Friday, 18 June 2021 06:26

## MATHEMATICS GRADE 12 - EXAMINATION GUIDELINES 2021

MATHEMATICS
EXAMINATION GUIDELINES
2021

 CONTENTS Page Chapter 1: Introduction 3 Chapter 2: Assessment in Grade 122.1 Format of question papers for Grade 12       2.2 Weighting of topics per paper for Grade 12 2.3 Weighting of cognitive levels 44    5 Chapter 3: Elaboration of Content for Grade 12 (CAPS) 6 Chapter 4: Acceptable reasons: Euclidean Geometry4.1 Acceptable Reasons: Euclidean Geometry (ENGLISH) 4.2 Aanvaarbare redes: Euklidiese Meetkunde (AFRIKAANS) 1 912 Chapter 5: Information sheet 15 Chapter 6: Guidelines for marking 16 Chapter 7: Conclusion 16

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1. INTRODUCTION
The Curriculum and Assessment Policy Statement (CAPS) for Mathematics outlines the nature and purpose of the subject Mathematics. This guides the philosophy underlying the teaching and assessment of the subject in Grade 12.
The purpose of these Examination Guidelines is to:
• Provide clarity on the depth and scope of the content to be assessed in the Grade 12 National Senior Certificate Examination in Mathematics
• Assist teachers to adequately prepare learners for the examinations
This document deals with the final Grade 12 external examinations. It does not deal in any depth with the school-based assessment (SBA), performance assessment tasks (PATs) or final external practical examinations as these are clarified in a separate PAT document which is updated annually.
These guidelines should be read in conjunction with:
• The National Curriculum Statement (NCS) Curriculum and Assessment Policy Statement (CAPS): Mathematics
• The National Protocol of Assessment: 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 (Grades R–12)
• National policy pertaining to the programme and promotion requirements of the National Curriculum Statement, Grades R to 12
Included in this document is a list of Euclidean Geometry reasons, both in English and Afrikaans, which should be used as a guideline when teaching learners Euclidean Geometry.
The Information Sheet for Paper 1 and 2 is included in this document.
All candidates will write two external papers as prescribed.
2.1 Format of Question Papers for Grade 12
 Paper Topics Duration Total Date Marking 1 Patterns and sequencesFinance, growth and decayFunctions and graphsAlgebra, equations and inequalitiesDifferential CalculusProbability 3 hours 150 October/November Externally 2 Euclidean GeometryAnalytical GeometryStatistics and regressionTrigonometry 3 hours 150 October/November Externally
Questions in both Papers 1 and 2 will assess performance at different cognitive levels with an emphasis on process skills, critical thinking, scientific reasoning and strategies to investigate and solve problems in a variety of contexts.
An Information Sheet is included on p. 15.
2.2 Weighting of Topics per Paper for Grade 12
 PAPER 1 MARKS PAPER 2 MARKS Algebra, Equations and InequalitiesNumber PatternsFunctions and GraphsFinance, Growth and DecayDifferential CalculusCounting Principle and Probability 252525252525 Statistics and RegressionAnalytical GeometryTrigonometryEuclidean Geometry 20405040 TOTAL 150 TOTAL 150
2.3 Weighting of Cognitive Levels
Papers 1 and 2 will include questions across four cognitive levels. The distribution of cognitive levels in the papers is given below.
 Cognitive Level Description of Skills to be Demonstrated Weighting Approximate Number of Marks in a 150-mark Paper Knowledge Recall Identification of correct formula on the information sheet (no changing of the subject) Use of mathematical facts Appropriate use of mathematical vocabulary Algorithms Estimation and appropriate rounding of numbers 20% 30 marks Routine Procedures Proofs of prescribed theorems and derivation of formulae 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 35% 52–53 marks Complex Procedures 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 Learners are expected to solve problems by integrating different topics. 30% 45 marks Problem Solving Non-routine problems (which are not necessarily difficult) Problems are mainly unfamiliar Higher-order reasoning and processes are involved Might require the ability to break the problem down into its constituent parts Interpreting and extrapolating from solutions obtained by solving problems based in unfamiliar contexts. 15% 22–23 marks
3. ELABORATION OF CONTENT/TOPICS
The purpose of the clarification of the topics is to give guidance to the teacher in terms of depth of content necessary for examination purposes. Integration of topics is encouraged as learners should understand Mathematics as a holistic discipline. Thus questions integrating various topics can be asked.
FUNCTIONS
• Candidates must be able to use and interpret functional notation. In the teaching process learners
must be able to understand how f (x) has been transformed to generate f (-x) , - f (x) ,
f (x + a) f (x) + a , af (x) and x = f (y) where a∈R.
• Trigonometric functions will ONLY be examined in PAPER 2.

NUMBER PATTERNS, SEQUENCES AND SERIES
• The sequence of first differences of a quadratic number pattern is linear. Therefore, knowledge of linear patterns can be tested in the context of quadratic number patterns.
• Recursive patterns will not be examined explicitly.
• Links must be clearly established between patterns done in earlier grades.

FINANCE, GROWTH AND DECAY
• Understand the difference between nominal and effective interest rates and convert fluently between them for the following compounding periods: monthly, quarterly and half-yearly or semi-annually.
• With the exception of calculating i in the Fv and Pv formulae, candidates are expected to calculate the value of any of the other variables.
• Pyramid schemes will NOT be examined in the examination.

ALGEBRA
• Solving quadratic equations by completing the square will NOT be examined.
• Solving quadratic equations using the substitution method (k-method) is examinable.
• Equations involving surds that lead to a quadratic equation are examinable.
• Solution of non-quadratic inequalities should be seen in the context of functions.
• Nature of the roots will be tested intuitively with the solution of quadratic equations and in all theprescribed functions.

DIFFERENTIAL CALCULUS
• The following notations for differentiation c an be used: f '(x) , x Dx ,dy or y '
dx
• In respect of cubic functions, candidates are expected to be able to:
• Determine the equation of a cubic function from a given graph.
• Discuss the nature of stationary points including local maximum, local minimum and points of inflection.
• Apply knowledge of transformations on a given function to obtain its image.
• Candidates are expected to be able to draw and interpret the graph of the derivative of a function.
• Surface area and volume will be examined in the context of optimisation.
• Candidates must know the formulae for the surface area and volume of the right prisms. These formulae will NOT be provided on the formula sheet
• If the optimisation question is based on the surface area and/or volume of the cone, sphere and/or pyramid, a list of the relevant formulae will be provided in that question. Candidates will be expected to select the correct formula from this list.

PROBABILITY
• Dependent events are examinable but conditional probabilities are not part of the syllabus.
• Dependent events in which an object is not replaced are examinable.
• Questions that require the learner to count the different number of ways that objects may be arranged in a circle and/or the use of combinations are not in the spirit of the curriculum.
• In respect of word arrangements, letters that are repeated in the word can be treated as the same (indistinguishable) or different (distinguishable). The question will be specific in this regard.

EUCLIDEAN GEOMETRY AND MEASUREMENT
• Measurement can be tested in the context of optimisation in calculus and two- and three-dimensional trigonometry.
• Composite shapes could be formed by combining a maximum of TWO of the stated shapes.
• The following proofs of theorems are examinable:
• The line drawn from the centre of a circle perpendicular to a chord bisects the chord;
• The line drawn from the centre of a circle that bisects a chord is perpendicular to the chord;
• The angle subtended by an arc at the centre of a circle is double the size of the angle subtended by the same arc at the circle (on the same side of the chord as the centre);
• The opposite angles of a cyclic quadrilateral are supplementary;
• The angle between the tangent to a circle and the chord drawn from the point of contact is equal to the angle in the alternate segment;
• A line drawn parallel to one side of a triangle divides the other two sides proportionally;
• Equiangular triangles are similar.
• Corollaries derived from the theorems and axioms are necessary in solving riders:
• Angles in a semi-circle
• Equal chords subtend equal angles at the circumference
• Equal chords subtend equal angles at the centre
• In equal circles, equal chords subtend equal angles at the circumference
• In equal circles, equal chords subtend equal angles at the centre.
• The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle of the quadrilateral.
• If the exterior angle of a quadrilateral is equal to the interior opposite angle of the quadrilateral, then the quadrilateral is cyclic.
• Tangents drawn from a common point outside the circle are equal in length.
• The theory of quadrilaterals will be integrated into questions in the examination.
• Concurrency theory is excluded.

TRIGONOMETRY
• The reciprocal ratios cosec θ, sec θ and cot θ can be used by candidates in the answering of problems but will not be explicitly tested.
• The focus of trigonometric graphs is on the relationships, simplification and determining points of intersection by solving equations, although characteristics of the graphs should not be excluded.

ANALYTICAL GEOMETRY
• Prove the properties of polygons by using analytical methods.
• The concept of collinearity must be understood.
• Candidates are expected to be able to integrate Euclidean Geometry axioms and theorems into Analytical Geometry problems.
• The length of a tangent from a point outside the circle should be calculated.
• Concepts involved with concurrency will not be examined.

STATISTICS
• Candidates should be encouraged to use the calculator to calculate standard deviation, variance and the equation of the least squares regression line.
• The interpretation of standard deviation in terms of normal distribution is not examinable.
• Candidates are expected to identify outliers intuitively in both the scatter plot as well as the box and whisker diagram. In the case of the box and whisker diagram, observations that lie outside the interval (lower quartile – 1,5 IQR; upper quartile + 1,5 IQR) are considered to be outliers. However, candidates will not be penalised if they did not make use of this formula in identifying outliers.
4. ACCEPTABLE REASONS: EUCLIDEAN GEOMETRY
In order to have some kind of uniformity, the use of the following shortened versions of the theorem statements is encouraged.
4.1 ACCEPTABLE REASONS: EUCLIDEAN GEOMETRY (ENGLISH) 