LOWER SECONDARY
MATHEMATICS SYLLABUS
SPECIAL/EXPRESS
NORMAL ACADEMIC
NORMAL TECHNICAL
CURRICULUM PLANNING AND DEVELOPMENT DIVISION
MINISTRY OF EDUCATION
SINGAPORE
© MINISTRY OF EDUCATION
ALL RIGHTS RESERVED
YEAR OF IMPLEMENTATION FROM 2001
1
CONTENTS
1
INTRODUCTION ……………………………………………………..…………..
2
AIMS OF MATHEMATICS EDUCATION .……………………………………... 4
3
FRAMEWORK OF THE MATHEMATICS PROGRAMME ..….……………...
5
4
GENERAL ASSESSMENT GUIDELINES ...…………………………………...
8
5
OBJECTIVES OF THE LOWER SECONDARY MATHEMATICS
PROGRAMME ..…………………………………………………………………..
10
6
CONTENT CHART
•
7
27
Secondary One (Normal Academic Course)... .……………..…………….
Secondary Two (Normal Academic Course) ...…………………………….
29
36
Secondary One & Two (Normal Technical Course) ..……..…………..….
41
SYLLABUS
•
•
12
Secondary One & Two (Normal Academic Course) ..……..…………..….
CONTENT CHART
•
11
13
21
SYLLABUS
•
•
10
Secondary One (Special/Express Course) ..…………………..…………..
Secondary Two (Special/Express Course) ………………………………...
CONTENT CHART
•
9
Secondary One & Two (Special/Express Course) …………..……………. 11
SYLLABUS
•
•
8
3
Secondary One (Normal Technical Course)... .……………..…………….
Secondary Two (Normal Technical Course) ...…………………………….
43
50
APPENDIX: DEFINITION OF SUGGESTED THINKING SKILLS ………….
55
2
INTRODUCTION
The Ministry of Education’s vision of “Thinking School, Learning Nation” gives impetus for
the infusion of three initiatives: Thinking Skills, Information Technology (IT) and National
Education into the curriculum. As we move towards a knowledge-based society that is
powered by IT, the need to prepare our people for the challenges and opportunities of the
future becomes obvious. Besides being proficient in the use of IT, pupils will need to be able
to think creatively, learn independently and work successfully in teams. Above all, as
Singapore’s economy moves towards globalisation, they need to have a strong feeling for
home and remain Singaporean in heart, mind and being. Against this background and with
the Desired Outcomes of Education as the overarching aim, the mathematics syllabus was
revised.
This revised mathematics syllabus better reflects the recent developments in mathematics
education. The focus of the syllabus is mathematical problem solving. The emphasis is on
the development of concepts, skills and its underlying processes. This, together with the
explication of thinking skills and the integration of IT in mathematics teaching and learning,
will give leverage to the development of mathematical problem solving.
This syllabus consists of two parts.
Part A explains the philosophy of the syllabus and the spirit in which it should be
implemented. It also spells out the objectives of the mathematics programme.
The framework of the mathematics programme summarises the essence of mathematics
teaching and learning in schools. The learning of mathematics at all levels involves more
than the basic acquisition of concepts and skills. It also involves an understanding of the
mathematical thinking and general problem-solving strategies, having positive attitudes to
and an appreciation of mathematics as an important and powerful tool in everyday life. This
framework forms the basis for mathematics teaching and learning in schools.
Part B gives the syllabus content for each level. Care has been taken to ensure that there is
continuity from the primary to the secondary level. In the syllabus, the spiral approach is
adopted to ensure that each topic is covered at appropriate levels in increasing depth to
enable pupils to consolidate the concepts and skills learnt and to further develop them. All
topics come with ‘Learning Outcomes’ to enable teachers to monitor pupils' progress. The
‘Remarks’ column provides teachers with guidance in interpreting the syllabus.
This syllabus is a guide for teachers to plan their mathematics programmes. Teachers need
not be bound by the sequence of topics presented here but should ensure that the hierarchy
and linkages are maintained. Teachers should exercise flexibility and creativity when using
the syllabus.
3
PART A
AIMS OF MATHEMATICS EDUCATION IN SCHOOLS
Mathematics education aims to enable pupils to:
•
acquire and apply the skills and knowledge relating to number, measure and space in
mathematical situations that they will meet in life.
•
acquire mathematical concepts and skills necessary for a further study in mathematics
or other disciplines.
•
develop the ability to make logical deduction and induction as well as to explicate their
thinking and reasoning skills through the solving of mathematical problems.
•
use mathematical language to communicate mathematical ideas and arguments
precisely, concisely and logically.
•
develop positive attitudes towards mathematics including confidence, enjoyment and
perseverance.
•
appreciate the power and structure of mathematics, including patterns and relationships,
and to enhance their intellectual curiosity.
4
PART A
FRAMEWORK OF THE MATHEMATICS PROGRAMME
The conceptualisation of the mathematics syllabus is based on the following framework:
Appreciation
Interest
Confidence
Perseverance
Estimation and
Approximation
Mental calculation
Communication
Use of mathematical tools
Arithmetic manipulation
Algebraic manipulation
Handling data
Monitoring one’s
own thinking
Thinking skills
Heuristics
Numerical
Geometrical
Algebraic
Statistical
The primary aim of the mathematics programme is to enable pupils to develop their ability in
mathematical problem solving. Mathematical problem solving includes using and applying
mathematics in practical tasks, in real life problems and within mathematics itself. In this
context, a problem covers a wide range of situations from routine mathematical problems to
problems in unfamiliar context and open-ended investigations that make use of the relevant
mathematics and thinking processes.
The attainment of this mathematical problem solving ability is dependent on five inter-related
components - Concepts, Skills, Processes, Attitudes and Metacognition
1
Concepts
Concepts refer to the basic mathematical knowledge needed for solving mathematical
problems. They cover the following:
•
Numerical concepts
•
Geometrical concepts
•
Algebraic concepts
•
Statistical concepts
5
PART A
2
Skills
Skills refer to the topic-related manipulative skills that pupils are expected to use when
solving problems. They include:
3
•
Estimation and approximation
•
Mental calculation
•
Communication
•
Use of mathematical tools
•
Arithmetic manipulation
•
Algebraic manipulation
•
Handling data
Processes
Processes refer to the thinking and heuristics involved in mathematical problem solving.
Some thinking skills and heuristics, which are applicable to problem solving at the
secondary level, are listed below.
Thinking skills:
•
Classifying
•
Comparing
•
Identifying Attributes & Components
•
Sequencing
•
Induction
•
Deduction
•
Generalising
•
Justifying
•
Verifying
•
Spatial visualisation
Heuristics for problem solving:
•
Act it out
•
Use a diagram/model
•
Use guess-and-check
•
Make a systematic list
6
PART A
•
Look for pattern(s)
•
Work backwards
•
Use before-after concept
•
Make suppositions
•
Restate the problem in another way
•
Simplify the problem
•
Solve part of the problem
•
Think of a related problem
•
Use equations
(Refer to Appendix A for the definitions of the suggested thinking skills.)
4
Attitudes
Attitudes refer to the affective aspects of mathematics learning such as:
5
•
enjoying mathematics
•
appreciating the beauty and power of mathematics
•
showing confidence in using mathematics
•
persevering in solving a problem
Metacognition
Metacognition refers to the ability to monitor one's own thinking processes in problem
solving. This includes:
•
constant and conscious monitoring of the strategies and thinking processes used
in carrying out a task
•
seeking alternative ways of performing a task
•
checking the appropriateness and reasonableness of answers
This framework encompasses the whole mathematics programme for primary and
secondary schools. The secondary curriculum is a continuation of the primary
curriculum.
7
PART A
GENERAL ASSESSMENT GUIDELINES
Assessment is an integral part of the teaching-learning process. The main purpose of
mathematical assessment should be to improve the teaching and learning of mathematics.
Therefore we need assessment that:
• is continual and provides accurate and useful information about pupils’ learning
• supports the programme in its aim to enable pupils to develop their problem solving
ability and in its emphasis on developing mathematical concepts, skills and thinking, as
well as positive attitudes towards mathematics
Teachers should assess different aspects of thinking, learning and behaviour. They should
try to incorporate, where appropriate, the following aspects in the assessment tasks set as
these are the key features of the syllabus:
•
•
•
•
•
•
•
•
mental calculation
mathematical communication
practical uses of mathematics
investigative work
problem solving
critical thinking
creative work
use of information technology
Continual Assessment
It is important to carry out assessment on a continual basis. Teachers should use continual
assessment to:
• obtain information about pupils’ learning, suitability of teaching methods and materials
and effectiveness of the teaching programme
• identify any learning difficulties that pupils may encounter or misconceptions that they
may have so as to plan effective remedial help
• provide prompt feedback to pupils on their progress and attainment
• promote pupils’ confidence in doing mathematics by focusing on what they can achieve
rather than what they cannot
In continual assessment, teachers should use a variety of assessment modes as different
modes provide different kinds of feedback about pupils. The modes used should yield the
feedback required, and collectively, they give teachers a more comprehensive profile of the
pupils.
Assessment can be carried out through:
•
•
•
•
classroom observations;
oral communication
written assignments and tests
practical and investigative tasks
8
PART A
From observations of pupils engaged in a task and through their responses in oral
communication, immediate feedback can be obtained on their understanding of concepts,
thinking processes, mastery of skills and even their attitudes towards mathematics. Written
assignments and tests help teachers to track the progress of pupils and are useful in the
assessment of content and procedures, mathematical thinking and problem solving.
Through practical and investigative tasks, teachers can assess pupils’ problem solving skills,
creative and critical thinking as well as their attitudes. If the tasks are done in groups,
teamwork can also be assessed.
Semestral Assessment
Semestral assessment comprises the mid-year and end-of-year examinations, which are
common school-based examinations administered under formal conditions. The purpose of
semestral assessment is to determine how far pupils have achieved the overall programme
objectives for the semester or year. As such, it should be broad in coverage, test a
representative sample of the syllabus taught and reflect its emphasis.
9
PART A
OBJECTIVES OF THE LOWER SECONDARY MATHEMATICS PROGRAMME
The objectives of the lower secondary mathematics programme are to enable pupils to:
•
develop an understanding of mathematical concepts - numerical, geometrical, algebraic,
statistical;
•
perform calculations by using suitable methods, including mental calculation, with and
without a calculating aid;
•
develop the ability to estimate and to make approximations and also to be alert to the
reasonableness of results and measurements;
•
apply systems of measurement in everyday use and in the solutions of problems;
•
use geometrical instruments;
•
collect and analyse data;
•
interpret, use and present information in written, graphical, diagrammatic and
tabular forms;
•
understand and use mathematical language, symbols and diagrams to represent and
communicate mathematical ideas effectively;
•
recognise and apply spatial relationships in two and three dimensions;
•
recognise the appropriate mathematical procedures for a given situation;
•
recognise patterns and structures in a variety of situations and form, and/or justify
generalisations;
•
apply and interpret mathematical concepts learnt in familiar and unfamiliar contexts,
including daily life;
•
think logically and derive conclusions deductively and apply these processes in
mathematical situations;
•
analyse problems; formulate them into mathematical terms and use the appropriate
strategies to solve them; verify and interpret the solutions; and present their
mathematical arguments and solutions in a logical and clear fashion;
•
recognise the relationships between topics in the syllabus;
•
become aware of the application of mathematics in other subjects;
•
develop an inquiring mind through investigative activities;
•
enjoy learning mathematics through a variety of activities.
10
SPECIAL/EXPRESS
PART B
Content Chart - Secondary One & Two (Special/Express Course)
Secondary One
1
2
Secondary Two
Whole numbers
1
Arithmetic problems
• the four operations
• ordering
• factors and multiples
2
Standard form
3
Number sequences
Fractions and decimals
• concept and notation
• ordering
• the four operations
3
Approximation and estimation
• rounding off
• estimation
4
Use of a scientific calculator
5
Squares, square roots, cubes and
cube roots
6
Number sequences
7
Measures and money
8
Ratio, proportion and rate
9
Percentage
10 Simple financial transactions
11 Real numbers
• integers
• rational and irrational numbers
1
Perimeter and area
1
• square, rectangle, triangle,
parallelogram, trapezium, circle
• sphere, pyramid and cone
2
2
Volume and surface area
Arc length and sector area
Volume and surface area
• cube, cuboid, prism, cylinder
1
Algebraic expressions and
formulae
1
Algebraic manipulation and
formulae
2
Simple algebraic manipulation
2
Solutions of equations
3
Simple linear equations
• simultaneous linear equations
• simple fractional equations
• quadratic equations
11
SPECIAL/EXPRESS
PART B
Content Chart - Secondary One & Two (Special/Express Course)
Secondary One
1
Simple plane figures
2
Simple solid figures
3
AngIe properties
• angles formed with a common
vertex
• angles formed with parallel
lines
• angle properties of triangles
• angle properties of squares,
rectangles, parallelograms and
rhombuses
4
Geometrical construction
of simple geometrical
figures
1
Handling data
•
•
•
•
•
•
•
•
Secondary Two
1
Graphs of linear and quadratic
functions
2
Graphs in practical situations
1
Motion geometry
•
•
•
•
reflection
rotation
translation
enlargement
2
Similar and congruent figures
3
Angle properties of polygons
4
Scale drawing
5
Symmetry
• line symmetry
• rotational symmetry
1
Averages
• mean
• mode
• median
table and chart
pictogram
dot diagram
bar graph
line graph
pie chart
stem and leaf diagram
histogram
1
2
Pythagoras’ theorem
Trigonometrical ratios:
• sine
• cosine
• tangent
1
Problem solving heuristics
1
Problem solving heuristics
2
Practical uses of Mathematics
2
Practical uses of Mathematics
12
LEVEL : SEC 1 (SPECIAL/EXPRESS)
PART B
TOPICS/OUTCOMES
REMARKS
ARITHMETIC
Pupils should be able to
1
Whole numbers
The four operations
• use the four operations for calculations
with whole numbers
•
•
•
•
Include combined operations, i.e. correct
ordering of operations and the use of
brackets
Include mental calculation and estimation
Include awareness of the following
(i) commutative law
(ii) associative law
(iii) distributive law
Exclude tedious calculations when the use
of a calculator is not allowed
Ordering
• order numbers
• use the following symbols correctly:
=, ≠, >, <, ≥, ≤
•
Include the use of the number line
•
Include the use of index notation and the
terms “prime factorization”, “ HCF” and
“LCM”
Include odd and even numbers
Factors and multiples
• use prime numbers, common factors
and common multiples
•
2
Fractions and decimals
Concept and notation
• use fractions and decimals
• convert fractions to decimals, and vice
versa
•
Include the use of facts such as
1
3
1
0.25 =
, 0.75 =
, 0.125 =
4
4
8
1
2
0.333 ≈
, 0.667 ≈
3
3
•
Exclude conversion of recurring decimals
to fractions
•
Include comparison of fractions with
decimals
Ordering
• compare and order fractions and
decimals
13
LEVEL : SEC 1 (SPECIAL/EXPRESS)
PART B
TOPICS/OUTCOMES
REMARKS
The four operations
• use the four operations for calculations
with fractions and decimals
•
•
3
Include
i) combined operations
ii) rounding off decimals to a specific
degree of accuracy
iii) mental calculation and estimation
Exclude tedious calculations when the use
of a calculator is not allowed
Approximation and estimation
Rounding off
• round off numbers and measures to a
specified degree of accuracy
•
Include
i)
decimal places and significant figures
ii) the use of the approximation sign “≈”
•
Include use of estimation to check the
reasonableness of answers
Include mental estimation
Estimation
• make estimates of numbers and
measures
4
•
Use of a scientific calculator
• use the relevant keys of a scientific
calculator
•
•
5
Include appropriate checks of accuracy by
estimation, e.g. in evaluating 47 600 ÷ 85
as 560, pupils should recognize that
47 600 ÷ 85 ≈ 600 (not 6 or 6 000)
Pupils should be able to round off the
answer in the context of a given problem,
e.g. pupils should realize the absurdity of
giving the speed of a car to 5 decimal
places
Squares, square roots, cubes and cube
roots
• find squares, square roots, cubes and
cube roots of numbers
•
Include the use of the square root sign
"
" and cube root sign " 3 "
•
•
Include estimation
Include the use of the following keys of a
calculator:
x 2 , x , x y , x1 / y
14
LEVEL : SEC 1 (SPECIAL/EXPRESS)
PART B
TOPICS/OUTCOMES
6
Number sequences
• continue a given number sequence
7
REMARKS
•
Recognize simple number patterns and
state the rules for the patterns
•
Include
i) use of the term: “capacity”
ii) practical activities to reinforce concept
of units of measures
iii) conversion of measures and currency
iv) calculation of time in terms of the 24hour and 12-hour clocks
reading of clocks, dials and timetables
Measures and money
Mass, length, time and money
• solve problems involving the use of units
of mass, length, time, money
•
8
Ratio, proportion and rate
Ratio and proportion
• find the ratio of two or more quantities
•
Include expressing a ratio in its lowest
terms
• state the relationship between ratio and
fraction
•
Include rewriting x : y = a : b
a
as x = ( ) y
b
• use direct and inverse proportion
•
Include the concept of reciprocals
• solve problems involving ratio and
proportion
•
Include scales
• recognize and use common measures of
rate
•
Include conversion such as km/h to m/s
and vice versa
• solve problems involving rate
•
Include calculation of average speed
Percentage
•
Include the use of facts such as
1
i)
25%
=
0.25
=
4
1
ii) 50%
=
0.5
=
2
3
iii) 75%
=
0.75
=
4
1
iv) 20%
=
0.2
=
5
4
v) 80%
=
0.8
=
5
Rate
9
• convert between
− percentage and fraction
− percentage and decimal
15
LEVEL : SEC 1 (SPECIAL/EXPRESS)
PART B
TOPICS/OUTCOMES
• calculate a given percentage of a
quantity
REMARKS
•
Include reverse problems
Example:
40% of a class are boys. If there are 16
boys, find the number of pupils in the class.
•
Include reverse problems such as finding
the original salary given the new salary and
the percentage increase
•
Include
i)
earnings, simple interest, compound
interest (without use of formulae), hirepurchase, discount, commission, profit
and loss, money exchange and
taxation
ii) reverse problems such as finding the
cost price given the selling price and
the percentage profit
•
Include ordering on the number line
•
Include understanding of the following
terms and their relationships:
• express one quantity as a percentage of
another
• calculate percentage increase/decrease
• solve problems involving percentage
10
Simple financial transactions
• solve problems on personal and
household finance, and simple financial
transactions
• extract data from tables and charts to
solve problems
11
Real numbers
Integers
• manipulate integers (positive, negative
and zero)
Rational and irrational numbers
• recognize rational and irrational numbers
Real numbers
irrational numbers
rational numbers
fractions
(+ve, -ve)
integers
(+ve, 0, -ve)
16
LEVEL : SEC 1 (SPECIAL/EXPRESS)
PART B
TOPICS/OUTCOMES
• manipulate rational numbers
REMARKS
•
Exclude tedious computations
•
Include figures related to these shapes
•
•
Include nets of these solids
Include finding the volumes of
composite solids
Exclude oblique prisms and oblique
cylinders
• find an approximate value of an irrational
number with a calculator
MENSURATION
Pupils should be able to
1
Perimeter and area
• find the perimeters and areas of squares,
rectangles, triangles, parallelograms,
trapeziums and circles
• solve problems involving the perimeters and
areas of squares, rectangles, triangles,
parallelograms, trapeziums and circles
2
Volume and surface area
•
find the volumes and surface areas of
cubes, cuboids, prisms and cylinders
•
•
solve problems involving the volumes and
surface areas of cubes, cuboids, prisms
and cylinders
•
Include problems involving density
•
Include collecting like terms and
removing brackets
ALGEBRA
Pupils should be able to
1
Algebraic expressions and formulae
• use letters to represent numbers
• express basic arithmetic processes
algebraically
• substitute numbers for words and letters in
formulae and expressions
2
Simple algebraic manipulation
• manipulate simple algebraic expressions
17
LEVEL : SEC 1 (SPECIAL/EXPRESS)
PART B
TOPICS/OUTCOMES
3
REMARKS
Simple linear equations
• solve simple linear equations
•
Include simple cases involving fractional
and decimal coefficients
Examples:
1
1
x+5=xi)
2
3
ii)
2 + 0.6x = 2x
•
Include the use of the following geometrical
terms:
point, line, plane, parallel,
perpendicular, right angles, acute,
obtuse and reflex angles,
complementary and
supplementary angles, base
angles, interior and exterior
angles, regular and irregular
polygons, diagonal and vertex
•
Include tiles and tessellations
•
Include sketching of these solids
• solve problems involving linear
equations
GEOMETRY
Pupils should be able to
1
Simple plane figures
• identify the following plane figures:
− triangles: isosceles triangles,
equilateral triangles, right-angled
triangles, acute-angled triangles,
obtuse-angled triangles and
scalene triangles
− special quadrilaterals: squares,
rectangles, parallelograms,
rhombuses, trapeziums and kites
− polygons: pentagons, hexagons,
octagons and decagons
2
Simple solid figures
• identify the following simple solid figures:
cubes, cuboids, prisms, cylinders,
pyramids, cones and spheres
3
Angle properties
Angles formed with a common vertex
• calculate unknown angles involving:
− adjacent angles on a straight line
− vertically opposite angles
− angles at a point
Angles formed with parallel lines
• calculate unknown angles involving:
− corresponding angles
− alternate angles
− interior angles between parallel lines
18
LEVEL : SEC 1 (SPECIAL/EXPRESS)
PART B
TOPICS/OUTCOMES
REMARKS
Angle properties of a triangle
• calculate unknown angles involving:
− angle sum of a triangle
− base angles of an isosceles triangle
− angles of an equilateral triangle
− exterior angle of a triangle
Angle properties of squares, rectangles,
parallelograms and rhombuses
•
Include angle properties related to their
diagonals
• measure line segments and angles
•
Use rulers, set-squares, protractors and
compasses
• draw line segments, angles, parallel
lines and perpendicular lines
•
Include cases where the following are
given:
i)
the distance from a point to a line
ii)
the distance between two parallel
lines
• construct angle bisectors and
perpendicular bisectors
•
Use protractors, set squares, compasses
and straight edges/rulers
• construct simple geometrical figures
from given data
•
Include drawing the nets of cubes, cuboids,
prisms and pyramids
•
Include the use of tally marks
• calculate unknown angles using the
angle properties of
− squares
− rectangles
− parallelograms
− rhombuses
4
Geometrical construction of simple
geometrical figures
STATISTICS
Pupils should be able to
1
Handling data
• collect, classify and tabulate data
• read and interpret tables and statistical
diagrams
• construct a bar graph, pie chart,
pictogram, dot diagram, line graph, stem
and leaf diagram, and histogram with
equal intervals
19
LEVEL : SEC 1 (SPECIAL/EXPRESS)
TOPICS/OUTCOMES
PART B
REMARKS
PROBLEM SOLVING
Pupils should be able to
1
Problem solving heuristics
•
2
use appropriate heuristics to solve
problems
Practical uses of mathematics
• solve mathematical problems in
everyday life
20
LEVEL : SEC 2 (SPECIAL/EXPRESS)
PART B
TOPICS/OUTCOMES
REMARKS
ARITHMETIC
Pupils should be able to
1
Arithmetic problems
• solve problems involving measures, money,
ratio, proportion, rate and speed,
percentage, personal and household finance
and simple financial transactions
2
Standard form
• read and write numbers in standard form A x
n
10 where n is a positive or negative integer,
and 1 ≤ A < 10
3
•
•
Include significant figures of numbers
written in a standard form
Include estimation and approximation
Number sequences
• formulate the algebraic expression for the
general term of a number sequence
•
Exclude direct application of arithmetic
progression and geometric progression
formulae
•
Include the use of the term
“hemisphere”
Include nets where applicable
Exclude oblique pyramids and cones
MENSURATION
Pupils should be able to
1
Volume and surface area
• find the volumes and surface areas of
spheres, pyramids and cones
•
•
• solve problems involving the volumes and
surface areas of spheres, pyramids and
cones
2
Arc length and sector area
• express arc length as a fraction of
circumference and sector area as a fraction
of the area of a circle
• find arc length and sector area
•
Include finding the area of a segment of
a circle
• solve problems involving arc length and
sector area
21
LEVEL : SEC 2 (SPECIAL/EXPRESS)
PART B
TOPICS/OUTCOMES
REMARKS
ALGEBRA
Pupils should be able to
1
Algebraic manipulation and formulae
• expand products of simple algebraic
expressions
•
Examples:
i)
(ax + b) (cx + d)
ii) (ax + by) (cx + dy)
(a, b, c and d are integers)
•
Include only simple expressions such as
the following:
x
x−4
i)
+
3
2
2x
3(x − 5)
ii)
3
2
3a
5ab
) (
)
iii) (
4
3
3a
9a
÷
iv)
4
10
1
2
v)
+
x−2
x−3
1
5
Expressions such as
and
+
2
x −4 x−2
x2 − 4
will be introduced at the upper
2
x + 4x + 4
secondary level
• factorize algebraic expressions of the
form:
ax + ay
ax + bx + kay + kby
2 2
2 2
ax − by
2
2
a ± 2ab + b
2
ax + bx + c
• manipulate simple algebraic fractions
•
• transform simple formulae
2
•
Exclude formulae involving square roots
•
Include both the elimination and
substitution methods
Solutions of equations
Simultaneous linear equations
• solve simultaneous linear equations in
two unknowns
• solve problems involving simultaneous
linear equations
22
LEVEL : SEC 2 (SPECIAL/EXPRESS)
PART B
TOPICS/OUTCOMES
REMARKS
Simple fractional equations
•
Examples:
x
x−2
i)
+
= 3
3
4
3
ii)
= 6
x
3
iii)
= 6
x−2
•
Exclude fractional equations such as
1
2
1
+
= 2
x−2
x−3
2
•
Include finding the value of y from the
graph given the value of x, and vice versa
• interpret and use graphs in practical
situations
•
Include travel graphs (distance-time
graphs) and conversion graphs
• draw graphs using data from practical
situations
•
Include the choice of appropriate scales
• solve fractional equations involving
numerical and linear algebraic
denominators
• solve problems involving simple
fractional equations
Quadratic equations
• solve quadratic equations by
factorization
• solve problems involving quadratic
equations
GRAPHS
Pupils should be able to
1
Graphs of linear and quadratic functions
• use cartesian coordinates in two
dimensions
• draw graphs of linear and quadratic
functions
• use graphical methods to solve
simultaneous linear equations
2
Graphs in practical situations
23
LEVEL : SEC 2 (SPECIAL/EXPRESS)
PART B
TOPICS/OUTCOMES
REMARKS
GEOMETRY
Pupils should be able to
1
Motion Geometry
Reflection, rotation, translation,
enlargement
2
• draw the reflection of a simple plane
figure in horizontal or vertical lines
•
Include “line of reflection” (mirror line)
• draw the rotation of a simple plane
figure, about the origin of the x-y plane,
or a vertex or a mid-point of an edge of
the figure, through multiples of 90°
•
Include ”centre of rotation” and “angle of
rotation”
• draw the translation of a simple plane
figure in the x-y plane
•
Include only movements parallel to the x
and y axes
• draw an enlargement of a given plane
figure
•
Include “centre of enlargement” and “scale
factor” (scaling up or down)
Exclude negative scale factors
•
• identify reflection, rotation, translation
and enlargement of a given plane figure
•
Exclude problems involving finding “line of
reflection”, “centre of rotation”, “angle of
rotation”, "centre of enlargement” and
“scale factor”
• draw the image of a figure involving
combined movements
•
Exclude more than 2 movements
•
Exclude tests for similarity and congruency
between two triangles
• calculate
− the sum of interior angles of a
polygon
− the sum of exterior angles of a
polygon
•
Include regular polygons
• calculate unknown angles of a polygon
•
Include finding the number of sides of a
polygon
Similar and congruent figures
• recognise similar and congruent figures
• find unknown sides/angles of
similar/congruent figures
3
Angle properties of polygon
24
LEVEL : SEC 2 (SPECIAL/EXPRESS)
PART B
TOPICS/OUTCOMES
4
REMARKS
Scale drawing
• read and make scale drawings
5
Symmetry
Line symmetry and rotational symmetry
• identify line and rotational symmetry of
plane figures
•
•
Include ‘line of symmetry’ and ‘centre and
order of rotational symmetry’
Exclude the use of the term ‘point
symmetry’
• use symmetrical properties of triangles,
quadrilaterals and regular polygons
•
Include properties of these figures directly
related to their symmetries
• use symmetrical properties of prisms,
cylinders, pyramids and cones
•
Include ‘plane of symmetry’ and ‘axis of
rotational symmetry’
•
•
Distinguish between the purposes for which
mean, median and mode are used
Exclude grouped data
•
•
Include the converse of the theorem
Proving of the theorem is not required
STATISTICS
Pupils should be able to
1
Averages
• find mean, median and mode
TRIGONOMETRY
Pupils should be able to
1
Pythagoras' theorem
• state Pythagoras’ theorem
• apply Pythagoras’ theorem to find a side
of a right-angled triangle
• solve problems applying Pythagoras’
theorem
2
Trigonometrical ratios: sine, cosine and
tangent
• state the sine, cosine and tangent ratios
for an acute angle of a right-angled
triangle
25
LEVEL : SEC 2 (SPECIAL/EXPRESS)
PART B
TOPICS/OUTCOMES
REMARKS
• find the values of the trigonometrical
ratios with a calculator
•
Include the use of inverse trigonometrical
function keys of the calculator
• apply the sine, cosine or tangent ratio for
an acute angle to find
− a side of a right-angled triangle
− an angle of a right-angled triangle
•
Exclude expressing angles in degrees and
minutes
• solve trigonometrical problems in two
dimensions
•
Include angles of elevation and depression
PROBLEM SOLVING
Pupils should be able to
1
Problem solving heuristics
•
2
use appropriate heuristics to solve
problems
Practical uses of mathematics
• solve mathematical problems in
everyday life
26
NORMAL ACADEMIC
PART B
Content Chart - Secondary One & Two (Normal [Academic] Course)
Secondary One
1
2
Whole numbers
1
Simple financial transactions
• the four operations
• ordering
• factors and multiples
2
Arithmetic problems
3
Standard form
4
Number sequences
Fractions and decimals
• concept and notation
• ordering
• the four operations
3
Secondary Two
Approximation and estimation
• rounding off
• estimation
4
Use of a scientific calculator
5
Squares, square roots, cubes and
cube roots
6
Number sequences
7
Measures and money
8
Ratio, proportion and rate
9
Percentage
10 Real numbers
• integers
• rational and irrational numbers
1
Perimeter and area
1
• square, rectangle, triangle,
parallelogram, trapezium, circle
• sphere, pyramid and cone
2
2
Volume and surface area
Arc length and sector area
Volume and surface area
• cube, cuboid, prism, cylinder
1
Algebraic expressions and
formulae
1
Algebraic manipulation and
formulae
2
Simple algebraic manipulation
2
Solutions of equations
• simple linear equations
• simultaneous linear equations
• simple fractional equations
27
NORMAL ACADEMIC
PART B
Content Chart - Secondary One & Two (Normal [Academic] Course)
Secondary One
Secondary Two
1
Graphs of linear and quadratic
functions
2
Graphs in practical situations
1
Simple plane figures
1
Similar and congruent figures
2
Simple solid figures
2
Angle properties of polygons
3
AngIe properties
3
Scale drawing
4
Symmetry
• angles formed with a common
vertex
• angles formed with parallel lines
• angle properties of triangles
• angle properties of squares,
rectangles, parallelograms and
rhombuses
4
Geometrical construction of
simple geometrical figures
1
Handling data
•
•
•
•
•
•
•
•
• line symmetry
• rotational symmetry
1
Averages
• mean
• mode
• median
table and chart
pictogram
dot diagram
bar graph
line graph
pie chart
stem and leaf diagram
histogram
1
Problem solving heuristics
1
Problem solving heuristics
2
Practical uses of Mathematics
2
Practical uses of Mathematics
28
LEVEL : SEC 1 (NORMAL ACADEMIC)
PART B
TOPICS/OUTCOMES
REMARKS
ARITHMETIC
Pupils should be able to
1
Whole numbers
The four operations
• use the four operations for calculations
with whole numbers
•
•
•
•
Include combined operations, i.e. the
correct ordering of operations and the use
of brackets
Include mental calculation and estimation
Include awareness of the following
(i) commutative law
(ii) associative law
(iii) distributive law
Exclude tedious calculations when the use
of a calculator is not allowed
Ordering
• order numbers
• use the following symbols correctly:
=, ≠, >, <, ≥, ≤
•
Include the use of the number line
•
Include the use of index notation and the
terms “prime factorization”, “ HCF” and
“LCM”
Include odd and even numbers
Factors and multiples
• use prime numbers, common factors
and common multiples
•
2
Fractions and decimals
Concept and notation
• use fractions and decimals
• convert fractions to decimals, and vice
versa
•
•
Include the use of facts such as
1
3
1
0.25 =
, 0.75 =
, 0.125 =
4
4
8
1
2
0.333 ≈
, 0.667 ≈
3
3
Exclude conversion of recurring decimals
to fractions
Ordering
• compare and order fractions and
decimals
•
Include comparison of fractions with
decimals
29
LEVEL : SEC 1 (NORMAL ACADEMIC)
PART B
TOPICS/OUTCOMES
REMARKS
The four operations
• use the four operations for calculations
with fractions and decimals
•
•
3
Include
i) combined operations
ii) rounding-off decimals to a specific
degree of accuracy
iii) mental calculation and estimation
Exclude tedious calculations when the use
of a calculator is not allowed
Approximation and estimation
Rounding off
• round off numbers and measures to a
specified degree of accuracy
•
Include
i)
decimal places and significant figures
ii) the use of the approximation sign “≈”
•
Include use of estimation to check the
reasonableness of answers
Include mental estimation
Estimation
• make estimates of numbers and
measures
4
•
Use of a scientific calculator
•
Use the relevant keys of a scientific
calculator
•
•
5
Include appropriate checks of accuracy by
estimation, e.g. in evaluating 47 600 ÷ 85
as 560, pupils should recognize that
47 600 ÷ 85 ≈ 600 (not 6 or 6 000)
Pupils should be able to round off the
answer in the context of a given problem,
e.g. pupils should realize the absurdity of
giving the speed of a car to 5 decimal
places
Squares, square roots, cubes and cube
roots
• find squares, square roots, cubes and
cube roots of numbers
•
Include the use of the square root sign
‘
‘, and cube root sign ‘ 3 ‘
•
•
Include estimation
Include the use of the following keys of a
calculator:
x 2 , x , x y , x1 / y
6
Number sequences
• continue a given number sequence
•
Recognise simple number patterns and
state the rules for the patterns
30
LEVEL : SEC 1 (NORMAL ACADEMIC)
PART B
TOPICS/OUTCOMES
7
REMARKS
Measures and money
Mass, length, time and money
•
Include
i) use of the term “capacity”
ii) practical activities to reinforce concept
of units of measures
iii) conversion of measures and currency
iv) calculation of time in terms of the 24hour and 12-hour clocks
v) reading of clocks, dials and timetables
find the ratio of two or more quantities
•
Include expressing a ratio in its lowest
terms
• state the relationship between ratio and
fraction
•
Include rewriting x : y = a : b
a
as x = ( ) y
b
• use direct and inverse proportion
•
Include the concept of reciprocals
• solve problems involving ratio and
proportion
•
Include scales
• recognize and use common measures of
rate
•
Include conversion such as km/h to m/s
and vice versa
•
•
Include calculation of average speed
•
Include the use of facts such as
1
i)
25%
=
0.25
=
4
1
ii) 50%
=
0.5
=
2
3
iii) 75%
=
0.75
=
4
1
iv) 20%
=
0.2
=
5
4
v) 80%
=
0.8
=
5
• solve problems involving the use of units
of mass, length, time, money
8
Ratio, proportion and rate
Ratio and proportion
•
Rate
9
solve problems involving rate
Percentage
• convert between
− percentage and fraction
− percentage and decimal
31
LEVEL : SEC 1 (NORMAL ACADEMIC)
PART B
TOPICS/OUTCOMES
• calculate a given percentage of a
quantity
REMARKS
•
Include reverse problems
Example:
40% of a class are boys. If there are 16
boys, find the number of pupils in the class
•
Include reverse problems such as finding
the original salary given the new salary and
the percentage increase
•
Include ordering the number line
•
Include understanding the following terms
and their relationships:
• express one quantity as a percentage of
another
• calculate percentage increase/decrease
• solve problems involving percentage
10
Real numbers
Integers
• manipulate integers (positive, negative
and zero)
Rational and irrational numbers
• recognize rational and irrational numbers
Real numbers
irrational numbers
rational numbers
fractions
(+ve, -ve)
• manipulate rational numbers
integers
(+ve, 0, -ve)
•
Exclude tedious computations
•
Include figures related to these shapes
• find an approximate value of an irrational
number with a calculator
MENSURATION
Pupils should be able to
1
Perimeter and area
• find the perimeters and areas of
squares, rectangles, triangles,
parallelograms, trapeziums and circles
•
solve problems involving the perimeters
and areas of squares, rectangles,
triangles, parallelograms, trapeziums
and circles
32
LEVEL : SEC 1 (NORMAL ACADEMIC)
PART B
TOPICS/OUTCOMES
2
REMARKS
Volume and surface area
•
find the volumes and surface areas of
cubes, cuboids, prisms and cylinders
•
•
•
•
solve problems involving the volumes
and surface areas of cubes, cuboids,
prisms and cylinders
Include nets of these solids
Include finding the volumes of composite
solids
Exclude oblique prisms and oblique
cylinders
•
Include problems involving density
•
Include collecting like terms and removing
brackets
•
Include the use of the following geometrical
terms:
point, line, plane, parallel,
perpendicular, right angles, acute,
obtuse and reflex angles,
complementary and
supplementary angles, base
angles, interior and exterior
angles, regular and irregular
polygons, diagonal and vertex
•
include tiles and tessellations
ALGEBRA
Pupils should be able to
1
Algebra expression and formulae
• use letters to represent numbers
• express basic arithmetic processes
algebraically
• substitute numbers for letters in
formulae and expressions
2.
Simple Algebraic Manipulation
• manipulate simple algebraic expressions
GEOMETRY
Pupils should be able to
1
Simple Plane Figures
• identify the following plane figures:
− triangles: isosceles triangles,
equilateral triangles, right-angled
triangles, acute-angled triangles,
obtuse-angled triangles and
scalene triangles
− special quadrilaterals: squares,
rectangles, parallelograms,
rhombuses, trapeziums and kites
− polygons: pentagons, hexagons,
octagons and decagons
33
LEVEL : SEC 1 (NORMAL ACADEMIC)
PART B
TOPICS/OUTCOMES
2
Simple solid figures
• identify the following simple solid figures:
cubes, cuboids, prisms, cylinders,
pyramids, cones and spheres
3
REMARKS
•
Include sketching these solid figures
•
Include angle properties related to their
diagonals
•
Use rulers, set-squares, protractors and
compasses
Angle properties
Angles formed with a common vertex
• calculate unknown angles involving:
− adjacent angles on a straight line
− vertically opposite angles
− angles at a point
Angles formed with parallel lines
• calculate unknown angles involving:
− corresponding angles
− alternate angles
− interior angles between parallel lines
Angle properties of a triangle
• calculate unknown angles involving:
− angle sum of a triangle
− base angles of an isosceles triangle
− angles of an equilateral triangle
− exterior angle of a triangle
Angle properties of squares, rectangles,
parallelograms and rhombuses
• calculate unknown angles using the
angle properties of
− squares
− rectangles
− parallelograms
− rhombuses
4.
Geometrical construction of simple
geometrical figures
• measure line segments and angles
34
LEVEL : SEC 1 (NORMAL ACADEMIC)
PART B
TOPICS/OUTCOMES
REMARKS
• draw line segments, angles, parallel
lines and perpendicular lines
•
Include cases where the following are
given:
i)
the distance from a point to a line
ii) the distance between two parallel lines
• construct angle bisectors and
perpendicular bisectors
•
Use protractors, set squares, compasses
and straight edges/rulers
• construct simple geometrical figures
from given data
•
Include the drawing of nets of cubes,
cuboids, prisms and pyramids
•
Include the use of tally marks
STATISTICS
Pupils should be able to
1
Handling data
• collect, classify and tabulate data
• read and interpret tables and statistical
diagrams
• construct a bar graph, pie chart,
pictogram, dot diagram, line graph, stem
and leaf diagram, and histogram with
equal intervals
PROBLEM SOLVING
Pupils should be able to
1
Problem solving heuristics
•
2
use appropriate heuristics to solve
problems
Practical uses of mathematics
• solve mathematical problems in
everyday life
35
LEVEL : SEC 2 (NORMAL ACADEMIC)
PART B
TOPICS/OUTCOMES
REMARKS
ARITHMETIC
Pupils should be able to
1
Simple financial transactions
• solve problems on personal and
household finance, and simple financial
transactions
•
Include
i)
earnings, simple interest, compound
interest (without use of formulae), hirepurchase, discount, commission, profit
and loss, money exchange and
taxation
ii) reverse problems such as finding the
cost price given the selling price and
the percentage profit
•
Include significant figures of numbers
written in standard form
Include estimation and approximation
• extract data from tables and charts to
solve problems
2
Arithmetic problems
• solve problems involving measures,
money, ratio, proportion, scale, rate,
speed and percentage
3
Standard form
• read and write numbers in standard form
n
A x 10 where n is a positive or negative
integer, and 1 ≤ A < 10
4
•
Number sequences
• formulate the algebraic expression for
the general term of a number sequence
•
Exclude direct application of arithmetic
progression and geometric progression
formulae
•
•
•
Include the use of the term “hemisphere”
Include nets where applicable
Exclude oblique pyramids and cones
MENSURATION
Pupils should be able to
1
Volume and surface area
• find the volumes and surface areas of
spheres, pyramids and cones
•
solve problems involving the volumes
and surface areas of spheres, pyramids
and cones
36
LEVEL : SEC 2 (NORMAL ACADEMIC)
PART B
TOPICS/OUTCOMES
2
REMARKS
Arc length and sector area
• express arc length as a fraction of
circumference and sector area as a
fraction of the area of a circle
• find arc length and sector area
•
Include finding the area of a segment of a
circle
•
Examples:
i)
(ax + b) (cx + d)
ii) (ax + by) (cx + dy)
(a, b, c and d are integers)
• manipulate simple algebraic fractions
•
Include only simple expressions such as
the following:
x
x−4
i)
+
3
2
2x
3(x − 5)
ii)
3
2
3a
5ab
) (
)
iii) (
4
3
3a
9a
÷
iv)
4
10
1
2
v)
+
x−2
x−3
• transform simple formulae
•
Exclude formulae involving square roots
• solve problems involving arc length and
sector area
ALGEBRA
1
Algebraic manipulation and formulae
• expand products of simple algebraic
expressions
• factorize algebraic expressions of the
form:
ax + ay
ax + bx + kay + kby
2 2
2 2
ax − by
2
2
a ± 2ab + b
2
ax + bx + c
37
LEVEL : SEC 2 (NORMAL ACADEMIC)
PART B
TOPICS/OUTCOMES
2
REMARKS
Solutions of equations
Simple linear equations
• solve simple linear equations
•
Include simple cases involving fractional
and decimal coefficients
Examples:
1
1
x+5 = xi)
2
3
ii)
2 + 0.6x = 2x
•
Include both the elimination and
substitution methods
•
Examples:
x
x−2
i)
+
= 3
3
4
3
ii)
= 6
x
3
iii)
= 6
x−2
Exclude fractional equations such as
1
1
2
+
= 2
2
x−2
x−3
• solve problems involving linear
equations
Simultaneous linear equations
• solve simultaneous linear equations in
two unknowns
• solve problems involving simultaneous
linear equations
Simple fractional equations
• solve fractional equations involving
numerical and linear algebraic
denominators
•
• solve problems involving simple
fractional equations
GRAPHS
Pupils should be able to
1
Graphs of linear and quadratic functions
• use cartesian coordinates in two
dimensions
• draw graphs of linear and quadratic
functions
•
Include finding the value of y from the
graph given the value of x, and vice versa
38
LEVEL : SEC 2 (NORMAL ACADEMIC)
PART B
TOPICS/OUTCOMES
REMARKS
• use graphical methods to solve
simultaneous linear equations
2
Graphs in practical situations
• interpret and use graphs in practical
situations
•
Include travel graphs (distance-time
graphs) and conversion graphs
• draw graphs using data from practical
situations
•
Include choice of appropriate scales
•
Exclude tests for similarity/congruency
between two triangles
• calculate
- the sum of interior angles of a
polygon
- the sum of exterior angles of a
polygon
•
Include regular polygons
• calculate unknown angles of a polygon
•
Include finding the number of sides of a
polygon
•
Include ‘line of symmetry’ and ‘centre and
order of rotational symmetry’
Exclude the use of the term ‘point
symmetry’
GEOMETRY
Pupils should be able to
1
Similar and congruent figures
• recognise similar and congruent figures
• find unknown sides/angles of
similar/congruent figures
2
3
Angle properties of a polygon
Scale Drawing
• read and make scale drawings
4
Symmetry
Line symmetry and rotational symmetry
• identify line and rotational symmetry of
plane figures
•
• use symmetrical properties of triangles,
quadrilaterals and regular polygons
•
Include properties of these figures directly
related to their symmetries
• use symmetrical properties of prisms,
cylinders, pyramids and cones
•
Include ‘plane of symmetry’ and ‘axis of
rotational symmetry’
39
LEVEL : SEC 2 (NORMAL ACADEMIC)
PART B
TOPICS/OUTCOMES
REMARKS
STATISTICS
Pupils should be able to
1
Averages
• find mean, median and mode
•
•
Distinguish between the purposes for which
mean, median and mode are used
Exclude grouped data
PROBLEM SOLVING
Pupils should be able to
1
Problem solving heuristics
•
2
use appropriate heuristics to solve
problems
Practical uses of mathematics
•
solve mathematical problems in
everyday life
40
NORMAL TECHNICAL
PART B
Content Chart - Secondary One & Two (Normal [Technical] Course)
Secondary One
1
2
Secondary Two
Whole numbers
1
Simple financial transactions
• the four operations
• ordering
• factors and multiples
2
Arithmetic problems
3
Directed numbers
• concept and notation
• ordering
• the four operations
4
Number sequences
Fractions and decimals
• concept and notation
• ordering
• the four operations
3
Approximation and estimation
• rounding off
• estimation
4
Use of a scientific calculator
5
Squares, square roots, cubes and
cube roots
6
Measures and money
7
Ratio, proportion, scale, rate and
speed
8
Percentage
1
Perimeter and area
•
•
•
•
•
•
1
1
Algebraic expressions and
formulae
• concept and notation
• substitution
• simplification
Volume and surface area
•
•
•
•
square
rectangle
triangle
parallelogram
trapezium
circle
1
cube
cuboid
prism
cylinder
Algebraic expressions and
formulae
• substitution
• manipulation
2
Solutions of simple linear
equations
41
NORMAL TECHNICAL
PART B
Content Chart - Secondary One & Two (Normal [Technical] Course)
Secondary One
1
Simple plane figures
2
Simple solid figures
3
AngIe properties
• angles formed with a common
vertex
• angles formed with parallel
lines
• angle properties of triangles
• angle properties of squares,
rectangles, parallelograms and
rhombuses
4
Construction of triangles
1
Handling data
•
•
•
•
•
•
•
Secondary Two
1
Linear graphs
2
Graphs in practical situations
1
Symmetry
• line symmetry
• rotational symmetry
2
Angle properties of polygons
3
Geometrical construction
• simple four-sided figures
• scale drawing
1
Averages
• mean
• mode
• median
table and chart
pictogram
dot diagram
bar graph
line graph
pie chart
histogram
1
Problem solving heuristics
1
Problem solving heuristics
2
Practical uses of Mathematics
2
Practical uses of Mathematics
3
Mathematics for leisure and
recreation*
3
Mathematics for leisure and
recreation*
*Non-examination topic
42
LEVEL : SEC 1 (NORMAL TECHNICAL)
PART B
TOPICS/OUTCOMES
REMARKS
ARITHMETIC
Pupils should be able to
1
Whole numbers
The four operations
• use the four operations for calculations
with whole numbers
•
•
•
•
Ordering
• order numbers
• use the following symbols correctly:
=, ≠, >, <, ≥, ≤
Include combined operations, i.e. correct
ordering of operations and the use of
brackets
Include mental calculation and estimation
Include awareness of the following
(i) commutative law
(ii) associative law
(iii) distributive law
Exclude tedious calculations when the use
of a calculator is not allowed
•
Include the use of the number line
•
Include the use of index notation and the
terms “prime factorization”, “ HCF” and
“LCM”
Include odd and even numbers
Factors and multiples
• use prime numbers, common factors
and common multiples
•
2
Fractions and decimals
Concept and notation
• use fractions and decimals
• convert fractions to decimals, and vice
versa
•
Include the use of facts such as
1
3
1
0.25 =
, 0.75 =
, 0.125 =
4
4
8
1
2
0.333 ≈
, 0.667 ≈
3
3
•
Exclude conversion of recurring decimals
to fractions
•
Include comparison of fractions with
decimals
Ordering
• compare and order fractions and
decimals
43
LEVEL : SEC 1 (NORMAL TECHNICAL)
PART B
TOPICS/OUTCOMES
REMARKS
The four operations
• use the four operations for calculations
with fractions and decimals
•
•
3
Include
combined operations
rounding off decimals to a specific
degree of accuracy
iii) mental calculation and estimation
Exclude tedious calculations when the use
of a calculator is not allowed
i)
ii)
Approximation and estimation
Rounding off
• round off numbers and measures to a
specified degree of accuracy
•
Include
i) decimal places and significant figures
ii) the use of the approximation sign “≈”
•
Include use of estimation to check the
reasonableness of answers
Include mental estimation
Estimation
• make estimates of numbers and
measures
4
•
5
•
Use of a scientific calculator
use the relevant keys of a scientific
calculator
•
•
Include appropriate checks of accuracy by
estimation, e.g. in evaluating 47 600 ÷ 85
as 560, pupils should recognize that
47 600 ÷ 85 ≈ 600 (not 6 or 6 000)
Pupils should be able to round off the
answer in the context of a given problem,
e.g. pupils should realize the absurdity of
giving the speed of a car to 5 decimal
places
•
Include the use of the square root sign ‘
Squares, square roots, cubes and cube
roots
• find squares, square roots, cubes and
cube roots of numbers
and the cube root sign ‘
•
•
3
‘
'
Include estimation
Include the use of the following keys of a
calculator:
x 2 , x , x y , x1 / y
6
Measures and money
Mass, length, time and money
• use common instruments to measure
quantity
•
•
Include measuring capacity/volume of a
liquid
Common instruments refer to rulers,
measuring cylinders, weighing scales and
stop-watches
44
LEVEL : SEC 1 (NORMAL TECHNICAL)
PART B
TOPICS/OUTCOMES
• solve problems involving the use of units
of mass, length, time and money
7
REMARKS
•
Include
(i) conversion of measures and currency
(ii) calculation of time in terms of 24-hour
and 12-hour clocks
(iii) reading of clocks, dials, timetables and
charts
• find the ratio of two or more quantities
•
Include expressing a ratio in its lowest
terms
• state the relationship between ratio and
fraction
•
Include rewriting x : y = a : b
a
as x = ( ) y
b
• use ratio, proportion and scale
•
Include direct proportion and inverse
proportion
Include map reading and estimating
distance from the map
Include the concept of reciprocals
Ratio, proportion and rate
Ratio and proportion
•
•
• solve problems involving ratio, proportion
and scale
Rate and speed
8
• use common measures of rate and
speed
•
Include conversion such as km/h to m/s and
vice versa
• solve problems involving rate and speed
•
Include calculation of average speed
• convert between
− percentage and fraction
− percentage and decimal
•
• calculate a given percentage of a
quantity
•
Include the use of facts such as
1
i)
25%
=
0.25
=
4
1
ii) 50%
=
0.5
=
2
3
iii) 75%
=
0.75
=
4
1
iv) 20%
=
0.2
=
5
4
v) 80%
=
0.8
=
5
Include reverse problems
Example:
40% of a class are boys. If there are 16
boys, find the number of pupils in the class.
PERCENTAGE
45
LEVEL : SEC 1 (NORMAL TECHNICAL)
PART B
TOPICS/OUTCOMES
REMARKS
• express one quantity as a percentage of
another
• calculate percentage increase/decrease
•
Include reverse problems such as finding
the original salary given the new salary and
the percentage increase
•
Include figures related to these shapes
•
•
Exclude expressions with brackets
Exclude expressions involving squares and
higher powers
•
•
•
Include collecting like terms
Exclude removing of brackets at this level
Exclude expressions involving squares and
higher powers
MENSURATION
Pupils should be able to
1
Perimeter and area
• find the perimeters and areas of
squares, rectangles, triangles,
parallelograms, trapeziums and circles
• solve problems involving the perimeters
and areas of squares, rectangles,
triangles, parallelograms, trapeziums
and circles
ALGEBRA
Pupils should be able to
1
Algebraic expressions and formulae
Concept and notation
• use letters to represent numbers
• express basic arithmetic processes
algebraically
Substitution
• substitute numbers for letters in
expressions and formulae
Simplification
• simplify simple algebraic expressions
46
LEVEL : SEC 1 (NORMAL TECHNICAL)
PART B
TOPICS/OUTCOMES
REMARKS
GEOMETRY
Pupils should be able to
1
Simple plane figures
• identify the following plane figures:
− triangles: isosceles triangles,
equilateral triangles, right-angled
triangles, acute-angled triangles,
obtuse-angled triangles and
scalene triangles
− special quadrilaterals: squares,
rectangles, parallelograms,
rhombuses, trapeziums and kites
− polygons: pentagons, hexagons,
octagons and decagons
2
Include the use of the following geometrical
terms:
point, line, plane, parallel,
perpendicular, right angle, acute,
obtuse and reflex angles,
complementary and
supplementary angles, base
angle, interior and exterior angles,
regular and irregular polygons,
diagonal and vertex
•
Include tiles and tessellations
•
Include sketching these solid figures
Simple solid figures
• identify the following simple solid figures:
cubes, cuboids, prisms, cylinders,
pyramids, cones and spheres
3
•
Angle properties
Angles formed with a common vertex
• calculate unknown angles involving:
− adjacent angles on a straight line
− vertically opposite angles
− angles at a point
Angles formed with parallel lines
• calculate unknown angles involving:
− corresponding angles
− alternate angles
− interior angles between parallel lines
Angle properties of a triangle
• calculate unknown angles involving:
− the angle sum of a triangle
− the base angles of an isosceles
triangle
− the angles of an equilateral triangle
− the exterior angle of a triangle
47
LEVEL : SEC 1 (NORMAL TECHNICAL)
PART B
TOPICS/OUTCOMES
REMARKS
Angle properties of squares, rectangles,
parallelograms and rhombuses
•
Include angle properties related to their
diagonals
• measure line segments and angles
•
Use rulers, set-squares, protractors and
compasses
• draw line segments, angles, parallel
lines and perpendicular lines
•
Include cases where the following are
given:
(i) the distance from a point to a line
(ii) the distance between two parallel lines
• construct angle bisectors and
perpendicular bisectors
•
Use protractors, set squares, compasses
and straight edges/rulers
•
Include the use of tally marks
• calculate unknown angles using the
angle properties of a
− square
− rectangle
− parallelogram
− rhombus
4
Construction of triangles
• construct triangles from given data
STATISTICS
Pupils should be able to
1
Handling data
• collect, classify and tabulate data
• read and interpret tables and statistical
diagrams
• construct a bar graph, pie chart,
pictogram, dot diagram, line graph, and
histogram with equal intervals
48
LEVEL : SEC 1 (NORMAL TECHNICAL)
PART B
TOPICS/OUTCOMES
REMARKS
PROBLEM SOLVING
Pupils should be able to
1
Problem solving heuristics
•
2
use appropriate heuristics to solve
problems
Practical uses of mathematics
• solve mathematical problems in
everyday life
3
Mathematics for leisure and recreation*
• appreciate solving non-routine problems
•
Non-routine problems in this topic refer to
simple puzzles and simple problems for
leisure and recreation
* Non-examination topic
49
LEVEL : SEC 2 (NORMAL TECHNICAL)
PART B
TOPICS/OUTCOMES
REMARKS
ARITHMETIC
Pupils should be able to
1
Simple financial transactions
•
solve problems on personal and
household finance, and simple financial
transactions
•
Include
i)
earnings, simple interest, compound
interest (without the use of formulae),
hire-purchase, discount, commission,
profit and loss, money exchange and
taxation
II) reverse problems such as finding the
cost price given the selling price and
the percentage profit
•
Include the use of the term "integers"
(positive, negative, zero)
•
Include ordering on the number line
•
Include combined operations and the use of
brackets
•
Recognise simple number patterns and
state the rules for the patterns
• extract data from tables and charts to
solve problems
2
Arithmetic problems
•
3
solve problems involving measures,
money, ratio, proportion, scale, rate,
speed and percentage
Directed numbers
Concept and notation
• use directed numbers
Ordering
• compare and order directed numbers
The four operations
• use the four operations for calculation
with directed numbers
4
Number sequences
• continue a given number sequence
50
LEVEL : SEC 2 (NORMAL TECHNICAL)
PART B
TOPICS/OUTCOMES
REMARKS
MENSURATION
Pupils should be able to
1
Volume and surface area
• find the volumes and surface areas of
cubes, cuboids, prisms and cylinders
•
•
•
• solve problems involving the volumes
and surface areas of cubes, cuboids,
prisms and cylinders
Include nets of these solids
Include finding the volumes of composite
solids
Exclude oblique prisms and oblique
cylinders
•
Include problems involving density
• simplify simple algebraic expressions
•
Include collecting like terms and removing
brackets
• expand products of simple algebraic
expressions
•
Examples:
(i) (ax + b) (cx + d)
(ii) (ax + by) (cx + dy)
(a, b, c and d are integers)
•
Exclude formulae involving square roots
ALGEBRA
Pupils should be able to
1
Algebraic expressions and formulae
Substitution
• substitute numbers for letters in
expressions and formulae
Manipulation
• factorize algebraic expressions of the
form:
ax + ay
ax + bx + kay + kby
2 2
2 2
ax -by
2
2
x ± 2xy + y
2
ax + bx + c
• transform simple formulae
51
LEVEL : SEC 2 (NORMAL TECHNICAL)
PART B
TOPICS/OUTCOMES
2
REMARKS
Solutions of equations
Solution of simple linear equations
• solve simple linear equations
•
Include cases involving fractional and
decimal coefficients
Examples:
1
1
x+5 = x(i)
2
3
(ii) 2 + 0.6x = 2x
• solve problems involving linear
equations
GRAPHS
Pupils should be able to
1
Graphs
Linear graphs
•
use cartesian coordinates in two
dimensions
• draw linear graphs
•
Include finding the value of y from the graph
given the value of x, and vice versa
• interpret and use graphs in practical
situations
•
Include travel graphs (distance-time graphs)
and conversion graphs
• draw graphs using data from practical
situations
•
Include choice of appropriate scales
• use graphical methods to solve
simultaneous linear equations
2
Graphs in practical situations
GEOMETRY
Pupils should be able to
1
Symmetry
Line symmetry and rotational symmetry
• identify line and rotational symmetry of
plane figures
•
•
Include ‘line of symmetry’ and ‘centre and
order of rotational symmetry’
Exclude the use of the term ‘point
symmetry’
52
LEVEL : SEC 2 (NORMAL TECHNICAL)
PART B
TOPICS/OUTCOMES
2
REMARKS
• use symmetrical properties of triangles,
quadrilaterals and regular polygons
•
Include properties of these figures directly
related to their symmetries
• use symmetrical properties of prisms,
cylinders, pyramids and cones
•
Include ‘plane of symmetry’ and ‘axis of
rotational symmetry'
• calculate
- the sum of interior angles of a
polygon
- the sum of exterior angles of a
polygon
•
Include regular polygons
• calculate the unknown angles of a
polygon
•
Include finding the number of sides of a
polygon
•
Distinguish between the purposes for which
mean, median and mode are used
Exclude grouped data
Construction
Simple four-sided figures
•
Construct simple four-sided figures
Scale drawing
• read and make scale drawings
3
Angle properties of polygon
STATISTICS
Pupils should be able to
1
Averages
• find the mean, median and mode
•
53
LEVEL : SEC 2 (NORMAL TECHNICAL)
PART B
TOPICS/OUTCOMES
REMARKS
PROBLEM SOLVING
Pupils should be able to
1
Problem solving heuristics
•
2
use appropriate heuristics to solve
problems
Practical uses of mathematics
• solve mathematical problems in
everyday life
3
Mathematics for leisure and recreation*
• appreciate solving non-routine problems
•
Non-routine problems in this topic refer to
simple puzzles and simple problems for
leisure and recreation
* Non-examination topic
54
APPENDIX
DEFINITION OF SUGGESTED THINKING SKILLS
• Classifying - using relevant attributes to sort, organise and group information
• Comparing - using common attributes to identify commonalities and
discrepancies across numerous sets of information
• Identifying Attributes & Components - recognising and articulating the parts
that together constitute a whole
• Sequencing - placing items in a hierarchical order according to a quantifiable
value
• Induction - drawing a general conclusion from clues gathered (from specific to
general)
• Deduction - inferring various specific situations or examples from given
generalisations (from general to specific)
• Generalising - using repeated, controlled and accurate observations to develop
a rule, principle or formula that explains a number of related situations
• Verifying - checking or confirming the truth of an idea, using specific standards
or criteria of evaluation
• Spatial Visualisation - visualising a situation or an object and mentally
manipulating various alternatives for solving a problem related to a situation or
object without the benefit of concrete manipulatives
55