Download i. problem solving - European School Luxembourg

European Schools
Office of the Secretary-General of the Board of Governors
Pedagogical Unit
Ref.: 4611-D-88
Orig.: FR
Version: EN
Mathematics Syllabus – 3rd Year
Approved by the Board of Governors on 31st January and 1st February 1989
?? 1989
1/22
Mathematics Syllabus – 3rd Year
INTRODUCTION
The main themes of the 3rd. year syllabus extend those taught during the second year,
but presenting the syllabus as a continuation of these subjects in no way indicates that
they will then be completed.
The teacher is free to teach the subjects in any order that is pedagogically sound.
The calculator should not be used solely as a calculating aid but, by thoughtful use, help
the pupil have a better understanding of the structures behind mathematical operations.
That is, it should contribute to developing understanding rather than acquiring
techniques.
The teacher should take advantage of any opportunity, arising during lessons, to
introduce pupils to algorithmic procedures.
Claude BOUCHER,
Chairman of the Mathematics Committee
4611-D-88-EN
2/22
Mathematics Syllabus – 3rd 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.
4611-D-88-EN
3/22
Mathematics Syllabus – 3rd Year
II. NUMBERS
It is important that pupils are made aware of the coherence of the number system (
1° The Introduction of the additive inverse of a natural number leads to the met
2° The introduction of the multiplicative inverse of a natural number leads to



).
.
.
3° As well as being shown that the set of rational& Is equal to the set of numbers written in decimal form, some of which have an unlimited
but periodic form, the pupils will also meet other numbers which are written as non-terminating, non-periodic decimals.
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
1° Decimal form
4611-D-88-EN
Convert a traction into decimal form which may be
unlimited and periodic
Examples:
6
 0,857142 857142 857142...
7
1
 0,11111111...
9
1
 0,55555555...
5
1
 0,01 01 01...
99
37
 0,37 37 37...
99
 0,857142
 0,1
 0,5
 0,01
 0,37
4/22
Mathematics Syllabus – 3rd Year
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
Convert a non-terminating, periodic decimal into a
fraction
Examples:
0,777... 
7
9
0,5252... 
52
99
1
1 4
4
1
 0, 444... 
 

100
100 9 900 225
1
3,75353...  3,7  0,05353...  3,7   0,5353...
10
37 53 3663 53
 


10 990 990 990
3663  53 3716 1858



990
990
495
29
2,9  2,9000... 
10
0,00444... 
Another method:
3,171717...
100 x  317,1717...
x
3,1717...
99 x  314  x 
4611-D-88-EN
314
99
5/22
Mathematics Syllabus – 3rd Year
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
Consider the question: do non-rational numbers
exist, namely numbers which, in decimal form, are
non-terminating and non-periodic?
E.g.: 0,123456...
1,248163264...
1,357911131517...
(where the rule for continuing the digits
precludes the existence of a repeating pattern)
Likewise, consider the existence of lengths whose
magnitude in non-rational
Can these be drawn?
Link with Pythagoras
2° Order and bounds
- order and addition
- order and multiplication
- bounds
Give the moat appropriate approximation to a
number
Example: 4,53576 m
Replace a  b by
Remember that multiplying by a negative quantity
changes the order
ac bc
a.c  b.c when c  0
a.c  b.c when c  0
Place a rational number between bounds which:
1° are two consecutive integers
2° get progressively nearer together
Estimate the order of magnitude or a result
4,536 m
Use a number line
Example:
If a 
2
3
a  [0 ;1]
a  [0,6 ;0,7]
a  [0,66 ;0,67]
…
4611-D-88-EN
6/22
Mathematics Syllabus – 3rd Year
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
3° Operations
Calculate using rational numbers
Quotient of two natural
numbers (recap)
Determine the (Euclidean) quotient of two numbers
Use results which arise from this relationship
D  d .q  r where r  d and d  0
D d
r
   q  and n  \ {0}
n n
n
or it could be expressed as
d .q  D  d .( q  1) where {D; d ; q} 
Multiplicative inverse or a
non-zero rational number
Calculate the multiplicative inverse of
- a number
- the additive Inverse of a number
- the multiplicative inverse of a number
- the product of two numbers
- the quotient of two numbers
Quotient of two rational
numbers
Determine the quotient of two rational numbers
correct to one decimal place
a:b 
a
 a.b-1
b
\ {0}
b0
Express the quotient exactly in fraction form
Powers (natural indices)
Calculate:
a n .a m
a
( a.b) m
(a n )m
a
 
b
am
an
4611-D-88-EN
m
b0
a0
7/22
Mathematics Syllabus – 3rd Year
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
Powers (integers)
Interpret a n when n  0 and use scientific notation
Simple cases only
Examples :
1 1

23 8
10-2  0,01
2-3 
0,025  2,5  10-2
4° Ratio and proportion
Recognise and determine quantities which are in
- direct proportion
- inverse proportion
Review measures
(See Chapter I “Problems”)
Possible applications:
Numerical tables could be used
a c
a b
    a.d  b.c
b d
c d
This could be extended to:
1. the sine and cosine ratios
2. the ratio of the circumference of a circle to its
diameter
4611-D-88-EN
8/22
Mathematics Syllabus – 3rd Year
III. ALGEBRA
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
Algebraic expressions
Apply the rules about brackets and simplify an
algebraic expression
Calculate the numerical value of an expression by
substitution
Polynomials
Simplify and order a polynomial in one variable
State the degree of a polynomial
Add, subtract, multiply polynomials in one variable
Use particular products such as
(a  b)2
Factorisation
(a  b)(a  b)
Generalise arithmetic techniques and use them with
simple algebraic fractions
Find the degree of a sum or product of polynomials
in one variable
Verify this geometrically
Pick out a common factor in an expression
Factories expressions such as
a 2  b2
a 2  2ab  b2
Algebraic fractions
Use factorisation for simplifying algebraic fractions
Example :
3x  3 y
x  2 xy  y 2
2
4611-D-88-EN
9/22
Mathematics Syllabus – 3rd Year
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
Equations, inequations in
one unknown and one
degree
Solve equations and inequations relative to a given
set of elements using the rules of arithmetic
Do not neglect the following cases:
Represent these solutions on a number line
0x  4
Replace an equation or inequation by an equivalent
equation or inequation
0x  0
Use a formula to calculate the value of one of its
elements
Solve problems which involve more than one
inequality
Relations
Define a relation
Represent a relation graphically
Functions
Define a function
Numerical functions of the
first degree
Establish the domains and range of the function
Represent the ordered pairs of a function on a
cartesian graph
0 x  -3
...
Example: calculate one of the parallel sides of a
trapezium given its other parallel side, its area and
its height
Define clearly the set of objects, the set of images
and the verbal link
A function can be defined as a special case of a
relation
Start from a practical situation
Other functions could be introduced such as
a
x
x  x2
x
4611-D-88-EN
10/22
Mathematics Syllabus – 3rd Year
IV. DESCRIPTIVE STATISTICS
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
Probability
Enumerate all the possible results arising from an
experiment involving random variables
Restrict thin work to practical situations
Enumerate the results fulfilling given conditions
Calculate the probability of an event
Compare this with the relative frequency of an event
Collection and ordering of
data
Group data in intervals and draw the appropriate
histogram
Interpréter des histogrammes
4611-D-88-EN
Recall 2nd year work
Use practical situations
Example: different histograms having the same
mean value
11/22
Mathematics Syllabus – 3rd Year
V. GEOMETRY
In the third Year, the study of transformations of the plane which was begun in the second year, will be extended. The emphasis will be on
learning to reason deductively. The geometry course should contribute to a better knowledge Of plane figures through discovering their
properties, which can then be proved either using transformations or previously known properties. The pupil should be led, through
practise, to be able to choose the most suitable method of proof. Although this Course does not explicitly mention the geometry of space
the teacher should extend concepts about the plane to three dimensions, where appropriate, and also point out where properties which are
valid for the plane are not applicable in space.
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
Parallels and
perpendicularity
State Euclid's parallel postulate and the theorem
about unique perpendicularity, and use them to
prove that
- 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
- …
Explain why
- a line and a plane
- two planes
are parallel or perpendicular in given situations
This might be the opportunity to introduce
- "reductio ad absurdum"
- transitivity
Example:
if a b and b c , then a c
Counter-examples :
either a and c intersect

if a  b and b  c , then or
a and c do not intersect

if a  b and b  c , then a is not perpendicular to c
Limit this topic to an intuitive and explorative
approach
4611-D-88-EN
12/22
Mathematics Syllabus – 3rd Year
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
Distance
State the triangular inequality
Construct triangles from different data
Use this in constructions and examples
demonstrating inequalities
The triangle inequality can be applied in the following
cases:
triangle :
-
B
A
P
C
for  ABC where P is a point inside or outside the
triangle, show that
AB  BC  CA
 AP  BP  CP
2
4611-D-88-EN
13/22
Mathematics Syllabus – 3rd Year
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
-
quadrilateral:
C
·X
P·
D
B
A
AB  BC  CD  DA
 AP  BP  CP  DP
2
or
(1) AC  BD  AB  BC  CD  DA
(2) AC  BD  AX  BX  CX  DX
Another example is to compare paths :
AB  BD  AC  CD
B
A
4611-D-88-EN
C
D
14/22
Mathematics Syllabus – 3rd Year
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
Consider some open-ended problems
Examples:
- construire un losange connaissant soit une
diagonale et un côté, soit un côté
- construct a rhombus given a diagonal and a side
- construct a rhombus given a aide
- construct a rectangle given a aide and the
distance between the mid-points of opposite
aides (2 cases)
- construct a rectangle given a diagonal and one
side
- construct a rectangle given a diagonal
- construct a rectangle…
Define a circle, its interior and exterior
Express using the correct vocabulary the relative
positions of a line and a circle
4611-D-88-EN
Construct loci defined by unequal distances from
fixed points
15/22
Mathematics Syllabus – 3rd Year
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
Transformations of the
plane
Define the transformation by the process of
determining the image of a point and know that this
image is unique
State the elements which define the transformation
(in particular using a vector to define a translation)
It is strongly recommended that gridded paper is
used
Transformations may be determined with the aid of a
co-ordinate system
Construct the image of a point using given
equipment:
- compasses
- unmarked ruler
- la règle à parallèles
- …
and given data
Construct the image of any plane figure
Construct the axis, the centre of symmetry, given a
point and its image
4611-D-88-EN
16/22
Mathematics Syllabus – 3rd Year
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
Determine possible fixed points
These are some properties which could be proved:
- the image of a pair of parallel lines is a pair of
Enumerate the principal invariants:
parallel lines
- alignment
- the image of a pair of intersecting lines is a pair
- intersection
of intersecting lines
- parallels and perpendiculars
- the image of a parallelogram is a parallelogram
- line-segments
- the image of a pair of perpendicular lines is a
- angles
pair of perpendicular lines
- length of a line-segment
- the image of the mid-point of a line-segment is
- amplitude of en angle
the mid-point of the image of the line-segment
Use invariance to construct images of lines and line- - the image of a circle is a circle with the same
segments
radius
the image of a perpendicular bisector of a lineRecognise whether a figure's orientation is changed
segment
is the perpendicular bisector of the
or not
image of the line-segment
- the image or the bisector of an angle is the
bisector of the image of the angle
- the image of the tangent to a circle is the tangent
to the image of the circle
4611-D-88-EN
17/22
Mathematics Syllabus – 3rd Year
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
The following are examples which could be used as
applications of transformation work:
-
Reflection
1° reflection
-
construct a triangle given an axis of symmetry,
show that this axis is a perpendicular bisector, an
angle bisector, height and median
-
construct a quadrilateral given that

a diagonal is an axis of symmetry

the line joining the mid-points of a pair of
opposite sides is an axis of symmetry

both the lines joining mid-points of opposite
sides are axes of symmetry

the diagonals are axes of symmetry
-
use reflection

in mirrors and billiards

find the shortest path from A to B touching
the line d en route (A and B on the same side
of d)
B
A
d
A’
-
4611-D-88-EN
locus problems
ex. find the locus of the image of a point when
the axis of symmetry turns about one of its own
points
18/22
Mathematics Syllabus – 3rd Year
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
-
-
Half-turn symmetry
Translation
4611-D-88-EN
2 ° half-turn symmetry
-
show that in a parallelogram

opposite sides are equal in length

opposite angles are equal

inside angles are supplementary in pairs

the diagonals bisect each other
-
show that if the diagonals of a convex
quadrilateral bisect each other, then the
quadrilateral is a parallelogram
-
prove that vertically opposite angles are equal
-
…
3° translation
-
find the translation which maps

a line onto a given line

a line-segment onto a given line-segment

a circle onto a circle
and be able to express the conditions necessary
for each of these to be possible
-
demonstrate some properties of angles and
parallels
19/22
Mathematics Syllabus – 3rd Year
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
-
Rotation
Define a rotation by the process of determining the
image of the point
This work should be limited to an intuitive and
explorative approach
State the elements which define a rotation
Tracing paper is recommended
Recognise invariance, such as: alignment, distance, Find the rotation which maps a line onto another line
amplitude and sense of an angle, parallels,
Find the rotation which maps
orientation

an equilateral triangle onto itself
Construct the image of any plane figure

a regular hexagon onto itself
(Combinations of rotations is a possibility)
-
Enlargement
Construct the image of a point
This work should be limited to an intuitive and
explorative approach
State the elements which define a dilatation (positive
scale. factors)
Find the image of a circle
Construct the image of any plane figure
Compare the length of a line-segment with that of its
image
Compare the area of a figure with that of its image
4611-D-88-EN
20/22
Mathematics Syllabus – 3rd Year
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
Applications of
transformations of the
plane: axis of symmetry,
centre of symmetry
Give the definition of an axis or centre of symmetry
Use practical examples
Give the definition of the perpendicular bisector and
the angle bisector
Construct figures given their axes or centres of
symmetry
Construct these with justification
Prove the properties of the perpendicular bisector
and the angle bisector (use transformations)
Demonstrate the existence of the inscribed and
circumscribed circles of a triangle
Find the centre of a given circle
Prove that 3 collinear points cannot lie on the same
circle
Show that a right-angled triangle is circumscribed by
a semi-circle
Classify figures according to their centres and/or
axes of symmetry
Angles
4611-D-88-EN
State and use the following:
- vertically opposite angles
- alternate angles
- corresponding angles
- angles either side of perpendiculars
- the sum of angles of a triangle and a convex
polygon
Having discovered these properties, they can be
justified using transformations of the plane
21/22
Mathematics Syllabus – 3rd Year
SUBJECTS
KNOWLEDGE & SKILLS
POSSIBLE TEACHING APPROACHES
Pupils must be able to:
Pythagoras’ theorem
State and use this theorem
This could be justified using the equivalence of
areas
Calculate the approximate value of one of the sides
of a right-angled triangle knowing the lengths of the
other two
Calculate the approximate area of a right-angled
triangle knowing the lengths of the hypotenuse and
one other side
(See Chapter II “Numbers” for approximation
techniques)
4611-D-88-EN
22/22