Download MATHEMATICS CURRICULUM FOR PHYSICS

Mathematics Curriculum for PCM (elective)
REPUBLIC OF RWANDA
MINISTRY OF EDUCATION
RWANDA EDUCATION BOARD (REB)
P.O.BOX, 3817 KIGALI
www.reb.rw
MATHEMATICS CURRICULUM FOR PHYSICS - CHEMISTRY BIOLOGY (PCB) COMBINATION (ELECTIVE)
ADVANCED LEVEL
Kigali, January 2014
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Mathematics Curriculum for PCM (elective)
Foreword
In the line with Rwandan Education system reform, where combinations system has been introduced
in schools in 2009, the leaver’s profiles for secondary education have been revised. At this time after
the curriculum of Mathematics as core subject have been revised, the Curriculum Department of
Rwanda Education Board is at the stage of designing of the Curriculum of Mathematics as elective
subject in accordance with the revised leaver’s profile, to the pedagogical approach progress and the
new time table allocation.
It is in this regard, this Curriculum has been developed for Mathematics as elective and non
examinable subject in upper secondary Physics-Chemistry-Biology combination.
Starting from the 2013 academic year this curriculum will replace the curriculum which had been
proposed in 2009 by the instructions No 426/12.00/2009 of the Minister of State in charge of Primary
and Secondary Education, dated 02/03/2009.
Dr John RUTAYISIRE
Director General of REB (Rwanda Education Board)
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Mathematics Curriculum for PCM (elective)
List of participants in elaboration of this curriculum
The following were involved in the development of this curriculum:
Coordinator:
Dr. Joyce MUSABE, Deputy Director General of CPMD (Curriculum and Pedagogical Material
Department),
Supervisor:
RUTAKAMIZE Joseph, Director of Science and Art Unit
Curriculum Specialists:
1.KAYINAMURA Aloys : Mathematics Specialist/ CPMD, team leader
2.NYIRANDAGIJIMANA Anathalie: Specialist in charge of Pedagogic Norms/ CPMD;
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Mathematics Curriculum for PCM (elective)
Teachers :
1. HABIMANA Joseph : Mathematics Teacher, G.S.O. de Butare
2. HABINEZA NSHUTI Jean Clément :Mathematics teacher, Ecole Secondaire de Nyanza
3. KAMUHANDA James Kant: Mathematics teacher, Nyagatare Secondary School
4. MUSENGIMANA Théophile: Mathematics teacher, Kiziguro Secondary School
Textbook Approval Committee (TAC)
1.Dr MUSABE Joyce, Member and Chairperson
2.Mr. NHIMIYIMANA Alexis, Member and Secretary
3.Mr. RWAMBONERA Francois, Member
4.Mr. NTAGANZWA Damian, Member
5.Mr. GASANA Janvier, Member
6.Mr. BUTERA Anastase, Non permanent member
7.Mr. RUTAKAMIZE Joseph, Non permanent member
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Mathematics Curriculum for PCM (elective)
Table of contents
List of participants.................................................................................................................................... ii
Table of contents ..................................................................................................................................... iii
1. Introduction .......................................................................................................................................... 1
2. General objectives ............................................................................................................................... 3
3. METHODOLOGICAL NOTES .......................................................................................................... 7
4. EVALUATION APPROACH ............................................................................................................. 9
5. PROGRAMs ...................................................................................................................................... 12
5.1. PROGRAM FOR SENIOR 4.......................................................................................................... 12
Chapter 1.ELEMENTARY LOGIC....................................................................................................... 13
Chapter 2: COMBINATORIAL ............................................................................................................ 43
5.3 PROGRAM FOR Sinior six............................................................................................................. 45
Chapter 1: Complex numbers ................................................................................................................ 46
Chapter 2: Analysis ................................................................................................................................ 52
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Mathematics Curriculum for PCM (elective)
Chapter.3: Differential equations .......................................................................................................... 60
References .............................................................................................................................................. 62
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Mathematics Curriculum for PCM (elective)
I.INTRODUCTION
After completing the curriculum of Mathematics for ordinary level, the Curriculum of Mathematics for
advanced level: Elective and non examinable subject comes for capacity building of secondary
students, especially in PCB combination where Mathematics is elective non examinable subject.
This curriculum is new and deals learning ally with Logic, Trigonometry, Analysis, Algebra,
Statistics, Combinatorial Analysis, Probability and Differential Equations. The chapters are developed
in a logical progressive sequence enabling the learner to have a good comprehension of the subject
matter.
This Mathematics curriculum is prepared in a format which helps teachers to teach a particular topic
effectively. The structure of each chapter is presented in three columns as follow:
•
Learning objectives;
•
Contents
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Mathematics Curriculum for PCM (elective)
•
Suggested Teaching and Learning Activities. Note that at the end of detailed program for
each grade, there is a proposal of lesson distribution.
To avoid the areas of Mathematics to be continually seen as separate and learners acquiring concepts
and skills in isolation, Mathematics is linked to everyday life and experiences in and out of school.
Learners will have the opportunity to apply Mathematics in different contexts, and see the relevance of
Mathematics in natural sciences.
This curriculum is prepared for PCB combination and has to be taught in two periods a week.
This curriculum also helps learners to use ICT tools to support the mastery and achievement of the
desired learning outcomes. Technology used in the teaching and learning of Mathematics, for example
calculators, are to be regarded as tools to enhance the teaching and learning process and not to replace
teachers.
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Mathematics Curriculum for PCM (elective)
II. GENERAL ORIENTATION
The proposed Mathematics Curriculum has content and subject matter specific to PhysicsChemistry-Biology (PCB) Combination. As Math subject is an unavoidable support for
learning sciences and many other subjects, this curriculum aims at preparing the learner to
follow and understand studies offered at higher levels of learning in various fields such as,
Human Medicine, Veterinary medicine, Sciences, Allied Health sciences, Nursing sciences,
Agriculture and rural development.
In order to create a link between this curriculum and the leaver’s profile the following skills have to be
improved through different teaching and learning activities.
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Mathematics Curriculum for PCM (elective)
Skills
Main learning activities
1. ICT skills as tools for learning

Using computers and projectors in presenting individual
or group activities
2. Communication skills

Using calculators in operations

Discussion in group, oral and writing presentations of
findings (results),
3. Logical thinking skills

Using formulae to solve problems related to natural
sciences

Interpretation of graphs (curves)

Relating the solution of a problem to the real life
4. Critical and interpretation skills 
Collecting data, analyzing data, synthesizing data,
interpreting data and presenting data by using tables,
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Mathematics Curriculum for PCM (elective)
charts, diagrams, graphs,…
5. Individual and group learning

skills
Organize and conduct individual and group activities in
a given time
6. Social skills

Working in groups through exchange
7. Problem solving skills

Activities related to natural sciences ( Physics,
Chemistry, Biology)
8. Motivation and self confidence

Activities related to the use of Mathematics in real life
skills
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Mathematics Curriculum for PCM (elective)
III. GENERAL OBJECTIVES
After the completion of Mathematics curriculum for advanced level: Elective and non examinable
subject, a learner should be able to:
1. Develop clear, logical, creative and coherent thinking;
2. Master basic mathematical concepts and use them correctly in solving problems related to
natural sciences.
3. Understand the useful role played by Mathematics in the learning of natural sciences.
4. Use the acquired mathematical concepts and skills to follow easily higher studies (Colleges,
Higher Institutions and Universities);
5. Use acquired mathematical skills to develop work spirit, team work, self-confidence and time
management without supervision;
6. Use ICT tools to explore Mathematics (examples: calculators, computers, mathematical
software, etc.).
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Mathematics Curriculum for PCM (elective)
IV. METHODOLOGICAL APPROACHES
The use of teaching resources and teaching materials like: reference books, worked examples
presented on charts, diagrams/ graphs/curves on wall charts, geometrical instruments, is crucial in
guiding learners to develop mathematical ideas.
Teachers should use real or concrete materials to help learners gain experience, construct abstract
ideas, make inventions, build self confidence, encourage independence and inculcate the spirit of
cooperation.
In order to assist learners in having positive attitudes towards Mathematics, confidence and thinking
systematically, learners have to be involved into the teaching and learning process. Learning in groups
should be emphasized to help learners to develop social skills, encourage cooperation and build self
confidence.
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Mathematics Curriculum for PCM (elective)
Various teaching strategies and approaches such as direct instruction, learner-centered learning
(Problem solving, group work learning, individualization, etc.) must be incorporated.
While teaching and learning, the following point should be taken into consideration:
•
Usage of relevant, suitable and effective teaching materials;
•
Formative evaluation to determine the effectiveness of teaching and learning process.
The choice of a suitable approach will stimulate the teaching and learning environment inside or
outside the classroom.
In this curriculum, suggested various exercises in all chapters may be done in groups or individually.
In implementation of this curriculum, some activities to be done should be related to the main courses
(core subjects like Physics, Chemistry and Biology) in order to establish the relationship between
Mathematics and the above mentioned subjects.
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Mathematics Curriculum for PCM (elective)
V. EVALUATION APPROACH
Evaluation or assessment has to be planned and carried out as a part of the classroom activities.
Different methods of assessment can be conducted. These may be in the form of formative
assessments (oral questioning and answering, assignments: Individual and group work activities),
summative assessments (General tests at the end of one or several topics and exams).
The harmonization of group findings and group presentations is required and some of the group
activities should be marked. The making guides of general test and exams must be given to the
learners.
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Mathematics Curriculum for PCM (elective)
VI. IMPORTANT FACTORS
The teachers of Mathematics should make sure that the material taught and the examples given are
drawn from everyday life such that the learner sees clearly the application of Mathematics involving
problems dictated by nature. In this case Mathematics will take on a form of practical applicability
rather than being abstract.
Bearing in mind the difficulties encountered by teacher of Mathematics in some areas, it is
recommended that teachers have regular meetings and consultations among teachers of the same
school or between neighboring schools.
IT in general and particularly ICT should be used as a pedagogical tool to facilitate teaching and
learning of Mathematics. Nowadays using ICT we would like to encourage teachers of Mathematics
and students to share experiences and good practices with their colleagues in the country and around
the world.
For the better harmonization of the Mathematics program it is recommended that trainings be held
frequently in which teachers can be exposed to any new methods of teaching and latest information
relevant in their subject or discipline. The teacher should try to motivate the learner to like
Mathematics. This may be done through showing him/her the usefulness of the subject its
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Mathematics Curriculum for PCM (elective)
applicability in everyday life and in other disciplines. The approach should be such that the learner
doesn’t see the subject simply as an abstract difficult subject.
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Mathematics Curriculum for PCM (elective)
VII. CURRICULUM DEVELOPMENT
VII.1. SENIOR 4
General objectives
At the end of senior 4 the learner should be able to:
1. Develop clear, logical and coherent thinking.
2. Solve algebraically and graphically given problems in the set of real numbers.
3. Solve problems in physics using trigonometric concepts and trigonometric formulas.
4. Solve problems in physics using algebra and vectors
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Mathematics Curriculum for PCM (elective)
CHAPTER 1.ELEMENTARY LOGIC (6 PERIODS)
General objective: At the end of this chapter, learners should be able to develop clear, logical and
coherent thinking.
Learning objectives
Contents
Suggested teaching-learning activities
At the end of this topic,
1.Elementary Logic
- After giving various examples which are
learner should be able
/ or not propositions, learners should be
to:
asked to form their own sentences and
- Define a statement/
1.1.Definitions:
lead them to determine whether the
proposition
- Propopositions (simple
given
- Formulate his/her
and compound statement)
statements or not.
own statement
- Construct a truth
sentences
are
mathematical
- The teacher should give learners various
- truth tables and truth
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exercises to construct the truth table of
Mathematics Curriculum for PCM (elective)
table of a
values
one or several propositions
statement
- Give the negation
of a proposition
- Through
various
exercises,
teacher
- Negation of a
should emphasize on the use of the word
proposition
“not” or “no” and their symbols to
change a true statement into a false
statement and vice versa
1.2. logical connectors
- Form a compound Logical connectives and
statement using the their properties:
- Through group works, teacher should
helpp learners to discuss about the use of
logical connectives “or”;
“
“and”
conjunction “and”
Conjunction (∧)
- Lead learners to differentiate the use of
- Determine the truth
Disjunction (∨)
“If p, then q” and “p
“ if and only if q”
value of the
implication (
)
through various exercises
compound
equivalence (
)
Statement
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Mathematics Curriculum for PCM (elective)
- Form a compound
statement using
-Truth tables of
the disjunction “or”
propositions including
-Determine the truth
connectives
value of the
compound
Statement
- Use correctly
implication and
equivalence signs in
mathematical
expressions
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Mathematics Curriculum for PCM (elective)
CHAPTER 2: ALGEBRA (28 PERIODS)
General objective: At the end of this chapter the learner should be able to solve algebraically and
graphically given problems in the set of real numbers.
Learning objectives
Contents
Suggested teaching-learning
activities
At the end of this
2. Algebra
topic, learner should be
able to:
- Carry out correctly
2.1. Operations in the
- Define absolute value by means of
the operations on
Set of Real numbers :
a concrete example (for example:
sets of numbers.
- Define absolute value
considering the number line, the
2.1.1. Absolute Value:
-
Definition and
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distance from zero to positive A
is the same from zero to the
Mathematics Curriculum for PCM (elective)
-Solve problems
involving absolute
value by applying
properties of the
absolute value.
properties
negative A where A is a non zero
Absolute value of a
real number)
product, of a
The teacher should prepare various
quotient, of a sum
group works on absolute value
and of a difference
2.2. Indices or Powers
and radicals (Surds):
- The teacher should make a review
- Use indices and
Indices or Powers
on indices and powers by giving
surds’ (radicals)
and its properties
several examples and exercise as
properties to simplify
Surds:
group works.
algebraic
- Square roots
- Facilitate learners to understand
expressions.
- Cube root
surds and their properties through
- nth root of a positive
examples.
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Mathematics Curriculum for PCM (elective)
real number and its
properties
- Give various group activities to be
presented by students
Conversion from
indices to Surds
notation and viceversa
2.3. Quadratic
- Identify the
- Lead learners to use the expression
equations
quadratic equations
Definition of a
with one unknown
quadratic equations
among other
Solution of a
equations.
quadratic equation
- Solve the quadratic
equations
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2

b   b 2  4ac  
ax 2  bx  c  a  x    

2a   4a 2  

Mathematics Curriculum for PCM (elective)
- Determine the Sum
Determine the sum
for determining
and product of
and the product of
of quadratic equation
roots of quadratic
roots of quadratic
equations
equations
- Solve the quadratic
the solution
- The teacher should give various
inequalities
exercises involving the use of sum
and product of roots of quadratic
equations
- Emphasize the role of variation
- Study the Sign of
table in Solving quadratic
quadratic function
inequalities
- Sketch the graph of
- To plot a curve of a quadratic
function
, teacher
should help learners to determine
the concavity by considering the
a quadratic
function
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Mathematics Curriculum for PCM (elective)
2.4. QUADRATIC
FUNCTIONS
sign of a vertex, axis of symmetry
and intersection with axes.
Graphical
representation of a
quadratic function
- Understand the
concept of
Basic concepts of the
logarithm by
definition
- Relate definition of logarithm
2.5. Logarithms
y  a x  log a y  x to the
logarithms to:
-
The base “a” and
- ( 8 23  log 2 8  3 )
its properties
-
examples of powers
The base “10” and
- Help students to look for the
its properties
- Natural / Napierian
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properties of logarithms, by
Mathematics Curriculum for PCM (elective)
logarithms (to the
- Solve problems
involving
base “e”)
means of examples.
- Give various exercises to solve
Logarithmic equations
systems of logarithmic equations
logarithmic
by using the acquired methods.
equations.
- While solving equations involving
logarithms help students to
determine the domain of a
solution /existence conditions,
before computing.
- The teacher should give examples
of application of logarithm in real
life.
with
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Mathematics Curriculum for PCM (elective)
Examples:
1. The Population growth of a
country at a rate R
2. The compound interest in the
Bank
3. Half life time in radioactivity
- During activities, only simple
logarithmic equations should be
given.
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Mathematics Curriculum for PCM (elective)
CHAPTER 3: TRIGONOMETRY (12 PERIODS)
General objective: At the end of this chapter the learner should be able to use the trigonometric concepts
and formulas in solving Physics problem involving Trigonometry
Learning objectives
Contents
Suggested teaching-learning
activities
At the end of this topic, 3. Trigonometry
the learner will be able
- The teacher should make a
to:
3.0. Revision of angles and
review on angles by giving
- Represent the angle
their measurements
several examples and
in a trigonometric
circle.
exercises as group works
3.1. Trigonometric circle:
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(Examples: Exercises
Mathematics Curriculum for PCM (elective)
 Definitions :
leading to conversion of
- Define: sine, cosine,
Sine, cosine, tangent,
angle measurements,
tangent and cotangent
secant and cosecant of an
constructing angles,….)
of any oriented angle.
oriented angle.
- Use trigonometric circle to
determine the sine, cosine,
tangent and cotangent of an
-Verify trigonometric
identities using
fundamental formula
 Fundamental formula
sin2x + cos2x =1
 Trigonometric identities
- Simplify the
oriented angle
- Teacher should use
trigonometric circle to show
the fundamental
trigonometric
trigonometric formula
expressions
- Teacher should lead learners to
3.2.Graphical representation:
Graphical representation
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use basic identities and
fundamental trigonometric
Mathematics Curriculum for PCM (elective)
of trigonometric
- Represent
functions: ( sine and
graphically the sine,
cosine) by using
cosine functions by
coordinates
plotting
- Apply trigonometric
formulae in various exercises
trigonometric functions, the
3.3. Trigonometric ratios

- In graphical representation of
Trigonometric ratios
method of plotting is
recommended.
ratios to solve
in a right angled
problems related to
triangle
and cotangent of principal
Trigonometric ratios
angles (0°,30°,45°,60°,90°)
the right angled

triangle.
- Solve problems
related to
in any other triangle

Solution of right
trigonometric
angled triangle and
equations
any other triangle
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- The use of sine, cosine, tangent
through group works should be
emphasized
Mathematics Curriculum for PCM (elective)
-
The teacher should lead
3.3.Transformation
learners to use transformation
formulae
formulae in solving
 Addition formulae
 Double angle (duplication)
trigonometric equations.
-
The teacher should provide
group activities which use
formulae
 Simpson’s formulae
addition formulae, double
angle (duplication).
-
The teacher should guide
learners to obtain Simpson’s
formulae using addition
formulae (these activities
should be done as group
work).
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Mathematics Curriculum for PCM (elective)
CHAPTER 4: LINEAR ALGEBRA (14 PERIODS)
General objective: At the end of this chapter the learner should be able to solve problems in physics
using algebra and vectors
Learning objectives
Contents
Suggested teaching-learning
activities
At the end of this topic,
4. Linear algebra
the learner should be able
4.1. Vectors
to:

Add vectors and
multiply a vector with

Operations on
-
Vectors:
The teacher should make a
a real number
Revision on vectors in
review on vectors in
(scalar).
Cartesian
Cartesian plane
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Mathematics Curriculum for PCM (elective)
plane : addition of vectors
and multiplication of
vectors by a real
number

Calculate the Scalar

product of two
vectors


product of vectors

Determine the
magnitude of a vector
Scalar products or dot
modulus or magnitude
-The teacher should prepare
(length) of a vector
various exercises on:

Unit vector

relations in a triangle

Conditions for two lines

distance between two
Apply the scalar
to be parallel or
product to solve given
perpendicular
problems.
points
-The use of geometrical
instruments by the teacher and
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Mathematics Curriculum for PCM (elective)
learners is recommended
-To visualize some concepts,

Application of scalar
teacher should use drawings.
product:
-
relations in a triangle
-
distance between two
- Apply scalar product to

points
Find the area of a
triangle,

4.2. Matrices of order 2
of two vectors,

and 3

Definition of a matrix

Operations on matrices:
Addition, subtraction
and multiplication
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Show the orthogonality
Calculate the angle
between two vectors …
Mathematics Curriculum for PCM (elective)
- The teacher should define the



Determinants of
matrix using any rectangular
order 2 and 3by
matrices of order 2 and
array of numbers to help
means of examples
order 3:
learners to understand easily
Define matrices of
Add matrices and
-
with a scalar
-
Calculation /
subtraction and multiplication
computing
of matrices should be given in


Inverse of matrices
of order 2
Calculate the

group activities.
- During solving system of 2
Solution of
linear equations, the teacher
matrices of order 2
simultaneous
should emphasize on the use
and order 3
equations using
of inverse method
Determine the inverse
matrices
determinants of

the definition of matrix.
- Various exercises on addition,
Properties
multiply matrices,
multiply matrix
Definition and
- While solving system of 3
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Mathematics Curriculum for PCM (elective)
of a matrix of order 2

-
System of 2 or 3
linear equations, the teacher
equations
should emphasize on the use
Problems leading to the
of Cramer’s Rule and/ or
2 or 3 linear
system of 2 or 3 linear
elimination method.
equations by using
equations
Solve problems
related to a system of
-
matrices
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Mathematics Curriculum for PCM (elective)
VII.2. PROGRAM FOR SENIOR FIVE
General objectives
At the end of senior 5 the learner should be able:
1. To study and to represent graphically a numerical function.
2. To apply the properties of real numbers, to solve problems in physics
3. To determine and interpret the dispersion parameters of statistical series in one variable
4. To represent statistical data by scatter diagram and determine the linear regression line
5. To solve problems on combinations and permutations
6. To apply combinatorial concepts to solve problems related to Elementary probability
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Mathematics Curriculum for PCM (elective)
Chapter 1: ANALYSIS (36 PERIODS)
General objective: At the end of this chapter the learner should be able to study and to represent
graphically a numerical function.
Learning objectives
Contents
Suggested teaching-learning
activities
At the end of this
topic, the learner
1. ANALYSIS
1.1.Numerical functions
should be able:

Determine if a
 Definition of a numerical
- Through worked examples
given function
function: examples, parity
involving different functions, the
is odd function,
of a numerical function: odd
teacher helps learners to master the
even function,
function, even function and
use of compound concepts in this
periodic
periodic functions,
unit such as: domain of a function,
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Mathematics Curriculum for PCM (elective)


function
increasing and decreasing
odd function, even function,
Find the period
functions, domain of
periodic function.
of a given
definition of a function.
- The teacher should prepare various
function
exercises to help learners to identify
Determine if a
if a function decreases or increases
given function
on a given interval.
decreases or
increases on a
1.2. Limits :
given interval


Concept of a limit with
examples
Determine the
domain of

Right hand limits and left
- Introduce the concept of limit
hand limits
definition of
through examples in real life (for
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Mathematics Curriculum for PCM (elective)
different

Operations on limits
example, if you draw different
functions

Extension of the concept
concentric circles, the limit will be
of limit: when values of
the centre, etc.)
the variable or those of

Evaluate
the function tend towards
correctly the
infinity,
given limits

- Before introducing the notation
lim f ( x )  b the teacher should give
xa
Indeterminate cases:
examples showing that if x takes
values very closed to a then
 0
, ,   , 0  
 0
f (x)
takes values which are very closed to
b.
- Worked examples and various
exercises are needed in order to help
learners to use correctly the concept of
limit.
- Each of the following indeterminate
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Mathematics Curriculum for PCM (elective)
 0
, ,   , 0   , will be
 0
illustrated using one example and the
rest of exercises will be reserved to
group activities
forms:

Study the
continuity of a
given function
1.3.Continuity of a function at
-
and discontinuity, examples in real
a point

To introduce the concept of continuity
life should be used (a broken bridge
Continuity and
at a given point
discontinuity of a
or on a given
function at a point,
should be used as an example of
discontinuity, etc.)
-
interval
The teacher should help learners to
interpret graphical representation of
curves showing the discontinuity and
continuity of functions at points using
wall charts.
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Mathematics Curriculum for PCM (elective)
-
Various activities on the study of
continuity of different functions
should be given in groups.
1.4.Asymptotes:

Define an

asymptote


Find different

The use of limit concept is required to
curve
help learners to determine vertical,
Determination of vertical,
horizontal and oblique asymptotes
asymptotes of a
horizontal and oblique
given function
asymptote
Define the
derivative of a
Find the first
through examples and then exercises
in groups should be done.
1.5.Differentiation :

function at a point

Definition: asymptotes on a -

Derivative of a function
-
The teacher should lead learners to
at a point,
identify geometrical interpretation of
Geometric interpretation
derivative of a function at a point as
37
Mathematics Curriculum for PCM (elective)
derivative of a
of a derivative of a
given function
function at a point,
using definition


Kinematical meaning of
To determine
a derivative: velocity at a
derivatives using
time t
formulae

follow.
Derivative of a function:
derivative of a constant,
of a sum, of a product,
and a quotient of
While denoting derivatives, the
teacher should help learners to be
familiar with the notations
functions

-
Differentiation of some
dx
y    f ( x )  and
, so that the
dy
important functions:
dx
can be used to
dy
power function,
second notation
composite function,
explain the velocity, acceleration at a
38
Mathematics Curriculum for PCM (elective)
circular functions,
given time
parametric functions and
(v 
implicit functions.

-
Successive
-
Application of derivative:
given in groups.
-
Tangent and normal at a
application of derivatives
Hospital’s theorem (proof
not required)

Explain to the learners that the
determine the maximum and
-
tangent and
normal at a point
-
first derivative can be used to
Determine the
equation of
Various activities concerning the use
of Hospital’s rule should be given as
point of a function
-
Various activities on differentiation of
functions in different forms should be
differentiation,

d 2s
ds
, a 2 )
dt
dt
-
Minimum and maximum
minimum of a function at a point
points of a function,
and the interval where that
Increasing and decreasing
function is increasing and
39
Mathematics Curriculum for PCM (elective)
on a curve.

interval of a function.
decreasing. Then, the second
Concavity, inflection point /
derivative is used to determine the
Hospital’s rule for
Turning point on a curve by
concavity and inflection point.
eliminating the
use of the second derivative,
Use correctly
-
indeterminate
forms in
calculating limits.

Determine
whether a turning
point of a given
function is a
maximum or a
minimum point.

solve problems
40
Mathematics Curriculum for PCM (elective)
involving
maximum or
minimum values

Study the
concavity of a
given function
and find its
inflection points.
1.6.Study of a function and
curve sketching

Study different
functions and

Types of functions to
-
Ensure that each required form of
function is sketched through one
consider :
41
Mathematics Curriculum for PCM (elective)
sketch their
-
Polynomial functions
example and various exercises
curves.
-
Rational functions
should be given as group
x
ax  b
with a  0 and
cx  d
c0 x
ax  bx  c
with
dx  c
2
a  0 and d  0

activities.
-
The teacher should help learners to
sketch different functions related
to: Sinusoidal motion, projectile
motion , harmonic motion, etc.
Circular functions
(tangent and cotangent)
42
Mathematics Curriculum for PCM (elective)
CHAPTER 2: COMBINATORIAL (16 PERIODS)
General objective: At the end of this chapter the learner should be able to solve problems on
combinations and permutations.
Learning objectives
Contents
Suggested teaching-learning
activities
At the end of this topic,
learner should be able
3. Combinatorial
to:
3.1. Permutations and
combinations
 Solve problems
involving factorial
notation

Factorial notation

Calculation of the number
learners to be familiar with
of permutations
factorial
(arrangements) of n
calculations through different
43
-
The
teacher
should
notation
help
and
Mathematics Curriculum for PCM (elective)
 Determine the
number of
elements taken r at a time.

permutations of n
-
in a house, random selection of
elements taken n at a time
taken r at a time.
a football team, election of
(n!)

By use of examples from real
life, (arrangement of furniture
of permutations of n
different objects
 Determine the
Calculation of the number
examples and group activities.
representative committee, etc.)
Simple combinations
-
number of
permutations
-
44
The teacher should introduce
the concepts of permutation and
combination. It is also advisable
to go out of classroom for
visualizing permutation of n
elements through different
activities
By the help of the teacher,
various exercises and problems
on
permutations
and
combinations will be done in
groups.
Mathematics Curriculum for PCM (elective)
VII.3 PROGRAM FOR SINIOR SIX
General objectives
At the end of senior 6 the learner should be able:
1. To utilize the algebraic, trigonometric and exponential forms of a non-zero complex number to
solve problems in trigonometry and geometry.
2. To study and represent logarithmic and exponential functions and to apply them in other
scientific domains
3. To calculate the integrals of functions and apply them in various domains
4. To solve simple differential equations of 1st and 2nd order with constant coefficients
45
Mathematics Curriculum for PCM (elective)
CHAPTER 1: COMPLEX NUMBERS (14 PERIODS)
General objective: At the end of this chapter the learner should be able to utilize the algebraic,
trigonometric and exponential forms of a non-zero
zero complex number to solve problems in trigonometry
and geometry.
Learning objectives
Contents
Suggested teaching-learning
teaching
activities
1. Complex numbers
At the end of the topic the
1.1.The set of complex
learner will be able to:

Define a complex number
numbers

in algebraic form

Calculate the sum /
Definition of a complex
number

Notation of the set of
46
- Before introducing complex
numbers, the teacher should
give different equations in
leading to the negative
discriminant to help learners to
discover the necessity
neces
of a new
set, set of complex numbers.
Mathematics Curriculum for PCM (elective)
difference and the product
of complex numbers


properties
Determine the real and
imaginary parts of a


Determine the conjugate
of complex numbers
Apply conjugate
properties of complex
numbers in solving simple
equations
Multiplication in

Algebraic form of a
complex number
-
real and imaginary
parts
-
conjugate and
modulus of complex
numbers,
-
notation and
properties
47
(Example:
)
- Various exercises should be
done in groups to help learners to
, and
its properties
complex number

complex numbers
Addition in
and its
master different operations in
Mathematics Curriculum for PCM (elective)
1.2. Geometric
representation of a
complex number:

Represent a complex

Location of points and
- While teaching, establish a
number in Argand
vectors in a complex
relationship between the
diagram
plane or Argand diagram.
representation of a point in
Cartesian plane and the
representation of complex
number in complex plane or
Argand diagram
1.3.Trigonometric form of
complex number
- Define a complex number
in trigonometric form

Modulus and Argument
of a complex number
48
- The teacher should help
learners to identify that
Mathematics Curriculum for PCM (elective)
- Determine a modulus and

an argument of power of
complex number

- Convert a complex
number from a
trigonometric form to an

Trigonometric form of a
considering the sign of real
complex number
part and imaginary part will
Trigonometric form of
facilitate to determine the
product, and quotient of
position of argument of a
two complex numbers
complex number in Argand
Power of complex
diagram
algebraic form and vice
number in trigonometric
versa
form
- Define a complex number
in Exponential form
-
Activities on trigonometric form
of product, and quotient of two

DeMoivre’s theorem.

Changing a complex
complex numbers will be done
in groups , do not insist on them
number from a
trigonometric form to an
algebraic form and vice
versa
49
-
While teaching exponential
Mathematics Curriculum for PCM (elective)
form of a complex number,

Exponential form of a
give only definition and one
complex number
worked example. Most
activities will be done
do as
group works
1.4.
Calculations in the field
of complex numbers
- Calculate a square root of
-
a complex number
- Solve quadratic equations
Square root of a complex
While calculating the square root
of
number
-
-
Quadratic equations in
z
a
bi , it is advisable to
solve the system
in
, Where
50
Mathematics Curriculum for PCM (elective)
x  iy is the square root of z
and make sure that the choice
of x and y depends on the sign
of b
1.5.Application of complex
-The teacher should prepare
numbers
- Apply De Moivre’s
 Calculation of sine, cosine,
various exercises related to the
formula in calculating sine,
tangent, cotangent of
cosine, tangent and
multiple angles of a given
cotangent of multiple
angle and demonstrating the
-While solving the simple
angles of a given angle
trigonometric identities by
trigonometric equations, make
- Factorize polynomials in C
application of complex numbers.
use of De Moivre’s theorem sure that the equations of the
 Linearization of
trigonometric polynomials.
51
form : a cos x  b sin x  c are
also solved
Mathematics Curriculum for PCM (elective)
CHAPTER 2: ANALYSIS (36 PERIODS)
General objective: At the end of this chapter the learner should be able:
-
To study and represent logarithmic and exponential functions and to apply them in natural
sciences
To calculate the integrals of functions and apply them in natural sciences
Learning objectives
Contents
Suggested teachinglearning activities
At the end of the topic the
learner should be able to:
2. Calculus (Analysis)
2.1.Logarithmic and
-Define Napierian logarithm
function
exponential functions

Napierian logarithmic
functions :
-Represent graphically the
-
Definition and properties,
52
- Through Various
Mathematics Curriculum for PCM (elective)
function y  ln x
-
-Solve logarithmic equations
-Calculate the derivative of
-
logarithmic functions in base a
-Define an exponential function
in base e
Equation of a logarithmic
exercises, teacher
teache
function
should help learners
Differentiation
to apply logarithmic
(derivation) of the
properties to any base
functions of the form
(The group activities
ln u , where u is
should be done as
differentiable over an
revision).
interval I.
-Study and represent graphically
- While teaching
Differentiation
(derivation) of the
-Solve exponential equations
function ln x , the
-
Study of logarithmic
teacher should
functions of the forms
emphasize on
log a x and y  ln x
polynomial functions
only and the other
53
Mathematics Curriculum for PCM (elective)
functions should be
done by students as
Exponential functions in
homework, group
exponential functions in base e
base e ; definition,
activities, etc.
- Solve logarithmic equations
notation, properties, study
-Define an exponential function
of functions y  e x
in
,exponential equations,
base a
differentiation of eu ,
-Calculate the derivative of
- Study and represent

where u
graphically
- Solve exponential equations
y  a x in base a
- The teacher facilitates
learners to establish the
relationship between
logarithm to base a and
logarithm to base
e through change of
basis.
log e x ln x
log a x 

log e a ln a
where a 0 and a
-After
After the definition of
Napiierian logarithm,
-derivative of exponential
54
Mathematics Curriculum for PCM (elective)
functions in base a
the graphical
representation of loga x
- Establish the relationship
,
between
and
exponential functions
will be done by
learners following
procedures of the study
of functions (see content
of Senior five).
five)
y  a x and y  e x
- Study and represent
graphically
logarithmic and exponential
- The teacher should
lead learners to solve
equations involving
logarithmic and
exponential expressions
through various
exercises.
functions in
base a
55
Mathematics Curriculum for PCM (elective)
2.2.CALCULUS INTEGRALS
2.2.1. Anti-derivative
integrals
- Define a primitive function

- Apply immediate primitives in
Definition and Properties
of :
exercises
-
Primative of a function
- Calculate integrals by
-
set of primitive functions
decomposition
-
immediate primitives
method

Techniques of integration:
- Calculate integrals by
-
Integration by
substitution
decomposition: integration
-The teacher should lead
method
of a sum of functions,
learners to calculate
integration of a product of
indefinite integrals by
functions by a real number
decomposition, by
-Calculate integrals by parts
56
Mathematics Curriculum for PCM (elective)
- Calculate integrals of rational
-
functions
Integration by change of
change of variable and
variable,
by parts through one
worked example on each
- Calculate integrals of
-
Integration by parts
irrational functions
-
Integration of trigonometric method and various
- Calculate integrals of
functions: in
trigonometric
in
and exercises will be done in
group activities.
, where
Functions
, and
is a
non-zero real number.
2.2.2. Definite
- Define a definite integral
- Calculate definite integrals by
applying their properties
integrals

- The teacher should
lead learners to calculate
Definition and Properties
definite integrals by
of definite integrals
decomposition, by
57
Mathematics Curriculum for PCM (elective)
- Calculate the area of a plane

Methods of integration,
change of variable and
surface by use of definite

Application of definite
by parts through one
integrals on the:
worked example on each
integrals
- Calculate the volume of a
-
solid of
revolution by use of definite
-
Calculation of area
method and various
of a plane surface
exercises will be done in
Calculation of Work group activities.
integrals
-The teacher should lead
- Calculate the length of a
learners to determine the
curved surface by the use of
area of: square,
definite integral
rectangle, trapezium,
- Calculate the work done by a
triangle using definite
given force
integrals.
58
Mathematics Curriculum for PCM (elective)
- Explain that the
calculation of the work
done depends on the
application of definite
integrals
b
( W   F .dr where r is
a
the distance travelled).
59
Mathematics Curriculum for PCM (elective)
CHAPTER.3: DIFFERENTIAL EQUATIONS (10 PERIODS)
General objective: At the end of this chapter the learner should be able to solve simple differential
equations of 1stand 2ndorder with constant coefficients
Learning objectives
Contents
Suggested teaching-learning
activities
At the end of the unit
the learner will be able
3. Differential
to :
- Define a differential

equation of the 1st
order
- Solve differential

equations
-Through exercises, the teacher should
Definition and
help learners to verify whether a given
terminology
function is a solution of a given
Examples of
differential equation.
differential equations
60
Mathematics Curriculum for PCM (elective)
of the 1st
equations with
separable variables
3.1.Differential equations
homogeneous
of the 1st order it is advisable to use
of the 1st order
- Solve simple

-In solving linear differential equations
Differential equations
natural sciences examples (For
differential equations
with separable
example: Growth and Decay Problems
- Solve linear
variables
differential equations
st
of the 1 order

leading to
Simple homogeneous
differential equations
dN
 kN , where N(t) is the
dt
amount of substance at a given time,
Electrical Circuits
cuits leading to The basic
equation governing the amount of
current I (in amperes) in a simple RLRL
circuit
61
Mathematics Curriculum for PCM (elective)
VIII. REFERENCES
1. Arthur Adam, Freddy Goossens, Francis Lousberg, Mathématisons 65, DeBoeck, 3e edition 1991
2. Christopher Claphan and James Nicholson: Concise Dictionary of Mathematics, Oxford University
Press, Fourth Edition, 2009
3. Curriculum Development Center, Ministry of Education Malaysia, Integrated Curriculum for
Secondary School, Mathematics form 4 and 5, Ministry of Education Malaysia, 2006
4. DPES-RWANDA, Complexes 5eme, Livre de l’élève, IMPRISCO – Kigali, Février 1990
5. DPES-RWANDA, Coniques, Livre de l’élève, IMPRISCO – Kigali, 1988
6. J.K.Backhouse, SPT Houldsworth B.E.D. Cooper and P.J.F.Horril, Pure mathematics 1, Longman,
Third edition 1985, fifteenth impression 1998
7. J.K.Backhouse, SPT Houldsworth B.E.D. Cooper and P.J.F.Horril, Pure mathematics 2, Longman,
Third edition 1985, fifteenth impression 1998
8. JAMES STEWART, Mc. MASTER UNIVERSITY, Calculus Early Transcendental, sixth edition,
United States of America, 2008.
62
Mathematics Curriculum for PCM (elective)
9. John A. Grahan Robert. Sorgenfrey. Trigonometry with application, Hounghton Mifflin Campany,
Boston 1993
10. Manjeet Singh: Pioneer Mathematics, Dhanpat Rdi & Co. (Pvt) Ltd, Educational and Technical
Publishers 2003
11. MINEDUC/NCDC: Advanced Level Mathematics Currriculum for science combinations, Kigali,
April 2010
12. Richard G. Brown, David P. Robbins, Advanced Mathematics: A precalculus course, Houghton
Miffin Company, Boston 1994
13. SCHAUM’S OUTLINE OF “Theory and Problems of COLLEGE MATHEMATICS” third edition,
McGraw-Hill Companies, 1976.
14. Shampiyona Aimable, Mathématiques 6, Kigali, Juin 2005
15. Teach yourself Mathematics, UK, 2003, Trevor Johnson and Hugh Neil
63
Mathematics Curriculum for PCM (elective)
IX. APPENDICES
IX.1. Weekly time allocation for A’ level Physics- Chemistry-Biology combination (PCB)
Number of periods (1 period = 50 min)
S4
S5
S6
Core subjects (all compulsory and examinable)
Physics
7
7
7
Chemistry
7
7
7
Biology
7
7
7
Entrepreneurship
5
5
5
General paper
2
2
2
SUB-TOTAL
28
28
28
Subject
64
Mathematics Curriculum for PCM (elective)
Elective skills subjects
(school may choose one or two non examinable subjects below but students must choose one not
both subjects)
Fine arts
2
2
2
Mathematics
2
2
2
Computer science
2
2
2
English language skills communication 2
2
2
French
2
2
2
Co-curricular activities
Sport, culture activities, Clubs, spiritual
activities, study, research in library…
2
2
2
TOTAL
36
36
36
65
Mathematics Curriculum for PCM (elective)
IX.2. ADVANCED LEVEL/ PHYSICS-CHEMISTRY- BIOLOGY (PCB) COMBINATION
LEAVER’S PROFILE
Upon completion of advanced level secondary education in Physics-Chemistry-Biology (PCB), the
student should have acquired basic knowledge, skills and attitudes which will enable him/her to:
 Apply experimental, prospective and axiomatic processes;
 Analyse, explain facts and practical applications of phenomena relating to daily life;
 Posses appropriate attitude in usual scientific and professional situations, by improving
knowledge, being realistic and self motivated;
 Apply ordinary skills, mathematical techniques and operational methods in the resolution of
problems related other subjects;
 Collect, interpret statistic data and present the results;
 Have access to higher studies in higher institutions of learning and universities mainly in the
following faculties:
 Human Medicine
 Veterinary medicine
 Sciences
 Allied Health sciences
 Nursing sciences
 Agriculture and rural development
66