Definition - St John`s College

Definition - St John`s College
National Curriculum Statement
Grades 10–12
(General)
Advanced Programme Mathematics
(previously known as Additional Mathematics)
A subject in addition to the NSC requirements
Department of Education
2006
Department of Education
Sol Plaatje House
123 Schoeman Street
Private Bag X895
Pretoria 0001
South Africa
Tel: +27 12 312-5911
Fax: +27 12 321-6770
120 Plein Street
Private Bag X9023
Cape Town 8000
South Africa
Tel: +27 21 465-1701
Fax: +27 21 461-8110
http://education.pwv.gov.za
© 2006 Department of Education
ISBN [to be inserted]
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CONTENTS
HOW TO USE THIS BOOK
4
ACRONYMS
6
CHAPTER 1: INTRODUCING THE NATIONAL CURRICULUM STATEMENT
Principles
The kind of learner that is envisaged
The kind of teacher that is envisaged
Structure and design features
Learning programme guidelines
7
7
10
10
11
12
CHAPTER 2: Advanced Programme Mathematics
Definition
Purpose
Scope
Educational and career links
Learning outcomes
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15
CHAPTER 3: LEARNING OUTCOMES AND ASSESSMENT STANDARDS
Learning Outcome 1: Calculus
Learning Outcome 2: Algebra
Learning Outcome 3: Statistics
Learning Outcome 4: Mathematical Modelling
Learning Outcome 5: Matrices and Graph Theory
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24
28
32
35
CHAPTER 4: ASSESSMENT
Introduction
Why assess?
Types of assessment
What assessment should be and do
How to assess
Methods of assessment
Methods of collecting assessment evidence
Recording and reporting
Subject competence descriptions
Competence Descriptors
Promotion
What report cards should look like
Assessment of learners who experience barriers to learning
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HOW TO USE THIS BOOK
This document is a policy document divided into four chapters. It is important for the reader to
read and integrate information from the different sections in the document. The content of each
chapter is described below.
‰
Chapter 1 - Introducing the National Curriculum Statement
This chapter describes the principles and the design features of the National Curriculum
Statement Grade 10–12 (General). It provides an introduction to the curriculum for the
reader.
‰
Chapter 2 - Introducing Advanced Programme Mathematics
This chapter describes the definition, purpose, scope, career links and Learning
Outcomes of Advanced Programme Mathematics. It provides an orientation to the
subject.
‰
Chapter 3 - Learning Outcomes, Assessment Standards and Content and Contexts
This chapter contains the Assessment Standards for each Learning Outcome, as well as
content and contexts for the subject. The Assessment Standards are arranged to assist
the reader to see the intended progression from Grade 10 to Grade 12. The Assessment
Standards are consequently laid out in double page spreads. At the end of the chapter
is the proposed content and contexts, which may be used to teach, learn and attain
Assessment Standards.
‰
Chapter 4 – Assessment
This chapter deals with the Advanced Programme Mathematics approach to assessment
being suggested by the National Curriculum Statement. At the end of the chapter is a
table of subject-specific competence descriptions. Codes, scales and competence
descriptions are provided for each grade. The competence descriptions are arranged to
demonstrate progression from Grade 10 to Grade 12.
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Symbols:
The following symbols are used to identify Learning Outcomes, Assessment Standards, grades,
codes, scales, competence description and content and contexts:
= Learning Outcome
= Assessment Standard
= Grade
= Code
= Scale
= Competence Description
= Content and Contexts
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ACRONYMS
AIDS
CASS
FET
GET
HIV
IKS
OBE
NCS
NQF
SAQA
Acquired Immune Deficiency Syndrome
Continuous Assessment
Further Education and Training
General Education and Training
Human Immunodeficiency Virus
Indigenous Knowledge Systems
Outcomes-Based Education
National Curriculum Statement
National Qualifications Framework
South African Qualifications Authority
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CHAPTER 1
INTRODUCING THE NATIONAL CURRICULUM STATEMENT
The adoption of the Constitution of the Republic of South Africa (Act 108 of 1996) provided a
basis for curriculum transformation and development in South Africa. The Preamble states that the
aims of the Constitution are to:
• heal the divisions of the past and establish a society based on democratic values, social
justice and fundamental human rights;
• improve the quality of life of all citizens and free the potential of each person;
• lay the foundations for a democratic and open society in which government is based on the
will of the people and every citizen is equally protected by law; and
• build a united and democratic South Africa able to take its rightful place as a sovereign
state in the family of nations.
The Constitution further states that ‘everyone has the right … to further education which the State,
through reasonable measures, must make progressively available and accessible’.
The National Curriculum Statement Grades 10 – 12 (General) lays a foundation for the
achievement of these goals by stipulating Learning Outcomes and Assessment Standards, and by
spelling out the key principles and values that underpin the curriculum.
PRINCIPLES
The National Curriculum Statement Grades 10 – 12 (General) is based on the following principles:
• social transformation;
• outcomes-based education;
• high knowledge and high skills;
• integration and applied competence;
• progression;
• articulation and portability;
• human rights, inclusivity, environmental and social justice;
• valuing indigenous knowledge systems; and
• credibility, quality and efficiency.
Social transformation
The Constitution of the Republic of South Africa forms the basis for social transformation in our
post-apartheid society. The imperative to transform South African society by making use of various
transformative tools stems from a need to address the legacy of apartheid in all areas of human
activity and in education in particular.
Social transformation in education is aimed at ensuring that the educational imbalances of the past
are redressed, and that equal educational opportunities are provided for all sections of our
population. If social transformation is to be achieved, all South Africans have to be educationally
affirmed through the recognition of their potential and the removal of artificial barriers to the
attainment of qualifications.
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Outcomes-based education
Outcomes-based education (OBE) forms the foundation for the curriculum in South Africa. It
strives to enable all learners to reach their maximum learning potential by setting the Learning
Outcomes to be achieved by the end of the education process. OBE encourages a learner-centred
and activity-based approach to education. The National Curriculum Statement builds its Learning
Outcomes for Grades 10 – 12 on the Critical and Developmental Outcomes that were inspired by
the Constitution and developed through a democratic process.
The Critical Outcomes require learners to be able to:
• identify and solve problems and make decisions using critical and creative thinking;
• work effectively with others as members of a team, group, organisation and community;
• organise and manage themselves and their activities responsibly and effectively;
• collect, analyse, organise and critically evaluate information;
• communicate effectively using visual, symbolic and/or language skills in various modes;
• use science and technology effectively and critically showing responsibility towards the
environment and the health of others; and
• demonstrate an understanding of the world as a set of related systems by recognising that
problem solving contexts do not exist in isolation.
The Developmental Outcomes require learners to be able to:
• reflect on and explore a variety of strategies to learn more effectively;
• participate as responsible citizens in the life of local, national and global communities;
• be culturally and aesthetically sensitive across a range of social contexts;
• explore education and career opportunities; and
• develop entrepreneurial opportunities.
High knowledge and high skills
The National Curriculum Statement Grades 10 – 12 (General) aims to develop a high level of
knowledge and skills in learners. It sets up high expectations of what all South African learners can
achieve. Social justice requires the empowerment of those sections of the population previously
disempowered by the lack of knowledge and skills. The National Curriculum Statement specifies
the minimum standards of knowledge and skills to be achieved at each grade and sets high,
achievable standards in all subjects.
Integration and applied competence
Integration is achieved within and across subjects and fields of learning. The integration of
knowledge and skills across subjects and terrains of practice is crucial for achieving applied
competence as defined in the National Qualifications Framework. Applied competence aims at
integrating three discrete competences – namely, practical, foundational and reflective
competences. In adopting integration and applied competence, the National Curriculum Statement
Grades 10 – 12 (General) seeks to promote an integrated learning of theory, practice and reflection.
Progression
Progression refers to the process of developing more advanced and complex knowledge and skills.
The Subject Statements show progression from one grade to another. Each Learning Outcome is
followed by an explicit statement of what level of performance is expected for the outcome.
Assessment Standards are arranged in a format that shows an increased level of expected
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performance per grade. The content and context of each grade will also show progression from
simple to complex.
Articulation and portability
Articulation refers to the relationship between qualifications in different National Qualifications
Framework levels or bands in ways that promote access from one qualification to another. This is
especially important for qualifications falling within the same learning pathway. Given that the
Further Education and Training band is nested between the General Education and Training and the
Higher Education bands, it is vital that the Further Education and Training Certificate (General)
articulates with the General Education and Training Certificate and with qualifications in similar
learning pathways of Higher Education. In order to achieve this articulation, the development of
each Subject Statement included a close scrutiny of the exit level expectations in the General
Education and Training Learning Areas, and of the learning assumed to be in place at the entrance
levels of cognate disciplines in Higher Education.
Portability refers to the extent to which parts of a qualification (subjects or unit standards) are
transferable to another qualification in a different learning pathway of the same National
Qualifications Framework band. For purposes of enhancing the portability of subjects obtained in
Grades 10 – 12, various mechanisms have been explored, for example, regarding a subject as a 20credit unit standard. Subjects contained in the National Curriculum Statement Grades 10 – 12
(General) compare with appropriate unit standards registered on the National Qualifications
Framework.
Human rights, inclusivity, environmental and social justice
The National Curriculum Statement Grades 10 – 12 (General) seeks to promote human rights,
inclusivity, environmental and social justice. All newly-developed Subject Statements are infused
with the principles and practices of social and environmental justice and human rights as defined in
the Constitution of the Republic of South Africa. In particular, the National Curriculum Statement
Grades 10 – 12 (General) is sensitive to issues of diversity such as poverty, inequality, race,
gender, language, age, disability and other factors.
The National Curriculum Statement Grades 10 – 12 (General) adopts an inclusive approach by
specifying minimum requirements for all learners. It acknowledges that all learners should be able
to develop to their full potential provided they receive the necessary support. The intellectual,
social, emotional, spiritual and physical needs of learners will be addressed through the design and
development of appropriate Learning Programmes and through the use of appropriate assessment
instruments.
Valuing indigenous knowledge systems
In the 1960s, the theory of multi-intelligences forced educationists to recognise that there were
many ways of processing information to make sense of the world, and that, if one were to define
intelligence anew, one would have to take these different approaches into account. Up until then
the Western world had only valued logical, mathematical and specific linguistic abilities, and rated
people as ‘intelligent’ only if they were adept in these ways. Now people recognise the wide
diversity of knowledge systems through which people make sense of and attach meaning to the
world in which they live. Indigenous knowledge systems in the South African context refer to a
body of knowledge embedded in African philosophical thinking and social practices that have
evolved over thousands of years. The National Curriculum Statement Grades 10 – 12 (General) has
infused indigenous knowledge systems into the Subject Statements. It acknowledges the rich
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history and heritage of this country as important contributors to nurturing the values contained in
the Constitution. As many different perspectives as possible have been included to assist problem
solving in all fields.
Credibility, quality and efficiency
The National Curriculum Statement Grades 10 – 12 (General) aims to achieve credibility through
pursuing a transformational agenda and through providing an education that is comparable in
quality, breadth and depth to those of other countries. Quality assurance is to be regulated by the
requirements of the South African Qualifications Authority Act (Act 58 of 1995), the Education
and Training Quality Assurance Regulations, and the General and Further Education and Training
Quality Assurance Act (Act 58 of 2001).
THE KIND OF LEARNER THAT IS ENVISAGED
Of vital importance to our development as people are the values that give meaning to our personal
spiritual and intellectual journeys. The Manifesto on Values, Education and Democracy
(Department of Education, 2001:9- 10) states the following about education and values:
Values and morality give meaning to our individual and social relationships. They are the common
currencies that help make life more meaningful than might otherwise have been. An education
system does not exist to simply serve a market, important as that may be for economic growth and
material prosperity. Its primary purpose must be to enrich the individual and, by extension, the
broader society.
The kind of learner that is envisaged is one who will be imbued with the values and act in the
interests of a society based on respect for democracy, equality, human dignity and social justice as
promoted in the Constitution. The learner emerging from the Further Education and Training band
must also demonstrate achievement of the Critical and Developmental Outcomes listed earlier in this
document. Subjects in the Fundamental Learning Component collectively promote the achievement
of the Critical and Developmental Outcomes, while specific subjects in the Core and Elective
Components individually promote the achievement of particular Critical and Developmental
Outcomes.
In addition to the above, learners emerging from the Further Education and Training band must:
• have access to, and succeed in, lifelong education and training of good quality;
• demonstrate an ability to think logically and analytically, as well as holistically and
laterally; and
• be able to transfer skills from familiar to unfamiliar situations.
THE KIND OF TEACHER THAT IS ENVISAGED
All teachers and other educators are key contributors to the transformation of education in South
Africa. The National Curriculum Statement Grades 10 – 12 (General) visualises teachers who are
qualified, competent, dedicated and caring. They will be able to fulfil the various roles outlined in
the Norms and Standards for Educators. These include being mediators of learning, interpreters and
designers of Learning Programmes and materials, leaders, administrators and managers, scholars,
researchers and lifelong learners, community members, citizens and pastors, assessors, and subject
specialists.
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STRUCTURE AND DESIGN FEATURES
Structure of the National Curriculum Statement
The National Curriculum Statement Grades 10 – 12 (General) consists of an Overview Document,
the Qualifications and Assessment Policy Framework, and the Subject Statements. The subjects in
the National Curriculum Statement Grades 10 – 12 (General) are categorised into Learning Fields.
What is a Learning Field?
A Learning Field is a category that serves as a home for cognate subjects, and that facilitates the
formulation of rules of combination for the Further Education and Training Certificate (General).
The demarcations of the Learning Fields for Grades 10 – 12 took cognisance of articulation with
the General Education and Training and Higher Education bands, as well as with classification
schemes in other countries. Although the development of the National Curriculum Statement
Grades 10 – 12 (General) has taken the twelve National Qualifications Framework organising
fields as its point of departure, it should be emphasised that those organising fields are not
necessarily Learning Fields or ‘knowledge’ fields, but rather are linked to occupational categories.
The following subject groupings were demarcated into Learning Fields to help with learner subject
combinations:
• Languages (Fundamentals);
• Arts and Culture;
• Business, Commerce, Management and Service Studies;
• Manufacturing, Engineering and Technology;
• Human and Social Sciences and Languages; and
• Physical, Mathematical, Computer, Life and Agricultural Sciences.
What is a subject?
Historically, a subject has been defined as a specific body of academic knowledge. This
understanding of a subject laid emphasis on knowledge at the expense of skills, values and
attitudes. Subjects were viewed by some as static and unchanging, with rigid boundaries. Very
often, subjects mainly emphasised Western contributions to knowledge.
In an outcomes-based curriculum like the National Curriculum Statement Grades 10 – 12
(General), subject boundaries are blurred. Knowledge integrates theory, skills and values. Subjects
are viewed as dynamic, always responding to new and diverse knowledge, including knowledge
that traditionally has been excluded from the formal curriculum.
A subject in an outcomes-based curriculum is broadly defined by Learning Outcomes, and not only
by its body of content. In the South African context, the Learning Outcomes should, by design,
lead to the achievement of the Critical and Developmental Outcomes. Learning Outcomes are
defined in broad terms and are flexible, making allowances for the inclusion of local inputs.
What is a Learning Outcome?
A Learning Outcome is a statement of an intended result of learning and teaching. It describes
knowledge, skills and values that learners should acquire by the end of the Further Education and
Training band.
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What is an Assessment Standard?
Assessment Standards are criteria that collectively describe what a learner should know and be able
to demonstrate at a specific grade. They embody the knowledge, skills and values required to
achieve the Learning Outcomes. Assessment Standards within each Learning Outcome collectively
show how conceptual progression occurs from grade to grade.
Contents of Subject Statements
Each Subject Statement consists of four chapters and a glossary:
• Chapter 1, Introducing the National Curriculum Statement: This generic chapter introduces
the National Curriculum Statement Grades 10 – 12 (General).
• Chapter 2, Introducing the Subject: This chapter introduces the key features of the subject.
It consists of a definition of the learning field, its purpose, scope, educational and career
links, and Learning Outcomes.
• Chapter 3, Learning Outcomes, Assessment Standards, Content and Contexts: This chapter
contains Learning Outcomes with their associated Assessment Standards, as well as content
and contexts for attaining the Assessment Standards.
• Chapter 4, Assessment: This chapter outlines principles for assessment and makes
suggestions for recording and reporting on assessment. It also lists subject-specific
competence descriptions.
• Glossary: Where appropriate, a list of selected general and subject-specific terms are briefly
defined.
LEARNING PROGRAMME GUIDELINES
A Learning Programme specifies the scope of learning and assessment for the three grades in the
Further Education and Training band. It is the plan that ensures that learners achieve the Learning
Outcomes as prescribed by the Assessment Standards for a particular grade. The Learning
Programme Guidelines assist teachers and other Learning Programme developers to plan and
design quality learning, teaching and assessment programmes.
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CHAPTER 2
DEFINITION
Advanced Programme Mathematics is an extension of Mathematics and is similarly based on the
following view of the nature of the discipline.
Advanced Programme Mathematics enhances mathematical creativity and logical reasoning about
problems in the physical and social world and in the context of Mathematics itself. All mathematics
is a distinctly human activity developed over time as a well-defined system with a growing number
of applications in our world. Knowledge in the mathematical sciences is constructed through the
establishment of descriptive, numerical and symbolic relationships. Advanced Programme
Mathematics also observes patterns and relationships, leading to additional conjectures and
hypotheses and developing further theories of abstract relations through rigorous logical thinking.
Mathematical problem solving in Advanced Programme Mathematics enables us to understand the
world in greater depth and make use of that understanding more extensively in our daily lives. The
Mathematics presented in Advanced Programme Mathematics has been developed and contested
over time through both language and symbols by social interaction, and continues to develop, thus
being open to change and growth.
PURPOSE
In a society that values diversity and equality, and a nation that has a globally competitive
economy, it is imperative that within the Further Education and Training band learners who
perform well in Mathematics or who have a significant enthusiasm for mathematics are offered an
opportunity to increase their knowledge, skills, values and attitudes associated with mathematics,
and so put them into a position to contribute more significantly as citizens of South Africa. The
study of Advanced Programme Mathematics contributes to the personal development of high
performing mathematics learners by providing challenging learning experiences; feelings of
success and self-worth, and to the development of appropriate values and attitudes through the
successful application of its knowledge and skills in context, and through the collective
engagement with mathematical ideas.
SCOPE
Advanced Programme Mathematics is aimed at increasing the number of learners who through
competence and desire enter Higher Education to pursue careers in mathematics, engineering,
technology and the sciences. Advanced Programme Mathematics is an extension and challenge
for learners who demonstrate a greater than average ability in, or enthusiasm for mathematics.
The greater breadth of mathematical knowledge gained and the deepening of mathematical process
skills developed through being exposed to Advanced Programme Mathematics enhances the
learner’s understanding of mathematics both as a discipline and as a tool in society. This broadens
the learner’s perspective on possible careers in mathematics and develops a passion for and a
commitment to the continued learning of mathematics amongst mathematically talented learners.
This assists in meeting their needs and encourages more mathematically talented learners to pursue
careers and interests in mathematically related fields. Studying Advanced Programme Mathematics
will also further the appreciation of the development of Mathematics over time, establishing a
greater understanding of its origins in culture and in the needs of society.
Advanced Programme Mathematics enables learners to:
• extend their mathematical knowledge to solve new problems in the world around them and
grow in confidence in this ability;
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•
•
•
•
•
•
•
•
•
use sophisticated mathematical processes to solve and pose problems creatively and
critically;
demonstrate the patience and perseverance to work both independently and cooperatively
on problems that require more time to solve;
contribute to quantitative arguments relating to local, national and global issues;
focus on the process of science and mathematics, rather than on right answers;
view science and mathematics as valuable and interesting areas of learning;
become more self-reliant and validate their own answers;
learn to value mathematics and its role in the development of our contemporary society and
explore relationships among mathematics and the disciplines it serves;
communicate mathematical problems, ideas, explorations and solutions through reading,
writing and mathematical language;
enable students to become problem solvers and users of science and mathematics in their
everyday lives.
The study of Advanced Programme Mathematics should encourage students to talk about
mathematics, use the language and symbols of mathematics, communicate, discuss problems and
problem solving, and develop competence and confidence in themselves as mathematics students.
EDUCATIONAL AND CAREER LINKS
Advanced Programme Mathematics is valuable in the curriculum of any learner who intends to
pursue a career in the physical, mathematical, financial, computer, life, earth, space and
environmental sciences or in technology. Advanced Programme Mathematics also supports the
pursuance of careers in the economic, management and social sciences. The knowledge and skills
attained in Advanced Programme Mathematics provide more appropriate tools for creating,
exploring and expressing theoretical and applied aspects of the sciences. .
The subject Additional Mathematics in the Further Education and Training band provides the ideal
platform for linkages to Mathematics in Higher Education institutions. Learners proceeding to
institutions of Higher Education with Advanced Programme Mathematics, will be in a strong
position to progress effectively in whatever mathematically related discipline they decide to follow.
The added exposure to modelling encountered in Advanced Programme Mathematics provides
learners with deeper insights and skills when solving problems related to modern society,
commerce and industry.
Advanced Programme Mathematics, although not required for the study of mathematics,
engineering, technology or the sciences in Higher Education, is intended to provide talented
mathematics learners an opportunity to advance their potential, competence, enthusiasm and
success in mathematics so that it is more likely that they will follow mathematically related careers.
In particular the following are some of the career fields that demand the use of high level
mathematics:
• Actuarial science
• Operations research
• Mathematical modelling
• Economic and Industrial sciences
• Movie and video game special effects
• Engineering
• Computational mathematics
• Theoretical and applied physics
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•
•
Statistical applications
Academic research and lecturing in mathematics, applied mathematics, actuarial science
and statistics
LEARNING OUTCOMES.
Learning Outcome 1: Calculus
The learner is able to establish, define, manipulate, determine and represent the derivative and
integral, both as an anti-derivative and as the area under a curve, of various algebraic and
trigonometric functions and solve related problems with confidence.
Learning Outcome 2: Algebra
The learner is able to represent, investigate, analyse, manipulate and prove conjectures about
numerical and algebraic relationships and functions, and solve related problems.
Learning Outcome 3: Statistics
The learner is able to organise, summarise, analyse and interpret data to identify, formulate and test
statistical and probability models, and solve related problems.
Learning Outcome 4: Mathematical modelling
The learner is able to investigate, represent and model growth and decay problems using formulae,
difference equations and series.
Learning Outcome 5: Matrices and Graph Theory
The learner is able to identify, represent and manipulate discrete variables using graphs and
matrices, applying algorithms in modeling finite systems.
COURSE REQUIREMENTS
Compulsory
Grade 10
Calculus
Algebra
Statistics
Compulsory
Grade 11 and 12
Calculus
Matrices &
applications
Mathematical
modelling
Options (pick one topic)
Algebra
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Statistics
Matrices &
applications
Mathematical
modelling
CHAPTER 3
LEARNING OUTCOMES AND ASSESSMENT STANDARDS
Learning Outcome 1: Calculus
The learner is able to establish, define, manipulate, determine and represent the derivative and integral, both as an anti-derivative and as the area under
a curve, of various algebraic and trigonometric functions and solve related problems with confidence.
Grade 10
We know this when the learner is able
to:
Grade 11
We know this when the learner is able
to:
11.1.1 (a) Sketch the graphs of mathematical
functions including split domain and
composite functions (comprised of
linear, quadratic, hyperbolic, absolute
value and exponential functions).
(b) Manipulate and analyse split domain
and composite functions using the
definition of a function and the graph of
the function.
11.1.2 (a) Define and use trigonometric and
reciprocal trigonometric functions
to:
- solve problems in right-angled
triangles,
- prove basic trigonometric
identities,
- manipulate trigonometric
statements,
- solve trigonometric problems in
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Grade 12
We know this when the learner is able
to:
realistic and mathematical
contexts.
(b) convert between angles measured in
degrees and radians
(c) use trigonometric functions defined
in terms of a real variable (angle in
radians) and the x, y and r-definition
to:
- calculate the lengths of arcs of
circles,
- calculate the area of sectors and
segments of circles, and
- find the general solution of
trigonometric equations.
11.1.3 (a) Compare the graphical, numerical
12.1.3 (a) Use first principles and graphs to
and symbolic representations of the
determine the continuity and
limit of a function.
differentiability at a given point of
algebraic functions, including split
(b) Determine the limit of a function at
domain functions
a point, including from the right and
left, and to infinity algebraically.
(b) Without proof, apply the theorem
and its converse, "A function that is
(Note: The limit at a point is defined as
differentiable at a point is
the limit from the left and from the
continuous at that point", and deal
right)
with examples to indicate that the
converse is not valid.
(c) Illustrate the continuity of a function
graphically and apply the definition
(c) Demonstrate the derivative of a
of continuity at a point to simple
function at a point as the rate of
algebraic functions, including split
change, by graphical, numerical and
domain functions.
symbolic representations.
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11.1.4
(a) Illustrate the differentiability of a
function graphically and determine
the derivative as the gradient of a
function at a point using limits.
(b) Establish the derivatives of
functions of the form ax2 + bx + c ,
x,
ax + b ,
1
ax + b
1
ax + b
,
from first principles.
12.1.4
(a) Use the following rules of
differentiation
d
[ f ( x).( gx)] = g ( x). d [ f ( x)] + f ( x). d [g ( x)]
dx
dx
dx
d
d
g ( x). [ f ( x)] − f ( x). [g ( x)]
d ⎡ f ( x) ⎤
dx
dx
⎢
⎥=
dx ⎣ g ( x) ⎦
[g ( x)]2
d
[ f ( g ( x))] = f ' ( g ( x)).g ' ( x) or
dx
dy dy dt
=
×
dx dt dx
applied to
- polynomial, rational, radical and
trigonometric functions,
(Learners may assume without
proof that the derivative of sin x
is cos x),
- higher order derivatives,
- functions in two variables, using
implicit differentiation, and
- Newton’s method
(b) Calculate maximum and minimum
values of a function using calculus
methods. Determine both absolute
and relative maximum and
minimum values of a function over
a given interval.
(c) Use calculus methods to sketch the
curves of polynomial and rational
functions determining:
- intervals over which a function
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-
is increasing or decreasing
y- intercepts and x-intercepts,
using Newton’s Method if
necessary.
the coordinates and nature of
stationary points
any vertical, horizontal or
oblique asymptotes
(d) Use methods learned above to solve
practical problems involving
optimisation and rates of change in
real, realistic and abstract
mathematical contexts. Verify the
results of the calculus modelling by
referring to the practical context.
10.1.5 (a) Represent irregular shapes found in
the world using scales diagrams
(b) Estimate, measure, calculate and
evaluate the approximate area of
irregular shapes found in the field
using squares; equilateral, rightangled and isosceles triangles;
circles.
(c) Approximate the area between
familiar curves, such as straight
lines, parabolas, hyperbolae and
exponential graphs, and the x-axis
using the:
- Rectangle Method
- Trapezoidal Rule.
11.1.5
(a) Investigate and develop a formula
for the upper and lower sums method
of approximating area under the
curve of y = x n , for n ∈ N and x ≥
0 on the interval [a; b] (i.e.
∫
b
a
x n dx
=
[
1
n +1
x n +1
]
b
a
)
(b) Estimate the margin of error of the
approximate area determined by the
upper and lower sums method.
(c) Use available technology to
manipulate the width of subintervals and the accuracy of the
approximate area under a
polynomial function.
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12.1.5
(a) Recognise anti-differentiation as the
reverse of differentiation
(b) Demonstrate an understanding of
the Fundamental Theorem of
Calculus and its significance.
(c) Manipulate and then integrate
algebraic and trigonometric
functions of the form:
- ∫ ax n dx , a is a constant and n ∈ Q,
-
p( x)
∫ p( x) dx and ∫ q( x) dx , p(x) and
q(x) are polynomials or radicals,
-
∫ f (g(x)).g′( x) dx
on an interval of the x-axis.
(d) Experiment with the accuracy of the
approximation by varying the width
and number of rectangles or
trapeziums.
(d) Investigate and intuitively develop
the Riemann Sum as an
approximate of the area under the
curve of a polynomial function.
Formulae used for this purpose
should include (without proof):
n
n
∑i =
∑1 = n
i =1
1=1
2
n
n
+
2 2
-
n
n2 n
2
=
+
+
i
∑
3
2 6
i =1
n
n4 n3 n2
3
+
+
i =
∑
4
2
4
i =1
(f) Develop graphically the following
intuitive rules of the definite
integral for polynomials over any
interval:
∫
∫
b
a
b
f ( x) dx
b
∫
c
b
∫
f ( x)dx =
b
b
a
a
c
a
f ( x)
c . f ( x ) dx = c . ∫ f ( x ) dx ,
∫ ( f ( x) + g ( x))dx
b
a
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=
n
∫ sin mx dx ; ∫ sin mx cos nxdx
and
using only the following methods:
direct anti-derivatives
- simplification of trigonometric
functions using appropriate squares,
compound angle and product-sum
formulae given
- integration by t-substitution
- integration with a given
trigonometric substitution
- integration by parts
(d) Perform both definite and indefinite
integration
a
− ∫ f ( x )dx ,
=
f ( x)dx +
a
n
similar functions
(e) Define the Riemann (definite)
integral as the approximating
rectangles are made narrower and
the number of strips n → ∞ .
∫
∫ sec x tan xdx )
∫ sin xdx and ∫ cos xdx where n ∈ N
and n ≤ 3
3
n
-
where the anti-derivative of a
trigonometric function can be
directly determined from the
derivative (e.g. ∫ cos xdx , ∫ sec 2 xdx ,
∫
b
a
b
f ( x)dx + ∫ g ( x)dx
a
(e) Apply the definite integral and
techniques of integration to solve
area and volume problems by:
- Calculating the area under or
between curves using the
manipulation of intervals.
- Calculating the volume generated by
(g) Determine the area between simple
polynomials and the x-axis or
between two simple polynomials
using the definite integral as
developed above.
rotating a function about the x-axis
in mathematical and real world
contexts.
Content: Learning Outcome 1: Calculus
The learner is able to establish, define, manipulate, determine and represent the derivative and integral, both as an anti-derivative and as the area under
a curve, of various algebraic and trigonometric functions and solve related problems with confidence.
Grade 10
Grade 11
► Functions
• Graph mathematical functions
including split domain and composite
functions.
• Trigonometric and reciprocal
trigonometric functions
- Solve problems in right-angled
triangles
- Prove basic trigonometric
identities
- Manipulate trigonometric
statements
- Solve trigonometric problems in
realistic and mathematical
contexts
- Solve the general solution of
trigonometric equations.
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Grade 12
• Radian measure
• Trigonometric functions using radian
measure
• Solve arc lengths, areas of sectors and
segments of circles
► Differentiation
• Limits of functions
- Limit of a function at a point,
from the right and left,
- Limits to infinity
• Continuity of a function, including
split domain functions.
• Differentiability of a function,
including split domain functions.
• Solve derivatives of functions of the
form ax² + bx + c ,
1
ax + b
► Integration
• Represent irregular shapes found in
the world using scale diagrams
• Approximate area of polygons and
irregular shapes found in the field
• Approximate the area between
familiar curves and the x-axis using
the:
x,
ax + b ,
1
ax + b from first principles.
,
► Integration
• Approximate the area under the curve
of y = x , for n ∈ N and x ≥ 0
using the upper and lower sums
method
• Estimate margins of error of an
approximate area
n
22 of 48
► Differentiation
• Continuity and differentiability at a
given point of algebraic functions,
including split domain functions
• Graphical representation of the
derivative of a function at a point as
the gradient of the tangent at the point.
• Use the product, quotient and chain
rules to determine the derivatives of
- polynomial, rational and radical
functions,
- higher order derivatives,
- trigonometric functions where
the angle is in radian measure
- functions in x and y using
implicit differentiation, and
- Newton’s method
• Optimisation and rates of change
• Sketch the curves of polynomial and
rational functions
► Integration (anti-differentiation)
• Fundamental Theorem of Calculus
• Anti-derivatives
• Basic properties of the indefinite
integral
• Simplify and then integrate algebraic
and trigonometric functions using only
- Rectangle Method
- Trapezoidal Rule.
on an interval of the x-axis.
• The Riemann Sum as an approximate
of the area under a curve
• The Riemann (definite) integral
• Basic properties of the definite
integral
• Determine the area between simple
polynomials and the x-axis or between
two simple polynomials.
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the following methods:
- direct anti-derivatives
- simplification of trigonometric
functions
- integration by t-substitution
- integration with a given
trigonometric substitution
- integration by parts
• Area under or between curves using
the manipulation of intervals.
• Volume generated by rotating a
function about the x-axis.
Learning Outcome 2: Algebra
The learner is able to represent, investigate, analyse, manipulate and prove conjectures about numerical and algebraic relationships and functions, and
solve related problems.
Grade 10
Grade 11
We know this when the learner is able
to:
10.2.1 (a) Characterise and discuss the nature
and relevance of the roots of
x2 + 1 = 0
Grade 12
We know this when the learner is able
to:
We know this when the learner is able
to:
11.2.1
Determine the real and complex roots of
quadratic and cubic equations using:
- factorisation,
- the quadratic formula, and
- the factor theorem to find the
first real root of cubic equations
12.2.1
11.2.2
Demonstrate an understanding of the
absolute value of an algebraic
expression as a distance from the origin.
12.2.2
(b) Classify numbers using sub-fields of
the Complex numbers.
(c) Determine the roots of equations of
the form ax 2 + bx + c = 0 and
classify the roots as real or
imaginary.
(c) Perform the four basic operations
(+, -, /, x) on complex numbers
without the use of a calculator.
10.2.2 Manipulate algebraic expressions by
(a) factorising third degree polynomials
using the factor theorem to find an
initial factor
(a) Simplify and manipulate algebraic
expressions using the laws of
exponents and logarithms.
(b) Demonstrate an understanding of e
and its role in exponents and
logarithms by using e freely in
(b) simplifying algebraic fractions with
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polynomial denominators of at most
degree three, where the polynomials
can be factorised.
problem solving.
(c) Decomposing algebraic fractions
into partial fractions when the
denominator is of the form:
- (a1x+b1) (a2x+b2) …(anx+bn), using
the ‘cover up’ method.
- (ax+b)²(cx+d)², by comparing
coefficients.
10.2.3 Solve quadratic inequalities.
11.2.3
Solve
(a) equations containing multi-term
algebraic fractions using algebraic
methods.
12.2.3
(b) polynomial and rational inequalities.
(b) simple logarithmic equations using
the laws of logarithms and algebraic
manipulation.
(c) absolute value equations of the form
a|mx – p| = q
11.2.4
(a) Draw absolute value graphs given
the equation of the function in the
form y = a |x – p| + q
(b) Find the equation of the absolute
value function, in the form y = a |x
– p| + q, given the graph and
necessary points on the graph
(c) Interpret the graphs of absolute
value functions to determine the:
- domain and range;
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Solve
(a) simple exponential equations using
the laws of exponents and algebraic
manipulation.
12.2.4
(a) Draw exponential graphs, including
y = ex.
(b) Draw logarithmic graphs, including
y = ln x.
(c) Find the equation for the reflections
of the exponential or logarithmic
functions in the lines x=0, y=0 and
y = x, the inverse of the function.
-
intercepts with the axes;
turning points, minima and
maxima;
shape and symmetry;
(d) Draw the absolute value graph of
other simple functions by inference.
12.2.5
Use mathematical induction to prove:
(a) statements of summation of series.
(b) statements about factors and
factoring with Natural numbers.
Content: Learning Outcome 2: Algebra
The learner is able to represent, investigate, analyse, manipulate and prove conjectures about numerical and algebraic relationships and functions, and
solve related problems.
Grade 10
Grade 11
► Complex numbers
• Determine and classify the roots of
quadratics equations
• Add subtract, multiply and divide
complex numbers
► Complex numbers
• Determine the real and complex roots
of quadratic and cubic equations
► Algebraic manipulation:
• Factorise third degree polynomials
• Simplify algebraic fractions
• Partial fractions
► Absolute values
• Define the absolute value as a distance
from the origin.
• Draw and solve the absolute value
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Grade 12
► Exponents and logarithms
• Simplify and manipulate algebraic
expressions using the laws of
exponents and natural logarithms
function
• Interpret the graphs of absolute value
functions
• Graph the absolute value of other
simple functions by inference.
► Solve quadratic inequalities.
► Solve
• Equations containing multi-term
algebraic fractions
• Polynomial and rational inequalities
• Absolute value equations
• Draw and solve exponential and
logarithm functions, including y = ex
and y = ln x
• Reflections and the inverse of
exponential or logarithmic functions
• Use e in problem solving.
► Solve
• Exponential equations
• Logarithmic equations
► Mathematical induction
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Learning Outcome 3: Statistics
The learner is able to organise, summarise, analyse and interpret data to identify, formulate and test statistical and probability models, and solve related
problems.
Grade 10
Grade 11 (optional)
We know this when the learner is able
to:
10.3.1 Organise and interpret univariate
numerical data in order to:
Grade 12 (optional)
We know this when the learner is able
to:
11.3.1
We know this when the learner is able
to:
12.3.1
(a) group data in a way that projects the
underlying distribution
(b) represent ungrouped and grouped
data in graphs that facilitate
interpretation (including graphs
learned in the FET; histograms and
cumulative frequency curves)
(a) Perform a one-tail and/or two-tail
hypothesis test on bivariate data,
- Distinguishing between one-tail and
two-tail events,
- Establishing a null hypothesis based
on the prevalent conditions, and
- Using statistical methods to accept
or reject the null hypothesis in
decision making
(b) Generate a predictive model using
linear regression and correlation
- Calculating, with available
technology, and interpreting the
Correlation coefficient
- using the least-squares method to
calculate a predictive linear
regression function, and
- predicting through interpolation and
extrapolation
(c) calculate, using formulae, the mean,
median and mode of grouped data
(d) calculate the quartiles, deciles and
percentiles of ungrouped and
grouped data, and solve associated
problems
(e) calculate, using available
technology, and interpret the
variance and standard deviation of
ungrouped and grouped data
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(f) use the techniques listed above to
evaluate data and so identify
potential sources of bias, errors in
measurement, and potential uses and
misuses of statistics and charts.
10.3.2 (a) Use Venn diagrams as an aid to
solving probability problems of
random events.
(b) Use Tree diagrams as an aid to
solving probability problems of
random events.
(c) Use Geometric diagrams to solve
probability problems
(d) Identify and determine the
probability of mutually exclusive and
independent events.
(e) Use the Laws of Probability to
evaluate simple random events.
11.3.2
(a) Recognise and then determine the
probability of conditional events
using diagrams and the formula
P( A B) =
P( A ∩ B)
P (B )
(b) Count arrangements and choices
using permutations (including those
where repetition occurs) and
combinations. (Available
technology may be used to perform
the necessary calculations)
(c) Identify, apply and calculate the
probability of the following
distributions of discrete random
events
- Hypergeometric distribution model
- Binomial distribution model
(d) Identify and apply the Normal
distribution model to the probability
of continuous random events, using
statistical tables and calculations as
necessary
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12.3.2
(a) Formulate a probability mass or
density function for a
- Hypergeometric distribution
- Binomial distribution
- Simple continuous probability
models
- Normal distribution
(b) Apply the Normal distribution model
to a sample to estimate a population
mean or proportion, using statistical
tables to deal with various
confidence levels.
Content: Learning Outcome 3: Statistics
The learner is able to organise, summarise, analyse and interpret data to identify, formulate and test statistical and probability models, and solve related
problems.
Grade 10
Grade 11 (optional)
► Descriptive Statistics
Grouped data
• Suitable graphical representations
• Histograms
• Cumulative frequency curves,
including estimating the median and
the quartiles
• Calculate mean, median, mode
For example,
Grade 12 (optional)
► Descriptive Statistics
• One-tail and two-tail hypothesis
testing on bivariate data,
• Linear regression and correlation
⎛1
⎞
⎜ 2n− f ⎟
Median = b + ⎜
⎟×c
f
c
⎜
⎟
⎝
⎠
• Calculate various percentiles. For
example,
⎛1
⎞
⎜ 4n− f ⎟
Q1 = b + ⎜
⎟×c
f
c
⎜
⎟
⎝
⎠
⎛ 6
⎞
⎜ 10 n − f ⎟
D6 = b + ⎜
⎟×c
f
c
⎜
⎟
⎝
⎠
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• Variance and standard deviation of
ungrouped and grouped data
Study bias and errors in measurement
► Probability
• Venn diagrams
• Tree diagrams
• Mutually exclusive and independent
events.
• Calculations involving basic Laws of
Probability
► Probability
• Conditional probability solved using
diagrams and the formula
• Counting Techniques
- Permutations
- Combinations
- Repetitions
• Probability of discrete random events:
- Hypergeometric distribution
model
- Binomial distribution model
• Continuous random events
- Normal distribution model
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► Probability
• Probability mass or density functions
- Hypergeometric distribution
- Binomial distribution
- Simple continuous probability
models
- Normal distribution
• Apply the Normal distribution model
to a sample to estimate a population
mean or proportion
Learning Outcome 4: Mathematical modelling
The learner is able to investigate, represent and model growth and decay problems using formulae, difference equations and series.
Grade 10
Grade 11 (optional)
We know this when the learner is able
to:
10.4.1
(a) Generalise number patterns using
first order linear difference equations
of the form un = k.un-1 + c.
Grade 12 (optional)
We know this when the learner is able
to:
11.4.1
(b) Use appropriate technology to solve
higher terms in first order linear
difference equations.
(a) Generalise or produce number
patterns using second order
homogenous linear difference
equations
(un = p.un-1 + q.un-2).
We know this when the learner is able
to:
12.4.1
(b) Use appropriate technology to solve
higher terms in second order
homogenous linear difference
equations.
(c) Convert first order linear difference
equations into a general solution in
explicit form.
(d) With the aid of appropriate
technology, use first order linear
difference equations to solve future
and present value annuities.
10.4.2
(a) Use simple and compound growth
formulae to solve problems in
various contexts including but not
limited to:
- simple interest and straight line
depreciation,
- compound interest and reducing
balance depreciation,
(a) Model simple population growth and
decay problems using
- a discrete Malthusian population
model of the form Pn+1 = (1 + r).Pn.
- a discrete Logistic population model
of the form Pn+1 = Pn + a(1–b.Pn).Pn.
- a discrete two species LotkaVolterra predator-prey population
model written in difference equation
form
Rn+1 = Rn + a.Rn (1– Rn/k) – b.Rn.Fn
Fn+1 = Fn + e.b.Rn.Fn – c.Fn
(b) Evaluate a realistic population
scenario and apply the most suitable
model for a given scenario.
11.4.2
Formulate timelines and apply future
and present value annuity formulae to:
(a) Convert between effective and
nominal interest rates when solving
problems with different
accumulation periods.
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12.4.2
Formulate timelines and apply future
and present value annuity formulae to:
(a) Determine the number of repayment
periods using logarithms.
(b) Calculate the number of payments
and the final payment when a loan
- compound growth and decay
problems
(b) Investigate and derive the future
value annuity formula using first
order linear difference equations in
explicit form.
(b) Calculate the present value or future
value of an annuity, or the termly
payment.
(c) Calculate the balance outstanding on
a loan at a specified point in the
amortisation period.
is repaid by fixed instalments.
(c) Solve annuity problems involving
changing circumstances such as
changes to time periods, repayments
(including missed payments),
withdrawals and interest rates.
(d) Determine the scrap value of
existing equipment or an asset, the
future cost of the replacement
equipment or an asset, and the equal
instalments required to establish a
sinking fund in a given context.
(e) Calculate the value of a deferred
annuity.
(f) Convert effective and nominal
interest rates to solve problems with
different accumulation periods.
Content: Learning Outcome 4: Mathematical modelling
The learner is able to investigate, represent and model growth and decay problems using formulae, difference equations and series.
Grade 10
► First order linear difference equations of
the form un = k.un-1 + c.
Grade 11 (optional)
► Second order homogenous linear
difference equations (un = p.un-1 + q.un-2)
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Grade 12 (optional)
► Model simple population growth and
decay problems using
• discrete Malthusian population model
• discrete Logistic population model
• a discrete two species Lotka-Volterra
predator-prey population model
► Simple and compound growth and
decay
► • Effective and nominal interest rates
• Formulate timelines and apply future
and present value annuity formulae to:
- solving the future value, present
value or termly instalment
- solving the balance outstanding
- sinking funds
- deferred annuities
- solve problems with different
accumulation periods.
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► Formulate timelines and apply future
and present value annuity formulae to:
• the number of repayment periods
• fixed instalments
• annuity problems involving changing
circumstances
Learning Outcome 5: Matrices and Graph Theory
The learner is able to identify, represent and manipulate discrete variables using graphs and matrices, applying algorithms in modeling finite systems.
Grade 10
Grade 11 (optional)
We know this when the learner is able
to:
10.5.1 (a) Arrange numbers in a suitable
rectangular array or matrix to
facilitate problem solving.
Grade 12 (optional)
We know this when the learner is able
to:
11.5.1
(b) Knowing when a matrix operation is
possible, perform the following
operations on a matrix or matrices
- addition,
- multiplication of a matrices, and
- multiplication by a scalar.
Use 2 × 2 matrices to transform points
and figures in the Cartesian Plane by:
⎛a⎞
(a) A translation, given in the form ⎜⎜ ⎟⎟ ,
⎝b⎠
(b) Rotation through any given angle
about the origin,
12.5.1
(c) Solve systems of two variable linear
equations using the method of
diagonalisation.
(d) Determine the inverse of 2 x 2
matrices by a sequence of row
transformation
[A: Inxn] = [Inxn: A-1]
(e) Shear and stretch with the x or y axis
as the invariant line using negative
or positive shear/stretch factors
(a) Determine the number of different
graphs that can be drawn on n ≤ 6
vertices.
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Use matrices to:
(a) Solve systems of three variable
linear equations using the method of
diagonalisation.
(b) determine the inverse matrix by a
sequence of row transformations
using [A: Inxn] = [Inxn: A-1].
(c) Reflection in any given line through
the origin,
(d) Enlargement, using construction,
with positive or negative scale
factors and the centre of
enlargement at the origin,
(e) Solve systems of linear equations
using the inverse matrix.
10.5.2 (a) Define simple, regular and connected 11.5.2
graphs, their vertices, edges and the
degree of the graph.
We know this when the learner is able
to:
(c) calculate the determinant of the
matrix.
(d) Solve systems of linear equations
using the inverse matrix.
12.5.2
(a) Solve minimum connector problems
using graphs, matrices and the
Kruskal and Prim algorithms.
(b) Identify isomorphic graphs.
(b) Represent simple network problems
using a weighted graph
(c) Define walks, paths and circuits.
(d) Evaluate and determine Eulerian
paths within a graph.
(e) Evaluate and classify graphs,
intuitively and algorithmically, as
Eulerian Circuits or Hamiltonian
Circuits
(f) Use Euler’s, Fleury’s and Dirac’s
algorithms to test the nature of the
paths and circuits in a graph.
(c) Solve simple optimisation problems
using weighted graphs
(d) Determine the shortest path of a
network or weighted using the
shortest path algorithm.
(e) Optimise route inspection (Chinese
postman) problems using Eulerian
circuits, paths and the shortest path
algorithm.
(b) Solve by finding an upper bound for
simple travelling salesman problems
using graphs and matrices and the
nearest-neighbour algorithm.
(c) Solve simple travelling salesman
problems using simple algorithms
researched in the literature.
(d) Use matrices to represent graphs
and to solve optimisation problems.
Content: Learning Outcome 5: Matrices and Graph Theory
The learner is able to identify, represent and manipulate discrete variables using graphs and matrices, applying algorithms in modelling finite systems.
Grade 10
► Matrices
• Arrange numbers in a matrix to
facilitate problem solving.
• Matrix operations
• Solve systems of two variable linear
equations
• Inverse of 2 x 2 matrices
Grade 11 (optional)
► Transform points and figures in the
Cartesian Plane using 2 × 2 matrices
36 of 48
Grade 12 (optional)
► Matrices
• Solve systems of three variable linear
equations
• Inverse matrices
• Determinant of a matrix.
► Graph Theory
• Define the basic qualities of a regular
graph.
• Identify isomorphic graphs.
• Walks, paths and circuits.
• Eulerian paths
• Eulerian Circuits or Hamiltonian
Circuits
► Graph Theory
• Define and use appropriate algorithms
to classify various graphs.
• Represent and solve simple
optimisation problems using a
weighted graph
• Shortest path algorithm
• Optimise route inspection (Chinese
postman) problems
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► Graph Theory
• Solve travel problems using graphs,
matrices and simple algorithms.
• Minimum connector problems
• Kruskal and Prim algorithms.
• Simple travelling salesman problems
• Nearest-neighbour algorithm
• Matrices to solve graph problems
CHAPTER 4
ASSESSMENT
INTRODUCTION
Assessment is a critical element of the National Curriculum Statement Grades 10 – 12 (General). It
is a process of collecting and interpreting evidence in order to determine the learner’s progress in
learning and to make a judgement about a learner’s performance. Evidence can be collected at
different times and places, and with the use of various methods, instruments, modes and media. To
ensure that assessment results can be accessed and used for various purposes at a future date, the
results have to be recorded. There are various approaches to recording learners’ performances.
Some of these are explored in this chapter. Others are dealt with in a more subject-specific manner
in the Learning Programme Guidelines.
Many stakeholders have an interest in how learners perform in Grades 10 – 12. These include the
learners themselves, parents, guardians, sponsors, provincial departments of education, the
Department of Education, the Ministry of Education, employers, and higher education and training
institutions. In order to facilitate access to learners’ overall performances and to inferences on
learners’ competences, assessment results have to be reported. There are many ways of reporting.
The Learning Programme Guidelines and the Assessment Guidelines discuss ways of recording
and reporting on school-based and external assessment as well as giving guidance on assessment
issues specific to the subject.
WHY ASSESS
Before a teacher assesses learners, it is crucial that the purposes of the assessment be clear and
unambiguous established. Understanding the purposes of assessment ensures that an appropriate
match exists between the purposes and the methods of assessment. This, in turn, will help to ensure
that decisions and conclusions based on the assessment are fair and appropriate for the particular
purpose or purposes.
There are many reasons why learners’ performance is assessed. These include monitoring progress
and providing feedback, diagnosing or remediating barriers to learning, selection, guidance,
supporting learning, certification and promotion. In this curriculum, learning and assessment are
very closely linked. Assessment helps learners to gauge the value of their learning. It gives them
information about their own progress and enables them to take control of and to make decisions
about their learning. In this sense, assessment provides information about whether teaching and
learning is succeeding in getting closer to the specified Learning Outcomes. When assessment
indicates lack of progress, teaching and learning plans should be changed accordingly.
TYPES OF ASSESSMENT
This section discusses the following types of assessment:
• baseline assessment;
• diagnostic assessment;
• formative assessment; and
• summative assessment.
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Baseline assessment
Baseline assessment is important at the start of a grade, but can occur at the beginning of any
learning cycle. It is used to establish what learners already know and can do. It helps in the
planning of activities and in Learning Programme development. The recording of baseline
assessment is usually informal.
Diagnostic assessment
Any assessment can be used for diagnostic purposes – that is, to discover the cause or causes of a
learning barrier. Diagnostic assessment assists in deciding on support strategies or identifying the
need for professional help or remediation. It acts as a checkpoint to help redefine the Learning
Programme goals, or to discover what learning has not taken place so as to put intervention
strategies in place.
Formative assessment
Any form of assessment that is used to give feedback to the learner is fulfilling a formative
purpose. Formative assessment is a crucial element of teaching and learning. It monitors and
supports the learning process. All stakeholders use this type of assessment to acquire information
on the progress of learners. Constructive feedback is a vital component of assessment for formative
purposes.
Summative assessment
When assessment is used to record a judgement of the competence or performance of the learner, it
serves a summative purpose. Summative assessment gives a picture of a learner’s competence or
progress at any specific moment. It can occur at the end of a single learning activity, a unit, cycle,
term, semester or year of learning. Summative assessment should be planned and a variety of
assessment instruments and strategies should be used to enable learners to demonstrate
competence.
WHAT ASSESSMENT SHOULD BE AND DO
Assessment should:
• be understood by the learner and by the broader public;
• be clearly focused;
• be integrated with teaching and learning;
• be based on pre-set criteria of the Assessment Standards;
• allow for expanded opportunities for learners;
• be learner-paced and fair;
• be flexible;
• use a variety of instruments; and
• use a variety of methods;
HOW TO ASSESS
Teachers’ assessment of learners’ performances must have a great degree of reliability. This means
that teachers’ judgements of learners’ competences should be generalisable across different times,
assessment items and markers. The judgements made through assessment should also show a great
degree of validity; that is, they should be made on the aspects of learning that were assessed.
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Because each assessment cannot be totally valid or reliable by itself, decisions on learner progress
must be based on more than one assessment. This is the principle behind continuous assessment
(CASS). Continuous assessment is a strategy that bases decisions about learning on a range of
different assessment activities and events that happen at different times throughout the learning
process. It involves assessment activities that are spread throughout the year, using various kinds of
assessment instruments and methods such as tests, examinations, projects and assignments. Oral,
written and performance assessments are included. The different pieces of evidence that learners
produce as part of the continuous assessment process can be included in a portfolio. Different
subjects have different requirements for what should be included in the portfolio. The Learning
Programme Guidelines discuss these requirements further.
Continuous assessment is both classroom-based and school-based, and focuses on the ongoing
manner in which assessment is integrated into the process of teaching and learning. Teachers get to
know their learners through their day-to-day teaching, questioning, observation, and through
interacting with the learners and watching them interact with one another. Continuous assessment
should be applied both to sections of the curriculum that are best assessed through written tests and
assignments and those that are best assessed through other methods, such as by performance, using
practical or spoken evidence of learning.
METHODS OF ASSESSMENT
Self-assessment
All Learning Outcomes and Assessment Standards are transparent. Learners know what is expected
of them. Learners can, therefore play an important part, through self-assessment, in ‘pre-assessing’
work before the teacher does the final assessment. Reflection on one’s own learning is a vital
component of learning.
Peer assessment
Peer assessment, using a checklist or rubric, helps both the learners whose work is being assessed
and the learners who are doing the assessment. The sharing of the criteria for assessment empowers
learners to evaluate their own and others’ performances.
Group assessment
The ability to work effectively in groups is one of the Critical Outcomes. Assessing group work
involves looking for evidence that the group of learners co-operate, assist one another, divide work,
and combine individual contributions into a single composite assessable product. Group assessment
looks at process as well as product. It involves assessing social skills, time management, resource
management and group dynamics, as well as the output of the group.
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METHODS OF COLLECTING ASSESSMENT EVIDENCE
There are various methods of collecting evidence. Some of these are discussed below.
Observation-based assessment
Observation-based assessment methods tend to be less structured and allow the development of a
record of different kinds of evidence for different learners at different times. This kind of
assessment is often based on tasks that require learners to interact with one another in pursuit of a
common solution or product. Observation has to be intentional and should be conducted with the
help of an appropriate observation instrument.
Test-based assessment
Test-based assessment is more structured, and enables teachers to gather the same evidence for all
learners in the same way and at the same time. This kind of assessment creates evidence of learning
that is verified by a specific score. If used correctly, tests and examinations are an important part of
the curriculum because they give good evidence of what has been learned.
Task-based assessment
Task-based or performance assessment methods aim to show whether learners can apply the skills
and knowledge they have learned in unfamiliar contexts or in contexts outside of the classroom.
Performance assessment also covers the practical components of subjects by determining how
learners put theory into practice. The criteria, standards or rules by which the task will be assessed
are described in rubrics or task checklists, and help the teacher to use professional judgement to
assess each learner’s performance.
RECORDING AND REPORTING
Recording and reporting involves the capturing of data collected during assessment so that it can be
logically analysed and published in an accurate and understandable way.
Methods of recording
There are different methods of recording. It is often difficult to separate methods of recording from
methods of evaluating learners’ performances.
The following are examples of different types of recording instruments:
• rating scales;
• task lists or checklists; and
• rubrics.
Each is discussed below.
Rating scales
Rating scales are any marking system where a symbol (such as A or B) or a mark (such as 5/10 or
50%) is defined in detail to link the coded score to a description of the competences that are
required to achieve that score. The detail is more important than the coded score in the process of
teaching and learning, as it gives learners a much clearer idea of what has been achieved and where
and why their learning has fallen short of the target. Traditional marking tended to use rating scales
without the descriptive details, making it difficult to have a sense of the learners’ strengths and
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weaknesses in terms of intended outcomes. A six-point scale of achievement is used in the National
Curriculum Statement Grades 10 – 12 (General).
Task lists or checklists
Task lists or checklists consist of discrete statements describing the expected performance in a
particular task. When a particular statement (criterion) on the checklist can be observed as having
been satisfied by a learner during a performance, the statement is ticked off. All the statements that
have been ticked off on the list (as criteria that have been met) describe the learner’s performance.
These checklists are very useful in peer or group assessment activities.
Rubrics
Rubrics are a combination of rating codes and descriptions of standards. They consist of a
hierarchy of standards with benchmarks that describe the range of acceptable performance in each
code band. Rubrics require teachers to know exactly what is required by the outcome. Rubrics can
be holistic, giving a global picture of the standard required, or analytic, giving a clear picture of the
distinct features that make up the criteria, or can combine both. The Learning Programme
Guidelines give examples of subject-specific rubrics.
To design a rubric, a teacher has to decide the following:
• Which outcomes are being targeted?
• Which Assessment Standards are targeted by the task?
• What kind of evidence should be collected?
• What are the different parts of the performance that will be assessed?
• What different assessment instruments best suit each part of the task (such as the process
and the product)?
• What knowledge should be evident?
• What skills should be applied or actions taken?
• What opportunities for expressing personal opinions, values or attitudes arise in the task
and which of these should be assessed and how?
• Should one rubric target all the Learning Outcomes and Assessment Standards of the task
or does the task need several rubrics?
• How many rubrics are, in fact, needed for the task?
It is crucial that a teacher shares the rubric or rubrics for the task with the learners before they do
the required task. The rubric clarifies what both the learning and the performance should focus on.
It becomes a powerful tool for self-assessment.
Reporting performance and achievement
Reporting performance and achievement informs all those involved with or interested in the
learner’s progress. Once the evidence has been collected and interpreted, teachers need to record a
learner’s achievements. Sufficient summative assessments need to be made so that a report can
make a statement about the standard achieved by the learner.
The National Curriculum Statement Grades 10 – 12 (General) adopts a six-point scale of
achievement. The scale is shown in Table 4.1.
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Table 4.1 Scale of achievement for the National Curriculum Statement Grades 10 – 12
(General): Rating Description of Competence Marks
Code
7
6
5
4
3
2
1
Outstanding achievement
Meritorious achievement
Substantial achievement
Adequate achievement
Moderate achievement
Elementary achievement
Not achieved
%
80 – 100
70 - 79
60 – 69
50 – 59
40 – 49
30 - 39
0 - 29
SUBJECT COMPETENCE DESCRIPTIONS
To assist with benchmarking the achievement of Learning Outcomes in Grades 10 – 12, subject
competences have been described to distinguish the grade expectations of what learners must know
and be able to achieve.
Seven levels of competence have been described for each subject for each grade. These
descriptions will assist teachers to assess learners and place them in the correct rating. The
descriptions summarise what is spelled out in detail in the Learning Outcomes and the Assessment
Standards, and give the distinguishing features that fix the achievement for a particular rating. The
various achievement levels and their corresponding percentage bands are as shown in Table 4.1.
In line with the principles and practice of outcomes-based assessment, all assessment – both
school-based and external – should primarily be criterion-referenced. Marks could be used in
evaluating specific assessment tasks, but the tasks should be assessed against rubrics instead of
simply ticking correct answers and awarding marks in terms of the number of ticks. The statements
of competence for a subject describe the minimum skills, knowledge, attitudes and values that a
learner should demonstrate for achievement on each level of the rating scale. When
teachers/assessors prepare an assessment task or question, they must ensure that the task or
question addresses an aspect of a particular outcome. The relevant Assessment Standard or
Standards must be used when creating the rubric for assessing the task or question. The
descriptions clearly indicate the minimum level of attainment for each category on the rating scale.
The competence descriptions for this subject appear at the end of this chapter.
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COMPETENCE DESCRIPTORS
•
•
•
•
•
Grade 10
Grade 11
Grade 12
Level 7
Level 7
Level 7
set up mathematical models to solve
problems in straightforward situations
investigate and mathematise problem
situations creatively
communicate mathematical information
using suitable representations and commonly
used mathematical notation in logically
constructed arguments
produce clear geometric or algebraic
solutions of non-routine problems
validate results and justify solutions with
logical argument
•
•
•
•
•
•
evaluate mathematical calculations and
manipulations accurately
set up mathematical models to solve real life
problems and draw appropriate conclusions
use appropriate mathematical symbols and
representations (graphs, sketches, tables,
equations) to communicate ideas clearly and
creatively
produce clear geometric, algebraic or
trigonometric solutions and proofs to multistep non-routine problems
justify conclusions as to the validity of selfformulated conjectures with logical
argument
ask “what if” questions to extend simple
investigations, concepts or solutions
•
•
•
•
•
•
•
•
Level 6
•
•
•
simplify and calculate accurately using
efficient methods
attempt to set up mathematical models to
solve problems in straightforward situations
investigate problem situations methodically
Level 6
•
•
•
evaluate mathematical information and data
to hypothesise about situations
simplify and calculate by choosing the
correct strategy or method
attempt to set up mathematical models to
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differentiate between various techniques or
methods and develop the appropriate
technique for given problems
evaluate mathematical calculations and
manipulations accurately
set up mathematical models to solve more
complex real life problems and draw
appropriate conclusions
think creatively and laterally on a broad
range of complex mathematical concepts
use appropriate mathematical symbols and
representations (graphs, sketches, tables,
equations) to communicate ideas clearly,
concisely and creatively
produce clear, logical, geometric, algebraic
or trigonometric solutions and proofs to
multi-step non-routine problems
justify conclusions as to the validity of selfformulated conjectures with logical
argument and proof
ask “what if” questions to extend
investigations, concepts or solutions
Level 6
•
•
•
evaluate mathematical information and data
to hypothesise about situations
simplify and calculate by choosing the
correct strategy or method
attempt to set up mathematical models to
•
•
and make suitable generalisations
communicate mathematical information
using suitable representations in logically
constructed arguments
draw conclusions and attempt to justify
solutions with logical argument
•
•
•
Level 5
•
•
•
•
•
•
•
•
Make good estimates in straightforward
situations
Complete difficult calculations accurately
Simplify difficult expressions correctly
Apply correctly the techniques, algorithms
and formulae learned in this and lower
grades to arrive at correct solutions to
routine problems related to everyday life
Apply given mathematical models to
straightforward situations
Communicate mathematical information
using suitable representations
investigate problem situations methodically
make conjectures after investigation of
simple mathematical problems
solve real life problems
investigate more complex problem situations
methodically and make suitable
generalisations
communicate mathematical information
using suitable representations in clear
logically constructed arguments
attempt to justify conclusions with logical
argument
•
•
•
Level 5
•
•
•
•
•
work accurately to simplify and solve
mathematical problems
Apply given mathematical models to real life
situations
correctly apply learned techniques,
algorithms and formulae to solve routine
mathematical and real life problems
communicate mathematical information
using suitable representations and notation
make conjectures after investigation of
routine problems
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solve more complex real life problems and
draw appropriate conclusions
investigate novel problem situations
methodically and make suitable
generalisations
communicate mathematical information
using suitable representations in efficient and
logically constructed arguments
justify conclusions with logical argument
Level 5
•
•
•
•
•
work accurately to simplify and solve
mathematical problems
Apply given mathematical models to more
complex situations
correctly apply learned techniques,
algorithms and formulae to solve more
complex mathematical and real life problems
communicate mathematical information
efficiently using suitable representations and
notation
make conjectures after investigation of real
life situations
Level 4
•
•
•
•
•
•
•
make estimates in straightforward situations
classify, organise and represent
mathematical information and data
complete routine numerical calculations
accurately
simplify basic expressions correctly
demonstrate understanding of the techniques,
algorithms and formulae learned in this and
lower grades to arrive at correct solutions to
simple routine problems related to everyday
life
check solutions and detect errors in
calculations
communicate mathematical information
using some form of representation
Level 4
•
•
•
•
•
•
Level 3
•
•
•
•
•
•
•
show evidence of an attempt to estimate
recognise and define mathematical
constructs
complete simple calculations accurately
using learned rules
simplify basic expressions correctly using
learned rules
show evidence of the techniques, algorithms
and formulae learned in this and lower
grades
follow instructions on how to represent
mathematical information eg drawing graphs
communicate using memorised mathematical
terminology
make estimates in routine situations
classify, organise and represent
mathematical information and data
complete numerical calculations accurately
demonstrate understanding of the techniques,
algorithms and formulae learned in this and
lower grades to arrive at correct solutions to
simple routine problems related to everyday
life
evaluate solutions and calculation
communicate mathematical information
using suitable representations
Level 4
•
•
•
•
•
•
Level 3
•
•
•
•
•
•
complete simple calculations accurately
using learned rules
demonstrate a conceptual understanding of
learned rules and methods in solving
problems
recognise and identify mathematical
terminology and the associated concepts or
rules
visualise information graphically to aid in
problem solving
follow instructions on how to represent
mathematical information
communicate using memorised mathematical
terminology
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make estimates in real life situations
classify, organise and represent
mathematical information and data
complete routine and non-routine numerical
calculations accurately
demonstrate understanding of the techniques,
algorithms and formulae learned in this and
lower grades to arrive at correct solutions to
simple routine problems related to everyday
life
critically evaluate solutions and calculations
communicate mathematical information
using suitable representations and notation
Level 3
•
•
•
•
•
•
complete simple calculations accurately
using learned rules
demonstrate a conceptual understanding of
learned rules and methods in solving
problems
recognise and describe mathematical
concepts, ideas and connections
use graphical or pictorial images to aid in
problem solving
follow instructions on how to represent
mathematical information
communicate using memorised mathematical
terminology
Level 2
•
•
•
•
•
attempt to estimate
initiate calculations using learned rules
partially simplify expressions using learned
rules
apply in a rote manner the techniques,
algorithms and formulae learned in this and
lower grades
attempt to follow instructions on how to
represent mathematical information
Level 2
•
•
•
Level 1
•
attempt to apply learned methods and
techniques, not necessarily in the correct
context
attempt simple calculations using learned
rules
apply in a rote manner learned techniques,
algorithms and formulae to solve problems
attempt to follow instructions on how to
represent mathematical information
Level 2
•
•
•
Level 1
•
•
do investigations in an unstructured,
arbitrary manner
attempt to apply learned methods and
techniques, not necessarily in the correct
context
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attempt simple calculations using learned
rules
apply in a rote manner learned techniques,
algorithms and formulae to solve problems
attempt to follow instructions on how to
represent mathematical information
Level 1
•
•
do investigations in an unstructured,
arbitrary manner
attempt to apply learned methods and
techniques, not necessarily in the correct
context
PROMOTION
Promotion at Grade 10 and Grade 11 level will be based on internal assessment only, but must be
based on the same conditions as those for the Further Education and Training Certificate. The
requirements, conditions, and rules of combination and condonation are spelled out in the
Qualifications and Assessment Policy Framework for Grades 10 – 12 (General). This subject is in
addition to the normal package of subjects. Hence performance in this subject does not affect
promotion of the learner. However continued participation in the course depends on adequate
performance by the learner in meeting the outcomes at each grade.
WHAT REPORT CARDS SHOULD LOOK LIKE
There are many ways to structure a report card, but the simpler the report card the better, providing
that all important information is included. Report cards should include information about a
learner’s overall progress, including the following:
• the learning achievement against outcomes;
• the learner’s strengths;
• the support needed or provided where relevant;
• constructive feedback commenting on the performance in relation to the learner’s previous
performance and the requirements of the subject; and
• the learner’s developmental progress in learning how to learn.
In addition, report cards should include the following:
• name of school;
• name of learner;
• learner’s grade;
• year and term;
• space for signature of parent or guardian;
• signature of teacher and of principal;
• date;
• dates of closing and re-opening of school;
• school stamp; and
• school attendance profile of learner.
ASSESSMENT OF LEARNERS WHO EXPERIENCE BARRIERS TO LEARNING
The assessment of learners who experience any barriers to learning will be conducted in
accordance with the recommended alternative and/or adaptive methods as stipulated in the
Qualifications and Assessment Policy Framework for Grades 10 – 12 (General) as it relates to
learners who experience barriers to learning. Refer to the White Paper 6 on Special Needs
Education: Building an Inclusive Education and Training System.
The theoretical knowledge is easily adapted to accommodate learners who have barriers to
learning.
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