European Schools
Office of the Secretary-General
Ref.: 2010-D-581-en-1
Orig.: EN
MATHEMATICS SYLLABUS SECONDARY 4th YEAR
4 period/week course
JOINT TEACHING COMMITTEE
Meeting on 4 and 5 February 2010
Proposal: entry into force in September 2010
2010
1/8
ALGEBRA (for guidance: 60 periods)
TOPIC
KNOWLEDGE & SKILLS
USE OF TECHNOLOGY
Basic Calculations
Pupils must be able to and/or understand:
Pupils must be able to and/or understand:
do basic calculations (+, −,
x and / ) over the sets
handle basically a calculator
-
-
transform a fraction into a decimal and vice versa
-
» »
,
and »
manage approximate and exact mode calculation
-
verify calculation rules and properties established in
-
calculate lcm and hcf
-
years 1, 2 & 3 and use them in simple algebraic
simplify and factorise numerical expressions
-
expressions
use the calculator for controlling results
-
Square Roots and a Pupils must be able to and/or understand:
Pupils must be able to and/or understand:
new set of numbers
recognise problems leading to square roots
-
solve 2
-
x =
a ;
a ∈ ;
»
x ∈ »
solve by trial 2
x
,
a
-
a
+
=
∈ »
give the definition of
-
a
understand the idea of roots
-
recognise that squaring and square rooting are inverse
-
operations
calculate the square roots of perfect squares (between
calculate the square root of a number
-
-
approximate a number in the decimal form
1 and 400) without a calculator
-
approximate a number as a fraction
-
understand that
-
2 ∉ » and recognise other irrational
numbers
understand that it is necessary to define the set of real
-
numbers »
realise that » ⊂ » ⊂ » ⊂ »
-
know that all arithmetic rules in » apply in »
show properties of square roots
-
-
apply the following:
-
a
a
•
a b =
ab ;
=
for
a
+
» ,
b
+
∈
∈ »
b
b
simplify expressions involving square roots
-
2
n m =
n m for
n ∈ ,
»
m ∈ »
2010-01-22
2/8
TOPIC
KNOWLEDGE & SKILLS
USE OF TECHNOLOGY
• e.g. 12 = 2 3
•
2
a =
a for
a ∈ R
calculate also more difficult expressions like
-
(
+
)2
3 2
12
; ( 2 + 3 5 )( 1
− +
2 )
Linear dependency Pupils must be able to and/or understand:
Pupils must be able to and/or understand:
and proportionality:
1st degree functions
recognise that one value depends on another value
draw the graph of a linear function
-
-
and equations
and define a function accordingly
operate the transformation
-
ax +
by =
c ⇔
y =
mx +
p
write the equation of and recognise a linear function
-
find the equation of a linear function given its graph
-
y =
mx +
p
use sliders to vary
m and
p
-
transform an equation
ax +
by =
c, (
b ≠ 0) in the
find the intersection point with the x-axis given the
-
-
form
y
graph and given the equation of a linear function
=
mx +
p and the converse
plot a set of (
x, y) values and the graph of a linear
-
recognise that the graphical representation of
-
function according to them
ax +
by =
c is a straight line and the converse
solve equations
-
understand the meaning of
m and
p
-
solve equations step by step and check solutions
-
define geometrically
m and
p
-
verify results by use of a calculator
-
find algebraically and geometrically the zero (root) of a
-
linear function
recognise real problems which lead to such functions
-
make equations from simple problems
-
solve linear equations
-
understand that the equation
, with
a and
-
ax +
by =
c
b non-zero, has an infinite number of solutions
give a geometric interpretation of such equations
-
recognise real problems which lead to such equations
-
and solve them without a calculator
2010-01-22
3/8
Simultaneous
Pupils must be able to and/or understand:
Pupils must be able to and/or understand:
equations of the
type:
recognise real problems which lead to simultaneous
draw the graph of two linear functions
-
-
ax +
by =
c
equations
solve simultaneous equations
-
solve simultaneous equations geometrically
solve simultaneous equations step by step
-
-
dx +
ey =
f
solve simultaneous equations by substitution and/or
check solutions
-
-
elimination methods
check solutions
-
solve real problems without a calculator
-
Polynomials
Pupils must be able to and/or understand:
Pupils must be able to and/or understand:
recognise polynomial expressions and calculate their
calculate the numerical value of an algebraic
-
-
value
expression by substituting given numerical values to
handle algebraic expressions with powers and
the variables
-
recognise equivalent expressions
simplify expressions involving powers
-
add and multiply algebraic expressions with powers
simplify algebraic expressions
-
-
simplify and order polynomial expressions
factorise algebraic expressions
-
-
add and multiply polynomials in one variable
-
expand the expression ( + )
n
a
b and calculate
-
handle simple factorisation
-
apply the special identities:
coefficients of Pascal’s triangle
-
(
a ±
b)² =
a² ± 2
ab +
b²
(
a +
b)(
a −
b) =
a² −
b²
3
3
2
2
3
(
a ±
b) =
a ± 3
a b + 3
ab ±
b
2010-01-22
4/8
STATISTICS (for guidance: 18 periods)
TOPIC
KNOWLEDGE & SKILLS
USE OF TECHNOLOGY
Collect, organise
Pupils must be able to and/or understand:
Pupils must be able to and/or understand:
and analyse data
recognise populations and samples in everyday life
insert data into a spreadsheet
-
-
situations
order data in a table of frequencies
-
recognise discrete and continuous data
use a calculator to convert frequencies into
-
-
determine frequencies from collected raw data
percentages and the converse
-
establish a table of frequencies
find the minimum and the maximum value of a
-
-
convert frequencies into percentages and the
numerical set of data
-
converse
define and name a variable
-
establish the range of a set of data
calculate cumulative frequencies
-
-
form equal classes intervals
calculate the arithmetic mean and the median
-
-
establish a table of cumulative frequencies
-
calculate the arithmetic mean and the median
-
Representation of
Pupils must be able to and/or understand:
Pupils must be able to and/or understand:
data
draw a frequency graph
use a calculator to plot a graph
-
-
represent data by bar charts and histograms
represent data on pie charts, by bar charts and
-
-
histograms
Interpret and
Pupils must be able to and/or understand:
Pupils must be able to and/or understand:
compare data
read off data from a diagram
calculate the IQR
-
-
interpret a data diagram and determine the arithmetic
discuss dispersion by comparing graphs
-
-
mean , median, mode, interquartile range (IQR) from a
frequency graph
use percentages to compare data
-
develop caution in interpreting data and misuse of
-
statistics
2010-01-22
5/8
GEOMETRY (for guidance: 50 periods)
TOPIC
KNOWLEDGE & SKILLS
USE OF TECHNOLOGY
Basic Geometry
Pupils must be able to and/or understand:
Pupils must be able to and/or understand:
draw geometric figures such as triangles,
draw geometric figures on the calculator
-
-
quadrilaterals and regular polygons on paper
measure lengths and angles
-
verify that the perpendicular bisectors of a triangle
define a variable
-
-
meet in one point
construct perpendicular and parallel lines
-
show the same for the altitudes (heights), the angle
verify properties in specific geometric situations
-
-
bisectors and the medians of a triangle
verify the expected sum of the angles of a triangle, a
-
quadrilateral and a polygon
Right-Angled
Pupils must be able to and/or understand:
Pupils must be able to and/or understand:
Triangles
recognise the following 4 properties of right-angled
verify these properties using construction and
-
-
triangles:
measurements
a) they possess two complementary angles
b) the length of the median line extended from the
right-angle is equal to half the hypotenuse
c) the centre of the hypotenuse is the centre of the
circumscribed circle of the triangle
d) the hypotenuse is the diameter of the
circumscribed circle
2010-01-22
6/8
Pythagoras
Pupils must be able to and/or understand:
Pupils must be able to and/or understand:
Theorem
recognise Pythagoras' Theorem
verify the theorem using
-
-
prove Pythagoras' Theorem
-
• sides length measurements
understand that there are different ways to prove
-
• area measurements
Pythagoras' Theorem
apply Pythagoras' Theorem to problems in two
-
dimensions
recognise real problems and apply Pythagoras'
-
Theorem
Circles
Pupils must be able to and/or understand:
Pupils must be able to and/or understand:
give the definition of a circle
calculate the length of a circle
-
-
draw a circle, a sector, an arc, a chord
calculate the area of a disc
-
-
define the sector of a disc, the arc of a circle and a
find out by dynamical procedure some properties of
-
-
chord
triangles inscribed in a circle
apply the formulae to calculate the circumference of a
verify by measuring angles
-
-
circle and the length of an arc
verify by measuring angles
-
apply the formulae to calculate the area of a disc and
-
of a sector of a disc
recognise tangents to circles and how lines intersect or
-
completely avoid circles
recognise properties of right-angled triangles inscribed
-
in a circle
recognise that the angle from a chord to the centre of
-
a circle is double than the angle made at the
circumference
recognise that all angles made from a chord to the
-
circumference of a circle are equal
2010-01-22
7/8
Enlargement
Pupils must be able to and/or understand:
Pupils must be able to and/or understand:
enlarge (and reduce) geometrically
draw simple enlargements
-
-
determine the scale factor of an enlargement
find out the scale factor using variables and sliders
-
-
explain and recognise invariants of enlargements
verify invariants by measurements
-
-
determine the effect of enlargement on angles
-
Similar triangles
Pupils must be able to and/or understand:
Pupils must be able to and/or understand:
recognise congruent and similar triangles
use constructions and measurements to verify lengths
-
-
recognise the connection between parallel lines and
use dynamical procedures to prove the theorem using
-
-
the constant ratio of the segments of all lines cutting
variables and sliders
these parallel lines (intercept Theorem)
recognise the intercept theorem when working with
-
triangles
link the intercept theorem with enlargements
-
recognise the converse of the intercept theorem
-
apply the intercept theorem to calculate lengths and to
use constructions and measurements to verify lengths
-
-
prove lines are parallel
Plane Sections of
Pupils must be able to and/or understand:
Solids
apply Pythagoras’ Theorem and the intercept theorem
-
to plane sections of solids
calculate the internal diagonal of a cube or cuboid, the
-
edges of a pyramid or the height of a cone with
particular angles
recognise and solve real problems
-
2010-01-22
8/8