2. Los planes
2.2. Plan de seguridad y análisis de riesgos
Contents
1. Introduction ... 2
1.1 Why choose Cambridge? 1.2 Why choose Cambridge International AS and A Level? 1.3 Why choose Cambridge International AS and A Level Mathematics? 1.4 Cambridge AICE (Advanced International Certifi cate of Education) Diploma 1.5 How can I fi nd out more?
2. Teacher support ... 6
2.1 Support materials 2.2 Resource lists 2.3 Training
3. Assessment at a glance ... 7
4. Syllabus aims and assessment objectives ... 10
4.1 Syllabus aims 4.2 Assessment objectives
5. Syllabus content ... 11
6. List of formulae and tables of the normal distribution (MF9) ... 28
7. Mathematical notation ... 33
8. Other information ... 37
Introduction
1. Introduction
1.1 Why choose Cambridge?
Recognition
Cambridge International Examinations is the world’s largest provider of international education programmes and qualifi cations for learners aged 5 to 19. We are part of Cambridge Assessment, a department of the University of Cambridge, trusted for excellence in education. Our qualifi cations are recognised by the world’s universities and employers.
Cambridge International AS and A Levels are recognised around the world by schools, universities and employers. The qualifi cations are accepted as proof of academic ability for entry to universities worldwide, though some courses do require specifi c subjects.
Cambridge International A Levels typically take two years to complete and offer a fl exible course of study that gives learners the freedom to select subjects that are right for them.
Cambridge International AS Levels often represent the fi rst half of an A Level course but may also be taken as a freestanding qualifi cation. The content and diffi culty of a Cambridge International AS Level examination is equivalent to the fi rst half of a corresponding Cambridge International A Level. Cambridge AS Levels are accepted in all UK universities and carry half the weighting of an A Level. University course credit and advanced standing is often available for Cambridge International AS and A Levels in countries such as the USA and Canada.
Learn more at www.cie.org.uk/recognition
Excellence in education
Our mission is to deliver world-class international education through the provision of high-quality curricula, assessment and services.
More than 9000 schools are part of our Cambridge learning community. We support teachers in over 160 countries who offer their learners an international education based on our curricula and leading to our qualifi cations. Every year, thousands of learners use Cambridge qualifi cations to gain places at universities around the world.
Our syllabuses are reviewed and updated regularly so that they refl ect the latest thinking of international experts and practitioners and take account of the different national contexts in which they are taught.
Cambridge programmes and qualifi cations are designed to support learners in becoming:
• confi dent in working with information and ideas – their own and those of others
• responsible for themselves, responsive to and respectful of others
Introduction
Support for teachers
A wide range of support materials and resources is available for teachers and learners in Cambridge schools.
Resources suit a variety of teaching methods in different international contexts. Through subject discussion forums and training, teachers can access the expert advice they need for teaching our qualifi cations. More details can be found in Section 2 of this syllabus and at www.cie.org.uk/teachers
Support for exams offi cers
Exams offi cers can trust in reliable, effi cient administration of exam entries and excellent personal support from our customer services. Learn more at www.cie.org.uk/examsoffi cers
Not-for-profi t, part of the University of Cambridge
We are a not-for-profi t organisation where the needs of the teachers and learners are at the core of what we do. We continually invest in educational research and respond to feedback from our customers in order to improve our qualifi cations, products and services.
Our systems for managing the provision of international qualifi cations and education programmes for learners aged 5 to 19 are certifi ed as meeting the internationally recognised standard for quality management, ISO 9001:2008. Learn more at www.cie.org.uk/ISO9001
1.2 Why choose Cambridge International AS and A Level?
Cambridge International AS and A Levels are international in outlook, but retain a local relevance. The syllabuses provide opportunities for contextualised learning and the content has been created to suit a wide variety of schools, avoid cultural bias and develop essential lifelong skills, including creative thinking and problem-solving.
Our aim is to balance knowledge, understanding and skills in our programmes and qualifi cations to enable candidates to become effective learners and to provide a solid foundation for their continuing educational journey. Cambridge International AS and A Levels give learners building blocks for an individualised curriculum that develops their knowledge, understanding and skills.
Schools can offer almost any combination of 60 subjects, and learners can specialise or study a range of subjects, ensuring a breadth of knowledge. Giving learners the power to choose helps motivate them throughout their studies.
Through our professional development courses and our support materials for Cambridge International AS and A Levels, we provide the tools to enable teachers to prepare learners to the best of their ability and work with us in the pursuit of excellence in education.
Cambridge International AS and A Levels have a proven reputation for preparing learners well for university,
Introduction
Learners studying Cambridge International AS and A Levels have the opportunities to:
• acquire an in-depth subject knowledge
• develop independent thinking skills
• apply knowledge and understanding to new as well as familiar situations
• handle and evaluate different types of information sources
• think logically and present ordered and coherent arguments
• make judgements, recommendations and decisions
• present reasoned explanations, understand implications and communicate them clearly and logically
• work and communicate in English.
Guided learning hours
Cambridge International A Level syllabuses are designed on the assumption that learners have about 360 guided learning hours per subject over the duration of the course. Cambridge International AS Level syllabuses are designed on the assumption that learners have about 180 guided learning hours per subject over the duration of the course. This is for guidance only and the number of hours required to gain the qualifi cation may vary according to local curricular practice and the learners’ prior experience of the subject.
1.3 Why choose Cambridge International AS and A Level Mathematics?
Cambridge International AS and A Level Mathematics is accepted by universities and employers as proof of mathematical knowledge and understanding. Successful candidates gain lifelong skills, including:
• a deeper understanding of mathematical principles
• the further development of mathematical skills including the use of applications of mathematics in the context of everyday situations and in other subjects that they may be studying
• the ability to analyse problems logically, recognising when and how a situation may be represented mathematically
• the use of mathematics as a means of communication
• a solid foundation for further study.
The syllabus allows Centres fl exibility to choose from three different routes to AS Level Mathematics – Pure Mathematics only or Pure Mathematics and Mechanics or Pure Mathematics and Probability and Statistics.
Centres can choose from three different routes to Cambridge International A Level Mathematics depending on the choice of Mechanics, or Probability and Statistics, or both, in the broad area of ‘applications’.
Prior learning
We recommend that candidates who are beginning this course should have previously completed a Cambridge O Level or Cambridge IGCSE course in Mathematics or the equivalent.
Introduction
Cambridge International AS Level Mathematics constitutes the fi rst half of the Cambridge International A Level course in Mathematics and therefore provides a suitable foundation for the study of Mathematics at A Level and thence for related courses in higher education.
1.4 Cambridge AICE (Advanced International Certifi cate of Education) Diploma
Cambridge AICE Diploma is the group award of the Cambridge International AS and A Level. It gives schools the opportunity to benefi t from offering a broad and balanced curriculum by recognising the achievements of candidates who pass examinations in three different curriculum groups:
• Mathematics and Science (Group 1)
• Languages (Group 2)
• Arts and Humanities (Group 3)
A Cambridge International A Level counts as a double-credit qualifi cation and a Cambridge International AS Level counts as a single-credit qualifi cation within the Cambridge AICE Diploma award framework.
To be considered for an AICE Diploma, a candidate must earn the equivalent of six credits by passing a combination of examinations at either double credit or single credit, with at least one course coming from each of the three curriculum groups.
Mathematics (9709) is in Group 1, Mathematics and Sciences.
Credits gained from Cambridge AS Level Global Perspectives (8987) or Cambridge Pre-U Global
Perspectives and Independent Research (9766) can be counted towards the Cambridge AICE Diploma, but candidates must also gain at least one credit from each of the three curriculum groups to be eligible for the award.
Learn more about the Cambridge AICE Diploma at www.cie.org.uk/qualifi cations/academic/uppersec/aice The Cambridge AICE Diploma is awarded from examinations administered in the June and November series each year. In India, the Cambridge AICE Diploma is also awarded from examinations administered in the March series each year.
1.5 How can I fi nd out more?
If you are already a Cambridge school
You can make entries for this qualifi cation through your usual channels. If you have any questions, please contact us at [email protected]
Teacher support
2. Teacher support
2.1 Support materials
Cambridge syllabuses, past question papers and examiner reports to cover the last examination series are on the Syllabus and Support Materials DVD, which we send to all Cambridge schools.
You can also go to our public website at www.cie.org.uk/alevel to download current and future syllabuses together with specimen papers or past question papers and examiner reports from one series.
For teachers at registered Cambridge schools a range of additional support materials for specifi c syllabuses is available from Teacher Support, our secure online support for Cambridge teachers. Go to http://teachers.cie.org.uk (username and password required).
2.2 Resource lists
We work with publishers providing a range of resources for our syllabuses including textbooks, websites, CDs, etc. Any endorsed, recommended and suggested resources are listed on both our public website and on Teacher Support.
The resource lists can be fi ltered to show all resources or just those which are endorsed or recommended by Cambridge. Resources endorsed by Cambridge go through a detailed quality assurance process and are written to align closely with the Cambridge syllabus they support.
2.3 Training
We offer a range of support activities for teachers to ensure they have the relevant knowledge and skills to deliver our qualifi cations. See www.cie.org.uk/events for further information.
Assessment at a glance
3. Assessment at a glance
The 7 units in the scheme cover the following subject areas:
• Pure Mathematics (units P1, P2 and P3);
• Mechanics (units M1 and M2);
• Probability and Statistics (units S1 and S2).
Centres and candidates may:
• take all four Advanced (A) Level components in the same examination series for the full Cambridge International A Level;
• follow a staged assessment route to the Cambridge International A Level by taking two Advanced Subsidiary (AS) papers (P1 & M1 or P1 & S1) in an earlier examination series;
• take the Advanced Subsidiary (AS) qualifi cation only.
Cambridge International AS Level candidates take:
Paper 1: Pure Mathematics 1 (P1)
1 hour 45 minutes
About 10 shorter and longer questions 75 marks weighted at 60% of total
plus one of the following papers:
Paper 2: Pure Mathematics 2 (P2)
Paper 4: Mechanics 1 (M1) Paper 6: Probability and Statistics 1 (S1)
1 hour 15 minutes
About 7 shorter and longer questions
50 marks weighted at 40%
of total
1 hour 15 minutes
About 7 shorter and longer questions
50 marks weighted at 40%
of total
1 hour 15 minutes
About 7 shorter and longer questions
50 marks weighted at 40%
of total
Assessment at a glance
Cambridge International A Level candidates take:
Paper 1: Pure Mathematics 1 (P1) Paper 3 Pure Mathematics 3 (P3)
1 hour 45 minutes
About 10 shorter and longer questions 75 marks weighted at 30% of total
1 hour 45 minutes
About 10 shorter and longer questions 75 marks weighted at 30% of total
plus one of the following combinations of two papers:
Paper 4: Mechanics 1 (M1) Paper 6: Probability and Statistics 1 (S1)
1 hour 15 minutes
About 7 shorter and longer questions 50 marks weighted at 20% of total
1 hour 15 minutes
About 7 shorter and longer questions 50 marks weighted at 20% of total
or
Paper 4: Mechanics 1 (M1) Paper 5: Mechanics 2 (M2)
1 hour 15 minutes
About 7 shorter and longer questions 50 marks weighted at 20% of total
1 hour 15 minutes
About 7 shorter and longer questions 50 marks weighted at 20% of total
or
Paper 6: Probability and Statistics 1 (S1) Paper 7: Probability and Statistics 2 (S2)
1 hour 15 minutes
About 7 shorter and longer questions 50 marks weighted at 20% of total
1 hour 15 minutes
About 7 shorter and longer questions 50 marks weighted at 20% of total
Assessment at a glance
Question papers
There is no choice of questions in any of the question papers and questions will be arranged approximately in order of increasing mark allocations.
It is expected that candidates will have a calculator with standard ‘scientifi c’ functions available for use for all papers in the examination. Computers, graphical calculators and calculators capable of algebraic manipulation, are not permitted.
A list of formulae and tables of the normal distribution (MF9) is supplied for the use of candidates in the examination. Details of the items in this list are given for reference in Section 6.
Relationships between units
Units P2, M2, S2 are sequential to units P1, M1, S1 respectively, and the later unit in each subject area may not be used for certifi cation unless the corresponding earlier unit is being (or has already been) used.
Unit P3 is also sequential to unit P1, and may not be used for certifi cation unless P1 is being (or has already been) used. The subject content of unit P2 is a subset of the subject content of unit P3; otherwise, the subject content for different units does not overlap, although later units in each subject area assume knowledge of the earlier units.
Availability
This syllabus is examined in the May/June and October/November examination series. The syllabus is also available for examination from March 2016 for India only.
This syllabus is available to private candidates.
Detailed timetables are available from www.cie.org.uk/examsoffi cers
Centres in the UK that receive government funding are advised to consult the Cambridge website www.cie.org.uk for the latest information before beginning to teach this syllabus.
Combining this with other syllabuses
Candidates can combine this syllabus in an examination series with any other Cambridge syllabus, except:
• syllabuses with the same title at the same level.
Syllabus aims and assessment objectives
4. Syllabus aims and assessment objectives
4.1 Syllabus aims
The aims of the syllabus are the same for all students. These are set out below and describe the educational purposes of any course based on the Mathematics units for the Cambridge International AS and A Level examinations. The aims are not listed in order of priority.
The aims are to enable candidates to:
• develop their mathematical knowledge and skills in a way which encourages confi dence and provides satisfaction and enjoyment
• develop an understanding of mathematical principles and an appreciation of mathematics as a logical and coherent subject
• acquire a range of mathematical skills, particularly those which will enable them to use applications of mathematics in the context of everyday situations and of other subjects they may be studying
• develop the ability to analyse problems logically, recognise when and how a situation may be represented mathematically, identify and interpret relevant factors and, where necessary, select an appropriate mathematical method to solve the problem
• use mathematics as a means of communication with emphasis on the use of clear expression
• acquire the mathematical background necessary for further study in this or related subjects.
4.2 Assessment objectives
The abilities assessed in the examinations cover a single area: technique with application.
The examination will test the ability of candidates to:
• understand relevant mathematical concepts, terminology and notation
• recall accurately and use successfully appropriate manipulative techniques
• recognise the appropriate mathematical procedure for a given situation
• apply combinations of mathematical skills and techniques in solving problems
• present mathematical work, and communicate conclusions, in a clear and logical way.
Syllabus content
5. Syllabus content
The mathematical content for each unit in the scheme is detailed below. The order in which topics are listed is not intended to imply anything about the order in which they might be taught.
As well as demonstrating skill in the appropriate techniques, candidates will be expected to apply their knowledge in the solution of problems. Individual questions set may involve ideas and methods from more than one section of the relevant content list.
For all units, knowledge of the content of Cambridge O Level/Cambridge IGCSE Mathematics is assumed.
Candidates will be expected to be familiar with scientifi c notation for the expression of compound units, e.g. 5 m s–1 for 5 metres per second.
Unit P1: Pure Mathematics 1 (Paper 1)
Candidates should be able to:
1. Quadratics • carry out the process of completing the square for a quadratic
polynomial ax2 + bx + c, and use this form, e.g. to locate the vertex of the graph of y = ax2 + bx + c or to sketch the graph
• fi nd the discriminant of a quadratic polynomial ax2 + bx + c and use the discriminant, e.g. to determine the number of real roots of the equation ax2 + bx + c = 0
• solve quadratic equations, and linear and quadratic inequalities, in one unknown
• solve by substitution a pair of simultaneous equations of which one is linear and one is quadratic
• recognise and solve equations in x which are quadratic in some function of x, e.g. x4 – 5x2 + 4 = 0.
2. Functions • understand the terms function, domain, range, one-one function, inverse function and composition of functions
• identify the range of a given function in simple cases, and fi nd the composition of two given functions
• determine whether or not a given function is one-one, and fi nd the inverse of a one-one function in simple cases
• illustrate in graphical terms the relation between a one-one function and its inverse.
Syllabus content
3. Coordinate geometry
• fi nd the length, gradient and mid-point of a line segment, given the coordinates of the end-points
• fi nd the equation of a straight line given suffi cient information (e.g. the coordinates of two points on it, or one point on it and its gradient)
• understand and use the relationships between the gradients of parallel and perpendicular lines
• interpret and use linear equations, particularly the forms y = mx + c and y – y1 = m(x – x1)
• understand the relationship between a graph and its associated algebraic equation, and use the relationship between points of intersection of graphs and solutions of equations (including, in simple cases, the correspondence between a line being tangent to a curve and a repeated root of an equation).
4. Circular measure • understand the defi nition of a radian, and use the relationship between radians and degrees
• use the formulae s = rθ and A = 2
1r2θ in solving problems concerning the arc length and sector area of a circle.
5. Trigonometry • sketch and use graphs of the sine, cosine and tangent functions (for angles of any size, and using either degrees or radians)
• use the exact values of the sine, cosine and tangent of 30°, 45°, 60°, and related angles, e.g.cos150° =
2
• fi nd all the solutions of simple trigonometrical equations lying in a specifi ed interval (general forms of solution are not included).
6. Vectors
• use standard notations for vectors, i.e.
• carry out addition and subtraction of vectors and multiplication of a vector by a scalar, and interpret these operations in geometrical terms
• use unit vectors, displacement vectors and position vectors
• calculate the magnitude of a vector and the scalar product of two vectors
• use the scalar product to determine the angle between two directions
• use the scalar product to determine the angle between two directions