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MATHEMATICS CURRICULUM ASSESSMENT POLICY STATEMENT ORIENTATION WORKSHOP FOR TEACHERS GR EDWARDS 22 OCTOBER 2011

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ADMINISTRATION Attendance register Claim forms S & T Claim Form Z43

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GENERIC TRAINING

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INTRODUCTION The Curriculum and Assessment Policy Statement (CAPS) is a revision of the National Curriculum Statement (NCS). In developing the CAPS, a key aim has been to have just one document providing guidelines for planning, content and assessment for each subject. The CAPS also continue to support the key principles that underline the NCS, including: social transformation; high knowledge and high skills; integration and applied competence; progression; articulation and portability; human rights, inclusivity, environmental and social justice; valuing of indigenous knowledge systems (IKS) and credibility, quality and efficiency.

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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. Axiomatic is one approach to establishing mathematical truth. Mathematics is distinctly human activity practiced by all cultures. Mathematical problem solving enables us to understand the world around us, and most of all, to teach us to think creatively.

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PRINCIPLES APPLY ACROSS ALL GRADES: No calculators with programmable functions, graphical or symbolic facilities should be allowed. Calculators should only be used to perform standard numeric computations and to verify calculations by hand. Mathematical modelling is an important focal point of the curriculum. Real life problems should be incorporated into all sections whenever possible. 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. Learners need to reflect on the process and not be concerned only with getting the answer(s). Appropriate approximation and rounding skills should be taught 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. Teaching should not be limited to how but should also 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.

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PRINCIPLES APPLY ACROSS ALL GRADES: Mixed ability teaching requires teachers to challenge the most able learners and at the same time provide remedial support for those who are not competent yet. Teachers need to design questions to rectify misconceptions that are exposed by tests and examinations. 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.

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GENERIC TRAINING Policy documents Critical outcomes Time allocation Focus of Content Areas Weighting of Content Areas Allocation of Teaching Time Programme of Assessment

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Policy Documents National Curriculum and Assessment Policy Statement (CAPS) for Mathematics.CAPS National Policy pertaining to the Programme and Promotion requirements of the National Curriculum Statements Grades R – 12(NPR).National Policy pertaining to the Programme and Promotion requirements National Protocol for Assessment Grades R – 12(NPA).National Protocol for Assessment Grades R – 12(NPA).

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Critical Outcomes The National Curriculum Statement Grades R - 12 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;

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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.

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Time Allocation 4½ hours per week. Six(6) periods of 45 minutes each. Make sure you get the time on your roster.

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Focus of Content Areas The Main Topics in the FET Mathematics Curriculum 1. Functions 2. Number Patterns, Sequences, Series 3. Finance, growth and decay 4. Algebra 5. Differential Calculus

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The Main Topics in the FET Mathematics Curriculum 6. Probability 7. Euclidean Geometry and Measurement 8. Analytical Geometry 9. Trigonometry 10. Statistics

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The Main Topics in the FET Mathematics Curriculum OUT : Linear Programming Transformations IN : Probability Euclidean Geometry

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Weighting of Content Areas CAPS p. 12

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Content clarification with teaching guidelines Allocation of Teaching Time Sequencing and Pacing of Topics Grade 10 Grade 11 Grade 12

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Daily Lesson Plan Example Voorbeeld

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Programme of Assessment

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WHAT IS ASSESSMENT Assessment is a planned process of identifying (selecting learner response items) gathering (learner responses) interpreting (marking learner responses) information about the knowledge and skills demonstrated by learners.

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ASSESSMENT INFORMAL no need to be recorded by teacher, could be marked by learner or peer, usually used to develop skills, demonstrate knowledge and skills and for learners to practice, not used for promotion purposes(solely developmental purpose) FORMAL marked and recorded by teacher, 7 tasks in Programme of Assessment, used for promotion purposes(mainly promotion purpose)

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ASSESSMENT Why? What? Who? How? When? Where?

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WHY? PURPOSE Developmental: to assist learner to learn (i.e. apply knowledge, etc) Promotion: to make a summative judgement of the learners ability (e.g. use the marks for any of the tasks in the programme of assessment)

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WHO Teacher (formal assessment task e.g. project, controlled test) Learner (informal assessment task e.g. homework, classwork) Peer (informal assessment task e.g. homework) SMT (e.g. controlled/standardised test) External (e.g. controlled test set by district, etc)

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WHAT What content, skills, values E.g. ability to collect data, make observations, one-step problem-solving, multi-step problem solving, interpreting and drawing graphs, recalling laws and definitions, converting units, etc Use taxonomy

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HOW FORMAL TASK: project, investigation, assignment, controlled test, June examination, Final examination INFORMAL TASK: homework, classwork, class test

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When Last 15 minutes of a period (e.g. classwork) First 5 minutes of a lesson (e.g. oral Q & A) End of term, week, year (e.g. control test, june exam, class test, final exam, project, etc) End of unit of work, concept, section (e.g. class test, homework) Weekly (e.g. assignment) Daily (homework)

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Where Classroom Home

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GRADE 10: FORMAL ASSESSMENT GradesFormal School-based assessmentsEnd-of –Year examinations R-3100%n/a 4-675%25% 7-940%60% 10 & 1125% including a midyear examination 75% 1225% including midyear and trial examinations External examination: 75%

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FORMAL ASSESSMENT: GRADE 10 Test 10%

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Cognitive levels(p.55) Knowledge20% Routine Procedures35% Complex Procedures30% Problem Solving15%

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Knowledge (20%) Straight recall. Identification of correct formula on the information sheet(no changing of the subject). Use of mathematical facts. Appropriate use of mathematical vocabulary.

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Routine Procedures (35%) Estimation and appropriate rounding of numbers. Proofs of prescribed theorems and derivation of formulae. Identification and direct use of correct formula on the information sheet(no changing of the subjects). Perform well known procedures. Simple applications and calculations which might involve few steps. Derivation from given information may be involved. Identification and use (after changing the subject) of correct formula. Generally similar to those encountered in class.

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Complex Procedures (30%) Problems involve complex calculations and/or higher order 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.

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Problem Solving (15%) Non-routine problems (which are not necessarily difficult). Higher order reasoning and processes are involved. Might require the ability to break the problem down into its constituent parts.

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Number of Assessment Tasks and Weighting Term 1Test Project/Assignment Term 2Assignment/Test Examination Term 3Test 10 Term 4Test10

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In general Although the project/investigation is indicated in the first term, it could be scheduled in term 2. Only ONE project/investigation should be set per year. Tests should be at least ONE hour long and count at least 50 marks. Project or investigation must contribute 25% of term 1 marks while the test marks contribute 75% of the term 1 marks. The combination (25% and 75%) of the marks must appear in the learners report. None graphic and none programmable calculators are allowed (for example, to factorise a² - b² = (a-b)(a+b), or to find roots of equations) will not be allowed. Calculators should only be used to perform standard numerical computations and to verify calculations by hand. Formula sheet must not be provided for tests and for final examinations in Grades 10 and 11.

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Mark distribution for Mathematics NCS end-of-year papers: Grades PAPER 1Paper 2 DescriptionGrade 10 Statistics15 ± 3 Algebra and equations (and inequalities) 30 ± 3Analytical Geometry15 ± 3 Patterns and sequences15 ± 3Trigonometry and measurement50 ± 3 Finance, growth and decay10 ± 3Euclidean Geometry20 ± 3 Functions and graphs30 ± 3 Probability15 ± 3 TOTAL100TOTAL100 No calculators with programmable functions, facilities or symbolic facilities (for example, to factorise, or to find roots of equations) will be allowed. Calculators should only be used to perform standard numerical computations and to verify calculations by hand.

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Bookwork Grade 12 o Paper 1 – maximum 6 marks o Paper 2 – Theorems and/or trigonometric proofs – maximum 12 marks Grade 11 o Paper 2 - Theorems and/or trigonometric proofs – maximum 12 marks Grade 10 o No bookwork

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Annual Assessment Plan Annual Assessment Plan should be given to learners at beginning of year. Include dates, task, maximum mark

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Exam and Test templates End of year exam Tests

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The 7 Formal Assessment Tasks 5 tests 1 project / investigation June examination Final examination Or 4 tests 1 assignment 1 project / investigation June examination Final examination

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Moderation Internal and External Internal HoD (or subject head) if HoD did not specialise in subject All Programme of Assessment (PoA) tasks: HoD moderates before task is given to learners HoD moderates 10% of learner responses to Programme of Assessment tasks External Moderation by Subject advisors

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Moderation: What to look for? Duration, length Weighting of content Weighting of cognitive levels Language – mathematical vs layman Use of pictures, diagrams, graphs to facilitate language Level of difficulty

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CONTENT TRAINING

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IN GENERAL Each content area has been broken down into topics. The sequencing of topics within terms gives an idea of how content areas can be spread and re-visited throughout the year. 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 order of topics is not prescriptive, but ensure that in the first two terms, more than six topics are covered/taught so that assessment is balanced between paper 1 and 2.

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CONTENT TRAININGCONTENT TRAINING(p.22) Term 1: Algebraic expressions Exponents Numbers and Patterns Equations and Inequalities Trigonometry

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ALGEBRAIC EXPRESSIONS (3 WEEKS) Real numbers Surds Appropriate rounding Multiply a binomial by trinomial Factorising o Trinomials o Grouping in pairs o Sum and difference of two cubes Use factorising to simplify algebraic fractions Add and subtract algebraic fractions

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Different cognitive levels in factorisation

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EXPONENTS (2 WEEKS) Revise the laws of exponents Use the laws of exponents to simplify expressions and solve equations

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Examples

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NUMBERS AND PATTERNS (1 WEEK) Arithmetic sequence is done in Grade 12, hence is not used in Gr. 10 Consider the linear number pattern 3; 5; 7; 9;....

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Examples

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EQUATIONS AND INEQUALITIES (2 WEEKS) Solve linear equations Solve quadratic equations Solve simultaneous equations Solve literal equations Solve linear inequalities

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EXAMPLES

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TRIGONOMETRY (3 WEEKS) Special angles and reciprocal ratios θ ε { 0º; 30º; 45º; 60º; 90º} without using a calculator cosec θ; sec θ ; cot θ (examined only in gr. 10) Solving right angles triangles Solving simple trigonometric equations (angles between 0º and 90º ) Defining and plotting trigonometric functions

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EXAMPLES Similarity of triangles is fundamental to the trigonometric ratios sinθ; cosθ and tanθ;

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Triangles for special angles

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CONTENT TRAINING Term 2: Functions Activity Euclidean Geometry Activity (Revision)Activity

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FUNCTIONS (4 WEEKS) The concept of a function Plot basic graphs defined by: Investigate the effect of a and q on the graphs defined by Investigate the effect of a and q on the trigonometric graphs Sketch, use and interpret the graphs

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Functions Activity

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Euclidean Geometry(3 weeks) Activity

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Euclidean Geometry(3 weeks) Similar and congruent triangles Detailed revision of basic geometry Revision of triangles and parallel lines Concepts of similarity and congruence Investigate special polygons Properties of quadrilaterals Making conjectures and then proving them

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CONTENT TRAINING Term 3: Analytical Geometry Finance and growth Statistics Euclidean Geometry Trigonometry Measurement

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Analytical geometry(2 weeks) The distance between two points The gradient of a line segment The midpoint of a line segment

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Examples

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Finance and Growth(2 weeks) Simple and compound growth Foreign exchange rates

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STATISTICS (2.5 WEEKS) Measures of central tendency grouped and ungrouped data Measure of dispersion Five number summaries and box and whiskers diagram Analysing and interpreting statistical summaries

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Euclidean Geometry(1 week) Solve problems and prove riders using the properties of parallel lines, triangles and quadrilaterals.

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Trigonometry(2 weeks) Problems in two dimensions

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Example At the base of an electricity pylon, two poles and the cross wire are positioned as in the diagram: (a)What angle does each wire make with the ground? (b)What is the length of each wire? (c)At what height above the ground do the two wires intersect? Platinum Mathematics Gr. 10,Maskew Miller Longman, J Campbell S McPetrie

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Example(cont.)

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Measurement(1 week) Revise the volume and surface area of right-prisms and cylinders Study the effect on volume and surface area when multiplying any dimension by a constant factor k Calculate the volume and surface area of sphere, right pyramids and right cones (In case of pyramids, bases must either be equilateral triangle or a square. Problem types must include composite figures)

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VOLUME AND SURFACE AREA

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VOLUME AND SURFACE

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CONTENT TRAINING Term 4: Probability

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Probability(2 weeks) Use of probability models. Use of Venn diagrams to solve probability problems.

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THATS IT TOWARDS GREATER PERFORMANCE IN MATHEMATICS

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