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Skills available for South Africa grade 12 maths curriculum

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National Curriculum Statement (CAPS) National Curriculum Statement (CAPS)

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12.1 Functions

12.1.a 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.
12.1.b The inverses of prescribed functions and be aware of 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.
12.1.c Problem solving and graph work involving the prescribed functions (including the logarithmic function).

12.2 Number Patterns, Sequences and Series

12.2.a Identify and solve problems involving number patterns that lead to arithmetic and geometric sequences and series, including infinite geometric series.

12.3 Finance, Growth and Decay

12.3.a Calculate the value of n in the formulae A = P(1 + i)ⁿ and A = P(1 - i)ⁿ
- Compound interest: word problems (12-F.17)
12.3.b Apply knowledge of geometric series to solve annuity and bond repayment problems.
12.3.a Critically analyse different loan options.

12.4 Algebra

12.4.a Demonstrate an understanding of the definition of a logarithm and any laws needed to solve real life problems.
12.4.b Take note and understand, the Remainder and Factor Theorems for polynomials up to the third degree.
12.4.c Factorise third-degree polynomials (including examples which require the Factor Theorem).
- Factorise by grouping (11-D.4)
- Factorise sums and differences of cubes (12-D.12)

12.5 Differential Calculus

12.5.a An intuitive understanding of the concept of a limit.
12.5.b Differentiation of specified functions from first principles.
12.5.c Use of the specified rules of differentiation.
12.5.d The equations of tangents to graphs.
- Find equations of tangent lines using limits (12-Y.7)
12.5.e The ability to sketch graphs of cubic functions.
12.5.f Practical problems involving optimization and rates of change (including the calculus of motion).
- Velocity as a rate of change (12-Y.4)

12.6 Probability

12.6.a Generalisation of the fundamental counting principle.
- Counting principle (11-X.3)
- Combinations and permutations (12-Q.3)
12.6.b Probability problems using the fundamental counting principle.
- Find probabilities using combinations and permutations (12-Q.4)

12.7 Euclidean Geometry and Measurement

12.7.a Revise earlier (Grade 9) work on the necessary and sufficient conditions for polygons to be similar.
12.7.b Prove (accepting results established in earlier grades):
- 12.7.b.1 that a line drawn parallel to one side of a triangle divides the other two sides proportionally (and the Mid-point Theorem as a special case of this theorem);
- 12.7.b.2 that equiangular triangles are similar;
- 12.7.b.3 that triangles with sides in proportion are similar;
- 12.7.b.4 the Pythagorean Theorem by similar triangles; and
- 12.7.b.5 riders.

12.8 Trigonometry

12.8.a Proof and use of the compound angle and double angle identities
12.8.b Solve problems in two and three dimensions.

12.9 Analytical Geometry

12.9.a Use a two-dimensional Cartesian co-ordinate system to derive and apply:
- 12.9.a.1 the equation of a circle (any centre); and
- 12.9.a.2 the equation of a tangent to a circle at a given point on the circle.

12.10 Statistics

12.10.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.
12.10.b Use a calculator to calculate the linear regression line which best fits a given set of bivariate numerical data.
12.10.c Use a calculator to calculate the correlation co-efficient of a set of bivariate numerical data and make relevant deductions.
- Match correlation coefficients to scatter plots (12-S.6)
- Calculate correlation coefficients (12-S.7)

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12.1 Functions

12.1.b The inverses of prescribed functions and be aware of 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.

12.1.c Problem solving and graph work involving the prescribed functions (including the logarithmic function).

12.2 Number Patterns, Sequences and Series

12.2.a Identify and solve problems involving number patterns that lead to arithmetic and geometric sequences and series, including infinite geometric series.

12.3 Finance, Growth and Decay

12.3.a Calculate the value of n in the formulae A = P(1 + i)n and A = P(1 - i)n

12.3.b Apply knowledge of geometric series to solve annuity and bond repayment problems.

12.3.a Critically analyse different loan options.

12.4 Algebra

12.4.a Demonstrate an understanding of the definition of a logarithm and any laws needed to solve real life problems.

12.4.b Take note and understand, the Remainder and Factor Theorems for polynomials up to the third degree.

12.4.c Factorise third-degree polynomials (including examples which require the Factor Theorem).

12.5 Differential Calculus

12.5.a An intuitive understanding of the concept of a limit.

12.5.b Differentiation of specified functions from first principles.

12.5.c Use of the specified rules of differentiation.

12.5.d The equations of tangents to graphs.

12.5.e The ability to sketch graphs of cubic functions.

12.5.f Practical problems involving optimization and rates of change (including the calculus of motion).

12.6 Probability

12.6.a Generalisation of the fundamental counting principle.

12.6.b Probability problems using the fundamental counting principle.

12.7 Euclidean Geometry and Measurement

12.7.a Revise earlier (Grade 9) work on the necessary and sufficient conditions for polygons to be similar.

12.7.b Prove (accepting results established in earlier grades):

12.7.b.1 that a line drawn parallel to one side of a triangle divides the other two sides proportionally (and the Mid-point Theorem as a special case of this theorem);

12.7.b.2 that equiangular triangles are similar;

12.7.b.3 that triangles with sides in proportion are similar;

12.7.b.4 the Pythagorean Theorem by similar triangles; and

12.7.b.5 riders.

12.8 Trigonometry

12.8.a Proof and use of the compound angle and double angle identities

12.8.b Solve problems in two and three dimensions.

12.9 Analytical Geometry

12.9.a Use a two-dimensional Cartesian co-ordinate system to derive and apply:

12.9.a.1 the equation of a circle (any centre); and

12.9.a.2 the equation of a tangent to a circle at a given point on the circle.

12.10 Statistics

12.10.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.

12.10.b Use a calculator to calculate the linear regression line which best fits a given set of bivariate numerical data.

12.10.c Use a calculator to calculate the correlation co-efficient of a set of bivariate numerical data and make relevant deductions.

12.3.a Calculate the value of n in the formulae A = P(1 + i)ⁿ and A = P(1 - i)ⁿ