SA NCS Mathematics:Learning Outcomes, Assessment Standards, Content and Contexts

---

---

Learning Outcome 1: Number and Number Relationships

When solving problems, the learner is able to recognise, describe, represent and work confidently with numbers and their relationships to estimate, calculate and check solutions.

The learner will use the following content in order to calculate and estimate accurately in solving standard problems, as well as those that are nonroutine and unseen. The problems will be taken from mathematical and real-life contexts such as health and finance.

Grade 10

Assessments Standards

We know this when the learner is able to:

• Identify rational numbers and convert between terminating or recurring decimals and the form :

$\frac{a}{b}$; $a,b,\in Z$; $b\ne 0$

• • Simplify expressions using the laws of exponents for integral exponents.

• • Establish between which two integers any simple surd lies.

• • Round rational and irrational numbers to an appropriate degree of accuracy.

• Investigate number patterns (including but not limited to those where there is a constant difference between consecutive terms in a number pattern, and the general term is therefore linear) and hence:
  • make conjectures and generalisations;
  • provide explanations and justifications and attempt to prove conjectures.

• Use simple and compound growth formulae ($A=P(1+ni)$ and $A=P(1+i{)}^n$ ) to solve problems, including interest, hire-purchase, inflation, population growth and other reallife problems.

• Demonstrate an understanding of the implications of fluctuating foreign exchange rates (e.g. on the petrol price, imports, exports, overseas travel).

• Solve non-routine, unseen problems.

Proposed content

• Conversion of terminating and recurring decimals to the form: a b ; a, b,  ; b 0.
• The laws of exponents for integral exponents.
• Rational approximation of surds.
• Number patterns, including those where there is a constant difference between consecutive terms indicating that the general term is linear.
• Simple and compound growth formulae $A=P(1+ni)$ and $A=P(1+i)n$; solving for any variable except in the compound growth formula.
• Foreign exchange rates.

Grade 11

Assessments Standards

We know this when the learner is able to:

• Understand that not all numbers are real. (This requires the recognition but not the study of non-real numbers.)
  • Simplify expressions using the laws of exponents for rational exponents.
  • Add, subtract, multiply and divide simple surds (e.g. $\sqrt{3}+\sqrt{12}=3\sqrt{3}$ and $\frac{\sqrt{2}}{2}=\frac{1}{\sqrt{2}}$ )
  • Demonstrate an understanding of error margins.

• Investigate number patterns (including but not limited to those where there is a constant second difference between consecutive terms in a number pattern, and the general term is therefore quadratic) and hence:
  • make conjectures and generalisations;
  • provide explanations and justifications and attempt to prove conjectures.

• Use simple and compound decay formulae ($A=P(1-ni)$ and $A=P(1-i{)}^n$) to solve problems (including straight line depreciation and depreciation on a reducing balance) (link to Learning Outcome 2).

• Demonstrate an understanding of different periods of compounding growth and decay (including effective compounding growth and decay and including effective and nominal interest rates.
• Solve non-routine, unseen problems.

Proposed content

• Recognition of non-real numbers.
• Use of the laws of exponents for rational exponents.
• Add, subtract, multiply and divide simple surds.
• Error margins.
• Number patterns, including those where there is a constant second difference between consecutive terms indicating that the general term is quadratic
• Simple and compound decay formulae $A=P(1-ni)$ and $A=P(1-i{)}^n$. Calculation of all variables in $A=P(1-i{)}^n$ (for n by trial and error using a calculator).
• Different periods of compounding growth and decay.

Grade 12

Assessments Standards

• Demonstrate an understanding of the definition of a logarithm and any laws needed to solve real-life problems (e.g. growth and decay see
  • Identify and solve problems involving number patterns, including but not limited to arithmetic and geometric sequences and series.
  • Correctly interpret sigma notation.
  • Prove and correctly select the formula for and calculate the sum of series, including:
    • $\sum _{i=1}^N=n$
    • $\sum _{i=1}^{n}i=\frac{n(n+1)}{2}$
    • $\sum _{i=1}^{n}a+(i-1)d=\frac{n}{2}[2a(n-1)d]$
    • $\sum _{i=1}^{n}a.r^{i-1}=\frac{a(r^n-1)}{r-1};r\ne 1$
    • $\sum _{i=1}^{\mathrm{\infty }}a.r^{i-1}=\frac{a}{1-r}$ for$-1<r<1$

• • Correctly interpret recursive formulae: (e.g. $T_{n+1}=T_n+T_{n-1}$ )

• • Calculate the value of $n$ in the formula $A=P(1\pm i{)}^n$
  • Apply knowledge of geometric series to solving annuity, bond repayment and sinking fund problems, with or without the use of the formulae:
    • $F=\frac{x[(1+i{)}^n-1]}{i}$

and

• • • $F=\frac{x[(1+i{)}^n]-1}{i}$

• Critically analyse investment and loan options and make informed decisions as to the best option(s) (including pyramid and micro-lenders schemes).
• Solve non-routine, unseen problems.

Proposed content

• Definition of a logarithm and any laws needed to solve real-life problems (e.g. growth and decay).
• The calculation of n using the growth and decay formulae.
• Number patterns, including arithmetic and geometric sequences and series.
• Sigma notation.
• Proof and application of the formulae for the sum of series, including
  • $\sum _{i=1}^N=n$
  • $\sum _{i=1}^{n}i=\frac{n(n+1)}{2}$
  • $\sum _{i=1}^{n}a+(i-1)d=\frac{n}{2}[2a(n-1)d]$
  • $\sum _{i=1}^{n}a.r^{i-1}=\frac{a(r^n-1)}{r-1};r\ne 1$
  • $\sum _{i=1}^{\mathrm{\infty }}a.r^{i-1}=\frac{a}{1-r}$ for$-1<r<1$
• Recursive formulae (e.g.$T_{n+1}=T_n+T_{n-1}$ )
• Annuities, bond repayments and sinking funds, with or without the use of the formulae:
  • $F=\frac{x[(1+i{)}^n-1]}{i}$

and

• • $F=\frac{x[(1+i{)}^n]-1}{i}$
• Loan options.

Learning Outcome 2: Functions and Algebra

The learner is able to investigate, analyse, describe and represent a wide range of functions and solve related problems.

The approach to the content of this Learning Outcome should ensure that learning occurs through investigating the properties of functions and applying their characteristics to a variety of problems. Functions and algebra are integral to modelling and so to solving contextual problems. Problems which integrate content across Learning Outcomes and which are of a non-routine nature should also be used. Human rights, health and other issues which involve debates on attitudes and values should be involved in dealing with models of relevant contexts.

Grade 10

Assessments Standards

We know this when the learner is able to:

• Demonstrate the ability to work with various types of functions, including those listed in the following Assessment Standard.
• Recognise relationships between variables in terms of numerical, graphical, verbal and symbolic representations and convert flexibly between these representations (tables, graphs, words and formulae).

• 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 to generalise the effects of the parameters a and q on the graphs of functions including:
  • $y=ax+q$
  • $y=a{x}^2+q$
  • $y=\frac{a}{x}+q$
  • $y=a{b}^x+q;b>0$
  • $y=asin(x)+q$
  • $y=acos(x)+q$
  • $y=atan(x)+q$

• Identify characteristics as listed below and hence use applicable characteristics to sketch graphs of functions including those listed above:
  • domain and range;
  • intercepts with the axes;
  • turning points, minima and maxima;
  • asymptotes;
  • shape and symmetry;
  • periodicity and amplitude;
  • average gradient (average rate of change);
  • intervals on which the function increases/decreases;
  • the discrete or continuous nature of the graph.

• Manipulate algebraic expressions:
  • multiplying a binomial by a trinomial;
  • factorising trinomials;
  • factorising by grouping in pairs;
  • simplifying algebraic fractions with monomial denominators.

• Solve:
  • linear equations;
  • quadratic equations by factorisation;
  • exponential equations of the form kax+p = m (including examples solved by trial and error);
  • linear inequalities in one variable and illustrate the solution graphically;
  • linear equations in two variables simultaneously (numerically, algebraically and graphically ).

• Use mathematical models to investigate problems that arise in real-life contexts:
  • making conjectures, demonstrating and explaining their validity;
  • expressing and justifying mathematical generalisations of situations;
  • using the various representations to interpolate and extrapolate;
  • describing a situation by interpreting graphs, or drawing graphs from a description of a situation, with special focus on trends and features. (Examples should include issues related to health, social, economic, cultural, political and environmental matters.)

• Investigate the average rate of change of a function between two values of the independent variable, demonstrating an intuitive understanding of average rate of change over different intervals (e.g. investigate water consumption by calculating the average rate of change over different time intervals and compare results with the graph of the relationship).

Proposed content

• Study of functions including
  • $y=ax+q$
  • $y=a{x}^2+q$
  • $y=\frac{a}{x}+q$
  • $y=a{b}^x+q;b>0$
  • $y=asin(x)+q$
  • $y=acos(x)+q$
  • $y=atan(x)+q$
• Conversion between numerical, graphical, verbal and symbolic representations.
• Investigation of the effects of the parameters a and q on the above functions.
• Sketch graphs of the above functions using the following characteristics:
  • domain and range;
  • intercepts with the axes;
  • turning points, minima and maxima;
  • asymptotes;
  • shape and symmetry;
  • periodicity and amplitude;
  • average gradient (average rate of change);
  • intervals on which the function increases/decreases;
  • the discrede or continuous nature of the graph;
  • the discrete or continuous nature of the graph.

• Algebraic manipulation:
  • multiplying a binomial by a trinomial;
  • factorising trinomials;
  • factorising by grouping in pairs;
  • simplifying algebraic fractions with monomial denominators.

• Solution of:
  • linear equations;
  • quadratic equations by factorisation;
  • exponential equations of the form kax+p = m (including examples solved by trial and error);
  • linear inequalities in one variable and graphical illustration of the solution;
  • linear equations in two variables simultaneously (numerically, algebraically and graphically).
  • the discrete or continous nature of graph.

• Average rate of change of a function between two values of the independent variable.

Grade 11

Assessments Standards

We know this when the learner is able to:

• Demonstrate the ability to work with various types of functions including those listed in the following Assessment Standard.
• Recognise relationships between variables in terms of numerical, graphical, verbal and symbolic representations and convert flexibly between these representations (tables, graphs, words and formulae).

• Generate as many graphs as necessary, initially by means of point-by-point plotting, supported by available technology, to make and test conjectures about the effect of the parameters k, p, a and q for functions including:
  • $y=sin(kx)$
  • $y=cos(kx)$
  • $y=tan(kx);$
  • $y=sin(x+p)$
  • $y=cos(x+p)$
  • $y=tan(x+p)$
  • $y=a(x+p{)}^2+q$
  • $y=\frac{a}{x+p}+q$
  • $y=a{b}^{x+p}+q;b>0$

• Identify characteristics as listed below and hence use applicable characteristics to sketch graphs of functions including those listed above:
  • domain and range;
  • intercepts with the axes;
  • turning points, minima and maxima;
  • asymptotes;
  • shape and symmetry;
  • periodicity and amplitude;
  • average gradient (average rate of change);
  • intervals on which the function increases/decreases;
  • the discrete or continuous nature of the graph.

• Manipulate algebraic expressions:
  • by completing the square;
  • simplifying algebraic fractions with binomial denominators.

• Solve:
  • quadratic equations (by factorisation, by completing the square, and by using the quadratic formula) and quadratic inequalities in one variable and interpret the solution graphically;
  • equations in two unknowns, one of which is linear and one of which is quadratic, algebraically or graphically.

• Use mathematical models to investigate problems that arise in real-life contexts:
  • making conjectures, demonstrating and explaining their validity;
  • expressing and justifying mathematical generalisations of situations; using various representations to interpolate and extrapolate;
  • describing a situation by interpreting graphs, or drawing graphs from a description of a situation, with special focus on trends and pertinent features. (Examples should include issues related to health, social, economic, cultural, political and environmental matters.)

• 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.

• Solve linear programming problems by optimising a function in two variables, subject to one or more linear constraints, by numerical search along the boundary of the feasible region.
• Solve a system of linear equations to find the co-ordinates of the vertices of the feasible region.

Proposed content

• Study of functions including:
  • $y=sin(kx)$
  • $y=cos(kx)$
  • $y=tan(kx);$
  • $y=sin(x+p)$
  • $y=cos(x+p)$
  • $y=tan(x+p)$
  • $y=a(x+p{)}^2+q$
  • $y=\frac{a}{x+p}+q$
  • $y=a{b}^{x+p}+q;b>0$

• Conversion between numerical,using graph and verbal and symbolic representations.

• Investigation of the effects of the parameters k, p, a, and q on the above functions.

• Sketch graphs of the above functions using the following characteristics:
  • domain and range;
  • intercepts with the axes;
  • turning points, minima and maxima;
  • asymptotes;
  • shape and symmetry;
  • periodicity and amplitude;
  • average gradient (average rate of change);
  • intervals on which the function increases/decreases;
  • the discrete or continuous nature of the graph.

• Algebraic manipulation:
  • completing the square;
  • simplifying algebraic fractions with binomial denominators.

• Solution of:
  • quadratic equations (by factorisation, by completing the square, and by using the quadratic formula);
  • quadratic inequalities in one variable and graphical interpretation of the solution;
  • equations in two unknowns, one of which is linear and one which is quadratic, algebraically or graphically.

• Average gradient between two points on a curve and the gradient of a curve at a point.

• Linear programming:
  • optimising a function in two variables subject to one or more linear constraints, by numerical search along the boundary of the feasible region;
  • solving a system of linear equations to find the co-ordinates of the vertices of the feasible region.

Grade 12

Assessments Standards

We know this when the learner is able to:

• Demonstrate the ability to work with various types of functions and relations including the inverses listed in the following Assessment Standard.
• Demonstrate knowledge of the formal definition of a function.

• Investigate and generate graphs of the inverse relations of functions, in particular the inverses of:
  • $y=ax+q$
  • $y=a{x}^2$
  • $y=a^x;a>0$
• Determine which inverses are functions and how the domain of the original function needs to be restricted so that the inverse is also a function.

• Identify characteristics as listed below and hence use applicable characteristics to sketch graphs of the inverses of the functions listed above:
  • domain and range;
  • intercepts with the axes;
  • turning points, minima and maxima;
  • asymptotes;
  • shape and symmetry;
  • average gradient (average rate of change);
  • intervals on which the function increases/decreases.

• Factorise third degree polynomials (including examples which require the factor theorem).

• Investigate and use instantaneous rate of change of a variable when interpreting models of situations:
  • demonstrating an intuitive understanding of the limit concept in the context of approximating the rate of change or gradient of a function at a point;
  • establishing the derivatives of the following functions from first principles: ***$f(x)=b$
    • $f(x)=x$
    • $f(x)=x^2$
    • $f(x)=x^3$
    • $f(x)=\frac{1}{x}$

and then generalise to the derivative of

• • • $f(x)=x^n$

Use the following rules of differentiation:

• • $\frac{d}{dx}[f(x)\pm g(x)]=\frac{d}{dx}[f(x)]\pm \frac{d}{dx}[g(x)]$
  • $\frac{d}{dx}[k.f(x)]=k\frac{d}{dx}[f(x)]$

• Determine the equations of tangents to graphs.

• Generate sketch graphs of cubic functions using differentiation to determine the stationary points (maxima, minima and points of inflection) and the factor theorem and other techniques to determine the intercepts with the x-axis.

• Solve practical problems involving optimisation and rates of change.

• Solve linear programming problems by optimising a function in two variables, subject to one or more linear constraints, by establishing optima by means of a search line and further comparing the gradients of the objective function and linear constraint boundary lines.

Proposed content

• Study of functions:
  • formal definition of a function;
  • the inverses of:
  • $y=ax+q$
  • $y=a{x}^2$
  • $y=a^x;a>0$

• Sketch graphs of the inverses of the functions above using the characteristics:
  • domain and range;
  • intercepts with the axes;
  • turning points, minima and maxima;
  • asymptotes;
  • shape and symmetry;
  • average gradient (average rate of change);
  • intervals on which the function increases/decreases.

• Factorise third-degree polynomials (including examples which require the factor theorem).

• Differential calculus:
  • an intuitive understanding of the limit concept in the context of approximating the rate of change or gradient of a function at a point;
  • the derivatives of the following functions from first principles:
    • $f(x)=b$
    • $f(x)=x$
    • $f(x)=x^2$
    • $f(x)=x^3$
    • $f(x)=\frac{1}{x}$

• • the derivative of $f(x)=x^n$ (proof not required);
  • the following rules of differentiation:
    • $\frac{d}{dx}[f(x)\pm g(x)]=\frac{d}{dx}[f(x)]\pm \frac{d}{dx}[g(x)]$
    • $\frac{d}{dx}[k.f(x)]=k\frac{d}{dx}[f(x)]$
  • the equations of tangents to graphs;
  • sketch graphs of cubic and other suitable polynomial functions using differentiation to determine the stationary points (maxima, minima and points of inflection) and the factor theorem and other techniques to determine the intercepts with the x-axis;
  • practical problems involving optimisation and rates of change.

• Linear programming:
  • optimisation of a function in two variables, subject to one or more linear constraints, by means of a search line and further comparing the gradients of the objective and constraint functions.

Learning Outcome 3: Space, Shape and Measurement

The learner is able to describe, represent, analyse and explain properties of shapes in 2-dimensional and 3-dimensional space with justification.

An important aspect of this Learning Outcome is the use of the content indicated in the representation of contextual problems in two and three dimensions so as to arrive at solutions through the measurement and calculation of associated values. Powerful mathematical tools which enable the investigation of space are embedded in the content. The treatment of formal Euclidean geometry is staged through the grades so as to assist in the gradual development of proof skills and an understanding of local axiomatic systems. Opportunities for making connections across the geometries involved in this Learning Outcome as well as with the Mathematics of other Learning Outcomes should be sought in requiring the solution to standard as well as non-routine unseen problems.

Grade 10

Assessments Standards

We know this when the learner is able to:

• Understand and determine the effect on the volume and surface area of right prisms and cylinders, of multiplying any dimension by a constant factor k.

• Through investigations, produce conjectures and generalisations related to triangles, quadrilaterals and other polygons, and attempt to validate, justify, explain or prove them, using any logical method (Euclidean, co-ordinate and/or transformation).

• Disprove false conjectures by producing counter-examples.

• Investigate alternative definitions of various polygons (including the isosceles, equilateral and right-angled triangle, the kite, parallelogram, rectangle, rhombus and square).

• Represent geometric figures on a Cartesian co-ordinate system, and derive and apply, for any two points $(x_1;y_1)$ and $(x_2;y_2)$, 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.

• Investigate, generalise and apply the effect of the following transformations of the point (x ; y):
  • a translation of p units horizontally and q units vertically;
  • a reflection in the x-axis, the y-axis or the line $y=x$.

• Understand that the similarity of triangles is fundamental to the trigonometric functions $sin\theta$ ,$cos\theta$ and $tan\theta$, and is able to define and use the functions.

• Solve problems in two dimensions by using the trigEonometric functions ($sin\theta$ ,$cos\theta$ and $tan\theta$) in right-angled triangles and by constructing and interpreting geometric and trigonometric models (examples to include scale drawings, maps and building plans).

• Demonstrate an appreciation of the contributions to the history of the development and use of geometry and trigonometry by various cultures through a project.

Proposed content

• The effect on the volume and surface area of right prisms and cylinders, of multiplying one or more dimensions by a constant factor k.
• Investigation of polygons, using any logical method (Euclidean, co-ordinate and/or transformation).
• Alternative definitions of polygons.
• Co-ordinate geometry: for any two points ($x_1$ ; $y_1$) and ($x_2$ ; $y_2$), derive and use the 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.

• Transformation geometry:
  • translation of p units horizontally and q units vertically;
  • reflection in the x-axis, the y-axis and the line y = x.

• Trigonometry:
  • introduction through the similarity of triangles and proportion;
  • scale drawing and the interpretation of maps and building plans;
  • definition and use of the definitions $sin\theta$ ,$cos\theta$ and $tan\theta$ ;
  • the periodicity of trigonometric functions.

• Research into the history of the development of geometry and trigonometry in various cultures.

Grade 11

Assessments Standards

We know this when the learner is able to:

• Use the formulae for surface area and volume of right pyramids, right cones, spheres and combinations of these geometric objects.

• Investigate necessary and sufficient conditions for polygons to be similar.

• Prove and use (accepting results established in earlier grades):
  • that a line drawn parallel to one side of a triangle divides the other two sides proportionally (the 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.

• 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 point and parallel or perpendicular to a given line;
  • the inclination of a line.

• Investigate, generalise and apply the effect on the co-ordinates of:
  • the point (x ; y) after rotation around the origin through an angle of $90^{\circ }$ or $180^{\circ }$  ;
  • the vertices ($x_1$ ; $y_1$), ($x_2$ ; $y_2$), . . ., ($x_n$ ; $y_n<math>)ofapolygonafterenlargementthroughtheorigin,byaconstantfactor<math>k$.

• Derive and use the values of the trigonometric functions (in surd form where applicable) of $30^{\circ }$, $45^{\circ }$ and $60^{\circ }$.
• Derive and use the following identities:
  • $tan\theta =\frac{sin\theta }{cos\theta }$
  • $si{n}^2\theta +co{s}^2\theta =1$
• Derive the reduction formulae for $sin(90^{\circ }\pm \theta ),cos(90^{\circ }\pm \theta ),sin(180^{\circ }\pm \theta ),cos(180^{\circ }\pm \theta ),tan(180^{\circ }\pm \theta ),sin(360^{\circ }\pm \theta ),cos(360^{\circ }\pm \theta ),tan(360^{\circ }\pm \theta ),sin(-\theta ),cos(-\theta )$ and $tan(-\theta )$.
• Determine the general solution of trigonometric equations.
• Establish and apply the sine, cosine and area rules.

• Solve problems in two dimensions by using the sine, cosine and area rules; and by constructing and interpreting geometric and trigonometric models.

• Demonstrate an appreciation of the contributions to the history of the development and use of geometry and trigonometry by various cultures through educative forms of assessment (e.g. an investigative project).

Proposed content

• Apply the formulae for the surface area and volume of right prisms, right cones, spheres and combinations of these shapes.

• Euclidean geometry:
  • necessary and sufficient conditions for polygons to be similar;
  • the 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);
  • equiangular triangles are similar;
  • triangles with sides in proportion are similar;
  • Theorem of Pythagoras by similar triangles.

• Co-ordinate geometry:
  • the equation of a line through two points;
  • the equation of a line through one point and parallel or perpendicular to a given line;
  • the inclination of a given line.

• Transformation geometry:
  • rotation around the origin through an angle of $90^{\circ }$ or $180^{\circ }$ ;
  • the enlargement of a polygon, through the origin, by a factor of k.

• Trigonometry:
  • function values of the special angles $30^{\circ }$ , $45^{\circ }$ and $60^{\circ }$ (in surd form where applicable);
  • derivation and use of the identities $tan\theta =\frac{sin\theta }{cos\theta }$

and $si{n}^2\theta +co{s}^2\theta =1$;

• • derivation and use of reduction formulae for $sin(90^{\circ }\pm \theta ),cos(90^{\circ }\pm \theta ),sin(180^{\circ }\pm \theta ),cos(180^{\circ }\pm \theta ),tan(180^{\circ }\pm \theta ),sin(360^{\circ }\pm \theta ),cos(360^{\circ }\pm \theta ),tan(360^{\circ }\pm \theta ),sin(-\theta ),cos(-\theta )$ and $tan(-\theta )$.;
  • the general solution of trigonometric equations;
  • proof and application to problems in two dimensions, of the sine, cosine and area rules.

• Research into the history of the development of geometry and trigonometry in various cultures.

Grade 12

Assessments Standards

We know this when the learner is able to:

• Accept the following as axioms:
  • results established in earlier grades;
  • a tangent is perpendicular to the radius, drawn at the point of contact with the circle, and then investigate and prove the theorems of the geometry of circles:
  • the line drawn from the centre of a circle, perpendicular to a chord, bisects the chord and its converse;
  • the perpendicular bisector of a chord passes through the centre of the circle;
  • 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;
  • angles subtended by a chord at the circle on the same side of the chord are equal and its converse;
  • the opposite angles of a cyclic quadrilateral are supplementary and its converse;
  • two tangents drawn to a circle from the same point outside the circle are equal in length;
  • the angles between a tangent and a chord, drawn to the point of contact of the chord, are equal to the angles which the chord subtends in the alternate chord segments and its converse.

• Use the theorems listed above to:
  • make and prove or disprove conjectures;
  • prove riders.

• 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 given a point on the circle.

• Use the compound angle identities to generalise the effect on the co-ordinates of the point (x ; y) after rotation about the origin through an angle .
• Demonstrate the knowledge that rigid transformations (translations, reflections, rotations and glide reflections) preserve shape and size, and that enlargement preserves shape but not size.

• Derive and use the following compound angle identities:
  • $sin(\alpha \pm \beta )=sin\alpha cos\beta \pm cos\alpha sin\beta$
  • $cos(\alpha \pm \beta )=cos\alpha cos\beta \mp sin\alpha sin\beta$
  • $sin2\alpha =2sin\alpha cos\alpha$
  • $cos2\alpha =co{s}^2-si{n}^2\alpha$
  • $=2co{s}^2\alpha -1$
  • $=1-2si{n}^2\alpha$

• Solve problems in two and three dimensions by constructing and interpreting geometric and trigonometric models.

• Demonstrate a basic understanding of the development and uses of geometry through history and some familiarity with other geometries (e.g. spherical geometry, taxi-cab geometry, and fractals).

Proposed content

• Euclidean geometry: accepting as axioms all results established in earlier grades and the fact that the tangent to a circle is perpendicular to the radius, drawn to the point of contact, prove the following theorems:
  • the line drawn from the centre of a circle perpendicular to a chord bisects the chord and its converse;
  • the perpendicular bisector of a chord passes through the centre of the circle;
  • 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;
  • angles subtended by a chord at the circle on the same side of the chord are equal and its converse;
  • the opposite angles of a cyclic quadrilateral are supplementary and its converse;
  • two tangents drawn to a circle from the same point outside the circle are equal in length;
  • the tangent-chord theorem and its converse.

• Co-ordinate geometry:
  • the equation of a circle (any centre);
  • the equation of a tangent to a circle given a point on the circle.

• Transformation geometry:
  • the compound angle formula in generalising the effect on the co-ordinates of the point (x ; y) after rotation about the origin through an angle  ;
  • rigid transformations (translations, reflections, rotations and glide reflections) and enlargement.

• Trigonometry:
  • compound angle identities:
    • $sin(\alpha \pm \beta )=sin\alpha cos\beta \pm cos\alpha sin\beta$
    • $cos(\alpha \pm \beta )=cos\alpha cos\beta \mp sin\alpha sin\beta$
    • $sin2\alpha =2sin\alpha cos\alpha$
    • $cos2\alpha =co{s}^2-si{n}^2\alpha$
    • $=2co{s}^2\alpha -1$
    • $=1-2si{n}^2\alpha$
  • Problems in two and three dimensions.
• Research into history and one or more other geometries such as:
  • spherical geometry;
  • taxi-cab geometry;
  • fractals.

Learning Outcome 4: Data Handling and Probability

The learner is able to collect, organise, analyse and interpret data to establish statistical and probability models to solve related problems.

The content indicated below for this Learning Outcome really only becomes meaningful and alive when used to address issues of importance to the learner and to society. Activities should involve learners in the completion of the statistical cycle of formulating questions, collecting appropriate data, analysing and representing this data, and so arriving at conclusions about the questions raised.

Grade 10

Assessments Standards

We know this when the learner is able to:

• Collect, organise and interpret univariate numerical data in order to determine:
  • measures of central tendency (mean, median, mode) of grouped and ungrouped data, and know which is the most appropriate under given conditions;
  • measures of dispersion: range, percentiles, quartiles, interquartile and semi-inter-quartile range.

• Represent data effectively, choosing appropriately from:
  • bar and compound bar graphs;
  • histograms (grouped data);
  • frequency polygons;
  • pie charts;
  • line and broken line graphs.

• Use probability models for comparing the relative frequency of an outcome with the probability of an outcome (understanding, for example, that it takes a very large number of trials before the relative frequency of throwing a head approaches the probability of throwing a head).

• Use Venn diagrams as an aid to solving probability problems, appreciating and correctly identifying:
  • the sample space of a random experiment;
  • an event of the random experiment as a subset of the sample space;
  • the union and intersection of two or more subsets of the sample space;
  • $P(S)=1$ (where S is the sample space);
  • $P(AorB)=P+P-P(AandB)$ (where A and B are events within a sample space);
  • disjoint (mutually exclusive) events, and is therefore able to calculate the probability of either of the events occurring by applying the addition rule for disjoint events: $P(AorB)=P+P$;
  • complementary events, and is therefore able to calculate the probability of an event not occurring: $P(notA)=1-P.$

• Identify potential sources of bias, errors in measurement, and potential uses and misuses of statistics and charts and their effects (a critical analysis of misleading graphs and claims made by persons or groups trying to influence the public is implied here).
• Effectively communicate conclusions and predictions that can be made from the analysis of data.

• Use theory learned in this grade in an authentic integrated form of assessment (e.g. in an investigative project).

Proposed content

• Data handling (calculations):
  • measures of central tendency (mean, median, mode) of grouped and ungrouped data;
  • measures of dispersion: range, percentiles, quartiles, interquartile and semi-interquartile range;
  • errors in measurement;
  • sources of bias.

• Data handling (representation):
  • bar and compound bar graphs;
  • histograms (grouped data);
  • frequency polygons;
  • pie charts;
  • line and broken line graphs.

• Probability:
  • definition in terms of equally likely outcomes;
  • relative frequency after many trials approximating the probability;
  • Venn diagrams as an aid to solving probability problems:
    • the sample space of a random experiment (S),
    • an event of the random experiment as a subset of the sample space,
    • the union and intersection of two or more subsets of the sample space,
    • $P(S)=1$ (where S is the sample space),
    • $P(AorB)=P+P$ P(A and B);
  • (where Aand B are events within a sample space),
    • disjoint (mutually exclusive) events: $P(AorB)=P+P,$
    • complementary events: $P(notA)=1P$;
  • potential uses and misuses of statistics and charts.

Grade 11

Assessments Standards

We know this when the learner is able to:

• Calculate and represent measures of central tendency and dispersion in univariate numerical data by:
  • five number summary (maximum, minimum and quartiles);
  • box and whisker diagrams;
  • 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 using histograms and frequency polygons.

• Represent bivariate numerical data as a scatter plot and suggest intuitively whether a linear, quadratic or exponential function would best fit the data (problems should include issues related to health, social, economic, cultural, political and environmental issues).

• Correctly identify dependent and independent events (e.g. from two-way contingency tables or Venn diagrams) and therefore appreciate when it is appropriate to calculate the probability of two independent events occurring by applying the product rule for independent events: $P(AandB)=P.P$.

• Use tree and Venn diagrams to solve probability problems (where events are not necessarily independent).

• Identify potential sources of bias, errors in measurement, and potential uses and misuses of statistics and charts and their effects (a critical analysis of misleading graphs and claims made by persons or groups trying to influence the public is implied here).
• Effectively communicate conclusions and predictions that can be made from the analysis of data.

• Differentiate between symmetric and skewed data and make relevant deductions.

• Use theory learned in this grade in an authentic integrated form of assessment (e.g. in an investigative project).

Proposed content

• Calculations and data representation:
  • measures of central tendency and dispersion in univariate numerical data by:
    • five number summary (maximum, minimum and quartiles),
    • box and whisker diagrams,
    • ogives,
    • calculating the variance and standard deviation sets of data manually (for small sets of data) and using available technology (for larger sets of data), and representing results graphically using histograms and frequency polygons;
  • scatter plot of bivariate data and intuitive choice of function of best fit supported by available technology;
  • symmetric and skewed data.

• Probability:
  • dependent and independent events;
  • two-way contingency tables;
  • the product rule for independent events: $P(AandB)=P.P$;
  • Venn diagrams and other techniques to solve probability problems (where events are not necessarily independent).

Grade 12

Assessments Standards

We know this when the learner is able to:

• Demonstrate the ability to draw a suitable sample from a population and understand the importance of sample size in predicting the mean and standard deviation of a population.

• Use available technology to calculate the regression function which best fits a given set of bivariate numerical data.

• Use available technology to calculate the correlation co-efficient of a set of bivariate numerical data to make relevant deductions.

• Generalise the fundamental counting principle (successive choices from $m_1$ then $m_2$ then $m_3$ ... options create $m_1.m_2.m_3$ ... different combined options) and solve problems using the fundamental counting principle.

• Identify potential sources of bias, errors in measurement, and potential uses and misuses of statistics and charts and their effects (a critical analysis of misleading graphs and claims made by persons or groups trying to influence the public is implied here).

• Effectively communicate conclusions and predictions that can be made from the analysis of data.

• Identify data which is normally distributed about a mean by investigating appropriate histograms and frequency polygons.

• Use theory learned in this grade in an authentic integrated form of assessment (e.g. in an investigative project).

Proposed content

• Content from previous grades as used in statistical investigations.
• Sampling.
• An inituitive understanding of the least squares method for linear regression.
• Regression functions and correlation for bivariate data by the use of available technology.
• Identification of normal distributions of data.
• Probability problems using the fundamental counting principle.

About the Proposed Content sections

In this section content and contexts are provided to support the attainment of the Assessment Standards. The content needs to be dealt with in such a way as to assist the learner to progress towards the achievement of the Learning Outcomes. Content must serve the Learning Outcomes but not be an end in itself. The contexts suggested will enable the content to be embedded in situations which are meaningful to the learner and so assist learning and teaching. The teacher should be aware of and use local contexts, not necessarily indicated here, which could be more suited to the experiences of the learner. Content and context, when aligned to the attainment of the Assessment Standards, provide a framework for the development of Learning Programmes. The Learning Programme Guidelines give more detail in this respect.

Contexts: inclusivity, human rights and indigenous knowledge systems

Mathematics is often referred to as the `queen of sciences and yet the servant of all' due to the evident power that it has in concisely formulating the theoretical aspects of the sciences and in providing tools for solving problems. This power extends beyond the natural sciences to the engineering, computing, actuarial, financial, economic, business, social and other sciences. Mathematics is, however, a human endeavour. Through the continuing inventiveness of the human mind, new aspects of Mathematics have been created and recreated through social interaction over the centuries of human existence.

The mastery of Mathematics depends to a large extent on mathematical processes such as investigating patterns, formulating conjectures, arguing for the generality of such conjectures, and formulating links across the domains of Mathematics to enable lateral thinking. Mathematics is a cognitive science. It requires understanding before competence in the Learning Outcomes can be achieved.

Mathematics is thus a key subject in providing access to a wide variety of learning. This curriculum focuses on the development of mathematical process skills, and in so doing endeavours to unlock the power of Mathematics. At the same time, the tools to enable this power to become effective are not neglected. Mathematics will become a `pump' and not a `filter' for the learner.

Mathematics has often been used as a filter to block access to further or additional learning, not only in Mathematics itself but also in areas and careers related or even unrelated to Mathematics. The past political history of our country is a prime example of how the deliberate lack of provision of quality learning for all in Mathematics was used to stunt the development of the majority of our people. Being literate in Mathematics is an essential requirement for the development of the responsible citizen, the contributing worker and the self- managing person. Being mathematically literate implies an awareness of the manner in which Mathematics is used to format society. It enables astuteness in the user of the products of Mathematics such as hire-purchase agreements and mathematical arguments in the media, hence the inclusion of Mathematical Literacy as a fundamental requirement in the Further Education and Training curriculum. The development of literacy in Mathematics, in the sense outlined here, is also a fundamental responsibility of the Mathematics teacher and other educators. The requirements of the Assessment Standards in Mathematics ensure this.

Many local and international studies have shown the existence of a set of attitudes, described as `mathsphobia', in school-going learners and in the population at large. In implementing this curriculum, it is the responsibility of the teacher to endeavour to win learners to Mathematics. This will be ensured by complying with the Assessment Standards of the subject, not formalising into the abstract prematurely but first taking care to develop understanding and process skills. The teacher needs to be sensitive to the manner in which gendered attitudes towards Mathematics play themselves out in the classroom, particularly in co-educational schools. Stereotyping needs to be guarded against, as Mathematics is often seen to be a male preserve, leading to arrogance and domination by the boys in the class. The interests of all need to be taken into account in providing access to Mathematics.

Another aspect of providing access and affirmation for learners of Mathematics is to look at examples of Mathematics in the variety of cultures and societal practices in our country. Mathematics is embedded in many cultural artefacts which we experience in our daily lives: the murals of the Ndebele, the rhythm in the drums of the Venda, the beadwork of the Zulu and Vedic art, to name but a few. Architecture, games and music are rich fields to explore through the lens of Mathematics. Ethnomathematics provides a wealth of more recently developed materials, sensitive to the sacredness of culture, for use in the classroom. The flexibility allowed by the curriculum also promotes the incorporation of local practices as starting points for applications or investigations. Ethnomathematics also stresses that Mathematics originated in cultures other than the Greek, and that it continued to be developed in sophistication by many societies other than the European. Projects in the history of Mathematics can be used to explore this.