Two GCSEs in mathematics
MEI
Background
“Respondents to the Inquiry also report the universal perception among teachers and
pupils that the amount of effort required to obtain the single GCSE in mathematics is
similar to that needed to obtain the two awards in English or the double award in
science.” (2004 Smith Report) The Report included the recommendation that
consideration be given to re-designating GCSE Mathematics as a double award.
Subsequently, a decision was made that there would be two separate GCSEs in
Mathematics rather than one double award GCSE. Piloting work in England has focused
on piloting two GCSEs based on the same content. However, it has so far proved difficult
to make two such qualifications sufficiently distinct from each other. There has also been
concern about whether one, or both, GCSEs should be the gatekeeper for progress to A
Level Mathematics.
The new National Curriculum stresses the importance of mathematical process skills. It is
statutory for first teaching from September 2008 for KS3 and September 2010 for Key
Stage 4. This new National Curriculum will require new assessment instruments. In
addition to other changes to GCSE, Functional Skills qualifications are being introduced
and a pass in Mathematics Functional Skills at level 2 will be required for candidates to
qualify for the award of A* to C in Mathematics GCSE.
The new GCSEs offer the opportunity to make a significant improvement to students’
experience of mathematics at this level. This opportunity must not be missed.
This proposal for two GCSEs in Mathematics
The starting point for this proposal was to consider how the introduction of the two
GCSEs could result in improved learning of mathematics at this level.
It seems clear that there are several major problems with the current system.
• Many students dislike mathematics and consider it pointless and difficult.
• Some students who succeed in terms of getting grade C, or above, in GCSE
Mathematics still find it difficult to use mathematics in future study or
employment.
• Some students find the transition from GCSE to A Level Mathematics difficult and
so do not successfully complete the A Level.
• The most able students are not sufficiently challenged by current GCSE
Mathematics examination papers. This has a knock-on effect on teaching in some
classrooms so that students’ experience of mathematics is not particularly rich.
This proposal is designed to overcome these problems.
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The two GCSEs are called Contextual Mathematics and Mathematics. The vast majority
of students are expected to take both, either simultaneously or sequentially.
Design considerations
The criteria designed by QCA were constrained by a requirement that both GCSE
specifications should cover the same content. (The same was also true for those
subsequently devised by ACME.) However, Ofqual has raised concerns about whether
two sufficiently distinctive GCSEs in Mathematics can be developed along these lines.
A possible reaction to this is to conclude that it is not practicable to have two GCSEs in
mathematics and that we should therefore give up the idea altogether. However, to do so
would be to miss a significant opportunity to enable more of our young people to become
more successful at learning mathematics.
An alternative view is to say that, since the requirement for both GCSE specifications to
cover the same content has led to problems, it should be relaxed.
The design of the two GCSEs presented here is based on the two specifications
emphasising different process skills, with different, but overlapping, content.
GCSE Contextual Mathematics
Contextual Mathematics focuses on functionality, and to a greater depth than the
Functional Skills qualification. It includes the syllabus for the level 1 and 2 Functional
Skills qualifications (the Functional Skills Standards) but also has additional content and
assessment leading to a full GCSE. Much of the additional content builds on work
specified in the Functional Skills standards but there are also topics which are important
for those going into work or studying a broad range of other subjects. However,
Contextual Mathematics is not designed to be a complete preparation for A Level study of
Mathematics.
Contextual Mathematics explores ways of working with contextual problems in greater
depth than GCSE Mathematics does. At present, the assessment space available for the
single GCSE in Mathematics is very limited; it is not possible to examine topics to any
great depth while maintaining syllabus coverage. This impacts on teaching, discouraging
in-depth exploration of contextual problems in the classroom. By contrast, it will be
possible for the assessment of Contextual Mathematics to include work that is altogether
less trivial; thus the questions which students would be asked in Contextual Mathematics
examinations, especially at Higher tier, will often be more challenging than questions
addressing the same content and skills in the current GCSE examinations.
Contextual Mathematics is thus designed to prepare students to be responsible citizens
who understand and can use the mathematics which they are likely to encounter in their
other GCSEs and in their future lives. It covers a subset of the KS4 Programme of Study
appropriate to this aim. Many students, including the most able, are challenged and
motivated by using mathematics in context. It is, therefore, not intended or expected that
this GCSE will be seen as easier than GCSE Mathematics. GCSE Contextual
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Mathematics would be a good preparation for many, but not all, level 3 courses in other
subjects. As a GCSE, it would have currency in the world beyond school.
More students taking GCSE Contextual Mathematics would find that their studies are
purposeful and successful than is the case for the current GCSE. Consequently, they will
be more likely to want to go on to use and study mathematics further.
GCSE Mathematics
The proposed GCSE Mathematics focuses on smoothing the transition for students to take
A Level in mathematics and other related subjects. It is designed to ensure that students
taking it are better prepared than those taking the current GCSE. However, it does not
concentrate on preparing students to be able to use mathematics in the workplace and in
other courses in the same way that Contextual Mathematics does.
GCSE Mathematics covers the full KS4 programme of study in mathematics. It aims to
further develop students’ understanding of mathematics at this level. All students taking
Mathematics will also take either Contextual Mathematics or a separate Functional Skills
assessment. This allows the assessment of Mathematics GCSE to concentrate on those
topics and processes which are only assessed lightly, if at all, in Contextual Mathematics.
These include greater facility in working with algebra and greater rigour, including the
idea of proof. Students who are successful in GCSE Mathematics will be well prepared
to embark on A Level Mathematics, or equivalent courses, with confidence.
The two GCSEs
The purpose and emphasis of each GCSE is different. The two GCSEs are free standing;
it is possible for a student to take just one of them and then to progress to further study.
Although Contextual Mathematics does not cover the whole of the KS4 Programme of
Study, it does draw from the full range of content areas.
Students who are successful in both GCSEs will develop skills in using mathematics in
context and a greater fluency in the mathematical techniques which are required for
courses of further study that demand mathematical rigour. Many students find that one of
these aspects of mathematics comes more naturally to them than the other.
Taking both GCSEs will enable more students to experience success in mathematics and
this will give them greater confidence to work on areas which they find challenging.
Enabling students to study more mathematics
In a letter to Ken Boston at QCA dated 17/3/06, the Parliamentary Under Secretary of
State for Schools, expressed concern that a double award (i.e. two GCSE grades for one
examination) might result in a diminution of the amount of mathematics being studied:
“We share your view that faced with the choice between a double award and functional
skills which could be certificated in their own right, many students might not be entered
for maths GCSE at all.”
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Although a double award GCSE has been ruled out, the current consultation on GCSE
criteria makes it clear that a possible scenario is that of one Mathematics GCSE with a
separate Functional Skills qualification, which is a pre-requisite for GCSE grades A* to C.
Under this arrangement, Lord Adonis’s concerns would again be relevant. It would be
possible that some students might spend the whole of Key Stage 4 working towards the
Functional Skills qualification, with only those who pass it being given access to the full
programme of study.
By contrast, working towards Contextual Mathematics GCSE would allow these students
to study more mathematics. Having access to a wider curriculum and the motivation of
working towards a full GCSE would increase the likelihood of these students being
successful in achieving the Functional Skills qualification as well as ensuring that they
have access to opportunities to progress further if they wish to do so.
Status
The two GCSEs will be different, with each one a worthwhile qualification in its own
right. Neither of them should be regarded as the lesser qualification.
The achievement of students in both GCSEs should be recognised in schools’ performance
measures.
Possible regulatory concerns
Are the two GCSEs of similar size and demand?
To consider the size and demand of these two GCSEs, something of a new mind set is
required.
Because mathematics assessments tend to differentiate by task rather then by outcome, a
strong link exists between items of subject content and GCSE grades, with the result that
demand and content have come to be equated. However, it is possible to ask both easier
and harder questions on the same subject matter and questions set in context tend to fall at
the harder end of the spectrum. GCSE Contextual Mathematics, therefore, needs less
subject content in it than Mathematics: the difficulty which many students find in using
mathematics in context ensures that it is a full GCSE in demand. An examination
addressing all the content and skills in the programme of study and doing so in context
would be far too demanding for a GCSE.
Making Mathematics Count argued that GCSE Mathematics should be worth more than
one GCSE, on the grounds that students do more work for mathematics than for other
subjects with a single GCSE qualification. Indeed, this imbalance has been clear since the
inception of the National Curriculum, when mathematics, like English, was expected to
take up more classroom time than other subjects. Under these proposals, students will be
able to take both GCSEs and so gain two separate qualifications. If a school only enters
its students for GCSE Mathematics, rather than for both GCSEs, then its students might do
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more than one GCSE’s worth of work, but this is already the case under the current
system. It will clearly not be in schools’ best interest to fail to enter students for both
GCSEs.
Is there too much overlap between the two GCSEs?
There is widespread recognition that the current GCSE in Mathematics requires more
work than GCSEs in other subjects; however, it is not desirable to split the content
between two separate qualifications without having any overlap between them. To do so
would hinder the fostering of links in students’ minds between different areas in
mathematics. The recognition that these links exist is important for students’
understanding of mathematics and their ability to use it.
Assessment of the common content will have a different emphasis in the two GCSEs and
will often require the use of different methods. GCSE Contextual Mathematics will
concentrate on using mathematics in context. GCSE Mathematics will concentrate more
fully on mathematics for its own sake and, where contexts are used, they are more likely
to be drawn from mathematics itself. The types of problems set in the two examinations
would be very different, and the two examinations could use rather different assessment
structures.
How does GCSE Contextual Mathematics relate to functional mathematics?
GCSE Contextual Mathematics addresses all the skills within functional mathematics, and
considerably extends the content and techniques that may be assessed in context. In
GCSE Contextual Mathematics these are not restricted to material up to National
Curriculum Level 4, or Level 6, but include material across levels 1 and 2 of the National
Qualifications Framework. GCSE Contextual Mathematics will, therefore, be a much
larger and more challenging qualification than Functional Skills Mathematics at levels 1
and 2. Consideration should be given to making a pass in GCSE Contextual Mathematics
a full proxy for the Functional Skills in Mathematics qualification at the same level.
However, if students are required to sit a separate functional skills assessment, the
additional work they have done in preparing for GCSE Contextual Mathematics will be an
excellent preparation for it.
How is the programme of study covered?
GCSE Mathematics will assess the whole programme of study. This is consistent with the
current Mathematics GCSE and so ensures comparability of qualifications over time. It is
worth noting, however, that the proposal to introduce a GCSE in Contextual Mathematics
enables a proper assessment of the skills required by the programme of study in relation to
all the content covered by it. In this sense, this proposal offers a more complete
assessment of the programme of study than a single GCSE Mathematics can.
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Questions and answers
Is it possible for students to take just one of the GCSEs?
The expectation is that most students would take both GCSEs, either consecutively or
simultaneously. However, Contextual Mathematics only might be preferred by some
students taking vocational options; it could, for example, be appropriate for students
taking Level 1 diplomas in Key Stage 4. On the other hand, adults returning to education
might prefer to take GCSE Mathematics only as a precursor to going on to take AS
Mathematics. Other adult learners might prefer to take just Contextual Mathematics to
improve their understanding of the mathematics they use at work.
Students who take one GCSE and succeed in it have a range of options open to them so
that they can continue to study Mathematics if they wish to do so. These can be seen on
the Pathways flowchart on page 7.
Schools might enter students for just one of the two GCSEs. Won’t this restrict
students’ future options?
It is expected that by far the majority of students will take both GCSEs.
GCSE Mathematics is the gatekeeper to AS and A Level Mathematics and so any school
that did not offer this would indeed be restricting its students’ options. Since this is the
GCSE which covers the full programme of study, maintained schools would be required to
offer a course leading to it.
However, some students may choose to take only one of the two and it seems more likely
that the one selected would be Contextual Mathematics. Although students will have
studied less content in Contextual Mathematics than they do in the current GCSE, they are
likely to have a better understanding of it. Consequently, many of the students who
succeed in Contextual Mathematics can be expected to want to continue with further study
involving mathematics and their schools should allow them the opportunity to do so.
There are several pathways available to students from each of the GCSEs and, in any case,
it is expected that most students would take them both.
How will schools find the time and the teachers to teach two GCSEs?
The two GCSEs will not, in total, have any more content than the current GCSE but there
will be an increased emphasis on understanding. There may need to be changes to the way
students are taught in order to ensure that they are able to succeed in the demands of the
new examinations. This will require time and resources for staff development but it
should not require any more teaching time than the current system.
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Pathways in Mathematics proposed in this document
NB GCSE and A Level Use of Mathematics
could also be part of this system
AS
Mathematics
(+ Further
Maths)
Functional
Skills
KS3
GCSE
Contextual
Mathematics
(includes
functional
skills)
GCSE
Mathematics
STEM area
Diplomas
FSMQ
AS Use of
Mathematics
Other Diplomas
AS Statistics
Mathematics GCSEs Aug 2008
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A Level
Mathematics
(+ Further Maths)
MEI
A Level Statistics
Competences for GCSE Contextual Mathematics and GCSE Mathematics
Note Although in this draft all the Contextual Mathematics competences are included in Mathematics also, it would be possible to have a small
number of competences (e.g. those relating to the use of spreadsheets) which were in Contextual Mathematics but not also in Mathematics.
A student can …
Process
Representing
GCSE Contextual Mathematics Competences
GCSE Mathematics Competences
(Functional Skills standards are shown in italics)
(The Contextual Mathematics competences are also
included but they are not repeated here)
Key Processes
Recognise that a situation has aspects that can be
represented using mathematics
Make an initial model of a situation using suitable forms of
representation
Decide on the methods, operations and tools, including
ICT, to use in a situation
Select the mathematical information to use
Recognise the limitations and scope of a model or
representation
Compare and evaluate representations of a situation
Simplify a situation or problem in order to represent it
mathematically using appropriate variables, symbols,
diagrams and models
Note Foundation and Higher Tier topics are not separated out in this draft. However, Foundation Tier Contextual Mathematics would include all
the Functional Skills standards.
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Process
GCSE Contextual Mathematics Competences
GCSE Mathematics Competences
(Functional Skills standards are shown in italics)
(The Contextual Mathematics competences are also
included but they are not repeated here)
Analysing
Use appropriate mathematical procedures
Examine patterns and relationships
Change values and assumptions or adjust relationships to
see the effects on answers in the model
Find results and solutions
Make connections within mathematics
Use knowledge of related problems
Visualise, including work with dynamic images
Use accurate notation, including correct syntax when using
ICT
Make and justify conjectures and generalisations,
considering special cases and counter-examples
Reason inductively, deduce and prove
Interpreting
Interpret results and solutions
Draw conclusions in light of the situation
Consider the appropriateness and accuracy of the results
and conclusions
Choose appropriate language and forms of presentation to
communicate results and conclusions.
Make sense of someone else’s findings and judge their
value in the light of the evidence they present
Compare the elegance and efficiency of alternative
solutions
Note Foundation and Higher Tier topics are not separated out in this draft. However, Foundation Tier Contextual Mathematics would include all
the Functional Skills standards.
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Process
GCSE Contextual Mathematics Competences
GCSE Mathematics Competences
(Functional Skills standards are shown in italics)
(The Contextual Mathematics competences are also
included but they are not repeated here)
Number and Algebra
Working with numbers
Numbers
Understand and use positive and negative numbers of any
size in practical contexts
Carry out calculations with numbers of any size in
practical contexts
Use integers (positive and negative), fractions and decimals
whenever needed
Represent even, odd, prime, square, and triangular numbers
as dot patterns and understand their properties intuitively
Find factors and multiples of a given number
Fractions, decimals, percentages
Understand and use equivalences between fractions,
decimals and percentages
Add and subtract fractions; add, subtract, multiply and
divide decimals to a given number of decimal places
Multiply and divide fractions
Numbers
Explore the structure of the number system e.g change a
recurring decimal to a fraction
Represent odd, even and square numbers algebraically
Express any whole number as a product of prime
factors and use them in finding HCF, LCM of two, or
more, numbers
Note Foundation and Higher Tier topics are not separated out in this draft. However, Foundation Tier Contextual Mathematics would include all
the Functional Skills standards.
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Process
Working with numbers
Proportional reasoning
GCSE Contextual Mathematics Competences
GCSE Mathematics Competences
(Functional Skills standards are shown in italics)
(The Contextual Mathematics competences are also
included but they are not repeated here)
Powers
Work with integer powers
Work with square and cube roots
Interpret numbers in index form
Use standard form for large or small numbers
Perform standard form calculations on a calculator
Powers
Work with any powers, including negative and
fractional
Work with surds
Use the rules for multiplication and division with
numbers in index form
Perform standard form calculations without a calculator
Use of calculator
Use a calculator when appropriate
Estimate whether the answer from a calculator is sensible
Use of calculator
Use a calculator efficiently
Proportional reasoning
Understand, use and calculate ratio and proportion,
including problems involving scale
Use scaling up and down arguments (e.g. the unitary
method)
Work with percentages in context
Use multipliers for percentage change
Proportional reasoning
Use information on ratios and proportions to form
equations, and solve them to find required information
Work with direct proportion using a formula
Work with inverse proportion using a formula
Note Foundation and Higher Tier topics are not separated out in this draft. However, Foundation Tier Contextual Mathematics would include all
the Functional Skills standards.
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Process
Estimation
Using algebra
GCSE Contextual Mathematics Competences
GCSE Mathematics Competences
(Functional Skills standards are shown in italics)
(The Contextual Mathematics competences are also
included but they are not repeated here)
Estimation
Round figures to a given number of decimal places or
significant figures
Understand that measurements are always rounded and that
the actual value lies within a range
State the upper and lower bounds for a measurement
rounded to a particular degree of accuracy
Find upper and lower bounds for calculations where the
upper (or lower) bound of all measurements is used
Understand how accurately to give the final answer
Find approximate solutions to problems
Understand when it is appropriate to find an approximate
solution
Algebraic expressions
Understand the concept of a variable
Understand generalisation
Use variables in a spreadsheet
Multiply 2 brackets by using a grid or other appropriate
method
Estimation
Estimate the effect of possible errors on the outcome of
any calculation
Algebraic expressions
Use variables in algebraic expressions
Manipulate algebraic expressions
Note Foundation and Higher Tier topics are not separated out in this draft. However, Foundation Tier Contextual Mathematics would include all
the Functional Skills standards.
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Process
Using algebra
GCSE Contextual Mathematics Competences
GCSE Mathematics Competences
(Functional Skills standards are shown in italics)
(The Contextual Mathematics competences are also
included but they are not repeated here)
Formulae and equations
Understand and use simple equations and simple formulae
involving one- or two-step operations
Substitute values into a given formula
Construct simple formulae in words and enter them into a
spreadsheet
Use reasoned argument to solve a linear equation
Solve an equation graphically
Use trial and improvement to solve an equation to a given
level of accuracy
Solve quadratic equations, by a method of the student’s
choice e.g. factorising or graph
Formulae and equations
Re-arrange formulae and make them into equations
Use formal algebra to solve equations
Solve quadratic equations by factorising
Solve quadratic equations using the formula or
completing the square.
Use graphs to solve equations related to the original
graph eg use y = 2 x 2 − x to solve 2 x 2 + 3x = 1
Inequalities
Use reasoned argument to justify the solution of an
inequality
Inequalities
Use formal algebraic methods to solve inequalities
Note Foundation and Higher Tier topics are not separated out in this draft. However, Foundation Tier Contextual Mathematics would include all
the Functional Skills standards.
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Process
Graphical visualisation
GCSE Contextual Mathematics Competences
GCSE Mathematics Competences
(Functional Skills standards are shown in italics)
(The Contextual Mathematics competences are also
included but they are not repeated here)
Graphs
Plot given data
Construct a table of values from a formula giving one
variable in terms of another in cases where the formula
relates to a context
Interpret the information contained in a graph, including
points of intersection of different lines or curves
Deduce the equation of a relationship from a straight line
graph
Recognise how changing the details of a situation can affect
the graph.
Use a spreadsheet to draw a graph
Recognise graphs of direct proportion, inverse proportion
and exponential growth
Relate graphs of trigonometric functions to a point moving
round a circle
Sketch a single transformation of a given function e.g.
sketch y = 2 x 2 on the same axes as the graph of y = x 2
Graphs
Draw a graph of a polynomial or exponential function
from its equation
Sketch y = mx + c for any value of m and c.
Deduce the equation of a relationship from a quadratic
graph
Use and construct formulae for graphs of exponential
growth
Recognise graphs of trigonometric functions and their
properties
Sketch transformations of a general function eg given a
graph of y = f ( x) (function f not specified) be able to
sketch the graph of y = f (2 x) on the same axes
Finding the points of intersection of a circle and a
straight line
Recognise the graph of x 2 + y 2 = r 2 as the locus of points a
distance r from the origin
Note Foundation and Higher Tier topics are not separated out in this draft. However, Foundation Tier Contextual Mathematics would include all
the Functional Skills standards.
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Process
GCSE Contextual Mathematics Competences
GCSE Mathematics Competences
(Functional Skills standards are shown in italics)
(The Contextual Mathematics competences are also
included but they are not repeated here)
Geometry and Measures
Working with units
Units
Use, convert and calculate using metric and, where
appropriate, imperial measure
Use simple compound units
Decide what units are appropriate in a given situation
Units
Work confidently with a variety of units
Working with shapes
and angles
Properties of 2-D shapes
Use symmetry to deduce and describe the properties of a
plane shape
Mensuration of 2-D and 3-D shapes
Find area, perimeter and volume of common shapes
Use scale drawing and measurement to find the perimeter
of a plane shape with straight edges
Find the area of a plane shape by partitioning it into simple
shapes
Properties of 2-D shapes
Use formal geometrical language to describe and
deduce the properties of a plane shape
Angles
Measure angles up to 360o
Use a reasoned argument to find angles
Use angles in real-world contexts
Angles
Construct formal geometrical arguments and proofs
Mensuration of 2-D and 3-D shapes
Use formulae for area and volume to find other
measurements e.g. find the height of a triangle given its
base and area
Note Foundation and Higher Tier topics are not separated out in this draft. However, Foundation Tier Contextual Mathematics would include all
the Functional Skills standards.
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Process
Working with shapes
and angles
GCSE Contextual Mathematics Competences
GCSE Mathematics Competences
(Functional Skills standards are shown in italics)
(The Contextual Mathematics competences are also
included but they are not repeated here)
Calculations in triangles and other shapes
Use Pythagoras’ theorem in 2D
Use scale drawing and similarity to find unknown sides or
angles in triangles
Use Pythagoras’ theorem in 3D
Use trigonometric relationships in right angled triangles
Find the height of a triangle then its area
Calculations in triangles and other shapes
Do multi-stage calculations using Pythagoras’ theorem
Find distance between 2 points given in 2D coordinates
Know the relationship between gradient of a straight
line and the tangent of the angle it makes with the
x-axis
Use trigonometric graphs to help solve simple
equations such as cos x = 0.3
Do accurate calculations involving surds and
Pythagoras in 3D
Find the distance between 2 points in a 3D coordinate
system
1
2
Use the formula Area = ab sin C for area of a triangle.
Use sine and cosine rules
Properties of 3-D shapes
Recognise and use 2D representations of 3D objects
Draw plan and elevations of a given shape
Interpret plans and elevations
Write down the 3D coordinates of a given point, given a
diagram with the 3 axes on
Properties of 3-D shapes
Solve 3D problems which require the use of sections
through the shape
Note Foundation and Higher Tier topics are not separated out in this draft. However, Foundation Tier Contextual Mathematics would include all
the Functional Skills standards.
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Process
GCSE Contextual Mathematics Competences
GCSE Mathematics Competences
(Functional Skills standards are shown in italics)
(The Contextual Mathematics competences are also
included but they are not repeated here)
Working with shapes
and angles
Circles
Draw circles and related shapes
Find the circumference and area of a circle or sector
Circles
State, prove and use circle theorems
Transforming figures
Transformations
Use symmetry (translation, reflection and rotation) and
enlargement to describe a shape or diagram and to deduce
relationships within it
Understand the effect of enlargement on area and volume
Transformations
Describe transformations correctly and unambiguously
e.g. Rotation 90º clockwise, centre (1,2)
Use transformations to solve geometrical problems
Solve problems involving finding the length scale
factor given the volume or area scale factor between 2
similar shapes
Vectors
Perform translations using column vectors in 2D
Use column vectors to describe journeys
Vectors
Find a vector that represents the sum or difference of
two vectors using a diagram
Find the total vector and know that it is the overall
displacement
Find the length of a vector using Pythagoras’ theorem
Use vectors in geometry proofs in 2D
Note Foundation and Higher Tier topics are not separated out in this draft. However, Foundation Tier Contextual Mathematics would include all
the Functional Skills standards.
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Process
GCSE Contextual Mathematics Competences
GCSE Mathematics Competences
(Functional Skills standards are shown in italics)
(The Contextual Mathematics competences are also
included but they are not repeated here)
Statistics
Working with
uncertainty
Probability
Use a numerical scale from 0 to 1 to express and compare
probabilities
Describe situations using the language of chance
Discuss and start to quantify risk
Find probability theoretically by listing all possible events,
including the use of sample spaces
Estimate probability by finding relative frequency and
realise that a larger sample gives a better estimate of
probability
Use tree diagrams for events which are not conditional
Probability
Perform probability calculations involving one or more
chance events
Relate probability to a wide range of situations
Understand the idea of mutually exclusive and
independent events
Use Venn diagrams to represent number of possibilities
and hence find probabilities
Use tree diagrams for conditional events
Note Foundation and Higher Tier topics are not separated out in this draft. However, Foundation Tier Contextual Mathematics would include all
the Functional Skills standards.
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Process
GCSE Contextual Mathematics Competences
GCSE Mathematics Competences
(Functional Skills standards are shown in italics)
(The Contextual Mathematics competences are also
included but they are not repeated here)
Problem solving using
statistics
The statistical problem solving cycle
Use statistical methods to investigate situations
Collect and represent discrete and continuous data, using
ICT where appropriate
Use and interpret statistical measures, tables and
diagrams, for discrete and continuous data, using ICT
where appropriate
Recognise and deal with practical difficulties e.g. missing
or incorrect data
Use data appropriately
Draw conclusions from using the data
The statistical problem solving cycle
Appreciate the need to collect data to understand
situations and solve a range of problems
Determine what data need to be collected
Collect data
Use data appropriately
Draw conclusions based on the use of data
Recognise the best methods for solving a problem and
explain why other methods are not relevant in this case
Problem solving using
statistics
Data presentation
Present single variable data relating to simple situations
using appropriate display techniques
Comment on and interpret the information provided by a
data display
Interpret data using box plots and cumulative frequency
diagrams
Data presentation
Present single variable data relating to a range of
situations using a variety of appropriate display
techniques
Relate the display technique used to the information
sought from the data
Draw and interpret histograms
Note Foundation and Higher Tier topics are not separated out in this draft. However, Foundation Tier Contextual Mathematics would include all
the Functional Skills standards.
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MEI
Process
GCSE Contextual Mathematics Competences
GCSE Mathematics Competences
(Functional Skills standards are shown in italics)
(The Contextual Mathematics competences are also
included but they are not repeated here)
Data measures
Calculate simple measures of location and variability for a
data set
Interpret those measures in relation to a situation or
problem
Use median and interquartile range to compare large data
sets
Problem solving using
statistics
Bivariate data
Draw and interpret a scatter diagram
Draw a line of best fit by eye
Use a scatter diagram and line of best fit to solve a problem
Interpret correlation (or lack of it) in a given context
Data measures
Calculate a variety of appropriate measures of location
and variability for a data set
Interpret those measures in relation to a situation or
problem
Understand the effects of changing the data on the
measures
Decide which average is the most appropriate for the
circumstances
Understand that the interquartile range is not affected
by outliers
Recognise that two data sets can have similar medians
and interquartile ranges but still have different
characteristics
Bivariate data
Recognise when a line of best fit may not be
appropriate e.g. know that extrapolation may not be
valid
Identify situations where correlation is spurious
Note Foundation and Higher Tier topics are not separated out in this draft. However, Foundation Tier Contextual Mathematics would include all
the Functional Skills standards.
Mathematics GCSEs Aug 2008
Page 20
MEI