NUMERACY POLICY by HC12110514334

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									             NUMERACY POLICY


The application of this policy and procedure ensures that no employee receives less
favourable treatment on grounds of sex, trans-gender status, sexual orientation, religion or
belief, marital status, civil partnership status, age, race, colour, nationality, national origin,
ethnic origin, disability, part time status or trade union activities.




POLICY AGREED BY __________________________________________________              DATE _____________

POLICY REVIEW DATE ________________________________________________________________________




                                                                            VERSION 1 – JULY 2012




                                                  1
CONTENTS                                             PAGE

Numeracy Policy Aims                                 3

What is Numeracy?                                    4

The National Numeracy Strategy’s approach to
calculation                                          5

Introducing and developing algebra                   11

Shape, space and measures                            14

Handling Data                                        16

Using and applying mathematics and thinking skills   18

Our approach to teaching                             22

Key Objectives for different abilities               24

Key mathematical vocabulary                          28

Mathematical competencies required in other
subjects                                             37

Appendix                                             41

Audit                                                44




                                 2
NUMERACY POLICY AIMS

The aims of the numeracy policy are:

      To raise standards of numeracy and mathematics for all pupils throughout the
       school.

      To ensure that pupils do not experience and discontinuity from subject to
       subject, either in the way certain mathematical methods and strategies are
       taught or in relation to the level of difficulty of mathematics expected of
       them.

      To promote a positive attitude towards mathematics amongst all pupils.

      To promote a positive attitude towards mathematics amongst all staff and for
       all staff to actively promote its development across the curriculum.

      To ensure that all staff are aware of what numeracy is and what the National
       Numeracy Strategy’s approach to calculation is.

      To ensure that all staff, but particularly those in key departments (e.g.
       Science, Geography, Technology, ICT, P.E. and History), are aware of the
       range of mathematical skills that pupils bring to their lessons.

      To ensure that all staff in key departments are aware of the mathematical
       demands of their own subject in order that this is reflected in schemes of
       work.

      To ensure that staff in key departments provide opportunities for pupils to
       develop and apply their mathematical skills in their own subject.

      To promote a common approach to the teaching of key mathematical ideas
       and processes in all subjects which require them.




                                           3
WHAT IS NUMERACY?

Numeracy is a proficiency, which is developed mainly in mathematics but also in
other subjects. It is more than an ability to do basic arithmetic. It involves
developing confidence and competence with numbers and measures. It requires
and understanding of the number system, a repertoire of mathematical techniques
and an inclination and ability to solve quantitative or spatial problems in a range of
contexts. Numeracy also demands practical understanding of the ways in which
data are gathered by counting and measuring, and presenting in graphs,
diagrams, charts and tables.

As a teacher you can help children to acquire this proficiency by giving a sharp
focus to the relevant aspects of the programmes of study for mathematics. The
outcome should be numerate pupils who are confident enough to tackle
mathematical problems without going immediately to teachers or friends for help.

Your pupils should:

      Have a sense of the size of a number and where it fits into the number system;

      Know by heart number facts such as number bonds, multiplication tables,
       doubles and halves;

      Use what they know by heart to figure out answers mentally;

      Calculate accurately and efficiently, both mentally and with pencil and
       paper, drawing on a range of calculation strategies;

      Recognise when it is appropriate to use a calculator and be able to do so
       efficiently;

      Make sense of number problems, including non-routine problems, and
       recognise the operations needed to solve them;

      Explain their methods and reasoning using correct mathematical terms;

      Judge whether their answers are reasonable and have strategies for
       checking them where necessary;

      Suggest suitable units for measuring and make sensible estimates of
       measurements;

      Explain and make predictions from the numbers in graphs, diagrams, charts
       and tables.

                                          4
THE NATIONAL NUMERACY STRATEGY’S APPROACH TO CALCULATION

This section outlines what the strategy says about mental calculation, written
calculation and the role of calculators. It explains how the four operations are
taught in the mathematics department and how pupils are taught to record their
calculations.

It is important to stress here that pupils are not being taught to follow pencil and
paper algorithms blindly and without understanding. There are various pencil and
paper methods for each of the four operations; in particular it is intended to place
an emphasis on the grid method for multiplication and the “chunking” method for
division. Examples of the various methods are contained in the appendix.

Number: From KS2 to KS3

An ability to calculate mentally lies at the heart of numeracy. As a teacher,
whether of mathematics or another subject, you should stress the importance of
mental calculation methods and give all pupils regular opportunities to develop the
skills involved. The skills include an ability to:

      Remember number facts and recall them without hesitation;

      Use known facts to figure out new facts: for example, knowing that half of 250
       is 125 can be used to work out 250 – 123;

      Draw on a repertoire of mental strategies to work out calculations such as 326
       – 81, 223 x 4 or 2.5% of £3000, with some thinking time;

      Understand and use the relationships between operations to work out
       answers and check results: for example, 900 ÷ 15 = 60, since 6 x 150 = 900;

      Approximate calculations to judge whether or not an answer is about the
       right size: for example, recognise that ¼ of 57.9 is just under ¼ of 60, or 15;

      Solve problems such as: “How many CDs at £3.99 ech can I buy with “25?” or
       “Roughly how long will it take me to go 50 miles at 30 mph?”. An increased
       emphasis in KS1 and KS2 on mental calculation does not mean that written
       methods are not taught in primary years, but the balance between mental
       and written methods and the progression from one to the other become
       increasingly important in the later years of KS2 and in KS3.




                                             5
CALCULATION IN KS1 AND KS2

In the early years pupils work orally with numbers. Alongside their oral work they
learn to read, interpret and complete statements such as 5 + 8 = * or 13 = * + 5, and
then to record the results of their mental calculations in the correct way, using a
horizontal format like 43 – 8 = 35. As they progress to working with larger numbers
they learn more sophisticated mental methods and tackle more complex problems.
They develop some of these methods intuitively and some they are taught explicitly.
Through a process of regular explanation and discussion of their own and other
people’s methods they begin to acquire a repertoire of mental calculation
strategies. It can be hard to hold all the intermediate steps of a calculation in the
head and so informal pencil and paper notes, recording some or all the solution,
become part of a mental strategy. These personal jottings may not be easy for
someone else to follow but they are an important staging post to getting the right
answer and acquiring fluency in mental calculation.

The approach in the primary years builds on the use of the number line, first with
numbers marked and then a blank line, to record steps in calculations such as 47 +
26 or 261 – 174.

Jottings:
Pupils make jottings to assist their mental calculations, e.g. 47 + 26



                        +3          +20         +3

                   47        50                 70    73

They record steps so that they and others can see what they have done, as in:

36 + 27:

      36 + 20 = 56 and then 56 + 7 = 63
Or:
      30 + 20 = 50 and 6 + 7 = 13, so 50 + 13 = 63


Not everyone does a mental calculation like 81 – 26 in the same way (nor is it
necessary for them to do so) but some methods are more efficient and reliable
than others. Some methods only work for particular cases. Your role is not simply to
accept pupils’ personal methods but to help them adopt better ones. By
explaining, discussing and comparing different part-written, part-mental methods,
you can guide pupils towards choosing and using methods that are quicker, can
be applied more widely and are helpful for their future learning.


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STANDARD WRITTEN METHODS

Standard written methods are reliable and efficient procedures for calculating
which, once grasped, can be used in many different contexts. But they are of no
use to someone who applies them accurately and who cannot judge whether the
answer in reasonable. For each operation, a standard written method should be
taught to most pupils, then refined and practised.

The method chosen should fulfil several criteria:

      It should not be too time-consuming to carry out;
      Pupils should be able to explain why it works and apply it reliably;
      The way it is set out should help them to avoid mistakes;
      It should support their further learning of mathematics.

The progression towards written methods is crucial, since standard written methods
are based on steps which are done mentally and which need to be secured first.
For example, the calculation of 487 + 356, done by the traditional method in
columns, requires the mental calculations 7 + 6 = 13, 8 + 5 + 1 = 14 and 14 + 3 + 1 =
8, while a division calculation such as 987 ÷ 23 can involve mental experiment with
multiples of 23 before the correct multiple is chosen.

Many countries, and in particular those which are most successful at teaching
number, avoid the premature teaching of standard written methods in order not to
jeopardise the development of mental calculation strategies. The bridge from
recording part-written, part-mental methods to learning standard methods of
written calculations begins only when pupis can add or subtract reliably any pair of
two-digit numbers in their heads, usually when they are about nine years old.

Standard written methods for addition and subtraction should be well established
by Year 6 for nearly all pupils and will be used in KS3 with an increasing range of
whole numbers and decimals. Multiplication and division methods will need to be
developed further. When they transfer to KS3, some pupils may still use informal
written methods to record, support and explain their multiplication and division
calculations. For example, a few may use a “grid method” or Napiers’ bones
(examples are in the appendix) for multiplying two or three-digit numbers, which
you will need to consolidate and build on in KS3 and relate to work in algebra. For
division they may use a “chunking” method; this too will need to be developed and
refined to ensure efficiency. The aim is that where it is appropriate to do so all
pupils use standard written methods efficiently and accurately, and with
understanding.

When they have reached the stage of working out more complex calculations
using written methods, pupils in KS3 still need to practise and refine their mental
calculation strategies. Help them to develop estimation skills in all aspects of
calculation, but particularly in multiplication and division. When faced with any
calculation, no matter how large or how difficult the numbers may appear to be,

                                            7
encourage them first to ask themselves: “Can I do this in my head?”. They then
need to ask: “Do I know the approximate size of the answer?” and “Does the
answer make sense in the context of the question?” so that they can be reasonably
sure that their calculation is right.

CALCULATOR METHODS

The calculator is a powerful and efficient tool in the right hands. It has an important
part to play in subjects such as design and technology, geography, history and
science, since it allows pupils to make use of real data from their research or
experiments – often numbers with several digits. As with numeracy, the
appropriate use of calculators is a whole-school matter. All subjects need to adopt
a similar approach and agree when, how and for what purpose calculators are to
be used.
Before Year 5, the calculator’s main role in mathematics is not as a calculating tool,
since pupils are still developing the mental calculation skills and written methods
that they will need in later years. But, it does offer a unique way of learning about
mathematical ideas throughout all key stages. For example, pupils might use a
calculator to find two consecutive numbers with a given product and then discuss
their methods.

If pupils are to use the basic facilities of a calculator constructively and efficiently
for calculating purposes, you will need to teach them in KS3 the technical skills that
they will require. For example, during KS3 they need to learn:

      How to select from the display the number of figures appropriate to the
       context of the calculation;

      How to enter numbers and interpret the display when the numbers represent
       money, metric measurements, units of time or fractions;

      The order in which to use the keys for calculations involving more than one
       step;

      How to use facilities such as the memory, brackets, the square root and cube
       root keys, the sign change key, the fraction key, the constant facility, and so
       on.

By the end of key stage 3, pupils should have the knowledge and skills to use a
calculator to work out expressions such as 3250 x 1.05³ or √(7.82² - 2.91²).

All pupils need to continue to learn when it is, and when it is not, appropriate to use
a calculator, and their first-line strategy should involve mental calculations. They
should have sufficient understanding of the calculation in front of them to be able
to decide which method to use – mental, written or calculator, or a combination of
these. When they do use a calculator, they should be able to draw on well-
established skills of rounding numbers and calculating mentally to gain a sense of
the approximate size of the answer, and have strategies to check and repeat the

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calculation if they are not sure whether it is right. These skills do not happen by
accident. They need to be taught in mathematics and applied across the
curriculum whenever calculators are used.

PROPORTIONAL REASONING

Throughout KS3 pupils will extend their understanding of the number system to
positive and negative numbers and, in particular, to fraction and their
representations as terminating or recurring decimals.

Fractions, decimals, percentages, ratio and proportion are different ways of
expressing related ideas and relationships. The connections start to be established
in KS2, particularly the equivalence between fractions, decimals and percentages.
The ideas of ratio and proportion, and the relationship between them, should be a
strong feature of work in KS3. By the end of the key stage, pupils should be able to
solve problems involving fractions, decimals, percentages, ratio and proportion,
and their interconnections.

After calculation, the application of proportional reasoning is the most important
aspect of elementary number. Proportionality underlies key aspects of number,
algebra, shape, space and measures, and handling data. It is also central in
applications of mathematics in subjects such as science, technology, geography
and art. The study of proportion begins in KS2 but it is in KS3 where secure
foundations need to be established.

Problems involving proportion are often solved by informal methods, particularly
when the numbers involved are easy to deal with mentally. But it is important to
teach methods that can be applied generally. For example, the unitary method is
useful for solving problems involving proportion, and multiplicative methods
involving fractions or decimals are useful for solving percentage problems.

When you are teaching proportional reasoning:

      Emphasise the language and notation of ratio and proportion, and the links
       to fractions, decimals and percentages;

      Teach pupils specific methods for solving proportion problems so that they do
       not remain dependent on informal approaches;

      Help pupils to understand what they are calculating: for example, a distance
       divided by a time gives a speed – an example of a rate; but a distance
       divided by another distance gives a scale factor or multiplier – a
       dimensionless number;

      Make explicit links between ideas of proportionality in number, algebra,
       shape, space and measures, and handling data.



                                            9
In algebra, direct proportion is viewed as a linear relationship of the form y = mx.
The graphical representation of this equation helps pupils to visualise ideas such as
rate of change and gradient.

The algebraic representation of a proportion (e.g. a:b = c:d or a/b = c/d) underpins
a general method for solving problems.

In shape, space and measures, proportionality arises when enlargement by
different scale factors is considered. Scaling has a wide range of applications, for
example, in maps, plans and scale drawings. Similar figures have sides or
dimensions that are in proportion. Recognition of the similarity of all circles leads to
an understanding that the circumference is directly proportional to the diameter,
while awareness of the similarity of triangles with the same angles leads to an
understanding of trigonometry.

In statistics, proportions are often calculated when data is interpreted and
inferences drawn. Proportions are also used when probabilities are estimated or
calculated based on outcomes that, in theory, are equally likely.

FEATURES OF NUMBER IN KEY STAGE 3

To summarise, the distinctive features of number in KS3 are:

      Developing understanding of the number line;

      Building on the approach to calculation developed in Key Stages 1 and 2,
       which emphasises mental methods and gradually refined written methods,
       extending to calculations with fractions, decimals and percentages;

      Developing effective use of calculators, including choosing appropriate
       methods for estimating, calculating and checking;

      Developing proportional reasoning, including making links to algebra, shape,
       space and measures, and handling data.




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INTRODUCING AND DEVELOPING ALGEBRA

Algebra in Key Stage 3 is generalised arithmetic. Its origins lie in arithmetic, in the art
of manipulating sums, products and powers of numbers. The same rules are seen to
hold true for all numbers, of whatever type, so it becomes possible to generalise the
rules with letters in place of numbers. Indeed all numerical entities, coefficients as
well as unknowns, can be represented by letters. This insight releases in due course
the full power of algebra.

The diagram below is drawn form the Royal Society and Joint Mathematical
Council’s publication Teaching and learning algebra pre-19. The lines showing the
links between topics are the most significant aspect.


   Arithmetic with                                                 Functions
   numbers
                                                                           Y = f(x) form
            Understandin                                                  Non-linear
             g equivalent                                                   functional
             form                                                           relationships
            General           Relationships between                       Sequences
             terms             variables                                   Recursive
            Sequences                                                      functions
                                       Formulae
                                    -    in words
                                    -    in symbols
 Arithmetic with                       Equations                  Graphs
 symbols                               Inequalities
                                       Identities                         Coordinates
           Understanding                                                  Graphs of
            equivalent                                                      functions
            form                                                      -     linear
           General terms                                             -     Quadratic
           Sequences                                                      Graphical
           Manipulation                                                    solution of
            of symbols                                                      equations
           Factorising,                                                   Relationships
            changing the                                                    expressed
            subject…                                                        graphically




Algebra in Years 7 to 9 includes equations, formulae and identities, and sequences,
functions and graphs. You need to stress the links between these topics and with
arithmetic. Letters do not represent quantities like length or cost; they represent
numbers. Pupils will have spent much time manipulating numbers in Key Stages 1
and 2, and you can build on their experience.




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For example, in response to the question:

“6 x 8 = 48. What can you deduce from this?”, a year 6 class might suggest:

8 x 6 = 48          8 = 48/6          80 x 6 = 480      16 x 6 = 96
8 x 6 + 1 = 49      6 = 48/8          80 x 60 = 4800    and so on
In KS3, this can be extended into the simple manipulation of equations, referring
back regularly to the number examples.

For example. ab = c implies:

ba = c                     b = c/a                a = c/b       2ab = 2c
2ab + 1 = 2c + 1

You can draw other examples from pupils’ experience of multiplication based on
the distributive law. By encouraging pupils to record their mental calculations in
algebraic form by making use of brackets, you can help them to generalise.

For example, you can draw parallels between:

6 x 42 = 6(40 + 2) = 6 x 40 + 6 x 2 = 240 + 12 = 252

and: a(b + c) = ab + ac

or between: 147 – 99 = 147 – (100 – 1) = 147 – 100 + 1 = 48

and: a – (b – c) = a – b + c

EQUATIONS, FORMULAE AND IDENTITIES

The initial approach to manipulating number statements needs to extend into a set
of rules for solving equations, under a general heading: “do the same to both sides
of the equation”. Pupils’ developing skills in solving equations will be dependent on
their ability to add, subtract, multiply and divide directed numbers and to simplify
expressions and collect like terms. Although to begin with you will frequently refer to
number examples, in time pupils will appreciate that expressions and equations can
be manipulated in their own right according to given rules and conventions.

The generalised algebraic use of brackets, multiplying out a single term over a
bracket, and the inverse process of taking out a common factor, can also develop
from examples with numbers. Multiplying out a pair of brackets can be based on
the “grid method” of multiplication and developed into an algebraic process. The
most able pupils can attempt the reverse process of factorisation but for most pupils
this will be an activity for KS4.


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By the end of the Key Stage, pupils need to understand that a statement such as:

               (x + 1)(x – 1) = x² - 1
is an identity that is true for every value of x, unlike an equation, where the purpose
is to find the value(s) of x that alone satisfy the equation.

SEQUENCES, FUNCTIONS AND GRAPHS

Many applications of algebra involve finding a formula that generates the general
term of a sequence: for example, in predicting the number of matchsticks needed
for a certain pattern, or the number of paving slabs for the border of a rectangular
pond. It is important for pupils to justify their formulae from physical patterns, rather
than merely from number sequences, since this allows them to “prove” their
solutions, not just illustrate or verify them. The apparently different but equivalent
formulae that arise from alternative ways of looking at the problem can help pupils
to understand equivalent algebraic expressions.
Functions and graphs can be taught and learned in tandem. At KS3 the main
emphasis is on linear functions and their graphs. A graphical calculator, or graph
plotting software, has an important role since it helps pupils to learn from exploring
problems.

FEATURES OF ALGEBRA IN KEY STAGE 3

To summarise, the distinctive features of algebra in Key Stage 3 are:

      Developing understanding that algebra is a way of generalising from
       arithmetic, from particular cases or from patters and sequences;

      Providing regular opportunities to construct algebraic expressions and
       formulae and to transform one expression into another – collecting like terms,
       taking out common factors, working with inverses, solving linear equations;

      Using opportunities to represent a problem and its solution in tabular,
       graphical or symbolic form, using a graphical calculator or a spreadsheet
       where appropriate, and to relate solutions to the context of the problem;

      Developing algebraic reasoning, including an appreciation that while a
       number pattern may suggest a general result, a proof is derived from the
       structure of the situation being considered.




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SHAPE, SPACE AND MEASURES

Geometry in Key Stage 3 is the study of points, lines and planes and the shapes that
they can make, together with a study of plane transformations. A key aspect is the
use and development of deductive reasoning in geometric contexts. Geometrical
activities can be linked to accurate drawing, construction and loci, and work on
measures. By ensuring that pupils have a range of suitable experiences you can
develop their knowledge and understanding of shape and space and their
appreciation of the ways that properties of shapes enrich our culture and
environment.

GEOMETRICAL REASONING

Pupils can be aware of and use geometrical facts or properties that they have
discovered intuitively from practical work before they can prove them analytically.
The aim in KS3 is for pupils to use and develop their knowledge of shapes and
space to support geometrical reasoning. For example, they need to appreciate
that tearing the corners off a triangle and placing them side by side at best
indicates that the angle sum of a triangle is approximately 180°, and that however
many particular cases they can find of triangles with an angle sum of 180°, this does
not prove the general case.

In Key Stage, you can build on pupils’ experience and the practical demonstrations
and explanations that have sufficed in Key Stages 1 and 2. Teach them to
understand and use short chains of deductive reasoning and results about
alternate and corresponding angles to reach a proof. Later, pupils should be able
to explain why the angle sum of any quadrilateral is 360°, and to deduce formulae
for the area of a parallelogram and of a triangle from the formula for that area of a
rectangle. These chains of reasoning are essential steps towards the proofs that are
introduced in KS4.

APPRECIATION OF SHAPE AND SPACE

Geometry cannot be learned successfully solely as a series of logical results. Pupils
also need opportunities to use instruments accurately, draw shapes and appreciate
how they can link together, for example, in tessellations. In Key Stage 3, it is vital to
distinguish between the imprecision of constructions which involve protractors and
rulers, and the “exactness in principle” of standard constructions which use only
compasses and a straight edge. Geometrical reasoning can show pupils why
construction methods work, for example, the method to construct a perpendicular
bisector of a line segment.

Practical work with transformations will produce interesting problems to solve as well
as helping pupils to understand the topic more fully. Urge them to visualise solutions
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to problems such as: “When a triangle is rotated through 180° about the mid-point
of one side, what shape do the original and final triangles form?”

Linking geometry to subjects such as art, through symmetry or tessellations, or
religious education, perhaps through a study of the properties of Islamic patterns or
cathedral rose windows, offers good opportunities to develop creativity. By
encouraging pupils to speculate why the properties they have found hold true you
can strengthen their reasoning skills.

USE OF ICT

ICT offers good opportunities to develop geometrical reasoning and an
appreciation of shape and space. For example, pupils can use the programming
language Logo to explore properties of plane shapes, such as the exterior angles of
polygons. With dynamic geometry software, such as Geometers’ Sketchpad they
can use rapid geometric drawing to explore a condition such as “one pair of
opposite angles of a quadrilateral is equal”, and discover the special
circumstances under which condition is true. More able pupils may be able to
prove their conjectures analytically, but the formal use of congruent triangles is
often needed, and for most pupils this will be tackled in Key Stage 4.

MEASURES

Pupils in KS3 need to develop their awareness of the relative sizes of units,
converting between them, and using the rough equivalence of common imperial
and metric units. Towards the end of the key stage, they need to become familiar
with compound measures such as speed or density. Help them to appreciate the
imprecision of measurement and to recognise the accuracy to which
measurements can be stated. Draw as far as possible on their practical experience
of measures in other subjects, particularly design and technology, science,
geography and physical education.

Work on perimeter, area and volume will extend to a range of shapes, including
rectangles, parallelograms, circles, cuboids and prisms. A project such as “design a
swimming pool” allows pupils to exercise imagination, practise calculations of
length, area and volume, and experience working with larger numbers and units.
The heart of work on measure will be with triangles which, at the end of the key
stage, will extend to Pythagoras’ Theorem and similarity, leading on to trigonometry.
As far as possible, the relevant formulae for calculating perimeters, areas and
volumes should be explored and justified logically, not simply stated as facts. The
use of formulae can be linked to work in algebra, and can be enhanced by the use
of spreadsheets and graphical calculators.




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FEATURES OF SHAPE, SPACE AND MEASURES IN KS3

To summarise, the distinctive features of shape, space and measures in KS3 are:

      Developing geometrical reasoning and construction skills, and an
       appreciation of logical deduction;

      Developing visualisation and sketching skills, including a dynamic approach
       to geometry, making use of ICT and other visual aids;

      Developing awareness of the degree of accuracy of measurements.



HANDLING DATA

Handling data is described by the cycle shown in the diagram.


       Evaluate results         Specify the problem
                                      and plan


       Interpret and discuss                              Collect data from a
               data                                       variety of sources



                               Process and represent
                                        data


Data handling is best taught in a coherent way in the context of real statistical
enquiries so that teaching objectives arise naturally from the whole cycle. As an
enquiry develops, you will need to reinforce and develop certain skills by direct
teaching of particular objectives. It is easier to make sure that problems are
relevant if at least some of the enquiries are linked to other subjects. For example, a
question can be formulated and data collected in geography, with mathematics
lessons concentrating on processing, representing and interpreting the data. The
other subject can make further interpretations and consider the implications of
what has been discovered. The supplement of examples in the Framework
suggests a variety of possible projects, many of them linked directly to QCA’s
schemes of work for other subjects, including science, geography and physical
education, with further possibilities in history, religions education, languages and
personal, social and health education.



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As pupils move through KS3, the cross-curricular aspects of data handling become
more important. It is usually best for a cross-curricular enquiry to be defined in the
other subject, but good preparation is needed to check that the mathematical
skills, techniques and representations that pupils need to learn next are likely to
arise. In Year 7, much of the work may take place in mathematics lessons, with
small sets of data that pupils can generate readily from simple experiments and
easily accessible secondary sources. In Year 9, pupils should engage with large sets
of real data from a much wider range of sources and contexts. The experience of
working with real data in KS3 is an important preparation for controlled assessments
in GCSE Statistics.

PRIMARY AND SECONDARY SOURCES

Give pupils experience of collecting and using primary data from, for example,
questionnaires or results of an experiment, and secondary data from published
sources, including reference materials, ICT databases and the Internet.

Real data present problems that “textbook” or contrived data can skirt around,
such as the accuracy of recording or how to deal with data that are ambiguous.
The sizes of numbers can be problematic, either because they are large or, in the
case of a pie chart, because they are not factors of 360. The time to process and
represent real data is likely to be greater thatn with textbook examples but by using
them pupils will have gained useful skills that can be transferred to other
investigations.

PROBABILITY

Pupils will have met some of the language of probability in KS2 but will have had
little experience of quantitative probability. Pupils need to understand that
probability is a measure of what might happen. Help them to reason out what the
probability is for simple cases by considering all the possible outcomes for particular
events.
The data handling cycle illustrated above applies also to probability. There are two
aspects to develop in tandem through this cycle: probability described by the
proportion of successes in an experiment, and probability derived from theoretical
considerations.

As they compare their practical and theoretical results, pupils will begin to sense
that as the number of trials becomes very large the proportion of successes
converges to the theoretical probability.

FEATURES OF HANDLING DATA IN KEY STAGE 3

To summarise, the distinctive features of handling data in KS3 are:

      Basing work on purposeful enquiry, using situations of interest and relevance
       to pupils and making appropriate links to other subjects;

                                          17
      Placing an emphasis on making inferences from data, drawing on a range of
       secondary sources to ensure that samples are sufficiently large;

      Using ICT as a powerful source of data, and as a means of processing data
       and simulating situations.

USING AND APPLYING MATHEMATICS AND THINKING SKILLS
Thinking skills underpin using and applying mathematics and the broad strands of
problem solving, communication and reasoning. Well-chosen mathematical
activities will develop pupils’ thinking skills. For example, you might devote
occasional stand-alone lessons to an investigation of a problem. Used well, this
approach can focus pupils’ attention on the “using and applying” or thinking skills
that they have used so that they can apply these skills more generally in their
mathematics work. But the main approach to using and applying mathematics
and thinking skills is to integrate them within everyday teaching, thereby helping
pupils to make the connections in learning associated with success in mathematics.

You can introduce problem solving, applications of mathematics and the use of
reasoning and thinking skills at many points in a unit of work. A problem can serve
as an introduction, to assess pupils’ prior knowledge or to set a context for the work;
it can be used to provide motivation for acquiring a skill; or it can be set as a class
activity or as homework towards the end of a topic, so that pupils use and apply
the mathematics they have been taught.

Pupils need to be able to select the mathematics required to solve a problem and
recognise that an idea that they meet in one strand of mathematics can be
applied to another. A good “diet” will include:

      Problems and applications that extend content beyond what has just been
       taught;

      Familiar and unfamiliar problems in a range of numerical, algebraic and
       graphical contexts, some with a unique solution and some with several
       possible solutions;

      Activities that develop short chains of deductive reasoning and concepts of
       proof in algebra and geometry;

      Occasional opportunities to sustain thinking by tackling more complex
       problems.

There are five categories of thinking skills in the National Curriculum:

information processing skills, enquiry skills, creative thinking skills, reasoning skills and
evaluation skills. The contribution of mathematics to these skills is drawn directly
from using and applying mathematics.


                                              18
INFORMATION PROCESSING SKILLS

These enable pupils to find and organise relevant information, to compare and
contrast it, and to identify and analyse relationships. For example, pupils acquire
information processing skills when they process the information in a 2D picture of a
3D object, or when they investigate what factors influence the distribution of grass
and non-grass plants on the school field. They compare and contrast when, for
example, they classify quadrilaterals, order a set of decimals, or consider two
different representations of the same set of data.

The National Curriculum for using and applying mathematics states that pupils
should:

      Collect data from a variety of suitable sources, including experiments and
       surveys, and primary and secondary sources, including the Internet;

      Decide what statistical analyses are needed;

      Process and represent the data, using ICT as appropriate, turning raw data
       into usable information that gives insight into the problem or enquiry;

      Interpret data and draw conclusions from them.


ENQUIRY SKILLS

Enquiry lies at the heart of mathematics. Enquiry skills enable pupils to ask questions,
define questions for enquiry, plan research, predict outcomes, anticipate
consequences, and draw conclusions. Central to enquiry is an ability to see
connections between different aspects of mathematics and thus open up further
ways of tackling a problem.

Pupils develop enquiry skills when they modify a problem they have solved to
create a new problem: for example, when they find the smallest number with
exactly three factors, then extend this to four factors, or other numbers of factors.
They use enquiry skills when they find and eliminate alternatives: for example, when
they explore how many different shapes can be made with five identical squares,
and decide what to do with reflections or rotations. They practise enquiry skills in
cross-curricular statistical surveys: for example, by examining possible indicators for
economic development and deciding which of them are likely to highlight
differences between countries most clearly.




                                           19
The National Curriculum for using and applying mathematics requires pupils to:

      Pose problems and plan how to solve them;

      Predict outcomes;

      Decide what data or information to collect, the degree of precision or
       accuracy required, and the inferences they will be able to draw;

      Choose the appropriate mathematics and resources to use, including ICT;

      Interpret and discuss representations of data and results of analyses, looking
       for patterns and relationships, and explaining and justifying how they
       reached conclusions;

      Identify what further information is needed to pursue a supplementary
       enquiry.


CREATIVE THINKING SKILLS

These enable pupils to generate and develop ideas, to hypothesise, to apply
imagination, and to seek innovative alternatives. For example, pupils need to think
creatively when they visualise the path of a moving point or how a shape might
change as slices are taken from it, when they look for pairs of numbers whose sum
equals their product, or when they formulate algebraic expressions to make this
addition square work:


                ?                ?

   ?         3a + 2b          a + 4b

   ?         4a + 3b          2a + 5b


The National Curriculum for using and applying mathematics requires pupils to:

      Select and combine known facts and problem-solving strategies in creative
       ways to solve problems, using alternative approaches to overcome
       difficulties;

      Represent problems and solutions in numerical, algebraic, geometric or
       graphical form, moving from one to another to gain a different perspective
       on the problem and to identify similarities and differences;

                                           20
      Explore connections in their mathematical work;

      Explore, identify, and use pattern and symmetry in mathematical contexts;

      Visualise and use mathematical imagery;

      Conjecture, hypothesise, and ask questions such as” What is...?” or “Why?”;

      Investigate whether particular cases can be generalised further.


REASONING SKILLS

These enable pupils to give reasons for opinions and actions, to infer and deduce,
to make informed judgements and decisions, and to use precise language to
reason. Pupils apply reasoning skills when, for example, they investigate how to
draw two straight lines from the vertex of a square to divide the square into three
equal parts and then prove analytically that their construction holds, or when they
look for examples of pairs of unit fractions whose sum is another unit fraction. They
apply reasoning when they argue, for example, that it is necessary but not sufficient
that a multiple of 4 is an even number, or that a square is a trapezium is not
necessarily a square.

The National curriculum for using and applying mathematics states that pupils
should:

      Use step-by-step deduction and efficient techniques for solving a problem,
       including breaking down complex problems into simpler steps or a series of
       tasks, and working systematically;

      Present concise, reasoned arguments, explaining and justifying inferences,
       deductions and conclusions, using mathematical notation, symbols and
       diagrams correctly and consistently;

      Identify exceptional cases or counter-examples that do not accord with an
       argument and explain why;

      Distinguish between practical demonstration and proof, and between
       conventions, definitions and derived properties.

EVALUATION SKILLS

These enable pupils to develop and apply evaluation criteria and to judge the
value of information and ideas. For example, pupils use evaluation skills when they
compare mental methods of calculation to judge which is the most efficient, or
when they evaluate whether the use of a calculator is appropriate. They use
evaluation skills when they generate a number sequence in a practical context
                                         21
and then consider the best way to describe and express the general term. They
might evaluate the questions used in a statistical survey to compare attitudes to
fairly traded goods and decide that the questions used to explore attitudes are
crude compared with the complexity of personal beliefs.

The National curriculum for using and applying mathematics requires pupils to:

      Review progress as they work;

      Move from one form of representation to another to gain different
       perspectives on a mathematical problem;

      Evaluate the effectiveness of their chosen methods, techniques and
       problem-solving strategies, and the resources they have chosen to use.

      Check and evaluate their solutions, including the accuracy of their
       calculations, analyses and results;

      Consider anomalies in data or measurements and try to explain them;

      Examine critically, improve, then justify their choices of mathematical
       presentation.



OUR APPROACH TO TEACHING
This section provides a simple checklist of what teachers need to be aware of and
what pupils need to be encouraged to do in all lessons where there is a
mathematical element to the work.

As teachers we strive to:

      Be aware of the range of mathematical attainment that pupils bring to our
       lessons.
       (Note: Just because they are in, for example, the top set for your subject
       doesn’t mean that they will be in the top set for mathematics. The list of key
       objectives, see below, will support this).

      Build pupils’ confidence when they are struggling with a calculation.
       (Note: Encourage the pupils to make sense of the numbers involved, to take
       a step back and consider what is needed to answer the question before
       diving in with a set method. Encourage short cuts based on mental strategies
       if possible).

      Encourage pupils to understand the methods that they are using.
       (Note: Be wary of teaching tricks which have no meaning for the pupil. For
       example, rules like “to multiply by 10, add a nought” or when transforming
       formulae, “change the side, change the sign” can be very helpful if used by

                                           22
      pupils who understand the mathematics behind these mnemonics. However,
      if they don’t relate to any understanding on the pupils’ part they will be half
      remembered and mis-used).

     Use mathematical language accurately and consistently within the
      department and across the school.
      (Note: The exclusive use of “times” for multiply and “take-away” for
      subtraction can subtly limit pupils’ understanding of important concepts.
      Also, if pupils encounter different words for the same ideas in different lessons
      this will cause confusion).

     Value pupils’ different methods for calculation and regularly ask “How did
      you do that?” and “Did anyone do that a different way?”
      (Note: It is important to encourage pupils to see calculation as an act of
      problem-solving, something that they think about and make their own
      decisions about regarding the most appropriate and efficient method for
      them to use, rather than blindly rushing to a set pencil and paper procedure).

We encourage pupils to:

     See mental calculation as the first resort when faced with any calculation.
      (Note: It is uncommon for pupils to reach for the calculator or a rote learnt
      pencil and paper procedure for calculation that they could, with a little
      thought, do in their head. If all teachers are working together to discourage
      this whenever it arises in lessons; this will do much to encourage pupils’
      numeracy skills).

     Explain any calculation they have done by showing all their working out.
      Encourage the idea that “show your working” does not necessarily mean the
      standard pencil and paper method. It could mean any writing or other
      jotting which genuinely describes how they have arrived at the answer.

     Estimate before a calculation is done whenever possible.

     Consider the reasonableness of their answers after a calculation has been
      done.

     Know how to use all the relevant buttons on their calculator efficiently and
      effectively when it is appropriate and to be able to interpret the display
      sensibly.

     Use appropriate mathematical language confidently.
      (Note: In the same way that we want pupils to “show their working” in their
      writing, we want them to “show their thinking” in their talk.
      Teachers need to create regular opportunities for pupils to explain how they
      make calculations that arise in their lessons in order to encourage this use of
      language).


                                          23
KEY OBJECTIVES FOR DIFFERENT ABILITIES

This section shows what a below average, average and above-average pupils
should be able to cope with in Years 7, 8 and 9. It should be pointed out that the
Years 5 and 6 objectives are included since some of Years 7, 8 and 9 may still be
working at this level. This will enable colleagues in other departments to make sure
that they are not making undue mathematical demands of pupils in their own
subject.

Year 5

      Multiply and divide any positive integer up to 10000 by 10 or 100 and
       understand the effect.
      Order a given set of positive and negative integers.
      Use decimal notation for tenths and hundredths.
      Round a number with one or two decimal places to the nearest integer.
      Relate fractions to division and to their decimal representations.
      Calculate mentally a difference such as 8006 – 2993.
      Carry out column addition and subtraction of positive numbers less than
       10000.
      Know by heart all multiplication facts up to 10 x 10.
      Carry out short multiplication and division of a three-digit by a single-digit
       integer.
      Carry out long multiplication of a two-digit by a two-digit integer.
      Understand area measured in square centimetres (cm²); understand and use
       the formula in words “length x breadth” for the area of a rectangle.
      Recognise parallel and perpendicular lines, and properties of rectangles.
      Use all four operations to solve simple word problems involving numbers and
       quantities, including time, explaining methods and reasoning.

Year 6

      Multiply and divide decimals mentally by 10 or 100, and integers by 1000, and
       explain the effect.
      Order a mixed set of numbers with up to three decimal places.
      Reduce a fraction to its simplest form by cancelling common factors.
      Use a fraction as an operator to find fraction of numbers or quantities (e.g.
       5/8 of 32, 7/10 of 40, 9/100 of 400 centimetres).
      Understand percentage as the number of parts in every 100, and find simple
       percentages of small whole-number quantities.
      Solve simple problems involving ratio and proportion.
      Carry out column addition and subtraction of numbers involving decimals.
      Derive quickly division facts corresponding to multiplication tables up to 10 x
       10.
                                          24
     Carry out short multiplication and division of numbers involving decimals.
     Carry out long multiplication of a three-digit by a two-digit integer.
     Use a protractor to measure acute and obtuse angles to the nearest degree.
     Calculate the perimeter and area of simple compound shapes that can be
      split into rectangles.
     Read and plot coordinates in all four quadrants.
     Identify and use the appropriate operations (including combinations of
      operations) to solve word problems involving numbers and quantities, and
      explain methods and reasoning.
     Solve a problem by extracting and interpreting information presented in
      tables, graphs and charts.

Year 7

     Simplify fractions by cancelling all common factors; identify equivalent
      fractions.
     Recognise the equivalence of percentages, fractions and decimals.
     Extend mental methods of calculation to include decimals, fractions and
      percentages.
     Multiply and divide three-digit by two-digit whole numbers; extend to
      multiplying and dividing decimals with one or two places by single-digit
      whole numbers.
     Break a complex calculation into simpler steps, choosing and using
      appropriate and efficient operations and methods.
     Check a result by considering whether it is of the right order of magnitude.
     Use letter symbols to represent unknown numbers or variables.
     Know and use the order of operations and understand that algebraic
      operations follow the same conventions and order as arithmetic operations.
     Plot the graphs of simple-linear functions.
     Identify parallel and perpendicular lines; know the sum of angles at a point,
      on a straight line and in a triangle.
     Convert one metric unit to another (e.g. grams to kilograms); read and
      interpret scales on a range of measuring instruments.
     Compare two simple distributions using the range and one of the mode,
      median or mean.
     Understand and use the probability scale from 0 to 1; find and justify
      probabilities based on equally likely outcomes in simple contexts.
     Solve word problems and investigate in a range of contexts, explaining and
      justifying methods and conclusions.

Year 8

     Add, subtract, multiply and divide integers
     Use the equivalence of fractions, decimals and percentages to compare
      proportions; calculate percentages and find the outcome of a given
      percentage increase or decrease.
     Divide a quantity into two or more parts in a given ratio; use the unitary
      method to solve simple word problems involving ratio and direct proportion.
                                         25
     Use standard column procedures for multiplication and division of integers
      and decimals, including by decimals such as 0.6 or 0.06; understand where to
      position the decimal point by considering equivalent calculations.
     Simplify or transform linear expressions by collecting like terms; multiply a
      single term over a bracket.
     Substitute integers into simple formulae.
     Plot the graphs of linear functions, where y is given explicitly in terms of x;
      recognise that equations of the form y = mx + c correspond to straight-line
      graphs.
     Identify alternate and corresponding angles; understand a proof that the
      sum of the angles of a triangle is 180° and of a quadrilateral is 360°.
     Enlarge 2D shapes, given a centre of enlargement and a positive whole-
      number scale factor.
     Use straight edge and compasses to do standard constructions.
     Deduce and use formulae for the area of a triangle and parallelogram, and
      the volume of a cuboid; calculate volumes and surface areas of cuboids.
     Construct, on paper and using ICT, a range of graphs and charts; identify
      which are most useful in the context of a problem.
     Find and record all possible mutually exclusive outcomes for single events
      and two successive events in a systematic way.
     Identify the necessary information to solve a problem; represent problems
      and interpret solutions in algebraic, geometric or graphical form.
     Use logical argument to establish the truth of a statement.

Year 9

     Add, subtract, multiply and divide fractions.
     Use proportional reasoning to solve a problem, choosing the correct numbers
      to take as 100%, or as a whole.
     Make and justify estimates and approximations of calculations.
     Construct and solve linear equations with integer coefficients, using an
      appropriate method.
     Generate terms of a sequence using term-to-term and position-to-term
      definitions of the sequence, on paper and using ICT; write an expression to
      describe the nth term of an arithmetic sequence.
     Given values for m and c, find the gradient of lines given by equations of the
      form y = mx + c.
     Construct functions arising from real-life problems and plot their
      corresponding graphs; interpret graphs arising from real situations.
     Solve geometrical problems using properties of angles, of parallel and
      intersecting lines, and of triangles and other polygons.
     Knowing that translations, rotations and reflections preserve length and angle
      and map objects on to congruent images.
     Know and use the formulae for the circumference and area of a circle.
     Design a survey or experiment to capture the necessary data from one or
      more sources; determine the sample size and degree of accuracy needed;
      design, trial and if necessary refine data collection sheets.


                                         26
      Communicate interpretations and results of a statistical enquiry using
       selected tables, graphs and diagrams in support.
      Know that the sum of probabilities of all mutually exclusive outcomes is 1 and
       use this when solving problems.
      Solve substantial problems by breaking them into simpler tasks, using a range
       of efficient techniques, methods and resources, including ICT; give solutions
       to an appropriate degree of accuracy.
      Present a concise, reasoned argument, using symbols, diagrams, graphs and
       related explanatory text.

YEAR 9 OBJECTIVES FOR ABLE PUPILS

      Know and use the index laws for multiplication and division of positive integer
       powers.
      Understand and use proportionality and calculate the result of any
       proportional change using multiplicative methods.
      Square a linear expression and expand the product of two linear expressions
       of the form x ± n; establish identities.
      Solve a pair of simultaneous linear equations by eliminating one variable; link
       a graphical representation of an equation or a pair of equations to the
       algebraic solution.
      Change the subject of a formula.
      Know that if two 2D shapes are similar, corresponding angles are equal and
       corresponding angles are in the same ratio.
      Understand and apply Pythagoras’ Theorem.
      Know from experience of constructing them that triangles given SSS, SAS, ASA
       or RHS are unique, but that triangles given SSA or AAA are not; apply these
       conditions to establish the congruence of triangles.
      Use measures of speed and other compound measures to solve problems.
      Identify possible sources of bias in a statistical enquiry and plan how to
       minimise it.
      Examine critically the results of a statistical enquiry and justify choice of
       statistical representation in written presentations.
      Generate fuller solutions to mathematical problems.
      Recognise limitations on the accuracy of data and measurements.

KEY MATHEMATICAL VOCABULARY
This section contains lists of keywords for each of Years 7, 8 and 9, which pupils
should know and be encouraged to use in lessons. For definitions of the words you
will need to refer to a mathematical dictionary or to the National Curriculum
glossary. The glossary can be found on the QCA website at www.qca.org.uk.

Year 7

This list contains the key words used in the Year 7 teaching programme and
supplement of examples. Some words will be familiar to pupils in Year 7 from earlier
work.
                                          27
APPLYING MATHEMATICS AND SOLVING PROBLEMS

      Answer                         Evidence
      Explain                        Explore
      Investigate method             Problem
      Reason                         Results
      Solution (of a problem)        Solve
      True, False

NUMBERS AND THE NUMBER SYSTEM

PLACE VALUE, ORDERING AND ROUNDING

      Approximate                             Approximately equal to (≈)
      Between                                 Compare
      Decimal number                          Decimal place
      Digit                                   Equals (=)
      Greater than (>), less than (<)         Greatest value, least value
      Most/least significant digit            Nearest
      Order                                   Place value
      Round                                   Tenth, hundredth, thousandth
      To one decimal place (to 1 d.p.)        Value
      Zero place holder

Integers, powers and roots

       Classify                            Common factor
       Consecutive                         Divisible, divisibility
       Divisor                             Factor
       Factorise                           Highest common factor (HCF)
       Integer                             Lowest common multiple (LCM)
       Multiple                            Negative (e.g. – 6)
       Plus, minus                         Positive (e.g. + 6)
       Prime                               Prime factor
       Property                            Sign
       Square number, squared              Square root
Fractions, decimals, percentages, ratio and proportion

      Cancel, cancellation                    Convert
      Decimal fraction                        Equivalent
      Fraction                                Lowest terms
      Mixed number                            Numerator, denominator
      Percentage (%)                          Proper/improper fraction
      Proportion                              Ratio, including notation 3 :2
      Simplest form



                                         28
Calculations

      Add, addition                         Amount
      Brackets                        Calculate, calculation
      Calculator: clear, display, enter     Key, memory
      Change (money)                  Commutative
      Complements (in 10, 100)              Currency
      Difference                      Discount
      Divide, division                Double, halve
      Estimate                        Exact, exactly
      Exchange rate                   Factor
      Increase, decrease                    Inverse
      Multiply, multiplication        Nearly
      Operation                       Order of operations
      Partition                       Product
      Quotient                        Remainder
      Rough, roughly                  Sale price
      Sign                            Subtract, subtraction
      Sum                             Total

Algebra

Equations, formulae and identities

      Algebra                        Brackets
      Commutative                           Equals (=)
      Equation                       Expression
      Evaluate                       Prove
      Simplify, simplest form        Solution (of an equation)
      Solve (an equation)                   Squared
      Substitute                     Symbol
      Term                           Unknown
      Value                          Variable
      Verify




Sequences, functions and graphs

      Axis, axes                     Consecutive
      Continue                       Coordinate pair
      Coordinate point               Coordinates
      Equations (of a graph)         Finite, infinite
      Function                       Function machine
                                         29
      Generate                       Graph
      Increase, decrease                     Input, output
      Mapping                        n th term

      Origin                         Predict
      Relationship, rule             Sequence
      Straight-line graph            Term
      X-axis, y-axis                 X-coordinate, y-coordinate

Shape, space and measures

Geometrical reasoning: lines, angles and shapes

      Adjacent (side)                Angle: acute, obtuse, right, reflex
      Angles at a point              Angles on a straight line
      Base (of plane shape or solid) Base angles
      Centre                                Circle
      Concave, convex                Degree (°)
      Diagonal                       Diagram
      Edge (of solid)                Equal (sides, angles)
      Face                           Horizontal, vertical
      Identical (shapes)             Intersect, intersection
      Line, line segment             Opposite (sides, angles)
      Parallel                       Perpendicular
      Plane                          Point
      Polygon: pentagon, hexagon Octagon
      Quadrilateral: arrowhead, delta       Kite, parallelogram, rectangle
      Rhombus, square, trapezium Regular, irregular
      Shape                                 Side (of 2D shape)
      Solid (3D) shape: cube, cuboid        Cylinder, hemisphere, prism
      Pyramid, square-based pyramid         Sphere, Tetrahedron
      Three-dimensional (3D)         Triangle: equilateral, isosceles
      Scalene, right-angled          Two-dimensional (2D)
      Vertex, vertices               Vertically opposite angles

Transformations

      Axis of symmetry               Centre of rotation
      Congruent                      Line of symmetry
      Line symmetry                  Mirror line
      Object, image                  Order of rotational symmetry
      Reflect, reflection            Reflection symmetry
      Rotate, rotation               Rotational symmetry
      Symmetrical                    Transformation
      Translate, Translation

Coordinates

      Axis, axes                     Coordinates

                                         30
      Direction                    Grid
      Intersecting, intersection   Origin
      Position                     Quadrant
      Row, column                         X-axis, y-axis
      X-coordinate, y-coordinate

Construction and loci

      Construct                    Draw
      Measure                      Net
      Perpendicular                      Protractor (angle measurer)
      Ruler                        Set square
      Sketch

Measures

      Area: mm², cm², m², km²      Capacity: millilitre, centilitre, litre
      Pint, gallon                 Length: millimetre, centimetre, metre, km
      Mile                         Mass: gram, kilogram, ounce
      Pound                              Time: second, minute, hour, day
      Week, month, year, decade    Century, millennium
      Temperature: °C, °F          Depth
      Distance                     Height, high
      Perimeter                    Surface, surface area
      Width

Handling data

      Average                      Bar chart
      Bar-line graph               Class interval
      Data, grouped data                   Data collection sheet
      Database                     Experiment
      Frequency                    Frequency chart
      Frequency diagram                    Interpret
      Interval                     Label
      Mean                         Median
      Mode, modal class/group              Pie chart
      Questionnaire                Range
      Represent                    Statistics
      Survey                               Table
      Tally                        Title


Probability

      Certain, uncertain           Chance, no chance, good chance
      Poor chance, 50-50                Even chance
      Dice                         Doubt

                                       31
      Equally likely                         Fair, unfair
      Likelihood                     Likely, unlikely
      Outcome                        Possible, impossible
      Probability                    Probability scale
      Probable                       Random
      Risk                           Spin, spinner

Year 8

This list contains the new key words introduced in the Year 8 teaching programme
and supplement of examples. Words from earlier years are also used.

Applying mathematics and solving problems

      Conclude, conclusion           Counter-example
      Deduce                         Exceptional case
      Justify                        Prove, proof

Numbers and the number system

Place value, ordering and rounding

      Ascending, descending          Billion
      Index                          Power

Integers, powers and roots

      Cube, cube number                    Cube root (e.g. 3√8)
      Cubed (e.g. 2³)                Prime factor decomposition
      To the power of n (e.g. 6^4)

Fractions, decimals, percentages, ratio and proportion

      Direct proportion              Recurring decimal
      Terminating decimal                  Unit fraction
      Unitary method




Calculations

      Associative                    Best estimate
      Degree of accuracy                     Distributive
      Interest                       Profit, loss
                                         32
      Service charge                 Sign change key
      Tax                            Value added tax (VAT)

Algebra

Equations, formulae and identities

      Algebraic expression           Collect like terms
      Formula, formulae              Linear equation
      Linear expression              Multiply out (expressions)
      Prove, proof                   Transform
      Verify

Sequences, functions and graphs

      Arithmetic sequence                  Difference pattern
      Flow chart                     General term
      Gradient                       Intercept
      Linear function                Linear relationship
      Linear sequence                Notation T(n)
      Slope                          Steepness

Shape, space and measures

Geometrical reasoning: lines, angles and shapes

      Alternate angles               Bisect, bisector
      Complementary angles           Congruent, congruence
      Corresponding angles           Elevation
      Equidistant                    Exterior angle
      Heptagon                       Interior angle
      Isometric                      Mid-point
      Plan view                      Prove, proof
      Supplementary angles           Tessellate, tessellation
      Triangular prism               View

Transformations

      Centre of enlargement          Enlarge, enlargement
      Map                            Plan
      Scale, scale factor                  Scale drawing



Constructions and loci

      Compasses                      Construction lines
      Locus, loci                    Perpendicular bisector
                                         33
      Straight edge

Measures

      Bearing, three-figure bearing   Displacement
      Foot, yard                      Hectare
      Tonne                           Volume: mm³, cm³, m³

Handling data

      Continuous                      Data log
      Discrete                        Distance-time graph
      Distribution                    Interrogate
      Line graph                      Population pyramid
      Primary source                  Sample
      Scatter graph                          Secondary source
      Stem-and-leaf diagram           Two-way table

Probability

      Biased                                Event
      Experimental probability        Sample
      Sample space                    Theoretical probability
      Theory

Year 9

This list contains the new key words introduced in the Year 9 teaching programme
and supplement of examples. Words from earlier years are also used.

Applying mathematics and solving problems

      Generalise                      Trial and improvement

Numbers and the number system

Place value, ordering and rounding

      Exponent                        Greater that or equal to (≥)
      Less than or equal to (≤)       Significant figures
      Standard (index) form           Upper bound, lower bound



Integers, powers and roots

      Index, indices                  Index law

                                          34
      Index notation

Fractions, decimals, percentages, ratio and proportion

      Proportional to                Proportionality

Calculations

      Compound interest                    Constant
      Cost price, selling price      Reciprocal

Algebra

Equations, formulae and identities

      And, or                       Common factor
      Cubic equation                Cubic expression
      Expand the product (of two linear expressions)
      Factorise                     Identically equal to
      Identity                      Index law
      Inequality                    Quadratic equation
      Quadratic expression          Region
      Simultaneous equations        Subject of the formula
      Take out common factors

Sequences, functions and graphs

      Cubic function                 Curve
      First/second difference        Identity function
      Inverse function               Inverse mapping
      Quadratic function                    Quadratic sequence
      Maximum/ minimum point                Maximum/ minimum value
      Self-inverse

Shape, space and measures

Geometrical reasoning: lines, angles and shapes

      Arc                            Centre (of circle)
      Chord                                Circumference
      Convention                     Cross-section
      Definition                     Derived property
      Diameter                       Hypotenuse
      Pi (Π)                         Plane
      Projection                     Pythagoras’ Theorem
      Similar, similarity            Region
      Radius                               Section
      Sector                               Segment
                                         35
      Tangent (to a curve)


Transformations

      Axis of rotational symmetry     Plane symmetry
      Plane of symmetry

Construction and loci

      Circumcentre                           Circumcircle
      Circumscribed                   Inscribed

Measures

      Adjacent, opposite, hypotenuse        Angle of depression
      Angle of elevation             Density
      Pressure                       Sine (sin), cosine (cos), tangent (tan)
      Speed: miles per hour, metres per second

Handling data

      Bias                            Census
      Cumulative frequency            Estimate of the mean/median
      Interquartile range             Line of best fit
      Quartiles                       Raw data
      Representative (sample)

Probability

      Exhaustive                       Independent
      Limit                            Mutually exclusive
      Notation: P(n) for probability of event “n”
      Relative frequency                     Tree diagram




Mathematical competencies required in other subjects
                                          36
The link between Numeracy and Literacy

The role of language is important in numeracy and there will be regular contact
between the Numeracy and Literacy Co-ordinators to ensure that both are aware
of developments in their respective areas.

      Language is an important tool for learning mathematics. Explaining to
       oneself, or someone else “putting it into words”, can be a powerful means of
       working through and clarifying ideas.

      Pupils should use language as a tool for reflecting on their mathematical
       experiences and hence for their own mathematical learning.

      Pupils also need to develop the skills of recording their mathematics. The first
       forms of recording are likely to be in everyday language or in pictures or
       diagrams. Gradually these representations may be shortened, leading to the
       need to use symbols.

Pupils should develop spatial language in much the same way as they learn to talk
about various animals and objects – by hearing it used appropriately by others and
being encourages to use progressively more sophisticated language in describing
their experiences.




Links to other subjects

English

English lessons can help to develop and support pupils’ numeracy skills, for
example, by use of mathematical vocabulary and technical terms, by asking
children to read and interpret problems to identify the mathematical content, and
by encouraging them to explain, argue and present their conclusions to others.
Pupils use statistical analysis in the study of advertising and of the media. They learn
to distinguish between fact and opinion by looking at differences that are backed
up by data. Sequencing plays an important part in placing events in order when
compiling a writing frame. Use of metre, syllables and rhythm in poetry and
Shakespeare.




                                           37
Science

Almost every scientific investigation or experiment is likely to require one or more of
the mathematical skills of classifying, counting, measuring, calculating, estimating
and recording in tables and graphs. In science pupils will, for example, order
numbers, including decimals, calculate simple means and percentages, use
negative numbers when taking temperatures and relate these to boiling and
melting points, collect, display and interpret data, decide whether it is more
appropriate to use a line graph or bar chart, and plot, interpret and predict from
graphs. They need to understand the scale on the axes of a graph, draw and
interpret lines of best fit, use and manipulate formulae, convert units for example
change minutes into seconds, use the correct units for measuring length, area,
volume, mass, charge, voltage, time etc. Pupils will meet the concept of variation,
squares and square roots, compound measures such as speed and density. They
will need to know how to use a calculator properly and will use standard form for
very large and very small numbers.

Speed = distance ÷ time

Force = mass x acceleration

Acceleration = change in velocity ÷ time taken

Density = mass ÷ volume

Work done = force x distance moved in direction of force

Energy transferred = work done

KE = ½ mass x speed²      (KE = ½ mv²)

Change in potential energy = mass x gravitational field strength x change in height

Weight = mass x gravitational field strength

Pressure = force ÷ area

Moment = force x perpendicular distance from pivot

Charge = current x time

Voltage = current x resistance

Electrical power = voltage x current

Wave speed = frequency x wavelength



                                           38
Art and Technology

Measurements are often needed in Art and Technology. Many patterns and
constructions are based on spatial ideas and properties of shapes, including
symmetry and tessellations. Pupils use the concepts of perspective in landscape
drawing and proportion in various topics. A lot of work is also undertaken using
estimation of measurements and quantities. Design may need enlarging or
educing, introducing ideas of multiplication and ratio. Trigonometry and
Pythagoras’ Theorem may be used in the calculation of forces, moments and stress
and strain and Pi is used when there are calculations involving circles. Equations
may need to be rearranged and a calculator has to be used properly. The
understanding of angles and measurements is used in drawing and nets arise in
surface development. When dealing with recipes and cooking pupils will carry out
a great deal of measurement calculations that include working out times and
calculating cost. Pupils may need to change the units of measurement from one
unit to another. Surveys are carried out and questionnaire design results in the
analysis of data. Pupils draw graphs and need to be able to interpret them.

History, Geography and Religious Education

In History and Geography, pupils will collect data by counting and measuring and
make use of measurements of many kinds. In Geography pupils statistical analysis
includes pie charts, bar and line graphs, Spearman’s Rank Correlation Co-efficient
(beyond the scope of GCSE mathematics) and mean chloropleth maps. They look
at patterns of numbers and interpret their significance, which uses the concept of
proportion. The study of maps includes the use of co-ordinates and ideas of angle,
direction, position, scale and ratio. In History, the study of maps and distances
includes the effect on people and events. In particular they look at Roman miles
when considering distance to put events into perspective. Pupils look at the
percentage of people surviving a battle and the percentage of trade across the
Roman Empire. Roman numerals are studied in History and pupils examine how
shape affects the strength of a castle. The pattern of the days of the week, the
calendar and recurring annual festivals all have a mathematical basis and pupils
look at the lunar calendar and Ramadan in RE. Historical ideas require
understanding of the passage of time, which can be illustrated on a time line,
similar to the number line that they already know. Symmetry is important in RE,
especially when looking at Islamic design and pupils consider the importance of
symbols such as the Star of David. Pupils use measurements when they study
Noah’s Ark and they interpret statistics and graphs in the topic of abortion. They
look at mathematics in the workplace and study poverty and wealth, using
payment and economics in speculating, for example world wide depression.

ICT

Pupils will apply and use mathematics in a variety of ways when they solve
problems using ICT. For example, they will collect and classify data, enter it into
data handling software, produce graphs and tables, and interpret and explain their
results. Pupils enter formulae into spreadsheets using mathematical operators, but
                                        39
substitute * for x and type = at the beginning of the formula. Their work in control
includes the measurement of distance and angle, using uniform non-standard then
standard measures. When they use computer models and simulations they will
draw on their abilities to manipulate numbers and identify patterns and
relationships.
Modern Foreign Languages

Pupils study foreign currency and numbers within a country. They perform simple
addition, subtraction and multiplication. They tell the time in a foreign language
and have to be able to use analogue and 24 hour notation. Percentages occur
when they analyse surveys and graphs and the terms “more than”, “less than” and
“the most popular answer was…” are used.

Physical Education and Music

Athletic activities require measurement of height, distance, angle and time, while
ideas of counting, time, symmetry, movement, position and direction are used
extensively in music, dance, gymnastics and ball games.




                                         40
Appendix

This appendix contains different approaches to calculation that teachers may be
unfamiliar with.

Often pupils will partition numbers to make the numbers easier to handle. The
following are examples where the numbers have been partitioned into hundreds,
tens and units.

           345 + 47 = (300 + 40 + 5) + (40 + 7) = 300 + 80 + 12 = 392

           53 – 27 = 53 – 20 – 7 = 33 – 7 = 26

           Half of 238 = half of (200 + 30 + 8) = 100 + 15 + 4 = 119

This is the grid method for multiplication:

242 x 3

   x            200         40          2
   3            600        120          6        = 726



36 x 48

       x              30            6
    40              1200          240        = 1440 +

    8                240           48        = 288
                                                  1728




The following are examples of counting on in multiples of 100, 10 or 1 (counting on
from the smaller number to the larger number).

           86 + 57 = 86 + 50 + 7 = 136 + 7

                                                   41
                +50                        +4                +3
      86                      136                    140          143




       84 – 56
This problem is changed to “How much do i add to get from 56 to 84?”

      56 + 4 + 20 + 4 = 84


           +4        +20              +4
      56        60                         80        84


Change from £5 after spending £2.47
Count on from £2.47: 3p + 50p + £2 = £2.53


Compensation

      34 + 19 = 34 + 20 – 1 = 54 – 1 = 53

      98 + 32 = 100 + 32 – 2 = 132 – 2 = 130

      56 – 38 = 56 – 40 + 2 = 18

      £3.98 x 5 = (£4 x 5) – (2p x 5) = £20 – 10p = £19.90

      13 x 21 = (13 x 20) + 13 = 260 + 13 = 273

Chunking

Considering division as repeated subtraction. Estimation skills are needed for
division in order for students to make a sensible choice of multiple to subtract.


      72 ÷ 5

                            72
                           - 50     (10 X 5)
                            22
                           - 20     (4 X 5)
                              2                            Answer: 14 remainder 2


                                                42
There is a problem that pupils encounter with this method and that iis the
subtraction, hence the following is preferred:

In the following 72 ÷ 5 the 72 is seen as the target number and it is made up by
adding multiples of 5.

     50         10 x 5
     20          4x5
     70         14 x 5
We need 2 to make 72, thus the answer is 14 remainder 2.

The standard method for 72 ÷ 5 is:

            14 remainder 2
        5   72


Napier’s bones or Gelosia

Napier’s bones method for multiplying three digits by two digits:



                                3               2                   7
                            1               0                   2
                                    2               8                   8   4
                          2                 1                   5
                                    4               6                   6   8

1           5         6                 9               6

327 x 48 = 15626

Explanation: The numbers outside the grid are multiplied and the answer goes inside
one of the cells with the tens digit above and the units digit below.

                                                7
                                            2
                                                     8      4

This cell shows how 7 x 4 = 28 is entered.
When all of the cells have been filled, the numbers are then added along the
diagonals, starting from the right hand side.
The first number is 6, since it is the only number in that diagonal.
To make the answer in the next diagonal we do 6 + 5 + 1 = 12. 2 is written down
and the 1 from the tens is carried to the next diagonal.
                                                43
     This method is continued until the end.

     The standard method for 327 x 48 is long multiplication:

                   327
                   x48
                  2616               (327 x 8)
                 13080               (327 x 40)
                 15696

     Mathematical Association Audit
     Reproduced here is the Maths Association audit. It has been modified to fit with the
     Key Stage 3 Curriculum – adaptations could be made for KS4.



        Mathematical Skills               N.C.    NNS Year   Year 7   Year 8   Year 9   Other
                                          Level
       Number and Algebra

Understand place value and
order numbers up to 100                     2        2

Know when to add or subtract
when solving problems                       2      2, 3, 4


Recognise odd and even number               2        2

Read and write numbers in figures
and words, up to a million                          4, 5

Round numbers to the nearest 10,
100 or 1000                                 3      3, 4, 5


Use <, >, <, > and = symbols                        4, 5

Mental addition and subtraction
of 2 digit numbers                          3        4

Written addition and subtraction
of three digit numbers                      3        4

Multiplication tables – 2, 3, 4, 5
and 10 and associated division              3        4
facts

Use simple fractions and
recognise when two simple                  3,4       4
                                                    44
fractions are equivalent

Use decimal notation – money            3         4

Recognise and use negative              3         4
numbers in context

Multiplication tables up to 10 X 10     4         5
and associated division facts

Multiply and divide whole               4        4,5,6
numbers by 10 and 100

                                       N.C.    NNS Year   Year 7   Year 8   Year 9   Other
                                       Level
Written addition and subtraction         4        5
of whole numbers

Short multiplication and division of    4         5
whole numbers

Add and subtract decimals to            4        5,6
two decimal places

Order decimals to three decimal         4         6
places

Recognise and use simple                4         5
percentages

Use simple formulae expressed in        4        4,5,6
words

Use co-ordinates in the first           4        4,5
quadrant

Multiply and divide decimals by         5
10, 100, 1000

Order, add and subtract negative        5        5,6
numbers in context

+, -, X, ÷ with up to 2 decimal         5        5, 6
places

Calculate fractions of quantities       5        5, 6

Use simple ratio and proportion         5        5, 6

Long multiplication                     5        5, 6

Long division                           5

                                                 45
Check answers using inverse          5       4, 5, 6
operations

Estimate and check answers using     5       4, 5, 6
approximations

Construct simple formulae            5        5, 6

Use co-ordinates in four             5         6
quadrants

Calculate a number as a fraction     6       4, 5, 6
of another


                                    N.C.    NNS Year   Year 7   Year 8   Year 9   Other
                                    Level
Calculate a number as a               6        6
percentage of another

Understand and use fraction,         6        5, 6
decimal and percentage
equivalence

Add and subtract fractions           6         7

Calculate using ratios               6         7

Solve linear equations               6         7

Use co-ordinates for                 6         7
geographical representation

Calculate percentages of             5        5, 6
quantities

Use simple formulae                  5        5, 6

Use significant figures              7         9

Multiply and divide fractions and    7         9
decimals

Use a calculator efficiently and     7         9
appropriately

Use proportional change              7         9

Solve simultaneous linear            7         9
equations graphically

Solve simultaneous linear            7         9
equations algebraically
                                              46
Solve simple inequalities               7         9

Calculate power and roots               8

Use standard form                       8

Use formulae involving fractions,       8
decimals or negative numbers

Calculate the original quantity         8
given the result of proportional
change

Solve problems involving                8
repeated proportional change


                                       N.C.    NNS Year   Year 7   Year 8   Year 9   Other
                                       Level
Interpret graphs modelling real life     8
situations

Draw graphs modelling real life         8
situations

Transform formulae                      8

Solve inequalities in two variables     8

Determine the bounds of intervals       EP

Find formulae that approximately        EP
connect data and express
general laws in symbolic form
Use direct proportion                   EP

Use indirect proportion                 EP

Use the rules of indices                EP

Solve problems using intersections      EP
or gradients of graphs




                                                 47
     Mathematical Skills         N.C. Level   NNS Year   Year 7   Year 8   Year 9   Other

Shape, Space and Measures

Mathematical names for 2-D           2         4, 5, 6
and 3-D shapes

Understand angle as a                2         4, 5, 6
measure of turn

Recognise right angles               2           4

Understand reflective                3         4, 5, 6
symmetry

Use metric units of length,          3         4, 5, 6
capacity, mass and time

Solve problems involving                        4, 5
time or timetables

Make simple 3-D models               4         4, 5, 6
from nets

Draw 2-D shapes in different         4
orientations

Understand rotational                4         4, 5, 6
symmetry

Reflect simple shapes in a           4         4, 5, 6
mirror line

Measure and read scales              4         4, 5, 6
using appropriate units and
accuracy

Find perimeters of simple            4         4, 5, 6
shapes

Find areas by counting               4          4, 5
squares

Measure and draw angles              5          5, 6

Know the angle sum of a              5           6
triangle

Know the sum of angles at a          5           6
point
Identify all the symmetries of       5         4, 5, 6
2-D shapes
Convert one metric unit to           5          5, 6
                                                   48
another
Know rough metric/imperial            5           6
equivalence of common
units
Estimate measures                     5         4, 5, 6


                                  N.C. Level   NNS Year   Year 7   Year 8   Year 9   Other

Know and use the formula              5           5
for the area of a rectangle

Draw and interpret simple
scale drawings

Recognise 2-D                         6         4, 5, 6
representations of 3-D
shapes

Know and use properties of            6           7
quadrilaterals

Use angle and symmetry                6           7
properties of polygons

Use angle properties of               6           8
intersecting and parallel lines

Devise instructions for a             6          7,8,9
computer to generate and
transform shapes

Use the formula for the               6           9
circumference of a circle

Use the formula for the area          6           9
of a circle

Find the areas of plane               6           7
rectilinear figures

Use the formulae for the              6           8
volume of a cuboid

Enlarge shapes by a positive          6           8
whole number scale factor

Use Pythagoras' Theorem in            7           9
2D

Calculate lengths and areas           7
in plane shapes

Calculate volumes of prisms           7           9
                                                    49
Enlarge shapes by a                             9
fractional scale factor

Determine the locus of a            7          8,9
moving object

Understand the limitations of       7
accuracy of measurements
                                                                                   Other
                                                                          Year 9
                                N.C. Level   NNS Year   Year 7   Year 8
Understand and use                  7
compound measures

Understand and use similarity       8
and congruence

Use trigonometry in 2-D             8

Distinguish between                EP
formulae for perimeter, area
and volume by considering
dimensions
Use Pythagoras' Theorem in         EP
3D

Use trigonometry in 3-D            EP

Calculate lengths of circular      EP
arcs

Calculate areas of sectors         EP

Calculate surface area of          EP
Cylinders

Calculate volume of cones          EP
and spheres




                                                 50
                                                                                     Other
     Mathematical Skills          N.C. Level   NNS Year   Year 7   Year 8   Year 9

        Handling data

Sort and classify objects by          2           4
more than one criterion

Record results in simple lists,       2           2
tables and block graphs

Interpret simple tables and           3           3
lists

Interpret pictograms                  3          3, 4

Draw pictograms                       3          3, 4

Interpret bar graphs                  3         4, 5, 6

Draw bar graphs                       3         4, 5, 6

Collect data and record               4         4, 5, 6
them using frequency tables

Understand and use the                4          4, 5
mode, the median and the
range of a set of data
Group collected data into             4
equal class intervals

Draw frequency diagrams               4           6
using grouped data

Interpret line graphs                 4          5, 6

Select and use appropriate
scales for axes

Draw line graphs                      4          5, 6

Understand and use the                5           6
mean of a set of data

Use averages and ranges to            5           7
compare two sets of data

Interpret pie charts                  5           6

Understand and use the                5           7
probability scale from 0 to 1
Find probabilities using           5           7
equally likely outcomes or
experiment

                               N.C. Level   NNS Year   Year 7   Year 8   Year 9   Other
Create frequency tables with       6           8
equal class intervals to
record continuous data
Interpret frequency diagrams       6           8

Draw frequency diagrams            6           8

Draw pie charts                    6           8

Draw scatter diagrams              6           8

Understand simple                  6           9
correlation

Use two-way tables to record       6           8
all the possible outcomes of
two events
Use the fact that the total        6           8
probability of all mutually
exclusive outcomes of an
experiment is 1
Specify and test hypotheses        7           9
using appropriate methods
and taking account of
variability and bias
Find modal class of grouped        7           9
data

Estimate the mean, median          7           9
and range of grouped data

Use averages and ranges            7           9
and frequency polygons to
compare two sets of data
Draw a line of best fit on a       7           9
scatter diagram

Use relative frequency to          7           9
estimate probability

Interpret cumulative               8
frequency tables and
diagrams

Construct cumulative               8
frequency tables and
diagrams


                                              52
Estimate the median,               8
quartiles and inter-quartile
range from a cumulative
frequency diagram
Interpret histograms with         EP
unequal class intervals


                               N.C. Level   NNS Year   Year 7   Year 8   Year 9   Other
Understand and use                EP
sampling

Draw histograms with              EP
unequal class intervals

Use Spearman's coefficient        EP
of correlation




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