2000-D-385
MATHEMATICS SYLLABUS
5 PERIOD PER WEEK
Years 6 and 7.
Appoved by the Board of Governors on 17 and 18 may 2000 in Rethymnon - Crete
Will enter into application : in year 6 in September 2000
ref. BG:2000-D-214
Mathematics Syllabus. Secondary Level.
5 periods per week in years 6 and 7.
1.INTRODUCTION
This course is designed for pupils who will use mathematics as part of their studies in higher education and for this reason need a sound foundation and good knowledge of Mathematics
Emphasis should be placed on modern problem solving methods and appropriate technology.
The course is divided into four main areas: - Analysis
-Algebra
-Geometry
-Probability
- OBJECTIVES
2.1 General objectives
Secondary education in the European Schools has a dual purpose: development within particular disciplines, and the personal development of the pupils within a broader social and cultural context. The former comprises the acquisition within each discipline of appropriate understanding, knowledge and application thereof. Pupils should acquire the skills of description, interpretation, and discrimination, while also developing the capacity to demonstrate their understanding in practical terms. As well as its particular didactic aspect, educational development takes place in a whole series of contexts: spiritual, moral, social and cultural. This entails the development of pupils’ own sense of self both as individuals and as part of the wider social context in which they find themselves; in school and further afield.
These two purposes, which are inseparable in practice, are rooted in a European context, emphasising the richness and diversity of the separate cultures which give rise to the whole. This reference, the experience of living together as true Europeans within the Schools should lead pupils to deep respect for the traditions characterising each separate European nation, while still allowing them to develop and preserve their own identity.
2.2Objectives particular to mathematics.
2.2.1General Skills.
The mathematics course will provide the opportunity for pupils to develop the following skills:
Analysis of problems
-Understand the formulation, analyse the structure and identify the core ideas,
-Pick out relevant information from within different modes of expression.
Manipulation, argumentation, reasoning
-Translate information from one form to another, for example from ordinary language to graphical or algebraic representation and vice-versa,
-make observations based on previous knowledge and as a function of the intended goal,
-formulate a conjecture, chose an appropriate strategy,
-gather arguments for the construction of a proof,
-choose a suitable procedure and see it through to its conclusion,
-where appropriate, use Mathematics to deal with questions arising from other branches of knowledge.
Communication
-master vocabulary, symbols and connectives “if … then”, “hence”, “so”, “is equivalent to”, “thus”, “therefore”, “and”, “or”, …
-draw up an explanation, a proof,
-present results clearly, concisely, devoid of ambiguity,
-produce a diagram, graph or table which clarifies or sums up a situation.
Generalisation, structuring, synthesising
-recognise commonality in difference,
-extend the application of a rule, theorem or property to cover a larger domain,
-formulate generalisations and check their validity,
-construct a theoretical model from previously discovered items of mathematical knowledge.
2.2.2Specific Skills.
Algebra. Complex Numbers
- Carry out calculations involving complex numbers, determine the argument, modulus, conjugate of a complex number and give a geometrical interpretation.
- Transform a complex number from algebraic form to trigonometric form and vice versa.
- Solve second order and binomial equations over .
Analysis.
- Describe the general characteristics of a function from its graph using precise terms.
- Given the properties of two functions, find the properties of their sum, difference, product, quotient or composition.
- Decompose a function into standard functions.
- Construct the graph of the reciprocal of a standard function, and determine its domain, roots, …
- Use the graph of a function to help represent a problem.
- Find a limit of a function.
- Determine the equations of the asymptotes on the graph of a function.
- Find general solutions, in particular for trigonometric functions.
- Anticipate the existence of asymptotes by considering the graph and the algebraic definition of a function.
- Check the plausibility of results and calculations, particularly when using a calculator.
- Find the derivative of a function.
- Interpret the derivative of a function at a point in geometric and physical terms
- When finding a derivative, check whether the result is realistic by using numerical, algebraic and graphical properties.
- Use the properties of derivatives in diverse applications.
- Compare the graph of a function with that of its derivative and vice-versa.
- Find indefinite and definite integrals using standard methods: substitution and integration by parts.
- Justify the fundamental theorem of calculus.
- Apply integration to resolve problems emanating from the domains of mathematics, science, economics.
- Model problems so that they may be dealt with using integration.
- Make use of modern technology: calculators, computers and specific programs.
- Recognise the continuity or discontinuity of a function.
Geometry.
- Find the vector, parametric and cartesian equations of a line and a plane, from defining properties thereof.
- Find analytically the distance between two points, between a point and a line, between a point and a plane.
Probability. Numerical analysis of data.
- Recognise the difference between permutation and combination, with and without repetition.
- Prove and apply formulae for the calculation of permutations, combinations.
- Define the terms: random variable, probability relating thereto, expected value (mean), variance and standard deviation.
- Solve problems of probability using an exhaustive list, a spreadsheet, a calculator or computer program.
- Recognise the appropriate conditions for applying rules of probability.
- Use probability in the understanding, analysis and criticism of numerically based information.
3.CONTENTS AND METHOD.
The content of the 5 period mathematics course in years 6 and 7 is given in the tables in point5.
The first column describes the general subject areas.
The second column establishes the content and depth of study.
The third column describes possible approaches to teaching.
4.ASSESSMENT
4.1 General Principles
Assessment may be formative or summative.
Formative assessment is a continuous process. Its object is to provide information on what has been learnt by the pupil and in so doing, to help the pupil progress. As much for pupils as for parents and the school, it plays a substantial rôle in such things as career choice and related matters. This kind of assessment should not necessarily be formally graded and must on no account be seen in a punitive light.. By concentrating on pupils’ achievements in themselves, such assessment encourages the raising of standards through self-criticism. For teachers, such assessment illuminates their specific objectives, their methods and the results of their work.
Summative assessment should produce a formal grading of the knowledge and understanding gained by a pupil at the moment of assessment..
The following principals should be observed throughout:
- Pupils’ work should be assessed in the light of the objectives laid down in the syllabus and the content and capacities therein specified.
- Assessment is based on material studied in class.
- All types of work undertaken by pupils in class - oral participation, written and practical work - may be the subject of assessment:.
- Pupils should be made aware of the work necessary to achieve a certain level, and of the criteria used in assessment at any given level.
- Pupils should be able to measure their own achievement against that attained by others in the same or other sections. This requires co-ordination between teachers both within and across sections.
4.2 Written examinations
Written examinations are set according to the current regulations of the schools.
For year 6,the final markconsists of the A-mark (based on oral participation and written work)andthe B-mark (established by the end of semester examinations).
The assessment will follow the principles set out in point 4.1 above.
4.3 The Baccalaureate
In year 7, the preliminary mark consists of the A-mark based on oral participation and written work, and the B-mark established by the written examination in January.
For the Baccalaureate, the pupil will take a written examination (4 hours) consisting of four compulsory “short” questions (50 points in total) of which two questions on analysis, one on geometry and one on probability, and two optional “long” questions (50 points in total) to be chosen from three questions: the first on analysis, the second on geometry and the third on probability.
Questions will be based on the syllabus for year 7, but may require knowledge of materiel studied in year 6.
Marking should take into account method and presentation.
4.4Content of the written examination
Examinations are identical for candidates in all language sections. The content of the end of year examination is as follows:
A.ANALYSIS
1.1 Study of functions of the following types:
+ f
(limited to and )
(x) = a polynomial of degree i<3;
[a, b, c, d, f . , and –2 2]
1.2Integration as specified in the syllabus where the functions under consideration will be polynomial, trigonometric functions and their reciprocals, or functions as specified in point 1.1 above, on condition that the methods to be tested will be limited to integration by inspection, integration by parts or integration by substitution.
1.3Calculation of volumes of solids of revolution generated by rotation about the x-axis.
1.4Solution of differential equations of the type . Simple cases.
B.GEOMETRY
The entire syllabus of Geometry.
C.PROBABILITY
Questions based on all the material covered in the syllabus. For Bayes’ Theorem, cases will be restricted to n=2.
Questions will be put forward in the author’s language, and also in the three working languages provided that the course is taught in these languages at the school.
The original language should be specified.
4.6Permitted material.
The formula booklet provided by the school.
A calculator conforming to current regulations for examinations in mathematics and sciences.
4.7Critera for correction and marks awarded.
Each of questions 1 and 2 is to be marked out of 12, 6 marks being awarded for each of two parts.
Each of questions 3 and 4 is to be marked out of 13, 6 marks being awarded for the first part and 7 for the second.
Questions I, II and III are each to be marked out of 25.
5. Syllabus for years 6 and 7.
5.1 Syllabus for year 6
Complex NumbersTOPIC
1) Complex Numbers / CONTENT
- Introduction to complex numbers.
Real and imaginary parts of a complex number.
- Complex conjugates.
- Operations on complex numbers:
sum, product, quotient of two complex numbers.
Reciprocal of a non-zero complex number.
- Square roots of a complex number.
Solution over of quadratic equations with complex coefficients.
- Geometric representation of a complex number. (Argand diagram)
- Trigonometric form.
- Modulus of a complex number. Modulus of a product, of a quotient,
of the reciprocal.
Argument of a non-zero complex number. Argument of a product, of a quotient. Argument of the reciprocal of a non-zero complex number.
Powers, nth roots.
de Moivre’s Theorem. / REMARKS
No particular style of introduction is imposed. Concepts should be put forward, but a derivation from first principals is not desirable.
Coefficients of these equations should be free of parameters
Analysis
TOPIC1) General introduction to real functions of a real variable.
2) Continuity and limits.
TOPIC
Continuity and limits / CONTENT
- Definition of a real function of a real variable.
- Domain of a function.
- Zeros of a function. Sign of a function.
- Even and odd functions. Periodic functions.
- Composition of two functions.
- Inverse of a bijection.
- Increasing and decreasing functions, constant, monotonic, over an interval. Local and global extrema.
- Graph of a function.
Continuity
- Notion of continuity of a function at a point;
examples and counter-examples.
- Continuity of a function from the right [left] of a point.
- Continuity of a function over an open [closed] interval.
- Statement (without proof) of theorems concerning continuity:
-of the absolute value of a continuous function.
CONTENT
- of the product of a continuous function with a real number.
- of the sum, product, quotient, composition of two continuous functions.
- Continuity over of polynomial functions.
- Continuity of rational functions over their domain. / REMARKS
The order in which the two notions of continuity and limit will be studied is left to the teacher. The study should largely rely on intuition.
REMARKS
TOPIC
3) Differentiation. / Limits
- Notion of the limit of a function at a point;
examples and counter-examples.
- removable discontinuity.
- Right-hand [left-hand] limit of a function at a point.
- Extension of the notion of limit: infinite limit, limit as the
variable tends to + or - .
- Statement (without proof) of theorems concerning limits:
- of the absolute value of a function.
- of the product of a function with a real number.
- of the sum, product, quotient, composition of two functions.
-Indeterminant forms.
CONTENT
- Value of the derivative of a function at a point. Geometrical interpretation.
- Equation of the tangent at a point on the graph of a function.
- If a function is differentiable at a point, it is also continuous at that point.
- Value of the right-hand [left-hand] derivative at a point.
- Derivative of a function. Successive derivatives. / REMARKS
4) Study of real functions of a real variable / - Derivative of a product of a differentiable function with a real number,
Derivative of the sum, product, quotient and compostition of two differentiable functions
Derivative of the inverse of a function.
- l’Hospital’s rule.
- Application of the notions of limit and derivatives to the analysis of a function:
Increase and decrease of a function and identification of any extremum, asymptotes on the graph of a function.
Concave/convex nature of the graph of a function, points of inflection; tangents at such points.
- Application of these ideas, and those of previous paragraphs,
to the study of polynomial, rational, circular ( sine, cosine, tangent ) functions. / Inclusion of the proof of these results is at the teacher’s discretion.
The result is to be stated and used.
The functions studied should be limited to those for which the use of the derivative is straightforward
GEOMETRY IN 3-DIMESIONAL SPACE
TOPIC
1) Common objects in 3-D space.
2) Vectors in 3-D space.
3) Analytical geometry of the point, the plane, the line. / CONTENT
- Points, lines, planes, spheres.
- Relative positions of these.
- Definition.
- Sum of vectors.
- Product of a vector with a scalar. Collinear vectors.
- Vector equation of a line
- Linear combination of two vectors. Coplanar vectors.
- Vector equation of a plane.
- Scalar product of two vectors
- Magnitude of a vector, distance between two points.
- Orthogonal vectors.
- Orthogonal, normalised, orthonormal basis.
- Application of these concepts to problems in analytical geometry
- Parametric and cartesian equations of a plane.
- Parametric and cartesian equations of a line. / REMARKS
Study of this section could be based on revision of vectors in 2 dimensional space and the obvious extension into 3-D space.
In this section, the basis will always be orthonormal.
PROBABILITY
TOPIC
1) The counting of arrangements and selections.
2) Introduction to the theory of probability. / CONTENT
- Permutations and combinations of a finite set with or without repetition
- random outcomes, possibilities, possibility space.
- events, simple events.
- certainty, impossibility
- the negation of an event.
- P(AB), P(AB), P(A notB) .
- mutually exclusive events.
- The relation between probability and relative frequency.
- Probability defined on a finite possibility space.
- Elementary properties.
- General idea of a probability distribution. / REMARKS
Note that different languages differ in the interpretation to be given to the terms here
5.2 7th Year Syllabus.
AnalysisTOPIC
1) Analysis of real functions
of a real variable
TOPIC / CONTENT
Application of the following concepts:
- domain of a function.
- zeros of a function. Sign of a function.
- even and odd functions, Periodic functions.
- limit of a function at a point.
- continuity at a point, over an interval.
- differentiability of a function at a point, over an interval.
- l’Hospital’s rule.
- increase and decrease of a function. Extremum.
- graph of a function, tangent at a point on the graph of a function, asymptotes, concavity, points of inflection.
To the following functions:
- absolute value.
- polynomials.
- rational functions.
- functions of the type ,
where P(x) is a 1st or 2nd degree polynomial.
CONTENT
- circular functions: sine, cosine, tangent.
- natural logarithm function, exponential function with base e.
- functions obtained by addition, multiplication, division or
composition of the preceding functions. / REMARKS
The majority of this paragraph is studied in year 6. It is not necessary to repeat the entire content, but simply to revise and where necessary to extend understanding.
The study of these functions puts into practice the concepts studied in the previous section and those introduced in year 6.
The functions studied should be limited to those for which the use of the derivative is straightforward
REMARKS
In liaison with the teaching of other subjects, mention base 10 logarithms
2) Integration.
TOPIC
Integration continued / - Integral of a function defined on a closed and bounded interval
- Graphical interpretation of such integrals as area.
- Properties of integrals :
;
Linearity:
CONTENT
Lower and upper rectangle sums, enclosure thereby
-
/ REMARKS
TOPIC
Integration /
- Mean value of a function f on an interval :
given by
- Primitives (indefinite integral) of a function continuous over an interval
CONTENT
- If f is continuous over an interval I, and aI , then the function F
defined on I by is the primitive of f over I which is zero when x = a.
- Indefinite integral
- Evaluation of integrals by the following methods :
- Integration by inspection.
- Integration by parts.
- Integration by substitution.
- Application of these methods to the functions studied in
section §1.
- Application of the theory of integration to finding plane areas and volumes of revolution generated by rotation about the x-axis.
- First order differential equations with variables separable leading to the form y’.f(y) = g(x). / Practical examples should be used to illustrate this point.
REMARKS
The integration of rational functions may require their transformation to the sum of simple functions. The form of the decomposition should be given in all cases other than .
May be applied to problems emanating from physics, biology, economics, etc.
GEOMETRY OF THREE DIMENSIONAL SPACE
TOPIC1) Vectors in 3-D Space
2) Analytical geometry of the point, the plane, the line. / CONTENT
Throughout the chapter on Geometry, the basis will always be chosen as orthonormal.
- Collinear vectors. Vector equation of a line.
Coplanar vectors. Vector equation of a plane.
- Scalar product of two vectors in 3-D.
- magnitude of a vector; distance between two points.
Orthogonal vectors. Orthonormal basis.
- Vector product of two vectors.
- Triple scalar product.
- Application of these concepts to problems in analytical geometry.
- The use of the preceding concepts :
- in the calculation of areas of common plane figures:
triangle, trapezium, parallelogram.
- in the calculation of volumes of common solids:
prism, parallelepiped, cylinder, pyramid.
- Parametric and cartesian equations of a plane.
- Parametric and cartesian equations of a line. / REMARKS
The majority of this paragraph is studied in year 6. It is not necessary to repeat the entire content, but simply to revise and where necessary to extend understanding.
The use of matrices and determinants of order 3 is left to the discretion of the teacher.
TOPIC2) Analytical geometry of the point, the plane, the line (continued)
3) Analytic geometry of the sphere / CONTENT
Relative position of two planes, of a line and a plane, of two
lines.
Orthogonal projection of a point onto a plane.
Distance between a point and a plane.
Distance between two parallel planes
Distance between a plane and a parallel line.
Orthogonal projection of a point on a line.
Distance of a point from a line.
Distance between two lines.
Angle between two vectors in 3-D.
Angle between two lines.
Angle between two planes.
Angle between a line and a plane.
- Cartesian equation of a sphere.
- Relative positions of a point and sphere; of a plane and sphere; of a line and sphere.
- Volume and surface area of the sphere. / REMARKS
PROBABILITY
TOPIC1) The counting of arrangements and selections.
2) Probability.
3) Conditional probability.
TOPIC
3) Conditional probability (continued) / CONTENT
- Permutations and combinations of a finite set with or without repetition
- Events.
- Probability defined on a finite possibility space.
Elementary properties
Probability distribution.
- Probability conditional upon an event with non-zero probability; notation .
- Joint probability :
Use of this result within tree diagrams for random trials with several outcomes
CONTENT
- Independent events:
- total probability:
/ REMARKS
The concepts of paragraphs 1) and 2) are introduced in years 6. What is required here is revision.
Note that different languages differ in the interpretation to be given to the terms here
REMARKS
4) Discrete random variables
.
TOPIC
4) Discrete random variables (continued)
5) Continuous random variables. / - Bayes’ Theorem:
General properties
- discrete random variables.
- sample space.
- Probability function of a discrete random variable.
- Cumulative distribution function of a discrete random variable.
- Expected value, variance and standard deviation of a discrete random variable.
CONTENT
Binomial variates.
- Binomial variates: Bernoulli trials,
distribution and probability of a binomial variate, use of
tables.
- Expected value, variance and standard deviation of a binomial variate.
Poisson distribution
- Poisson probability distribution, properties, use of tables.
- Expected value, variance and standard deviationof a Poisson variate.
- Poisson distribution as an approximation to the binomial distribution for n>50 and p<0.1.
General
- Concept of continuous random variables.
- Probability density function of a continuous random variable.
- Cumulative distribution function of a continuous random variable. / In application, avoid situations in which the relevant contingency table cannot be partitioned as 2x2.
Adopting the convention:
REMARKS
The formulae for these three parameters are given without proof.
The formulae for these three parameters are given without proof.
TOPIC
5) Continuous random variables. (continued) / CONTENT
Normal (or Gaussian) distribution
- Definition.
- Expected value, variance and standard deviation of a Normal distribution.
- Normal curve and cumulative Normal curve.
- Standardised Normal distribution, use of tables.
- Normal approximation to the binomial distribution given npq>9. / REMARKS
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