraic expressions;

  • factorise expressions of the form

  • addition and subtraction of algebraic fractions with linear or quadratic denominator

  • multiplication and division of simple algebraic fractions

Indices


  • use and interpret positive, negative, zero and fractional indices.

Solutions of equations and inequalities


  • solve simple linear equations in one unknown;

  • solve fractional equations with numerical and linear algebraic denominators;

  • solve simultaneous linear equations in two unknowns by:

∗  substitution and elimination methods

∗  graphical method

  • solve quadratic equations by factorisation and either by use of the formula or by completing the square;

  • solve simple linear inequalities.

Graphical representation of inequalities


  • represent linear inequalities in one or two variables graphically.


Congruence and similarity


Include:

  • congruent figures and similar figures

  • properties of similar polygons:

∗  corresponding angles are equal

∗  corresponding sides are proportional

  • enlargement and reduction of a plane figure by a scale factor

  • scale drawings

  • determining whether two triangles are 
 congruent similar

  • ratio of areas of similar plane figures

  • ratio of volumes of similar solids

  • solving simple problems involving similarity and congruence


Geometrical constructions


  • measure lines and angles;

  • construct simple geometrical figures from given data, angle bisectors and perpendicular bisectors using protractors or set squares as necessary;

  • read and make scale drawings. 
(Where it is necessary to construct a triangle given the three sides, ruler and compasses only must be used.)


Bearings


  • interpret and use three-figure bearings measured clockwise from the north (i.e. 000°–360°).


Symmetry


  • recognise line and rotational symmetry (including order of rotational symmetry) in two dimensions, and properties of triangles, quadrilaterals and circles directly related to their symmetries;

  • recognise symmetry properties of the prism (including cylinder) and the pyramid (including cone);

Properties of circles


Include:

  • 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

∗  the line joining an external point to the centre of the circle bisects the 
angle between the tangents

  • angle properties of circles:

∗  angle in a semicircle is a right angle

∗  angle between tangent and radius of a circle is a right angle

∗  angle at the centre is twice the angle at the circumference

∗  angles in the same segment are equal

∗  angles in opposite segments are supplementary


Angle, triangles and polygons


Include:

  • right, acute, obtuse and reflex angles, complementary and supplementary angles, vertically opposite angles, adjacent angles on a straight line, adjacent angles at a point, interior and exterior angles

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

  • properties of triangles and special quadrilaterals

  • classifying special quadrilaterals on the basis of their properties

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

  • properties of regular pentagon, hexagon, octagon and decagon

  • properties of perpendicular bisectors of line segments and angle bisectors

  • construction of simple geometrical figures from given data (including perpendicular bisectors and angle bisectors) using compasses, ruler, set square and protractor, where appropriate


Locus


  • use the following loci and the method of intersecting loci:

(a)  sets of points in two or three dimensions

    1. (i)  which are at a given distance from a given point,

    2. (ii)  which are at a given distance from a given straight line,

(iii) which are equidistant from two given points;

(b)  sets of points in two dimensions which are equidistant from two given intersecting straight lines.


Mensuration


solve problems involving

(i)  the perimeter and area of a rectangle and triangle,

(ii)  the circumference and area of a circle,

(iii)  the area of a parallelogram and a trapezium,

(iv)  the surface area and volume of a cuboid, cylinder, prism, sphere, pyramid and cone (formulae will be given for the sphere, pyramid and cone),

  1. conversion between cm2 and m2 , and between cm3 and m3

  2. problems involving volume and surface area of composite solids

  3. arc length and sector area as fractions of the circumference and area of a circle

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

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


Trigonometry


  • 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 of a degree to one decimal place);

  • solve trigonometrical problems in two dimensions including those involving angles of elevation and depression and bearings;

  • extend sine and cosine functions to angles between 90° and 180°; solve problems using the sine and cosine rules for any triangle and the formula for the area of a triangle;

  • solve simple trigonometrical problems in three dimensions. (Calculations of the angle between two planes or of the angle between a straight line and plane will not be required.)


Statistics


  • 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, dot diagram, stem and leaf diagram, and frequency polygons;

  • use frequency density to construct and read histograms with equal and unequal intervals;

  • calculate the mean, median and mode for individual data and distinguish between the purposes for which they are used;

  • range, interquartile, and standard deviation as a measure of spread of data

  • construct and use cumulative frequency diagrams; estimate the median, percentiles, quartiles and interquartile range;

  • calculate the mean for grouped data; identify the modal class from a grouped frequency distribution.

  • 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


  • calculate the probability of a single event as either a fraction or a decimal (not a ratio);

  • 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.)


Matrices


  • display information in the form of a matrix of any order;

  • solve problems involving the calculation of the sum and product (where appropriate) of two matrices, and interpret the results;

  • calculate the product of a scalar quantity and a matrix;

  • use the algebra of axb matrices including the zero and identity matrices;

  • calculate the determinant and inverse of a non-singular matrix. (A–1 denotes the inverse of A.)


Transformations


  • use the following transformations of the plane: reflection (M), rotation (R), translation (T), enlargement (E), shear (H), stretching (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 coordinates and matrices. (Singular matrices are excluded.)


Vectors in two dimensions


  • describe a translation by using a vector represented by ,

  • add vectors and multiply a vector by a scalar;

  • calculate the magnitude of a vector as

(Vectors will be printed as and their magnitudes denoted by modulus signs, e.g.. ln all their answers to questions candidates are expected to indicate a in some definite way, e.g. by an arrow or by underlining, thus 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.