European Schools
Office of the Secretary-General of the Board of Governors
Pedagogical Unit
Ref.: 88-D-72
Orig.: FR
Version: EN
Mathematics Syllabus – 2nd Year
Approved by the Board of Governors on 26 and 27 April 1988
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Mathematics Syllabus – 2nd Year
INTRODUCTION
During the first year pupils were studying numbers and discovering spatial properties, as
a prelude to geometry, in order to revise and consolidate their skills in mathematics. The
second year syllabus has the following aims:
NUMBERS
a) Concepts: - to extend the concept of a number in a convincing manner.
- to make pupils progressively aware of the similarities between different
structures which they will meet in their number work, without a
systematic study as such.
b) Skills:
- a skill should be understood as a reasoned use of methods which one
is capable of justifying. It must not be limited to a blind repetition of
techniques.
GEOMETRY
The development of the ability to use abstract ideas, which was initiated in the first year,
should be extended during the second year with the aim that the pupil relies less and
less upon physical examples. However models are a useful aid when introducing new
ideas.
Geometrical facts should be organised more coherently in the second year, at least
within the context under discussion. This means that when studying a geometrical
situation, a distinction is made between the known properties, particular hypotheses and
the deductions one wants to make. But this does not imply that the whole geometry
course should be axiomatically based.
The study of geometry is limited to the plane. However the teacher should take every
opportunity to extend ideas about characteristics of the plane which are applicable in
space, and, conversely, present properties of the plane as particular cases of general
properties of space.
Transformations should be used in the discovery and justification of properties of figures
and in resolving problems. However, some properties may be verified from
demonstrations based on general premises, which are already known or have been
established, without recourse to transformations.
Using instruments and handling models play an important part in stimulating intuition
and reinforcing reasoning powers while elaborating upon basic ideas.
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Mathematics Syllabus – 2nd Year
PROBLEMS
The paragraph about problem-solving for the first year course is applicable here too. It
is recommended that during the second year pupils tackle some open-ended
investigations. Without a formal development of the subject, pupils should become
aware of the concept of a function through practical situations, whether arising
numerically or geometrically, so that eventually they recognise a functional relationship
whenever it appears.
Claude BOUCHER,
Chairman of the Mathematics Committee
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Mathematics Syllabus – 2nd Year
English interpretation of the official version in French
I. PROBLEM SOLVING
Problem solving has an important role in mathematical development which motivates
pupils and encourages reasoning skills.
Examples and problems can be taken from the real and the physical world. In addition
artificial and closed situations, as well as explorations and experiments, can be created
which will enable pupils to:
-
be able to use the usual operations appropriately in problems;
-
recognise in concrete situations, arising from numerical, geometrical or algebraic
contexts, a functional relationship;
-
interpret and describe given situations (e.g. tables, diagrams, graphs, ...);
-
choose appropriate methods leading to solutions;
-
explain solutions in writing, by formulas and orally;
-
estimate and check possible answers.
Such skills can be developed by allowing pupils to:
-
experience puzzles and games;
-
organise explorations;
-
deal with data for statistical interpretation;
-
interpret graphical representations;
-
evaluate problems and situations.
It is recommended that pupils be given some investigations which are open-ended.
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Mathematics Syllabus – 2nd Year
II. NUMBERS
Mental arithmetic skills, estimation and an understanding of number size should be developed especially through oral, exercises and
approximations.
The use of calculators should be encouraged for
-
checking results;
-
solving more complicated numerical problems.
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
Natural (counting) Numbers
(including zero)
Addition, multiplication;
their properties
The neutral and absorbent
roles of 0 and 1
Commutative, associative
and distributive laws
Subtraction, division
Powers
Recognise the properties of these operations,
formulate and use them in mental and written
calculations
The application of these properties provides practise
in developing arithmetic skills
Quantifiers may be used without a systematic study
being made
Apply the rules of priority in a sequence of operations Show that the properties valid for multiplication and
and handle brackets
division are not always applicable for subtraction and
division
Calculate:
Approach the formulas through numerical examples
a n .a m , (a n )m , (a.b)m ,
a m : a n (m  n)
(m  0 , n  0 , a  0)
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Mathematics Syllabus – 2nd Year
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
L.C.M. and H.C.F.
Determine the L.C.M. and H.C.F. by factorising
The intersection of sets could be used to introduce
these ideas
Number systems
Change a number written in base 10 into (i) base 2
(ii) Roman numerals, and vice-versa
Show the advantages of using base 10
Add two numbers in base 2
Compare the units used for measuring angles,
surfaces and time
Integers
The integer, its sign, its
absolute value, its additive
inverse
Use the following definition:
a  a if a  0
a  0 if a  0
There are different ways of introducing and
representing whole numbers (coloured numbers,
arrows...)
a  -a if a  0
Order in
Order a set of integers and place them on a number
line
Conservation of order
under addition
One could use 7  3 instead of -7  (-3) initially to
distinguish between the sign of the integer and the
sign of the operation
Conservation of equality
under addition
Show and use the properties of the group [ ; +]
without going into details about the theory of groups
Addition and subtraction
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Mathematics Syllabus – 2nd Year
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
Multiplication, division
Calculate using the rules about signs
Multiplication and equality
Apply in
Multiplication and order
Using the above notation:
Apply the rules of priority in a sequence of operations
-(a  b)  a  b  a  b  -a  b
and handle brackets
Properties of addition and
multiplication
the rules seen in
Apply the rules of signs with brackets
A flow-chart can clarify the hierarchy of the
operations
a  (b  c )  a  (b  c )
 a  (b  c )
 abc
 abc
a  (b  c )  a  (b  c )
 a  (b  c )
abc
 abc
Powers (natural indices)
Apply the formulas found when working in
Use examples of the type:
22 ; -22 ; (-2)2 ; -(-2)2 ; -(-22 )
23 ; -23; (-2)3; -(-2)3; -(-23 )
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Mathematics Syllabus – 2nd Year
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
Rational Numbers
Equivalent fractions,
simplification of fractions,
irreducible fractions
Apply the rules about signs
The aim in the 2nd year is to improve arithmetic
skills
The following types of example can be used:
3 6
... -9
 

4 ... -24 ...
-3 6
... -12 3
3
 

 4 ... -16 ... -4
4
Fractions and decimals on
the number line
Nesting decimals and
fractions
Arithmetic using rationals
Change a fraction into a decimal
Note the inclusive sets
Nest a fraction between two numbers in decimal form
Multiply fractions
Add and subtract fractions using the L.C.M.
Apply the properties used with
Using the L.C.M. is not always the most efficient
method
The L.C.M. can be found in simple cases without
factorisation
Divide fractions
and
to
Raise a fraction or decimal to a power, which is a
simple, natural number
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
Change a decimal (with a finite number of digits) into
a fraction
Calculate the multiplicative inverse of a fraction
Percentages

The following examples could be used:
1
- 
2
Use percentages in simple examples
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2
2
2
 1 2 2
;  -  ; - ; 2 ; (-0,1) 3 ; (0, 2) 2 ; etc.
3 3
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Mathematics Syllabus – 2nd Year
III. ALGEBRA
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
Algebraic expressions
Read and recognize the operations mentioned in
algebraic expressions
Express a sequence of calculations orally and in
writing
Calculate the value of algebraic expressions by
substitution, using the properties of operations and
the rules of priority
The priorities of operations can be shown through
flow-charts
Recognise sums and products
Use the rules appropriate to the additive inverse
This can be extended to the products of sums and
differences including (a  b)2 , ( a  b).( a  b) and
simple cases of factorisation
Multiply a sum, a difference by a number
Propositions, open
sentences
Recognize when a proposition is true
Recognize the functional nature of an open sentence
Equations, inequations in
one unknown and one
degree
Solve equations and inequations relative to a given
set of elements using the properties of arithmetic
The idea of equivalence of equations and of
inequations can be used
Use a formula to calculate the value of one of its
elements
Ex.: Calculate one of the parallel sides of a
trapezium given its other parallel side, its area and
height
Plotting points on a plane
Plot points using coordinates
Some situations can be represented by sets of
isolated points, others by a line of points
Find the solution set which makes an open sentence
true
Interpret graphs
Represent practical situations by sets of points
Functions
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Recognise a function (see chapter 1)
Practical examples seen during the course of the
year give rise to the notion of a function
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Mathematics Syllabus – 2nd Year
IV. DESCRIPTIVE STATISTICS
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
Collection and ordering of
data
Collect and order data in:
- tables,
- bar charts,
- histograms
Extend the work done in the 1st year
Interpret these types of representations
Arithmetic mean
Calculate the mean
Relative frequency
Calculate the relative frequency
Distinguish clearly between the absolute and relative
frequency
Pie charts may be used (restrict to 5 divisions)
V. GEOMETRY
Experience has shown that if pupils are to gain a true visual sense of geometry and appreciate objects' shapes and their properties then
the teaching of geometry must begin with a study of space. To this end it is vital that pupils handle and observe objects in practical
situations.
However, geometry is more than a series of observations. An appreciation of the mathematical value of these observations is gained if:
-
skills, such as evaluation, recordings measuring and manipulation are developed
-
pupils make discoveries concerning relationships between properties
-
systematic ordering and application of properties is encouraged
-
d'établir une construction cohérente sans pour autant vouloir élaborer une géométrie axiomatique à ce niveau.
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Mathematics Syllabus – 2nd Year
L'initiation à la géométrie est un moyen privilégié pour apprendre à manier correctement et avec précision des instruments usuels tels que
la règle graduée, l'équerre géo-dreieck, le compas. L'élève apprend progressivement à reproduire des figures, en utilisant les instruments
de dessin, les quadrillages… ; ensuite il dessine en respectant un certain nombre de consignes données (compréhension d'un énoncé
géométrique) ; et enfin il construit en énonçant des propriétés à respecter.
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
Plane, line, half-line, line
segment, point
a) Express orally and/or in writing that:
- the plane, the line are infinite sets of points
- every line is an infinite part of a plane
- each pair of points defines a unique line
- every line segment, every half-line is a proper,
yet infinite, subset of a line
b) Use appropriately the following symbols:
,, , , , , \, , , , , 
Use the environment to find physical examples of
planes, lines and points
The teacher should make his pupils aware that the
plane is an infinite set of elements called points; that
every line is an infinite part of a plane; that through a
point pass an infinite number of lines; that through
two distinct points only one line can be drawn; that
through three points…
The notation for a segment, half-line and line can be
used [a , b] , ]a , b[ , etc.
Euclid's parallel axiom
Express orally and/or in writing these properties
Perpendicularity properties
One could use these two properties to initiate pupils
in the skills of logical reasoning
For example one could prove:
- if a b and if b c , then a c
- if a b and if b  c , then a  c
- if a  b and if b  c , then a c
- …
Distances (point - point,
point - line; parallel lines)
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Measure, copy, compare lengths
State the distance properties of the perpendicular
bisector and angle bisector
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Mathematics Syllabus – 2nd Year
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
The Circle
Express orally and/or in writing the definition of a
circle
Set notation could be used to define the
circumference of a circle, its interior, its exterior...
Introduction to
transformations:
Experiment personally to show how a shape moves
under
- reflection
- half-turn symmetry
- translation
Other transformations such as rotation and parallel
projection may be introduced according to the needs
and levels of the pupils
a) axial symmetry
(reflection)
b) half-turn symmetry
c) translation
Express that
- reflection is defined by an axis,
- a half-turn is defined by a centre,
- translation is defined by a point and its image
A vector can be used to denote translation
The notion of an ordered pair formed from the point
and its image could be used
The angle bisector may be constructed using only a
graduated ruler
Construct the image of a point and a part of the
plane under one of these transformations
Use squared paper and coordinates to find images
and discover properties of figures
Construct axes, centres from figures and images
Recognise invariance
Construct using compasses and ruler
- perp. bisector
- angle bisector
Reconnaître des points fixes et des figures
globalement invariantes
Angles
Find equal angles using the properties of the above
transformations
Examples using vertically opposite angles, alternate
angles... can be used
Construct and copy angles
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Mathematics Syllabus – 2nd Year
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
Plane surfaces
1) Quadrilaterals, triangles
Special lines of a triangle
Construct quadrilaterals and triangles meeting given
criteria of symmetry
Classify quadrilaterals and triangles according to
properties of symmetry
The construction of a circumcircle and inscribed
circle is justified here
Note that 3 collinear points do not belong to the
same circle
Express orally and/or in writing the definitions of
- the perp. bisector,
- the altitude (height),
- the angle bisector,
- the median
Construct these lines using compasses and ruler
2) Areas
Distinguish between a surface and its area
Recognise figures which have the same area
Calculate the area of quadrilaterals and triangles
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Use transformations or calculation; (the notion of
shearing could be used)
For calculating the area of a parallelogram or triangle
use "side times appropriate altitude"
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