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] 2 of 48 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 13 13 13 13 14 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 16 16 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 38 38 38 38 39 39 40 41 41 43 44 48 48 48 3 of 48 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. 4 of 48 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 5 of 48 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 6 of 48 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. 7 of 48 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 8 of 48 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 9 of 48 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. 10 of 48 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. 11 of 48 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. 12 of 48 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; 13 of 48 • • • • • • • • • 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 14 of 48 • • 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 15 of 48 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 16 of 48 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. 17 of 48 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 18 of 48 - 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. 19 of 48 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 20 of 48 = 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. 21 of 48 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. 23 of 48 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 24 of 48 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; 25 of 48 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 26 of 48 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 27 of 48 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 28 of 48 (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 29 of 48 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 ⎜ ⎟ ⎝ ⎠ 30 of 48 • 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 31 of 48 ► 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. 32 of 48 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) 33 of 48 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. 34 of 48 ► 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. 35 of 48 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 37 of 48 ► 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. 38 of 48 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. 39 of 48 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. 40 of 48 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 41 of 48 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. 42 of 48 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. 43 of 48 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 44 of 48 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 45 of 48 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 46 of 48 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 47 of 48 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. 48 of 48

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