MATHEMATICS

ACCOUNTING PROFESSION OPTION

Senior
5 Teacher's Guide

Experimental Version

Kigali, 2023
 © 2023 Rwanda Basic Education Board
All rights reserved.
This document is the property of the government of Rwanda.
Credit must be provided to REB when the content is quoted.
FOREWORD
Dear Teachers,
Rwanda Basic Education Board is honoured to present the teacher’s guide
for Mathematics in the Accounting Profession Option. This book serves as
a guide to competence-based teaching and learning to ensure consistency
and coherence in the learning of the Mathematics Subject. The Rwandan
educational philosophy is to ensure that students achieve full potential at
every level of education which will prepare them to be well integrated in
society and exploit employment opportunities.

Specifically, the curriculum for Accounting Profession Option was reviewed

to train quality Accountant Technicians who are qualified, confident and
efficient for job opportunities and further studies in Higher Education in
different programs under accounting career advancement.

In line with efforts to improve the quality of education, the government

of Rwanda emphasizes the importance of aligning teaching and learning
materials with the syllabus to facilitate their learning process. Many factors
influence what students learn, how well they learn and the competences
they acquire. Those factors include the relevance of the specific content, the
quality of teachers’ pedagogical approaches, the assessment strategies and
the instructional materials.

High Quality Technician Accounting program is an important component

of Finance and Economic development of the Rwanda Vision 2050, “The
Rwanda We Want” that aims at transforming the country’s socioeconomic
status. The qualified Technicians accountant will significantly play a major
role in the mentioned socioeconomic transformation journey. Mathematics
textbooks and teacher’s guide were elaborated to provide the mathematical
operations, algebraic functions and equations, and basic statistics that are
necessary to train a Technician Accountant capable of successfully perform
his/her duties.

The ambition to develop a knowledge-based society and the growth of

regional and global competition in the jobs market has necessitated the shift
to a competence-based curriculum.

Mathematics | Teacher's Guide | Senior Five | Experimental Version iii

The Mathematics teacher’s guide provides active teaching and learning
techniques that engage students to develop competences. In view of this,
your role as a Mathematics teacher is to:

• Plan your lessons and prepare appropriate teaching materials.

• Organize group discussions for students considering the importance of
social constructivism suggesting that learning occurs more effectively
when the students work collaboratively with more knowledgeable
and experienced people.
• Engage students through active learning methods such as inquiry
methods, group discussions, research, investigative activities and
group or individual work activities.
• Provide supervised opportunities for students to develop different
competences by giving tasks which enhance critical thinking, problem
solving, research, creativity and innovation, communication and
cooperation.
• Support and facilitate the learning process by valuing students’
contributions in the class activities.
• Guide students towards the harmonization of their findings.
• Encourage individual, pair and group evaluation of the work done in
the classroom and use appropriate competence-based assessment
approaches and methods.

To facilitate you in your teaching activities, the content of this book is self-
explanatory so that you can easily use it. It is divided in 3 parts:
The part I explains the structure of this book and gives you the methodological
guidance;
The part II gives a sample lesson plan;
The part III details the teaching guidance for each concept given in the
student book.
Even though this Teacher’s guide contains the guidance on solutions for all
activities given in the student’s book, you are requested to work through
each question before judging student’s findings.

I wish to sincerely express my appreciation to the people who contributed

towards the development of this book, particularly, REB staff, UR Lecturers,

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Teachers from TTC and General Education and experts from different
Education partners for their technical support. A word of gratitude goes also
to the administration of Universities, Head Teachers and TTCs principals who
availed their staff for various activities.

Dr. MBARUSHIMANA Nelson

Director General, REB.

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ACKNOWLEDGEMENT
I wish to express my appreciation to the people who played a major role in
the development of this teacher`s guide for Mathematics in the Accounting
profession option. It would not have been successful without active
participation of different education stakeholders.

I owe gratitude to different universities and schools in Rwanda that allowed

their staff to work with REB in the in-house textbooks production initiative.

I wish to extend my sincere gratitude to lecturers and teachers whose efforts

during writing exercise of this teacher`s guide was very much valuable.

Finally, my word of gratitude goes to the Rwanda Basic Education Board

staffs who were involved in the whole process of in-house textbook writing.

Joan MURUNGI
Head of Curriculum, Teaching and Learning Resources Department/REB

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TABLE OF CONTENT
FOREWORD...................................................................................................... iii
ACKNOWLEDGEMENT..................................................................................... vi
PART I. GENERAL INTRODUCTION................................................................... x
1.1 The structure of the guide........................................................................x
1.2 Methodological guidance.........................................................................x
1.2.1 Developing competences..................................................................x

1.2.2 Addressing cross cutting issues......................................................xii

1.2.3 Guidance on how to help students with special education needs in

classroom...............................................................................................xiv

1.2.4. Guidance on assessment ............................................................xvii

1.2.5. Teaching methods and techniques that promote active learning ..... xix
PART II: SAMPLE OF A LESSON PLAN......................................................... xxiv
PART III: UNIT DEVELOPMENT .........................................................................1
UNIT 1: MATRICES AND DETERMINANTS ....................................................1
1.1. Key unit competence............................................................................... 1
1.2. Prerequisite ............................................................................................. 1
1.3. Cross-cutting issues to be addressed..................................................... 1
1.4 Guidance on introductory activity .......................................................... 1
1.5 List of lessons...........................................................................................2
Lesson 1: Definitions and notations..........................................................3

Lesson 2: Equality of matrices..................................................................5

Lesson 3: Addition and subtraction of matrices..................................... 6

Lesson 4: Scalar multiplication................................................................ 8

Lesson 5: Multiplication of matrices....................................................... 9

Lesson 6: Inversion of matrices...............................................................11

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 Lesson 7: Definition and calculations of determinants of orders 2 × 2 and 3 × 3 ....13

Lesson 8 : Properties of determinants................................................... 15

Lesson 9: Inverse of a matrix.................................................................. 16

Lesson 10: Solving simultaneous linear equations by using inverse of

matrix...................................................................................................... 19

Lesson 11: Solving simultaneous linear equations by Cramer’s rule..... 21

1.6. Summary of unit 1..................................................................................23

1.7. Additional information for the teacher................................................27
1.9. Additional activities.............................................................................. 30
1.9.1. Remedial activity........................................................................... 30

1.9.2. Consolidation activity................................................................... 30

1.9.3. Extended activity........................................................................... 31

UNIT 2: DIFFERENTIATION/DERIVATIVES .................................................33
2.1. Key unit competence ............................................................................33
2.2. Prerequisite ..........................................................................................33
2.3. Cross-cutting issues to be addressed...................................................33
2.4. Guidance on introductory activity .......................................................33
2.5. List of lessons........................................................................................34
Lesson1: Average rate of change............................................................. 36

Lesson2: Instantaneous rate of change.................................................37

Lesson3: Differentiation of polynomial functions................................. 38

Lesson4: Differentiation of a product function..................................... 40

Lesson5: Differentiation of a power function........................................ 41

Lesson6: Differentiation of the composite function (The chain rule).43

Lesson7: Differentiation of a quotient function.................................... 44

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 Lesson8: Differentiation of a logarithmic function............................... 46

Lesson9: Differentiation of an exponential function............................ 48

Lesson10: Equation of the tangent to the graph of a function at a point ....49

Lesson11: Hospital’s rule ........................................................................ 51

2.6. Summary of unit 2.................................................................................52

2.7. Additional information for the teacher............................................... 56
2.8. Answers of End unit assessment.........................................................57
2.9. Additional activities............................................................................. 59
2.9.1. Remedial activity........................................................................... 59

2.9.2. Consolidation activity................................................................... 59

2.9.3. Extended activity.......................................................................... 60

UNIT 3: APPLICATIONS OF DERIVATIVES IN FINANCE AND IN ECONOMICS..61

3.1. Key unit competence............................................................................. 61

3.2. Prerequisite .......................................................................................... 61
3.3. Cross-cutting issues to be addressed................................................... 61
3.4 Guidance on introductory activity ........................................................62
3.5. List of lessons........................................................................................62
Lesson 1: Marginal cost.......................................................................... 63

Lesson 2: Marginal revenue................................................................... 64

Lesson 3: Minimization of the total cost function................................ 65

Lesson 4: Maximization of the total revenue function........................ 67

Lesson 5 : Price elasticity of demand...................................................... 68

Lesson 6: Price elasticity of supply........................................................ 70

3.6. Summary of unit3.................................................................................. 71

3.7. Additional information for the teacher................................................73

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 3.8. Answers of End unit assessment.........................................................74
3.9. Additional activities..............................................................................75
3.9.1. Remedial activity............................................................................75

3.9.2. Consolidation activity....................................................................75

3.9.3. Extended activity.......................................................................... 76

UNIT 4: UNIVARIATE STATISTICS AND APPLICATIONS .........................77
4.1 Key unit competence............................................................................77
4.2 Prerequisite ...........................................................................................77
4.3 Cross-cutting issues to be addressed....................................................77
4.4 Guidance on introductory activity ........................................................77
4.5 List of lessons ....................................................................................... 79
Lesson 1: Statistical concepts: descriptive and inferential statistics,
population, sample, statistic, and parameter...................................... 80

Lesson 2: Variables and types of variables.............................................82

Lesson 3: Data and types of data ......................................................... 83

Lesson 4: Levels of measurement scale................................................ 84

Lesson 5: Sampling and sampling methods.......................................... 85

Lesson6: Bar graph................................................................................ 88

Lesson 7: Histogram............................................................................... 90

Lesson 8: Time series graph................................................................... 91

Lesson 9: Pie chart..................................................................................92

Lesson 10: Graph interpretation............................................................ 94

Lesson 11: Describing data using mean, median, and mode................ 95

Lesson 12: Summarizing data using variance, standard deviation, range,

mean deviation, and coefficient of variation........................................ 96

Lesson 13: Determining the position of data value using quartiles..... 98

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 Lesson 14: Skewness.............................................................................100

Lesson 14: Chebyshev’s theorem and Empirical rule ...........................101

Lesson 14................................................................................................101

Lesson 15: Examples of the applications of univariate statistics in solving

problems related to finance, accounting, and economics................. 102

4.6 Summary of the unit ........................................................................... 103

4.7 Additional Information for Teacher.................................................... 107
4.8. Answers of End unit assessment....................................................... 107
4.9 Additional activities.............................................................................108
4.9.1 Remedial activities........................................................................108

4.9.2 Consolidation activity...................................................................108

4.9.3 Extended activity.........................................................................108

Bivariate statistics and Applications............................................................110
5.1 Key unit competence............................................................................110
5.2 Prerequisite ..........................................................................................110
5.3 Cross-cutting issues to be addressed. .................................................110
5.4 Guidance on introductory activity .......................................................110
5.5 List of lessons ....................................................................................... 112
Lesson 1: Key concepts of bivariate statistics.......................................113

Lesson 2: Covariance and correlation...................................................115

Lesson 3: Regression lines and analysis ...............................................116

Lesson 4: Spearman’s coefficient of correlation..................................118

Lesson 5: Applications of bivariate statistics in accounting related

subjects or any other area: Relationship between advertising
products and total revenue, the relationship between attendance
in class and the marks scored, etc................................................... 120
5.6 Summary of the unit ............................................................................121

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 5.7 Additional Information for Teacher.................................................... 122
5.9 Additional activities............................................................................. 123
5.2.1 Remedial activities........................................................................ 123

5.9.2 Consolidation activities................................................................ 123

5.2.3 Extended activities....................................................................... 125

REFERENCES ...............................................................................................126

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PART I. GENERAL INTRODUCTION

1.1 The structure of the guide

The teacher’s guide of Mathematics is composed of three parts:
The Part I concerns general introduction that discusses methodological
guidance on how best to teach and learn Mathematics, developing
competences in teaching and learning, addressing cross-cutting issues in
teaching and learning and Guidance on assessment.

Part II presents a sample lesson plan. This lesson plan serves to guide the
teacher on how to prepare a lesson in Mathematics.

The Part III is about the structure of a unit and the structure of a lesson. This
includes information related to the different components of the unit and
these components are the same for all units. This part provides information
and guidelines on how to facilitate students while working on learning
activities. More other, all application activities from the textbook have
answers in this part.

1.2 Methodological guidance

1.2.1 Developing competences
Since 2015 Rwanda shifted from a knowledge based to a competence-based
curriculum for pre-primary, primary, secondary education and recently
the curriculum for profession options such as TTC, Associate Nurse and
Accounting options . This called for changing the way of learning by shifting
from teacher centred to a learner centred approach. Teachers are not only
responsible for knowledge transfer but also for fostering students’ learning
achievement and creating safe and supportive learning environment. It
implies also that students have to demonstrate what they are able to transfer
the acquired knowledge, skills, values and attitude to new situations.

The competence-based curriculum employs an approach of teaching and

learning based on discrete skills rather than dwelling on only knowledge or
the cognitive domain of learning. It focuses on what learner can do rather
than what learner knows. Students develop competences through subject
unit with specific learning objectives broken down into knowledge, skills and
attitudes/ values through learning activities.

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 In addition to the competences related to Mathematics, students also develop
generic competences which should promote the development of the higher
order thinking skills and professional skills in Mathematics teaching. Generic
competences are developed throughout all units of Mathematics as follows:

Generic
Ways of developing generic competences
competences
All activities that require students to calculate,
Critical thinking convert,interpret, analyse, compare and contrast, etc have a
common factor of developing critical thinking into students
All activities that require students to plot a graph of a given
algebraic data, to organize and interpret statistical data
Creativity and
collected and to apply skills in solving problems of production/
innovation
finance/ economic have a common character of developing
creativity into students
All activities that require students to make a research and
Research and apply their knowledge to solve problems from the real-
problem solving life situation have a character of developing research and
problem solving into students.
During Mathematics class, all activities that require students
to discuss either in groups or in the whole class, present
Communication
findings, debate …have a common character of developing
communication skills into students.
Co-operation,
All activities that require students to work in pairs or in
interpersonal
groups have character of developing cooperation and life
relations and life
skills among students.
skills
All activities that are connected with research have a
common character of developing into students a curiosity of
applying the knowledge learnt in a range of situations. The
purpose of such kind of activities is for enabling students
Lifelong learning
to become life-long students who can adapt to the fast-
changing world and the uncertain future by taking initiative
to update knowledge and skills with minimum external
support.
Specific instructional activities and procedures that a teacher
may use in the class room to facilitate, directly or indirectly,
students to be engaged in learning activities. These include a
range of teaching skills: the skill of questioning, reinforcement,
Professional skills
probing, explaining, stimulus variation, introducing a lesson;
illustrating with examples, using blackboard, silence and non-
verbal cues, using audio – visual aids, recognizing attending
behaviour and the skill of achieving closure.

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The generic competences help students deepen their understanding
of Mathematics and apply their knowledge in a range of situations. As
students develop generic competences they also acquire the set of skills
that employers look for in their employees, and so the generic competences
prepare students for the world of work.

1.2.2 Addressing cross cutting issues

Among the changes brought by the competence-based curriculum is the

integration of cross cutting issues as an integral part of the teaching learning
process-as they relate to and must be considered within all subjects to be
appropriately addressed. The eight cross cutting issues identified in the
national curriculum framework are: Comprehensive Sexuality Education,
Environment and Sustainability, Financial Education, Genocide studies, Gender,
Inclusive Education, Peace and Values Education, and Standardization Culture.
Some cross-cutting issues may seem specific to particular learning areas/
subjects but the teacher need to address all of them whenever an opportunity
arises. In addition, students should always be given an opportunity during
the learning process to address these cross-cutting issues both within and
out of the classroom.

Below are examples of how crosscutting issues can be addressed:

Cross-Cutting Issue Ways of addressing cross-cutting issues

Comprehensive Sexuality Education: Using different charts and their
The primary goal of introducing interpretation, Mathematics teacher
Comprehensive Sexuality Education should lead students to discuss the
program in schools is to equip children, following situations: “Alcohol abuse
adolescents, and young people with and unwanted pregnancies” and advise
knowledge, skills and values in an age students on how they can fight against
appropriate and culturally gender them.
sensitive manner so as to enable them
to make responsible choices about Some examples can be given when
their sexual and social relationships, learning statistics, powers, logarithms
explain and clarify feelings, values and and the related graphical interpretation.
attitudes, and promote and sustain risk
reducing behaviour.

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 Environment and Sustainability: Using Real life models or students’
Integration of Environment, Climate experience, Mathematics Teachers
Change and Sustainability in the should lead students to illustrate the
curriculum focuses on and advocates situation of “population growth” and
for the need to balance economic discuss its effects on the environment
growth, society well-being and and sustainability.
ecological systems. Students need
basic knowledge from the natural
sciences, social sciences, and
humanities to understand to interpret
principles of sustainability.

Financial Education:
The integration of Financial Education Through different examples and
into the curriculum is aimed at a calculations on interest (simple and
comprehensive Financial Education compound interests), interest rate
program as a precondition for problems, total revenue functions and
achieving financial inclusion targets total cost functions, supply and demand
and improving the financial capability functions Mathematics Teachers can
of Rwandans so that they can make lead students to discuss how to make
appropriate financial decisions that appropriate financial decisions.
best fit the circumstances of one’s life.
Gender: At school, gender Mathematics Teachers should
will be understood as family address gender as cross-cutting issue
complementarities, gender roles through assigning leading roles in the
and responsibilities, the need for management of groups to both girls and
gender equality and equity, gender boys and providing equal opportunity in
stereotypes, gender sensitivity, etc. the lesson participation and avoid any
gender stereotype in the whole teaching
and learning process.

Inclusive Education: Inclusion is Firstly, Mathematics Teachers need to

based on the right of all students to identify/recognize students with special
a quality and equitable education needs. Then by using adapted teaching
that meets their basic learning needs and learning resources while conducting
and understands the diversity of a lesson and setting appropriate tasks to
backgrounds and abilities as a learning the level of students, they can cater for
opportunity. students with special education needs.
They must create opportunity where
students can discuss how to cater for
students with special educational needs.

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Peace and Values Education: Peace Through a given lesson, a teacher should:
and Values Education (PVE) is defined Set a learning objective which is
as education that promotes social addressing positive attitudes and values,
cohesion, positive values, including Encourage students to develop the
pluralism and personal responsibility, culture of tolerance during discussion
empathy, critical thinking and action in and to be able to instil it in colleagues
order to build a more peaceful society. and cohabitants;
Encourage students to respect ideas
from others.
Standardization Culture: With different word problems related
Standardization Culture in Rwanda to the effective implementation of
will be promoted through formal Standardization, Quality Assurance,
education and plays a vital role in terms Metrology and Testing, students can
of health improvement, economic be motivated to be aware of health
growth, industrialization, trade and improvement, economic growth,
general welfare of the people through industrialization, trade and general
the effective implementation of welfare of the people.
Standardization, Quality Assurance,
Metrology and Testing.

1.2.3 Guidance on how to help students with special education

needs in classroom

In the classroom, students learn in different way depending to their learning

pace, needs or any other special problem they might have. However, the
teacher has the responsibility to know how to adopt his/her methodologies
and approaches in order to meet the learning need of each student in the
classroom. Also teachers need to understand that student with special needs,
need to be taught differently or need some accommodations to enhance the
learning environment. This will be done depending to the subject and the
nature of the lesson.
In order to create a well-rounded learning atmosphere, teachers need to:
• Remember that students learn in different ways so they have to offer
a variety of activities (e.g. role-play, music and singing, word games
and quizzes, and outdoor activities);
• Maintain an organized classroom and limits distraction. This will help
students with special needs to stay on track during lesson and follow
instruction easily;
• Vary the pace of teaching to meet the needs of each student. Some
students process information and learn more slowly than others;

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 • Break down instructions into smaller, manageable tasks. Students
with special needs often have difficulty understanding long-winded
or several instructions at once. It is better to use simple, concrete
sentences in order to facilitate them understand what you are asking.
• Use clear consistent language to explain the meaning (and demonstrate
or show pictures) if you introduce new words or concepts;
• Make full use of facial expressions, gestures and body language;
• Pair a student who has a disability with a friend. Let them do things
together and learn from each other. Make sure the friend is not over
protective and does not do everything for the one with disability.
Both students will benefit from this strategy;
• Use multi-sensory strategies. As all students learn in different ways,
it is important to make every lesson as multi-sensory as possible.
Students with learning disabilities might have difficulty in one area,
while they might excel in another. For example, use both visual and
auditory cues.
Below are general strategies related to each main category of
disabilities and how to deal with every situation that may arise in the
classroom. However, the list is not exhaustive because each student
is unique with different needs and that should be handled differently:
Strategy to help students with developmental impairment:
• Use simple words and sentences when giving instructions;
• Use real objects that students can feel and handle. Rather than just
working abstractly with pen and paper;
• Break a task down into small steps or learning objectives. The student
should start with an activity that she/he can do already before moving
on to something that is more difficult;
• Gradually give the student less help;
• Let the student with disability work in the same group with those
without disability.
Strategy to help students with visual impairment:
• Help students to use their other senses (hearing, touch, smell and
taste) and carry out activities that will promote their learning and
development;
• Use simple, clear and consistent language;

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 • Use tactile objects to help explain a concept;
• If the student has some sight, ask him/her what he/she can see;
• Make sure the student has a group of friends who are helpful and who
allow him/her to be as independent as possible;
• Plan activities so that students work in pairs or groups whenever
possible;
Strategy to help students with hearing disabilities or communication
difficulties
• Always get the student’s attention before you begin to speak;
• Encourage the student to look at your face;
• Use gestures, body language and facial expressions;
• Use pictures and objects as much as possible.
• Keep background noise to a minimum.
Strategies to help students with physical disabilities or mobility difficulties:
• Adapt activities so that students who use wheelchairs or other
mobility aids, can participate.
• Ask parents/caregivers to assist with adapting furniture e.g the height
of a table may need to be changed to make it easier for a student to
reach it or fit their legs or wheelchair under
• Encourage peer support when needed;
• Get advice from parents or a health professional about assistive
devices if the student has one.
Adaptation of assessment strategies:
At the end of each unit, the teacher is advised to provide additional activities
to help students achieve the key unit competence. These assessment
activities are for remedial, consolidation and extension designed to cater for
the needs of all categories of students; slow, average and gifted students
respectively. Therefore, the teacher is expected to do assessment that fits
individual students.
Remedial activities After evaluation, slow students are provided with lower
order thinking activities related to the concepts learnt to
facilitate them in their learning.
These activities can also be given to assist deepening
knowledge acquired through the learning activities for
slow students.

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 Consolidation After introduction of any concept, a range number of
activities activities can be provided to all students to enhance/
reinforce learning.
Extended activities After evaluation, gifted and talented students can be
provided with high order thinking activities related to the
concepts learnt to make them think deeply and critically.
These activities can be assigned to gifted and talented
students to keep them working while other students are
getting up to required level of knowledge through the
learning activity.

1.2.4. Guidance on assessment

Assessment is an integral part of teaching and learning process. The
main purpose of assessment is for improvement of learning outcomes.
Assessment for learning/ Continuous/ formative assessment intends to
improve students’ learning and teacher’s teaching whereas assessment
of learning/summative assessment intends to improve the entire school’s
performance and education system in general.
Continuous/ formative assessment
It is an on-going process that arises during the teaching and learning process.
It includes lesson evaluation and end of sub unit assessment. This formative
assessment should play a big role in teaching and learning process. The
teacher should encourage individual, pair and group evaluation of the work
done in the classroom and uses appropriate competence-based assessment
approaches and methods.
Formative assessment is used to:
• Determine the extent to which learning objectives are being achieved
and competences are being acquired and to identify which students
need remedial interventions, reinforcement as well as extended
activities. The application activities are developed in the student
• book and they are designed to be given as remedial, reinforcement,
end lesson assessment, homework or assignment
• Motivate students to learn and succeed by encouraging students to
read, or learn more, revise, etc.
• Check effectiveness of teaching methods in terms of variety,
appropriateness, relevance, or need for new approaches and
strategies. Mathematics teachers need to consider various aspects

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 of the instructional process including appropriate language levels,
meaningful examples, suitable methods and teaching aids/ materials,
etc.
• Help students to take control of their own learning.
In teaching Mathematics, formative or continuous assessment should
compare performance against instructional objectives. Formative assessment
should measure the student’s ability with respect to a criterion or standard.
For this reason, it is used to determine what students can do, rather than
how much they know.
Summative assessment
The assessment can serve as summative and informative depending to its
purpose. The end unit assessment will be considered summative when it is
done at end of unit and want to start a new one.
It will be formative assessment, when it is done in order to give information
on the progress of students and from there decide what adjustments need
to be done.
The assessment done at the end of the term, end of year, is considered as
summative assessment so that the teacher, school and parents are informed
of the achievement of educational objective and think of improvement
strategies. There is also end of level/ cycle assessment in form of national
examinations.
When carrying out assessment?
Assessment should be clearly visible in lesson, unit, term and yearly plans.

• Before learning (diagnostic): At the beginning of a new unit or a

section of work; assessment can be organized to find out what
students already know / can do, and to check whether the students
are at the same level.
• During learning (formative/continuous): When students appear to be
having difficulty with some of the work, by using on-going assessment
(continuous). The assessment aims at giving students support and
feedback.
• After learning (summative): At the end of a section of work or a
learning unit, the Mathematics Teacher has to assess after the
learning. This is also known as Assessment of Learning to establish
and record overall progress of students towards full achievement.
Summative assessment in Rwandan schools mainly takes the form of

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 written tests at the end of a learning unit or end of the month, and
examinations at the end of a term, school year or cycle.
Instruments used in assessment.
• Observation: This is where the mathematics teacher gathers
information by watching students interacting, conversing, working,
playing, etc. A teacher can use observations to collect data on
behaviours that are difficult to assess by other methods such as
attitudes, values, and generic competences and intellectual skills.
It is very important because it is used before the lesson begins and
throughout the lesson since the teacher has to continue observing
each and every activity.
• Questioning
a) Oral questioning: a process which requires a student to respond
verbally to questions
b) Class activities/ exercise: tasks that are given during the learning/
teaching process
c) Short and informal questions usually asked during a lesson
d) Homework and assignments: tasks assigned to students by their
teachers to be completed outside of class.
Homework assignments, portfolio, project work, interview, debate, science
fair, Mathematics projects and Mathematics competitions are also the
different forms/instruments of assessment.

1.2.5. Teaching methods and techniques that promote active

learning
The different learning styles for students can be catered for, if the teacher
uses active learning whereby students are really engaged in the learning
process.
The main teaching methods used in Mathematics are the following:
• Dogmatic method (the teacher tells the students what to do, What to
observe, How to attempt, How to conclude)
• Inductive-deductive method: Inductive method is to move from
specific examples to generalization and deductive method is to move
from generalization to specific examples.
• Analytic-synthetic method: Analytic method proceeds from unknown
to known, ’Analysis’ means ‘breaking up’ of the problem in hand so

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 that it ultimately gets connected with something obvious or already
known. Synthetic method is the opposite of the analytic method.
Here one proceeds from known to unknown.
• Skills lab method: Skills lab method is based on the maxim “learning by
doing.” It is a procedure for stimulating the activities of the students
and to encourage them to make discoveries through practical
activities.
• Problem solving method, Project method and Seminar Method.
The following are some active techniques to be used in Mathematics:
• Group work
• Research
• Probing questions
• Practical activities (drawing, plotting, interpreting graphs)
• Modelling
• Brainstorming
• Quiz Technique
• Discussion Technique
• Scenario building Technique
What is Active learning?
Active learning is a pedagogical approach that engages students in doing
things and thinking about the things they are doing. Students play the key
role in the active learning process. They are not empty vessels to fill but
people with ideas, capacity and skills to build on for effective learning. Thus,
in active learning, students are encouraged to bring their own experience
and knowledge into the learning process.

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 The role of the teacher in active learning The role of students in active learning
• The teacher engages students • A learner engaged in active learning:
through active learning methods
• Communicates and shares relevant
such as inquiry methods, group
information with fellow students
discussions, research, investigative
through presentations, discussions,
activities, group and individual work
group work and other learner-
activities.
centred activities (role play, case
• He/she encourages individual, studies, project work, research and
peer and group evaluation of the investigation);
work done in the classroom and
• Actively participates and takes
uses appropriate competence-
responsibility for his/her own
based assessment approaches and
learning;
methods.
• Develops knowledge and skills in
• He provides supervised
active ways;
opportunities for students to
develop different competences • Carries out research/investigation by
by giving tasks which enhance consulting print/online documents
critical thinking, problem solving, and resourceful people, and
research, creativity and innovation, presents their findings;
communication and cooperation. • Ensures the effective contribution
• Teacher supports and facilitates the of each group member in assigned
learning process by valuing students’ tasks through clear explanation
contributions in the class activities. and arguments, critical thinking,
responsibility and confidence in
public speaking
• Draws conclusions based on the
findings from the learning activities.

Main steps for a lesson in active learning approach

All the principles and characteristics of the active learning process highlighted
above are reflected in steps of a lesson as displayed below. Generally, the
lesson is divided into three main parts whereby each one is divided into
smaller steps to make sure that students are involved in the learning process.
Below are those main part and their small steps:
1) Introduction
Introduction is a part where the teacher makes connection between the
current and previous lesson through appropriate technique. The teacher

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opens short discussions to encourage students to think about the previous
learning experience and connect it with the current instructional objective.
The teacher reviews the prior knowledge, skills and attitudes which have a link
with the new concepts to create good foundation and logical sequencings.
2) Development of the new lesson
The development of a lesson that introduces a new concept will go through
the following small steps: discovery activities, presentation of students’
findings, exploitation, synthesis/summary and exercises/application
activities.
• Discovery activity
Step 1:
• The teacher discusses convincingly with students to take responsibility of their
learning
• He/she distributes the task/activity and gives instructions related to the tasks
(working in groups, pairs, or individual to prompt / instigate collaborative
learning, to discover knowledge to be learned)
Step 2:
• The teacher let students work collaboratively on the task;
• During this period the teacher refrains to intervene directly on the knowledge;
• He/she then monitors how the students are progressing towards the
knowledge to be learned and boosts those who are still behind (but without
communicating to them the knowledge).
• Presentation of students’ findings/productions
• In this part, the teacher invites representatives of groups to present their
productions/findings.
• After three/four or an acceptable number of presentations, the teacher decides
to engage the class into exploitation of students’ productions.
• Exploitation of students’ findings/ productions
• The teacher asks students to evaluate the productions: which ones are correct,
incomplete or false
• Then the teacher judges the logic of the students’ products, corrects those
which are false, completes those which are incomplete, and confirms those

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Mathematics | Teacher's Guide | Senior Five | Experimental Version xxv
 which are correct.
• Institutionalization or harmonization (summary/conclusion/ and
examples)
• The teacher summarizes the learned knowledge and gives examples which
illustrate the learned content.
• Application activities
• Exercises of applying processes and products/objects related to learned unit/
sub-unit
• Exercises in real life contexts
• Teacher guides students to make the connection of what they learnt to real life
situations.
• At this level, the role of teacher is to monitor the fixation of process and
product/object being learned.
3) Assessment
In this step the teacher asks some questions to assess achievement of
instructional objective. During assessment activity, students work individually
on the task/activity. The teacher avoids intervening directly. In fact, results
from this assessment inform the teacher on next steps for the whole class
and individuals. In some cases, the teacher can end with a homework/
assignment. Doing this will allow students to relay their understanding on
the concepts covered that day. Teacher leads them not to wait until the
last minute for doing the homework as this often results in an incomplete
homework set and/or an incomplete understanding of the concept.

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PART II: SAMPLE OF A LESSON PLAN
School Name: …… Teacher’s name: …….
Term Date Subject Class Unit Lesson Duration Class size
Number
Term …. ……. Mathematics S5 Acc. 35 3 of 6 40 ….
minutes
Type of Special Educational needs ….
to be
catered for and the number of
students in each category
Unit title Applications of derivatives in Finance and in Economics
Key unit Apply differentiation in solving Mathematical problems that involve
competence financial quantities such as marginal cost, revenues and profits, elasticity
of demand and supply
Title of the Minimization of the cost function
lesson
Instructional Given a cost function by an equation, students will be able to use derivative
Objectives to find the value of the independent variable for which the cost function
is minimum. Hence, they will be able to precise the minimum value of the
cost function.

Plan for this This lesson will take place indoor. The teacher ensures that the students’
Class desks are arranged in a shape allowing group discussion, and the
environment is free of disturbance and free of any material not related to
the lesson.
Learning Chalkboard, calculator, manila paper, markers, pens, Mathematics note
Materials books, rulers
References Mathematics Student text book 5 for Accounting option

Timing for Description of teaching and learning Activity: Generic

each step the teacher organizes the students into groups Competences and
and gives clear instruction about discussion Cross-Cutting Issues
of Learning activity 3.2.1. The students discuss to be addressed
the activity for a while, then follows the
presentation by a student, from a group, the
whole class interact, the teacher helps the
students to capture the key point: the necessity
of finding a proper means for solving a problem
about minimization of a function in general, and
minimization of the cost function, in particular,
without trial and error. Individual activity,
whereby the students find, from daily life,
quantities to minimize, will close the lesson

Teacher activities Students’ activities

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 I n t r o d u c t i o n • Teacher distributes the • Students receive the • Students acquire
8 minutes flash cards, one per cards and start the positive attitude by
group, and invites the discussion, under the discussing without
students to discuss organization of the confrontation,
Learning activity 3.2.1., group task manager communication
already written on the skills, initiative
flash card. taking, leadership

Discovery • Teacher walks around • Students participate

activity, to provide assistance effectively in the
Skills:
presentation where necessary, and discussion, the group
and summary ensures that any group reporter writes
20 minutes member participates down, on a manila
effectively paper, the findings of Students acquire
the group knowledge, skills,
• Teacher invites the
positive attitude by
reporter of a group to • The group reporter
present, for the whole presents the discussing without
class, the findings of findings, with the confrontation,
the group support of all group communication skills,
members initiative taking, team
• Teacher invites the
students to interact • The whole class work spirit, value the
interacts, in an view of others’;
• Teacher invites the
organized way
students through
well-chosen questions • Students come with Problem solving;
to highlight the
the following:
main point from the Financial education:
discussion -What minimization is
-Necessity of finding good management
• Teacher requests the proper means to of money, through
students to write the solve a problem about minimizing the
Learning activity in minimization expenditure, the cost
their Mathematics note
• The students copy,
book from the flash cards,
the Learning activity
in their Mathematics
note book
3.Conclusion of the lesson
Application Teacher requests students Students find examples
activity and task to find, from daily life, in of situations where
for the next Economics field, examples minimization is
lesson of situations where the necessary
12 minutes minimization is necessary
Teacher requests students
to think about means of The students keep in
solving a minimization mind the task for the
problem and make next lesson
research in the library, or
internet

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PART III: UNIT DEVELOPMENT

Unit 1 MATRICES AND DETERMINANTS

1.1. Key unit competence

Use matrices and determinants notations and properties to solve simple
production, financial economical, and mathematical related problems

1.2. Prerequisite
The students will perform well in this unit if they have a good background on:
• The use of simple and correct terminology in the description of a fact;
• Carrying out numerical calculations correctly;
• Classifying a group of quantities according to specific criteria.
1.3. Cross-cutting issues to be addressed
• Inclusive education: promote the participation of all students while
teaching and learning
• Peace and value Education: During group activities, the teacher will
encourage the students to help each other and to respect opinions of
their classmates.
• Gender: Give equal opportunities to both girls and boys to participate
actively in all learning and application activities from the beginning to
the end of the lesson.

1.4 Guidance on introductory activity

• Form small groups of students, guide them to work on the introductory
activity;
• Through class discussions, let students think of different possible
criteria to classify given data and justify their validity to the entire class;
• The teacher should walk around to all groups and provide pieces of
advice where necessary;
• After a given time, invite students to present their findings and

Mathematics | Teacher's Guide | Senior Five | Experimental Version 1

 harmonize them.
• Try to arouse students’ curiosity about the content of this first unit.

1.5 List of lessons

Lesson title/ Number
Headings # Learning objectives
sub-headings of periods
Introductory activity Arouse the curiosity of
students on the content
1.1. Generalities of unit 1. 1
on matrices. 1. Definitions and Apply definition of
notations matrices to determine
types and order of
matrices
2. Equality of Apply the conditions 1
matrices for two matrices to be
equal in solving related
problems
1.2 Operations 3. Addition and Perform addition and 1
on matrices subtraction subtraction on matrices
of order less than or
equal to 3.

4. Scalar Perform scalar 1

multiplication multiplication of matrices
5. Multiplication Perform multiplication 2
of matrices, for on matrices of order less
n≤3 than or equal to 3.

6. Inversion of Determine the inverse 3

matrices of a matrix using row
operations
7. Definition and Compute the 2
calculation of determinants of order n
determinants (n ≤ 3) using expansion
1.3 Determinants of matrices of by cofactors
of matrices order two or
n≤3 order three

8. Properties of Evaluate the properties 2

determinants of determinants of
matrices

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 9. Inverse of a Determine the inverse 3
square matrix of a matrix of order less
for n ≤ 3 than or equal to 3.

1.4 Applications 10. Solving linear Apply the inverse matrix 3

of matrices and simultaneous and multiplication of
determinants equations using matrices to solve systems
inverse of a of linear equations
matrix
11. Solving system Apply the determinants 3
of linear in solving systems of
equation using linear equations
Cramer’s rule
1.5 End unit assessment 2

Answer to Introductory activity

From the analysis of the table, students will come with the following answers:
a) 450,000
b) 10,000

Lesson 1 : Definitions and notations

a) Learning objectives:
Apply definition of matrices to determine types and order of matrices

b) Teaching resources:
the following are possible resources that the teacher and the students can
use in the teaching –learning process:
–– Calculator;
–– Poster;
–– Manilla paper;
–– Student’s book;
–– Eventual reference text book about matrices
c) Prerequisite:
Students will perform better in this lesson if:
–– They can perform numerical calculations correctly, mentally or using a
calculator.
–– They can sort easily objects according to given criteria.

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Mathematics | Teacher's Guide | Senior Five | Experimental Version 3
 –– They have good background on tables with double entries.
d) Learning activities
–– Invite students to work in groups and do the learning activity 1.1.1 found
in their Mathematics Student books;
–– Move around in the class for facilitating students where necessary and
give more clarification on eventual challenges they may face during
their work, verify and identify groups with different working steps;
–– Through well-chosen questions, bring the students to discover the
components of a matrix.
–– Give time for individual thinking, then allow the students to exchange
their view, under your vigilance.
–– Invite one member from each group with different working steps to
present their work where they must explain the working steps;
–– As a teacher, harmonise the findings from presentations of students.
–– Use different probing questions and guide them to explore the content
and examples given in the student’s book and lead them to discover
how to define and differentiate types of matrices
–– After this step, guide student-teachers to do the application activity
1.1.1 and evaluate whether lesson objectives are achieved

Answers of learning activity 1.1.1

a) - Type of shirts;
-Week period
12 8 5 
b)  
 9 3 0
c) A matrix consists of entries arranged in rows and columns
e) Answers of application activity 1.1.1.
1) a) 2 × 3
b) 1× 3
c) 2 ×1
 7 15 
2) M =   , where rows stand for weeks, and columns stand for
4 9 
men’s shoes and ladies’ shoes.

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 Lesson 2 : Equality of matrices

a) Learning objectives:
Apply the conditions for two matrices to be equal in solving related problems

b) Teaching resources:
The following are possible resources that the teacher and the students can
use in the teaching –learning process about equality of matrices
–– Calculator;
–– Manilla paper;
–– Student’s book;
–– Eventual reference text book about matrices

c) Prerequisite:
Students will perform better in this lesson if:
–– They can easily solve simple equations;
–– From lesson1 of this unit, they understood the concept of order of a
matrix, the rows and columns of a matrix;
–– They can perform numerical calculations correctly, mentally or using a
calculator;
–– They can discover easily similarity and difference between two objects;
–– They have good English background on terminologies, such as
“corresponding”.
d) Learning activities
–– Invite students to work in groups and do the learning activity 1.1.2
found in their Mathematics Student books;
–– Move around in the class for facilitating students where necessary and
give more clarification on eventual challenges they may face during
their work, verify and identify groups with different working steps;
–– Through well-chosen questions, bring the students to discover the
conditions for two matrices to be equal;
–– Give time for individual thinking, then allow the students to exchange
their view, under your vigilance.
–– Invite one member from each group with different working steps to
present their work where they must explain the working steps;

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Mathematics | Teacher's Guide | Senior Five | Experimental Version 5
 –– As a teacher, harmonise the findings from presentations of students.
–– Use different probing questions and guide them to explore the content
and examples given in the student’s book and lead them to discover
how to define and differentiate types of matrices
–– After this step, guide student-teachers to do the application activity
1.1.2 and evaluate whether lesson objectives are achieved

Answers of learning activity 1.1.2

a) The three matrices are of the same order 2 × 2
b) The corresponding entries are the same
c) The nature of elements is the same in matrices B and C
d) The matrices B and C are equal
e) For two matrices to be equal, the entries of the two matrices must be the
same, and the corresponding entries must be equal.

f) Answers of application activity 1.1.2.

1.a) A ≠ B ;
b) A = B
−1 1
2. x = 1 or x = ; y = ; z = −5 ; t = 0 or t = −1
2 3

Lesson 3 : Addition and subtraction of matrices

a) Learning objectives:
Perform addition and subtraction on matrices of order less than or equal to
3.
b) Teaching resources:
The following are possible resources that the teacher and the students can
use in the teaching –learning process about addition and subtraction of
matrices:
–– Calculator;
–– Manilla paper;
–– Student’s book;
–– Eventual reference text book about matrices

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c) Prerequisite:
Students will perform better in this lesson if:
–– They can easily solve simple equations;
–– From lesson 1 of this unit, they understood the concept of order of a
matrix, the rows and columns of a matrix;
–– They can perform numerical calculations correctly, mentally or using a
calculator;
d) Learning activities
–– Invite students to work in groups and do the learning activity 1.2.1
found in their Mathematics Student books;
–– Through well-chosen questions, bring the students to discover the
conditions for two matrices to be added;
–– Give time for individual thinking, then allow the students to exchange
their view, under your vigilance.
–– As students are working in groups, overcome their possible challenge
by providing relevant assistance.
–– Use different probing questions and guide them to explore the content
and examples given in the student’s book and lead them to discover
how to find the sum and difference of matrices.
–– After this step, guide student-teachers to do the application activity
1.2.1 and evaluate whether lesson objectives are achieved.

Answers of learning activity 1.2.1

a) S and T have the same order 2 × 3

 13 13 15 
b)  
18 30 30 
c) - The same order;
-Adding the corresponding entries

e) Answers of application activity 1.2.1.

1.a) No, they cannot be added because they have different orders;
 12 46   −10 −12 
b) A + B 
= = ; A − B  
 22 9   6 −3 

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Mathematics | Teacher's Guide | Senior Five | Experimental Version 7
2.a) Addition;
 894 812 
b)  
 971 799 
c) i) 1, 611 day scholars;
ii) 894 girls are boarders.

Lesson 4 : Scalar multiplication

a) Learning objectives:
Perform scalar multiplication of matrices
b) Teaching resources:
the following are possible resources that the teacher and the students can
use in the teaching –learning process about scalar multiplication of matrices:
–– Calculator;
–– Manilla paper;
–– Student’s book;
–– Eventual reference text book about matrices
c) Prerequisite:
Students will perform better in this lesson if:
–– They can easily solve simple equations;
–– From lesson 1 of this unit, they understood the concept equality of
matrices;
–– They can perform numerical calculations correctly, mentally or using a
calculator;
d) Learning activities
–– Use example from daily life, such as the VAT (value added tax), to
introduce the lesson;
–– Invite students to work in groups and do the learning activity 1.2.2
found in their Mathematics Student books;
–– Through well-chosen questions, bring the students to discover how to
multiply a matrix by a scalar;
–– Let students think individually, first, then allow the students to
exchange their view in their respective groups, under your vigilance.
–– Encourage each member to participate actively in the group;
–– As students are working in groups, overcome their possible challenge
by providing relevant assistance.

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 –– Use different probing questions and guide them to explore the content
and examples given in the student’s book and lead them to discover
how to find the sum and difference of matrices.
–– After this step, guide student-teachers to do the application activity
1.2.2 and evaluate whether lesson objectives are achieved.

Answers of learning activity 1.2.2

a) By multiplying the item by 0.18
b) Multiplying the matrix by 1.18
c) Each entry of the matrix is multiplied by 1.18
d) M ' =(150 ×1.18 120 ×1.18 300 ×1.18 ) =(177 141.6 354 )

e) Answers of application activity 1.2.2

 22 11 
1.a) 4 A − 5 B =
 
 −5 −15 
 2 10 
b) 2( A + B) = 
 20 6 
 30 42 
2. D =  
 38 56 
Note: After doubling the number of each item to be sold, the number of
customers is going to be increased.

Lesson 5 : Multiplication of matrices

a) Learning objectives:
Perform multiplication on matrices of order less than or equal to 3.
b) Teaching resources:
The following materials may be used in the teaching-learning process:
–– Calculator,
–– Poster,
–– Manilla paper,
–– Student’s book;
–– Any reference text book containing multiplication of matrices

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Mathematics | Teacher's Guide | Senior Five | Experimental Version 9
c) Prerequisites:
Students will perform better in this lesson if:
–– They can easily solve simple equations;
–– They can multiply two matrices;
–– They can perform numerical calculations correctly, mentally or using a
calculator;
d) Learning activities
–– Invite students to work in groups and do the activity 1.2.3 found in their
Mathematics Student books
–– Use example from daily life, such as shopping in different supermarkets,
to introduce the lesson;
–– Encourage each member to participate actively in the group;
–– Ensure that all the students are given opportunity to communicate
through presentation of the findings to the whole class;
–– As a teacher, harmonize the findings from presentation: Two matrices
A and B can be multiplied together if and only if the number of columns

of A is equal to the number of rows of B. Am×n × Bm× p =

M m× p
–– Use different probing questions and guide them to explore the content
and examples given in the student’s book and lead them to discover
how to find the product of matrices of order 2 and order 3.
–– After this step, guide student-teachers to do the application activity
1.2.3 and evaluate whether lesson objectives are achieved.

Answers of learning activity 1.2.3

a) The number of columns of M equals the number of rows of P
b) Shopping bill of Agnes:
In supermarket S1 : 1(1800) + 3(2500) + 2(1500) =
12300 FRW
In supermarket S2 : 1(1600) + 3(2000) + 2(1200) =10000 FRW

Shopping bill of Agnes:

In supermarket S1 : 2(1800) + 4(2500) + 3(1500) =
18100 FRW
In supermarket S2 : 2(1600) + 4(2000) + 3(1200) 14800 FRW
=

12300 10000 
C =  , where row one shows the bills for Agnes
18100 14800 

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 in supermarkets S1 and S2 ;
row two shows the bills for Agnes in supermarkets S1 and S2
c) Two rows and two columns
 1800 1600 
1 3 2   12300 10000 
d) M .P =  .  2500 2000   
 2 4 3   1500 1200  18100 14800 
 
e) Answers of Application activity 1.2.3.
1. a) A and B are not conformable for multiplication
b) A.B = ( 29 22 )

 13 5 
2. M .N =  
 64 20 

Lesson 6 : Inversion of matrices

a) Learning objectives:
Determine the inverse of a matrix using row operations
b) Teaching resources:
The following materials may be used in the teaching-learning process:
–– Calculator,
–– Poster,
–– Manilla paper,
–– Student’s book;
–– Any reference text book containing inversion of matrices
c) Prerequisites:
Students will perform better in this lesson if:
–– They can easily solve simple equations;
–– They can multiply two matrices;
–– They can perform numerical calculations correctly, mentally or using a
calculator;
d) Learning activities
–– Invite students to work in groups and do the activity 1.2.4 found in their
Mathematics Student books;
–– From the product of two matrices yielding to the unit matrix, introduce
the concept of the inverse of a matrix;

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Mathematics | Teacher's Guide | Senior Five | Experimental Version 11
 –– Suggest, for discussion by students, many simple examples for the
students to grasp the technique of finding the inverse of a matrix;
–– Encourage each member to participate actively in the group;
–– Ensure that all the students are given opportunity to communicate
through presentation of the findings to the whole class;
–– As a teacher, harmonize the findings from presentation by helping
students to realise that calculating matrix inverse of matrix A , is to
−1
find matrix A such that A= . A−1 A=−1
. A I ,where I is identity matrix.
–– Use different probing questions and guide them to explore the content
and examples given in the student’s book and lead them to discover
how to determine the inverse of a matrix of order two and order three.
–– After this step, guide student-teachers to do the application activity
1.2.4 and evaluate whether lesson objectives are achieved.

Answers of learning activity 1.2.4

1 0 2  3 2 2  1 0 0
a)= 1     ;
A.B  0 1 −3  .  3 2=
−3  0 1 0
5   0 0 1
 1 −1 0   1 −1 −1   
order: 3 × 3
e) Each entry on the main diagonal (from top left to bottom right) is 1 and
any entry outside the main diagonal is 0
f) In (ii) and (iii)
g) M . X = I , where I is the identity matrix.
h) Answers of Application activity 1.2.4.
i) 1 0 0 1 1 1 1 0 0 
   
a)  0 1 0  ; the augmented matrix is 1 2 1 0 1 0 
0 0 1 1 1 2 0 0 1 
   
b) Applying the following row operations:
(1): R2 → R2 − R1 and R3 → R3 − R1
(2): R1 → R1 − R3 from the result in (1)
(3): R1 → R1 − R2 from the result in (2), we find:

 1 0 0 3 −1 −1
 
 0 1 0 −1 1 0 
 0 0 1 −1 0 1 
 

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  3 −1 −1
 
c) M −1 =  −1 1 0 
 −1 0 1 
 
1 1 1   3 −1 −1  1 0 0   3 −1 −1 1 1 1 1 0 0
         
d) 1 2 1  .  −1 1 0  =
 0 1 0  and 
−1 1 0  . 1 2 1 = 0 1 0
1 1 2   −1 0 1   0 0 1   −1 0 1  1 1 2   0 0 1 
      

Lesson 7 : Definition and calculations of determinants of

orders 2 × 2 and 3 × 3
a) Learning objectives:
Find the value of the determinant of order n (n ≤ 3) using expansion by
cofactors
b) Teaching resources:
The following materials may be used in the teaching-learning process:
–– Calculator,
–– Poster,
–– Manilla paper,
–– Student’s book;
–– Any reference text book containing determinants of matrices
c) Prerequisites:
Students will perform better in this lesson if:
–– They can perform numerical calculations correctly, mentally or using a
calculator
–– They can calculate determinants 2 × 2 without problems.
Knowing to perform elementary operations using calculator will ensure the
students to perform better in this lesson
d) Learning activities
–– Through learning activity 1.3.1, bring students to distinguish between
singular matrix and non-singular matrix;
–– Move around in class for facilitating students where necessary and
give more clarification on eventual challenges they may face during the
calculation of the determinant 2 × 2 ;
–– After defining the determinant of order 3 × 3 , ask the students if it is
easy to master the sum and difference of six terms, each with3 factors.
–– Let them work out to discover the necessity of mnemonic techniques
for calculation of a 3 × 3 determinant.

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Mathematics | Teacher's Guide | Senior Five | Experimental Version 13
 –– Ensure that the discussion is done without confrontation.
–– Invite one member from each group with different working steps to
present their work where they must explain the working steps;
–– As a teacher, harmonize the findings from presentation
–– As a teacher, use different probing questions, guide the students to
explore the content and examples given in the student’s book, and
bring them to master Sarrus’ method and expansion by cofactors.
–– After this step, guide student-teachers to do the application activity
1.3.1 and evaluate whether lesson objectives are achieved.

Answers of learning activity 1.3.1

1. a) Each entry is multiplied by 4
b) The entries of a row or column are scalar multiples of the
corresponding entries of another
row or column
2. Matrix A is singular
e) Answers of Application activity 1.3.1.
a c  a c
1.   is a matrix (a table of values in array form) but, is a
b d  b d
determinant, the number ad − bc
2. Sarrus’ method:

1 1 1
1 2 3 =(4 + 3 + 3) − (2 + 9 + 2) =−3
1 3 2

Expansion by cofactors:
Expanding, for example along the first row:

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14 Mathematics | Teacher's Guide | Senior Five | Experimental Version
1 1 1
2 3 1 3 1 2
1 2 3 = − + = (4 − 9) − (2 − 3) + (3 − 2) − 3
3 2 1 2 1 3
1 3 2

5 0
3. a) det X= = 45 ≠ 0 ; X is not singular.
−3 9

4k −20k
b) det X = = 20k 2 − 20k 2 = 0 ; X is singular
−k 5k

Lesson 8 : Properties of determinants

a) Learning objectives:
Use the properties of determinants of matrices to simplify the evaluation of
a determinant
b) Teaching resources:
The following materials may be used in the teaching-learning process:
–– Calculator,
–– Poster,
–– Manilla paper,
–– Student’s book;
–– Any reference text book containing determinants of matrices
c) Prerequisite:
Students will perform better in this lesson if:
–– They can easily calculate the determinant by using the Sarrus’ method
or expansion by cofactor;
–– They have mastered the multiplication of matrices
–– They can perform numerical calculations correctly, mentally or using a
calculator;
d) Learning activities
–– Guide the students through learning activity 1.3.2 to discover the
properties of a determinant;
–– Give for students to practice, many examples.

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Mathematics | Teacher's Guide | Senior Five | Experimental Version 15
 –– As they are discussing the problems, check the participation of each
member in his/her group.;
–– Help students to elaborate the main points of the lesson
–– Guide the students to do the application activity 1.3.2 and evaluate
whether lesson objectives are achieved.

Answers of learning activity 1.3.2

1. a) det A = 0 ;
b) detB = 0
2. a) rows 1 and 2 of matrix A are interchanged to obtain matrix B
−1 2 3 −5
b) = −1 ; =1
3 −5 −1 2
c) We conclude that the determinant of the matrix B is the opposite of
the determinant of the matrix
e) Answers of Application activity 1.3.2.
1. a) False
b) True
1 1 1   1 1 1 
2. a) 1 2 1  →  0 1 0 
1 1 2   0 0 1 
   

1 1 1 1 1 1
b) 1=2 1 0=1 0 1
1 1 2 0 0 1

Lesson 9 : Inverse of a matrix

a) Learning objectives:
Determine the inverse of a square matrix of order less than or equal to 3.
b) Teaching resources:
The following materials may be used in the teaching-learning process:
–– Calculator,
–– Poster,
–– Manilla paper,
–– Student’s book;
–– Any reference text book containing determinants of matrices

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16 Mathematics | Teacher's Guide | Senior Five | Experimental Version
c) Prerequisites:
Students will perform better in this lesson if:
–– They can easily perform elementary row operations;
–– They have mastered the multiplication of matrices
–– They are able to calculate the determinant of a matrix of order two and
order three
–– They can perform numerical calculations correctly, mentally or using a
calculator;
d) Learning activities
–– Guide students, through learning activity 1.4.1 to use elementary row
operations to find the inverse of a matrix.
–– Then guide the students of finding the inverse of a matrix by using the
method of cofactors.
–– Ask the students to compare the two results
–– From the two results, ask the students to conclude about how to find
the inverse of a matrix
–– Use different probing questions and guide them to explore the content
and examples given in the student’s book and lead them to discover
how to apply properties of the inverse of a matrix of order 2 and order 3.
–– After this step, guide student-teachers to do the application activity
1.4.1 and evaluate whether lesson objectives are achieved.

Answers of learning activity 1.4.1

1 1 1 1 0 0   1 1 1 1 0 0 
   
a) 1 2 1 0 1 0  →  0 1 0 −1 1 0 
1 1 2 0 0 1   0 0 1 −1 0 1 
   

 1 1 0 2 0 −1  1 0 0 3 −1 −1
   
→  0 1 0 −1 1 0  →  0 1 0 −1 1 0 
 0 0 1 −1 0 1   0 0 1 −1 0 1 
   

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Mathematics | Teacher's Guide | Senior Five | Experimental Version 17
  3 −1 −1
b) The inverse is −1  
A =  −1 1 0 
 −1 0 1 
 

1 1 1 1 1 1
c) i) 1=2 1 0=1 0 1
1 1 2 0 0 1

 2 1 1 1 1 2
+ − + 
 1 2 1 2 1 1
 1  3 −1 −1
1 1 1 1 1  
−
ii) C = + − = −1 1 0 
 1 2 1 2 1 1  

   −1 0 1 
+1 1
−
1 1
+
1 1
 2 1 1 1 1 2 


 3 −1 −1
 
iii) Adj ( A) = C T =  −1 1 0 
 −1 0 1 
 

 3 −1 −1
1  
iv) X = Adj ( A) =  −1 1 0 
det A  −1 0 1 
 

d) A−1 = X

e) Answers of Application activity 1.4.1.

1  2 −6 
1) a ) A−1 =
 
10  −5 20 
 3 2 −8 
1  
b) M −1
=  4 6 −19 
5 
 −5 −5 20 

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18 Mathematics | Teacher's Guide | Senior Five | Experimental Version
  5 1 13 
 2 5 − 14   − 12 4
−
12 
1   
i) A−1
2)=  1 −3 4  , −1  3 1 3
11  B = − −
 −1 3 7   4 4 4
 
 1 0
2 

 3 3 
 3 7 2 
 
ii) ( A ) =  1 0 2 
−1 −1

0 1 1
 

 2 5 −14 
1  
(10 A)
−1
iii)=  1 −3 4 
110 
 −1 3 7 

Lesson 10 : Solving simultaneous linear equations by using

inverse of matrix

a) Learning objectives:
Use the inverse matrix and the multiplication of matrices to solve systems of
linear equations.
b) Teaching resources:
The following materials may be used in the teaching-learning process:
–– Calculator,
–– Poster,
–– Manilla paper,
–– Student’s book;
–– Any reference text book containing determinants of matrices
c)Prerequisites:
Students will learn easily in this lesson, if they have a good background on:
• definition, size and types of matrices
• operations (Addition, subtraction and multiplication) on matrices.
• how to calculate the determinant of a matrix of order 2and order 3.
• how to calculate the inverse of the matrix

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Mathematics | Teacher's Guide | Senior Five | Experimental Version 19
 d)Learning activities
–– Through well-chosen questions, help the students to model the
problem in learning activity 1.4.2 by matrices and simultaneous linear
equations
–– From the product of two matrices, help the students using inverse
matrix to make a matrix the subject of the formula.
–– Finally, use the product and the equality of matrices to draw the
conclusion.
–– As students are discussing in groups, circulate to provide assistance,
where necessary.
–– Let a student chosen at random present his/her findings,
–– Finally, help the students to capture the main points of the lesson

Answers of learning activity 1.4.2

4 x + 3 y =
53
a) 
11x + y = 37

 4 3  53   x
b) A  =
=  ; B = ; X  
11 1   37   y
c)  4 3  x  =  53 
    
11 1  y   37 
1  1 −3 
d) A−1 = −  ;
29  −11 4 
1  1 −3  53   2  ;
−    =  
29  −11 4  37  15 
x=2
e) 
 y = 15

e) Answers of Application activity 1.4.2.

 4 7  x   34 
a)the system is equivalent to    =  
 3 −2  y   11 

4 7  −1 −1  −2 −7   x  −1  −2 −7  34   5 
Let A =   . Then A =  =  and   =     
 3 −2  29  −3 4   y  29  −3 4  11   2 

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20 Mathematics | Teacher's Guide | Senior Five | Experimental Version
 x = 5

y = 2
 1 −2 1  x   −2 
    
b) The system can be expressed as  3 1 −2  y  =
7
 1 3 −1  z   2 
    

 1 −2 1  5 1 3
  −1 1 
Let A  3 1 −2  ; then
= = A  1 −2 5 
 1 3 −1  11  
   8 −5 7 
 3 
 
 x  5 1 3  −2   11 
  1    −6 
y =
  11  1 −2 5  7  = 
z  8 −5 7  2   11 
    
 −37 
 11 

Lesson 11 : Solving simultaneous linear equations by

Cramer’s rule
a) Learning objectives:
Determine whether a system is a Cramer’s system or not, and solve the
system by Cramer’s rule
b) Teaching resources:
The following materials may be used in the teaching-learning process:
–– Calculator,
–– Poster,
–– Manilla paper,
–– Student’s book;
–– Any reference text book containing determinants of matrices
c) Prerequisite:
Students will perform better in this lesson if:
–– They can easily solve simple simultaneous linear equations by
elimination method;
–– They have mastered the calculation of determinants;
–– They can perform numerical calculations correctly, mentally or using a
calculator;

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Mathematics | Teacher's Guide | Senior Five | Experimental Version 21
 d) Learning activities
–– Through well-chosen questions, bring the students to derive the
Cramer’s rule
–– Provide, for discussion by students, many simple examples for the
students to master the Cramer’s rule;
–– Ensure each member participates actively in the group;
–– Give to all the students the same opportunity to communicate through
presentation of the findings to the whole class;

Answers of learning activity 1.4.3

b ' ax + b ' by =b 'c
1.a)
−ba ' x − bb ' y = −bc '
b) (ab '− ba ') x =
cb '− bc '

cb '− bc '
c) x = ;
ab '− ba '
c b
c' b'
d) x =
a b
a ' b'
− a' ax − a ' by =−a ' c
2. a)
aa ' x + ab ' y =ac '
b) (−a ' b + ab ') y =
− a ' c + ac '

ac '− ca '
c) y
= ;ab'− ba' ≠ 0
ab '− ba '
a c
a' c'
d) y =
a b
a' b'

4 2 3 4
14 −4 5 14
3. x = 2 ; y =
= = −1
3 2 3 2
5 −4 5 −4

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22 Mathematics | Teacher's Guide | Senior Five | Experimental Version
 e) Answers of Application activity 1.4.3.
34 7 4 34
a)
11 −2 3 11
=x == 5; y = 2
4 7 4 7
3 −2 3 −2
−2 −2 1 1 −2 1 1 −2 −2
7 1 −2 3 7 −2 3 1 7
b)
2 3 −1 3 1 2 −1 6 1 3 2 −37
=x = ;y= = = − ;z =
1 −2 1 11 1 −2 1 11 1 −2 1 11
3 1 −2 3 1 −2 3 1 −2
1 3 −1 1 3 −1 1 3 −1

1.6. Summary of unit 1

Unit 1 dealt with matrices and determinants. We studied how data can be
recorded using matrices, and how information from the field of Economics
and Finance can be drawn from a matrix. Different operations can be
performed on matrices. We ended by determinants as means in solving
problems, such as simultaneous linear equations, that can be modeled by
matrices.
Therefore,
• A matrix is a rectangular arrangement of numbers, in rows and
columns, within brackets ( ) or [ ] . A matrix is denoted by a capital
letter: A, B, C, ….
• The numbers in the matrix are called entries or elements.
• If matrix A has n rows and p columns, then we say that the matrix A is
of order n × p , read n by p, where the product n × p is the number of
entries in the matrix.
• If A is a matrix of order n × p , then A can generally be written as
( )
A = aij ,where i and j are positive integers, and; 1 ≤ i ≤ n ; 1 ≤ j ≤ p .
• A matrix with only one row is said to be a row matrix; that is a matrix
of order 1× p . For example, ( 2 4 7 ) is a row matrix.
• A matrix with only one column is said to be a column matrix; that is a
1
 
matrix of order n ×1 . For example,  3  is a column matrix.
6
 

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Mathematics | Teacher's Guide | Senior Five | Experimental Version 23
 • A square matrix is a matrix in which the number of rows is equal to
the number of columns; that is, matrix A of order n × p is a square
matrix if and only if n = p ;
In this case, instead of saying a matrix of order n × n , we, sometimes, simply
say a matrix of order n .
( )
If A = aij is a square matrix of order 2 , then i and j assume values in the
a a12 
set {1, 2} .Therefore, ( aij ) =  11 .
 a21 a22 

( )
In the same way, if A = aij is a square matrix of order 3 , then i and j
 a11 a12 a13 
assume values in the set {1, 2, 3} .Therefore, ( aij ) =  a21 a22 a23  .
a a32 a33 
 31
 a11 a12 a13   b11 b12 b13   a11 b=
= 11 a12 b=
12 a13 b13 
•      
 a21 a22=a23   b21 b22 b23  , then=
 a21 b=
21 a22 b=
22 a23 b23 
a a32 a33  b b33  a a33 b33 
 31  31 b32  31 b=
= 31 a32 b=
32

 a11 a12 a13   b11 b12 b13 

   
=• If A =a21 a22 a23  and B  b21 b22 b23  , then
a  b b33 
 31 a32 a33   31 b32

 a11 a12 a13   b11 b12 b13   a11 + b11 a12 + b12 a13 + b13 
     
A + B=  a21 a22 a23  +  b21 b22 b23 =  a21 + b21 a22 + b22 a23 + b23 
a a32 a33   b31 b32 b33  a +b a32 + b32 a33 + b33 
 31  31 31

 a11 a12 a13   b11 b12 b13   a11 − b11 a12 − b12 a13 − b13 
     
A − B=  a21 a22 a23  −  b21 b22 b23 =  a21 − b21 a22 − b22 a23 − b23 
a a32 a33   b31 b32 b33  a −b a32 − b32 a33 − b33 
 31  31 31

 a11 a12 a13   b11 b12 b13 

  
A.B =a21 a22 a23  .  b21 b22 b23 
a  
 31 a32 a33   b31 b32 b33 
 a11b11 + a12b21 + a13b31 a11b12 + a12b22 + a13b32 a11b13 + a12b23 + a13b33 
 
=  a21b11 + a22b21 + a23b31 a21b12 + a22b22 + a23b32 a21b13 + a22b23 + a23b33 
 a31b13 + a32b33 + a33b33 
 a31b11 + a32b21 + a33b31 a31b12 + a32b22 + a33b32

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24 Mathematics | Teacher's Guide | Senior Five | Experimental Version
  a11 a12 a13   ka11 ka12 ka13 
   
• If A =a21 a22 a23  , then kA  ka21 ka22 ka23 
a   ka ka33 
 31 a32 a33   31 ka32
 a11 a12 a13   a11 a21 a31 
   
• If A =a21 a22 a23  , then AT  a12 a22 a32 
a  a a33 
 31 a32 a33   13 a23
The transpose of a matrix A of order n × p is the matrix denoted AT whose
rows are the columns of A and whose columns are the rows of A .
• A matrix, in which all the entries are zeros is said to be the null matrix
or the zero matrix.
• A square matrix with each element along the main diagonal (from
the top left to the bottom right) being equal to and with all other
elements being is said to be the identity matrix, it is denoted by I;
• For any square matrix A of order n × n , and the identity matrix I of
order n × n , we have:
• A.I = A and I .A = A , that is, I is the identity element for
multiplication of matrices.  a11 a12 a13 
 
• Consider an arbitrary 3 × 3 matrix, A =  a21 a22 a23  , the
a a33 
 31 a32
determinant of A is defined as follows:
a11 a12 a13
A = a21 a22 a23 = (a11a22 a33 + a12 a23 a31 + a13a 21a 32 ) − (a31a22 a13 + a32 a23 a11 + a33 a21a12 )
a31 a32 a33

• Let A be a square matrix of order 2 × 2 or 3 × 3 .Then the inverse of

A Can also be calculated through the following four steps:
1) Find the determinant of A , that is det A ;
2) Find the matrix C of cofactors of A : each entry of A is replaced by its
cofactor.
3) Find the adjoint of matrix A , denoted, Adj ( A) : the transpose of the
matrix of cofactors;

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Mathematics | Teacher's Guide | Senior Five | Experimental Version 25
 1
4) The inverse of matrix A is A−1 = Adj ( A) , where matrix A is
regular, or invertible. det A

• To solve the two simultaneous linear equations in two unknowns x

and y , Cramer’s rule requires to go through the following steps:

ax + by = c
1) Arrange the equations to get 
a′x + b′y =c′

a b
D
Write down and calculate the principal determinant = = ab′ − a′b
a ′ b′
If D = 0 , then the system has no solution or infinitely many solutions; the
system is not a Cramer’s system.
If D ≠ 0 , then the system is a Cramer’s system and has unique solution,
proceed to the next step:
c b a c
2) Write down and calculate: D=x c′ b=′ cb′ − c′b and D=y = ac′ − a′c
a ′ c′

D D
3) Write down and calculate x = x and y = y ; the solution set of
the D D

 D D  
simultaneous equations is S =  x , y  
 D D  

In the same way, for the three simultaneous linear equations in three
ax + by + cz = d

unknowns, x , y and z ,a′x + b′y + c′z = d′
a′′x + b′′y + c′′x = d ′′


a b c
The principal determinant is D = a′ b′ c′
a′′ b′′ c′′

If D = 0 , then the system is not a Cramer’s system, it may have zero solution
or infinitely many solutions.
If D ≠ 0 , then the system is a Cramer’s system and has unique solution; the

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26 Mathematics | Teacher's Guide | Senior Five | Experimental Version
  D Dy Dz  
solution set is S =  x , ,   , where
 D D D  
d b c a d c a b d
= Dx d= ′ b′ c′ , Dy a= ′ d ′ c′ , Dz a′ b′ d ′
d ′′ b′′ c′′ a′′ d ′′ c′′ a′′ b′′ d ′′

1.7. Additional information for the teacher

It is advisable for the teacher to read ahead and master the material about
the unit so as to not be embarrassed by students’ questions. In particular,
the teacher should explain clearly the difference between matrix and
determinant, how rows and columns are counted, differentiating the
main diagonal from the other diagonal, differentiating between minor and
cofactor of an entry in a square matrix, the expansion by cofactor along a
row or a column.
It is very necessary to help students to learn this unit and finish all given
activities in the given time. The teacher may prepare many activities to
students to be performed in groups at home and then present them in
written form to be marked after. This strategy will help teacher to cover all
required topics and concepts in this unit. The teacher will use one example
and one application task while teaching and let students do the remaining
tasks themselves in groups and after class.

27
Mathematics | Teacher's Guide | Senior Five | Experimental Version 27
 1.8. Answers of End unit assessment

1.a) 2 × 3 ; b) 1× 2

5 6 11
2. a) m = 2 b) x =
− ;k = − ;y=
−1; z = ; t =
−3
2 5 20

 −8 1   1 −4 
3.a)   b)  
 5 8 7 0 

4 −1 17 7 1 0
=
4.a) X B .A − I
= I   X
b)=
5 55 5 0 1
 3 1 2  7 1 2
   
5.a) A.B =
= 3 0 3  ; B. A 13 1 2 
7 3 6  5 0 1
   

 1 −1 1   1 2 3   −1 0 0 
   
b)  −3 2 −1 .  2 4 6  =

 1 0 0
 −2 1 0   0 2 3   0 0 0 
    
 0 1 0
1  3 −4  −1 1  
6.a) A−1 = −   b) M =  −3 −3 3 
11  −2 −1  3 
 3 −2 0 

7.a) x = −2 ;

a c  a b a c 
b)i) If A =  T
 , then A =  ( AT )T =
 , and=  A;
b d  c d  b d 

a c  a' c'
ii) If A =   and B =   then
b d   b ' d '

T
 aa '+ cb ' ac '+ cd '   aa '+ cb ' ba '+ db '   a ' b '   a b
 =  .
T
( A.B ) =   = 
 ba '+ b ' d bc '+ dd '   ac '+ cd ' cb '+ dd '   c ' d '   c d
= BT . AT

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28 Mathematics | Teacher's Guide | Senior Five | Experimental Version
 12 6 3 2 1
8.a) = 18 b) 0
5 4 2 −5 =63
−2 1 4

2 0 0
9.a) Row 1 or column3; 1 2 0 = 24
3 5 6

2 1 2 2 1 7
b) i) 1 2 0=−
1 2
+2
2 2
16
=
ii) C3 → C1 + 2C2 + C3 iii) 1 2 4 = 16
5 6 3 6 3 5 17
3 5 6

 1 1 −1  x   0 
    
10.a)The system can be expressed as  1 2 3   y  = 14 
 2 1 4   z  16 
    

 1 1 −1  5 −5 5 
  −1 1  
Let A =  1 =
2 3 ; A  2 6 −4  ;
2 1 4  10  
   −3 1 1 

 x  5 −5 5   0  1
  1    
 y  10  2 6 −4  =
= 14   2
z  −3 1 1  16   3
      
1 1 1 6 1 1
b) D =2 1 −1 =−1 ≠ 0; Dx =1 1 −1 =−1;
3 2 1 10 2 1

1 1 6 1 1 6
Dy =2 1 −1 =−2; Dz =2 1 1 =−3
3 10 1 3 2 10

Dx −1 Dy −2 Dz −3
x
= y
= = 1;= z
= = 2;= = = 3
D −1 D −1 D −1

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Mathematics | Teacher's Guide | Senior Five | Experimental Version 29
 1.9. Additional activities
1.9.1. Remedial activity
3 0 1 2
1. Given the matrices A =   and B =   , find A + B and A.B
1 −2  5 0
Solution:
3 0  1 2 4 2 
A=
+B  + =   ;
 1 −2   5 0   6 −2 
3 0  1 2  3 6
= A.B =  .   
 1 −2   5 0   −9 2 
2. Use matrices to solve the simultaneous linear equations:
3 x − 4 y =
−4

 4x + 3y = 8
Solution:
 3 −4  x   −4 
The system can be expressed as    =  
 4 3  y   8 
4
 3 −4  −1 1  3 4   x  1  3 4   −4  5
Let A =   , then A = =  ;   =     
4 3  25  −4 3   y  25  −4 3   8  8
 
5
1.9.2. Consolidation activity
x y  0 −1
1. Matrices A =   and B =   are such that A.B = B. A .Find
the 1 2 2 3 

values of x and y .
Solution:
 x y   0 −1  2 y − x + 3 y 
=A.B =  .   
1 2 2 3   4 5 
 0 −1  x y   −1 −2 
=B. A =  .   
 2 3   1 2   2x + 3 2 y + 6 

2y − x + 3 y   −1 −2 
A.B = B. A if and only if  = 
 4 5   2x + 3 2 y + 6 

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30 Mathematics | Teacher's Guide | Senior Five | Experimental Version
  1
 x = 2
From the equality of matrices, we find: 
y = − 1
 2
3 0 2 
2.Find the inverse of the matrix=  
M  0 −1 2 
 0 3 −2 
 
Solution:

3 0 2 ;
det M =
0 −1 2 =
−12
0 3 −2
 −1 2 0 2 0 −1 
+ − + 
 3 −2 0 −2 0 3 
 0  −4 0 0 
2 3 2 3 0  
Matrix of cofactors: C = − + −  = 6 −6 −9 
 3 −2 0 −2 0 3   

   2 −6 −3 
 + 0 2
−
3 2
+
3 0 
 −1 2 0 2 0 −1 

 −4 6 2 
 
Adjoint matrix: Adj (M) = CT =  0 −6 −6 
 0 −9 −3 
 

 −4 6 2 
1  
M −1 =
−  0 −6 −6 
12  
 0 −9 −3 
1.9.3. Extended activity
The table below shows, for three related markets, the demand function and
the corresponding supply function:
Demand Qd Supply Qs
Market 1 23 − 5P1 + P2 + P3 −8 + 6P1
Market 2 15 + P1 − 3P2 + 2 P3 −11 + 3P2
Market 3 19 + P1 + 2 P2 − 4 P3 −5 + 3P3

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Mathematics | Teacher's Guide | Senior Five | Experimental Version 31
 Find the prices P1 , P2 , P3 for equilibrium in each market.
For equilibrium to occur, we have:

 23 − 5 P1 + P2 + P3 =−8 + 6 P1

15 + P1 − 3P2 + 2 P3 =−11 + 3P2
 19 + P + 2 P − 4 P =−5 + 3P
 1 2 3 3

This is equivalent to:

−11P1 + P2 + P3 = −31

 P1 − 6 P2 + 2 P3 =
−26
 P + 2P − 7 P = −24
 1 2 3

Solving the simultaneous equations by either method, we find:

P1 4;=
= P2 7;= P3 6

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32 Mathematics | Teacher's Guide | Senior Five | Experimental Version
Unit 2 DIFFERENTIATION/DERIVATIVES

2.1. Key unit competence

Solve Economical, Production, and Financial related problems using
Derivatives

2.2. Prerequisite
The students will perform well in this unit if they have a good background on:
• Equation of a straight line
• The gradient (or slope) of a straight line;
• Carrying out numerical calculations correctly;
• Independent variable and dependent variable;
• Numerical functions.

2.3. Cross-cutting issues to be addressed

• Inclusive education: promote the participation of all students while
teaching and learning
• Peace and value Education: During group activities, the teacher will
encourage the students to help each other and to respect opinions of
their classmates.
• Gender: Give equal opportunities to both girls and boys to participate
actively in all learning and application activities from the beginning to
the end of the lesson.

2.4. Guidance on introductory activity

• Form small groups of students, guide them to work on the introductory
activity;
• Through class discussions, let students think of the limiting value of
the dependent variable as the independent variable approaches a
given value;

Mathematics | Teacher's Guide | Senior Five | Experimental Version 33

 • The teacher should check the work of different groups and provide
pieces of advice where necessary;
• After a given time, invite students to present their findings and
harmonize them.
• Try to arouse students’ curiosity about the content of this second unit.

2.5. List of lessons

Lesson title/sub- Number of
Headings # Learning objectives
headings periods
Introductory activity Arouse the curiosity of
students on the content
of unit 2. 2
1. Average rate of Interpret the gradient of
2.1. change a straight line as rate of
Differentiation change
from first 2. Instantaneous Use limit to find the 2
principles. rate of change gradient of a curve at a
point
3. Differentiation Use simple rules of 2
of a polynomial derivatives to differentiate
function a polynomial
4. Differentiation of Identify a product and use 2
a product function formula to differentiate it
5. Differentiation of Identify a power and use 2
a power function formula to differentiate it
6. Differentiation Resolve a function into 2
of the composite its components and use
function (the the chain rule to find the
2.2 Rules for
chain rule) derivative
differentiation
7. Differentiation Use the quotient rule to 2
of a quotient differentiate a quotient
function
8. Differentiation Perform differentiation of 2
of a logarithmic a logarithmic function
function
9. Differentiation of Perform differentiation of 2
an exponential exponential function
function

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34 Mathematics | Teacher's Guide | Senior Five | Experimental Version
 10. Equation of the Determine the gradient 2
tangent of a curve at a point, and
To the graph of then the equation of the
2.3. a function at a tangent
Applications of point.
the derivatives 11. Hospital’s rule Remove indeterminate 2
cases using hospital’s rule
2.4. End of unit assessment 2

Answer to Introductory activity

a) =y 3x − 2 :

x1 x2 ∆x = x2 − x1 y1 y2 ∆y = y2 − y1 ∆y
∆x
1 2 1 1 4 3 3
1 1.5 0.5 1 2.5 1.5 3
1 1.1 0.1 1 1.3 0.3 3
… … … … … … …

y x2 + 1:
For =

x1 x2 ∆x = x2 − x1 y1 y2 ∆y = y2 − y1 ∆y
∆x
1 2 1 2 5 3 3
1 1.5 0.5 2 3.25 1.25 2.5
1 1.1 0.1 2 2.21 0.21 2.1
… … … … … … …
∆y
b) is called gradient (slope)
∆x
∆y ∆y
c) For (1) is constant, for (2) is variable
∆x ∆x

d) ∆y
lim =2
∆x →0 ∆x

35
Mathematics | Teacher's Guide | Senior Five | Experimental Version 35
 Lesson 1 : Average rate of change

a) Learning objectives:
Interpret the gradient of a straight line as the rate of change.
b) Teaching resources:
The following materials may be used in the teaching-learning process:
–– Calculator,
–– Manilla paper,
–– Student’s book;
–– Any reference text book containing gradient of a straight line
c) Prerequisites:
Students will perform better in this lesson if:
–– They can easily find the gradient of a line through two given points;
–– They can perform numerical calculations correctly, mentally or using a
calculator;
d) Learning activities
–– Request students to organize themselves in groups under your
supervision and discuss on the activity 2.1.1.
–– Through well-chosen questions, bring the students to discover how
the average rate of change in a function is calculated.
–– Have students discussing in groups problems involving average rate of
change.
–– Encourage each member to participate actively in the group.
–– Ensure that all the students are given opportunity to communicate
through presentation of the findings to the whole class.
–– As a teacher, you have to help students to harmonize their answers.
–– Use different probing questions and guide them to explore the content
and examples related to the average rate of change of a function and
how to denote it.
–– After this step, guide students to do the application activity 2.1.1 and
evaluate whether lesson objectives were achieved.

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36 Mathematics | Teacher's Guide | Senior Five | Experimental Version
 Answers of learning activity 2.1.1
a) 5 − 2 =3
b)=P (2) 9;=P (5) 78

∆y 78 − 9
= = 23
c)
∆x 5 − 2
d) Rate
e) Answers of Application activity 2.1.1.
1. 107500 = 1075
100
f (4) − f (2)
2. = 6.125
4−2

Lesson 2 : Instantaneous rate of change

a) Learning objectives:
Use limit to find the gradient of a curve at a point.
b) Teaching resources:
The following materials may be used in the teaching-learning process:
–– Calculator,
–– Manilla paper,
–– Student’s book;
–– Any reference text book containing derivatives from first principles
c) Prerequisites:
Students will perform better in this lesson if:
–– They have mastered the average rate of change from lesson 2.1.1;
–– They can evaluate the limit of a function as the independent variable
approaches a given value
–– They can perform numerical calculations correctly, mentally or using a
calculator;
d) Learning activities
–– Ensure that the students are organized in pairs;
–– Request the students to discuss in pairs the learning activity 2.1.2.;
–– As they are discussing, circulate to monitor the effective participation
of each member of the group;

37
Mathematics | Teacher's Guide | Senior Five | Experimental Version 37
 –– After a while, request a group representative to present to the whole
class the group’s finding;
–– Have students interact, and make, under your guidance, the summary
of the lesson.
–– Use different probing questions and guide them to explore the content
and examples related to the instantaneous rate of change of a function.
–– After this step, guide students to do the application activity 2.1.2 and
evaluate whether lesson objectives were achieved.

Answers of learning activity 2.1.2

a)

2.1 2.01 2.001 etc

x1 x1 → 2
0.1 0.01 0.001 etc ∆x → 0
∆x = x1 − x0
0.41 0.0401 0.004 etc /////////////
∆y = y1 − y0
4.1 4.01 4.001 etc
∆y ∆y
→4
∆x ∆x

∆y
b) lim =b
∆x →0 ∆x

∆y
c) lim = 6x
∆x →0 ∆x

e) Answers of Application activity 2.1.2

1. 2Q + 7

1
2.
2 x

Lesson 3 : Differentiation of polynomial functions

a) Learning objectives:
Use simple rules of derivatives to differentiate a polynomial
b) Teaching resources:
The following materials may be used in the teaching-learning process:

38
38 Mathematics | Teacher's Guide | Senior Five | Experimental Version
 –– Calculator,
–– Manilla paper,
–– Student’s book;
–– Any reference text book containing rules for differentiation of
polynomials
c) Prerequisites:
Students will perform better in this lesson if:
–– They have knowledge and skills on identification of polynomial
functions
–– They have mastered the calculation of the derivative from first
principles,
–– from section 2.1.2;
–– From the observation of a pattern, they can predict the next value;
–– They can perform numerical calculations correctly, mentally or using a
calculator;
d) Learning activities
–– Request the students to organize themselves into small groups;
–– Present the learning activity 2.2.1.for students to discuss in groups;
–– Find, by reaching each group, whether the students are able to
distinguish a polynomial from another function;
–– Check whether each group is comfortable with differentiation from
first principles;
–– Have students interact, and make, under your guidance, the summary
of the lesson.
–– Use different probing questions and guide them to explore the content
and examples related to the differentiation of polynomial functions.
–– After this step, guide students to do the application activity 2.2.1 and
evaluate whether lesson objectives were achieved.

Answers of learning activity 2.2.1

a) g ( x)
b) A letter(variable) raised to different positive integral powers, the
powers are multiplied by constants, and finally, these products are
added

39
Mathematics | Teacher's Guide | Senior Five | Experimental Version 39
 dC
c) i) =0
dx
d du dv
ii) (u + v) = +
dx dx dx
d du
iii) (ku ) = k
dx dx
iv) 1; 2 x;3 x 2 ;...; nx n −1

d) a1 + 2a2 x + 3a3 x 2 + ... + nan x n −1

e) Answers of Application activity 2.2.1

dz
a) =−24t 2 + 15t 4
dt
dP
b) = 20Q 3 + 3
dQ

Lesson 4 : Differentiation of a product function

a) Learning objectives:
Use simple rules of derivatives to differentiate a product function

b) Teaching resources:
The following materials may be used in the teaching-learning process:
–– Calculator,
–– Manilla paper,
–– Student’s book;
–– Any reference text book containing rules for differentiation of product
functions
c) Prerequisites:
Students will perform better in this lesson if:
–– They are able to expand the product of any two given functions.
–– They have mastered differentiation of polynomials from section 2.2.1.
–– From the observation of a function, they can tell whether it is a product
or not.
–– They can perform numerical calculations correctly, mentally or using a
calculator.

40
40 Mathematics | Teacher's Guide | Senior Five | Experimental Version
d) Learning activities
–– Ensure that students sitting on the same desk work together;
–– Present the learning activity 2.2.2.for students to discuss in groups;
–– Find, by reaching each group, whether the students are able to
distinguish a product from another function;
–– Check whether each group is comfortable with differentiation of
products;
–– Have students interact, and make, under your guidance, the summary
of the lesson.
–– Use different probing questions and guide them to explore the content
and examples related to the differentiation of polynomial functions.
–– After this step, guide students to do the application activity 2.2.2 and
evaluate whether lesson objectives were achieved.

Answers of learning activity 2.2.2

a) i) uv =
−12 x 3 + 27 x 2 + 8 x − 18

d
ii) (uv) = −36 x 2 + 54 x + 8
dx
du dv
b) i) = −6 x; = 4
dx dx
du dv
ii) v +u = −36 x 2 + 54 x + 8
dx dx
d du dv
c) (= uv) v +u
dx dx dx

e) Answers of Application activity 2.2.2.

dz
a) −24t 2 (4 + 3t 5 ) + 15t 4 (2 − 8t 3 ) =
= −192t 7 + 30t 4 − 96t 2
dt
dP
b) = 20Q 3 (3Q − 7) + 3(5Q 4= ) 75Q 4 − 140Q 3
dQ

Lesson 5 : Differentiation of a power function

a) Learning objectives:
Use simple rules of derivatives to differentiate a power function
b) Teaching resources:
The following materials may be used in the teaching-learning process:

41
Mathematics | Teacher's Guide | Senior Five | Experimental Version 41
 –– Calculator,
–– Manilla paper,
–– Student’s book;
–– Any reference text book containing rules for differentiation of power
functions
c) Prerequisites:
Students will perform better in this lesson if:
–– They have mastered differentiation of polynomials and products from
sections 2.2.1.and 2.2.2;
–– From the observation of a function, they can tell whether it is a power
or not;
–– They can perform numerical calculations correctly, mentally or using a
calculator;
d) Learning activities
–– Ensure that students sitting on the same desk work together;
–– Present the learning activity 2.2.3.for students to discuss in groups;
–– Find, by reaching each group, whether the students are able to
distinguish a power from another function;
–– Have students interact and make, under your guidance, the summary
of the lesson.
–– Use different probing questions and guide them to explore the content
and examples related to the differentiation of power functions.
–– After this step, guide students to do the application activity 2.2.3 and
evaluate whether lesson objectives were achieved.

Answers of learning activity 2.2.3

dy
a) ( 2 x + 1) = 8 x 3 + 12 x 2 + 6 x + 1 ;
3
= 24 x 2 + 24 x + 6
dx
d
b) i) (2 x + 1) =
2
dx
ii)The exponent is lowered, the new exponent is the old one minus 1, and
finally this result is multiplied by the derivative of the base of the power.

d n du
c) (u ) = nu n −1
dx dx

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42 Mathematics | Teacher's Guide | Senior Five | Experimental Version
e) Answers of Application activity 2.2.3.
a) dy
= 14(7 x + 8)
dx
b) dy
= 12(4 x + 5) 2
dx

Lesson 6 : Differentiation of the composite function (The

chain rule)

a) Learning objectives:
Resolve a function into its components and use the chain rule to find its
derivative
b) Teaching resources:
The following materials may be used in the teaching-learning process:
–– Calculator,
–– Manilla paper,
–– Student’s book;
–– Any reference text book containing rules for differentiation of power
functions
c) Prerequisites:
Students will perform better in this lesson if:
–– They have mastered composition of two functions;
–– They are able to split a function into its components;
–– From the observation of a function, they can tell whether it is a power
or not;
–– They can perform numerical calculations correctly, mentally or using a
calculator;
d) Learning activities
–– Invite student-teachers to work in pairs and do the activity 2.2.4 found
in their Mathematics books;
–– Move around in the class for facilitating students where necessary and
give more clarification on eventual challenges they may face during
their work;
–– Find, by reaching each group, whether the students are able to split a
function into its components;

43
Mathematics | Teacher's Guide | Senior Five | Experimental Version 43
 –– Verify and identify groups with different working steps;
–– As they are discussing, concentrate on slow students for further
explanation and provide assistance to groups in need.
–– Invite students to present the findings, and help them to harmonize
the answer.
–– After presentation, the teacher will help the students to apply the
chain rule formula in the provided examples for better understanding.
–– After this step, guide student-teachers to do the application activity
2.2.4 and evaluate whether lesson objectives are achieved or not for
eventual improvement for the following lesson.

Answers of learning activity 2.2.4

dy
a) = 3(2 x + 1) 2 × (2)
dx
b) i) u ( x) = u3
2 x + 1; v(u ) =

du dy
ii)= 2;= 3u 2
dx du
dy du
iii) .= (3u 2 )(2)
= 6(2 x + 1) 2
du dx
dy dy du
c) = .
dx du dx

e) Answers of Application activity 2.2.4

dy
a)= 14(7 x + 8)
dx
dy
b)= 12(4 x − 5) 2
dx

Lesson 7 : Differentiation of a quotient function

a) Learning objectives:
Use the quotient rule to differentiate a quotient function
b) Teaching resources:
The following materials may be used in the teaching-learning process:
–– Calculator,

44
44 Mathematics | Teacher's Guide | Senior Five | Experimental Version
 –– Manilla paper,
–– Student’s book;
–– Any reference text book containing rules for differentiation of power
functions
c) Prerequisites:
Students will perform better in this lesson if:
–– They have mastered the differentiation of product functions from
lesson6 of this unit;
–– From the observation of a function, they can tell whether it is a quotient
or not;
–– They can perform numerical calculations correctly, mentally or using a
calculator;
d) Learning activities
–– Invite students to work in pairs and do the activity 2.2.5 found in their
Mathematics books;
–– As they are discussing, concentrate on slow students for further
explanation and provide assistance to groups where it is necessary.
–– Invite students to present their findings, and help them to harmonize
the answer.
–– After presentation, the teacher will help the students to generalize the
derivative of quotient function and guide them to re-work the provided
examples for better understanding.
–– After this step, guide students to do the application activity 2.2.5 and
evaluate whether lesson objectives are achieved or not for eventual
improvement for the following lesson.

Answers of learning activity 2.2.5

u 3x 2 , =
a) i) Let = v 2x +1
y = u.v −1 becomes
=y (3 x 2 )(2 x + 1) −1
dy (6 x)(2 x + 1) − (2)(3 x 2 )
ii) =
dx (2 x + 1) 2

(6 x)(2 x + 1) − (2)(3 x 2 )
b)
(2 x + 1) 2

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Mathematics | Teacher's Guide | Senior Five | Experimental Version 45
 c)They are equal
du dv
v −u
d) d  u  = dx dx
  2
dx  v  v

e) Answers of Application activity 2.2.5

dy 5(7 x − 4) − 7(5 x − 6) 22
=a) =
dx (7 x − 4) 2
(7 x − 4) 2

dy 12 x3 (2 + 5 x) − 5(3 x 4 )
b) =
dx (2 + 5 x) 2

Lesson 8 : Differentiation of a logarithmic function

a) Learning objectives:
Perform differentiation of a logarithmic function
b) Teaching resources:
The following materials may be used in the teaching-learning process:
–– Calculator,
–– Manilla paper,
–– Student’s book;
–– Any reference text book containing rules for differentiation of power
functions
c) Prerequisites:
Students will perform better in this lesson if:
–– They have mastered the properties of logarithms, from unit3 studied
in senior4;
–– They can state the restrictions on a logarithmic function
–– They can perform numerical calculations correctly, mentally or using a
calculator;
d) Learning activities
–– Let the students organize themselves in groups under your supervision;
–– Present the learning activity 2.2.6.for students to discuss in groups
–– As the students are discussing, find how effective the discussion is
carried out;

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46 Mathematics | Teacher's Guide | Senior Five | Experimental Version
 –– Check whether each group is able to find restrictions on the independent
variable in a logarithmic function
–– Invite students to present their findings, and help them to harmonize
the answer
–– After presentation, the teacher will help the students to generalize the
differentiation of logarithmic functions and guide them to re-work the
provided examples for better understanding.
–– After this step, guide students to do the application activity 2.2.6 and
evaluate whether lesson objectives are achieved or not for eventual
improvement for the following lesson.
1.
Answers of learning activity 2.2.6
Consider function f ( x) = ln x
b
a)
x0 1 2 3 …
2.
∆x
0.1 ∆y ∆y ∆y
∆x ∆x ∆x
ln(1 + 0.1) − ln1 ln(2 + 0.1) − ln 2 ln(3 + 0.1) − ln 3
= = = 3.
0.1 0.1 0.1
= 0.953 = 0.4879 = 0.3278

0.01 ∆y ∆y ∆y
∆x ∆x ∆x
ln(1 + 0.01) − ln1 ln(2 + 0.01) − ln 2 ln(3 + 0.01) − ln 3 4.
= = =
0.01 0.01 0.01
= 0.995 = 0.4987 = 0.3327

0.001 ∆y ∆y ∆y
∆x ∆x ∆x
ln(1 + 0.001) − ln1 ln(2 + 0.001) − ln 2 ln(3 + 0.001) − ln 3
= = =
0.001 0.001 0.001
= 0.9995 = 0.4998 = 0.33327
…
1 1 1 1
f '(1)
b)= =; f '(2) =; f '(3) ;...
= f '( x)
1 2 3 x

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Mathematics | Teacher's Guide | Senior Five | Experimental Version 47
e) Answers of Application activity 2.2.6.

dy 3
a) =
dx x
dy 5
b) =
dx (5 x + 6) ln10

Lesson 9 : Differentiation of an exponential function

a) Learning objectives:
Perform differentiation of an exponential function
b) Teaching resources:
The following materials may be used in the teaching-learning process:
–– Calculator,
–– Manilla paper,
–– Student’s book;
–– Any reference text book containing differentiation of exponential
functions
c) Prerequisites:
Students will perform better in this lesson if:
–– They have mastered the properties of exponentials, from unit3 studied
in senior4;
–– They can convert from logarithms to exponentials and vice versa;
–– They can perform numerical calculations correctly, mentally or using a
calculator;
d) Learning activities
–– Ensure that the class is ready for learning;
–– Present the learning activity 2.2.7.for students to discuss in groups;
–– As the students are discussing, find how effective the discussion is
carried out;
–– Check whether each group member is able to convert an expression
from logarithm to exponential and vice versa;
–– Request the students to make the summary of the lesson.
–– Use different probing questions and guide them to explore the content
and examples related to the differentiation of exponential functions.

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48 Mathematics | Teacher's Guide | Senior Five | Experimental Version
 –– After this step, guide students to do the application activity 2.2.7 and
evaluate whether lesson objectives were achieved.

Answers of learning activity 2.2.7

1
1.a) x = ( ) lny
ln a
1 y'
b) 1 = ( ).
ln a y
c) y ' (ln
= = a ) y a x (ln a )

d) i) (a x ) ' = a x ln a

ii) (e x ) ' = e x

2. i) (a u ) ' = u ' a u ln a

ii) (eu ) ' = u ' eu

e) Answers of Application activity 2.2.7

dy 3
a) = 12 x 2 e 4 x
dx
dy
b) = 5(105 x + 6 ) ln10
dx

Lesson 10 : Equation of the tangent to the graph of a function at a point

a) Learning objectives:
Determine the gradient of a curve at a point, and then the equation of the
tangent
b) Teaching resources:
The following materials may be used in the teaching-learning process:
–– Calculator,
–– Manilla paper,
–– Student’s book;
–– Any reference text book containing the determination of the equation
of the tangent to the graph of a function at a point.

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Mathematics | Teacher's Guide | Senior Five | Experimental Version 49
c) Prerequisite:
Students will perform better in this lesson if:
–– They have a good understanding of the gradient, or the slope, of a
straight line;
–– They are able to find the equation of the line through a point and with
a given gradient;
–– They can perform numerical calculations correctly, mentally or using a
calculator;
d) Learning activities
–– Ensure that the class is ready for learning;
–– Present the learning activity 2.3.1 for students to discuss in groups;
–– As the students are discussing, find how effective the discussion is
carried out;
–– Check whether each group member is able to find the equation of the
line through a point and with a given gradient;
–– Through guided questions, bring the students to make a summary of
the main points of the lesson.
–– Use different probing questions and guide them to explore the content
and examples related to the equation of the tangent to the graph of a
function at a point.
–– After this step, guide students to do the application activity 2.3.1 and
evaluate whether lesson objectives were achieved.

Answers of learning activity 2.3.1

f ( x0 + ∆x) − f (x 0 )
1. a) m =
∆x
b) i) ∆x → 0
ii) f '( x0 )
iii) tangent

c) y − f ( x0 )= f '( x0 )( x − x0 )

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50 Mathematics | Teacher's Guide | Senior Five | Experimental Version
e) Answers of Application activity 2.3.1.

15 180
a) y =
− 2 x+ 2
e e
10
b) y =
− x + 10
40

Lesson 11 : Hospital’s rule

a) Learning objectives:
Remove indeterminate cases using Hospital’s rule
b) Teaching resources:
The following materials may be used in the teaching-learning process:
–– Calculator,
–– Manilla paper,
–– Student’s book;
–– Any reference text book containing the determination of the equation
of the tangent to the graph of a function at a point.
c) Prerequisite:
Students will perform better in this lesson if:
–– They understand well what an indeterminate case is, in the calculation
of a limit;
–– They are able to differentiate functions as studied in the first nine
lessons of this unit;
–– They can find the equation of the tangent to a curve at a point;
–– They can perform numerical calculations correctly, mentally or using a
calculator;
d) Learning activities
–– Ensure that the students are settled for the lesson;
–– Present the learning activity 2.3.2. for students to discuss in groups;
–– Draw the attention of the students on the fact that f ( x0 ) = 0 and
g( x0 ) = 0 ,or f ( x0 ) = ∞ and g( x0 ) = ∞ , for the continuation to hold
–– Guide students to discover that in the neighbourhood of x0 , the graph
of the function and the tangent to the graph are almost the same;
–– In the discussion, ensure that they write correctly the equations of the
tangents to the graphs of the two functions;
–– As the students are discussing, find how effective the discussion is carried
out;

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Mathematics | Teacher's Guide | Senior Five | Experimental Version 51
 –– Check whether each group member is able to find the equation of the
line through a point and with a given gradient;
–– Invite students to present their findings, and help them to harmonize
the answer
–– Use different probing questions and guide them to explore the content
and examples related to the Hospital’s rule.
–– After this step, guide students to do the application activity 2.3.2 and
evaluate whether lesson objectives were achieved.

Answers of learning activity 2.3.2

1.a) y = f '( x0 ) x − x0 f '( x0 ) + f (x 0 ) , where a = f '( x0 ), b =

− x0 f '( x0 ) + f ( x0 ) = − x0 f '( x0 )

y = g '( x ) x − x g'( x ) + g (x ) , c =
0 0 0
g '( x ), d =
0
− x g'( x ) + g ( x ) =
0 0 0 0 − x g'( x )
0 0

f '( x0 ) x − x0 f '( x0 ) ( x − x0 ) f'( x0 )

b) lim = lim
x → x0 g '( x0 ) x − x0 g '( x0 ) x → x0 ( x − x ) g '( x )
0 0

f '( x0 ) f '( x)
= = lim
g '( x0 ) x → x0 g '( x)

c) lim f ( x) = lim f '( x) , provided f ( x0 ) = 0 and g( x0 ) = 0 , or f ( x0 ) = ∞ and g( x0 ) = ∞

x → x0 g ( x) x → x0 g '( x)

e) Answers of Application activity 2.3.2.

x −1 1
a) lim x
=
x →1 ( x − 1)e 2e

e2 x−2 − 1 2
b) lim =
x →1 ln(5 x − 4) 5

2.6. Summary of unit 2

• The average rate of change of function , y = f ( x) as the independent
variable x assumes values from x1 to x2 , is the quantity defined by
∆y f ( x2 ) − f ( x1 )
=
∆x x2 − x1
• The instantaneous rate of change of function y = f ( x) at x = x0 is the
∆y
value of lim for x = x0 . Equivalently, the instantaneous rate of
∆x →0 ∆x
f ( x0 + ∆x) − f ( x0 )
change of function y = f ( x) at x0 is the number, ∆lim
x →0 ∆x

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52 Mathematics | Teacher's Guide | Senior Five | Experimental Version
 This number is called the derivative of function y = f ( x) at x0 ; it is
denoted by f '( xo ) . More generally, the derivative of function y = f ( x) is
denoted by f '( x) or y ' or dy
dx
• The rules for differentiation:
1) Derivative of a constant function
df d
= =
If f is a constant function, f ( x) = c , for all x then (C ) 0
dx dx
Example: Calculate the derivative of f ( x) = 8

df d
= =
Solution: (8) 0
dx dx

2) Derivative of identity function

The derivative of the identity function is the constant function 1, that is

df dx
if f ( =
x) x , = = 1
dx dx

3) Multiplication by a scalar
The derivative of the product of a constant real number by a function equals
the product of the real number by the derivative of the function; that is:

d d
(ku ) = k (u ) , where k is a constant and u is a function of variable x
dx dx

4) Sum Rule

d d d
(u + =
v) (u ) + (v) : the derivative of a sum equals the sum of derivatives
dx dx dx
of the terms, where u and v are functions of variable x ;

5) The Difference Rule

d d d
(u − =
v) (u ) − (v)
dx dx dx

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Mathematics | Teacher's Guide | Senior Five | Experimental Version 53
 6) Derivative of a power
d n
( x ) = nx n −1
dx ; the derivative of the nth power of the variable equals the
product of the exponent by the (n-1)th power of the variable. From the
properties above, it follows that:
d
(a0 + a1 x + a2 x 2 + ... + an x n ) = a1 + 2a2 x + ... + nan x n −1
dx
d n du
This holds for any function with power: (u ) = nu n −1
dx dx

7) Product rule
d d d
= (uv) v (u ) + u (v)
dx dx dx

8) Quotient rule
du dv '
v −u  u  u 'v − v 'u
d u dx dx , that is,
 =   =
dx  v  v 2 v v2

9) Differentiation of a composite function: Chain rule

Consider the following diagram:

If function y = f ( x) can be expressed as y = v[u ( x)] , where u ( x) and v(u )

dy dy du
are functions to determine, then, it can be shown that: = . . This
dx du dx
formula is known as the chain rule.
10) Differentiation of logarithmic function

= y f=( x) ln x is = 1
The derivative of function f '( x) (ln
= x) ' , that is
x
d 1
( ln x ) = , where x > 0 . More generally, if y = ln[u ( x)] , where u ( x)
dx x

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54 Mathematics | Teacher's Guide | Senior Five | Experimental Version
is function of variable x , then from the chain rule, d [ lnu(x) ] = 1 . du
dx u dx
u'
, that is [ ln u ( x) ] ' = . In the same way, if log a x = 1 .ln x , then
u ln a
1 1
( log a x ) '
= = (ln x) ' , where a > 0 and a ≠ 1 , and x > 0 .More generally,
ln a x ln a
u'
If y = log a u ( x) , where u ( x) is function of variable x , then [ log a u ( x) ] ' =
u ln a

11) Differentiation of exponential function

) ' e x , that is d e x = e x .
( )
x
=
The derivative of function ( x) e x is f =
y f= '( x) (e=
dx
u ( x)
More generally, if y = e , where u ( x) is function of variable x , then from

the chain rule, d eu ( x )  = eu ( x ) du , that is eu ( x )  ' = u ' eu . In the same way, if
dx   dx  
x
( )x
a x , then a ' = a ln a , where a > 0 and a ≠ 1 , and x > 0 . More generally, If

y = a u ( x ) , where u ( x) is function of variable x , then  a  ' = u ' a ln a

u ( x) u

12) Hospital Rule

f ( x) f ( x0 ) 0 ∞
If numerical functions f ( x) and g ( x) are such that xlim = = or ,
→ x g ( x) g ( x0 ) 0
0 ∞
then
to remove the indetermination, we proceed as follows, through Hospital’s
rule:
1) Differentiate separately, the numerator and the denominator, to get
f '(x) and g '( x) ;

f '( x) f '( x0 )
2) Calculate xlim =
→x 0 g '( x ) g '( x0 )
f ( x) f '( x0 )
3) Then lim =
x → x0 g ( x ) g '( x0 )
Note that: - the process can be repeated if necessary;

–– Hospital’s rule is used only if we have indetermination 0 or ∞

0 ∞

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Mathematics | Teacher's Guide | Senior Five | Experimental Version 55
 –– Hospital’s rule, is not the quotient rule for differentiation, that is

'
f ( x)  f ( x) 
lim ≠ lim  
x → x0 g ( x ) x → x0 g ( x )
 
–– Before applying Hospital’s rule, ensure that you have indetermination

0 ∞
or
0 ∞
13) Equation of Tangent line
Recall that the equation of a straight line passing through a given point
A( x0 , y0 ) having finite slope m is given by y − y0 = m ( x − x0 )
dy
ory − f ( x0 )= f '( x0 )( x − x0 )where f ′ ( x0 ) = dx
( x0 , y0 )

2.7. Additional information for the teacher.

This unit being the foundation in the sense that it has so many applications
in Mathematics and in other sciences, the teacher is advised to not take this
unit as granted. The examples must be well chosen and well prepared. No
improvisation of data as far as modelling financial or economics problems is
concerned.
Therefore, the teacher is advised to mentally prepare ahead his lessons, in
order to not be embarrassed or meet a dilemma in class.
The teacher will make research in the school library or on internet so as to
widen his/her knowledge.

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56 Mathematics | Teacher's Guide | Senior Five | Experimental Version
 2.8. Answers of End unit assessment
ln1.2 − ln1
1.a) = 0.9116
1.2 − 1
1
f '( x) (ln
b) = =x) ' f '(1) 1
;=
x

3( x + ∆x) 2 − 3 x
2. a) f '( x) lim
= = 6x
∆x →0 ∆x
−5 −5
−
b) 5 + ∆x x 5
= f '( x) lim
=
∆x →0 ∆x x2

dy
3. a) = 30 − x
dx
dy −1 1
b) = +
dx x2 x
ds
4. a) = 12t 3 (2t − 5) + 2(3t 4 )= 30t 4 − 60t 3
dt
ds
b) = 7t 6 (t 5 + 11) + 5t 4 (t 7 − 4)= 12t11 + 77t 6 − 20t 4
dt
du
5.a) Let u = x 2 − x + 2 .Then = 2 x − 1 ;
dx
dy 1
The function becomes y = u , and =
du 2 u
dy dy du 2x −1
= =
From the chain rule, .
dx du dx 2 x 2 − x + 1
du
b) Let=u 3 x 4 + 7 .Then = 12 x3
dx
dy
The function becomes y = u 6 and = 6u 5
du
dy dy du
From the chain rule,= = . 72 x3 (3 x 4 + 7)5
dx du dx

dQ (10 P − 9)( P 2 + 1) − 2 P(5 P 2 − 9 P + 8) 9 P 2 − 6 P − 9

6. a) =
dP ( P 2 + 1) 2 ( P 2 + 1) 2

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Mathematics | Teacher's Guide | Senior Five | Experimental Version 57
 dQ 6(8 P − 5) − 8(6 P − 7) 26
=b) = 2
dP (8 P − 5) (8 P − 5) 2

dy 3(3 x + 4) 2 (10 x − 7)
7. a) =
dx (5 x − 1) 2

dy −26(2 x + 1)
b) =
dx (3 x − 5)3

dy
8. a) = −3(21−3 x ln 2)
dx
dy 3ln(5 x )
b) =
dx x
9. a) y =−6 x − 2

1 1
y
b)= x + e2
4 4
1 1
10. a) 2 − b)
e 2

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58 Mathematics | Teacher's Guide | Senior Five | Experimental Version
2.9. Additional activities
2.9.1. Remedial activity
1. Find the derivative, with respect to x ,of:
4x
a) y =
1− x

b) y = x 2 e − x

Solution:
dy 4
a) =
dx (1 − x) 2
dy
b) ( x 2 + 2 x )e − x
=−
dx
2. Find the equation of the tangent to the graph of:

a) y =4 − 2 x − 2 x 2 at x = −1

1
b) y = at x = 2
x

Solution:

a) =
y 2x + 6

1
b) y =
− x +1
4

2.9.2. Consolidation activity

1.Find the derivative, with respect to x ,of:
1
a) y =ln x − ln(1 + x 2 ) when x = 2
2
2 x −1
b) y = 3 when x = 1

Solution:

dy 1 x
a) = −
dx x 1 + x 2

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Mathematics | Teacher's Guide | Senior Five | Experimental Version 59
 1
The value of the derivative at x = 2 is
10
dy 3 2 x −1 ln 3
b) =
dx 2x −1
The value of the derivative at x = 1 is 3ln 3
2. Calculate the following limits:

ex −1
a) lim
x →0 ( x + 1)e x − 1

x − 1 − x ln x
b) lim
x →1 ( x − 1) ln x

Solution: x
e −1 0
a) lim x
= :indeterminate case. By Hospital’s rule, we have:
x →0 ( x + 1)e − 1 0
x
e −1 ex 1 1
lim = x
lim = x
lim
=
x →0 ( x + 1)e − 1 x →0 ( x + 2)e x →0 x + 2 2

x − 1 − x ln x 0
b) lim = : indeterminate case. By Hospital’s rule, we have:
x →1 ( x − 1) ln x 0

x − 1 − x ln x 1
lim = −
x →1 ( x − 1) ln x 2
2.9.3. Extended activity
x
Differentiate, from first principles, the function f ( x) = 2
x +1
Solution:
x+h x
− 2
2
( x + h) + 1 x + 1 − x2 + 1
lim = 2
h →0 h ( x + 1) 2

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60 Mathematics | Teacher's Guide | Senior Five | Experimental Version
 APPLICATIONS OF DERIVATIVES IN
Unit 3 FINANCE AND IN ECONOMICS

3.1. Key unit competence

Apply differentiation in solving Mathematical problems that involve financial
context such as marginal cost, revenues and profits, elasticity of demand
and supply

3.2. Prerequisite
The students will perform well in this unit if they have a good background on:
• The derivatives as studied in unit 2 preceding this unit;
• The understanding of English language in order to model a problem
by an equation;
• The economic and financial concepts such as cost, revenue, profit, etc.
• Carrying out numerical calculations correctly;
• Manipulation of calculators for computing data.

3.3. Cross-cutting issues to be addressed

• Inclusive education: promote the participation of all students while
teaching and learning
• Peace and value Education: During group activities, the teacher will
encourage the students to help each other and to respect opinions of
their classmates.
• Gender: Give equal opportunities to both girls and boys to participate
actively in all learning and application activities from the beginning to
the end of the lesson.

Mathematics | Teacher's Guide | Senior Five | Experimental Version 61

3.4 Guidance on introductory activity
• Form small groups of students, guide them to work on the introductory
activity;
• Through class discussions, let students think of how to model and
solve a problem;
• The teacher should walk around to all groups and provide pieces of
advice where necessary;
• After a given time, invite students to present their findings and
harmonize them.
• Try to arouse students’ curiosity about the content of this third unit.

3.5. List of lessons

Lesson title/sub- Number of
Headings # Learning objectives
headings periods
Introductory activity Arouse the curiosity of
students on the content of
unit 3. 3

3.1. Marginal 1. Marginal cost Interpret the marginal cost

quantities as rate of change in the cost
2. Marginal Interpret the marginal 3
revenue revenue as the rate of change
in the revenue
3. Minimization Determine the price that 3
of the total cost will minimize the total cost
3.2. function function
Minimization
and
maximization 4. Maximization Determine the price that will 3
of functions of the total maximize the total revenue
revenue function
function
3.3. Price 5. Elasticity of Measure the responsiveness 2
elasticity demand of demand to change in price

6. Elasticity of Measure the responsiveness 2

supply of supply to change in price

3.4. End of unit assessment 2

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62 Mathematics | Teacher's Guide | Senior Five | Experimental Version
Answer to Introductory activity

300
a) h = ;
π x2
12000
C ( x) 50π x 2 +
b)= ;
x
120
c) x
= 3 ≈ 3.367 cm; h ≈ 8, 427 cm
π

Lesson 1 : Marginal cost

a) Learning objectives:
Interpret the marginal cost as rate of change in the cost
b) Teaching resources:
The following materials may be used in the teaching-learning process:
–– Calculator,
–– Manilla paper,
–– Student’s book;
–– Any reference text book containing information about marginal cost
c) Prerequisites:
Students will perform better in this lesson if:
–– They have mastered the cost function as studied in Senior4, unit2;
–– They can perform numerical calculations correctly, mentally or using a
calculator;
d) Learning activities
–– Request students to organize themselves in groups under your
supervision;
–– Through well-chosen questions and discussion of learning activity 3.1.1,
bring the students to discover that the instantaneous rate of change in
the cost is called the marginal cost and how to calculate it;
–– Encourage each member to participate actively in the group;
–– Ensure that all the students are given opportunity to communicate
through presentation of the findings to the whole class;
–– Use different probing questions and guide them to explore the content
and examples related to the Marginal cost.
–– After this step, guide students to do the application activity 3.1.1 and
evaluate whether lesson objectives were achieved.

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 Answers of learning activity 3.1.1

dy
a) dy
= 6 x + 7 .For x = 3 , = 25
dx dx
b) Marginal total cost

e) Answers of Application activity 3.1.1.

dC
1. =4 − 2 x + 6 x 2
dx

dQ ;
2. = 40 + 6 P − P 2
dP

For P = 10 ,the marginal product is 0

Lesson 2 : Marginal revenue

a) Learning objectives:
Interpret the marginal total revenue as rate of change in the total revenue
b) Teaching resources:
The following materials may be used in the teaching-learning process:
–– Calculator,
–– Manilla paper,
–– Student’s book;
–– Any reference text book containing information about marginal
revenue
c) Prerequisites:
Students will perform better in this lesson if:
–– They have mastered the revenue function as studied in Senior4, unit2;
–– They can perform numerical calculations correctly, mentally or using a
calculator;
d) Learning activities
–– Request students to organize themselves in groups under your
supervision;

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64 Mathematics | Teacher's Guide | Senior Five | Experimental Version
 –– Through well-chosen questions and discussion of learning activity 3.1.2,
bring the students to discover that the instantaneous rate of change
in the total revenue is called the marginal total revenue and how to
calculate it;
–– Encourage each member to participate actively in the group;
–– Ensure that all the students are given opportunity to communicate
through presentation of the findings to the whole class;
–– Use different probing questions and guide them to explore the content
and examples related to the Marginal revenue.
–– After this step, guide students to do the application activity 3.1.2 and
evaluate whether lesson objectives were achieved.

Answers of learning activity 3.1.2

a) 100Q − Q 2
b) 100 − 2Q ; for Q = 11 , the value is 78

e) Application activity3.1.2.
1. x) 30 x − 2 x 2 ; the marginal revenue
The total revenue function if R(=
is dR= 30 − 4 x
dx
2. The total revenue function is R (Q) =Q 3 + 2Q 2 + Q , the total marginal
dR dR
revenue is = 3Q 2 + 4Q + 1 ;For Q = 10 , = 341
dQ dQ

Lesson 3 : Minimization of the total cost function

a) Learning objectives:
Determine the price that will minimize the total cost function
b) Teaching resources:
The following materials may be used in the teaching-learning process:
–– Calculator,
–– Manilla paper,
–– Student’s book;
–– Any reference text book containing information about marginal
revenue

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 c) Prerequisites:
Students will perform better in this lesson if:
–– They have mastered the cost function as studied in Senior4, unit2;
–– They are able to differentiate simple functions as covered in unit2 of
this year;
–– They can perform numerical calculations correctly, mentally or using a
calculator;
d) Learning activities
–– Request students to organize themselves in groups under your
supervision;
–– Through well-chosen questions and discussion of learning activity 3.2.1,
bring the students to discover how to model a problem and how to
express the quantity to minimize as function of one variable;
–– Ensure that the students can differentiate and set the conditions for a
minimum to occur;
–– Encourage each member to participate actively in the group;
–– Request a student, chosen at randomly, to present the findings of his/
her group to the whole class;
–– Use different probing questions and guide the students to explore the
content and examples related to the Minimization of the total cost
function.
–– After this step, guide students to do the application activity 3.2.1 and
evaluate whether lesson objectives were achieved.

Answers of learning activity 3.2.1

How do I choose the base radius and the height of the can so as to pay the
least money for purchasing the material?
e) Application activity 3.2.1.
f)
dP 2Q − 8
= a) = 0 if and only if Q = 4 and the derivative changes
dQ Q − 8Q + 20
2

its sign from negative to positive; therefore, the minimum occurs for
Q=4
dP
b) = Q 2 − 17Q + 60 = 0 if and only if Q = 5 or Q = 12
dQ

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 d
We have (Q 2 − 17Q + 60) = 2Q − 17 ; this value is positive for Q = 12 ;
dQ
therefore, the minimum value occurs for Q = 12

Lesson 4 : Maximization of the total revenue function

a) Learning objectives:
Determine the price that will maximize the total revenue function
b) Teaching resources:
The following materials may be used in the teaching-learning process:
–– Calculator,
–– Manilla paper,
–– Student’s book;
–– Any reference text book containing information about applied extrema
problems.
c) Prerequisites:
Students will perform better in this lesson if:
–– They have mastered the revenue function as studied in Senior4, unit2;
–– They can easily solve simple equations in one unknown;
–– They are able to find derivatives of simple functions as covered in unit2
of this year;
–– They can perform numerical calculations correctly, mentally or using a
calculator;
d) Learning activities
–– Request students to organize themselves in groups under your
supervision;
–– Through well-chosen questions and discussion of learning activity 3.2.2,
bring the students to discover how to model a problem and how to
express the quantity to maximize as function of one variable;
–– Ensure that the students can find the derivative of a function and set
the conditions for a maximum to occur;
–– Encourage each member to participate actively in the group;
–– As they are discussing, concentrate on slow students for further
explanation and provide assistance to groups in need

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 –– Request a student, chosen at randomly, to present the findings of his/
her group to the whole class, and help them to harmonize the answer
–– Use different probing questions and guide the students to explore the
content and examples related to the Maximization of the total revenue
function.
–– Ask students to work out examples under your guidance, and work
individually application activity 3.2.2 to check the skills they have
acquired.

Answers of learning activity 3.2.2

How do I choose the length and the width of the rectangular plot so as to
obtain a plot with the greatest area?
e) Application activity 3.2.2.
Q) 24Q − 3Q 2
The total revenue function is R(=

dR d
=24 − 6Q =0 if and only if Q = 4 .Since (24 − 6Q) =−6 < 0 , the
dQ dQ
maximum value of the total revenue occurs for Q = 4 , and it is equal to 48

Lesson 4 : Price elasticity of demand

a) Learning objectives:
Measure the responsiveness of demand to change in price
b) Teaching resources:
The following materials may be used in the teaching-learning process:
–– Calculator,
–– Manilla paper,
–– Student’s book;
–– Any reference text book containing information about applied extrema
problems.
c) Prerequisites:
Students will perform better in this lesson if:
–– They have mastered the demand function as studied in Senior4, unit2;
–– They can easily solve simple equations in one unknown;
–– They are able to find derivatives of simple functions as covered in unit
2 of this year;

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 –– They can perform numerical calculations correctly, mentally or using a
calculator;
d) Learning activities
–– Request students to organize themselves in groups under your
supervision;
–– Through well-chosen questions and discussion of learning activity 3.3.1,
bring the students to discover how to model a problem and how to
express the quantity to maximize as function of one variable;
–– Ensure that the students can find the derivative of a function and set
the conditions for a maximum to occur;
–– Encourage each member to participate actively in the group;
–– As they are discussing, concentrate on slow students for further
explanation and provide assistance to groups in need
–– Request a student, chosen at randomly, to present the findings of his/
her group to the whole class, and help the students to harmonize the
answer
–– Use different probing questions and guide the students to explore the
content and examples related to the Elasticity of demand.
–– Ask students to work out examples under your guidance, and work
individually application activity 3.3.1 to check the skills they have
acquired.

Answers of learning activity 3.3.1

If the price increases, then the quantity demanded will decrease.

e) Application activity3.3.1.

dQd −2
a) =
dP P
100 dQd −1
At P = 4=, Qd ln= 1.832;=
16 dP 2
dQd P −1 4
The price elasticity of demand is ε d = . = . = −1.09
dP Qd 2 1.832
dQd −20
b) =
dP ( P + 1) 2
dQd −20 −5
At P = 3 ,=Qd 5; = =
dP 16 4

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 dQd P −5 3 −3
The price elasticity of demand is ε d = . = . = = −0.75
dP Qd 4 5 4

Lesson 6 : Price elasticity of supply

a) Learning objectives:
Measure the responsiveness of supply to change in price
b) Teaching resources:
The following materials may be used in the teaching-learning process:
–– Calculator,
–– Manilla paper,
–– Student’s book;
–– Any reference text book containing information about price elasticity.
c) Prerequisites:
Students will perform better in this lesson if:
–– They have mastered the supply function as studied in Senior4, unit2;
–– They can easily solve simple equations in one unknown;
–– They are able to find derivatives of simple functions as covered in unit2
of this year;
–– They can perform numerical calculations correctly, mentally or using a
calculator;
d) Learning activities
–– Request students to organize themselves in groups under your
supervision;
–– Through well-chosen questions and discussion of learning activity
3.3.2, bring the students to discover how the price elasticity of supply
is calculated;
–– Ensure that the students can find the derivative of a function and they
can use the formula of price elasticity of supply.
–– Encourage each member to participate actively in the group;
–– Request a student, chosen at randomly, to present the findings of his/
her group to the whole class, and help the students to harmonize the
answer

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 –– Use different probing questions and guide the students to explore the
content and examples related to the Elasticity of supply.
–– Ask students to work out examples under your guidance, and work
individually application activity 3.3.2 to check the skills they have
acquired.
e) Answers to activities
Learning activity 3.3.2.
If the price increases, then the quantity supplied will increase.
Application activity 3.3.2.

dQs 1
a) =
dP 4
dQs 1
=
At P = 11 , Qs 2;=
dP 4
dQs P 1 11 11
=
The price elasticity of supply is εs .
= =.
dP Qs 4 2 8
dQs
b)= 5e −0.2 P − Pe −0.2 P
dP
dQ
P = 4 , Qs 20
At= = e −0.8 ; s e −0.8
dP
dQs P 4 1
=
The price elasticity of demand is ε s = . e −0.8 .=
dP Qs 20e −0.8
5
3.6. Summary of unit 3
In this unit, we focused on some examples of how derivatives can be used in
Finance and in Economics. The following points were considered:
1) Marginal cost
Suppose a manufacturer produces and sells a product. Denote C(q) to be the
total cost for producing and marketing q units of the product. Thus, C is a
function of q and it is called the (total) cost function. The rate of change of C
with respect to q is called the marginal cost, that is,

dC
Marginal Cost =
dq

2) Marginal revenue
If y = C ( x) is cost of producing x units of a product, then R ( x) ,the total
revenue generated by selling x units of the product, is given by R( x) = x.C ( x)

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: the product of the number of units produced by the cost of producing the
units. Then, the marginal revenue is the instantaneous rate of change in the

dR
total revenue, that is .
dx
3)Minimization of the total cost function
dy d  dy 
If function y = f ( x) is such that = 0 at x0 and   > 0 at x0 , then
dx dx  dx 
the function y = f ( x) has a local minimum at x0 ; the minimum value of the
function is f ( x0 ) .
dy
Function y = f ( x) is said to be increasing for the values of x such that >0
dx
dy
, and decreasing for the values of x such that <0
dx
4) Maximization of the total revenue function
If the total cost function is y = f ( x) , then the total revenue function is
R = xy .
dR d  dR 
Suppose = 0 at x0 and   < 0 at x0 , then the total revenue
dx dx  dx 
function R = xy has a local maximum at x0 ; the maximum value of the total
revenue function is R( x0 ) .

5) Elasticity of demand
In Economics, price elasticity ε d measures the percentage change in quantity
associated with a percentage change in price. If the quantity Qd is related to
price P by, Qd = f ( P) , then the elasticity of demand is defined by

dQd P
εd = . .
dP Qd
Price elasticity of demand indicates how consumers respond to the change
in the amount proposed by the producers.
If ε d < 0 , then Qd and P are such that the increase in P implies the decrease
in Qd
6) Elasticity of supply
If the quantity Qs is related to price P by Qs = f ( P) , then the elasticity of
dQs P
demand is defined by ε s = . . In some cases, P is given in terms of Qs .
dP Qs

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In this case, start by making Qs the subject of the formula.

dQs P
The price elasticity of supply is defined by, ε s = . ,
dP Qs
Where Qs : quantity supplied, and P : amount received from consumers.
Price elasticity of supply indicates how producers respond to the change in
the amount they receive from the consumers

3.7. Additional information for the teacher.

The teacher will make research in the school library or on internet so as to
widen his/her knowledge.
Examples and exercises must be realistic; therefore, the teacher is advised to
choose and prepare carefully the examples to discuss.

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 3.8. Answers of End unit assessment
1.The total revenue function is R = 12.5Qe −0.005Q ;

dR
= (12.5 − 0.0625Q) e −0.005Q = 0 if and only=
if Q 200;
= P 1
dQ
d  4 x  12(ln x − 1)
2. =   = 0 for x= e ≈ 2.718
dx  3ln x  9 ln 2 x
dC
3.The marginal cost is = 3Q 2 − 6Q + 15
dQ
R 30Q − Q 2
4.Total revenue is=

dR
The total marginal revenue is = 30 − 2Q
dQ
dQ 2
5. = −15 ; the price elasticity of demand is −
dP 3

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3.9. Additional activities
3.9.1. Remedial activity
2
1) Given the total revenue= R 80 P − 2 P , where P is the price of an
item, obtain the marginal total revenue.
2) The total cost of commodity is given by the total cost function
25
C
= + 0.1Q 2 .Find the value of Q that will minimize the total cost.
Q
Prove that for this value, the minimum cost occurs, and find the
minimum total cost.
Solution:
dR
1) The marginal total revenue is = 80 − 4 P
dP
2) dC −25
= + 0.2Q =0 if and only if Q = 5
dQ Q 2
d  −25  50
d  −25  50 .For Q = 5 ,  2 + 0.2Q  = 3 + 0.2 > 0
 + 0.2Q  = + 0.2 dQ  Q  5
dQ  Q 2  Q
3

.Therefore, the minimum total cost occurs for Q = 5 , and the minimum
25
total cost is C =+ 0.1(5) 2 = 7.5
5

3.9.2. Consolidation activity

1. The quantity demanded and the price Qd and the price P of a
commodity are related by the equation P
= 60 − 3Qd .Find the price
elasticity of the demand when P = 12
2. The total demand function of a firm is given by Q
=d 200 − 2Q .Find
the marginal total revenue
Solution:

−1
Qd
1.We have:= P + 20 ;
3
dQd −1
= ;
dP 3
dQd −1 −1
For P = 12 , = ;Q
=d (12) + =
20 16 ;
dP 3 3

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Mathematics | Teacher's Guide | Senior Five | Experimental Version 75
 dQd P −1 12 −1
is ε d
The price elasticity of the demand= =. = .
dP Qd 3 16 4
is R 200Q − 2Q 2 ;
2.The revenue function of the firm=

dR
He marginal total revenue is = 200 − 4Q
dQ

3.9.3. Extended activity

The total cost function and the total revenue of a firm are given respectively

1 1
4 + 97Q − 8.5Q 2 + Q 3 and=
by C = R 58Q − Q 2
3 2

a) Find the total profit

b) Find the value of Q for which the total profit is maximum, and show that
for this value, the profit is, indeed, maximum
c) Find the maximum profit.
Solution:
a) The maximum profit is

1 1 1
π =(58Q − Q 2 ) − (4 + 97Q − 8.5Q 2 + Q 3 ) =−4 − 39Q + 8Q 2 − Q 3
2 3 3
dπ
b) = −39 + 16Q − Q 2 =0 if and only if Q = 3 or Q = 13
dQ
d dπ d
For Q = 3 , we have: ( )= (−39 + 16Q − Q 2 ) = 16 − 2Q > 0 ,
dQ dQ dQ
thus, the maximum does not occur for Q = 3

d dπ d
For Q = 13 , we have: ( )= (−39 + 16Q − Q 2 ) = 16 − 2Q < 0 , showing
dQ dQ dQ
that the maximum value of the total profit occurs for Q = 13 ;

1
c)The maximum total profit is −4 − 39(13) + 8(13) 2 − (39)3 =108.7
3

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 UNIVARIATE STATISTICS AND
Unit 4 APPLICATIONS

4.1 Key unit competence

Apply univariate statistical concepts to collect, organise, analyse, present,
and interpret data to draw appropriate decisions.

4.2 Prerequisite
Students will perform well in this unit if they are familiar with statistics
learned in lower secondary school (S1 and S3) and skilled in using scientific
calculators.

4.3 Cross-cutting issues to be addressed

• Inclusive education (promoting education for all in teaching)
• Peace and value Education (respect the views and thoughts of others
during class discussions)
• Financial education (develop sprits of saving and investments
decisions)
• Gender (equal opportunity for boys and girls to participate in class)
• Standardized culture (quality of production)

4.4 Guidance on introductory activity

• Invite students to work in a group, discuss and find out the answers
for the introductory activity from the student’s book.
• Facilitate students’ discussions and ask them to avoid noise or other
unnecessary conversations.
• During discussions, let students think of different ways to solve the
given problem.
• Walk around in all groups to help if necessary.
• Invite group members to present their findings and encourage boys
and girls to actively participate in presentations.

Mathematics | Teacher's Guide | Senior Five | Experimental Version 77

 •
Basing on students’ experience, prior knowledge and abilities shown
in answering the questions for this introductory activity, use different
probing questions to facilitate them to give their predictions and
ensure that you arouse their curiosity on what is going to be leant in
this unit.
Answer for an introductory activity
Use the introductory activity to give an overview of the whole unit, the
key concepts that will be discussed, and concepts such as inferential and
descriptive statistics, univariate data, population, sample, mean, mode,
median, quartile, variance, standard deviation, sampling and sampling
methods, quantitative and qualitative data, continuous and discrete data,
etc.
1. (a) The more data you have at your disposal, the better your
position will be to make good decisions and take advantage of new
opportunities. Good data will also give you the justification and
evidence to back up these decisions so that you can feel confident
explaining your reasoning. Without reliable data, you’re much more
likely to make mistakes and reach incorrect conclusions.
(b) In finance, quantitative data can be collected to determine a
company’s business income, earnings, and revenue-generating
capacity. Qualitative data can also be collected to understand
customers’ satisfaction with the company’s products and services.
(c) A family can record daily the money spent on each item bought;
this information can help the family make a well-informed decision
on the family budget. Similarly, a country needs information on the
amount of money to allocate to each sector (e.g., education, security,
agriculture, etc.), which can help make a national budget.
2. i) Data related to people’s favorite food.
ii) Questionnaire, interview
iii) Mode
3. During the accounting exam, out of 10, ten students scored the
following marks: 3, 5,6,3,8,7,8,4,8, 6.
a) 50 = 5
10
b) 3 and 8
c) Among ten students, only three scored below the average mark. This
indicates that students performed well in accounting exams.

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4.5 List of lessons
Number
Headings Lesson title/sub-headings Learning objectives
of periods
4.1 Basic Introductory activity Arouse the curiosity 1
concepts in of students on the
univariate content of unit 4.
statistics. 1 Statistical concepts: Explain and 2
statistics, descriptive differentiate various
and inferential statistics, concepts used in
population, sample, statistics
statistic, and parameter.
2 Variables and types of Differentiate types of 2
variables variables
3 Data and types of data Differentiate types of 2
data
4 Levels of measurement Differentiate levels of 2
scale measurement scale
5 Sampling and sampling Differentiate different 3
methods sampling methods
and decide which one
is feasible depending
on the context under
study
4.2 Organizing 1 Frequency table Represent data 2
and graphing accurately using
data a frequency
distribution table
2 Bar graph Represent data 2
accurately using a bar
graph
3 Histogram Represent data 2
accurately using a
histogram
4 Time series graph Represent time series 2
data using a time
series graph
5 Pie chart Represent data 2
accurately using a Pie
6 chart
Graph interpretation Interpret a statistical 1
graph.

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 4.3 Numerical 1 Describing data using mean, Describe data using 1
descriptive median and mode central measures,
measures mean, median and
mode and interpret
the results
2 Summarizing data using Summarize data 2
variance, standard using spread
deviation, and coefficient of measures, variance,
variation standard deviation,
and coefficient
of variation and
interpret the results
3 Determining the position of Determine the 2
data value using quartiles position of data value
using quartiles
4.4 Measure of 1 Skewness Determine the 1
symmetry skewness of the data

2 Chebyshev’s theorem and Apply Chebyshev’s 1

Empirical rule theorem and
Empirical rule to
determine the
skewness of the data
4.5 Applications 1 Examples of the Apply univariate 1
of univariate applications of univariate statistics in solving
statistics in statistics in solving mathematical
mathematical
problems related to finance, problems that involve
problems
that involve accounting, and economics. finance, accounting,
finance, and economics
accounting,
and
economics.
4.6 End unit 1
assessment

Lesson 1 : Statistical concepts: descriptive and inferential

statistics, population, sample, statistic, and parameter
a) Learning objective
Explain and differentiate various concepts used in statistics
b) Teaching resources
Student’s book and other reference books to facilitate research, calculator,

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Manila paper, markers, pens, pencils...
c) Prerequisites/Revision/Introduction
Students will perform well in this lesson if they are familiar with statistics
learned in lower secondary school (S1 and S3).
d) Learning activities:
• Invite students to work in groups and do learning activity 4.1.1 from
Student’s book;
• Move around to facilitate students where necessary and give more
clarification on eventual challenges they may face during their work;
• Ask randomly some groups to present their findings to the whole
class;
• As a teacher, harmonize the group findings and use different probing
questions to help students to explore examples and the content given
in the student’s book to enhance skills on concepts and terminologies
used in statistics.
• Ask students to do the application activity 4.1.1. and evaluate whether
lesson objectives were achieved to assess their competences.

Answers of learning activity 4.1.1

1.
i) The gathering, presentation, categorization, analysis, and
interpretation of data.
ii) Basically, there are two branches of statistics. They are descriptive
and inferential.

iii) Data, variable, population, mean, sample, etc

2. A company may select a sample of students before supplying and ask

what students prefer between milk and fruits. Information gathered
from that sample can help the company to make a well-informed
decision about what more students are likely to choose between milk
and fruits.
e) Answers of application activity 4.1.1
1. Suppose the scores of 100 students belonging to a specific country are
available. The performance of these students needs to be examined.
This data by itself will not yield any valuable results. However, by

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 using descriptive statistics, the spread of the marks can be obtained,
thus, giving a clear idea regarding each student’s performance. Now,
suppose the scores of the students of an entire country need to be
examined. Using a sample of 100 students, inferential statistics is used
to make generalizations about the population.
2. In this example:
• The population is all senior five students attending Kiziguro secondary
school.
• The sample could be all students enrolled in one section (combination,
let say, Mathematics, Economics, and Geography) at Kiziguro
secondary school (although this sample may not represent the entire
population).
• The parameter is the average (mean) amount of money spent
(excluding books) by senior five students at Kiziguro secondary
school: the population mean.
• The statistic is the average (mean) amount of money spent (excluding
books) by senior five students in the sample.

Lesson 2 : Variables and types of variables

a) Learning objective: Differentiate and explain types of variables.

b) Teaching resources
Student’s book and other Reference books to facilitate research, calculator,
Manila paper, markers, pens, pencils...
c) Prerequisites/Revision/Introduction
Students will perform well in this lesson if they are familiar with statistics
learned in lower secondary school (S1 and S3).
d) Learning activities:
• Invite students to work in groups and do learning activity 4.1.2 from
the Student’s book;
• Move around to facilitate students where necessary and give more
clarification on eventual challenges they may face during their work;
• Ask randomly some groups to present their findings to the whole
class;

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 • As a teacher, harmonize the group findings and use different probing
questions to help students to explore examples and the content
given in the student’s book to enhance skills on variables and types
of variables.
• Ask students to do the application activity 4.1.2. and evaluate whether
lesson objectives were achieved to assess their competences.

Answers of learning activity 4.1.2

a) Clients’ satisfaction and the amount of money saved.

b) Clients’ satisfaction will give qualitative data, while the amount of money
saved will give quantitative information.
e) Answers of application activity 4.1.2
i) Clients’ satisfaction.
Since there is only a single variable of interest, this is univariate statistics.
ii) Clients’ satisfaction and educational level.
Since there is only a single variable of interest, this is not a univariate
statistics.

Lesson 3 : Data and types of data

a) Learning objective: Differentiate types of data.
b) Teaching resources
Student’s book and other Reference books to facilitate research, calculator,
Manila paper, markers, pens, pencils...
c) Prerequisites/Revision/Introduction
Students will perform well in this unit if they are familiar with statistics
learned in lower secondary school (S1 and S3).
d) Learning activities:
• Invite students to work in groups and do learning activity 4.1.3 from
Student’s book;
• Move around to facilitate students where necessary and give more
clarification on eventual challenges they may face during their work;
• Ask randomly some groups to present their findings to the whole
class;

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 • As a teacher, harmonize the group findings and use different probing
questions to help students to explore examples and the content given
in the student’s book to enhance skills on data and types of data.
• Ask students to do the application activity 4.1.3. and evaluate whether
lesson objectives were achieved to assess their competences.

Answers of learning activity 4.1.3

Data are individual items of information that come from a population or
sample. Data is also defined as a set of observations.
Example
You go to the supermarket and purchase three soft drinks (500ml soda, 1ml
milk and 300ml juice) at 5000frw, four different kinds of fruits (apple, mango,
banana and avocado) at 800frw, two different kinds of vegetables (broccoli
and carrots) at 500frw, and two desserts (ice cream and biscuits) at 1000frw.
The prices (5000frw, 800frw, 500frw, and 1000frw) are data in this example.
Types of soft drinks, vegetable, fruits, and desserts are also data.

e) Answers of application activity 4.1.3

65000Frw, 45000Frw, 65000Frw, 15000Frw, 55000Frw, 35000Frw, 25000Frw,
45000Frw, 85000Frw and 95000Frw, are data which are quantitative.

Lesson 4 : Levels of measurement scale

a) Learning objectives: Differentiate levels of measurement scale
b) Teaching resources
Student’s book and other Reference books to facilitate research, calculator,
Manila paper, markers, pens, pencils...
c) Prerequisites/Revision/Introduction
Students will perform well in this lesson if they are familiar with statistics
learned in lower secondary school (S1 and S3).
d) Learning activities:
• Invite students to work in groups and do learning activity 4.1.4 from
Student’s book;
• Move around to facilitate students where necessary and give more
clarification on eventual challenges they may face during their work;
• Ask randomly some groups to present their findings to the whole
class;

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 • As a teacher, harmonize the group findings and use different probing
questions to help students to explore examples and the content given
in the student’s book to enhance skills on levels of measurement scale.
• Ask students to do the application activity 4.1.4. and evaluate whether
lesson objectives were achieved to assess their competences.

Answers of learning activity 4.1.4

Variables involved in this research are types of places, satisfaction, and age.
Types of places and satisfaction are categorical, while age is numerical.
Types of places cannot be ranked or ordered, but satisfaction and age can
be ranked.
Only for the age you can find the difference between its data values.

e) Answer of Application activity 4.1.4

i) Ages of the company workers (in years) is a ratio measurement scale.
ii) Color of clothes in a shop is a nominal measurement scale.
iii) Temperatures inside the room (in Celsius) is a ratio measurement
scale.
iv) Nationalities of the company workers is a nominal measurement scale.
v) Salaries of the company employees is a ratio measurement scale.
vi) Weights of boxes of fruits is a ratio measurement scale.

Lesson 5 : Sampling and sampling methods

a) Learning objectives
Differentiate different sampling methods and decide which one is feasible
depending on the context under study.
b) Teaching resources
Student’s book and other Reference books to facilitate research, calculator,
Manila paper, markers, pens, pencils...
c) Prerequisites/Revision/Introduction
Students will perform well in this lesson if they are familiar with statistics
learned in lower secondary school (S1 and S3).

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d) Learning activities:
• Invite students to work in groups and do learning activity 4.1.5 from
Student’s book;
• Move around to facilitate students where necessary and give more
clarification on eventual challenges they may face during their work;
• Ask randomly some groups to present their findings to the whole
class;
• As a teacher, harmonize the group findings and use different probing
questions to help students to explore examples and the content given
in the student’s book to enhance skills in sampling methods.
• Ask students to do the application activity 4.1.5. and evaluate whether
lesson objectives were achieved to assess their competences.

Answers of learning activity 4.1.5

The school may select a group of students (sample) and collect information
on how much they spend.
When selecting students, the school should ensure the representativeness
of students of all categories.
Answer for application activity 4.1.5
a) Systematic
b) Simple random sampling
c) Cluster
d) Stratified
e) Convenience

Lesson 6 : Frequency table

a) Learning objective
Represent data accurately using a frequency table.
b) Teaching resources
Student’s book and other Reference books to facilitate research, calculator,
Manila paper, markers, pens, pencils...
c) Prerequisites/Revision/Introduction
Students will perform well in this lesson if they are familiar with statistics
learned in lower secondary school (S1 and S3).

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d) Learning activities:
• Invite students to work in groups and do learning activity 4.2.1 from
Student’s book;
• Move around to facilitate students where necessary and give more
clarification on eventual challenges they may face during their work;
• Ask randomly some groups to present their findings to the whole
class;
• As a teacher, harmonize the group findings and use different probing
questions to help students to explore examples and the content given
in the student’s book to enhance skills in representing data using a
frequency table.
• Ask students to do the application activity 4.2.1 and evaluate whether
lesson objectives were achieved to assess their competences.

Answers of learning activity 4.2.1

The revenue paid by many people is 27000 FRW which four people paid.
Frequency (number of people who paid
Revenue paid (in FRW)
each revenue)
21000 1
22000 1
23000 1
24000 2
25000 1
26000 1
27000 4
28000 1
29000 3
31000 2
34000 1
36000 1
39000 1

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Answer for application activity 4.2.1
Marital Status Frequency
Divorced 8
Married 11
Separated 10
Single 21
Total 50

Lesson 7 : Bar graph

a) Learning objective
Represent data accurately using a bar graph.
b) Teaching resources
Student’s book and other Reference books to facilitate research, calculator,
Manila paper, markers, pens, pencils...
c) Prerequisites/Revision/Introduction
Students will perform well in this lesson if they are familiar with statistics
learned in lower secondary school (S1 and S3).
d) Learning activities:
• Invite students to work in groups and do learning activity 4.2.2 from
Student’s book;
• Move around to facilitate students where necessary and give more
clarification on eventual challenges they may face during their work;
• Ask randomly some groups to present their findings to the whole
class;
• As a teacher, harmonize the group findings and use different probing
questions to help students to explore examples and the content given
in the student’s book to enhance skills in representing data using a
bar graph.
• Ask students to do the application activity 4.2.2 and evaluate whether
lesson objectives were achieved to assess their competences.

Answers of learning activity 4.2.2

a) Microsoft, other, Lotus Development, Novell, WordPerfect, Autodesk,
Ashton-Tate, Borland International, Adobe Systems, Software Publishing,
Aldus, Santa Cruz, Symantec

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 19.5
b) × 5.7 =1.11 billion.
100
46.3
c) × 5.7 =2.64 billion.
100

e) Answer of application activity 4.2.2

1)

2)

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 Lesson 8 : Histogram

a) Learning objective
Represent data accurately using a histogram.
b) Teaching resources
Student’s book and other Reference books to facilitate research, calculator,
Manila paper, markers, pens, pencils...
c) Prerequisites/Revision/Introduction
Students will perform well in this lesson if they are familiar with statistics
learned in lower secondary school (S1 and S3).
d) Learning activities:
• Invite students to work in groups and do learning activity 4.2.3 from
Student’s book;
• Move around to facilitate students where necessary and give more
clarification on eventual challenges they may face during their work;
• Ask randomly some groups to present their findings to the whole
class;
• As a teacher, harmonize the group findings and use different probing
questions to help students to explore examples and the content given
in the student’s book to enhance skills in representing data using a
histogram.
• Ask students to perform the application activity 4.2.3 and evaluate
whether lesson objectives were achieved to assess their competences.

Answers of learning activity 4.2.3

Nine people spend more hours at work. They spend between 20 and 25 hours
per week.
a) 25 people
b) Histogram

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e) Answer of application activity 4.2.3

a.

b.

Lesson 9 : Time series graph

a) Learning objective
Represent data accurately using a time series graph.
b) Teaching resources
Student’s book and other Reference books to facilitate research, calculator,
Manila paper, markers, pens, pencils...
c) Prerequisites/Revision/Introduction
Students will perform well in this lesson if they are familiar with statistics
learned in lower secondary school (S1 and S3).
d) Learning activities:
• Invite students to work in groups and do learning activity 4.2.4 from
the S4 Mathematics student’s book;
• Move around to facilitate students where necessary and give more
clarification on eventual challenges they may face during their work;

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 • Ask randomly some groups to present their findings to the whole
class;
• As a teacher, harmonize the group findings and use different probing
questions to help students to explore examples and the content given
in the student’s book to enhance skills in representing data using a
time series graph.
• Ask students to do the application activity 4.2.4 and evaluate whether
lesson objectives were achieved to assess their competences.

Answers of learning activity 4.2.4

a) In the fourth quarter of 2006
b) 24 people
c) 14+24+9+8+12+22+11+7=107 people

e) Answer for application activity 4.2.4

Lesson 10 : Pie chart

a) Learning objective
Represent data accurately using a pie chart.
b) Teaching resources
Student’s book and other Reference books to facilitate research, calculator,
Manila paper, markers, pens, pencils...
c) Prerequisites/Revision/Introduction
Students will perform well in this lesson if they are familiar with statistics
learned in lower secondary school (S1 and S3).
d) Learning activities:
• Invite students to work in groups and do learning activity 4.2.5 from
the S4 Mathematics student’s book;

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 • Move around to facilitate students where necessary and give more
clarification on eventual challenges they may face during their work;
• Ask randomly some groups to present their findings to the whole
class;
• As a teacher, harmonize the group findings and use different probing
questions to help students to explore examples and the content given
in the student’s book to enhance skills in representing data using a pie
chart.
• Ask students to do the application activity 4.2.5 and evaluate whether
lesson objectives were achieved to assess their competences.

Answers of learning activity 4.2.5

a) The highest numbers of teenagers drink when upset (41%)
b) (41% + 25%) = 66% of teenagers drink because they are bored or upset.

c) Answer of application activity 4.2.5

Number= of mangoes (100 / 360 ) ×144 40 ,
=
=
Number of oranges ( 70 / 360 ) ×144 28 ,
=
Number of pawpaws = ( 40 / 360 ) ×144 = 16 ,
=
Number of apples (150 / 360 ) ×144 60 .
=
Cost of mangoes = 40 × 30 FRW = 1200 FRW ,
cost of pawpaws = 16 ×160 FRW = 2560 FRW .
The total cost of mangoes and pawpaws is = 2560 FRW + 1200 FRW = 3760 FRW

a) The most unsold is Apple i.e. 60 apples

b) Frequency table
Type of fruits Frequency (number remaining)
Mangoes 40
Oranges 28
Pawpaws 16
Apples 60
Total 144

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 Lesson 11 : Graph interpretation
a) Learning objective:
Interpret a statistical graph.
b) Teaching resources:
Manila papers, calculators, markers, student’s book, pens, notebooks.
c) Prerequisites/Revision/Introduction:
Students will perform well in this unit if they make a good revision on the
content of statistics learnt in senior three and in previous lessons of this unit.
d) Learning activities
• Invite students to work in groups and do the learning activity 4.2.6
from Student’s books;
• Move around in the class for facilitating students where necessary
and give more clarification on eventual challenges they may face
during their work;
• Verify and identify groups with different working steps;
• Invite one member from each group with different working steps to
present their work where they must explain the working steps;
• As a teacher, harmonize the findings from presentation highlighting
elements to be verified on a graph of data;
• Use different probing questions and guide them to explore examples
given in the student’s book and lead them to discover the different
ways of interpreting statistical data given graphically in reports or
newspapers.
• After this step, guide students to do the application activity 4.2.6 and
evaluate whether lesson objectives were achieved.

Answers of learning activity 4.2.6

a) There are 5 students with small size (S);
b) There are 13 students with medium size, 8 Students with large size and 4
students with Extra-large size.

e) Answers of application activity 4.2.6

a) There are 240 bags of cement produced in 8 minutes;
There are 96 bags of cement produced in 3min 12 seconds
There are 150 bags of cement produced in 5 minutes;

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There 210 bags of cement produced in 7 minutes;
b) b. It will take 2 minutes 48 seconds to produce 78 bags of cement.
c)
Number of bags 96 150 210 240
Time in minutes 3min 12sec 5 7 8

Lesson 12 : Describing data using mean, median, and mode

a) Learning objective
Describe data using mean, median, and mode and interpret the results
accurately.
b) Teaching resources
Student’s book and other Reference books to facilitate research, calculator,
Manila paper, markers, pens, pencils...
c) Prerequisites/Revision/Introduction
Students will perform well in this lesson if they are familiar with statistics
learned in lower secondary school (S1 and S3).
d) Learning activities:
• Invite students to work in groups and do learning activity 4.3.1 from
the S5 Mathematics student’s book;
• Move around to facilitate students where necessary and give more
clarification on eventual challenges they may face during their work;
• Ask randomly some groups to present their findings to the whole
class;
• As a teacher, harmonize the group findings and use different probing
questions to help students to explore examples and the content given
in the student’s book to enhance skills in describing data using mean,
median, and mode.
• Ask students to do the application activity 4.3.1 and evaluate whether
lesson objectives were achieved to assess their competences.

Answers of learning activity 4.3.1

a) A portfolio’s mean return on investment is an arithmetic average of returns
achieved over specified periods. The statistic can easily be calculated by
adding together all returns for a portfolio per unit of time and dividing by
the number of observations. The mean return on investment would be
calculated as follows:

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 ( 0.1 − 0.03 + 0.08 + 0.12 + 0.12 − 0.07 + 0.03) = 0.05
7 .
This would give us a mean return of 5% over the seven quarters.
b) We can arrange the return of the portfolio in the following ascending
order: Q5 = −7%, Q2 =−3%, Q7 =
+3%, Q3 =
+8%, Q1 =
+10%, Q4 =
+12%, and Q6 =
+12%.
The middle value in this series is 8%, achieved in Q3. Therefore, the median
return of the portfolio would be 8%.
a) The return of the portfolio that has been achieved frequently is +12%,
achieved in Q4 and Q6. Therefore, the mode would be +12%.
e) Answer of application activity 4.3.1
This is a sample of n = 6 , where

=x1 104,
= x2 340,
= x3 140,
= x4 185,
= x5 and x6
270, = 258
.
We find the sample mean by adding all the observations and dividing by 6:

(104 + 340 + 140 + 185 + 270 + 258) / 6 =

216.166
.
As in the problem salaries are in thousands then the mean is 216167FRW
To find the median monthly salary, we need to arrange as follows: 104, 140,
185, 258, 270,340
As n is even (n=6), Median is given by

1  6   6   1 rd
th th
1
  +  + 1  = 3 + 4 =
th
[185 + 258]= 221.5
2  2   2   2 2
As in the problem salaries are in thousands then median is 221500FRW

Lesson 13 : Summarizing data using variance, standard deviation,

range, mean deviation, and coefficient of variation

a) Learning objective
Summarize data using variance, standard deviation, range, mean deviation,
and coefficient of variation and interpret the results accurately.
b) Teaching resources
Student’s book and other Reference books to facilitate research, calculator,
Manila paper, markers, pens, pencils...
c) Prerequisites/Revision/Introduction
Students will perform well in this unit if they are familiar with statistics
learned in lower secondary school (S1 and S3).

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d) Learning activities:
• Invite students to work in groups and do learning activity 4.3.2 from
the S5 Mathematics student’s book;
• Move around to facilitate students where necessary and give more
clarification on eventual challenges they may face during their work;
• Ask randomly some groups to present their findings to the whole
class;
• As a teacher, harmonize the group findings and use different probing
questions to help students to explore examples and the content given
in the student’s book to enhance skills in summarizing data using
variance, standard deviation, range, mean deviation, and coefficient
of variation.
• Ask students to do the application activity 4.3.2 and evaluate whether
lesson objectives were achieved to assess their competences.

Answers of learning activity 4.3.2

The heights (at the shoulders) are 600mm, 470mm, 170mm, 430mm, and
300mm.
5

∑x i
600 + 470 + 170 + 430 + 300 . so, the mean (average)
a) i =1
= 394
5 5
height is 394 mm. Let’s plot this on the chart:

b) Now we calculate each dog’s difference from the mean:

To calculate the Variance, take each difference, square it, and then average
the result:

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 2062 + 762 + ( −224 ) + 362 + ( −94 )
2 2

σ 2
= 21704
5
So the variance is 21704.
Standard deviation is just the square root of the variance, so:
=σ = 21704 147.32 .
And the good thing about the standard deviation is that it is useful. Now we
can show which heights are within one standard deviation (147mm) of the
mean:

So, using the Standard Deviation, we have a “standard” way of knowing

what is normal and what is extra-large or extra-small.
e) Answer of application activity 4.3.2
For pianos, the cost of the piano is 0.4 standard deviations BELOW the mean.
For guitars, the guitar cost is 0.25 standard deviations ABOVE the mean. For
drums, the cost of the drum set is 1.0 standard deviations BELOW the mean.
Of the three, the drums cost the lowest compared to other instruments of
the same type. The guitar costs the most compared to other instruments of
the same type.

Lesson 14 : Determining the position of data value using quartiles

a) Learning objective
Determine the position of data value using quartiles and interpret the results
accurately.
b) Teaching resources
Student’s book and other Reference books to facilitate research, calculator,
Manila paper, markers, pens, pencils...
c) Prerequisites/Revision/Introduction
Students will perform well in this unit if they are familiar with statistics
learned in lower secondary school (S1 and S3).

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d) Learning activities:
• Invite students to work in groups and do learning activity 4.3.3 from
the S5 Mathematics student’s book;
• Move around to facilitate students where necessary and give more
clarification on eventual challenges they may face during their work;
• Ask randomly some groups to present their findings to the whole
class;
• As a teacher, harmonize the group findings and use different probing
questions to help students to explore examples and the content given
in the student’s book to enhance skills in determining the position of
the data value using quartiles.
• Ask students to do the application activity 4.3.3 and evaluate whether
lesson objectives were achieved to assess their competences.

Answers of learning activity 4.3.3

i) The median, M, lies at the 6th position and divides the set into two
halves; thus, M = 5600.
ii) There are five items with prices below M and five above M.
iii) The middle value of the lower half lies in the third position, which is
4800.
iv) The middle value of the upper half lies in the 9th position, which is 7700.
v) The two middle prices and the median divide the set into four quarters.
In short, the lower middle value is called the first (lower) Quartile, denoted
by Q1. The middle value of the upper half is called the third (upper) Quartile,
denoted by Q3. Thus: M = 56, Q1 = 48, Q3 = 77.
e) Answer of application activity 4.3.3
We begin by arranging the heights in ascending rank or order. The middle
height lies in the 7th position. median = 163 cm.

13 + 1
To obtain median, we take = 7 . This tells us that the median is in the
2
7th position which is 163 cm.

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 To obtain the lower quartile, we take 13 + 1 = 3.5 .
4
This means the lower quartile, Q1 is between the 3rd and the 4th position i.e.
3.5th position.
= Hence, Q 1= (160 +161 ) 160.5
1
2
To obtain Q3, we take ( 13 + 1 ) × 3 = 10.5 .
4
This means Q3 lies between the 10th and the 11th items i.e. 10.5th position.

1
Hence, Q1
= =(164 +165 ) 164.5 .
2

Lesson 15 : Skewness

a) Learning objective
Determine the skewness of data.
b) Teaching resources
Student’s book and other Reference books to facilitate research, calculator,
Manila paper, markers, pens, pencils...
c) Prerequisites/Revision/Introduction
Students will perform well in this lesson if they are familiar with statistics
learned in lower secondary school (S1 and S3).
d) Learning activities:
• Invite students to work in groups and do learning activity 4.4.1 from
the S5 Mathematics student’s book;
• Move around to facilitate students where necessary and give more
clarification on eventual challenges they may face during their work;
• Ask randomly some groups to present their findings to the whole
class;
• As a teacher, harmonize the group findings and use different probing
questions to help students to explore examples and the content given
in the student’s book to enhance skills in determining the skewness
of the data.
• Ask students to do the application activity 4.4.1 and evaluate whether
lesson objectives were achieved to assess their competences.

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 Answers of learning activity 4.4.1
a.

b. Dataset 1: mean=75.9, median=76, and mode=76

Dataset 2: mean=60.4, median=60, and mode=58
c. Dataset 1: median=mode and both greater than mean
Dataset 2: mean> median> mode.
For dataset 1, all measures of a central tendency (mean, median, and
mode) lie in the middle.

e) Answer of application activity 4.4.1

For histogram A, the data is skewed right, or there is a positive skewness.
The data is skewed left for histogram B, or there is a negative skewness.
For histogram A, the data is symmetric, or there is zero skewness.

Lesson 16 : Chebyshev’s theorem and Empirical rule

a) Learning objective
Apply Chebyshev’s theorem and Empirical rule to determine the skewness
of the data.
b) Teaching resources
Student’s book and other Reference books to facilitate research, calculator,
Manila paper, markers, pens, pencils...
c) Prerequisites/Revision/Introduction
Students will perform well in this lesson if they are familiar with statistics
learned in lower secondary school (S1 and S3).

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 d) Learning activities:
• Invite students to work in groups and do learning activity 4.4.2 from
the S5 Mathematics student’s book;
• Move around to facilitate students where necessary and give more
clarification on eventual challenges they may face during their work;
• Ask randomly some groups to present their findings to the whole
class;
• As a teacher, harmonize the group findings and use different probing
questions to help students to explore examples and the content
given in the student’s book to enhance skills in applying Chebyshev’s
theorem and Empirical rule to determine the skewness of the data.
• Ask students to do the application activity 4.5.2 and evaluate whether
lesson objectives were achieved to assess their competences.

Answers of learning activity 4.4.2

Mean =73.4 and std=1.17
a. Eight prices fall within one standard deviation from the mean.
b. Two prices fall within two standard deviations from the mean.
c. None of the price that falls within three standard deviations from the
mean.
Answer of application activity 4.4.2
a) According to the scale with measurements being given in grams:
mean=74.06 and s=1.166=1.17
b) The correct readings are 79.2, 77.4, 78.3, 75.2, 77.5, 78.6, 77.2, 76.3, and 76.
Therefore mean=77.26 and s=1.166=1.17.
If each reading is increased by 3.2, then the mean is increased by 3.2. The
standard deviation, however, remains unaltered.

Lesson 17 : Examples of the applications of univariate statistics in

solving problems related to finance, accounting, and
economics
a) Learning objective
Apply univariate statistics in solving problems related to finance, accounting,
and economics.
b) Teaching resources
Student’s book and other Reference books to facilitate research, calculator,
Manila paper, markers, pens, pencils...

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c) Prerequisites/Revision/Introduction
Students will perform well in this lesson if they are familiar with statistics
learned in lower secondary school (S1 and S3).
d) Learning activities:
• Invite students to work in groups and do learning activity 4.5.1 from
the S5 Mathematics student’s book;
• Move around to facilitate students where necessary and give more
clarification on eventual challenges they may face during their work;
• Ask randomly some groups to present their findings to the whole
class;
• As a teacher, harmonize the group findings and use different probing
questions to help students to explore examples and the content given
in the student’s book to enhance skills in applying univariate statistics
in solving problems related to finance, accounting, and economics.
• Ask students to do the application activity 4.5.1 and evaluate whether
lesson objectives were achieved to assess their competences.

Answers of learning activity 4.5.1

Answers will depend on what the student has provided.
Answer for application activity 4.5.1
1) a. 8 b. 535880FRW
2) 698000FRW

4.6 Summary of the unit

Statistics is the branch of mathematics that deals with the collection,
organization, analysis, interpretation, and drawing of conclusions from
numerical facts or data.
Statistics plays a vital role in nearly all businesses and forms the backbone for
all future development strategies. Every business plan starts with extensive
research, which is all compiled into statistics that can influence a final
decision. Statistics helps the businessman to plan production according to
the taste of customers.
There are two branches of statistics, namely descriptive statistics and
inferential statistics. Descriptive deals with describing the population under
study. Inferential focuses on drawing conclusions about the population
based on sample analysis and observation.

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 In statistics, we generally want to study a population. Because it takes a lot
of time and money to examine an entire population, we select a sample to
represent the whole population. The population is a collection of persons,
things, or objects under study. A sample is the portion of the population that
is available or to be made available for analysis.

For example, collecting the monthly savings data of every family that
constitutes your population may be challenging if you are interested in the
savings pattern of an entire country. In this case, you will take a small sample
of families from across the country to represent the larger population of
Rwanda. You will use this sample data to calculate its mean and standard
deviation.
In statistics, we collect information on individual items from a population or
sample, called statistical data. The collected data can be either qualitative or
quantitative.

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 Data

Categorical (Qualitative) Numerical (Quantitative)

Nominal Ordinal Discrete Continuous

This unit also introduced some tabular and graphical representations of the data,
such as frequency tables, histograms, bar graphs, time series graphs, and pie charts.
We also looked at numerical descriptive measures: mean, median, mode, range,
quartiles, variance, standard deviation, mean deviation, and coefficient of variation.
Measures of symmetry were also discussed.
The following formulas are very important to describe data:

• Mean , , where n is the number of observations in the

dataset, are observations.

Or x = 1 ∑ xf i
by multiplying each distinct value by its frequency and then dividing
n
the sum by the total number of data values.
• If n is odd, Median = x n +1  , or
 2 
th
 n +1 
 
median is given by  2  number which is located on this position
x n  + x n 
• If n is even, Median  
2
 +1
2  , or
=
2
1  n   n  
th th

Median is given by   +  + 1  , then the median is a half of the sum

2  2   2  
of number located on those two positions.
• The variance
Σ ( x − µ ) , where x is individual value, µ is the population mean, and
2

σ =
2

N
N is the population size or

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 2
 −
 −
Σ  x − x  , where x is individual value, x is the sample mean, and n is
S2 =  
n −1
the sample size.

• Standard deviation

Σ(x − µ)
2

σ
= σ
= 2
, where x is individual value, µ is the population
N
mean, and N is the population size, or

2
 −
 −
Σ  x − x  , where x is individual value, x is the sample mean,
=S = S2  
n −1
and n is the sample size.
• Coefficient of variation

σ
CV= ×100% , for population
µ
S
CV= − ×100% , for sample data
x
• Quartiles th
1 
In general, the lower quartile, Q1 takes the  ( n + 1)  position from the
4  th
3 
lower end on the rank order. The upper quartile, Q3 takes the  ( n + 1) 
4 
th
position on the rank order. For large population, it is enough to use  1 (n) 
 4 
th
3 
and  ( n )  positions for the lower and upper quartiles respectively.
4 

• Measure of skewness
One measure of skewness, called Pearson’s first coefficient of skewness, is
to subtract the mean from the mode, and then divide this difference by the
standard deviation of the data.

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 Mean − Mode
Skewness =
σ
Pearson’s second coefficient of skewness is also used to measure the
asymmetry of a data set. For this quantity, we subtract the mode from
the median, multiply this number by three and then divide by the standard
deviation.
3 ( Median − Mode )
Skewness =
σ
4.7 Additional Information for Teacher
While facilitating this unit, focus on interpreting the results obtained after
computing statistical measures. Also, emphasize graphical interpretations.
When explaining Chebyshev’s theorem and Empirical, lower the concept to
secondary school students by focusing on symmetrical data. Don’t talk too
much about normal distribution because students will learn distribution in
S6.
In this unit, while explaining, use many examples related to finance,
accounting, economics as much as you can to help students understand
concepts easily.

4.8. Answers of End unit assessment

This part provides the answers of end unit assessment activities designed
in an integrative approach to assess the key unit competence with cross-
reference to the student book.
1. Descriptive statistics.
2. Inferential statistics because you wish to generalize information
collected from a sample to the whole population.
3. (a) Inferential statistics
(b) Descriptive statistics
4. The variance is 5, and the standard deviation is 2.23.
5. The variance is 2.5, and the standard deviation is 1.58.
6. (a) 6 observations
(b) 8 variables
(c) Qualitative are 0 and quantitative are 8

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 4.9 Additional activities
4.9.1 Remedial activities
1. Eleven fish are sampled from a lake and their lengths were measured.
The lengths, in centimeters, of the fish are: 17, 16, 10, 17, 17, 16, 14, 14,
16, 10, and 14. Calculate the variance and standard deviation.
Answers: Sample variance is 6.65 and sample standard deviation is
2.58
2. Seven males are chosen randomly from a gym class in high school and
are asked their shoe size. They are: 9, 12, 11, 9, 13, 10, and 10.What are
the variance and standard deviation?
Answers: Sample variance is 2.28 and sample standard deviation is 1.51.

4.9.2 Consolidation activity

The mean of two samples of sizes 250 and 350 were 20 and 12, respectively.
Their standard deviations were 2 and 5, respectively. Find the variance of
combined sample of size 650.
Answer:
n1 x1 + n2 x2 250 × 20 + 320 ×12
=
Combined mean= = 13.6
n1 + n2 650
Let d1 =x1 − combinedMean =20 − 13.6 =6.4 , d 2 =
x2 − combinedMean =
12 − 13.6 =
−1.6

 250 ( 6.4 + 40.96 ) + 320(13.6 + 2.56 

var iance = 26.17
650
4.9.3 Extended activity
Suppose that a group of 100 students has an average age of 23 years, with
a standard deviation of 2 years. What information does the empirical rule
allow to obtain?
Answer: Construct the interval of the rule
Since the mean is 23 and the standard deviation is 2, then the intervals are:

• [ µ − σ , µ + σ ] = [ 23 − 2, 23 + 2] = [ 21, 25]
• [ µ − 2σ , µ + 2σ ] = [ 23 − 4, 23 + 4] = [19, 27]
• [ µ − 3σ , µ + 3σ ] = [ 23 − 6, 23 + 6] = [17, 29]

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Calculate the number of students in each interval according to the
percentages.
(100) × 68.27% = 68 students approximately.
(100) × 95.45% = 95 students approximately.
(100) × 99.73% = 100 students approximately.
Age intervals are associated with the numbers of students and interpret.
At least 68 students are between the ages of 21 and 25.
At least 95 students are between the ages of 19 and 27.
Almost 100 students are between 17 and 29 years old.

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 BIVARIATE STATISTICS AND
Unit 5 APPLICATIONS

5.1 Key unit competence

Apply bivariate statistical concepts to collect, organise, analyse, present, and
interpret data to draw appropriate decisions.

5.2 Prerequisite
Students will perform well in this unit if they are familiar with univariate
statistics learned in the previous unit (Unit 4) and they are skilled in using
scientific calculators.

5.3 Cross-cutting issues to be addressed.

• Inclusive education (promoting education for all in teaching)
• Peace and value Education (respect the views and thoughts of others
during class discussions)
• Financial education (develop sprits of saving, and investments
decisions)
• Gender (equal opportunity for boys and girls to participate in class)
• Environment sustainability (growth, etc)
• Standardized culture (quality of production)

5.4 Guidance on introductory activity

• In groups, facilitate students to read and do introductory activity from
the student book.
• Guide students to read and analyse the questions insisting on the
analysis of statistical data with two variables (x, y) and how they
can interpret the bivariate data using correlation coefficients and
regression lines.
• Facilitate any discussions, ensure that the work is done without noise
and without unnecessary conversations.

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 • Walk around the classroom to assist students in need.
• Invite group representatives to present their findings and encourage
gender in the presentation.
• In the lesson discussion, let students think of different ways to solve
the problem.
• Basing on students’ experience, prior knowledge and abilities shown
in answering the questions for this activity, use different questions to
facilitate them to give their predictions and ensure that you arouse
their curiosity on what is going to be leant in this unit.
Answer of Introductory activity:
Xi Yi X2i Xi Y i
1 4 1 4
2 8 4 16
3 2 9 6
4 12 16 48
5 10 25 50
6 14 36 84
7 16 49 112
8 6 64 48
9 18 81 162
Σ 45 90

1
X
=
1
∑ X=
45
X
= 5, = ∑ y=i 90= 10
n
i
9 n 9

Yi δ X ,,Y (X − X I )
Y −=
Y − 10
= 1.33(X − 5)
The equation of regression line of
Y =+10 1.33X − 6.65
=Y 1.33 X + 3.35
Scatter diagram: plotting the 9 sample points (1,4),(2,8),(3,4),(4,12),
(5,10),(6,14),(7,16) (8,6),(9,18).

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 The first point on the line is (2,6) . Another point on the line is (x,y) = (5,10)
so the regression line of y on x passes through the two points (2,6) and (4,10)
plot these points and join them the required line of regression of is obtained.

5.5 List of lessons

Headings # Lesson title/ Learning objectives Number of
sub-headings periods (9)
5.1 Introductory To arouse the curiosity of 1
Introduction activity student-teacher on the
to bivariate content of unit 5.
statistics 1 Key concepts Differentiate between 1
of bivariate univariate data and bivariate
statistics data, independent variable
and dependent variable,
and represent graphically
bivariate data in a scatter
diagram.

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 5.2 1 Covariance Apply covariance and 2
Measures and correlation coefficient in
of the linear correlation finding the relationship
relationship between two variables.
between two 2 Regression Apply regression line and 2
variables and line and analysis in estimating the
Applications: analysis value of one variable when
covariance, one is known.
Correlation, 3 Spearman’s Apply Spearman’s 1
regression coefficient of coefficient of correlation
line and correlation to measure statistical
analysis, and dependence between two
spearman’s variables.
coefficient of 4 Application Apply bivariate statistics in 1
correlation. of bivariate accounting-related subjects.
statistics
5.3 End unit assessment 1

Lesson 1 : Key concepts of bivariate statistics

a) Learning objective:
Differentiate between univariate data and bivariate data, independent
variable and dependent variable, and represent graphically bivariate data in
a scatter diagram.
b) Teaching resources
Student’s book and other Reference books to facilitate research, calculator,
Manila paper, markers, pens, pencils...
c) Prerequisites/Revision/Introduction
Students will learn better in this lesson if they have a good background
on univariate data and representation of points in x and y plane (cartesian
plane).
d) Learning activities:
• Invite students to work in groups and do learning activity 5.1.1 in their
Mathematics books.
• Move around in the class for facilitating students where necessary
and give more clarification.
• Verify and identify groups with different usable terminology.

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 • Invite one member from each group to present their work where they
must explain what they have done in learning activity 5.1.1.
• As a teacher, harmonize the findings from presentation and guide
students to answer the learning activity 5.1.1.
• Use different probing questions and guide students to explore the
content and examples given in the student’s book and lead them to
discover how to define univariate data, bivariate data, independent
variable and dependent variable. Use examples to differentiate
univariate from bivariate statistics, independent variable from
dependent variable.
• After this step, guide students to do the application activity 5.1.1 and
evaluate whether lesson objectives were achieved.

Answers of learning activity 5.1.1

Use the learning activity to give an overview of the whole unit, the key
concepts that will be discussed, and concepts such as Bivariate data, Scatter
diagram, Covariance, Correlation, and Regression lines.
i) Statistical measure is a correlation. Variables are temperature and ice
cream sales.
ii) Ice cream sales depend on temperature.
iii) Dependent variable (s)
iv)

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 v) According to the scatter diagram on iv, a positive relationship exists
between temperature and ice cream sales. It shows an increase in
temperature increased ice cream sales. This means that shops could
use this information to buy more ice cream for hotter spells during the
summer.
Answers for Application activity 5.1.1
1. Univariate statistics deals with one variable of interest whereas
bivariate statistics deals with two variables of interest and how the
two are related. If you are interested in finding out whether the more
people save their income, the more financially stable they become, in
this case you will need bivariate statistics. But if you are only interested
in finding out the average income people save, in this case you will
need univariate data.
2. Dependent variable is a variable that depends on the other.
Independent variable is a variable that affect the other. From the
example above on question 1, income will be an independent variable
whereas becoming financially stable will be the dependent variable.
In other words, to become financially stable depends on how much
income you save.

Lesson 2 : Covariance and correlation

a) Learning objective:
Apply covariance and correlation coefficient in finding the relationship
between two variables.
b) Teaching resources
Student’s book and other Reference books to facilitate research, calculator,
Manila paper, markers, pens, pencils...
c) Prerequisites/Revision/Introduction
Students will learn this lesson better if they have a good background in
univariate statistics.
d) Learning activities:
• Invite students to work in groups and do the activity 5.2.1 in their
Mathematics books.
• Move around in the class for facilitating students where necessary and
give more clarification to apply formula of covariance, and correlation
in finding the relationship between two variables.

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 • Invite one member from each group to present and explain how to find
relationship between two variables using covariance and correlation.
• As a teacher, harmonize the findings from presentation.
• Use different probing questions and guide students to explore the
content and examples given in the student’s book and lead them to
discover how to apply formula of covariance and correlation in finding
relationship between two variables.
• After this step, guide students to the application activity 5.2.1 and
evaluate whether lesson objectives were achieved.

Answers of learning activity 5.2.1

i) Correlation
ii) Covariance
Answers for application activity 5.2.1

There is a negative correlation.

Lesson 3 : Regression lines and analysis

a) Learning objective:
Apply regression line and analysis in estimating the value of one variable
when one is known.
b) Teaching resources
Student’s book and other Reference books to facilitate research, calculator,
Manila paper, markers, pens, pencils...

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c) Prerequisites/Revision/Introduction
Students will learn better in this lesson if they have a good background in
univariate statistics and linear equations.
d) Learning activities:
• Invite students to work in groups and do the learning activity 5.2.2 in
their Mathematics books.
• Move around in the class to facilitate students where necessary and
give more clarification about how to apply the regression line and
analysis in finding the unknown value of one variable when the other
one is known.
• Monitor the work of different groups.
• Invite one member from each group to present their work, where
they must explain the provided steps to find the regression lines.
• As a teacher, harmonize the findings from the presentation and guide
students to find the regression lines and prediction of values of one
variable when the values of the other one are known.
• Use different probing questions and guide students to explore the
content and examples given in the student’s book and lead them to
discover how to find the regression lines.
• After this step, guide students to application activity 5.2.2. and
evaluate whether lesson objectives were achieved.
Answers for activity 5.2.2

a.

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 b.
c. Non-connected points are far from the straight line, they are scattered.
d. There is a positive relationship between the number of cows and a number
of goats.
Answers for Application activity 5.2.2
a.
= y 0.19 x − 8.098
b. y = 4.06

Lesson 4 : Spearman’s coefficient of correlation

a) Learning objectives
Apply Spearman’s coefficient of correlation to measure statistical dependence
between two variables.
b) Teaching resources
Student’s book and other Reference books to facilitate research, calculator,
Manila paper, markers, pens, pencils...
c) Prerequisites/Revision/Introduction
Students will learn better in this lesson if they have a good background in
univariate statistics covered in the previous unit.
d) Learning activities:
• Invite students to work in groups and do the learning activity 5.2.3 in
their Mathematics books.
• Move around in the class to facilitate students where necessary and
give more clarification to define and apply the formula of Spearman’s
coefficient of correlation.
• Invite one member from each group to present their work, where
they must explain the provided steps to define and apply the formula
of Spearman’s coefficient of correlation in solving mathematical
problems.

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• As a teacher, harmonize the findings from the presentation and guide
students to find Spearman’s coefficient of correlation and apply that
formula in solving mathematical problems.
• Use different probing questions and guide students to explore the
content and examples given in the student’s book and lead them to
discover how to apply Spearman’s coefficient correlation in solving
mathematical problems.
• After this step, guide students to the application activity 5.2.3 and
evaluate whether lesson objectives were achieved.

Answers of learning activity 5.2.3

i) The death for underdeveloped countries (x) is 3 5 6 8 9 11.
The death rate for developed countries (y) is 2 3 4 5 6 8.
ii) See Rank (x) and Rank (y) in the table below.
iii)
Year x y Rank (x) Rank (y) Rank (x)- Rank d2
(y)=d
1995 3 2 1 1 0 0
1996 5 3 2 2 0 0
1997 6 4 3 3 0 0
1998 8 6 4 5 -1 1
1999 9 5 5 4 1 1
2000 11 8 6 6 0 0
n=6 n

∑d
i =1
i
2
=2

6× 2 2
1− 1
=− =0.94 .
6 ( 6 − 1)
2
35

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 Answer for Application activity 5.2.3
X y Rank (x) Rank (y) Rank (x)- Rank (y)=d d2
12 6 6.5 3.5 3 9
8 5 2 2 0 0
16 7 9.5 5.5 4 16
12 7 6.5 5.5 1 1
7 4 1 1 0 0
10 6 4 3.5 0.5 0.25
12 8 6.5 7 0.5 0.25
16 13 9.5 10 0.5 0.25
12 10 6.5 8.5 2 4
9 10 3 8.5 5.5 30.25
10

∑d
i =1
i
2
= 61

6 × 61 366
1− 1−
= 0.63 .
=
10 (10 − 1)
2
990

Lesson 5 : Applications of bivariate statistics in accounting-

related subjects or any other area: Relationship between advertising
products and total revenue, the relationship between attendance in
class and the marks scored, etc.
a) Learning objectives
Apply bivariate statistics in accounting-related subjects or any other area.
b) Teaching resources
Student’s book and other Reference books to facilitate research, calculator,
Manila paper, markers, pens, pencils...
c) Prerequisites/Revision/Introduction
Students will learn better in this lesson if they have a good background in
univariate statistics covered in the previous unit.
d) Learning activities:
• Invite students to work in groups and do the learning activity 5.2.4 in
their Mathematics books.
• Move around in the class to facilitate students where necessary
and give more clarification on applications of bivariate statistics in
economics, accounting, biology, medicine, etc.

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 • Invite one member from each group to present their work, where
they must explain the application of bivariate statistics.
• As a teacher, harmonize the findings from the presentation and
explain how bivariate statistics is applied to different areas. You can
refer to some examples provided in the student book.
• After this step, guide students to the application activity 5.2.4 and
evaluate whether lesson objectives were achieved.

Answers of learning activity 5.2.4

Bivariate statistics can help predict the value for one variable if we know the
value of the other.
Bivariate statistics can help find the relationship between two variables.
Answer for application activity 5.2.4
a) i. -0.976 ii. -0.292
b) The transport manager’s order is more profitable for the seller,
saleswomen is unlikely to try to dissuade.
c) (i) No, maximum value is 1. (ii) Yes, higher performing cars generally do
less mileage to the gallon. (iii) No, the higher the engine capacity, the
dearer the car.
d) When only two rankings are known; when relationship is non-linear.

5.6 Summary of the unit

Bivariate statistics deals with the collection, organization, analysis,
interpretation, and drawing of conclusions from bivariate data.
Bivariate statistics plays a vital role in predicting one variable’s value when the
other one is known. We normally collect bivariate data to try and investigate
the relationship between the two variables and then use this relationship
to inform future decisions. In the case of bivariate data sets, we are often
interested in whether elements with high values of one of the variables also
have high values of other variables.

Data sets that contain two variables, such as wage and gender, and consumer
price index and inflation rate data are said to be bivariate. For example, we
may collect the monthly savings and number of family members data of
every family that constitutes the population if we are interested in finding
the relationship between savings and the number of family members. In
this case, we will take a small sample of families from across the country to
represent the larger population of Rwanda. We will use this sample to collect

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 data on family monthly savings and the number of family members.
This unit also introduced the graphical representation of bivariate data, called
a scatter diagram. We also looked at numerical measures of the relationship
between two variables: covariance and correlation. Regression lines were
also discussed.
In this unit, each concept was introduced using a learning activity that
aroused students’ interest and curiosity about what was going to be learnt
in the lesson, and the lesson ended with an application activity to check
whether learning objectives were achieved or not.

5.7 Additional Information for Teacher

While facilitating this unit, focus on the difference between covariance and
correlation and their interpretations.
The coefficient of correlation between two variables x and y is given by
cov ( x, y )
r= , where cov ( x, y ) is covariance of x and y , σ x is the standard
σ xσ y
deviation for x , σ y is the standard deviation for y . Correlation describes
how the two variables are related, whereas correlation describes how the
two variables covary. The correlation coefficient ranges from -1 to +1.
If the linear coefficient of correlation takes values closer to -1, the correlation
is strong and negative and will become stronger the closer r approaches -1.
Focus also on interpreting correlation based on how data are scattered in a
scatter diagram.

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 5.8. Answers of End unit assessment
This part provides the answers of end unit assessment activities designed
in an integrative approach to assess the key unit competence with cross-
reference to the student book.
1. 0.26
2. 0.43. Some agreement between average attendance ranking a
position in the league and a high position in the league correlating
with high attendance.

3. =y 0.61x + 10.5
= , x 1.47 y − 1.14 , y = 28.83

4.= y 0.94 x + 92.26 , blood pressure=134.56

5.9 Additional activities

5.9.1 Remedial activities
Identify positive and negative correlation situation in the following cases:
a) The more times people have unprotected sex with different partners, the
more the rates of HIV in a society.
b) The more people save their incomes, the more financially stable they
become.
c) As weather gets colder, air conditioning costs decrease.
d) The more alcohol is consumed, the less the judgment one has.
e) The more one cleans the house, the less likely are to be pests problems.
Answer:
a) Positive correlation
b) Positive correlation
c) Negative correlation
d) Negative correlation
e) Negative correlation
5.9.2 Consolidation activities
The amount of government bursaries allocated to certain administration
regions in the country in a certain year is listed together with their population
sizes.

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 Region Population (10,000s)(x) Bursaries in millions (y)
1 29 8.0
2 58 16.8
3 108 33.9
4 34 10
5 115 34
6 19 6.5
7 136 40.5
8 33 10.2
9 25 8.8
10 47 12.5
11 49 17.3
12 33 12.6

Draw a scatter diagram and assess whether there appears to be a correlation

between the two measurements labeled x and y. Use a scale of 1 cm: 50 000
on the vertical axis and 1 cm: 10 m on the horizontal axis.
Answer:

The best line of fit in this scatter diagram follows the general trend of the
points. This line has a positive gradient. Thus, the relation in this data is called
a positive correlation.

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5.9.3 Extended activities
The following measurements were made, and the data was recorded to the
nearest cm.
Height of boys Height of his father
(cm) (cm)
164 171
168 186
150 164
162 180
159 176
165 177
187 192
152 167
180 189
166 180
a) Plot the data using coordinate axes.
b) Draw the line of best fit for the data
c) Find the equation of the line in (b) above.
d) Describe the correlation.
e) Would it be reasonable to use your graph to estimate the height of a
father whose son is 158 cm tall?
Answer:
(a) and (b)

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 c) Using points (152, 167) and (180, 189):

189 − 167 11
Gradient of line =
180 − 152 14
y mx + c
=
From the graph c=160.
11
=y y + 160
14
Equation of the line of best fit can be written as 14=y 11x + 2240.
d) The correlation in this data is positive or direct. The line of best fit has a
positive gradient and, therefore, a positive trend.
e) Yes Height of father is 175 cm.

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REFERENCES
Arem, C. (2006). Systems and Matrices. In C. A. DeMeulemeester, Systems
and Matrices (pp. 567-630). Demana: Brooks/Cole Publishing 1993 &
Addison-Wesley1994.
Kirch, W. (Ed.). (2008). Level of MeasurementLevel of measurement BT -
Encyclopedia of Public Health (pp. 851–852). Springer Netherlands.
https://doi.org/10.1007/978-1-4020-5614-7_1971
Crossman, Ashley. (2020). Understanding Levels and Scales of Measurement
in Sociology. Retrieved from https://www.thoughtco.com/levels-of-
measurement-3026703
Markov. ( 2006). Matrix Algebra and Applications. In Matrix Algebra and
Applications (pp. p173-208.).
REB, R.E (2020). Mathematics for TTCs Year 2 Social Studies Education.
Student’s book.
REB, R.E (2020). Mathematics for TTCs Year 3 Social Studies Education.
Student’s book
REB, R.E (2020). Mathematics for TTCs Year 1 Science Mathematics Education.
Students’ book
REB, R.E (2020). Mathematics for TTCs Year 1 Social Studies Education.
Student’s book
Rossar, M. (1993, 2003). Basic Mathematics For Economists. London and
New York: Taylor & Francis Group.

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