CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS) MATHEMATICS FET PHASE FINAL DRAFT

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1 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS) MATHEMATICS FET PHASE FINAL DRAFT Page 1 of 75

2 SECTION Background NATIONAL CURRICULUM AND ASSESSMENT POLICY STATEMENT FOR MATHEMATICS The National Curriculum Statement Grades R 1 (NCS) stipulates policy on curriculum and assessment in the schooling sector. To improve its implementation, the National Curriculum Statement was amended, with the amendments coming into effect in January 01. A single comprehensive National Curriculum and Assessment Policy Statement was developed for each subject to replace the old Subject Statements, Learning Programme Guidelines and Subject Assessment Guidelines in Grades R - 1. The amended National Curriculum and Assessment Policy Statements (January 01) replace the National Curriculum Statements Grades R - 9 (00) and the National Curriculum Statements Grades 10-1 (004). 1. Overview (a) (b) (c) (d) The National Curriculum Statement Grades R 1 (January 01) represents a policy statement for learning and teaching in South African schools and comprises the following: National Curriculum and Assessment Policy Statements for each approved school subject as listed in the policy document, National policy pertaining to the programme and promotion requirements of the National Curriculum Statement Grades R 1, which replaces the following policy documents: (i) National Senior Certificate: A qualification at Level 4 on the National Qualifications Framework (NQF); and (ii) An addendum to the policy document, the National Senior Certificate: A qualification at Level 4 on the National Qualifications Framework (NQF), regarding learners with special needs, published in the Government Gazette, No.9466 of 11 December 006. The National Curriculum Statement Grades R 1 (January 01) should be read in conjunction with the National Protocol for Assessment Grade R 1, which replaces the policy document, An addendum to the policy document, the National Senior Certificate: A qualification at Level 4 on the National Qualifications Framework (NQF), regarding the National Protocol for Assessment Grade R 1, published in the Government Gazette, No of 11 December 006. The Subject Statements, Learning Programme Guidelines and Subject Assessment Guidelines for Grades R - 9 and Grades 10-1 are repealed and replaced by the National Curriculum and Assessment Policy Statements for Grades R 1 (January 01). The sections on the Curriculum and Assessment Policy as contemplated in Chapters, and 4 of this document constitute the norms and standards of the National Curriculum Statement Grades R 1 and therefore, in terms of section 6A of the South African Schools Act, 1996 (Act No. 84 of 1996,) form the basis for the Minister of Basic Education to determine minimum outcomes and standards, as well as the processes and procedures for the assessment of learner achievement to be applicable to public and independent schools. Page of 75

3 1. General aims of the South African Curriculum (a) The National Curriculum Statement Grades R - 1 gives expression to what is regarded to be knowledge, skills and values worth learning. It will ensure that children acquire and apply knowledge and skills in ways that are meaningful to their own lives. In this regard, the curriculum promotes the idea of grounding knowledge in local contexts, while being sensitive to global imperatives. (b) The National Curriculum Statement Grades R - 1 serves the purposes of: equipping learners, irrespective of their socio-economic background, race, gender, physical ability or intellectual ability, with the knowledge, skills and values necessary for self-fulfilment, and meaningful participation in society as citizens of a free country; providing access to higher education; facilitating the transition of learners from education institutions to the workplace; and providing employers with a sufficient profile of a learner s competences. (c) The National Curriculum Statement Grades R - 1 is based on the following principles: Social transformation: ensuring that the educational imbalances of the past are redressed, and that equal educational opportunities are provided for all sections of our population; Active and critical learning: encouraging an active and critical approach to learning, rather than rote and uncritical learning of given truths; High knowledge and high skills: the minimum standards of knowledge and skills to be achieved at each grade are specified and sets high, achievable standards in all subjects; Progression: content and context of each grade shows progression from simple to complex; Human rights, inclusivity, environmental and social justice: infusing the principles and practices of social and environmental justice and human rights as defined in the Constitution of the Republic of South Africa. The National Curriculum Statement Grades 10 1 (General) is sensitive to issues of diversity such as poverty, inequality, race, gender, language, age, disability and other factors; Valuing indigenous knowledge systems: acknowledging the rich history and heritage of this country as important contributors to nurturing the values contained in the Constitution; and Credibility, quality and efficiency: providing an education that is comparable in quality, breadth and depth to those of other countries. (d) The National Curriculum Statement Grades R - 1 aims to produce learners that are able to: identify and solve problems and make decisions using critical and creative thinking; work effectively as individuals and with others as members of a team; organise and manage themselves and their activities responsibly and effectively; collect, analyse, organise and critically evaluate information; Page of 75

4 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. (e) Inclusivity should become a central part of the organisation, planning and teaching at each school. This can only happen if all teachers have a sound understanding of how to recognise and address barriers to learning, and how to plan for diversity. The key to managing inclusivity is ensuring that barriers are identified and addressed by all the relevant support structures within the school community, including teachers, District-Based Support Teams, Institutional-Level Support Teams, parents and Special Schools as Resource Centres. To address barriers in the classroom, teachers should use various curriculum differentiation strategies such as those included in the Department of Basic Education s Guidelines for Inclusive Teaching and Learning (010). 1.4 Time Allocation Foundation Phase (a) The instructional time for subjects in the Foundation Phase is as indicated in the table below: Subject I. Languages (FAL and HL) II. Mathematics III. Life Skills Beginning Knowledge Creative Arts Physical Education Personal and Social Well-being Time allocation per week (hours) 10 (11) 7 6 (7) 1 () 1 (b) (c) Instructional time for Grades R, 1 and is hours and for Grade is 5 hours. In Languages 10 hours is allocated in Grades R- and 11 hours in Grade. A maximum of 8 hours and a minimum of 7 hours are allocated for Home Language and a minimum of hours and a maximum of hours for Additional Language in Grades R. In Grade a maximum of 8 hours and a minimum of 7 hours are allocated for Home Language and a minimum of hours and a maximum of 4 hours for First Additional Language. (d) In Life Skills Beginning Knowledge is allocated 1 hour in Grades R and hours as indicated by the hours in brackets for Grade. Page 4 of 75

5 1.4. Intermediate Phase (a) The table below shows the subjects and instructional times in the Intermediate Phase. Subject Time allocation per week (hours) I. Home Language II. First Additional Language III. Mathematics IV. Science and Technology V. Social Sciences VI. Life Skills Creative Arts Physical Education Personal and Social Well-being Senior Phase (a) The instructional time in the Senior Phase is as follows: Time allocation per Subject week (hours) I. Home Language 5 II. First Additional Language 4 III. Mathematics 4.5 IV. Natural Sciences V. Social Sciences VI. Technology VII. Economic Management Sciences VIII. Life Orientation IX. Creative Arts Page 5 of 75

6 1.4.4 Grades 10-1 (a) The instructional time in Grades 10-1 is as follows: Subject Time allocation per week (hours) I. Home Language II. First Additional Language III. Mathematics IV. Life Orientation V. Three Electives (x4h) The allocated time per week may be utilised only for the minimum required NCS subjects as specified above, and may not be used for any additional subjects added to the list of minimum subjects. Should a learner wish to offer additional subjects, additional time must be allocated for the offering of these subjects. Page 6 of 75

7 SECTION CURRICULUM AND ASSESSMENT POLICY STATEMENT FOR MATHEMATICS (FET).1 What is Mathematics? Mathematics is the study of quantity, structure, space and change. Mathematicians seek out patterns, formulate new conjectures, and establish axiomatic systems by rigorous deduction from appropriately chosen axioms and definitions. 1 Mathematics is a distinctly human activity practised by all cultures, for thousands of years Mathematical problem solving enables us to understand the world (physical, social and economic) around us, and, most of all, to teach us to think creatively. The main topics in the Mathematics (FET) Curriculum 1. Functions. Number patterns, sequences, series. Finance, growth and decay 4. Algebra 5. Differential calculus 6. Probability 7. Euclidean geometry and measurement 8. Analytical geometry 9. Trigonometry 10. Statistics 1 See the Wikipedia definition in which truth is used instead of axiomatic systems. Truth does not indicate conformity to perceived physical reality as is usually assumed when this definition is used without reference to the philosophy of mathematics. Axiomatics is one approach to establishing mathematical truth. The Euclidean geometry topic in grade 1 is an example of an axiomatic system. Page 7 of 75

8 . Specific Aims IMPORTANT GENERAL PRINCIPLES WHICH APPLY ACROSS ALL GRADES 1. No calculators with programmable functions, graphical facilities or symbolic facilities (for example, to factorise a b ( a b)( a b), or to find roots of equations) should be allowed. Calculators should only be used to perform standard numerical computations and to verify calculations by hand.. Mathematical modeling is an important focal point of the curriculum. Real life problems should be incorporated into all sections whenever appropriate. Examples used should be realistic and not contrived.. Investigations provide the opportunity to develop in learners the ability to be methodical, to generalize, make conjectures and try to justify or prove them. It needs to be understood that learners need to reflect on the processes and not be concerned only with getting the answer/s. 4. Appropriate approximation and rounding skills should be taught so that the impression is not gained that all answers which are either irrational numbers or recurring decimals should routinely be given correct to two decimal places. 5. The history of mathematics should be incorporated into projects and tasks wherever possible. The aim of the inclusion of some history is to show mathematics as a human creation and still developing. 6. Contextual problems should include issues relating to health, social, economic, cultural, scientific, political and environmental issues whenever possible. 7. Teaching should not be limited to how but should feature the when and why of problem types: Finding the mean and standard deviation of a set of data has little relevance unless learners have a good grasp of why and when such calculations might be useful. 8. Mixed ability teaching requires teachers to challenge the most able learners and at the same time provide remedial support for those for whom mathematics is difficult. An appendix of challenging questions is provided after the year plan. Teachers need to design questions to rectify misconceptions that are exposed by tests and examinations. 9. Problem solving and cognitive development should be central to all mathematics teaching. Learning procedures and proofs without a good understanding of why they are important will leave learners ill-equipped to use their knowledge in later life. Page 8 of 75

9 . Time allocation for Mathematics: 4 hours and 0 minutes, e.g. six forty five minute periods, per week in grades 10, 11 and 1..4 Overview of topics 1. FUNCTIONS Grade 10 Grade 11 Grade Work with relationships between variables in terms of numerical, graphical, verbal and symbolic representations of functions and convert flexibly between these representations (tables, graphs, words and formulae). Include linear and some quadratic polynomial functions, exponential functions and some rational functions Generate as many graphs as necessary, initially by means of point-by-point plotting, supported by available technology, to make and test conjectures and hence generalise the effect of the parameter which results in a vertical shift and that which results in a vertical stretch and /or a reflection about the x axis Extend Grade 10 work on the relationships between variables in terms of numerical, graphical, verbal and symbolic representations of functions and convert flexibly between these representations (tables, graphs, words and formulae). Include linear and quadratic polynomial functions, exponential functions and some rational functions Generate as many graphs as necessary, initially by means of point-by-point plotting, supported by available technology, to make and test conjectures and hence generalise the effects of the parameter which results in a horizontal shift and that which results in a horizontal stretch and/or reflection about the y axis Introduce a more formal definition of a function and extend Grade 11 work on the relationships between variables in terms of numerical, graphical, verbal and symbolic representations of functions and convert flexibly between these representations (tables, graphs, words and formulae). Include linear, quadratic and some cubic polynomial functions, exponential and logarithmic functions, and some rational functions The inverses of prescribed functions and the fact that, in the case of many-to-one functions, the domain has to be restricted if the inverse is to be a function Problem solving and graph work involving the prescribed functions Problem solving and graph work involving the prescribed functions Problem solving and graph work involving the prescribed functions (including the logarithmic function).. NUMBER PATTERNS, SEQUENCES AND SERIES Page 9 of 75

10 10..1 Investigate number patterns leading to those where there is a constant difference between consecutive terms, and the general term is therefore linear Investigate number patterns leading to those where there is a constant second difference between consecutive terms, and the general term is therefore quadratic Identify and solve problems involving number patterns that lead to arithmetic and geometric sequences and series, including infinite geometric series.. FINANCE, GROWTH AND DECAY Use simple and compound growth formulae n A P( 1 in) A P 1 i to solve and problems (including interest, hire purchase, inflation, population growth and other real life problems) Use simple and compound decay formulae n A P( 1in) A P 1i to solve and problems (including straight line depreciation and depreciation on a reducing balance). Link to work on functions (a) Calculate the value of n in the formulae n n A P 1 i A P 1 i and (b) Apply knowledge of geometric series to solve annuity and bond repayment problems The implications of fluctuating foreign exchange rates The effect of different periods of compounding growth and decay (including effective and nominal interest rates). 1.. Critically analyse different loan options. 4. ALGEBRA (a) Identify rational numbers and convert between terminating or recurring decimals b 0. and the form : b a where a, b Z and Take note that there exist numbers other than those on the real line, the so-called complex numbers. It is possible to square certain complex numbers and obtain negative real numbers as answers. (b) Show that simple surds are not rational. Page 10 of 75

11 10.4. (a) Simplify expressions using the laws of exponents for integral exponents. (b) Establish between which two integers a given simple surd lies. (c) Round real numbers to an appropriate degree of accuracy (to a given number of decimal digits) (a) Apply the laws of exponents to expressions involving rational exponents. (b) Add, subtract, multiply and divide simple surds Demonstrate an understanding of the definition of a logarithm and any laws needed to solve real life problems Manipulate algebraic expressions by: multiplying a binomial by a trinomial; factorising trinomials; factorising the difference of two cubes; factorising by grouping in pairs; simplifying, adding and subtracting algebraic fractions with denominators of degree at most Solve: linear equations quadratic equations by factorisation literal equations (changing the subject of formulae) exponential equations (accepting that the laws of exponents hold for real exponents and solutions are not necessarily integral or even rational). linear inequalities in one variable and illustrate the solution graphically linear equations in two variables simultaneously (algebraically and graphically) Manipulate algebraic expressions by writing quadratic functions in the completed square form Solve: quadratic equations (by factorisation, by completing the square, and by using the quadratic formula); quadratic inequalities in one variable and interpret the solution graphically; equations in two unknowns, one of which is linear the other quadratic, algebraically or graphically Take note, and understand, the Remainder and Factor Theorems for polynomials up to the third degree. Factorise third degree polynomials (including examples which require the Factor Theorem). 5. DIFFERENTIAL CALCULUS Page 11 of 75

12 Investigate the average rate of change of a function between two values of the independent variable demonstrating an understanding of average rate of change over different intervals Investigate numerically the average gradient between two points on a curve and develop an intuitive understanding of the concept of the gradient of a curve at a point (a) An intuitive understanding of the concept of a limit. (b) Differentiation of specified functions from first principles. (c) Use of the specified rules of differentiation. (d) The equations of tangents to graphs. (e) Sketch graphs of cubic functions. (f) Practical problems involving optimisation and rates of change (including the calculus of motion). 6. PROBABILITY a) Compare the relative frequency of an experimental outcome with the theoretical probability of the outcome. (b) Venn diagrams as an aid to solving probability problems. (c ) Mutually exclusive events and complementary events. (d) The identity for any two events A and B: P( A or B) P( A) P( B) P( A and B) (a) Dependent and independent events. (b) Venn diagrams or contingency tables and tree diagrams as aids to solving probability problems (where events are not necessarily independent) (a) Generalise the fundamental counting principle. (b) Probability problems using the fundamental counting principle and other techniques. 7. EUCLIDEAN GEOMETRY AND MEASUREMENT (a) Investigate and form conjectures about the properties of special triangles, quadrilaterals and other polygons. Try to validate or prove conjectures using any logical method (Euclidean, analytical or transformation geometry from Grade 9). (b) Disprove false conjectures by producing (a) Investigate and prove theorems of the geometry of circles assuming results from earlier grades, together with one other result concerning tangents and radii of circles. (b) Solve circle geometry problems, providing reasons for statements when required (a) Revise earlier work on the necessary and sufficient conditions for polygons to be similar. (b) Prove (accepting results established in earlier grades): that a line drawn parallel to one side of a triangle divides the other two sides proportionally (and the Page 1 of 75

13 counter-examples (c) Investigate alternative (but equivalent) definitions of various polygons (including the isosceles, equilateral and right-angled triangle, the kite, parallelogram, rectangle, rhombus, square and trapezium) Solve problems involving volume and surface area of solids studied in earlier grades and combinations of those objects to form more complex shaped solids. Mid-point Theorem as a special case of this theorem) that equiangular triangles are similar that triangles with sides in proportion are similar the Pythagorean Theorem by similar triangles 8. TRIGONOMETRY a) Definitions of the trigonometric functions sin, cos and tan in right-angled triangles. b) Derive values of the trigonometric functions for the special cases { 0;0;45;60;90}. c) Take note that there are special names for the reciprocals of the trigonometric functions (these three reciprocals should be examined in grade 10 only): 1 cosec, sin 1 sec cot cos and 1 tan Solve problems in -dimensions by using the above trigonometric functions and by constructing and interpreting geometric and trigonometric models (a) derive and use the identities: sin tan ; cos sin cos 1 (b) derive the reduction formulae (c) determine the general solution of trigonometric equations (d) establish the sine, cosine and area rules Solve problems in - dimensions by constructing and interpreting geometric and trigonometric models Proof and use of the compound angle and double angle identities Solve problems in two and three dimensions by constructing and interpreting geometric and trigonometric models. Page 1 of 75

14 10.8. Extend the definitions of sin, cos and tan 0 0 to and know the graphs of y sin, y cos and y tan ) The effects of a and q on the graphs of y asin q, y acos q and y a tan q The effects of the parameters on the graphs of y asin( kx p) q, and y acos( kx p) q y a tan( kx p) q, at most two parameters taken at a time. 9. ANALYTICAL GEOMETRY Represent geometric figures in a Cartesian coordinate system, and derive and apply, for any two points (x 1 ; y 1 ) and (x ; y ), a formula for calculating: the distance between the two points; the gradient of the line segment joining the points; the co-ordinates of the mid-point of the line segment joining the points Use a Cartesian co-ordinate system to derive and apply : the equation of a line through two given points; the equation of a line through one poin and parallel or perpendicular to a given line; the inclination of a line Use a two-dimensional Cartesian co-ordinate system to derive and apply: the equation of a circle (any centre); the equation of a tangent to a circle at a given point on the circle. 10. STATISTICS (a) Collect, organise and interpret univariate numerical data in order to determine: measures of central tendency (mean, median, mode) of grouped and ungrouped data and represent these by five number summary (maximum, minimum, quartiles) and box and (a) Represent data effectively, choosing appropriately from : bar and compound bar graphs; histograms (grouped data); frequency polygons; pie charts; line and broken line graphs (a) Represent bivariate numerical data as a scatter plot and suggest intuitively and by simple investigation whether a linear, quadratic or exponential function would best fit the data. (b) Use of available technology to calculate the linear regression line which best fits Page 14 of 75

15 whisker diagrams, and know which is the most appropriate under given conditions; measures of dispersion: percentiles, quartiles, deciles, interquartile and semi-inter-quartile range. (b) Represent measures of central tendency and dispersion in univariate numerical data by: using ogives; calculating the variance and standard deviation of sets of data manually (for small sets of data) and using available technology (for larger sets of data) and representing results graphically. a given set of bivariate numerical data. (c) Use of available technology to calculate the correlation co-efficient of a set of bivariate numerical data and make relevant deductions Identify possible sources of bias and errors in measurements Skewed data in box and whisker diagrams and frequency polygons. Identify outliers..5 Cognitive Levels The four cognitive levels used to guide all assessment tasks are based on those suggested in the TIMSS study of Descriptors for each level and the approximate percentages of tasks, tests and examinations which should be at each level are given below: Cognitive levels Description of skills to be demonstrated Examples Knowledge Estimation and appropriate rounding of numbers Proofs of prescribed theorems and derivation of formulae y f x 0% Straight recall x Routine procedures Identification and direct use of correct formula on the information sheet (no changing of the subject) Use of mathematical facts Appropriate use of mathematical vocabulary 1. Write down the domain of the function (Grade 10). Prove that the angle AOB ˆ subtended by arc AB at the centre O of a circle is double the size of the angle ACB ˆ which the same arc subtends at the circle. (Grade 11) Perform well known procedures 1. Solve for x : x 5x 14 (Grade 10) Simple applications and calculations which might involve many steps 45% Derivation from given information may be involved. Determine the general solution of the equation Identification and use (after changing the subject) of correct formula 0 sin x (Grade 11) Generally similar to those encountered in class. Complex Problems involve complex calculations and/or higher order 1. What is the average speed covered on a round trip Page 15 of 75

16 procedures 5% Problem solving 10% reasoning There is often not an obvious route to the solution Problems need not be based on a real world context Could involve making significant connections between different representations Require conceptual understanding Unseen, non-routine problems (which are not necessarily difficult) Higher order understanding and processes are often involved Might require the ability to break the problem down into its constituent parts to and from a destination if the average speed going to the destination is100 km / h and the average speed for the return journey is 80 km / h? (Grade 11) x. Differentiate x with respect to x. (Grade 1) Suppose a piece of wire could be tied tightly around the earth at the equator. Imagine that this wire is then lengthened by exactly one metre and held so that it is still around the earth at the equator. Would a mouse be able to crawl between the wire and the earth? Why or why not? (Any grade) ANNUAL TEACHING PLAN SECTION 1. The examples discussed in the Clarification Column in the annual teaching plan which follows are by no means a complete representation of all the material to be covered in the curriculum. They only serve as an indication of some questions on the topic at different cognitive levels. Text books and other resources should be consulted for a complete treatment of all the material.. The cognitive levels of examples given are not absolute. What is a complex procedure in one grade may becomes routine or even knowledge in a higher grade (or even later in a year). Questions which are indicated as being problem solving cease to be problem solving once a learner has been taught how to solve that kind of problem. So teaching the techniques involved in the solution of the most demanding questions in the previous year s examination paper is unlikely to prepare candidates for the higher order questions that will be asked that year, but there is no better preparation for becoming a problem solver than being given plenty of challenging questions to tackle.. The order of topics is not prescriptive but it is recommended that care be taken to ensure that in the first two terms, some of the topics 1 to 6 as well as some of the topics 7 to 10 are taught so that assessment is balanced. Page 16 of 75

17 Algebraic expressions Exponents Number patterns Equations and inequalities Trigonometry Functions Trigonometric functions Euclidean Geometry EXAMS Analytical geometry Finance, growth and decay Statistics Trigonometry Euclidean geometry Measurement Probability Revision EXAMS ANNUAL TEACHING PLAN: SUMMARY Grade 10 Grade 11 Grade 1 No. of weeks No. of weeks Algebraic expressions Equations and inequalities Number patterns Analytical geometry Functions Trigonometry (reduction formulae, graphs, equations) EXAMS Measurement Euclidean Geometry Trigonometry (sine, area, cosine rules) Probability Finance, growth and decay Statistics Revision EXAMS Patterns, sequences and series Functions and inverse functions Exponential and logarithmic functions Finance, growth and decay Trigonometry compound angles Trigonometry D and D Polynomial functions Differential calculus Analytical geometry TESTS /EXAMS Geometry Statistics (regression and correlation) Counting and Probability Revision TESTS/EXAMS Revision EXAMS No. of weeks The detail which follows includes examples and numerical references to the Overview Page 17 of 75

18 MATHEMATICS: GRADE 10 PACE SETTER 1 TERM 1 Weeks WEEK 1 WEEK WEEK WEEK 4 WEEK 5 WEEK 6 WEEK 7 WEEK 8 WEEK 9 WEEK 10 WEEK 11 Topics Algebraic expressions Exponents Number patterns Equations and inequalities Trigonometry Assesment Test Investigation or project Test Page 18 of 75

19 Date completed TERM Weeks WEEK 1 WEEK WEEK WEEK 4 WEEK 5 WEEK 6 WEEK 7 WEEK 8 WEEK 9 WEEK 10 Topics Functions Trigonometric Euclidean geometry MID-YEAR EXAMINATION functions Assesment Assignment / Test Date completed TERM Weeks WEEK 1 WEEK WEEK WEEK 4 WEEK 5 WEEK 6 WEEK7 WEEK 8 WEEK9 WEEK 10 Topics Analytical geometry Finance, growth and decay Statistics Trigonometry Euclidean geometry Assesment Test Test Date completed 4 TERM 4 Paper 1 = hours Paper = hours Measure ment Weeks WEEK 1 WEEK WEEK WEEK 4 WEEK 5 WEEK 6 WEEK 7 WEEK 8 WEEK 9 WEEK 10 Algebraic expressions and equations (and 0 Euclidean geometry Analytical geometry 0 15 Topics Probability Revision Admin inequalities) Trigonometry and 50 Assesment Test Examinations exponents measurement Number Patterns Functions and graphs Finance, growth and decay Probability Statistics 15 Date completed Total marks 100 Total marks 100 Page 19 of 75

20 MATHEMATICS: GRADE 11 PACE SETTER 1 TERM 1 Weeks WEEK 1 WEEK WEEK WEEK 4 WEEK 5 WEEK 6 WEEK 7 WEEK 8 WEEK WEEK 10 WEEK 11 Page 0 of 75

21 9 Topics Algebraic expressions Equations and inequalities Number patterns Analytical geometry Assesment Investigation or project Test Date completed TERM Weeks WEEK 1 WEEK WEEK WEEK 4 WEEK 5 WEEK 6 WEEK 7 WEEK 8 WEEK 9 WEEK 10 Topics Functions Trigonometry (reduction formulae, graphs, equations) MID-YEAR EXAMINATION Assesment Assignment / Test Test Date completed TERM Weeks WEEK 1 WEEK WEEK WEEK 4 WEEK 5 WEEK 6 WEEK7 WEEK 8 WEEK9 WEEK 10 Topics Measurement Euclidean geometry Trigonometry (sine, cosine and area rules) Finance, growth and decay Probability Assesment Test Test Date completed 4 TERM 4 Paper 1 = hours Paper = hours Weeks WEEK 1 WEEK WEEK WEEK 4 WEEK 5 WEEK 6 WEEK 7 WEEK 8 WEEK 9 WEEK 10 Topics Statistics Revision FINAL EXAMINATION Admin Assesment Date completed Test Algebraic expressions and equations (and inequalities) Number Patterns Functions and graphs Finance, growth and decay Probability Euclidean geometry Analytical geometry Trigonometry and measurement Statistics 5 Total marks 150 Total marks Page 1 of 75

22 MATHEMATICS: GRADE 1 PACE SETTER 1 TERM 1 Weeks WEEK 1 WEEK WEEK WEEK 4 WEEK 5 WEEK 6 WEEK 7 WEEK 8 WEEK 9 WEEK 10 WEEK 11 Page of 75

23 Topics Number Patterns, Sequences And Series Functions: Formal definition; inverses Functions: exponential and logarithmic Finance, growth and decay Trigonometry: Assesment Test Investigation or project Assignment Date completed TERM Weeks WEEK 1 WEEK WEEK WEEK 4 WEEK 5 WEEK 6 WEEK 7 WEEK 8 WEEK 9 WEEK 10 Topics Trigonometry Functions: polynomials Differential calculus Analytical geometry MID-YEAR EXAMINATION / Test Assesment Assignment Date completed TERM Weeks WEEK 1 WEEK WEEK WEEK 4 WEEK 5 WEEK 6 WEEK7 WEEK 8 WEEK 9 WEEK 10 Topics Geometry Statistics Counting and Probability Revision TRIAL EXAMINATION Assesment Test Date completed 4 TERM 4 Paper 1 = hours Paper = hours WEE K 1 WEEK WEEK WEEK 4 WEEK 5 WEEK 6 WEEK 7 WEEK 8 WEEK 9 WEEK 10 Algebraic expressions and equations (and 5 Euclidean geometry 40 Revision FINAL EXAMINATION Admin inequalities) Analytical 40 Number Patterns Functions and graphs Finance, growth and decay Differential Calculus Counting and Probability Geometry Trigonometry & measurement Statistics 50 0 Total marks 150 Total marks 150 Page of 75

24 No of Weeks Topic Curriculum statement GRADE 10: TERM 1 Clarification Where an example is given, the cognitive demand is suggested: knowledge (K), routine procedure (R), complex procedure (C) or problem solving (P) 1. Understand that real numbers can be rational or irrational. Know the difference as far as the decimal expansions of the numbers are concerned. To recognise a real number as either rational or irrational the learner may use either the definition (A rational number is a number which can be written in the form a b where a, b Z and b 0) or the termination or recurring nature of the decimal expansion of the number.. Show that simple surds are irrational and establish between which two integers a given simple surd lies. A common misconception is that and is therefore rational. 7 (It is known that is an irrational number). Examples to illustrate the different cognitive levels involved in factorisation: Algebraic Expressions. Round decimal numbers to an appropriate degree of accuracy. 4. Multiplication of a binomial by a trinomial. 5. Factorisation to include types taught in grade 9 and: trinomials grouping in pairs difference between two cubes 6. Simplification of 1. Factorise fully: 1.1 m m 1 Since learners must recognize the simplest perfect squares : K 1. x x Since this type is routine and appears in all texts: R y 1y 18 Since one is required to work with fractions and identify when the expression has been fully factorised : C 1.4 a b a ( b) ( a ( b))( a a( b) ( b) ) ( a b)( a ab b ) Since one has to realize that a negative number raised to an odd power is negative: C Page 4 of 75

25 algebraic fractions using factorisation. 1. Simplify 1 x x 4 1 4x 1 x x 1 1 x (C) 1.5 Exponents 7. Addition and subtraction of algebraic fractions with denominators of degree at most. 1. Revise laws of exponents learnt in Grade 9 where x y 0; m, n, Z: x x x x x x m n m n m n m n m n mn x x m m x y xy m Also, by definition: n 1 x and n x 0 x. Use the laws of exponents to simplify expressions and solve equations, accepting that the rules also hold for n m, Q. 1. Show that n nis even for all nz and n n is divisible by 6 for all nz. It is not obvious (unless one has seen these questions before) that factorising the general form is the key to showing that the first expression always has an even factor and the second always has an even factor and has a factor divisible by. (P) Examples: Solve for x 1. x 0,15 Since 0,15 should be known to be : K x A simple two step procedure is involved: R. x 0 (correct to decimal places by trial and improvement) This requires conceptual understanding: identifying first the two integers between which the variable lies, then refining successive approximations: C By the end of the year this will probably have become routine. x x 8 1 Assuming this type of question has not been taught, spotting that the numerator can be factorised as a difference of squares requires insight: P The equation can also be solved by multiplying both sides by the denominator and then factorizing the resulting equation as a quadratic. Page 5 of 75

26 1 Numbers and patterns Equations and Inequalities Patterns: Investigate number patterns leading to those where there is a constant difference between consecutive terms, and the general term Is therefore linear. 1. Revise the solution of linear equations.. Solve quadratic equations (by factorisation).. Solve simultaneous linear equations in two unknowns. 4. Solve literal equations (Changing the subject of a formula ). 5. Solve linear inequalities (and show Examples: 1. Determine the 5 th and the n th terms of the number pattern 10 ;7 ;4 ;1;... Since there is an algorithmic approach to answering such questions: R. If the pattern MATHSMATHSMATHS is continued in this way, what will be the 67 th letter? Since it is not immediately obvious how one should proceed (unless similar questions have been tackled): P Examples: 1. Solve for x : x x x (R) 6. Solve for m : m m 1 (R) x y. Solve for x and y: xy 1 ; 1 (C) 4. Solve for r in terms of V, and h : V r h 5. Solve for x: 1 x 8 (C) x 6. Solve for x : 0, 15 (R) (R) solution graphically). Trigonometry 6. Solve exponential equations 1. Definitions of the trigonometric functions sin, cos and tan in right-angled triangles. Derive values of the trigonometric functions for the special cases { 0;0;45;60;90}.. Understand that the similarity of Comment: It is important to stress that 1. trigonometric ratios are independent of the lengths of the sides of a triangle and depend (uniquely) only on the angles, whence we consider them as functions of the angles;. doubling a ratio has a different effect from doubling an angle. For example, generally sin sin ; Page 6 of 75

27 triangles is fundamental to the trigonometric functions sin θ, cos θ and tan θ. Solve two dimensional problems involving right angled triangles. Example: Let ABCDbe a rectangle, with AB cm. Let E be on ADsuch that AB ˆE 45 and B E ˆC 75. Determine the area of the rectangle. (P) Comment: 4. Solve simple trigonometric equations for angles between 0 0 and A simple trigonometric equation is one that can be simplified in at most two steps to an equation of the form sin( ax b) c, or cos( ax b) c, or tan( ax b) c, where c is one of the standard trigonometric ratios. Example: 5. Extend the definitions of sin, cos 0 0 and tan to Plot the graphs of y sin, Solve for x: 4sin(x 10) 11. Examples: 1. Determine the length of the hypotenuse of a right triangle ABC, where ˆB 90, Â 0 and AB10 cm. (K) Sketch the graph of y sin x for x0 ;180 (C) y cos and y tan for [ 60;60]. Assessment Term 1: 1 Investigation or project (only one project in a year) (at least 50 marks) Example of an investigation: Imagine a cube of white wood which is dipped into red paint so that the surface is red, but the inside still white. If one cut is made, parallel to each face of the cube (and through the centre of the cube), then there will be 8 smaller cubes. Each of the Page 7 of 75

28 smaller cubes will have red faces and white faces. Investigate the number of smaller cubes which will have,, 1 and 0 red faces if //4/ /n equally spaced cuts are made parallel to each face. This task provides the opportunity to investigate, tabulate results, make conjectures and justify or prove them.. Test (at least 50 marks and 1 hour). Make sure all topics are tested. Two or three tests of at least 40 minutes would probably be better. Care needs to be taken to ask questions at all four cognitive levels: approximately 0% knowledge, approximately 45% routine procedures, 5% complex procedures and 10% problem solving. Page 8 of 75

29 GRADE 10: TERM Weeks Topic Curriculum statement Clarification 4 Functions 1. The concept of a function, where a certain quantity (output value) uniquely depends on another quantity (input value). Work with relationships between variables using tables, graphs, words and formulae. Convert flexibly between these representations.. Point by point plotting of basic graphs x of y x, y and y b ; b 0 x and b 1 to discover shape, domain (input values), range (output values), asymptotes, axes of symmetry, turning points and intercepts on the axes (where applicable). Notice that the graph of y xshould be known from Grade Investigate the effect of a and q on the graphs of y a. f x q, Comments: 1. A more formal definition of a function follows in Grade 1. At this level it is enough to investigate the way (unique) output values depend on how input values vary. The terms independent (input) and dependent (output) variables might be useful.. After summaries have been compiled about basic features of prescribed graphs and the effects of parameters a and q have been investigated: a : a vertical stretch (and/or a reflection about the x axis) and q a vertical shift, the following examples might be appropriate: a x. Sketched below are graphs of y b and y p. q k. x The horizontal asymptote of both graphs is the line y 1..1 Determine the values of a, b, p, q and k. (C) y O. Calculate the average gradient of each of the graphs sketched between x1 and x (R) (; 0) (1; -1) x Notice that average gradient is the gradient of the chord of a curve between two points (not the average of a number of gradients in the specified interval) 4. Remember that graphs in some practical applications may be either Page 9 of 75

30 where f x x, f x x, f x and x x f x b b b. 1, 0, 1 discrete or continuous. E.g. Two men do a job in 9 days. Draw a graph to illustrate the number of men required to do the job in a different number of days. Would it make sense for this graph to be continuous? Why or why not? (C ) 4. Study the effect of a and q on the graphs of: y asin q y acos q y a tan q for [ 60;60]. Euclidean Geometry 5. Sketch graphs, find the equations of given graphs, calculate average gradient, and interpret graphs. 1. Revise basic results established in earlier grades regarding lines, angles and polygons, especially the similarity and congruence of triangles.. Define the following special quadrilaterals: the kite, parallelogram, rectangle, rhombus, square and trapezium. Investigate and make conjectures about the properties of the sides, angles, diagonals and areas of these Comments: 1. Triangles are similar if their angles coincide, or if the ratios of their sides coincide: Triangles ABCand DEF are similar if A ˆ D ˆ, Bˆ Eˆ and C ˆ F ˆ AB BC CA. They are also similar if. DE EF FD. Teachers should start with only one definition to define the special quadrilaterals. The additional properties of the quadrilateral should be investigated and proved. For example, we could define a parallelogram as a quadrilateral with two pairs of opposite sides parallel. Then we investigate and prove that the opposite sides of the parallelogram are equal in length.. It must be explained that a single counter example disproves a conjecture but that numerous specific examples supporting a Page 0 of 75

31 quadrilaterals. Prove these conjectures. conjecture do not constitute a general proof. Example: In quadrilateral KITE, KI = KE and IT = ET. The diagonals intersect at M. Prove that: 1. IM = ME and. KT is perpendicular to IE. C: Since it is not obvious that one must first prove KIT KET Page 1 of 75

32 Mid-year examinations Assessment term : 1. Assignment / test (at least 50 marks) The following are good examples of assignments: Open book test Translation task Error spotting and correction Shorter investigation Journal Mind-map Olympiad (first round) Tutorial on an entire topic Tutorial on more complex / problem solving questions. Mid-year examination (at least 15 marks) One paper of hours (100 marks) or Two papers - one 1 hour (50 marks) and the other 1 hour (50 marks) Page of 75

33 Weeks Topic Analytical Geometry Curriculum statement Represent geometric figures on a Cartesian co-ordinate system. Derive and apply for any two points ; x ; y the formulae for x y and 1 1 calculating the: 1. distance between the two points;. gradient of the line segment joining the two points (and hence identify parallel and perpendicular lines);. coordinates of the mid-point of the line segment joining the two points. GRADE 10: TERM Example: Clarification Consider the points P(;5) and Q(;1 ) in the Cartesian plane. 1.1 Calculate the distance PQ. (K) 1. Find the coordinates of R if M1;0 is the mid-point of PR. (R) 1. Determine the coordinates of S if PQRS is a parallelogram. (C) 1.4 Is PQRS a rectangle? Why or why not? (R) Finance, growth and decay Use the simple and compound growth formulae A P( 1 in) n and A P( 1 i) to solve problems, including interest, hire purchase, inflation, population growth and other real life problems. Example: How long will it take a population to double if it is increasing at a rate of 1% p.a.? (C) This is considered complex because the value of n must be found by trial and improvement from the compound growth formula. A sensible answer (rounded to the nearest year) should be expected: months and days would not be sensible. Comment: An understanding must be developed of the fact that the foreign exchange rate influences petrol price, imports, exports and overseas travel. Page of 75

34 Weeks Topic Curriculum statement Clarification Comment: In grade 10, the intervals of grouped data should given using inequalities, that is, in the form 0 x<0 rather than in the form 0-19, 0-9,.5 Statistics 1. Measures of central tendency in grouped data: calculation of mean estimate of grouped data and Identification of modal interval and interval in which the median lies.. Revision of range as a measure of dispersion and extension to include percentiles, quartiles, interquartile and semiinterquartile range.. Five number summary (maximum, minimum and quartiles) and box and whisker diagram. 4. Use the statistical summaries (measures of central tendency and dispersion), and graphs to analyse and make meaningful comments on the context associated with the given data. Example: 1. The mathematics marks of 00 grade 10 learners at a school can be summarised as follows: Percentage obtained Number of candidates 0 x<0 4 0 x< x< x< x< x< x< x< Calculate the approximate mean mark for the examination.. Identify the interval in which each of the following data items lies:.1 the median;. the lower quartile;. the upper quartile..4 the thirtieth percentile. (R) Page 4 of 75

35 Weeks Topic Curriculum statement Comment: Clarification Using properties of quads; esp. parallelograms and congruence Example: EFGH is a parallelogram. Prove that MFNH is a parallelogram G M F 1 Euclidean Geometry Solve problems and prove riders using the properties of quadrilaterals; parallel lines and triangles. H Construct a parallelogram. N E E (R) F Construct the angle bisectors of each angle in a parallelogram (The bisector of F has been drawn for you). Investigate the nature of H G the internal quadrilateral formed by the 4 angle bisectors. Make a conjecture. Justify or prove your conjecture. Extension: What about the angle bisectors of any quadrilateral? Page 5 of 75

36 Trigonometry Problems in two dimensions. Example: Two flagpoles are 0 m apart. The one has height 10 m, while the other has height 15 m. Two tight ropes connect the top of each pole to the foot of the other. At what height above the ground do the two ropes intersect? What if the poles were a different distance apart? (P) 1 Measurement 1. Revise the volume and surface areas of right-prisms and cylinders.. Study the effect on volume andsurface area when multiplying any dimension by a constant factor k.. Calculate the volume and surface areas of spheres, triangular and rectangular pyramids and cones. Example: The height of a cylinder is 10 cm, and the radius of the circular base is cm. A hemisphere is attached to one end of the cylinder and a cone of height cm to the other end. Calculate the volume and surface area of the solid, correct to the nearest cm and cm respectively. (R) Assessment term : Two () Tests (at least 50 marks and 1 hour) covering all topics in approximately the ratio of the allocated teaching time. Page 6 of 75

37 No of Weeks Topic Probability Curriculum statement 1. The use of probability models to compare the relative frequency of events with the theoretical probability.. The use of Venn diagrams to solve probability problems, deriving and applying the following for any two events A and B in a sample space S: P( A or B) P( A) P( B) P( A and B) A and B are mutually exclusive if P( A and B) 0 GRADE 10: TERM 4 Clarification Comment: It generally takes a very large number of trials before the relative frequency of a coin falling heads up when tossed approaches 0,5. Example: A study was done to test how effective three different drugs, A, B and C were in relieving headache pain. Over the period covered by the study, 80 patients were given the chance to use all three drugs. The following results were obtained: 40 reported relief from drug A 5 reported relief from drug B 40 reported relief from drug C 1 reported relief from both drugs A and C 18 reported relief from drugs B and C 68 reported relief from at least one of the drugs 7 reported relief from all three drugs. Revision A and B are complementary if they are mutually exclusive and P( A) P( B) 1. Then P( B) P( not ( A)) 1 P( A). 1. Record this information in a Venn diagram. (C). How many of the subjects got relief from none of the drugs? (K). How many subjects got relief from drugs A and B but not C? (R) 4. What is the probability that a randomly chosen subject got relief from at least two of the drugs? (R) Comment: The value of working through past papers cannot be over-emphasised. 4 Examinations Page 7 of 75