Skills available for South Africa grade 12 maths curriculum
Objectives are in bold, followed by a list of the IXL maths skills that are aligned to that objective. Students can practise these skills online at www.ixl.com.
Objectives: National Curriculum Statement (CAPS)
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.
Domain and range (12-A.1)
Identify functions (12-A.2)
Find the gradient of a linear function (12-A.3)
Graph a linear function (12-A.4)
Write the equation of a linear function (12-A.5)
Linear functions over unit intervals (12-A.6)
Evaluate functions (12-A.7)
Find values using function graphs (12-A.8)
Complete a table for a function graph (12-A.9)
Characteristics of quadratic functions (12-C.1)
Find the maximum or minimum value of a quadratic function (12-C.2)
Graph a quadratic function (12-C.3)
Match quadratic functions and graphs (12-C.4)
Match polynomials and graphs (12-D.11)
Rational functions: asymptotes and excluded values (12-E.1)
Domain and range of exponential and logarithmic functions (12-F.1)
Identify linear and exponential functions (12-F.13)
Exponential functions over unit intervals (12-F.14)
Describe linear and exponential growth and decay (12-F.15)
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.
Composition of linear functions: find an equation (11-J.5)
Composition of linear and quadratic functions: find a value (11-J.6)
Add, subtract, multiply and divide functions (12-A.10)
Composition of functions (12-A.11)
Identify inverse functions (12-A.12)
Find values of inverse functions from tables (12-A.13)
Find values of inverse functions from graphs (12-A.14)
Find inverse functions and relations (12-A.15)
Check whether two rational functions are inverses (12-E.3)
12.1.c Problem solving and graph work involving the prescribed functions (including the logarithmic function).
Linear functions over unit intervals (12-A.6)
Exponential functions over unit intervals (12-F.14)
Describe linear and exponential growth and decay (12-F.15)
Exponential growth and decay: word problems (12-F.16)
Compound interest: word problems (12-F.17)
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.
Find terms of a sequence (12-P.1)
Find terms of a recursive sequence (12-P.2)
Identify a sequence as explicit or recursive (12-P.3)
Find a recursive formula (12-P.4)
Find recursive and explicit formulas (12-P.5)
Convert a recursive formula to an explicit formula (12-P.6)
Convert an explicit formula to a recursive formula (12-P.7)
Convert between explicit and recursive formulas (12-P.8)
Introduction to sigma notation (12-P.9)
Identify arithmetic and geometric series (12-P.10)
Find the sum of a finite arithmetic or geometric series (12-P.11)
Introduction to partial sums (12-P.12)
Partial sums of arithmetic series (12-P.13)
Partial sums of geometric series (12-P.14)
Partial sums: mixed review (12-P.15)
Convergent and divergent geometric series (12-P.16)
Find the value of an infinite geometric series (12-P.17)
Write a repeating decimal as a fraction (12-P.18)
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.
Convert between exponential and logarithmic form (12-F.2)
Evaluate logarithms (12-F.3)
Change of base formula (12-F.4)
Product property of logarithms (12-F.5)
Quotient property of logarithms (12-F.6)
Power property of logarithms (12-F.7)
Evaluate logarithms using properties (12-F.8)
Solve exponential equations using logarithms (12-F.10)
Solve logarithmic equations with one logarithm (12-F.11)
Solve logarithmic equations with multiple logarithms (12-F.12)
12.4.b Take note and understand, the Remainder and Factor Theorems for polynomials up to the third degree.
Evaluate polynomials using synthetic division (12-D.3)
Write a polynomial from its roots (12-D.4)
Find the roots of factorised polynomials (12-D.5)
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.
Find limits using graphs (12-T.1)
Find one-sided limits using graphs (12-T.2)
Determine if a limit exists (12-T.3)
Find limits at vertical asymptotes using graphs (12-V.1)
Determine end behaviour using graphs (12-V.2)
Determine end behaviour of polynomial and rational functions (12-V.3)
Find the limit at a vertical asymptote of a rational function I (12-W.1)
Find the limit at a vertical asymptote of a rational function II (12-W.2)
12.5.b Differentiation of specified functions from first principles.
Average rate of change I (12-Y.1)
Average rate of change II (12-Y.2)
Find instantaneous rates of change (12-Y.3)
Velocity as a rate of change (12-Y.4)
Find values of derivatives using limits (12-Y.5)
Find the gradient of a tangent line using limits (12-Y.6)
Find equations of tangent lines using limits (12-Y.7)
12.5.c Use of the specified rules of differentiation.
Sum and difference rules (12-Z.1)
Power rule I (12-Z.2)
Power rule II (12-Z.3)
Find derivatives of polynomials I (12-Z.4)
Find derivatives of polynomials II (12-Z.5)
Find second derivatives of polynomials (12-Z.6)
Find derivatives of rational functions (12-Z.7)
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.
Similarity ratios (12-K.1)
Similarity statements (12-K.2)
Side lengths and angle measures in similar figures (12-K.3)
Similarity rules for triangles (12-K.4)
Similar triangles and similarity transformations (12-K.5)
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.
Find trigonometric ratios using right triangles (12-H.5)
Find trigonometric ratios using the unit circle (12-H.6)
Trigonometric ratios: find a side length (12-H.10)
Trigonometric ratios: find an angle measure (12-H.11)
Solve a right triangle (12-H.12)
Law of Sines (12-H.13)
Law of Cosines (12-H.14)
Solve a triangle (12-H.15)
Area of a triangle: sine formula (12-H.16)
Area of a triangle: Heron's formula (12-H.17)
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
Find properties of circles (12-L.4)
Write equations of circles in standard form (12-L.5)
Graph circles (12-L.6)
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.
Find the equation of a regression line (12-S.8)
Interpret regression lines (12-S.9)
Analyse a regression line of a data set (12-S.10)
Analyse a regression line using statistics of a data set (12-S.11)
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)