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IGCSE MATHEMATICS
(0581)
COURSEWORK TRAINING HANDBOOK
© University of Cambridge International Examinations 2007
[BLANK PAGE]
2
Contents
Introduction
1
Setting Coursework Assignments
1.1
1.2
1.3
2
Administration of Coursework
2.1
2.2
2.3
2.4
3
Preparation for Successful Completion of Coursework
Choosing the Coursework Assignment
In the Classroom
Controlled Elements
Scheme of Assessment
3.1
3.2
3.3
3.4
3.5
3.6
4
What are the Advantages of Coursework?
Guidelines for Setting Coursework Assignments
Examples of Coursework Assignments
Guidance in Using the Scheme of Assessment
Overall Design and Strategy
Mathematical Content
Accuracy
Clarity of Argument and Presentation
Controlled Element
Specimen Coursework Assignments
4.1
4.2
Marked Assignments
Unmarked Assignments
APPENDIX A: Application for Accreditation and Assessment Forms
3
Introduction
The International General Certificate of Secondary Education (IGCSE) is an examination
designed for a wide ability range. It is a single system of examining which provides
opportunities for candidates of lower ability to demonstrate their knowledge, skills and
understanding whilst at the same time providing challenge to stimulate and stretch the more
able. To achieve this Mathematics is offered at two levels, referred to in the syllabus as the
Core curriculum and the Extended curriculum. As the names imply, candidates studying the
Extended curriculum cover all of the Core curriculum and a set of supplementary topics.
Much of the supplementary work extends topics met at the Core level, although some is
new.
The examination on the Extended curriculum is intended for mathematically able candidates
who are likely to proceed to further study in Mathematics or other related subjects. The
examination on the Core curriculum is intended to recognise achievement in areas of
Mathematics likely to be relevant to everyday life and employment. An eight-point scale of
grades is awarded: grades A*, A, B, C, D, E, F and G. Grades A* to E are available to
candidates studying the Extended curriculum and grades C to G are available to candidates
studying the Core curriculum. Those candidates failing to reach the required standard for
Grade E on the Extended curriculum or Grade G on the Core curriculum will be ungraded. It
is, therefore, important that candidates are entered at the appropriate level for their ability.
The abilities to be assessed in the IGCSE Mathematics examination are listed on page 2 of
the Syllabus for Mathematics 0580/0581 and are not repeated here. They cover a single
assessment objective, technique with application.
The emphasis in this examination is on positive achievement, so evidence of mastery of
most skills associated with each particular grade will be required for the award of that grade.
The examination will measure achievement in absolute (criterion referenced) terms, rather
than by relating to the performance of other candidates. The objectives should therefore be
clearer for those who teach and those who follow the examination course.
Two examination papers are set at each of the Core and Extended levels. The first contains
short-answer questions and the second contains structured questions. A Coursework option
at each level is also available.
Candidates may be entered for either Syllabus 0580 (without Coursework) or Syllabus 0581
(with Coursework). For both these specifications the written papers are identical; the
weighting of individual components, however, is different. Details of weightings are listed on
pages 3 and 4 of the Syllabus for Mathematics 0580/0581.
It is worth noting that the award rules are such that a candidate’s Coursework grade cannot
lower his or her overall result. This means that candidates entering for Syllabus 0581 (with
Coursework) are graded on their written papers together with Coursework and then graded
again on their written papers alone (as with Syllabus 0580). The higher of the two grades is
then awarded.
The following sections of this handbook focus on the Coursework component and provide
the training necessary for teachers to set, manage and assess this aspect.
In addition to this Coursework Training Handbook CIE has developed a book Teaching and
Assessing Skills in Mathematics. It is published by Cambridge University Press in the series
Cambridge Professional Development for Teachers. It is strongly recommended that you
4
read this book before completing this Handbook. The book can be ordered from CIE (via the
Publications Catalogue) or the publisher.
5
1
Setting Coursework Assignments
1.1
WHAT ARE THE ADVANTAGES OF COURSEWORK?
The Coursework component exists to provide candidates with an additional opportunity to
show their ability in Mathematics. This is especially valuable where an extended piece of
work can demonstrate ability more fully than an answer to a written question.
Coursework should aid development of the ability to solve problems, to use mathematics in a
practical way, to work independently, to apply mathematics across the curriculum, and if
suitable assignments are selected, it should enhance interest in, and enjoyment of, the
subject.
It is recommended that Coursework Assignments form an integral part of all IGCSE
Mathematics courses. Whether some of this Coursework should be submitted for
assessment, or not, is a matter for the teacher and the candidate to decide.
1.2
GUIDELINES FOR SETTING COURSEWORK ASSIGNMENTS
Coursework Assignments fall into two main categories: a mathematical investigation and an
application of mathematics. Both require a relatively structured approach. The mathematical
investigation will involve generalising a solution for a given open-ended situation and the
application of mathematics will involve consideration of a complex problem, providing a
solution, evaluating the work, recognising limitations and justifying valid improvements.
When setting Coursework Assignments it is worth considering the following points:
●
The assignment need not be lengthy or difficult. The important point is that the
students achieve something positive in mathematical terms, and that they
present their results intelligently.
●
Mathematical investigations must be open-ended in order for students to pursue
their own lines of enquiry.
●
The Internet provides a wealth of data which can be usefully exploited. Be
careful though not to allow students to spend so much time gathering information
that the mathematical analysis is superficial. Give students a limited amount of
time to collect their data.
●
Students’ assignments must be assessed using the criteria set out in the
Syllabus on pages 14 to 18. Before allowing students to engage in any work it is
worthwhile working through the assignment yourself to ensure that it is possible
to allocate the full range of marks in each strand for the work that students will
complete.
●
When setting Coursework Assignments keep the aims of this type of work in
mind. Some of these are:
○
○
○
to develop mathematical thinking, including logical reasoning;
to encourage discussion of mathematics;
to develop initiative in pursuing their own line of enquiry;
6
○
○
○
○
●
1.3
to use the mathematical skills they have learnt;
to identify and collect necessary information;
to formulate and interpret problems;
to write about the mathematics they have produced.
Above all remember that this is a mathematical assignment and emphasis must
be placed on mathematical content. High level techniques used correctly and
appropriately bring high marks.
EXAMPLES OF COURSEWORK ASSIGNMENTS
How Many?
There are 20 squares in this design.
How many squares are there in the design below?
Extend your investigation, making clear the rules and methods that you use.
This task allows candidates entered at both levels to score well against the assessment
criteria. It meets all of the relevant points given above in setting a good task. A likely
development of the task is given below.
It is fairly straightforward to draw a sequence of rectangles (3 by 1, 3 by 2, 3 by 3, etc.) and
count the number of squares. This leads to a sequence which can be examined to give an
algebraic rule connecting the number of squares to the number of columns for a rectangle
with 3 rows. At this point a candidate entered at the Extended level would score 1 mark in
each of the first 4 strands provided that a systematic approach has been adopted and a
suitable commentary on the work is given linking diagrams to tables and explaining how the
generalisation (nth term of the sequence) is found.
The next step is then up to the candidate to extend the investigation further - perhaps
considering sequences of rectangles with 4, 5 and 6 rows, each sequence providing a new
rule. Again, provided that the candidate describes the work completed and summarises each
section, a mark of 2 in each strand can be awarded at the Extended level.
If this candidate now goes on to link each of the rules together to find a rule which gives the
number of squares in an (n by m) rectangle then 3 marks can be awarded in each strand.
7
For 4 marks it would be expected that the candidate produces a more complex
generalisation, such as that obtained by considering the number of cubes within a bigger
cube, or cuboids within a bigger cuboid.
Connect 4
The game Connect 4 is a very popular one with all age groups. As shown in the
diagram above, there are 6 rows and 7 columns in a Connect 4 game.
This is a game for two players who take turns to drop their own coloured
counters into the Connect 4 framework. The counter drops vertically down the
column in which it is placed, and as far down as possible. The final position of
each counter will of course depend upon how many others are already in that
column. The winner is the first player to get 4 of their counters in a line; the line
may be horizontal, vertical or diagonal.
How many different winning lines of 4 counters are there in this game?
Extend your investigation, making the rules and methods clear.
This game usually provides a lot of interest and it is straightforward to count the number of
winning lines. Once candidates have understood the game, sequences of numbers can be
generated and rules found for specific grid sizes. The usual development is a rule which
gives the number of winning lines of length L on a grid of size (p x q).
8
Farmer’s Field
A farmer has 1000 metres of fencing. The fencing is to be used to enclose the
largest piece of land possible. The perimeter must be exactly 1000 metres.
Investigate the different shapes the farmer can consider and advise which will
give the largest area.
This investigation is accessible to all candidates and provides the opportunity for the most
able to explore the concept of limiting values.
Garden
You are asked to provide a design for a new garden. Your submission should
include a scale drawing and the total cost of materials used.
This practical application of mathematics gives candidates of all abilities the opportunity to
demonstrate their skills in a number of topic areas. It requires research into what features
might be included and provides many choices for candidates to make, stimulating interest
and engaging them in mathematics. The task brings together a whole range of topics from
accurate drawing to trigonometry in three dimensions.
Rich World, Poor World
Hypothesis: European countries are wealthier than African countries.
Write one, or more, hypotheses to compare aspects of life in two different
regions. Use appropriate data and techniques to test your hypotheses, planning
and specifying your methods carefully.
For Centres with access to the Internet this task provides a very good basis for candidates to
demonstrate their knowledge and use of statistical techniques. Experience shows that it
allows candidates to score well against the assessment criteria.
At the lowest level a candidate may record some data for European and African countries,
for example Gross Domestic Product (GDP) - which is a measure of the wealth of a country.
They may have drawn a graph or calculated the mean and made some comment based on
their data. At this point a candidate entered at the Core level would score 1 mark in each of
the first 4 strands.
9
For a mark of 2 across all strands (at the Core level) a candidate would have to write one or
more hypothesis and produce a clear plan, stating what statistical techniques are to be used
to analyse the data. The plan should then be followed with a commentary linking the various
techniques and summarising the results. Techniques of grade F or above are expected, for
example calculating the mean and range from a set of discrete data, interpreting simple
graphs. Note that drawing pictograms or bar charts, finding the mode and median for a small
set of discrete data without interpreting these are grade G techniques.
Candidates entered at the Core level who research two or more hypotheses and compare
the results (for example, considering GDP as one measure of wealth and life expectancy as
another) would score 3 marks provided that they use at least two different grade E, or
higher, techniques to analyse and comment on the data. For example pie charts and scatter
diagrams. There should be little redundancy in the calculations used to gain this mark.
For a mark of 4 across all strands (at the Core level) a candidate would be expected to be
using correct statistical terms in his or her plan, for example, ‘I am going to investigate
whether there is a negative correlation between GDP and death rate’. The write-up should
have a clear structure drawing together all components of the task. Techniques of at least
grade C should be employed to analyse the data, for example, comparing means from
grouped data, describing the strength and type of correlation, using a line of best fit and
drawing a frequency polygon for grouped data. Candidates should evaluate the task showing
an appreciation of the limitations of their data and the implications to their findings - how do
the findings relate to the whole population?
At Extended level for a mark of 4 across all strands a candidate would be expected to plan
a task with an over-arching hypothesis, breaking this down into 3 or more subtasks,
analysing the data using grade B, or better, techniques (for example, cumulative frequency
or histograms with unequal intervals) and provide a sophisticated evaluation, justifying valid
improvements.
Gulliver’s Problem
“... they measured my right thumb, and desired no more; for a mathematical
computation that twice round my thumb is once round the wrist and so on to the
neck and the waist...”
[extract from Gulliver’s Travels by Jonathan Swift]
Test Gulliver’s theory that twice the distance round a thumb is the same as the
distance around that person’s wrist.
This statistical task is accessible to all candidates. The collection of data is often completed
as a group exercise across the school (and perhaps in conjunction with a neighbouring
school). The data is usually sorted by age and gender and the analysis provides a
mathematically rich practical application.
10
2
Administration of Coursework
2.1
PREPARATION FOR SUCCESSFUL COMPLETION OF COURSEWORK
Students will be successful with Coursework Assignments if they are taught how to complete
them. This involves making problem solving and investigational work an integral part of the
scheme of work for all year groups in the school. Additionally, students need to be taught
how to meet the assessment criteria; what they need to include in their write-up and how the
write-up should be structured.
In order to achieve this it is important that all teachers in a department are able to adapt their
teaching methods to facilitate the different style of teaching necessary. Teachers need to be
given regular opportunities for feedback and evaluation so that problems can be shared.
This is particularly important if there are any non-specialists working in the department.
When several teachers in a Centre are involved in assessing Coursework Assignments for
submission as part of the IGCSE examination, arrangements must be made within the
Centre for all candidates to be assessed to a common standard. When a teacher has
assessed the work from his or her class, it is usual to select about 5 scripts and for the other
teachers to re-assess this work. Discussion about the marks awarded can then take place
and a common standard agreed. Each teacher should then modify the marks already
awarded to their class in the light of discussions.
2.2
CHOOSING THE COURSEWORK ASSIGNMENT
This relies on a teacher’s knowledge of the students’ ability. Choose assignments that
students will be interested in and can engage with. Be prepared for some surprises though.
It is often the case that students will exceed expectations given the independence and
freedom this type of work allows.
When an appropriate assignment has been chosen work through the assessment criteria in
the same way as How Many? and Rich World, Poor World have been described above. The
best Centres generate a marking guide which indicates what is expected for each mark in
the first 4 strands. This helps to ensure consistency of marking and enables a teacher to
know how well each candidate is doing as he or she completes the assignment.
Centres often offer a choice of assignments to candidates. In this case an initial discussion
with the teacher is beneficial and, indeed, often essential. Plan for this.
If a statistical assignment is chosen which requires the collection of data, allow adequate
time for candidates to do this. The data collection counts for few marks but often requires the
most time. Teachers in some Centres generate their own database of information from which
candidates can select the information they require. This speeds the process enormously and
allows candidates to concentrate on the most important part of the assignment, the analysis.
When planning which assignments to offer it is a good idea to make a note of the equipment
likely to be needed. Ensure that this is available in the classroom for students; it might
include different types of paper (lined, squared, graph, isometric, etc).
11
Experience shows that students who encounter coursework for the first time as an IGCSE
assignment do not score well against the criteria. Students often need help in producing a
systematic analysis of a problem. This is particularly apparent with a mathematical
investigation where students must show a clear, well structured path to obtain a
generalisation. Students who have experienced short mathematical assignments throughout
their time at school know what they must do to meet the assessment criteria and are
therefore at a distinct advantage.
Before starting assignments with groups of students it is good practice to inform parents that
this work is going to be completed over the next few weeks and that there is a deadline for
handing in work. In this way issues such as ‘what happens if a student is absent for part or
all of the time’ can be dealt with beforehand.
2.3
IN THE CLASSROOM
Students need not work in isolation on the assignment. The important point is that the writeup needs to be completed individually. Try working in small groups when the assignment is
first discussed. Manipulate the groups so that one student does not do all the work; this may
mean that some students work alone whilst others work in a group. Encourage discussion
(provided it is relevant).
Watch for conscientious students spending too much time on the assignment, both in class
and as homework. Remember that they may well have project work in several other subjects
as well.
Notwithstanding the comments above, stand back as much as possible. Part of the value of
this type of work is for the students to make their own mistakes and pursue ideas up blind
alleys; to arrive at a conclusion by a roundabout route. Restrict the amount of guidance you
give whilst students are completing their assignments. For example, in a mathematical
investigation students often want to know if a generalisation is correct. This is best answered
by asking the student to check his or her generalisation by showing that further examples fit
the pattern or to show that it must be true by examining the mathematical structure of the
work.
Be considerate to other teachers in the school, for example when an assignment involves
going out of the classroom to collect data (measuring, conducting opinion polls, surveying).
Plan for students to be able to access information, either in the library or from the Internet.
Some students may wish to use a computer as their work progresses. This is to be
encouraged, but they must realise that their work will be assessed on personal input, and not
on what the computer does for them. For example, statistical diagrams can be drawn easily
by a computer programme, but marks are awarded for the appropriateness of the diagram
and the written analysis, not for the ability to draw the diagram.
Students often find that the most demanding part of any assignment is presenting their
conclusions in an intelligible form. This can only be achieved well with practice and relates to
the point made earlier about completing assignments as an integral part of the mathematics
studied throughout school.
A good Coursework Assignment is normally between 8 and 15 sides of A4 paper in length.
These figures are only for guidance; the write-up for an assignment involving an application
of mathematics may need to be longer in order to present all the findings properly.
12
The time spent on a Coursework Assignment will vary, according to the candidate. As a
rough guide, between 10 and 20 hours would seem reasonable.
A final point to make here is that the write-up must show evidence of investigation by the
student, and must be his or her personal effort. Worksheets which include a high degree of
teacher direction are therefore inappropriate.
2.4
CONTROLLED ELEMENTS
If all the candidates in the class have completed the same Coursework Assignment then a
written test is a suitable form of controlled element. It should last for between 30 and 45
minutes. Each candidate should complete the test without assistance, and it should test all
the aspects of the task that the teacher hopes his or her students have learned. The level of
difficulty must also be appropriate and this will probably be different for some students in the
class. Written tests are most revealing if given a week or so after the completion of the
assignment, when ideas have had time to settle but should not have been forgotten.
If the candidates have been working on a series of different tasks, then a written summary
of the Coursework Assignment may be preferred. This need not be more than one A4 page,
but should be completed under supervised conditions without access to books or notes.
An oral exchange is a very good controlled element. The ability to respond orally to
mathematical questions, and to discuss mathematical ideas is an important skill. It is,
however, time-consuming and demanding on the teacher, who must be ready with plenty of
questions and prompts and yet (like a television interviewer) allow the candidate to do the
bulk of the talking. No more than five minutes with each candidate is necessary, unless the
candidate is very hesitant. [For external moderation purposes notes should be kept of the
interview and submitted alongside the candidate’s work.]
A parallel task should be completed under supervision, without help, and should be of such
a length that it can be completed in 30 to 45 minutes. It should involve, and assess, the
same mathematics as that involved in the main Coursework Assignment. For example, the
controlled element for a mathematical investigation might involve finding the nth-term of a
sequence, whilst the controlled element for an application of mathematics might involve
trigonometry.
13
3
Scheme of Assessment
The Scheme of Assessment for Coursework Assignments is divided into 5 strands: Overall
Design and Strategy; Mathematical Content; Accuracy; Clarity of Argument and
Presentation; Controlled Element.
In each strand there is a maximum of 4 marks available, awarded on a five-point scale: 0, 1,
2, 3, 4. The Coursework Assignment is, therefore, marked out of 20. The marks submitted
for each candidate will be scaled by CIE to a weighting of 20% of the total marks available
for the examination.
Participating schools will be provided with appropriate forms on which to enter the marks for
Coursework Assignments. Alternatively, the forms can be photocopied from the Syllabus.
Change to a candidate’s level of entry (from Extended to Core or vice versa) may be made
in some cases. Coursework Assignments already completed and assessed by the teacher
must be re-assessed at the new level before moderation. When a candidate transfers from
the Core level to the Extended level, opportunity must be given for the candidate to extend
his or her assignment and show competence at the higher level of mathematics.
The next five pages contain the detailed criteria for the assessment of Coursework. The
criteria are couched in general terms so as to be relevant to as many different types of work
as possible.
3.1
GUIDANCE IN USING THE SCHEME OF ASSESSMENT
It is important that students outline their own strategy and do not try to answer a question
posed without explaining what they are doing (and more importantly why they have chosen a
particular line of enquiry). Mathematics must be overt with a commentary to link the tables,
calculations, diagrams or derivation of algebraic formulae together, without this very few
marks will be scored.
Initially you must consider the mathematics employed. If it is low level, then the outcome
must be a low mark irrespective of the neatness and clarity of the written work. When
marking ask yourself the following questions:
Overall Design and Strategy -- Has the student summarised the task so that you know
what it is about? Is there a clear strategy for working on the task? Has the student followed
the plan and produced generalisations? Has the student extended the task in their own way?
Mathematical Content and Accuracy -- These strands are linked and usually have the
same mark - the mark for Accuracy will rarely exceed the mark for Mathematical Content.
What is the level of mathematics in the task? For example, at Extended level, grade E
techniques would be synonymous with a mark of 1 and grade A or B techniques with a mark
of 4; at Core level, grade F techniques would be synonymous with a mark of 1.
Clarity of Argument and Presentation -- Has the student linked the work together with a
commentary explaining the processes used? Is the interpretation correct? Are algebraic
statements correctly derived? Has the student outlined the limitations of the solution?
14
3.2
OVERALL DESIGN AND STRATEGY
Assessment Criteria
Much help has been received.
No apparent attempt has been made to plan the work.
Core
Extended
0
0
1
0
2
1
3
2
4
3
4
4
Help has been received from the teacher, the peer group or a
prescriptive worksheet.
Little independent work has been done.
Some attempt has been made to solve the problem, but only at
a simple level.
The work is poorly organised, showing little overall plan.
Some help has been received from the teacher or the peer
group.
A strategy has been outlined and an attempt made to follow it.
A routine approach, with little evidence of the student’s own
ideas being used.
The work has been satisfactorily carried out, with some
evidence of forward planning.
Appropriate techniques have been used; although some of
these may have been suggested by others, the development
and use of them is the student’s own.
The work is well planned and organised.
The student has worked independently, devising and using
techniques appropriate ot the task.
The student is aware of the wider implications of the task and
has attempted to extend it. The outcome of the task is clearly
explained.
The work is methodical and follows a flexible strategy to cope
with unforeseen problems.
The student has worked independently, the only assistance
received being from reference books or by asking questions
arising from the student’s own ideas.
The problem is solved, with generalisations where appropriate.
The task has been extended and the student has demonstrated
the wider implications.
15
3.3
MATHEMATICAL CONTENT
Assessment Criteria
Little or no evidence of any mathematical activity.
The work is very largely descriptive or pictorial.
A few concepts and methods relevant to the task have been
employed, but in a superficial and repetitive manner.
A sufficient range of mathematical concepts which meet the
basic needs of the task has been employed.
More advanced mathematical methods may have been
attempted, but not necessarily appropriately or successfully.
The concepts and methods usually associated with the task
have been used, and the student has shown competence in
using them.
Core
The student has used a wide range of Core syllabus
mathematics competently and relevantly, plus some
mathematics from beyond the Core syllabus.
Extended The student has developed the topic mathematically
beyond the usual and obvious. Mathematical
explanations are concise.
Core
Extended
0
0
1
0
2
1
3
2
4
3
4
4
A substantial amount of work, involving a wide range of
mathematical ideas and methods of Extended level standard or
beyond.
The student has employed, relevantly, some concepts and
methods not usually associated with the task in hand.
Some mathematical originality has been shown.
16
3.4
ACCURACY
Assessment Criteria
Very few calculations have been carried out, and errors have
been made in these.
Core
Extended
0
0
1
0
2
1
3
2
4
3 or 4*
Diagrams and tables are poor and mostly inaccurate.
Either correct work on limited mathematical content or
calculations performed on a range of Core syllabus topics with
some errors.
Diagrams and tables are adequate, but units are often omitted
or incorrect.
Calculations have been performed on all Core syllabus topics
relevant to the task, with only occasional slips.
Diagrams are neat and accurate, but routine; and tables
contain information with few errors.
The student has shown some idea of the appropriate degree of
accuracy for the data used.
Units are used correctly.
All the measurements and calculations associated with the task
have been completed accurately.
The student has shown an understanding of magnitude and
degree of accuracy when making measurements or performing
calculations.
Accurate diagrams are included, which support the written
work.
Careful, accurate and relevant work throughout. This includes,
where appropriate, computation, manipulation, construction and
measurement with correct units.
Accurate diagrams are included which positively enhance the
work, and support the development of the argument.
The degree of accuracy is always correct and appropriate.
*According to the mark for mathematical content.
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3.5
CLARITY OF ARGUMENT AND PRESENTATION
Assessment Criteria
Haphazard organisation of work, which is difficult to follow. A
series of disconnected short pieces of work. Little or no attempt
to summarise the results.
Core
Extended
0
0
1
0
2
1
3
2
4
3
4
4
Poorly presented work, lacking logical development.
Undue emphasis is given to minor aspects of the task, whilst
important aspects are not given adequate attention.
The work is presented in the order in which it happened to be
completed; no attempt is made to re-organise it into a logical
order.
Adequate presentation which can be followed with some effort.
A reasonable summary of the work completed is given, though
with some lack of clarity and/or faults of emphasis.
The student has made some attempt to organise the work into
a logical order.
A satisfactory standard of presentation has been achieved.
The work has been arranged in a logical order.
Adequate justification has been given for any generalisations
made.
The summary is clear, but the student has found some difficulty
in linking the various different parts of the task together.
The presentation is clear, using written, diagrammatic and
graphical methods as and when appropriate.
Conclusions and generalisations are supported by reasoned
statements which refer back to results obtained in the main
body of the work.
Disparate parts of the task have been brought together in a
competent summary.
The work is clearly expressed and easy to follow.
Mathematical and written language has been used to present
the argument; good use has been made of symbolic, graphical
and diagrammatic evidence in support.
The summary is clear and concise, with reference to the
original aims; there are also good suggestions of ways in which
the work might be extended, or applied in other areas.
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3.6
CONTROLLED ELEMENT
Assessment Criteria
Core
Extended
0
0
1
0
2
1
3
2
Little or no evidence of understanding the problem.
Unable to communicate any sense of having learned something
by undertaking the original task.
Able to reproduce a few of the basic skills associated with the
task, but needs considerable prompting to get beyond this.
Can answer most of the questions correctly in a straightforward
test on the project.
Can answer questions about the problem and the methods
used in its solution.
Can discuss or write about the problem, in some detail.
Shows competence in the mathematical methods used in the
work.
Little or no evidence of having thought about possible
extensions to the work or the application of methods to different
situations.
Can talk or write fluently about the problem and its solution.
4
Has ideas for the extension of the problem, and the applicability
of the methods used in its solution to different situations.
*Dependent on the complexity of the problem and the quality of the ideas.
3 or 4*
19
4
Specimen Coursework Assignments
The following specimen Coursework Assignments are included in this section. Assignments
1 – 5 have been assessed and a commentary is provided for each one. The remaining four
Assignments are included for you to assess. The marks for these four pieces of work and
comments as to why each of the marks has been awarded should be submitted for
accreditation purposes. Please write your comments and marks on the mark sheets provided
in appendix one and return it to CIE (see appendix A).
Assignment
Level
Title
1
Extended
Diagonals
2
Extended
Number Grid
3
Core
Noughts and Crosses
4
Extended
Estimation
5
Core
Designing a Swimming Pool
6
Extended
Anyone for T
7
Extended
Weekly Exercise
8
Core
Mystic Rose
9
Core
Probability
20
4.1
MARKED ASSIGNMENTS
Assignment 1
DIAGONALS
EXTENDED LEVEL IGCSE ASSIGNMENT
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
Assignment 1 Commentary and Marks
The work starts without introduction and launches straight into drawing diagrams and
collecting the results in tables. The work is systematic and the candidate considers
rectangles with 1, 2, 3, 4 and 5 rows.
On page 4 the candidate starts to analyse the sequences of numbers generated (confusing
columns with rows in the notation). The candidate gives a generalisation in words ‘... the
results are simply added by one’ and tests this on a new example which has not been used
to generate the rule.
On page 5 the candidate writes down an algebraic formula for the rule (note the confusion
with rows and columns). This rule simply appears without explanation or derivation. The
candidate then checks this rule with further examples.
The work is extended into 3 dimensions, starting on page 7. Again the work is systematic
and results are recorded in tables. The candidate explains that a diagnoal line through the
cuboid is too complicated so that a diagnoal line across the surface of the cuboid will be
considered.
On page 10 the candidate notes that the results for the extension will be similar to those for
the original 2 dimensional investigation and tries to spot a connection without success.
Classification of
Assessment
Comments
Mark
The work is structured with a minimal linking
commentary. Appropriate techniques have been
used and an attempt to extend the task has been
made. The extension, however, is essentially the
same as the original task.
2
The candidate understands what is needed, but
formulae simply appear and are tested. There is
no evidence to suggest competence in moving
from a sequence to an algebraic rule.
1
Diagrams are neat and accurate. Tests on
formulae are appropriate (not using data already
generated) and correct.
1
Clarity of argument and Adequate presentation is give and work is
presentation
summarised throughout. No justification is given
for formulae written down.
1
Overall design and strategy
Mathematical content
Accuracy
Controlled element
The candidate shows understanding and can
answer some basic questions, but has problems
applying the ideas in a new situation.
2
TOTAL
7
GRADE
E
36
Assignment 2
NUMBER GRID
EXTENDED LEVEL IGCSE ASSIGNMENT
37
38
39
40
41
42
43
44
45
46
47
48
49
Assignment 2 Commentary and Marks
The candidate sets out the problem and starts by examining a rule involving a 2 × 2 square
on a larger 10 × 10 grid. The rule is justified in terms of algebra, although the precise link
between the variable X and its position in the 2 x 2 square is not explained.
The rule is then applied to a 3 × 3 square and a 4 × 4 square (each on a 10 × 10 grid) and
similarly justified. By pattern spotting the candidate notices that there is a link between each
of these cases and suggests the formula 10(N-1)2 on page 3. Note that N is not defined. The
candidate then successfully checks this generalisation with a 5 × 5 square on a 10 × 10 grid.
The problem is now extended to squares on a 9 × 9 grid. By page 6 a formula has been
spotted which holds for this new size of grid. The formula is again checked with a 5 × 5
square. There is no attempt to link this with the previous formula.
A further extension to the work using rectangles on a 10 × 10 grid is now investigated. The
investigation is systematic and the candidate is able to spot a pattern from the results on
page 9.
Instead of introducing a variable grid size which would have introduced complex
mathematics, the candidate now decides to investigate a cross shape on a 10 × 10 grid. This
is essentially the same level of mathematics as previously seen.
Classification of
Assessment
Comments
Mark
Overall design and The work is planned and well structured. The work has
strategy
been extended, although not to a level which would
introduce more than a 2 variable formula. The outcome
of each separate investigation is summarised.
3
Mathematical
content
Algebra is manipulated correctly to produce
generalisations. The generalisations are not linked to
produce a higher order formula. The candidate shows
competence in using the appropriate mathematics.
3
Accuracy
Variables are not defined, nevertheless the work is
accurate and relevant, supporting the formulae
obtained.
3
Clarity of argument The candidate explains the work and uses tables and
and presentation
diagrams appropriately. Adequate justification is given
for developing individual rules which are linked
together by pattern spotting. Reasoning is implicit
rather than mathematical justification.
3
Controlled element
The Centre opted for an oral exchange to assess this
strand. The teacher commented that the candidate
could explain the work and show how a formula could
be found for variable sized square grids.
3
TOTAL
15
GRADE
B
50
Assignment 3
NOUGHTS AND CROSSES
CORE LEVEL IGCSE ASSIGNMENT
51
52
53
54
55
56
57
58
59
60
Assignment 3 Commentary and Marks
The candidate sets out the problem of Noughts and Crosses and starts by systematically
examining the number of winning lines on (3 × 3), (4 × 4) and (5 × 5) grids. The information
is gathered in a table on page 2 and the candidate explains why the number of winning lines
is 2n + 2.
The candidate then extends the problem by considering the number of winning lines of 3 on
(4 × 4) and (5 × 5) grids. The candidate considers the number of horizontal, vertical and
diagonal lines separately. These results are drawn together in a table on page 6. A formula
is found by pattern spotting which gives the number of winning lines of length 3 in an (n × n)
grid.
Classification of
Assessment
Comments
Mark
Overall design and
strategy
There is a systematic approach to collecting
data. Appropriate techniques are employed and
the candidate has extended the task.
4
Mathematical content
The candidate shows competence in using
methods which lead to algebraic generalisations.
There is no manipulation of algebra.
3
Accurate diagrams and tables are shown which
support the written work. n is defined implicitly in
the tables.
3
Clarity of argument and The presentation is clear. Generalisations are
presentation
explained by relating the formulae obtained to
the diagrams drawn.
4
Accuracy
Controlled element
Some of the more challenging questions are
answered correctly whilst mistakes are made
with the straightforward ones at the beginning of
the written test.
2
TOTAL
16
GRADE
C
61
Assignment 4
ESTIMATION
EXTENDED LEVEL IGCSE ASSIGNMENT
62
63
64
65
66
67
68
69
70
71
72
Assignment 4 Commentary and Marks
The candidate starts by outlining a plan to investigate whether the hypotheses ‘the older you
are the better you are at estimating’ and ‘boy are better than girls at estimating’ are true. A
random sample is taken from a large data set and the candidate dicusses some limitations of
the investigation on page 2.
After some discussion about how the mode, mean, range and standard deviation will help in
the analysis, age related results are presented on pages 3 and 4. These are then discussed
and a conclusion drawn that there is no link between age and the ability to make a good
estimate. A further comparison between age and estimating the area of a rectangle is made
through a scatter diagram. Correct use is made of statistical terms and there is little
redundancy in techniques used.
The candidate continues the task by considering the second hypothesis. Again the mode,
mean, range and standard deviation are used to provide an initial analysis of the data. A
cumulative frequency curve, together with the associated box and whisker plot is drawn to
help compare the data. Correct conclusions are drawn.
On pages 9 and 10 the candidate evaluates the work, outlining what improvements could be
made and why these improvements would make a difference.
Classification of
Assessment
Comments
Mark
Overall design and Two hypotheses are stated. The plan is well structured
strategy
and allows comparisons to be made across a number
of areas. All aspects of the work are drawn together.
Correct statistical terms are used throughout.
Sampling is used with justification.
4
Mathematical
content
A range of techniques is used to analyse the data. At
least two of these are grade B or better. The data is
chosen so as to be representative of the population.
Competence in using statistics is demonstrated.
4
Accuracy
Results are generally correct and there is little
redundancy in calculations used. Diagrams enhance
the understanding and support the arguments used.
4
Clarity of argument The work is expressed clearly and results summarised
and presentation
so that appropriate conclusions can be drawn, relating
them back to the original aims. The candidate
understands the limitations of the strategy used and
proposes
improvements,
saying
why
these
improvements might be effective.
4
Controlled element
The Centre opted for an oral exchange to assess this
strand. The teacher commented that the candidate
could explain the work well showing an appreciation of
the significance to be placed on the conclusions.
4
TOTAL
20
GRADE
A
73
Assignment 5
DESIGNING A SWIMMING POOL
CORE LEVEL IGCSE ASSIGNMENT
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
Assignment 5 Commentary and Marks
The candidate outlines a plan to tile a swimming pool, taking into account many aspects of
the work to be completed. The first section of the work considers different tiling patterns, all
using the same shape tile. The candidate suggests (on page 3) that, however the tiles are
positioned, approximately the same number will be needed to cover a given area. Costs of
different tiles are then listed on page 4.
The candidate then shows how to work out the number of tiles needed to cover a specific
area when grouting between tiles is not included. The next section of work explores formulae
to work out the area covered by a tile and its grouting depending on whether one, two, three
or four sides of the tile have grouting. All variables are defined. The candidate then repeats
the calculation to find the number of tiles needed to cover a specific area when grouting is
included.
Two different shapes of swimming pool are then considered (cuboid and cylinder) and their
nets sketched. No calculations are performed. Finally the candidate conducts a survey (with
11 people) to find out which of three tile types they prefer. The results are displayed in a bar
chart.
This is a case when some help from the teacher in designing the plan to follow would have
helped enormously. There is very little mathematical calculation or accurate drawing in
evidence.
Classification of
Assessment
Comments
Mark
Overall design and The work is organised and the candidate has made an
strategy
attempt to follow the initial plan, but only at a simple
level.
1
Mathematical
content
A simple general formula is found for the area of a tile
and its grouting and the candidate demonstrates a
knowledge of nets. This is not a sufficient range of
mathematical concepts for this task.
1
Accuracy
No accurate drawing, but accurate formulae produced.
Units are used correctly.
1
Clarity of argument A satisfactory standard of presentation and order of
and presentation
work. Undue emphasis is placed on minor aspects of
the task
1
Controlled element
Competent in calculating area, including changing
units. Mistakes made in substitution.
1
TOTAL
5
GRADE
U
92
4.2
UNMARKED ASSIGNMENTS
Assignment 6
Marks for this assignment should be submitted for accreditation purposes, using the forms in
Appendix A. Note that there is no controlled element for this assignment so no marks can be
allocated for the controlled element.
ANYONE FOR T
EXTENDED LEVEL IGCSE ASSIGNMENT
93
94
95
96
97
98
99
100
101
102
103
Assignment 7
Marks for this assignment should be submitted for accreditation purposes, using the forms in
Appendix A. Note that there is no controlled element for this assignment so no marks can be
allocated for the controlled element.
WEEKLY EXERCISE
EXTENDED LEVEL IGCSE ASSIGNMENT
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
Assignment 8
Marks for this assignment should be submitted for accreditation purposes, using the forms in
Appendix A.
MYSTIC ROSE
CORE LEVEL IGCSE ASSIGNMENT
121
122
123
124
125
126
127
128
129
Assignment 9
Marks for this assignment should be submitted for accreditation purposes, using the forms in
Appendix A.
PROBABILITY
CORE LEVEL IGCSE ASSIGNMENT
130
131
132
133
134
135
136
137
138
139
140
141
142
APPENDIX A: Application for Accreditation and Assessment Forms
Teachers wishing to gain accreditation for the assessment of coursework should complete
copies of the Individual Candidate Record Card for Assignments 6, 7, 8 and 9. The
completed forms should be sent to the address below along with the Application for
Accreditation Form:
IGCSE Accreditation Co-ordinator
PQAD Group
University of Cambridge Examinations
1 Hills Road
Cambridge
CB1 2EU
United Kingdom
Please allow between four and six weeks for CIE to assess the work that is submitted and to
inform you of an outcome.
Applicants will be informed of CIE’s decision regarding accreditation (the outcome) by post.
This will take the form of a Certificate in instances where accreditation is awarded, or in the
cases where accreditation cannot be awarded, a letter informing you of this. A report may
also be enclosed which will give guidance in the cases of accreditation failure and feedback
in the event that accreditation is awarded. It is not usual practice for CIE to inform teachers
or centres of accreditation outcomes over the telephone or by e-mail.
If accreditation is not awarded on one particular occasion this does not mean a teacher
cannot continue to teach, it simply restricts their ability to mark the practical examination until
accreditation is awarded. You may re-submit work for assessment as many times as is
necessary for accreditation to be awarded. There is, however, a charge each time for doing
so.
If you have any further questions, please contact CIE. The address to which your queries
should be sent is:
Customer Services
University of Cambridge International Examinations
1 Hills Road
Cambridge CB1 2EU
United Kingdom
Telephone: +44 1223 553554
Fax:+44 1223 553558
E-mail: international@ucles.org.uk
Website: www.cie.org.uk
143
APPLICATION FOR ACCREDITATION TO MARK
COURSEWORK IN IGCSE MATHEMATICS (0581)
Centre number
Centre name
Telephone
Centre address
Fax
Name of teacher
requesting
accreditation
Email address
Enclosed with this application form are the following documents:
Individual Candidate Record Cards completed for Assignments 6, 7, 8 and 9
I have marked coursework samples 6, 7, 8 and 9 using the information provided in the
training handbook and my own professional judgement
Signed
Name (in block capitals)
144
MATHEMATICS
Individual Candidate Record Card
IGCSE 2007
Please read the instructions printed overleaf and the General Coursework Regulations before completing this form.
Centre Number
Centre Name
June/November
Candidate Number
Candidate Name
Teaching Group/Set
2
0
0
7
Title(s) of piece(s) of work:
Classification of Assessment
Use space below for Teacher’s comments
Overall design and strategy
(max 4)
Mathematical content
(max 4)
Accuracy
(max 4)
Clarity of argument and presentation
(max 4)
Controlled element
(max 4)
Mark awarded
TOTAL
Mark to be transferred to Coursework Assessment Summary Form
WMS329
(max 20)
0581/05&06/CW/S/07
145
INSTRUCTIONS FOR COMPLETING INDIVIDUAL CANDIDATE RECORD CARDS
1.
Complete the information at the head of the form.
2.
Mark each item of Coursework for each candidate according to instructions given in the Syllabus and Training Manual. If a candidate
submits two assignments they should first be assessed separately in each category. The assessments should then be combined by
entering the higher of the two marks for each category in the ‘marks awarded’ column.
3.
Enter marks and total marks in the appropriate spaces. Complete any other sections of the form required.
4.
The column for teachers’ comments is to assist CIE’s moderation process and should include a reference to the marks awarded.
Comments drawing attention to particular features of the work are especially valuable to the Moderator.
5.
Ensure that the addition of marks is independently checked.
6.
It is essential that the marks of candidates from different teaching groups within each Centre are moderated internally. This
means that the marks awarded to all candidates within a Centre must be brought to a common standard by the teacher responsible for
co-ordinating the internal assessment (i.e. the internal moderator), and a single valid and reliable set of marks should be produced which
reflects the relative attainment of all the candidates in the Coursework component at the Centre.
7.
Transfer the marks to the Coursework Assessment Summary Form in accordance with the instructions given on that document.
8.
Retain all Individual Candidate Record Cards and Coursework which will be required for external moderation. Further detailed
instructions about external moderation will be sent in late March of the year of the June Examination and in early October of the year of
the November examination. See also the instructions on the Coursework Assessment Summary Form.
Note:
These Record Cards are to be used by teachers only for students who have undertaken Coursework as part of their IGCSE.
0581/05&06/CW/I/07
146