# Secondary Syllabus

Document Sample

```					Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

Contents

1. Rationale                                                        1
2. Aims                                                             1
3. Assessment Objectives                                            2
4. Syllabus Content:                                                3
Form 1                                     4
Form 2                                    10
Form 3                                    19
Form 4                                    28
Form 5                                    38
5. Information and Communication Technology (ICT) in Mathematics   48
Form 1                                    50
Form 2                                    54
Form 3                                    58
Form 4                                    62
Form 5                                    66
6. Implementing the Mathematics Syllabus                           70
7. Annual Examinations                                             73

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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

THE SECONDARY SCHOOLS MATHEMATICS CURRICULUM

1.       RATIONALE                                                              2.       AIMS

Mathematics furnishes the prime means by which information can be               Teachers should aim to enable candidates to
organised, communicated and manipulated. It is also an ever-expanding
   understand and appreciate the place and purpose of Mathematics in
body of facts, skills, concepts and strategies used in the solution of a wide
range of problems. As a consequence, when implementing this syllabus,                    society and apply mathematical concepts to situations arising in
teachers of Mathematics should emphasize that:                                           their own lives;

   apply mathematical knowledge and understanding to solve
(i)     Mathematics is useful. It equips children with the necessary                     problems;
knowledge to help them understand and interact with the world
around them. Moreover, it forms the basis of science, technology,               think and communicate mathematically - precisely, logically and
architecture, engineering, commerce, industry and banking. It is                 creatively;
also increasingly being used in the medical sciences, biological
sciences, economics and geography. This pervasiveness makes                     develop a positive attitude to Mathematics, including confidence
Mathematics one of the most important subjects in the school                     and perseverance;
curriculum.
(Utilitarian Aspect of Mathematics Teaching and Learning)                       develop an ability to work independently and co-operatively when
doing Mathematics;
(ii)    Mathematics is an evolving body of knowledge that is characterised
   appreciate the interdependence of        the different branches of
by its order, precision, conciseness and logic. It should offer the
Mathematics;
children intellectual challenge, excitement, satisfaction and wonder.
(Aesthetic Aspect of Mathematics Teaching and Learning)                         acquire a secure foundation for the further study of Mathematics;

   use Mathematics across the curriculum;

   make efficient, creative and effective use of appropriate technology
in Mathematics.

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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

3.       ASSESSMENT OBJECTIVES
Assessment is based on                                                              develop a feel for numbers;
    the candidate‟s ability to recall, understand and apply mathematical      develop and use a range of methods of computation, namely,
knowledge in a wide context;                                               mental, pencil-and-paper, calculator and computer methods, and
apply these to a range of problems;
    the candidate‟s ability to understand and analyse a problem, select
an appropriate strategy, apply suitable knowledge and techniques          develop and use a range of methods for approximation of numbers
to solve it, verify and interpret the results;                             and apply these to a range of problems;
    the candidate‟s ability to understand, interpret and evaluate             develop and use a range of methods for estimation of measures and
mathematical ideas that are presented in oral, written and visual          apply these to a range of problems;
forms.
   explore a variety of situations which lead to the expression of
In particular, the candidate will be required to demonstrate the ability to          relationships;
 communicate, conjecture, reason and prove mathematically;
   consider how relationships between number operations underpin the
   understand the nature of numbers and make use of them;                      techniques for manipulating algebraic expressions;

   understand the nature of algebraic relationships and make use of           consider how algebra can be used to model real-life situations and to
them;                                                                       solve problems;

   understand the nature and properties of shape, space and measures          explore shape and space through drawing and practical work;
and make use of them;
   use computers to generate and transform graphic images and to
   understand the nature of statistics and process, represent and              solve problems;
interpret data;
   formulate questions that can be solved using statistical methods;
   understand the nature of probability and calculate the probabilities
of events.                                                                 undertake purposeful inquiries based on data analysis;

During the course candidates should be given opportunities to                       engage in practical and experimental work in order to appreciate
principles which govern random events;
   use calculators and computer software including spreadsheets,
LOGO, a dynamic geometry package and a computer algebra                    use investigative approaches;
system;
   look critically at some of the ways in which representations of data
   use computers as a source of large samples, as a tool for exploring         can be misleading and conclusions can be uncertain.
graphical representations, and as a means for simulating events;
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

4.      THE SYLLABUS CONTENT

The Secondary Schools syllabus is divided into two parts; core and
extension topics. Form 1 syllabus has been kept at core level.

This facilitates differentiated teaching and learning and the possibility of
implementing setting.

“differentiated instruction provides students with multiple
options for taking in information, understanding ideas, and
expressing what they learn. A differentiated classroom
provides multiple approaches and is proactive, qualitative,
student centred, organic, and a mix of whole-class, group, and
individual instruction.”

(Tomlinson, C. A., 1995).

Core Topics              is the content to be covered by all secondary school
students. This level allows the student to acquire the
basic mathematical skills at a slower pace.

Extension Topics         is extension content to be covered by all those
students who intend to sit for the Mathematics
Matsec Paper IIB by the end of Form 5.

Revised Syllabus         Reversed-out sections in the September 2003
syllabus are the revised or the additional topics to
the September 2000 syllabus.

Core topics printed in italics are topics which
have already been covered in the Extension
part in one of the previous years.

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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

Secondary Schools Mathematics Syllabus
Foundation and Intermediate Levels

FORM 1
Core
Unit 1: Number

1.1     Natural Numbers
1.1.1   Properties of whole numbers
 Distinguish between even and odd numbers.                                It is important that students gain fluency with whole numbers. This
means that students should develop efficient mental and/or pencil and
 Find the factors and multiples of numbers.                               paper methods for manipulating whole numbers based on a
1.1.2   Place value                                                                well-understood knowledge of number properties and relationships.
 Express numbers less than 100 000 in words and figures and vice-versa.
 Arrange a set of numbers in ascending/descending order.                  Fluency with whole numbers can also be obtained if students are given
1.1.3   Four rules                                                                 opportunities
 Add and subtract natural numbers.                                         to use ICT to explore properties of numbers and the relationships
 Know multiplication facts up to 10  10.                                     between them,
 to carry out investigative work related to number.
 Multiply natural numbers by a single digit.
 Divide numbers by a single digit with remainder.
1.1.4   Rounding
 Round numbers to the nearest 10, 100, or 1000.
 Carry out rough estimates to check the accuracy of calculations.

1.2     Decimal Numbers
1.2.1   Notion of decimal numbers                                                  The notion of decimal number is introduced through the pupils‟
 Read decimal numbers from scales.                                        concrete experience with measurement, money and weight. For
1.2.2   Place value (up to two decimal places)                                     example, the length of a pencil can be expressed in ten centimetres and
4 tenths of a centimetre, i.e. 10.4 cm.
 Arrange a set of numbers in ascending/descending order.
1.2.3   Four rules
 Add and subtract decimal numbers.
 Multiply and divide decimal numbers by single digit natural numbers.
1.2.4   Rounding
 Round numbers up to two decimal places.

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1.3     Fractions
1.3.1 Notion of fractions
 Understand that a fraction is a part of a whole.
1.3.2 Fraction of a quantity
 Find a fraction of a whole number.                                           Restricted to proper fractions with denominator 2, 3, 4, … , 10, 100.
1.3.3 Ordering
 Identify two equivalent fractions.
 Compare the magnitude of two fractions.
1.3.4 Simplification
 Reduce fractions to their lowest terms.
 Change mixed to improper factions and vice-versa.
 Express one quantity as a fraction of another.
1.3.5 Four rules
 Add and subtract up to two proper fractions with the same denominator.
1.3.6 Relationship between fractions and decimals
 Convert decimals to fractions.                                               Decimals are restricted to two decimal places.
 Convert fractions to decimals.                                               Restricted to proper fractions with denominators being factors of 100.

Unit 2: Applications

2.1     Number
2.1.1   Simple problems
 Solve simple problems involving whole numbers, fractions, decimals and
percentages.

2.2       Metric Measures
2.2.1 Units
 Convert large units of length and weight to smaller units and vice-versa.
2.2.2 Four rules
 Add, subtract, multiply and divide quantities of length and weight.
 Solve simple problems involving four rules.

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2.3     Personal and household finance
2.3.1   The Maltese currency system
 Convert Maltese Liri to cents and vice-versa.
 Add, subtract, multiply and divide money quantities.
 Solve simple problems involving money.

2.4       Time
2.4.1   Units
 Distinguish between different units of time: sec., min., hour, day, week,
month, leap year.
 Read and use a calendar.
2.4.2   The clock
 Write times using 12-hour and 24-hour clock.
 Convert 12-hour to 24-hour clock and vice-versa.

2.5       Percentages
2.5.1   Notion of percentage
 Understand that a percentage is a fraction having 100 as denominator.       Restricted to proper fractions with denominators being factors of 100.
 Change fractions into percentages.
2.5.2   Percentage of a quantity
 Express a quantity as a percentage of another.
 Find the percentage of a quantity.                                          Restricted to simple percentages: 25%, 50% and 75%.

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Unit 3: Shapes, Space and Measurement

3.1       Measurement
3.1.1   Perimeter of simple shapes                                                       Students should appreciate that the concept of measurement goes
 Find the perimeter of simple shapes by adding the side lengths, which are      beyond simple notions such as length and width. It is important that
all given.                                                                     students learn to select the appropriate unit for the attribute being
3.1.2   Area of simple shapes.                                                               measured,
 the notions of measurement are linked to other areas of the
 Find the area of simple shapes drawn on squared paper.
mathematics curriculum such as number, shape and data handling.
 Use units for area: mm2 and cm2.
3.1.3   Area of rectangles and squares                                                   Experience with squared paper, LOGO and Dynamic Geometry
 Find the area of rectangles and squares using the formula:                     Software can enhance students‟ appreciation of concepts of
Area = length  breadth                                                       measurement, angles and the properties of basic shapes.
 Find the length/breadth in simple cases.
3.1.4   Volume of cube and cuboid                                                        Children should also be given opportunities to do extensive
investigative work involving both pencil and paper methods and ICT.
 Understand notions of faces, edges and vertices.
 Draw and distinguish the nets of cubes and cuboids.
 Find the volume by counting cubes.
 Find the volume of cubes and cuboids using the formula:
Volume = length  breadth  height.
 Use units of volume: mm3 and cm3.

3.2       Angles
3.2.1   Angles as a measure of turn.                                                     LOGO provides an ideal environment for children to experience angle
 Understand that a complete revolution is divided into 360.                    as a measure of turn, in both clockwise and anti-clockwise direction.
 Define a right angle as a quarter turn.                                        Besides turtle geometry gives children an opportunity to manipulate
angles of different sizes.
 Identify right, acute, obtuse and reflex angles.
 Estimate the size of an angle.                                                 A Dynamic Geometry Software can help pupils discover the properties
 Main compass directions (NSEW) linked to a fraction of a revolution            of vertically opposite angle, angles at a point and angles on a straight
(¼, ½, ¾).                                                                     line.
3.2.2   The protractor
 Use a protractor to measure and draw angles up to 180.
3.2.3   Calculations
 Make calculations involving angles at a point, angles on a straight line and
vertically opposite angles.

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3.3       Lines
3.3.1   Straight lines
 Construct a straight line using ruler and compasses.
 Estimate the length of a straight line.
3.3.2   Perpendicular lines
 Draw perpendicular lines using set squares and protractor.
3.3.3   Parallel lines
 Identify parallel lines in geometric figures.
3.3.4   Line symmetry
 Draw lines of symmetry.
 Complete a figure to make it symmetrical about a given line.
 Identify equal lines and angles through folding.

3.4       Shape
3.4.1   Tessellations                                                                      By drawing simple shapes such as squares, rectangles and triangles
 Cover a given area with tessellating shapes.                                     using simple LOGO commands such as FD, BK, RT, LT and
 Identify tessellating shapes.                                                    REPEAT, pupils get the opportunity to reflect upon the properties of
these shapes.
3.4.2   Squares and rectangles
 Identify equal lines and angles through line symmetry.                           A Dynamic Geometry Software also provides a perfect environment by
 Construct squares and rectangles using ruler and compasses for lengths and       which pupils can discover the properties of triangles, squares and
protractor for angles.                                                          rectangles. By drawing these shapes and measuring their sides and
 Identify and draw diagonals.                                                     angles, children can discover for themselves the properties of these
3.4.3   Triangles                                                                          shapes.
 Distinguish between scalene, isosceles and equilateral triangles.
 Construct triangles given the length of the sides using ruler and compasses
only.
3.4.4   The circle
 Identify parts of a circle: centre, radius, diameter, chord and circumference.
 Form patterns made up from a number of circles.

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Unit 4: Algebra

4.1       Patterns and sequences
4.1.1   Geometric and number patterns                                                   Children should be given opportunities to use a spreadsheet to generate
 Recognize geometric and number patterns.                                      sequences of numbers, which they can describe both verbally and
 Extend patterns and sequences of numbers.                                     symbolically.
 Describe simple patterns verbally.

4.2       Functions
4.2.1   Function machines                                                               At this stage pupils should realise that in algebra letters are no more
 Describe functions verbally.                                                  than a device to represent numbers. They should be encouraged to
 Represent simple functions symbolically.                                      describe functions, both verbally and symbolically.
 Construct simple function machines.

4.3       Graphs
4.3.1   Cartesian coordinates
 Mark points in the first quadrant using an ordered pair of numbers known
as the x-coordinate and the y-coordinate.
 Draw lines and shapes given the coordinates of their endpoints or vertices.

Unit 5: Data Handling

5.1       Statistics
Pupils should be given opportunities
5.1.1   Collection of data
 Collect data using observations, surveys and experiments.                         questions by collecting data and presenting it in meaningful ways,
5.1.2   Frequency tables                                                                 to use spreadsheets to display and analyse the collected data.
 Compile a frequency table for ungrouped data.
 Interpret data in simple frequency tables.
5.1.3   Charts and diagrams
 Draw and interpret bar charts.

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FORM 2
Core                                                 Extension
Unit 1: Number

1.1     Integers
1.1.1 Properties of whole numbers                         1.1.1 Properties of whole numbers                      Students should develop efficient mental
 Find the common factors and common                       Find the least common multiple of up to three   and/or pencil and paper methods for
manipulating whole numbers based on a
multiples of up to three numbers.                         numbers.
well-understood knowledge of number
 Identify a prime number as a number having 2                                                              properties and relationships.
distinct factors only (i.e. 1 and the number
itself)                                                                                                  Fluency with whole numbers can also be
 Write numbers as a product of their prime                                                                 obtained if students are given
factors.                                                                                                 opportunities
1.1.2 Notion of integers                                                                                          to use ICT to explore properties of
 Understand negative integers through practical                                                                numbers and the relationships
examples.                                                                                                    between them,
 to carry out investigative work
 Represent integers on the number line.                                                                        related to number.
 Arrange a set of integers in ascending/
descending order.
1.1.3 Four rules                                          1.1.3 Four rules                                       Division by a two-digit number using
 Add and subtract integers:                               Multiply numbers by a two-digit number.         the repeated subtraction method.
e.g. +2 + 6, 2 + 6, +2  6, 2  6                   Divide numbers by a two-digit number using                      e.g.     786  29
 Understand the precedence of mathematical                  repeated subtraction.
786
operations.
( by 10)             290    10  29
 Multiply positive integers by multiples of 10.                                                                                   496
1.1.4 Rounding                                                                                                                         290    10  29
 Round positive integers to the nearest 10, 100                                                                                   206
or 1000.                                                                                                 (5 is half of ten)    145     5  29
 Carry out rough estimates to check the                                                                                            61
accuracy of calculations.                                                                                ( by 1)               29     1  29
32
 29     1  29
3

2
Ans: 27 rem 3

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1.2     Decimal Numbers
1.2.1 Four rules                                          1.2.1 Four rules
 Multiply decimal numbers by 10 and 100.                  Multiply decimal numbers by two digit            Multiplication/division by 100 can be
 Divide decimal numbers by 10 and 100.                     numbers, e.g. 3.47  45, 5.28  7.1              seen a repeated multiplication/division
1.2.2 Rounding                                                                                                     by 10. A calculator can be used to help
children establish a relationship between
 Round numbers to a given number of decimal
multiplying/dividing by 10 and the
places.                                                                                                    manner in which the number is
changing.

1.3     Fractions
1.3.1 Four rules                                          1.3.1 Four rules
 Add and subtract up to two proper fractions               Add and subtract two mixed numbers.
with different denominators.                               e.g. 6 2  2 1 , 2 1  3 .
3    4    5   4
1.3.2 Quantities
 Find a fraction of a quantity,
e.g. 3 of Lm15.64.
4
1.3.3 Fractions and decimals                                                                                       Pupils should be aware of the decimal
 Use the calculator to change fractions to                                                                   equivalence    of      fractions with
decimals.                                                                                                  denominators of 2, 4, 5 and 10.

1.4     Indices
1.4.1 Notion of indices                                   1.4.1 Notion of indices
 Evaluate ax where a and x are positive                   Estimate the square roots of positive integers
integers.                                                  less than 100, e.g. 50  7 .
e.g. 53 = 125.
 Find the cube roots of perfect cubes less than
 Use appropriate language: “to the power of”,                100 without a calculator.
“square”, “cube”.
 Find the positive square root of perfect
squares less than or equal to 100.

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1.5     The calculator
1.5.1 Basic functions                                    1.5.1 Basic functions
 Make efficient use of the +, , , , = buttons.  Use the x2 and  button on the calculator.           The calculator should not be used only
 Encourage students to use approximations to                                                             as a numerical computation tool, but as
check the reasonableness of calculator results.                                                         a tool which can help pupils to gain
insight into mathematical concepts. It is
 Choose a reasonable approximation, e.g.                                                                 also important that children learn to use
Expressing Lm6.6666667 as Lm6.67.                                                                       the calculator appropriately and
efficiently while at the same time
employing suitable procedures to check
the result displayed on their calculator.

Unit 2: Applications

2.1                                                               Number
2.1.1 Problems
 Apply notions of integers, fractions and
decimals to practical situations.

2.2     Metric measures
2.2.1 Units                                               2.2.3 Problems
 Understand the notions of ml and l.                       Solve problems involving length, weight and
 Convert litres to millilitres and vice-versa.               capacity.
 Use the relationship 1000 cm3 = 1l.
2.2.2 Four rules
 Add, subtract, multiply and divide quantities
of capacity.

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2.3     Time
2.3.1 The clock
 Find time intervals in hours and minutes.
 Understand and use timetables.                     Encourage the use of the time-line to
work out intervals of time.

2.4     Percentages
2.4.1 Percentages, fractions and decimals
 Express a percentage as a fraction and as a
decimal.
2.4.2 Quantities
 Find the percentage of a quantity,
e.g. 15% of 1.36.

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Unit 3: Shape, Space and Measurement

3.1     Measurement
3.1.1 Perimeter and area of a rectangle                   3.1.1 Perimeter and area of a rectangle                     Pupils should appreciate that the concept
of measurement goes beyond simple
 Use of the formula A  l  b to find A.                    Find the area of
notions such as length and width. It is
 Find length and/or breadth in simple cases,                   a border,                                       important that
given either the area or the perimeter.                      the net of a cube and cuboid                     pupils learn to select the appropriate
3.1.2 Area of triangle.                                           Find the perimeter of compound shapes.                  unit for the attribute being
 Understand that the area of a triangle is half                                                                      measured;
the area of a rectangle.                                                                                       notions of measurement are linked
bh                                                                                         to other areas of the mathematics
 Use the formula A         .                                                                                       curriculum such as number, shape
2
and data handling.
3.1.3 The circle
 children        should    be      given
 Understand that  is the ratio between the                                                                          opportunities to do extensive
circumference and diameter of a circle.                                                                            investigative work involving both
 Use the formulae C   d and C  2 r to find                                                                       pencil and paper methods, LOGO
the circumference of a circle.                                                                                     and Dynamic Gometry Software.
 Use the  button on the calculator to find the
circumference.
3.1.4 Volume of a cube and a cuboid                       3.1.4 Volume of a cube and a cuboid
 Use the formula V  l  b  h to find the                 Use the formula V  l  b  h to find the length,
volume of a cube and a cuboid.                            breadth or height.

3.2     Angles
3.2.1 Angles of a triangle                                3.2.2 Angles of a Quadrilateral
 Understand that the sum of the angles of a               Use the angle sum property of a triangle to         A Dynamic Geometry Software is an
triangle is 180.                                         find the sum of the angles of a quadrilateral.      ideal tool by which pupils can discover
the angle properties of parallel lines,
 Find unknown angles in scalene, isosceles and            Find unknown angles in quadrilaterals.              triangles and quadrilaterals.
equilateral triangles.

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3.3     Lines
3.3.1 Parallel lines
 Understand the relationship between
corresponding, alternate and interior angles.
 Find unknown angles involving these
properties.
3.3.2 Symmetry                                            3.3.3 Scale drawing
 Revision of line symmetry.                               Understand and interpret scale drawings.
 Identify shapes having rotational symmetry.              Draw simple scale drawings of familiar
 Determine the order of rotational symmetry.                objects.

3.4     Shape
3.4.1 Triangles
 Construct a triangle given the length of one
side and two angles.
3.4.2 Parallelograms
 Identify the symmetrical properties of the
parallelogram, the rhombus and the kite.
 Draw these shapes using squared paper.
3.4.3 Cubes and cuboids                                   3.4.4 The circle
 Construct cubes and cuboids from their nets.             Draw the inscribed regular hexagon and
 Identify properties relating to faces, vertices            equilateral triangle in a given circle using
and edges.                                                ruler and compasses only.

3.4.5 Prism and pyramid
 Identify properties relating to faces, vertices
and edges.
 Construct prisms and pyramids from their
nets.

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Unit 4: Algebra

4.1     Patterns and sequences
4.1.1 Geometric and number patterns.
 Revise geometric and number patterns.                                                                        It is important that students not only
recognize, extend and verbally describe
 Extend/complete patterns and sequences of
geometric and numerical patterns, but
numbers.                                                                                                     they should represent these patterns
 Describe patterns verbally and represent                                                                     symbolically and graphically.
them with tables and symbols.
Children should also be given
opportunities to use a spreadsheet to
generate sequences of numbers, which
they can describe both verbally and
represent symbolically.

4.2     Functions
4.2.1 Function machines
 Write expressions for given functions.
e.g. writing 2x 1 for „double a number and

4.3     Equations
4.3.1 Linear equations                                     4.3.3 Linear equations
 Form and solve simple linear equations                    Solve simple linear equations involving two        Pupils should be given opportunities to
involving one operation.                                   operations, e.g. 2x  3  7 . (Exclude examples   make use of a Computer Algebra
x              involving collecting literal terms, e.g.          Software (CAS) and a spreadsheet to
e.g. x  3  7 , x  8  1 , 4x  24 ,     5                                                             introduce and reinforce the related
3               2x  3  7  3x .)
algebraic concepts. Pupils should
4.3.2     Formulae                                                                                                   become     familiar   with    authoring
 Evaluate simple formulae with 2 positive                                                                   expressions and equations and work
inputs.                                                                                                    with them in a CAS and a spreadsheet.
e.g. If p  3q  r find the value of p when
q = 4 and r = 5.

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4.4     Graphs
4.4.1 Cartesian coordinates
 Mark points using an ordered pair of              Students should be given opportunities
numbers in any quadrant.                         to use a spreadsheet and/or a CAS to
explore algebraic relationships both
 Draw lines and shapes given the coordinates       symbolically and graphically. For
of their endpoints or vertices.                  example by representing relationships of
4.4.2 Straight line graphs                                the form y  mx graphically using a
 Noting the relationship between the               CAS, pupils can appreciate that by
coordinates, e.g. y  2 x , y  x  3 .        changing values of m the gradient of the
 Draw a linear graph from a simple relation        line will change accordingly.
involving, for example:
 currency conversions,
 variations of cost and weight.

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Unit 5: Data Handling

5.1     Statistics
5.1.1 Tables                                                  Pupils should be given opportunities
5.1.1 Tables
 Compile a frequency table for discrete                Construct and interpret information tables.               relevant issues, and answer these
grouped and ungrouped data.                                                                                     questions by collecting data and
5.1.2 Charts and diagrams                                                                                                presenting it in meaningful ways,
 Draw and interpret bar charts for both grouped                                                                to use spreadsheets to display and
and ungrouped data.                                                                                             analyse the collected data.
5.1.3 The mean
 Understand that the mean is a value which
represents a set of data.
 Compute the mean from a set of raw data.
5.1.4 The range                                                                                                      It is important that pupils not only learn
 Understand that the range describes the                                                                      how to compute the mean and range but
variation of a set of data.                                                                                  understand its scope and its drawbacks.
 Compute the range of a set of raw data.

5.2                                                               Probability
5.2.1 Notion of probability
 Describe future events as certain, impossible,     Pupils should be given opportunities to
likely, unlikely, etc.                             use a spreadsheet to simulate randomly
 Understand that the likelihood of a future event   occurring events such as the throw of a
can be represented by a number from 0              die and the toss of a coin.
(impossibility) to 1 (certainty).
5.2.2 Probability of an event
 Understand that the probability of a successful
event is given by:
number of successful events
total number of possible outcomes
 Find the probability from statistical data.

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FORM 3
Core                                      Extension
Unit 1: Number

1.1     Real numbers
1.1.1 Properties of whole numbers
Students should develop efficient mental
 Find the least common multiple of up to three
and/or pencil and paper methods for
numbers.                                                              manipulating whole numbers based on a
1.1.2 Four rules                                                               well-understood knowledge of number
 Revise the four rules applied to whole numbers                          properties and relationships.
and decimals.
 Add and subtract integers, including the                               Fluency with whole numbers can also be
subtraction of negative integers,                                     obtained if students are given
e.g. 7(4).                                                          opportunities
 to use ICT to explore properties of
 Multiply numbers by a two-digit number.
numbers and the relationships
 Divide numbers by a two-digit number using                                  between them,
repeated subtraction.                                                  to carry out investigative work
 Understand the precedence of mathematical                                   related to number.
operations.
1.1.3 Accuracy of results
 Round numbers to a given number of decimal
places.
 Correct numbers to a given number of
significant figures.
1.1.4 Fractions
 Add and subtract two mixed numbers.
e.g. 6 2  2 1 , 2 1  3 .
3     4     5 4

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1.2     Indices
1.2.1 Notion of indices
 Estimate the square roots of positive integers
less than 100.
e.g. 50  7 .
 Find the square root using a calculator.
 Find the cube roots of perfect cubes less than
100, without using a calculator.
 Find cube roots using a calculator.
1.2.2 Integral indices
 Evaluate positive integral indices.
e.g. 2.53  15 .625 .
1.2.3 Standard form                               1.2.4 Rules of indices
 Express numbers in standard form and vice-       Evaluate integral indices with and without
versa.                                             calculator, e.g. 4 3  64 , 5 2  1 / 25 , 6 0  1
 Understand and use the rules for addition,
subtraction and opening brackets.
e.g. 3 2  35  37 , 4 5  4 2  4 3 , (5 3 ) 2  5 6 .

1.3     The calculator
1.3.1 Basic commands                                      1.3.1 Basic commands
 Use the x2 and  button on the calculator.               Use the xy, and x buttons.
y                               The calculator should not be used only
as a numerical computation tool, but as
 Use the bracket buttons.                        a tool which can help pupils to gain
insight into mathematical concepts. It is
also important that children learn to use
the calculator appropriately and
efficiently while at the same time
employing suitable procedures to check
the result displayed on their calculator.

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Unit 2: Applications

2.1     Number
2.1.1 Problems
 Apply notions of integers, fractions and
decimals to practical situations.

2.2     Metric measures
2.2.1 Problems
 Solve problems involving length, weight and
capacity.

2.3     Percentages
2.3.1 Revision
2.3.2 Expressing a decimal as a percentage
2.3.3 Percentage increase and decrease
 Apply percentages to problems of profit and
loss, discount, appreciation/ depreciation (year
by year up to 2 years), commission, v.a.t., etc.

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2.4     Ratio and proportion
2.4.1 Ratio                                               2.4.2 Map ratios
 Use ratio notation to compare two or more                Understand and use simple map ratios.
quantities.                                       2.4.3 Direct proportion
 Write ratios in their simplest form.                     Solve problems using the unitary method.
 Divide a quantity in a given ratio.                      Link cost with other measures.
 Apply proportion to common rates: km/litre,
kg/m2, etc.

2.5                                                               Distance and speed
2.5.1 Speed
 Understand that speed is a comparison of
distance with time.
 Express speed in km/h and m/s.
 Calculate time/distance given the speed and
distance/time.

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Unit 3: Shape, Space and Measurement

3.1     Mensuration
3.1.1 Perimeter and area of a rectangle                   3.1.2 Area of triangle
 Find the area of                                                               bh                          Pupils should appreciate that the concept
 Use the formula A        to find the base or
 a border,                                                                     2                           of measurement goes beyond simple
 the net of a cube and cuboid                            height.                                           notions such as length and width. It is
 Find the perimeter of compound shapes.                                                                      important that
 pupils should learn to select the
appropriate unit for the attribute
3.1.3 Area of parallelogram                              3.1.3 Area of parallelogram                                     being measured;
 Find the area of a parallelogram by dividing it          Use the formula A  bh to find the base or          notions of measurement are linked
into two equal triangles.                                height.                                                 to other areas of the mathematics
 Use the formula A  bh .                                                                                          curriculum such as number, shape
3.1.4 Area of trapezium                                                                                                  and data handling.
 Find the area of a trapezium by dividing it into                                                              children        should    be      given
two triangles.                                                                                                   opportunities to do extensive
3.1.5 Circumference and area of a circle                 3.1.5 Circumference and area of a circle                        investigative work involving both
pencil and paper methods, LOGO
 Revise the use of the formulae C   d and               Use C  2 r and A   r 2 to find the radius of        and Dynamic Geometry Software.
C  2 r to find the circumference of a circle.         a circle.
 Use A   r to find the area of a circle.
2                                         Use C   d to find the diameter.
3.1.6 Composite shapes
 Find the area of composite shapes by dividing
them into simple shapes.
3.1.7 Volume of a cube and a cuboid                      3.1.7 Volume of a cube and a cuboid
 Use the formula V  l  b  h to find the length,        Use V  Ah to find V, A or h.
breadth or height.                                      Find the side of a cube given the volume.
 Solve problems using the relations:
1 litre = 1000 cm3, 1 m3 = 1000 litres

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3.2     Angles
3.2.1 Revision                                            3.2.5 Bearings
3.2.2 Parallel lines                                             Use the eight main compass directions.
 Use the properties of corresponding, alternate           Understand and use the notions of clockwise
and interior angles to show that lines are                and anticlockwise.
parallel.                                                Use three-figure bearings measured clockwise
3.2.3 Angles of a triangle                                        from the north to describe the position of one
 Notion of an exterior angle of a triangle.                 point from another.
 Understand that the exterior angle of a triangle          Find the distance and bearing from a scale
is equal to the two interior opposite angles.             drawing or a map.
 Solve simple problems involving the exterior
angle theorem.
 Use the angle sum property of a triangle to
find the sum of the angles of a quadrilateral.
 Find unknown angles in quadrilaterals.

3.3     Shapes
3.3.1 Scale drawing                                       3.3.5 Polygons
 Understand and interpret scale drawings.                Recognize by name polygons up to 8 sides          Students should be given the
 Draw simple scale drawings of familiar                    excluding the heptagon.                          opportunity to use LOGO and a
objects.                                                Understand the symmetry properties of a           Dynamic Geometry Software to
regular polygon.                                 investigate     the     properties   of
3.3.2 The circle
 Construct a regular polygon.                      quadrilaterals and regular polygons.
 Draw the inscribed regular hexagon and
equilateral triangle in a given circle using            Find the sum of the interior/exterior angles of   Pupils are expected to use a formula
ruler and compasses only.                                 a polygon.                                       such as (2n  4) right angles to find the
3.3.3 Quadrilaterals                                      3.3.6 Pythagoras theorem                                 sum of the interior angles of a polygon.
 Identify the properties of the square,                   Understand the theorem of Pythagoras
rectangle, parallelogram, rhombus, kite and               through drawing and measurement.
trapezium through line and rotational                   Use this theorem to find the side/s of a right-
symmetry.                                                 angled triangle.
3.3.4 Triangle
 Construct a triangle given two sides and the
included angle.

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3.4                                                               Transformations
3.4.1 Enlargements
 Use squared paper to enlarge simple shapes    The lengths and areas of enlarged
by a positive integral scale factor without   shapes can be investigated by means of
referring to the centre of enlargement.       LOGO, a spreadsheet or a Dynamic
 Find the scale factor given two similar       Geometry Software.
shapes.
 Find the lengths of an enlarged shape.

Unit 4: Algebra

4.1     Patterns and sequences
4.1.1 Generate a sequence from a given rule.
4.1.2 Establish a rule given a sequence (Students will
not be expected to write the nth term.).
4.1.3 Complete geometric patterns.

4.2     Algebraic expressions
4.2.1 Simplify algebraic expressions
Pupils should be given opportunities to
 Collect like terms, e.g. 3x  5  2x  8 ,                                                                make use of a Computer Algebra
4c  2d  c  7d                                                                                       Software (CAS) and a spreadsheet to
 Multiply terms, e.g. 6 9x , 3z  2z                                                                     introduce and reinforce the related
4.2.2 Multiplication                                      4.2.3 Factorization                                    algebraic concepts. Pupils should
 Multiply a single term over a bracket,                   Taking out a single term common factor,        become     familiar   with   authoring
expressions and equations and work
e.g. 2 p(3 p  7)                                          e.g. 6p214p                                  with them in a CAS and a spreadsheet
4.2.4 Evaluation of expressions                                                                                  environment.
 Evaluate simple expressions by substitution.

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4.3     Equations and Formulae
4.3.1 Linear equations                                         4.3.1 Linear equations
 Form and solve simple linear equations                         Construct and solve simple linear equations
involving two operations.                                       from given situations.
e.g. 2x  3  7 , 2x  3  7  3x , 2( x  5)  21 .            e.g. A soft drink costs x cents and an 4x
cents. The total cost is 80 cents. Write down
an equation and find the value of x.
4.3.2 Formulae                                                 4.3.2 Formulae
 Revise evaluation of formulae by substitution.                 Construct simple formulae.

4.4     Graphs
4.4.1 Linear graphs
 Draw a straight line graph by                                                                                    Scales for both axes are given and equal.
 compiling a table of values from its
Students should be given opportunities
equation,
to use a spreadsheet and/or a CAS to
 plotting the points from the table,                                                                            explore algebraic relationships both
 joining the points.                                                                                            symbolically and graphically.
 Use the graph to find the value of one
coordinate given the other.

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Unit 5: Data Handling

5.1     Statistics
5.1.1 Information tables
 Construct and interpret information tables.                                                              Pupils should be given opportunities
5.1.2 Mean, Range and Mode                                                                                        to formulate questions about
 Revise Mean and Range.
questions by collecting data and
 Understand and find the Mode from raw data.                                                                  presenting it in meaningful ways,
5.1.3 Frequency tables                                     5.1.3 Frequency tables                                 to use spreadsheets to display and
 Collect, classify and tabulate statistical data.          Compile frequency tables for continuous           analyse the collected data.
 Revise compilation of a frequency table for                grouped and ungrouped data with equal class
discrete grouped and ungrouped data.                      intervals.
5.1.4 Charts and diagrams
 Draw and interpret pie charts.

5.2     Probability
5.2.1 Notion of probability
 Describe future events as certain, impossible,                                                           Pupils should be given opportunities to
likely, unlikely, etc.                                                                                   use a spreadsheet to simulate randomly
 Understand that the likelihood of a future event                                                         occurring events such as the throw of a
can be represented by a number from 0                                                                    die and the toss of a coin.
(impossibility) to 1 (certainty).
5.2.2 Probability of an event                              5.2.2 Probability of an event
 Understand that the probability of a successful           Find the probability that an event does not
event is given by:                                         occur.
number of successful events
total number of possible outcomes
 Find the probability from statistical data.
5.2.3 Possibility spaces
 Find the probability of two independent
events by constructing a possibility space.

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FORM 4
Core                                                              Extension
Unit 1: Number

1.1     Real numbers
1.1.1 Four rules
 Revise the four rules applied to whole numbers                                                                          Students should develop efficient mental
and decimals.                                                                                                           and/or pencil and paper methods for
1.1.2 Accuracy of results                                                                                                      manipulating whole numbers based on a
 Round numbers to a given number of decimal                                                                              well-understood knowledge of number
places.                                                                                                                 properties and relationships.
 Correct numbers to a given number of
Fluency with whole numbers can also be
significant figures.                                                                                                    obtained if students are given
 Round quantities to a given unit of measure,                                                                            opportunities
e.g. to the nearest cent.                                                                                               to use ICT to explore properties of
1.1.3 Estimation                                                                                                                   numbers and the relationships
 Estimate a result by rounding numbers to one                                                                                between them,
significant figure.                                                                                                      to carry out investigative work
1.1.4 Fractions                                                    1.1.4 Fractions                                                 related to number.
 Revise addition and subtractions of two                           Multiply and divide two fractions.
fractions.                                                          e.g.   3
4
 11 , 4 2  2 5 , 21  7 , 8 5  3 10
8
3
2 8      4     4      3

1.2     Indices
1.2.1 Rules of indices
 Evaluate integral indices,
e.g. 4 3  64 , 5 2  1 / 25 , 6 0  1
 Understand and use the rules for
multiplication, division and brackets raised to
a power.
e.g. 3 2  35  37 , 4 5  4 2  4 3 , (5 3 ) 2  5 6 .
1.2.2 Standard form
 Revise expressing numbers in standard form

4
and vice-versa.

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1.3     The calculator
1.3.1 Basic commands
 Use the xy, and x buttons.
y                                 The calculator should not be used as a
numerical computation tool, but as a
 Use the bracket buttons.                          tool which can help students to acquire
skills and insight in solving problems.
Pupils should be encouraged to check
the accuracy of the calculator result by:
 making rough estimates;
 repeating calculations;
 reversing the order of an operation.

Unit 2: Applications

2.1     Ratio and proportion
2.1.1 Notion of ratio
 Revision.
2.1.2 Map ratios
 Understand and use simple map ratios.
 Find the map distance given the scale and the
actual distance.
 Find the actual distance given the scale and
the map distance.
 Determine the map scale from map and actual
distances.
2.1.3 Direct proportion
 Solve problems using the unitary method.
 Link cost with other measures.
 Apply proportion to common rates:
km/litre, kg/m2, etc.

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2.2     Percentages
2.2.1 Revision
2.2.2 Percentage change                                    2.2.3 Interest
 Determine percentage change as a one-step                 Calculate simple interest.
process.                                                                         PTR
 Use the formula I         to find I.
 Revise percentages to problems of profit and                                      100
loss, discount, appreciation/ depreciation (year
by year up to 2 years), commission, v.a.t., etc.

2.3     Distance, time and speed
2.3.1 Speed                                                2.3.1 Speed
 Understand that speed is a comparison of                   Change units of speed.
distance with time.                                        e.g. km/h to m/s and vice-versa.
 Express speed in km/h and m/s.                             Calculate the average speed.
 Calculate time/distance given the speed and                Understand and interpret travel graphs.
distance/time.

2.4     Household and Personal Finance
2.4.1 Exchange rates                                       2.4.2 Personal finance
 Convert one currency into the other.                      Solve problems relating to salary,
 Solve simple problems involving exchange                    commission, overtime, income tax, etc.
rates.                                              2.4.3 Household finance
 Solve problems relating to budgeting,
holidays, best buys, invoices, bills, etc.

2.5     Metric measures
2.5.1 Length, mass, capacity
 Understand, convert and use metric units of
mass, length, capacity, area and volume to
solve practical problems.

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Unit 3: Shape, Space and Measurement

3.1     Measurement
3.1.1 Area of triangle.
bh
 Use the formula A        to find the base or                                                           Pupils should appreciate that the concept
2                                                                                of measurement goes beyond simple
height.                                                                                                 notions such as length and width. It is
3.1.2 Area of parallelogram                                                                                     important that
 Use the formula A  bh to find the base or                                                                pupils should learn to select the
height.                                                                                                      appropriate unit for the attribute
3.1.3 Circumference and area of a circle                                                                             being measured;
 notions of measurement are linked
 Use C  2 r and A   r 2 to find the radius of                                                              to other areas of the mathematics
a circle.                                                                                                    curriculum such as number, shape
 Use C   d to find the diameter.                                                                             and data handling.
3.1.4 Composite shapes                                                                                           children        should    be      given
opportunities to do extensive
 Find the area of composite shapes by dividing
investigative work involving both
them into simple shapes.                                                                                     pencil and paper methods, LOGO
3.1.5 Volume of a cube and a cuboid                     3.1.6 Area of trapezium                                      and Dynamic Geometry Software.
 Use V  Ah to find V, A or h.                          Use the formula A  1 (a  b)h to find A or h.
2
 Find the side of a cube given the volume.
 Solve problems using the relations:
1 litre = 1000 cm3, 1 m3 = 1000 litres      3.1.7 Solids
 Find the volume of a prism using
Volume = area of cross-section  length
 Find the surface area of a cylinder.
 Find the volume of a cylinder using
Volume =  r 2 h
3.1.8 Composite solids
 Find the surface area and volume of
composite solids.

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3.2     Angles
3.2.1 Bearings                                            3.2.2 Angles of elevation and depression
 Use the eight main compass directions.                   Use scale drawings to solve problems
 Understand and use the notions of clockwise                involving angles of elevation/depression in
and anticlockwise.                                        simple practical situations.
 Use three-figure bearings measured clockwise
from the north to describe the position of one
point from another.
 Find the distance and bearing from a scale
drawing or a map.

3.3     Shapes
3.3.1 Polygons
 Recognize by name polygons up to 8 sides                                                                 LOGO      and     Dynamic       Geometry
excluding the heptagon.                                                                                 Software      give      pupils      ample
 Understand the symmetry properties of a                                                                  opportunities to explore the properties of
regular polygon.                                                                                        polygons.
Pupils are expected to use a formula
 Construct a regular polygon.
such as (2n  4) right angles to find the
 Find the sum of the interior/exterior angles of                                                          sum of the interior angles of a polygon.
a polygon.
3.3.2 Scale drawing
 Determine lengths from a scale drawing.
 Make a scale drawing from a sketch.
3.3.3 Pythagoras theorem                                  3.3.3 Pythagoras theorem
 Understand the theorem of Pythagoras                     Use the converse of Pythagoras theorem to
through drawing and measurement.                          show that a triangle is right-angled.
 Use this theorem to find the side/s of a right-
angled triangle.

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3.4     Transformations
3.4.1 Enlargements                                        3.4.1 Enlargements
 Use squared paper to enlarge simple shapes by            Draw enlargements on squared paper given the     Pupils should be given opportunities to
a positive integral scale factor without                  centre of enlargement and a positive             use LOGO, a Dynamic Geometry
referring to the centre of enlargement.                   (integral/fractional) scale factor.              Software and a spreadsheet to explore
 Find the scale factor given two similar shapes.          Find the centre of enlargement and/or scale      and investigate transformations.
 Find the lengths of an enlarged shape.                    factor.
 Understand that similar figures occur when one
is an enlargement of the other.
3.4.2 Reflections
 Recognise, describe and construct reflections
in y   x , y  c , x  c
3.4.3 Rotations
 Recognise, describe and construct rotations
restricted to multiples of 90.
3.4.4 Translations
 Recognise, describe and construct translations
described by a simple column vector.

3.5     Trigonometry
3.5.1 Trigonometric ratios                                3.5.1 Trigonometric ratios
 Understand the trigonometric relationships              Use trigonometric ratios to solve problems
in a right-angled triangle.                              involving angles of elevation/depression.
 Use the sine, cosine and tangent ratios to
solve a right-angled triangle.

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3.6                                                               The Circle
3.6.1 Symmetry properties of the circle
 Understand that                                  Pupils should be given an opportunity to
 Equal chords are equidistant from the           use a Dynamic Geometry Software to
centre and its converse.                      explore the symmetry properties of the
 The perpendicular bisector of a chord           circle.
passes through the centre and its converse.
 A tangent is perpendicular to the radius at
the point of contact.
 Two tangents from an external point are
equal.
 Solve problems involving the symmetry
properties of the circle and their converse.

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Unit 4: Algebra

4.1     Algebraic Expressions
4.1.1  Expressions
 Revise algebraic expressions.
4.1.2 Factorization
 Taking out a single term common factor,
e.g. 6p214p

4.2     Equations and Formulae
4.2.1 Linear equations
 Construct and solve simple linear equations                                                               Pupils should be given opportunities to
from given situations.                                                                                   make use of a Computer Algebra
e.g. A soft drink costs x cents and an ice-                                                             Software (CAS) and a spreadsheet to
cream costs 4x cents. The total cost is 80                                                         introduce and reinforce the related
cents. Write down an equation and find                                                             algebraic concepts. Pupils should
the value of x.                                                                                    become     familiar   with   authoring
expressions and equations and work
4.2.2 Formulae
with them in a CAS and a spreadsheet
 Construct simple formulae.                                                                                environment.
 Substitute numerical values to find one of the
variables.
 Rearrange the formula with the subject
appearing only once.

4.3                                                                Indices
4.3.1 Rules of Indices
 Rules of indices applied to simple algebraic
terms.
e.g. a2  a3 = a5, a7  a3 = a4, (a2)3 = a6.
 Understand that a0 = 1.
 Understand that a1 is the reciprocal of a1.

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4.4     Graphs
 Understand that the gradient is a measure of
slope.
 Calculate the gradient of a line from the
coordinates of two points on it.
 Understand that parallel lines have equal
4.4.2 The equation of a straight line
 Revision
 Recognize that equations of the form
y  mx  c represent straight lines.
 Find the gradient and the y-intercept of a
straight line given its equation.
 Find the equation of a straight line from its
graph.

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Unit 5: Data Handling

5.1     Statistics
5.1.1 Frequency tables
 Revision                                                                                                  Pupils should be given opportunities
5.1.2 Charts and diagrams                                                                                          to formulate questions about
 Revise bar charts and pie charts.
questions by collecting data and
 Draw and interpret histograms with equal                                                                      presenting it in meaningful ways,
intervals.                                                                                                to use spreadsheets to display and
5.1.3 Measures of central tendency                        5.1.3 Measures of central tendency                          analyse the collected data.
 Compute the median for a set of raw data.               Compute the mean, median and mode from an
ungrouped frequency table.                 It is important that pupils not only learn
 Understand the advantages/disadvantages of how to compute the mean, mode, and
these statistics.                          median but understand the aim of these
statistics.

5.2     Probability
5.2.1 Probability of an event
 Find the probability that an event does not                                                                Pupils should be given opportunities to
occur.                                                                                                    use a spreadsheet to simulate randomly
5.2.2 Possibility spaces                                                                                          occurring events such as the throw of a
 Find the probability of two independent events                                                             die and the toss of a coin.
by constructing a possibility space.

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FORM 5
Core                                      Extension
Unit 1: Number

1.1     Real numbers
1.1.1 Four rules                                                               Students should develop efficient mental
 Revise the four rules applied to integers and                           and/or pencil and paper methods for
decimals.                                                              manipulating whole numbers based on a
well-understood knowledge of number
1.1.2 Fractions
properties and relationships.
 Revision
 Multiply and divide two fractions.                                      Fluency with whole numbers can also be
e.g. 3  11 , 4 2  2 5 , 21  7 , 8 5  3 10 .
4
8
3
2 8      4     4      3                           obtained if students are given
opportunities
1.1.3 Accuracy of results
 to use ICT to explore properties of
 Revise rounding numbers to a given number of                                numbers and the relationships
decimal places.                                                             between them,
 Revise correcting numbers to a given number                              to carry out investigative work
of significant figures.                                                     related to number.
 Revise rounding quantities to a given unit of
measure, e.g. to the nearest cent.
1.1.4 Estimation
 Revise estimating a result by rounding
numbers to one significant figure.

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1.2     The calculator
1.2.1 Basic functions
 Use a calculator efficiently.                     The calculator should not be used only
 Use the memory facility for mixed operations      as a numerical computation tool, but as
with numbers.                                     a tool which can help pupils to gain
insight into mathematical concepts. It is
also important that children learn to use
the calculator appropriately and
efficiently while at the same time check
the accuracy of the calculator result by:
 making rough estimates;
 repeating calculations;
 reversing the order of an operation.

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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

Unit 2: Applications

2.1     Ratio and proportion
2.1.1 Direct and inverse proportion
 Revise direct proportion.
 Understand and solve problems involving
inverse proportion.

2.2     Percentages
2.2.1 Revision
2.2.2 Percentage change
 Carry out calculations involving reverse
percentages.
e.g. to find the cost price given the selling price
and percentage profit.
2.2.3 Interest                                               2.2.3 Interest
 Calculate the simple interest.                                                      PTR
   Use the formula I        to find P, T or R.
PTR                                                         100
 Use the formula I            to find I.
100

2.3     Household and personal finance
2.3.1 Personal finance
 Solve problems relating to salary, commission,
overtime, income tax, etc.
2.3.2 Household finance
 Solve problems relating to budgeting,
holidays, best buys, invoices, bills, etc.

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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

Unit 3: Shape, Space and Measurement

3.1     Mensuration
3.1.1 Revision
3.1.2 Area of trapezium.                                  3.1.3  Circumference and area of a circle            It is important that
 Use the formula A  1 (a  b)h to find A or h.
2                                     Find the length of arc and area of sector as  pupils should learn to select the
appropriate unit for the attribute
fractions of the circumference and area of a
being measured;
circle.                                       notions of measurement are linked
3.1.4 Solids                                              3.1.4 Solids                                               to other areas of the mathematics
 Find the volume of a prism using                          Find the surface area of a pyramid                 curriculum such as number, shape
Volume = area of cross-section  length                                                                       and data handling.
 Find the surface area of a cylinder.                                                                          children     should   be    given
 Find the volume of a cylinder using                                                                            opportunities to do extensive
investigative work involving both
Volume =  r 2 h .                                                                                            pencil and paper methods, LOGO
3.1.5 Composite solids                                                                                                and Dynamic Geometry Software.
 Find the surface area and volume of
composite solids.

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3.2     Angles, lines and shape
3.2.1 Angles and lines                                                                                                Pupils should be given opportunities to
 Revise the basic facts about lines and angles.                                                                investigate the properties of shapes by
3.2.2 Constructions                                                                                                   means of LOGO and a Dynamic
 Carry out constructions based on                                                                              Geometry Software.
measurement.
 Estimate, measure and draw lines and
angles.
 Construct parallel lines.
 Construct simple geometrical figures from
given data.
 Use straight edges and compasses to
construct
 angles of 60 and 90,
 the perpendicular bisector of a line
segment,
 the perpendicular from a point to a line,
 the bisector of an angle.
3.2.3 Shape
 Revise the properties of the triangle,
3.2.4 Solution of right-angled triangles
 Revise Pythagoras theorem.
 Use the converse of Pythagoras theorem to
show that a triangle is right-angled.
 Use scale drawings to solve problems
involving angles of elevation/depression in
simple practical situations.
3.2.5 Similar figures                                     3.2.5    Similar figures*
 Understand that similar figures occur when                 Understand and use the AAA and common
one is an enlargement of the other.                          ratio property of sides to prove similarity of
 Identify similar figures.                                    triangles.
 Find the scale factor and lengths from similar
figures using models and plans drawn to
scale.

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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

3.2.6 Congruent figures
 Understand and identify congruent shapes.
 Appreciate the uniqueness of triangles
satisfying SSS, SAS, ASA and RHS.
 Use SSS, SAS, ASA and RHS to prove the
congruence of triangles.

3.3     Transformations
3.3.1 Enlargements                                        3.3.1 Enlargements                                    Pupils should be given opportunities to
 Draw enlargements on squared paper given                  Understand that enlargements preserve        use LOGO, a Dynamic Geometry
the centre of enlargement and a positive                                                                Software and a spreadsheet to explore
angle and not necessarily length.
and investigate transformations.
(integral/fractional) scale factor.
 Find the centre of enlargement and/or scale
factor.
3.3.2 Reflections
 Recognise, describe and construct
reflections in y   x , y  c , x  c
3.3.3 Rotations
 Recognise, describe and construct rotations
restricted to multiples of 90
3.3.4 Translations                                        3.3.5 Congruency
 Recognise, describe and construct
   Understand the congruency of shapes under
translations described by a simple column
reflection, rotation and translation
vector.

3.4     Trigonometry
3.4.1    Trigonometric ratios
 Revision
 Use trigonometric ratios to solve problems
involving:
 angles of elevation/depression;
 bearings.

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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

3.5     The Circle
3.5.1 Symmetry properties of the circle              3.5.2 Angle properties of the circle
Pupils should be given an opportunity to
 Understand that                                     Understand that
use a Dynamic Geometry Software to
 Equal chords are equidistant from the               The angle subtended at the centre is          explore the symmetry properties of the
centre and its converse.                              twice the angle subtended at the            circle.
 The perpendicular bisector of a chord                  circumference.
passes through the centre and its                  The angle in a semicircle is a right
converse.                                             angle.
 A tangent is perpendicular to the radius            Angles in the same segment are equal.
at the point of contact.                           Opposite angles of a cyclic quadrilateral
 Two tangents from an external point are                are supplementary.
equal.                                             The exterior angle of a cyclic
 Solve problems involving the symmetry                     quadrilateral is equal to the interior
properties of the circle and their converse.              opposite angle.
 Solve problems involving the angle
properties of the circle (Reasons justifying
the use of these angle facts in simple words
are expected).

3.6                                                               Loci
3.6.1 Loci in two dimensions
 Determine the locus of points which
 are at a fixed distance from a given
point.
 are equidistant from two given points.
 Apply these loci to practical situations.

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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

Unit 4: Algebra

4.1     Patterns and Sequences
4.1.1 Revision.
4.1.2 Use expressions to describe the nth term of a
simple sequence.

4.2     Functions, Equations and Formulae
4.2.1 Functions                                           Pupils should be given opportunities to
 Use function notation.                            make use of a Computer Algebra
e.g. f(x) = 7x  2                               Software (CAS) and a spreadsheet to
introduce and reinforce the related
4.2.2 Linear equations                                    algebraic concepts. Pupils should
 Revision                                          become     familiar   with   authoring
4.2.3 Simultaneous equations                              expressions and equations and work
 Solve two simultaneous linear equations in        with them in a CAS and a spreadsheet
two unknowns algebraically by elimination        environment.
and by substitution.
 Solve two simultaneous linear equations in
two unknowns graphically.
4.2.4 Formulae
 Revision.
 Change the subject of a formula.
 Use and construct formulae on a

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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

4.3     Indices
4.3.1 Rules of Indices
 Rules of indices applied to simple algebraic
terms.
e.g. a2  a3 = a5, a7  a3 = a4, (a2)3 = a6.
 Understand that a0 = 1.
 Understand that a1 is the reciprocal of a1.

4.4     Graphs
4.4.1 Linear graphs                                       4.4.3 Applications                                     Students should be given opportunities
 Revision                                                 Interpret information presented in a variety   to use a spreadsheet and/or a CAS to
of linear and non-linear graphs.               explore algebraic relationships both
symbolically and graphically. For
e.g. distance-time graphs, velocity-time
example by representing relationships of
graphs, conversion graphs, graphs of length    the form y  mx graphically using a
CAS, pupils can appreciate that by
 Construct table of values for quadratic                                                                  changing values of m the gradient of the
functions.                                                                                               line will change accordingly.
 Plot and draw graphs of such functions by
making use of
 pencil and paper methods,
 a graphing package.
 Read off values from graphs.

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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

Unit 5: Data Handling

5.1     Statistics
5.1.1 Information tables
 Understand and interpret conversion tables,                                                        Pupils should be given opportunities
time-tables, currency tables, etc.                                                                to formulate questions about
 Construct and use tables.
questions by collecting data and
5.1.2 Frequency tables                                                                                         presenting it in meaningful ways,
 Revision                                                                                            to use spreadsheets to display and
 Compile frequency tables from continuous                                                               analyse the collected data.
grouped and ungrouped data with equal class
intervals.                                                                                        It is important that pupils not only learn
5.1.3 Charts and diagrams                                                                                  how to compute the mean, mode, and
 Revision.                                                                                          median but understand the scope of
these statistics
 Pictograms and misleading bar charts.
5.1.4 Measures of central tendency                   5.1.4 Measures of central tendency
 Compute the mean, median and mode from              Revise computation of the range of a set of
an ungrouped frequency table.                      raw data.
 Compute the range from an ungrouped
frequency table.
5.1.5 Measures of variation
 Revise the range of a set of raw data.

5.2     Probability
5.2.1 Revision                                                                                             Pupils should be given opportunities to
use a spreadsheet to simulate randomly
occurring events such as the throw of a
die and the toss of a coin.

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5.    INFORMATION AND COMMUNICATION TECHNOLOGY (ICT) IN MATHEMATICS
“Students learn by asking questions and by establishing connections . . . Students learn from everyday experiences through observation,
listening, investigation, experimentation and the comparison of what has been discovered with what is already known . . . Learning is an
organic process of invention and mental structuring and not a mechanical process of gathering information . . . Students need to change
their modes of knowing in an active manner. Teachers or learning systems must facilitate this process. A healthy education therefore
encourages . . . a pedagogy based on questioning . . . (and) on learning by doing . . . Students are not empty receptacles to be filled in . . .”
(National Minimum Curriculum – Principle 3: Stimulation of Analytical, Critical and Creative Thinking Skills – Ministry of Education -
December 1999)

“. . . The computer is still being given lip service. Apart from a change in mentality, one must seriously consider which physical and
organizational changes are required of a school which seriously regards the computer as an indispensable learning vehicle.” (National
Minimum Curriculum – Principle 13: The Importance of Learning Environments – Ministry of Education - December 1999)

“Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students’
learning.” (Principles and Standards for School Mathematics– NCTM April 2000)

ICT has the potential to make a significant contribution to pupils‟ learning in                 Work with realistic sets of data (e.g. carry out experiments using
mathematics by helping them to:                                                                  random samples generated through simulation)
    Practise and consolidate number skills (e.g. use a spreadsheet to                      Explore, describe and explain patterns and relationships in
revise number skills)                                                                   sequences and tables of numbers (e.g. use a formula on a
    Experiment with, make hypotheses from, and discuss or explain                           spreadsheet to generate the given values)
relationships in shape and space (e.g. use dynamic geometry                            Develop skills of mathematical modelling through the exploration,
software to investigate angle properties in a circle)                                   interpretation and explanation of data (e.g. choose an appropriate
    Develop logical thinking and modify assumptions and strategies                          graphical representation to display information from a set of data)
through immediate feedback (e.g. plan a series of instructions to
draw a required shape or carry out a set of manipulations in a
When using ICT teachers need to keep in mind that:
    Make connections within and across the different areas of
mathematics (e.g. relate a function and its graph)
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

   ICT is employed because it is the most effective way to achieve          It cannot be emphasized strongly enough that the use of ICT in mathematics
lessons can only be effective if it is used within the contexts of good
teaching and learning objectives, not simply for reward or               mathematics teaching. Teachers therefore will have to set clear objectives
motivation.                                                              for pupils‟ learning in mathematics and understand how the ICT used will
help to support their teaching and their pupils‟ learning.
   They should avoid the use of ICT for simple or routine tasks which
would be better accomplished by other means.                             The following Internet sites are suitable for both students and teachers.
   Pupils are expected to use ICT to answer valid questions appropriate     learning resources:
to the subject matter being taught.
MSW Logo       http://www.softronix.com/logo.html
Teachers should therefore plan                                                   Dr Geo         http://www..drgeo.seul.org
WinGeom        http://math.exeter.edu/rparris/wingeom.html
   the ways in which ICT will be used to meet the teaching and              Graphmatica    http://www.pair.com/ksoft/
learning objectives
http://www.anglia.co.uk/educationa/mathsnet/software.html
   the key questions to ask and opportunities for teacher intervention in   http://forum.swarthmore.edu/dynamic.html
order to stimulate and direct pupils‟ learning                           http://www.nrich.maths.org.uk/
   the organization and conduct of the lesson and how the lesson is to
be managed.                                                              http://www.mathsyear2000.org
http://www.curriculum.gov.mt (the Curriculum Unit’s website – click
http://www.schoolnet.gov.mt     (the Education Division Schools’
website – for the mathematics’ section
click on Secondary…Maths…Maths
Resources for the Maths Section)

The following is a scheme for the employment of ICT in each Form.
The scheme relates directly to the mathematical content for that particular form.

Secondary Schools Mathematics Syllabus - September 2003                                                                                                 49
Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

FORM 1

Throughout the year students are expected to use EXCEL (a spreadsheet), LOGO (a programming language for students), DERIVE (Computer Algebra Software)
and CABRI GÉOMÈTRE II (Dynamic Geometry Software). The list outlined below gives an indication of what students are expected to know and do by the end
of the year.

Students should be given opportunities to interact with pre-written spreadsheets to become familiar with the spreadsheet environment and to consolidate particular
mathematical concepts.

   Pre-written spreadsheet lessons which can be used in Form 1:                   100 Square.xls                     Bqija.xls
Decimal + & .xls                  Fixed Perimeter.xls
Multiply Tables.xls                Number Pyramid.xls
Place Value.xls                    Grocer.xls
Number Bonds.xls                   Rounding.xls
Writing Numbers in Words.xls

Gradually students should be given opportunities to learn, through Excel, how to

   Construct simple bar charts.
(Jeans Program, which is sponsored by Sedqa, provides a similar exercise for Form 1 students.)

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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

MICROWORLDS LOGO

Logo contributes a lot towards the mathematics curriculum. It is a highly enjoyable experience; it helps students develop mathematical concepts and skills; it
enhances problem solving skills by encouraging them to think both algorithmically and procedurally; it gives rise to new learning and teaching styles, such as
group work, investigative work and mathematical communication (with the computer and with class mates).

Students should be given opportunities to become familiar with the following Logo primitives.

PenUp           PU              ShowTurtle       ST             Right          RT              Home        HOME
PenDown         PD              Forward         FD              Left           LT              Repeat      REPEAT
HideTurtle      HT              Back            BK              Clear Graphics CG

OPTIONAL: They may also learn how to use SetColor SETC, SetBackGround SETBG, SetShape SETSH, Heading (to change direction in which the turtle
points), FILL to colour a closed shape, CLEAN.

Typical activities using LOGO include:
 Drawing simple shapes or pictures made up of a number of simple primitives;
e.g. letters of the alphabet, squares, rectangles, equilateral triangles, sheds, houses, . . .;
 Angle rotation at a point;
 Estimating perimeters of simple shapes, in turtle steps;
 Drawing simple shapes or pictures requiring repeated patterns, using the REPEAT command;
 Predicting outcomes on the screen given a list of primitives;
 Debugging a list of primitives to produce a required shape or picture.

Secondary Schools Mathematics Syllabus - September 2003                                                                                                    51
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

DERIVE (Computer Algebra Software)

Used appropriately, Derive can enhance the learning of algebra. At this stage it should not be used to bypass pencil and paper methods but rather to reinforce
those methods.

The students should know how to
 Use the Author command;
 Use F3 and F4 to copy highlighted expressions when using the Author command;
 Edit expressions to correct mistakes or modify an expression;
 Use the Algebra and the Plot windows.

A typical form 1 activity using Derive include:
 Use of Cartesian coordinates to plot points.
e.g. Points can be authored:  Author…Expression…[[3,2],[3,6],[7,6],[7,2],[3,2]]

Secondary Schools Mathematics Syllabus - September 2003                                                                                                    52
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

CABRI GÉOMÈTRE II (Dynamic Geometry Software)

Pupils should be given opportunities to use CABRI to explore, discover and establish geometrical results. The exploratory work on CABRI should be the basis of
further work in the classroom.

The students should be given tasks that would enable them to learn know how to use the basic tools:

POINTER         (Pointer)
POINTS          (Point, Point on Object, Intersection Point)
LINES           (Line, Segment, Ray, Triangle, Regular Polygon)
CURVES          (Circle)
CONSTRUCT       (Perpendicular Line, Parallel Line, Midpoint, Compass)
TRANSFORM       (Reflection) – this tool may be used for line (reflective) symmetry
MEASURE         (Distance and Length, Area, Angle)
DISPLAY         (Label, Mark Angle)
Optional:       Any of the tools from the DRAW toolbox.

The students are expected to become familiar with the use of these tools in a natural way, through investigative tasks set by the teacher. Typical tasks include
discovering:
 The angles‟ sum properties at a point.
 The angles‟ sum properties on a straight line.
 The construction of a triangle given the three sides.
 The angles‟ sum property for any triangle.
 Symmetry properties, if any, of triangles, squares and rectangles.

The students should be encouraged to discuss, communicate and, whenever possible, explain their findings and record these in writing, by making use of the
appropriate mathematical language.

Secondary Schools Mathematics Syllabus - September 2003                                                                                                      53
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

FORM 2
Throughout the year students are expected to use EXCEL (a spreadsheet), LOGO (a programming language for students), DERIVE (Computer Algebra Software)
and CABRI GÉOMÈTRE II (Dynamic Geometry Software). The list outlined below gives an indication of what students are expected to know and do by the end
of the year.

Students should be given opportunities to interact with pre-written spreadsheets to become familiar with the spreadsheet environment and to consolidate particular
mathematical concepts.

   Pre-written spreadsheet lessons which can be used in Form 2:                   24h time.xls              Coins.xls
Percentages.xls           Place Value.xls
Rounding.xls

Gradually they should be given opportunities to learn how to

   Carry out simple calculations involving the four rules;
   Generate “function machines” (e.g. double a number and subtract 3 . . . =A1*2 – 3);
   Enter numerical data into the spreadsheet and analyze this data using the built in functions AVERAGE, MAX, MIN and construct formula to calculate the
RANGE;

Secondary Schools Mathematics Syllabus - September 2003                                                                                                        54
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

MICROWORLDS LOGO

Logo contributes a lot towards the mathematics curriculum. It is a highly enjoyable experience; it helps students develop mathematical concepts and skills; it
enhances problem solving skills by encouraging them to think both algorithmically and procedurally; it gives rise to new learning and teaching styles, such as
group work, investigative work and mathematical communication (with the computer and with class mates).

Students should be given opportunities to

       Use the basic Logo primitives encountered in Form 1
       Write simple procedures incorporating the Logo primitives.
e.g.    TO SQUARE
REPEAT 4 [FD 60 RT 90]
END
       Edit, save and load procedures.
       Document a procedure.

Students should be given opportunities to become familiar with the following Logo primitives.

Wait                WAIT               Set colour          SETC                Set background SETBG                 Set shape    SETSH
Fill                FILL

Typical activities using LOGO include:

   Write a procedure that will draw a particular given shape (e.g. square, rectangle, equilateral triangle);
   Debug a procedure until it draws the required shape;
   Predict what a given procedure will draw;

Secondary Schools Mathematics Syllabus - September 2003                                                                                                    55
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

DERIVE (Computer Algebra Software)

Used appropriately, Derive can enhance the learning of algebra. At this stage it should not be used to bypass pencil and paper methods but rather to reinforce
those methods. For example, to solve simple equations the SOLVE command should not be used and the balance method should be employed and reinforced.

Students should be given opportunities to

   Use the Author, Simplify Basic, Approximate and Substitute For commands;
   Use F3 and F4 to copy highlighted expressions when using the Author command;
   Edit expressions to correct mistakes or modify an expression;
   Solving simple linear equations involving two operations;
   Construct a simple formula or equation from a word problem and use it to solve the problem;
   Author a set of ordered pairs of points and plot these points in the PLOT WINDOW;
   Know how to view the ALGEBRA WINDOW and the PLOT WINDOW simultaneously or individually;
   Know how to adjust scales to obtain the desired graphical picture.

OPTIONAL: Know how to change the Plot Background, the colour of the Plots; know how to use other options related to the Plot Window.

Secondary Schools Mathematics Syllabus - September 2003                                                                                                    56
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

CABRI GÉOMÈTRE II (Dynamic Geometry Software)

Pupils should be given opportunities to use CABRI to explore, discover and establish geometrical results. The exploratory work on CABRI should be the basis of
further work in the classroom.

The students should be given tasks that would enable them to consolidate the use of the basic tools encountered in Form 1 and use other new tools, namely:

LINES           (Polygon)
CONSTRUCT       (Measurement transfer)
MEASURE         (Calculate)
DISPLAY         (Animation)
DRAW            (Show/Hide Axes, New Axes, Define Grid)

The students may become acquainted with these tools in a natural way when teachers set tasks requiring their use. Typical tasks include:
 Construction of lines, angles, parallels and perpendiculars;
 Construction of the perpendicular to a line;
 Constructing an equilateral triangle inscribed in a regular hexagon;
 Compound constructions involving basic ones;
 Discovering the relationship between the circumference and diameter of a circle to establish ;
 Plotting points satisfying particular rules;
 The angle sum property of a triangle.

The students should be encouraged to discuss, communicate and, whenever possible, explain their findings and record these in writing, by making use of the
appropriate mathematical language.

Secondary Schools Mathematics Syllabus - September 2003                                                                                                      57
2
Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

FORM 3
Throughout the year students are expected to use EXCEL (a spreadsheet), LOGO (a programming language for students), DERIVE (Computer Algebra Software)
and CABRI GÉOMÈTRE II (Dynamic Geometry Software). The list outlined below gives an indication of what students are expected to know and do by the end
of the year.

Students should be given opportunities to interact with pre-written spreadsheets to become familiar with the spreadsheet environment and to consolidate particular
mathematical concepts.

Pre-written spreadsheet lessons which can be used in Form 3:                       Decimal expansion.xls                Ladder.xls
Percentages.xls                      Pie Chart.xls
Ratio.xls                            Round.xls
Variables and F(x) Machines.xls

Gradually they should be given opportunities to learn how to

   Consolidate previously acquired knowledge and skills associated with the spreadsheet;
   Generate more complicated “function machines” (e.g. square a number, subtract 3 and divide the result by 5 . . . =(A1^2 – 3)/5);
   Carry out calculations involving percentages on a spreadsheet, for example, to model “what happens if . . .” situations;
   Generate a set of ordered pairs linearly related and construct a scatter and line graph to display this linear relationship.
   Construct a simple pie chart.

Secondary Schools Mathematics Syllabus - September 2003                                                                                                        58
3
Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

MICROWORLDS LOGO

Logo contributes a lot towards the mathematics curriculum. It is a highly enjoyable experience; it helps students develop mathematical concepts and skills; it
enhances problem solving skills by encouraging them to think both algorithmically and procedurally; it gives rise to new learning and teaching styles, such as
group work, investigative work and mathematical communication (with the computer and with class mates).

Students should be given opportunities to

       Revise and use the basic Logo primitives encountered in Form 1 and Form 2.
       Write simple procedures incorporating the Logo primitives.
e.g.       TO TRIANGLE
REPEAT 3 [FD 100 RT 120]
END
       Write procedures that call other procedures.
       Discover the “total trip” theorem (the sum of the exterior angles of a polygon is 360).
       Edit, save and load procedures.
       Document a procedure.

Typical activities using LOGO include:

   Write a procedure that will draw a particular given shape (e.g. square, rectangle, equilateral triangle, regular polygon, circle)
   Write a procedure that will draw a parallelogram (or a rhombus) given the sides and the included angle.
   Write procedures that draw composite shapes;
   Debug a procedure until it draws the required shape;
   Predict what a given procedure will draw.

Secondary Schools Mathematics Syllabus - September 2003                                                                                                    59
3
Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

DERIVE (Computer Algebra Software)

Used appropriately, Derive can enhance the learning of algebra. At this stage it should not be used to bypass pencil and paper methods but rather to reinforce
those methods. For example, to solve simple equations the SOLVE command should not be used and the balance method should be employed and reinforced.

Students should be given opportunities to

   Familiarize themselves with those facilities provided by the software they used in the previous two years.
   Use the EXPAND and SUBSTITUTE FOR VARIABLES features.
   Use the FACTORIZE feature to a) appreciate that factorization is the reverse process of expansion and
b) to develop paper and pencil strategies that will enable them to factorize simple binomials.
   Plot linear graphs, observe their characteristic properties and deduce information from them.
   Use the software as a problem solving tool.

OPTIONAL: Know how to change the Plot Background, the colour of the Plots; know how to use other options available for the Plot Window.

Typical form 3 activities using Derive include:
 Collecting like terms to check pencil and paper results;
 Multiplying terms to check pencil and paper results;
 Solving simple linear equations involving expanding brackets;
 Substitution to find a variable in a formula;
 Use of Cartesian coordinates to plot and join points.
 Finding the value of one coordinate given the other.

Secondary Schools Mathematics Syllabus - September 2003                                                                                                    60
3
Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

CABRI GÉOMÈTRE II (Dynamic Geometry Software)

Pupils should be given opportunities to use CABRI to explore, discover and establish geometrical results. The exploratory work on CABRI should be the basis of
further work in the classroom.

The students should be given tasks that would enable them to consolidate the use of the basic tools encountered in Form 1 and 2 and use other new features,
namely:

POINTER               (Dilate)
CHECK PROPERTY        (Collinear, Parallel, Perpendicular)
TRANSFORM             (Dilation)
DRAW                  (Thick, Dotted, Modify Appearance)

The students may become acquainted with these tools in a natural way when teachers set tasks requiring their use. Typical tasks include:
 Exterior angle of a triangle property;
 Establishing the sum of the interior and exterior angles of a polygon;
 Enlargements with a positive scale factor;
 Discovering Pythagoras‟ Theorem;
 Using the software as a problem solving tool.

The students should be given opportunities to discuss, communicate and, whenever possible, explain their findings and record these in writing, by making use of
the appropriate mathematical language.

Secondary Schools Mathematics Syllabus - September 2003                                                                                                     61
3
Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

FORM 4
Throughout the year students are expected to use EXCEL (a spreadsheet), LOGO (a programming language for students), DERIVE (Computer Algebra Software)
and CABRI GÉOMÈTRE II (Dynamic Geometry Software). The list outlined below gives an indication of what students are expected to know and do by the end
of the year.

Students should be given opportunities to use the spreadsheet as a problem solving tool. They should be given tasks that would help them

   Consolidate previously acquired knowledge and skills associated with the spreadsheet.
   Use the spreadsheet functions RAND(), INT, and IF to model and simulate real life situations.
   Carry out purposeful enquiries to answer “what happens if . . .” type questions.
   Carry out calculations involving percentages in real life contexts (e.g. related to tax, insurance, commission, simple interest, exchange rates, etc.).
   Use the GOAL SEEK command to investigate feasibility of a real life situations.
   Use INSERT NAME CREATE to create a variable, as a substitute to the usual cell referencing.
   Analyse data, draw histograms and interpret the results (e.g. examination results).
   Use the spreadsheet as a problem solving tool.

Rounding.xls               Straight line.xls
Translations

Secondary Schools Mathematics Syllabus - September 2003                                                                                                       62
4
Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

MICROWORLDS LOGO

Logo contributes a lot towards the mathematics curriculum. It is a highly enjoyable experience; it helps students develop mathematical concepts and skills; it
enhances problem solving skills by encouraging them to think both algorithmically and procedurally; it gives rise to new learning and teaching styles, such as
group work, investigative work and mathematical communication (with the computer and with class mates).

Students should be given opportunities to

       Revise and use the basic Logo primitives encountered in Form 1, Form 2 and Form 3.
       Write procedures that make use of variable inputs

Typical activities involving LOGO include:

       Write a procedure that will accept one or more variables to draw, for example, regular polygons having a different number of sides and varying side
lengths.
       Write procedures that perform simple transformations – reflection, enlargement, rotation and translation.
       Projects involving variables.

Secondary Schools Mathematics Syllabus - September 2003                                                                                                    63
4
Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

DERIVE (Computer Algebra Software)

Used appropriately, Derive can enhance the learning of algebra. At this stage it should not be used to bypass pencil and paper methods but rather to reinforce
those methods.

Students should be given opportunities to
 Familiarize themselves with the software features they were exposed to during the previous three years.
 Use the correct procedures to change the subject of formulae.
 Use the FACTORISE feature to a) appreciate that factorization is the reverse process of expansion and b) to develop paper and pencil strategies that will enable
them to factorise binomials and trinomials.
 Use the SOLVE feature to solve quadratic equations and appreciate that the algebraic solutions and the graphical solutions of a quadratic equation are identical.
 Plot linear and quadratic graphs, observe their characteristic properties and deduce information from them.
 Use the software as a problem solving tool.

OPTIONAL: Know how to change the Plot Background, the colour of the Plots; know how to use other options available for the Plot Window.

Secondary Schools Mathematics Syllabus - September 2003                                                                                                         64
4
Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

CABRI GÉOMÈTRE II (Dynamic Geometry Software)

Students should be given opportunities to use CABRI to explore, discover and establish geometrical results. The exploratory work on CABRI should be the basis
of further work in the classroom.

The students should be given tasks that would enable them to consolidate the use of the basic tools encountered in Form 1, 2 and 3 and use other new features,
namely:

POINTER   (Rotate, Dilate, Rotate and Dilate)
TRANSFORM (Reflection, Rotation, Translation)
MEASURE   (Calculate, Tabulate)

The students may become acquainted with these tools in a natural way when teachers set tasks requiring their use. Typical tasks include:
 Appreciating the properties of congruent triangles;
 Exploring properties of similar triangles;
 Discovering the symmetry properties of the circle;
 Discovering and establishing the angle properties of the circle;
 Using scale drawing to solve simple problems in 2-D;
 Transformation activities;
 Using the software as a problem solving tool.

The students should be given opportunities to discuss, communicate and, whenever possible, explain their findings and record these in writing, by making use of
the appropriate mathematical language.

Secondary Schools Mathematics Syllabus - September 2003                                                                                                     65
4
Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

FORM 5
Throughout the year students are expected to use EXCEL (a spreadsheet), LOGO (a programming language for students), DERIVE (Computer Algebra Software)
and CABRI GÉOMÈTRE II (Dynamic Geometry Software). The list outlined below gives an indication of what students are expected to know and do by the end
of the year.

Students should be given opportunities to use the spreadsheet as a problem solving tool. They should be given tasks that would help them

   Consolidate previously acquired knowledge and skills associated with the spreadsheet;
   Model and simulate real life situations;
   Carry out purposeful enquiries to answer “what happens if . . .” type questions (through use of logic gates IF, AND and OR);
   Construct travel graphs and deduce information from them;
   Analyse data and draw histograms, cumulative frequency curves and box-plots;
   Use the spreadsheet for modeling real-life situations and to solve problems.

By form 5 students should be able to carry out investigational work by writing down observations, recording and tabulating results, making predictions and carry
out tests on them (conjectures) and finally arrive at a generalised conclusion.

Secondary Schools Mathematics Syllabus - September 2003                                                                                                      66
5
Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

MICROWORLDS LOGO

Logo contributes a lot towards the mathematics curriculum. It is a highly enjoyable experience; it helps students develop mathematical concepts and skills; it
enhances problem solving skills by encouraging them to think both algorithmically and procedurally; it gives rise to new learning and teaching styles, such as
group work, investigative work and mathematical communication (with the computer and with class mates).

Students should be given opportunities to

       Revise and use the basic Logo primitives encountered in Form 1, 2, 3 and 4;
       Write procedures that make use of variable inputs.

Typical activities involving LOGO include projects involving variables.

Secondary Schools Mathematics Syllabus - September 2003                                                                                                    67
5
Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

DERIVE (Computer Algebra Software)

Used appropriately, Derive can enhance the learning of algebra. At this stage it should not be used to bypass pencil and paper methods but rather to reinforce
those methods.

Students should be given opportunities to
 Familiarize themselves with the software features they were exposed to during the previous four years.
 Carry out appropriate manipulations on expressions to change the subject of the formula.
 Plot linear and quadratic functions.
 Solve simultaneously linear equations graphically, using the Plot Window, and interpret the solutions.
 Use the software as a general problem solving tool.

Secondary Schools Mathematics Syllabus - September 2003                                                                                                    68
5
Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

CABRI GÉOMÈTRE II (Dynamic Geometry Software)

Students should be given opportunities to use CABRI to explore, discover and establish geometrical results. The exploratory work on CABRI should be the basis
of further work in the classroom.

The students should be given tasks that would enable them to consolidate the use of the basic tools encountered in Form 1, 2 , 3 and 4 and use other new features,
namely:

CONSTRUCT          (Measurement Transfer, Locus)
CHECK PROPERTY (Collinear, Parallel, Perpendicular, Equidistant, Member)
Optional - DISPLAY (Comments, Numerical Edit, Fix/Free, Trace On/Off, Animation, Multiple Animation)

The students may become acquainted with these tools in a natural way when teachers set tasks requiring their use.

 Constructing loci in two dimensions;
 Discovering symmetry properties;
 Transformation activities;
 Using the software as a general problem solving tool.

The students should be given opportunities to discuss, communicate and, whenever possible, explain their findings and record these in writing, by making use of
the appropriate mathematical language.

Secondary Schools Mathematics Syllabus - September 2003                                                                                                        69
5
Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

6.    IMPLEMENTING THE MATHEMATICS SYLLABUS
The teaching and learning process must reflect that the following               b) COMMUNICATION
considerations are being taken into account:
Communication is a necessary component for learning, doing and
understanding mathematics. Communication in mathematics means that
a) TEACHING AND LEARNING MATHEMATICS                                            one is able not only to use its vocabulary, notation and structure to express
ideas and relationships but also to think and reason mathematically. In
The quality of mathematics learning depends on the quality of the teaching.     fact, communication is considered the means by which teachers and
What students learn is a reflection of the learning experiences that teachers   students can share the processes of learning, doing and understanding
provide. On the other hand, teaching mathematics well is a complex task         mathematics. Students should express their thinking and problem solving
and no simple recipes for helping all students to learn are available.          processes in both written and oral formats, allowing code-switching when
Nevertheless effective mathematics teaching requires a serious commitment       necessary. The clarity and completeness of students‟ communication can
to the development of students‟ understanding of mathematics. Effective         indicate how well they understand the related mathematical concepts.
teachers know how to ask exploratory questions to reveal what the students      Because teaching is communicating, teachers play a central role in
already know and plan and design appropriate tasks that build on that           fostering students‟ mathematical communication and understanding.
knowledge. They reflect on their practice and engage themselves in              Teachers should therefore provide students with opportunities to discuss
continuous self-improvement.                                                    mathematics, particularly during activities which involve exploration,
conjecturing, analysis and application of mathematical ideas. This
Teachers must also ensure that students learn mathematics with                  approach gives way to the most common pedagogy prevalent in most
understanding in order to be able to apply procedures, concepts and             classes, namely that of “teaching by telling” and reinforces the alternative
processes effectively. Learning with understanding makes subsequent             approach, that of “constructivist teaching”. In constructivist teaching
learning easier. Mathematics makes more sense and is easier to remember         students construct new knowledge by connecting it with experience.
and to apply when students connect new knowledge to existing knowledge          Students should not be looked upon as empty vessels waiting to receive
in meaningful ways. Besides, learning with understanding creates                information but rather have a certain amount of stored knowledge which
autonomous learners who can take control of their learning, become              may be accessed and updated. Research has shown that students learn and
confident in their ability to tackle difficult problems, seek alternative       retain more information when they are able to articulate what they know to
solution methods and learn to persevere. Such learners develop a feeling of     others. Of more benefit is social constructivism which recognises that
accomplishment and eventually a willingness to pursue further engagement        students are part of the learning communities in which language and
with mathematics.                                                               meaning can be shared among peers and teachers. Social constructivism
therefore provides a framework in which teachers can facilitate student
communication and in which learning is both corporate and individual.

Secondary Schools Mathematics Syllabus - September 2003                                                                                                     70
Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

The ability to communicate mathematical ideas can be fostered if:                c) PROBLEM SOLVING
   Understanding of mathematics is developed through reflection and by
Learning to solve problems is one of the principal reasons for studying
organization and communication of ideas.
Mathematics. Problem solving therefore is the heart of the mathematics
curriculum. Consequently students should be capable of applying previously
   Understandings and relationships between and among mathematical              acquired knowledge to new and unfamiliar situations, to solve non-routine
concepts, procedures and symbols are communicated in writing and             problems, to pose questions, to analyse situations, translate results and apply
speaking. This must be done at each stage of conceptual development,         trial-and-error methods. Students should see alternative solutions to
whether concrete, pictorial or abstract, and in every area of mathematics.   problems and be aware that a particular problem may have more than one
solution. Many different activities, such as gathering data, exploring
   Mathematical situations are represented or described in a variety of ways    patterns, making and testing conjectures, and justifying conclusions through
(e.g. verbal, concrete, pictorial, graphical, algebraic).                    logical arguments, are necessary to develop the students‟ mathematical
reasoning and ability to communicate about mathematics.
   Positions on mathematical processes and solutions are defended through
sound argument.                                                              Problem Solving also develops critical thinking skills, particularly when:

   The need for mathematical symbolism is demonstrated.                            Checks for reasonableness and completeness of results form an integral
part of the problem solving process. Incorrect solutions are analysed to
identify common errors in the problem-solving process.
   The ability to read mathematics is emphasised.
   Multi-step solutions and non-routine problems are posed on a regular
   The ability to write mathematical problems from real-world situations is
basis.
emphasised.
   Activities that require collecting, organising, manipulating data and
   Proper and precise mathematical vocabulary and notation is stressed.
drawing inferences from that data are provided.
   Communication skills are developed in small groups working together,
   Group problem solving is encouraged so that the students will be able to
through listening, exploring, questioning, discussing and summarizing.
share responsibility for the product of the activity after having an
opportunity to discuss the results.

   Activities are structured so that several strategies or techniques are
available for use in the solution process.

   Inter-disciplinary projects are encouraged.

   Strategies such as top-down analysis and stepwise refinement are used to
analyse and solve complex problems.

Secondary Schools Mathematics Syllabus - September 2003                                                                                                       71
Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

d) INFORMATION AND COMMUNICATION TECHNOLOGY (ICT)                               They should also be able to understand the ways in which information is
gathered by counting and measuring and presented in the form of graphs,
Calculators and computers are essential tools for teaching, learning and        diagrams, charts and tables.
doing Mathematics. The potential that ICT has in making a significant
contribution to pupils‟ learning in mathematics has already been referred to    Students should therefore be provided with opportunities that help them
in Section 5 of this document and need not be amplified any further.             acquire a feel for the size of a number and know where it fits into the
However, as with any teaching tool, it can be used well or poorly. Teachers         number system
therefore have to select or create mathematical tasks that take advantage of     recall number facts (number bonds, multiplication tables, doubles and
what this new technology can do efficiently and well.                               halves)
 use a range of calculation strategies to calculate accurately and
e) DIFFERENTIATED TEACHING                                                          efficiently, both mentally and with pencil and paper
 appreciate when it is appropriate to use a calculator
All students are entitled to learn Mathematics. To achieve this goal
 check whether their answers are reasonable by using different strategies
expectations for students‟ learning must be raised. Teachers are therefore
required to develop effective methods to support the learning of mathematics     make sensible estimates of measurements
by all students. This does not mean that every student should receive            interpret graphs, diagrams, charts and tables and make predictions from
identical instruction; it means that reasonable and appropriate adjustments         the information these display.
be made to promote access and attainment for all students.

Some students may need further support. For example, some students may          g) ASSESSMENT
benefit from oral rather than written assessments. Others need more time to
complete certain tasks. Others may need additional resources and individual     The purpose of assessment is not simply to certify the students‟ attainment at
attention. Those students with special interests or exceptional talent in the   the end of a series of lessons. It also serves to inform teachers about their
subject may need enrichment material to challenge and engage them. The          effectiveness. Consequently assessment can help teachers make important
school must therefore take care to accommodate the special needs of some        decisions regarding their teaching and eventually adjust their teaching to
students without keeping back the learning of others by providing the           enhance the students‟ learning. Very often assessment is based on tests or
necessary human and material resources.                                         homeworks. Although these do contribute towards assessment, they are not
the only methods that can provide feedback. For example, through the use
of good tasks involving investigative work the students‟ level of proficiency
f) NUMERACY                                                                     can be determined. Classroom discussions in which students present and
evaluate different approaches to the solution of a complex problem may also
One of the outcomes of this curriculum is to produce citizens who have          be utilized to good effect. Other informal means, such as open-ended
confidence and competence with numbers and measures. They should be             questions, performance tasks, observations and conversations, journals and
able to understand the number system, possess a repertoire of computational     portfolios, can also give the teacher information about the students‟
skills and be capable of solving number problems in a variety of contexts.      progress. All the feedback that the teacher acquires from these different
forms of assessment can help the teacher to decide, for example, how and
Secondary Schools Mathematics Syllabus - September 2003                                                                                                    72
Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta

when to revisit a particular topic or how to adapt tasks for students who are   7.     ANNUAL EXAMINATIONS
either struggling or need enrichment or perhaps a challenge.
The Annual Examination will consist of two papers:
Formal assessments provide only one viewpoint on what students can do in a
very particular situation. Excessive reliance on such assessments may give      i)    Forms 1 to 3: A written non-calculator paper consisting of 10
an incomplete and perhaps a distorted picture of the students‟ performance.           questions to be answered in 10 minutes, carrying a total of 10 marks.
Teachers need to be aware that different students show what they know and
can do in different ways, so when various forms of assessment are used each           Forms 4 & 5: A written non-calculator paper consisting of 20
student will be allowed to show his or her best strengths.                            questions to be answered in 20 minutes, carrying a total of 20 marks.

Rulers, protractors and any other mathematical instrument will not be
allowed. Questions will typically involve number calculations,
approximations, estimations, data and graphical interpretations, application
of formulae, recall and application of properties of shapes and recall and
application of mathematical facts. To answer these questions, particularly
those involving numerical calculations, students are advised to choose and
use the more efficient techniques (mental and pencil and paper). They are
expected to have a range of strategies to aid mental calculations of unknown
facts from facts that can be rapidly recalled.

ii)   Forms 1 to 3: A written paper consisting of 15 questions to be
answered in 1 hour 50 minutes carrying a total of 90 marks. Five
questions will carry 4 marks each, another five questions will carry 6
marks each and the remaining five questions 8 marks each. The use of
calculators is only allowed in forms 2 and 3.

Forms 4 & 5: A written calculator paper consisting of not more
than 13 questions to be answered in 1 hour 40 minutes, carrying a total
of 80 marks.

Besides testing knowledge, skills and understanding, questions testing the
use of ICT may also be set (e.g. to write down or complete a set of LOGO
instructions to draw a rectangle.)

The difficulty levels of the questions will be roughly set as follows:
Low 25 30%;                 Medium 40 – 45%;                 High 25 – 35%.

The syllabus content in forms 2 to 5 papers will be roughly 70% from the
core and 30% from the extension part.
Secondary Schools Mathematics Syllabus - September 2003                                                                                                   73

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