The IGCSE

The University of Cambridge IGCSE is the examination system that we follow at St. Anne’s. It provides a curriculum and methods of assessment appropriate for a wide ability range. It is designed as a two-year curriculum programme leading to a certificate which is internationally recognised as equivalent in standard to the British GCSE and international GCE ‘O’ level examinations. It is administrated by the University of Cambridge Local Examinations Syndicate (UCLES). It has an 8 point scale of grades: A*, A, B, C, D, E, F and G. Grade A* and A is awarded for the highest level of achievement and grade G indicates minimum satisfactory performance. To take account of differing level of abilities there is a choice between Core and Extended curriculum papers. This allows teachers to decide on the most appropriate level of papers for their students. The Core curriculum in each subject is within the ability range of a large majority of students. it provides a full overview of the subject and is targeted at students expected to achieve grades D to G. The Extended curriculum, which comprises the Core curriculum and the Supplement, has been designed for the more academically able and leads naturally into higher education or professional training. It is targeted at those expected to achieve grades A to C.

Year 10 Key Objectives 

Quadratic Problems

Revision of factorising quadratics, solving quadratic equations by factorising and using formula. Problems involving quadratics.

Algebraic Fractions in expressions and equations

Revision of four operations with non-algebraic fractions. Simplification of a fraction by cancellation of a common factor. Factorising numerator and/or denominator and then cancelling common factors. Multiplying, dividing, adding and subtracting algebraic fractions.

Indices

Positive, negative, fractional indices. Standard form (scientific notation). Using a calculator. Solving simple exponential equations.

Statistics 1 - Cumulative Frequency

Mean, mode, median and range. Finding mean, mode, median from frequency tables and estimating them from grouped frequency tables. Running totals. Cumulative frequency curves. Median, quartiles and percentiles. Pie charts.

Formulae

Revision of expanding brackets, simplifying expressions, factorising, substitution. Constructing and using formulae. Transforming with squares, square roots and fractions.

Functions

Use function notation to describe simple functions and inverses. Form composite functions. The idea of a function of a variable. Domain and range.

Graphs

Revision of linear eqts. y = mx + c. Constructing tables of values for linear, quadratic, cubic and reciprocal functions. Solve associated equations approximately by graphical methods. Revise how to calculate gradient of a straight line from the co-ordinates of two points. Drawing tangents to curves to find gradients. Understand parabola, hyperbola.

Mensuration

Revision of areas of triangles and quadrilaterals. Parts of a circle. Arc and sector. Volume and surface area of prism, pyramid, cone and sphere. Dimensions of formulas . Nets of 3D shapes. Mass.

Similar Shapes

Similar triangles and figures. Relationship between areas and volumes of similar figures.

Trigonometry 1 - Trigonometry in RAT

Calculating sine, cosine and tangent of right-angled triangle. Finding angles and lengths from the calculations of acute angles. Angles of elevation and depression. Interpret and use three figure bearings.

Probability

Revision of probability of simple combined events. Possibility spaces. Tree diagrams. Calculate the probability of  independent and dependent events.

Year 11 Key Objectives

Trigonometry 2 - Trigonometry in any triangle

Revise the three basic trigonometry ratios of sine, cosine and tangent in right angled triangles up to 90º. Extend sine and cosine rules to angles between 90º and 360º. Sine and cosine rules to find missing sides or angles in triangles of any size. Sine rule for the area of any triangle. Applications to simple problems in 2 and 3D including angle between plane and line. Draw sine, cosine and tangent curves. Radian measure.

Matrices

Display information in the form of a matrix of any order. Calculate sum and product of two matrices. Calculate the product of matrix and scalar. Use algebra of 2 x 2 matrices including zero, identity of 2 x 2 matrices. Calculate determinant and inverse of  non-singular matrix.

Transformations of the Plane

Use the following transformations of the plane: reflection, rotation, translation, enlargement, shear, stretching and their combinations. Identify and give precise descriptions of transformations connecting given figures. Describe transformations by co-ordinates, matrices.

Loci

Constructions using ruler, compasses and protractor. Constructing triangles. Use the following loci and method of intersecting loci for sets of points in two dimensions which are:

• at a given distance from a given point
• at a given distance from a given straight line
• equidistant from two given points
• equidistant from two given intersecting straight lines

Graphs in Practical Situations

            Conversion graphs. Distance, speed, time. Distance-time and linear speed-time graphs. Finding acceleration and distance from a speed-time graph. Graphs of real life situations.

Limits of Accuracy

            Choosing appropriate degree of accuracy. Upper, lower limits and bounds for continuous and discrete data. Upper and lower bounds in calculations of addition, subtraction, division and multiplication. The effects of error on calculations involving measurement.

Angle Properties and Symmetry

Calculate unknown angles using following geometrical properties

• angle properties of regular and irregular polygons
• angle in a semi-circle
• angle between tangent and radius of circle (1st tan property)
• angle at centre of circle is twice angle at the circumference
• angles in same segment are equal
• angles in opposite segments are supplementary

Symmetry properties of polygons and use the following symmetry properties of circles

• equal chords are equidistant from the centre
• perpendicular bisector of a chord passes through the centre
• tangents from an external point are equal in length.
• Recognise symmetry properties of the prism and the pyramid.

Vectors in Two Dimensions

Describe a translation by using a vector. Add vectors and multiply a vector by a scalar. Calculate magnitude of vector using Pythagoras. Represent vectors by directed line segments. Use the sum and difference of two vectors to express vectors in terms of two coplanar vectors. Use position vectors.

Statistics 2 - Histograms

Difference between discrete and continuous data. Revision of frequency tables and bar-charts. Construct and read histograms with equal and unequal intervals.

Sets

Set language and notation. Universal set, empty set, complement, union, intersection, subsets, elements. Venn diagrams.

Linear Programming

Represent inequalities graphically and solving linear programming problems.

Other Information 

Calculators

            Students must buy a non-programmable scientific calculator to use throughout Y10/11. They must bring the calculator to all classes to improve their ability to use it correctly. Throughout the course, students will be shown how to use many of the functions on the calculator.

Evaluations

            Students will receive a mark on their assessment profile every 4 weeks. This will reflect the progress made by the student over that time. All assessments in Y10/11 are copies of old IGCSE questions. This allows parents, students and teachers to accurately predict the student’s level. This accuracy is demonstrated by the results over the years in the real IGCSE examination. Students will always be clearly told beforehand on what they will be assessed. At the end of each term there is an end of term evaluation which tests the material learnt during the term. A student’s end of term mark is weighted so that the term’s mark are worth 70% and the end of term evaluation is worth 30%. Although a 5 is a pass mark, a students should be aiming for at least a 6.5.

Spanish Mathematics

            In addition to the 4 periods of 45 – 40 minutes a week for the IGCSE course, our students have 2 periods a week of mathematics in Spanish. This allows us to cover differences in the two systems as well as reinforcing the major areas so that our students are accostomed to dealing with mathematics in Spanish.

Competitions

            Every year, Y6 to Y11 students from St. Anne’s participate in the Concurso Primavera de matemáticas. This is a mathematical competition organised by the Comunidad de Madrid and takes place in Mathematics Faculty at the Complutense University in Madrid. It intends to encourage students to answer mathematics problems in a different way to problems seen in class and to stimulate a student’s interest in mathematics. Although it is not competitive, St Anne’s students achieve excellent results.

Homework

            Our students have a demanding day. They are in school until 5:00 pm and often will not return home until 6 pm. Important relaxation time means that there is little time left to do a lot homework. The emphasis on homework must be on quality rather than quantity. If students work hard in class then we do not set much homework. Often homework will consist of finishing off class work or a 10/15 min problem to do. Revision before evaluations should not be excessive.

IGCSE results

            Our first students took the IGCSE in 1998 and since then 312 students have taken the examination and performed very well. While the International pass-rate is around 65%, at St. Anne’s we have a pass-rate of 93%.

Extended

Of our 312 students, 221 have taken the Extended exam with a pass-rate of 99%. The table shows the grades achieved up to 2009.

Core

Of our 312 students, 91 took the Core exam with a pass-rate of 78%. The table shows the grades achieved up to 2009.

Additional Mathematics

            For the last 6 years we have entered a small group of Y11 students in the University of Cambridge IGCSE Additional Mathematics course. This is an extremely high level of mathematics for students in Y10 and Y11. It is only for very able students who are on target for grade A or A* in the Extended course. Classes are small and informal compared to normal classes. Students have to do the significant extra material required for the Additional course as well as continuing their studies in all other classes. The course is taught using only 1 class a week (instead of the 4 suggested by the University of Cambridge) and students come out of a non-examination subject. This requires students to have an excellent level and to be highly self-motivated. Results are good and even results below a C are significant achievements for students who have had a valuable introduction to many areas of mathematics they will study successfully at higher levels.

Additional Course

Logarithmic and exponential functions

• know and use the laws of logarithms (including change of base of logarithms);
• solve equations of the form a^x = b.

Indices and surds

·         perform simple operations with indices and with surds, including rationalising the denominator.

Permutations and combinations

• use the Binomial Theorem for expansion of (a + b)ⁿ for positive integral n;
• use the general term

Factors of polynomials

• know and use the remainder and factor theorems;
• find factors of polynomials;
• solve cubic equations.

Simultaneous equations

• solve simultaneous equations in two unknowns with at least one linear equation.

Straight line graphs

• interpret the equation of a straight line graph in the form y = mx + c;
• transform given relationships, including y = axⁿ and y = Ab^x, to straight line form and hence determine unknown constants by calculating the gradient or intercept of the transformed graph;
• solve questions involving mid-point and length of a line;
• know and use the condition for two lines to be parallel or perpendicular.

Circular measure

• solve problems involving the arc length and sector area of a circle, including knowledge and use of radian measure.

Quadratic functions

• know the conditions for f(x) = 0 to have (i) two real roots, (ii) two equal roots, (iii) no real roots; and the related conditions for a given line to (i) intersect a given curve, (ii) be a tangent to a given curve, (iii) not intersect a given curve;
• solve quadratic equations for real roots and find the solution set for quadratic inequalities.

Functions

• understand the terms function, domain, range (image set), one-one function, inverse function and composition of functions;
• use the notation f(x) = sinx, f: x     lg x, (x > 0), f ^-1(x) and f ²(x) [= f (f(x))];
• understand the relationship between y = f(x) and y =  f(x) , where f(x) may be linear, quadratic or trigonometric;
• explain in words why a given function is a function or why it does not have an inverse;
• find the inverse of a one-one function and form composite functions;
• use sketch graphs to show the relationship between a function and its inverse.

Logarithmic and exponential functions

• know simple properties and graphs of the logarithmic and exponential functions including lnx and e^x.

Permutations and combinations

• recognise and distinguish between a permutation case and a combination case;
• know and use the notation n!, (with 0! = 1), and the expressions for permutations and combinations of n items taken r at a time;
• answer simple problems on arrangement and selection (cases with repetition of objects, or with objects arranged in a circle or involving both permutations and combinations, are excluded).

Trigonometry

• know the six trigonometric functions of angles of any magnitude (sine, cosine, tangent, secant, cosecant, cotangent);
• understand amplitude and periodicity and the relationship between graphs of e.g. sinx and sin2x ;
• draw and use the graphs of
  • y = a sin(bx) + c,
  • y = a cos(bx) + c,
  • y = a tan(bx) + c,
• where a, b are positive integers and c is an integer;
• use the relationships

                                                sin²A + cos²A = 1,

                                                sec²A = 1+ tan²A,

                                                cosec²A = 1+ cot²A

• solve simple trigonometric equations involving the six trigonometric functions and the above relationships (not including general solution of trigonometric equations);
• prove simple trigonometric identities.

Differentiation and Integration

• understand the idea of a derived function;
• use the notations f´(x), f´´(x),
• use the derivatives of the standard functions xⁿ (for any rational n), sin x, cos x, tan x, e^x, lnx together with constant multiples, sums and composite functions of these;
• differentiate products and quotients of functions;
• apply differentiation to gradients, tangents and normals, stationary points, connected rates of change, small increments and approximations and practical maxima and minima problems;
• discriminate between maxima and minima by any method;
• understand integration as the reverse process of differentiation;
• integrate sums of terms in powers of x excluding 1/x;
• integrate functions of the form (ax + b)ⁿ   (excluding n = –1), e^ax+b , sin(ax + b), cos(ax + b);
• evaluate definite integrals and apply integration to the evaluation of plane areas;
• apply differentiation and integration to kinematics problems that involve displacement, velocity and acceleration of a particle moving in a straight line with variable or constant acceleration and the use of x-t and v-t graphs.

Quadratic functions

• find the maximum or minimum value of the quadratic function f(x) = ax²+ bx + c by any method;
• use the maximum or minimum value of f(x) to sketch the graph or determine the range for a given domain;

Matrices

• display information in the form of a matrix of any order and interpret the data in a given matrix;
• solve problems involving the calculation of the sum and product (where appropriate) of two matrices and interpret the results;
• calculate the product of a scalar quantity and a matrix;
• use the algebra of 2 x 2 matrices (including the zero and identity matrix);
• calculate the determinant and inverse of a non-singular 2 x 2 matrix and solve simultaneous linear equations.

Set language and notation

• use set language and notation, and Venn diagrams to describe sets and represent relationships between sets as follows:

                        A = {x: x is a natural number}

                        B = {(x,y): y = mx + c}

                        C = {a, b, c, …}

• understand and use the following notation:

                        Union of A and B                                                A U B

                        Intersection of A and B                                       A n B

                        Number of elements in set A                               n(A)

                        “…is an element of…”                                        

                        “…is not an element of…”                                  

                        Complement of set A                                          A'

                        The empty set

                        Universal set                                                     

                        A is a subset of B                                              A  B

                        A is a proper subset of B                                    A  B

                        A is not a subset of B

                        A is not a proper subset of B

Vectors in 2 dimensions

• use vectors in any form, e.g. , p, ai – bj;

• know and use position vectors and unit vectors;
• find the magnitude of a vector. Add and subtract vectors and multiply vectors by scalars;
• compose and resolve velocities;
• use relative velocity including solving problems on interception (but not closest approach).

www.cie.org.uk