PHOTO REDACTED DUE TO THIRD PARTY RIGHTS OR OTHER …Year 10 2 tier UAM in exam 2 tier UAM in exam 2...

Renewing the Framework for secondary mathematics Spring 2008 subject leader development meeting: Sessions 2, 3 and 4

PHOTO REDACTED DUE TO THIRD PARTY RIGHTS OR OTHER LEGAL ISSUES

First published in 2008

Ref: 00124-2008DOM-EN

Renewing the Framework for secondary mathematics

Spring 2008 subject leader development meeting: Sessions 2, 3 and 4

Disclaimer

The Department for Children, Schools and Families wishes to make it clear that the Department and its agents accept no responsibility for the actual content of any materials suggested as information sources in this publication, whether these are in the form of printed publications or on a website.

In these materials icons, logos, software products and websites are used for contextual and practical reasons. Their use should not be interpreted as an endorsement of particular companies or their products.

The websites referred to in these materials existed at the time of going to print.

Please check all website references carefully to see if they have changed and substitute other references where appropriate.

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1The National Strategies | Secondary Spring 2008 subject leader development meeting for mathematics

© Crown copyright 2008 00124-2008DOM-EN

The National Strategies | Secondary Spring 2008 subject leader development meeting for mathematics


A timeline for change

Follow the progress of each cohort by means of a diagonal line starting from the year group as they begin the academic year 2008–09. For example, the line for the cohort comprising Year 7 in 2008–09 (top row) runs diagonally down, from left to right, to end as Year 11 in 2013 (bottom row). The year groups most affected by the new programmes of study are shown in dark blue. Other, lighter shades could be used in evolving plans as the department works through a phased implementation.

2007/08 2008/09 2009/10 2010/11 2011/12 2012/13 2013/14

Year 7 old KS3 newKS3

Year 8 old KS3 old or new KS3

newKS3

Year 9 old KS3 old or newKS3

old or new KS3

newKS3

Year 10 2 tier UAM in exam

2 tier UAM in exam

2 tier UAM in exam

Functionalskills

GCSE 1GCSE 2

Functionalskills

GCSE 1 GCSE 2

Year 11 2 tier with cwk

2 tier UAM in exam

2 tier UAM in exam

2 tier UAM in exam

Functionalskills

GCSE 1 GCSE 2

Functionalskills

GCSE 1GCSE 2

NB: The relationship of functional skills to GCSE will be informed by the pilot; however, to achieve grade C or above pupils will require functional skills level 2.



Mathematical processes and applications

In the new programmes of study at Key Stages 3 and 4 there is much greater emphasis on the key processes and attainment target 1 is different, now entitled ‘Mathematical processes and applications'. This reflects the importance given to key processes in the 2008 curriculum. It also parallels the emphasis on key processes in other subjects and on aspects of process that reach across the curriculum, enshrined in the personal, learning and thinking skills (PLTS). A key aim of the curriculum is that pupils should see themselves more explicitly as learners and become aware of their developing skills, central to their work in school and to all aspects of their lives.

‘Using and applying mathematics‘ was previously broadly described under the sub-headings of ‘problem-solving‘, ‘communicating‘ and ‘reasoning‘. Problem-solving lies at the heart of mathematics and involves a cycle of processes. These are elaborated in the key processes of the curriculum. By the inclusion of mathematical procedures, well-defined routines and algorithms, a more complete description of process is achieved:

Representing ●

Analysing ●

Use mathematical reasoning –

Use appropriate mathematical procedures –

Interpreting and evaluating ●

Communicating and reflecting ●

The process skills help pupils both to learn mathematics and to apply their mathematical subject knowledge to deal with situations from life and the world of work. To ensure that they make progress in developing these skills and can function mathematically, pupils need to experience a rich ‘diet’ of applications that includes:

increasingly ● complex applications, including non-routine or multi-step problems and extended enquiries, that require them to analyse a situation and sustain their thinking

situations that are ● unfamiliar (in the sense that they are different from the context where the mathematics was developed), including applications to other subjects or aspects of their lives, that requires them to make connections and transfer their skills, sometimes in creative ways

situations or problems that increase the ● technical demand of the mathematics required to solve them, including the application of more advanced concepts, more difficult procedures, or more rigorous argument and proof

opportunities to develop greater ● independence and autonomy in problem-solving skills, so that they can select and apply a higher level of mathematics for themselves.

In summary, it is the context, and the mathematics to be applied to it, that determines the nature of the processing skills that pupils need and the level of challenge they face. It is helpful to think of a ‘problem-solving cycle’ but, as the diagram overleaf shows, many contexts require movement in and out of the cycle. For example, the ‘representing’ phase of a more complex problem may require some ‘analysing’, ‘interpreting’ or ‘communicating’ in order to set up the model.

2 The National Strategies | Secondary Spring 2008 subject leader development meeting for mathematics

© Crown copyright 200800124-2008DOM-EN

Representing

InterpretingEvaluating

Use proceduresAnalysing

Use reasoning

CommunicatingReflectingCONTEXT MATHEMATICS

Mathematical processes should be embedded within the everyday teaching of the strands of number, algebra, geometry and measures, and statistics and in all cross-strand work. The related documents listed below give illustrative examples for each of the strands, including some observations of how ICT can be used to engage pupils with the key processes.

Key processes in number ●

Key processes in algebra ●

Key processes in geometry and measures ●

Key processes in statistics ●

ICT and the key processes ●



Launching the new programme of study in mathematics

IntroductionThese notes are designed to help you launch the new mathematics programme of study at Key Stage 3 with your department and to set out initial thoughts for a longer-term plan. The new secondary curriculum (phased in from September 2008) is based on some over arching principles, including:

Greater flexibility and coherenceThe curriculum is focused on the key concepts and processes that underlie each subject, with less detailed prescription of content. This makes it easier to see links between subjects and increase the coherence for pupils across the curriculum and school activities.

New focus on aims and skills The curriculum includes a framework for personal, learning and thinking skills (and functional skills for English, mathematics and ICT), embedded in the programmes of study. The key processes highlight the essential skills that learners need in order to make progress and achieve in each subject.

Emphasis on assessment for learningGreater flexibility in the curriculum will give teachers more time to focus on assessment for learning strategies and to provide more targeted assessments to meet individual learners’ needs.

In order to appreciate the scope for flexibility, it is essential first to recognise the impact that the key concepts and processes can have on pupils’ learning in mathematics. It is also important to appreciate how engaging pupils explicitly in the key processes will strengthen their skills in solving problems and applying their mathematical knowledge, much more effectively than over-emphasis on coverage of content.

So the first requirement is for your department to become familiar with the key processes in mathematics and how they expand on ‘using and applying mathematics’. The first five tasks NC1–5 will help you to launch the new curriculum with your department. They are best completed in short sequence, preferably in a departmental half-day or several shorter meetings not too far apart.

The tasks have specific, practical outcomes and should leave you in a good position to assess your current practice, particularly how you address the key processes in your teaching. You can then plan for evolutionary change in your Key Stage 3 scheme of work, over the period 2008–2011. Do not be tempted to go for a superficial rewrite of your existing scheme of work. More effectively, establish a collaborative approach to planning with a measured and sustained programme of revising and updating units of work.

To support your phased implementation of the new programmes of study further, a Renewed Framework for secondary mathematics will be available from summer 2008. The summer term subject leader development meetings will support the use of the renewed Framework by introducing a



Secondary mathematics planning toolkit, modelled on the Mathematics planning toolkit: Key Stage 4, which has been in use in many departments since summer 2007.

The documents needed for the launch tasks are included in a folder Launching the new KS3 curriculum (2008), on the CD-ROM. It includes a briefing note for an administrative assistant or technician to help with preparation of resources. One of the launch tasks involves reviewing and revising an early algebra unit and it would be helpful to bear this in mind in your preparatory work.

Suggested pre-reading for you and preferably all members of the mathematics department is:

NC programme of study for Key Stage 3 (2008) ●

Extracts from guidance in the renewed ● Framework for secondary mathematics (to be available from summer 2008):

Mathematical processes and applications –

Key processes in algebra –

Teaching and learning approaches –

You will need copies of the above for each colleague at your launch meeting.

Launching the new programme of studyFirst, complete a classifying task designed to begin to familiarise everyone with the key processes in mathematics. You will need to print copies of the document Key processes classifying task. Cut up sheets of the key process headings and statements so that each statement is on a separate slip of paper, one set per two teachers. You will also need, for each colleague:

Mathematical processes and applications ●

Note: This and the following four tasks are suitable for a half-day departmental meeting, or an equivalent sequence of shorter meetings.



Task NC1 Introducing the key mathematical processes

Use your pre-reading, the notes above and your knowledge of your department, to explain to colleagues:

the aims of the new national Curriculum programmes of study; ●

how you are proposing to respond as a first step. ●

Introduce the task, a card sort designed to familiarise everyone with the key processes in mathematics. Working in pairs, lay the five key process headings on the table:

Representing Communicating and reflecting Analysing – using mathematical reasoning Analysing – using appropriate procedures Interpreting and evaluating

Ask pairs to discuss each of the process statements in turn and assign them to one of the process headings. Expect a healthy debate, there is no right answer! It is a first step in becoming familiar with the new curriculum.

Allow time for classifying, then pose a couple of questions to the group:

Which cards were difficult to place? Why? ●

Can you find a set of three linked cards and explain the link? ●

Point out that the National Curriculum programme of study offers a categorisation, as a helpful prompt to thinking. However, there will always be room for debate about any description of processes.

The key processes are important when considering how pupils should engage with mathematics. You will consider this next, in the context of a particular example. Round off by asking colleagues to read (or re-read) the document Mathematical processes and applications, drawn from the guidance in the renewed Framework for secondary mathematics (available from summer 2008).

To get to grips properly with the key processes it helps to reflect on a mathematical task that is sufficiently rich and open. Exploring patterns and relationships on a hundred square, familiar to many, is an accessible context for algebraic generalisation and problems can be posed in many ways. You might find the ‘Hundred square’ prompt sheet useful when setting the task to colleagues. For personal preparation in leading the task you might also find it helpful to read Case study ‘matchstick shapes’ before the meeting. This gives an example of how a group of teachers built up a simple process map for themselves. For the meeting you will need the following as paper copies

For each colleague:

Key processes in algebra, highlighter pens ●

For pairs of colleagues:

100 number grid ● (for ‘hundred square’ task),

Key processes adaptable template ●

For the whole group:

Key processes blank template ● – a large hand-drawn version or the adaptable software version of the map to use on an interactive white board or projector.



Task NC2 Exploring the key processes

Provide pairs with a hundred square and give them 10 minutes to explore the problem posed. The context may be familiar, but it is helpful to explore the mathematics in order to identify the potential for engaging pupils with the key processes. Provide direction, if needed.

Following their previous reading, remind everyone that the key processes describe how pupils should engage with their learning, at all levels of mathematics. Allow time for individuals to read the document Key processes in algebra, having in mind particularly the task they have just been doing. It may be helpful to highlight particular sentences or sections which relate closely to the way pupils could be learning through this task.

Next draw everyone’s attention to the map Key processes adaptable template and allow a couple of minutes for pairs to peruse it. Perhaps refer back to the classification task and note differences and similarities. Emphasise that there is no perfect map or classification but that you and your colleagues need to have a shared vocabulary if you are to discuss the processes productively.

Explain that the adaptable map provides a graphic way of detailing general aspects of the processes. Mapping can provide a mechanism for you to get to know the processes by constructing your own version, based on a specific context. It is possible to start from the adaptable template however, your map will be considerably smaller and for this reason it is probably easier to begin from the blank template and create a few simple branches. Use a flip chart, board or interactive white board, putting the title ‘Hundred square’ in the centre of the map.

Hundred square

Reasoning

Procedures

Analysing

Interpreting and evaluating

Representing

Communicatingand re�ecting

Working together as a department, add some key processes that could be developed ●

through this task, drawing on ideas from the classifying task, from exploring the mathematics and from reading.

Begin to discuss and note the range of opportunities that could emerge for introducing ●

algebra through using this as an extended task across a number of lessons in Year 7.

The next task links very closely to NC2 and is best completed at the same time or very soon afterwards. You are beginning to consider the potential of this task as part of an early algebra unit in Year 7. Working with your department, the aim is to design one or two objectives relating to mathematical processes and application. The renewed Framework for secondary mathematics includes objectives adapted from the previous ‘using and applying’ objectives better to reflect the focus of the new programmes of study. Keeping the focus of the discussion on one specific example (the hundred square), the aim of task NC3 is to provide a ‘light touch‘ introduction to ways of working with these objectives. Process is crucially affected by the context and task so you will need to tailor the objectives with the learning in mind. A possible objective could be:



Pupils should learn to:

represent ‘hundred square’ patterns, using symbols and expressions, and work logically to produce and explain generalisations; compare different approaches and recognise where they are equivalent.

Task NC3 sets out the stages of thinking to help your colleagues design a similar objective based on your earlier work with the key processes. Don’t be tempted to short-cut this thinking or align too closely to the example above.

You will need, for pairs of colleagues:

Key processes map ‘hundred square’ (your agreed version from NC2) ●

Year 7 mathematical processes and applications objectives ●

Year 7 algebra objectives ●

Task NC3 Tailoring the mathematical process and application objectives

Remind colleagues about the last task. In particular mention the map of the cycle of key processes which you produced after exploring the algebra of the hundred square and the interrelated elements of the map which you noted as possible foci for an algebra unit plan in Year 7.

Introduce the table of ‘mathematical process and application’ (MPA) objectives and explain that these have been adapted from the ‘using and applying’ objectives to reflect the focus of the new programmes of study better. Say that you will work in more detail on these objectives at a later stage, for now you are simply ‘getting to know them’.

Ask pairs of colleagues to highlight phrases in the ● Year 7 MPA objectives which seem appropriate to the hundred square task. They may find that working from your Key process map (from NC2) helps them to do this.

Discuss the highlighted phrases, reach some consensus and design one or two composite ●

objectives that are simple enough to describe the learning opportunity presented by the hundred square problem.

It is important to see the MPA objectives as part of a collection of objectives in a unit. To complete this picture you need to consider which ‘range and content’ objectives you would select if you were to include the ‘hundred square’ as a major part of an algebra unit in Year 7.

Working as a group, consider the ● Year 7 algebra objectives and select a small collection which would be suitable, alongside the MPA objective, for a Year 7 algebra unit plan.

Summarise tasks NC2 and NC3 and emphasise that an extended and rich task of this kind can help to ensure that pupils learn through the key processes and understand the range and content in a more connected way.

The fourth task prepares you for a ‘stock-take’ on your approach to algebra in Year 7. For departments whose approach is well-aligned with the new curriculum, the main outcomes will be to introduce the language of key processes and to refine existing units. Departments that need to make a greater level of change might start by revising one or two units in Year 7, before setting out a manageable plan for the longer term. Think about how to draw in all members of the department. Enlarged or projected documents may help to do this. You will need:

copies of one or more algebra units from early in Year 7, including any resources and ●

textbooks that you use.

a unit planning template, either the one you use in school, or chosen from the three ●

provided in the folder on the CD-ROM.



Each colleague will also need a copy of:

Teaching and learning approaches ●

Task NC4 Revising a Year 7 algebra unit

Explain that the purpose of this task is to review an early algebra unit in Year 7. The Teaching and learning approaches document synthesises and interprets the aims, key concepts, key processes and curriculum opportunities in the new curriculum. It can help you reach a consensus on priorities for planning and teaching.

Ask everyone to read just the six sub-sections of ‘Some principles for effective learning’ (first main section of the document only). They should have in mind the context of teaching algebra to pupils in the first term of Year 7 and highlight text that they think is important when reviewing early algebra units.

Discuss individual suggestions and agree two or three priorities.

Now introduce the main task, which is to review and revise a Year 7 algebra unit. In order to address priorities in the new curriculum, your plan is likely to include a rich task developed over several lessons to:

engage pupils in particular aspects of the key processes ●

develop other aspects of effective learning. ●

Your unit might include a new task (e.g. ‘hundred square’) or an adaptation of an existing task. You might choose to:

Either adapt an existing unit,

Or drop notes into a unit planning template, if you want to start afresh.

Spend most of your time exploring how the task should be developed and incorporated in the unit, including:

how you might present the task to pupils who had not encountered algebraic representation ●

or used algebraic procedures before

ways in which pupils could develop or extend the problem and become more autonomous ●

in using the key processes.

Finally, identify what needs to be completed beyond this meeting in order for colleagues to prepare for teaching the unit. This will include new objectives adapted from the mathematical process and application strand and renewed objectives drawn from the algebra strand.

The final task in this sequence looks ahead to when you teach the unit. It will be essential to evaluate the unit, how you have adapted your teaching and the impact on pupils’ learning. It would be helpful if you could project the templates or work on enlarged paper versions of:

Teaching and learning review template: lessons/unit ●

Teaching and learning review template: pupils’ views ●



Task NC5 Preparing to review teaching and learning

Explain that the final task is to give further consideration as to how you will teach the revised Year 7 unit and to identify points to note, to help you review the impact of the changes made.

From the Teaching and learning approaches guidance, ask colleagues to read ‘Some principles for effective teaching’. Individually, highlight points in the text particularly relevant to teaching this unit (you are not trying to cover everything!). Then discuss and agree priorities for the department identifying important aspects to develop, without being over-ambitious at this stage.

Together, adapt the Teaching and learning review template: lessons/unit so that it is suitable as an observation or reflection sheet to help you review the unit later. On the basis of your agreed priorities, decide which sections of the template are relevant to copy and adapt as part of the agreed review prompt. Without being overambitious, your template should include:

the particular key processes with which you expect pupils to engage ●

other aspects of pupil learning you are seeking to develop ●

the particular teaching principles you are seeking to improve. ●

Having designed this review template you may wish to select matching prompts for gathering pupils’ views through small-group discussions based on the same priorities. To do this, use the adaptable template Teaching and learning review template: pupils’ views. Copy, paste and adapt the suggested questions for your chosen developments and agree when and how pupils’ views will be gathered.

Discuss how the adapted teaching and learning review sheet and the results of pupil discussions can be used in preparation for a departmental review meeting:

by all teachers as self-reflection on their lessons ●

for any lesson observations that may be possible ●

to inform discussions with small groups of pupils about their experiences in the lessons. ●

Agree when the unit will be taught and set a date for a review meeting.

Conclude by noting that starting on a small scale to establish the key processes in selected units of work will help all staff to move forward with the new curriculum and will inform the department’s long-term development plan.

Drafting a plan for the longer termIt would be appropriate to allow time to reflect on the outcomes of your launch meeting before setting out a longer-term plan. The timing of this next task will depend on such factors as whole- school planning for the new curriculum and whether you feel ready to set out a plan, or whether you would prefer to allow time for some trialling of one or two units of work, say in Year 7, before thinking ahead to the longer term.

For this task you will need:

A timeline for change ●



Task NC6 Drafting a development plan

This task is for the subject leader working with a colleague, such as the second in department or teacher with responsibility for Key Stage 3.

Reflect on and discuss:

the ‘big picture’ in the school and plans or points for consideration from the senior ●

leadership team as they seek to implement the new curriculum;

issues arising from the launch with your colleagues (and any subsequent classroom trialling), ●

related to implementing the new mathematics curriculum, particularly the incorporation of key processes.

Your discussion should help you to address questions about priorities and phasing. Use the chart A timeline for change to consider which year groups are most affected by the changes to the curriculum. In the light of this consider whether you will:

work on the planning and teaching for a particular year group or the whole key stage; ●

review the whole curriculum by working on critical units in all strands or review a larger number of ●

units in a selected strand.

When you have formulated your thoughts, it would be appropriate to discuss plans with a senior leader. This provides an opportunity to set out what you see as the challenges for your department and possible ways forward. Further work may be needed to arrive at an agreed plan that can lead to sustained phased development towards a scheme of work which fully reflects the new programmes of study.

Having launched the new programme of study with your department and considered some of the implications for your teaching, set out early thoughts on a development plan and discussed them with a school senior leader and the department, you should be ready to clarify your plans, identify priorities and start working on them.

In summer 2008 the renewed Framework for secondary mathematics will be available along with a Secondary mathematics planning toolkit, based on the structure of the Key Stage 4 planning toolkit which has been in use in many departments since summer 2007.

The toolkit will include a Key Stage 3 planning handbook which will offer guidance in the same format as this document to support the ongoing process of collaborative planning to implement the new curriculum at Key Stage 3. The handbook should enable you to step back and look more broadly at the issues of developing your scheme of work and help you to prioritise and firm up your plans for a phased implementation of the new programme of study. It will also help you to become familiar with the renewed Framework for secondary mathematics.



Hundred square grid

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100



Key processes in algebra

Algebra in Key Stages 3 and 4 is based on the generalisations of relationships familiar from basic number. It is developed to include the use of equations, formulae and identities, and sequences, functions and graphs. Algebra is purposeful when pupils encounter sufficiently complex situations where objects or relationships require representation in symbolic or graphical form. These occur frequently when describing generalisations underlying particular relationships. To use and make sense of algebra pupils need opportunities to relate it to their knowledge of the arithmetical operations. Suitable contexts for algebraic representation may come from within mathematics (for example, exploring number patterns and puzzles or finding areas of shapes), by linking with other subjects or from real-life applications. It should include use of ICT, such as graph-plotting and spreadsheet software to explore functions.

RepresentingRepresenting a situation places it into a mathematical form that enables it to be worked on. In algebra this might mean trying out and choosing between different diagrammatic, graphical and symbolic forms arising from looking at the problem or situation from different points of view. Aspects of representing within algebra include:

identifying assumptions, variables and relationships in order to create a mathematical model ●

developing understanding of algebraic conventions, for example, conventions of writing terms ●

and expressions, coordinate points and equations of lines, vectors and magnitude of vectors

constructing algebraic expressions, equations, formulae and identities, for example, ●

understanding and using signs such as =, < and > to represent relationships between variables

choosing appropriate algebraic representation of such relationships, using knowledge of ●

equivalence forms, for example, of tables, functions and graphs so that the context can be analysed and the solution communicated

choosing the tools most appropriate to represent the mathematics drawn from the ●

situation, for example, a graphical calculator or a spreadsheet.

As well as giving point to the subject, experience of algebraic representation is crucial if pupils are to understand and use precise algebraic language. Giving explicit attention to this helps them to understand the conventions for using letter symbols and constructing algebraic expressions. It can also give pupils insights into algebraic structure and order of operations, needed when transforming or interpreting symbolic and graphical representations.



Analysing – use mathematical reasoningAlgebra as a tool lies at the heart of much mathematical reasoning. Pupils need opportunities to experience the power of algebra in expressing generality. This includes:

identifying and describing numerical patterns and relationships, both symbolically and ●

graphically

making connections with arithmetical operations and with equivalent algebraic forms when ●

transforming expressions and equations

making connections between sequences, functions and graphs and exploring the effects of ●

varying values

making generalisations, explaining and proving, relating results to the context of the ●

problem.

Analysing – use appropriate mathematical proceduresUsing appropriate procedures involves manipulating expressions, equations and graphs, using and applying techniques and accurate notation and monitoring the accuracy of methods and solutions. Appropriate procedures in algebra include:

generating equivalent expressions and equations including a simplified form ●

factorising and expanding expressions and equations ●

solving equations exactly and approximately ●

manipulating formulae, including changing the subject of the formula ●

substituting values into equations and formulae, for example, evaluating a formula to convert ●

temperature in degrees C to degrees F.

Algebra at Key Stages 3 and 4 is generalised arithmetic. It requires understanding of the commutative, associative and distributive laws as they apply to the number operations, and of relationships between operations, including inverses. Pupils can be supported to generalise the rules with letters in place of numbers, for example, ab = c implies:

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

Taking an exploratory approach to transforming algebraic expressions and equations, where pupils are regularly asked to write expressions in different ways (‘find as many ways as you can’), builds their algebraic skills. They:

gain confidence in manipulating expressions into different equivalent forms ●

gain insights into which of a range of possible transformations will be both valid and ●

efficient as a next step, for example, in solving an equation or rearranging a formula

develop increasing fluency with algebraic manipulation without being rule-bound and, ●

when the steps in a procedure are not obvious, are able to resolve difficulties for themselves.



Interpreting and evaluatingAspects of interpreting and evaluating in algebra include:

relating numerical results, such as the solution of an equation, to the context under ●

consideration

interpreting general statements or conclusions expressed in algebraic form (e.g. an ●

expression or formula) and considering their significance

recognising the difference between numerical evidence and algebraic proof ●

interpreting graphs and graphical features such as points of intersection, gradients and the ●

general shape of a graph

evaluating different approaches, for example, where someone else has represented the ●

problem or approached its solution in a different way.

Communicating and reflectingAspects of communicating and reflecting in algebra include:

recognising and using the fact that algebraic language (symbolic and graphical) is a ●

powerful form of communication for expressing the steps in an argument or conclusions of an enquiry

considering alternative approaches, for example, comparing algebraic, graphical and ●

numerical approaches to tackling a problem

making links to related problems or to different problems with a similar structure. ●

Resources for algebra A range of resources to support the development of key processes in algebra are included in the ‘Ideas for rich tasks’ folder in the Secondary mathematics planning toolkit.

Interacting with mathematics in Key Stage 3 – algebra: ●

Constructing and solving linear equations – Year 7, Year 8 and Year 9 booklets

Teaching mental mathematics from level 5: ●

Algebra –

Measures and mensuration in algebra –

Standards Unit ● Improving learning in mathematics:

Mostly algebra – , sessions A1 – A14



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Re�e

ctin

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Com

mun

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ing



Case study: ‘Matchstick shapes’

Mapping the key processes: building on ‘using and applying mathematics’ in the current scheme of workThe context Department Y decided they would look at some of the rich tasks they have been using in Year 7 to cover ‘using and applying’ elements of the 2001 Framework to see how well they mapped to the new ‘key processes’. They wanted to get a sense of what they needed to do in order to begin to meet the demands of the new curriculum. They decided that each member of the Key Stage 3 team would take turns to bring a problem they liked, and that they thought had potential, to their regular meetings and look at how it could be adapted to address the key processes.

The storyM, one of the most experienced members of the department, started by sharing a problem she used with her Year 7 in the summer term every year. It was ’Matchstick shapes’ (p.32 of the Supplement of Examples).

M described briefly how she currently used and introduced the task. She had an introduction based on the image from page 32 of the Supplement of Examples, where she discussed how the class might tackle the problem. Together (normally with some gentle nudging from her) they would decide to start with one triangle and then build up the triangles one at a time and draw up a table on the board. They could then identify a pattern and from that generate a general rule. She knew that more was possible and she felt that the problem had lots of potential. She had recently been to a local network group meeting with teachers from other schools and one of them had described how they introduced the task:

Pupils arrive in the room and a line of triangles is already on the board.

Step 1

Step 2

Step 3

Step 1

Step 2

Step 3

. . . . .

Step 4

Pupils are asked to work in pairs – one of the pair draws the match design and the other watches how they do it. After a few minutes several pupils are asked to describe how their partner did it. This is modelled by the teacher who writes the associated arithmetic on the board (two examples below):

Step 1

Step 2

Step 3

Step 1

Step 2

Step 3

. . . . .

Step 4



Despite the different methods they all end up with the same answer. How about if there were 20 triangles? 45 triangles? 100 triangles? n triangles?

The pupils work on answers to each of these in turn – using their own methods and sharing them. The general case yields different algebraic expressions all of which can be simplified to show their equivalence (see below). Pupils then look back to make sense of each other’s expressions and see how they reflect the way they had drawn the image and that they are all equivalent.

11 triangles

(5+6)+(6+6) = 11 + 12 = 23

11 triangles

3+2+2+2+2+2+2+2+2+2+2= 3+10x2 = 23

20 triangles

(10+10)+(11+10) = 20 + 21= 41

20 triangles

3+19x2 = 41

n triangles

(n)+(n+1) = 2n+1

n triangles

3 + 2(n-1) =2n+1

M felt the power of this was in the lack of reliance on a table of results and a focus on the structure of the mathematics and how it really does reflect what it is representing.

There was lots of discussion about how the task could be presented and extended – either with more triangles or different situations (rows of squares) or other growing shapes.

The department sat down and produced the following mapping of the ’Matchstick shapes’ task against the processes. It was pretty obvious it covered an enormous range of opportunities but they decided that when they next used the tasks they would focus on three inter-related elements:

discussing methods ●

considering alternative solutions ●

engaging with others' methods. ●

Key Processes:-matchsticks

Reasoning

Procedures

Analysing


Representing

Communicatingand re�ecting

Justifying

Working logically

Making connections

Manipulating expressions

Collecting and analysing evidence

Recording

Trying out ideas

Creating representations

Selecting methods

Making connections between di�erent solutions

Discussing methods

Considering alternative solutions

Reasoning deductively

Di�erentiating between evidence and proof

Engaging with others’ methods

Making general statements

Considering alternative strategies



Year

7, 8

and

9 M

PA o

bjec

tive

s

Mat

hem

atic

al p

roce

sses

and

app

licat

ions

Solv

e pr

oble

ms,

ex

plor

e an

d in

vest

igat

e in

a

rang

e of

con

text

s

Incr

ease

the

chal

leng

e an

d bu

ild p

rogr

essi

on a

cros

s the

key

stag

e, a

nd fo

r gro

ups o

f pup

ils b

y:

incr

easi

ng th

e

●co

mpl

exit

y of

the

appl

icat

ion,

for e

xam

ple,

non

-rou

tine,

mul

ti-st

ep p

robl

ems,

ext

ende

d en

quiri

esre

duci

ng th

e

●fa

mili

arit

y of

the

cont

ext,

for e

xam

ple,

new

con

text

in m

athe

mat

ics,

con

text

dra

wn

from

oth

er su

bjec

ts, o

ther

asp

ects

of p

upils

’ liv

es

incr

easi

ng th

e

●te

chni

cal d

eman

d of

the

mat

hem

atic

s req

uire

d, fo

r exa

mpl

e, m

ore

adva

nced

con

cept

s, m

ore

diffi

cult

proc

edur

esin

crea

sing

the

degr

ee o

f

●in

depe

nden

ce a

nd a

uton

omy

in p

robl

em so

lvin

g an

d in

vest

igat

ion

Year

7Ye

ar 8

Year

9

Repr

esen

ting

Iden

tify

the

nece

ssar

y in

form

atio

n to

und

erst

and

or si

mpl

ify a

con

text

or p

robl

em; r

epre

sent

pr

oble

ms,

mak

ing

corr

ect u

se o

f sym

bols

, wor

ds,

diag

ram

s, ta

bles

and

gra

phs;

use

appr

opria

te

proc

edur

es a

nd to

ols,

incl

udin

g IC

T

Iden

tify

the

mat

hem

atic

al fe

atur

es o

f a c

onte

xt o

r pr

oble

m; t

ry o

ut a

nd c

ompa

re m

athe

mat

ical

re

pres

enta

tions

; sel

ect a

ppro

pria

te p

roce

dure

s an

d to

ols,

incl

udin

g IC

T

Brea

k do

wn

subs

tant

ial t

asks

to m

ake

them

mor

e m

anag

eabl

e; re

pres

ent p

robl

ems a

nd sy

nthe

sise

in

form

atio

n in

alg

ebra

ic, g

eom

etric

al o

r gr

aphi

cal f

orm

; mov

e fr

om o

ne fo

rm to

ano

ther

to

gai

n a

diffe

rent

per

spec

tive

on th

e pr

oble

m

Ana

lysi

ng –

use

m

athe

mat

ical

re

ason

ing

Clas

sify

and

vis

ualis

e pr

oper

ties a

nd p

atte

rns;

gene

ralis

e in

sim

ple

case

s by

wor

king

logi

cally

; dr

aw si

mpl

e co

nclu

sion

s and

exp

lain

reas

onin

g;

unde

rsta

nd th

e si

gnifi

canc

e of

a c

ount

er-

exam

ple;

take

acc

ount

of f

eedb

ack

and

lear

n fr

om m

ista

kes

Visu

alis

e an

d m

anip

ulat

e dy

nam

ic im

ages

; co

njec

ture

and

gen

eral

ise;

mov

e be

twee

n th

e ge

nera

l and

the

part

icul

ar to

test

the

logi

c of

an

argu

men

t; id

entif

y ex

cept

iona

l cas

es o

r cou

nter

-ex

ampl

es; m

ake

conn

ectio

ns w

ith re

late

d co

ntex

ts

Use

con

nect

ions

with

rela

ted

cont

exts

to

impr

ove

the

anal

ysis

of a

situ

atio

n or

pro

blem

; po

se q

uest

ions

and

mak

e co

nvin

cing

ar

gum

ents

to ju

stify

gen

eral

isat

ions

or

solu

tions

; rec

ogni

se th

e im

pact

of c

onst

rain

ts o

r as

sum

ptio

ns

Ana

lysi

ng –

use

ap

prop

riat

e m

athe

mat

ical

pr

oced

ures

With

in th

e ap

prop

riate

rang

e an

d co

nten

t: m

ake

accu

rate

mat

hem

atic

al d

iagr

ams,

gra

phs a

nd c

onst

ruct

ions

on

pape

r and

on

scre

en; c

alcu

late

acc

urat

ely,

sele

ctin

g m

enta

l met

hods

or c

alcu

latin

g de

vice

s as a

ppro

pria

te; m

anip

ulat

e nu

mbe

rs, a

lgeb

raic

exp

ress

ions

and

equ

atio

ns, a

nd a

pply

rout

ine

algo

rithm

s; us

e ac

cura

te n

otat

ion,

incl

udin

g co

rrec

t sy

ntax

whe

n us

ing

ICT;

reco

rd m

etho

ds, s

olut

ions

and

con

clus

ions

; est

imat

e, a

ppro

xim

ate

and

chec

k w

orki

ng

Inte

rpre

ting

and

ev

alua

ting

Inte

rpre

t inf

orm

atio

n fr

om a

mat

hem

atic

al

repr

esen

tatio

n or

con

text

; rel

ate

findi

ngs t

o th

e or

igin

al c

onte

xt; c

heck

the

accu

racy

of t

he

solu

tion;

exp

lain

and

just

ify m

etho

ds a

nd

conc

lusi

ons;

com

pare

and

eva

luat

e ap

proa

ches

Use

logi

cal a

rgum

ent t

o in

terp

ret t

he m

athe

mat

ics

in a

giv

en c

onte

xt o

r to

esta

blis

h th

e tr

uth

of a

st

atem

ent;

give

acc

urat

e so

lutio

ns a

ppro

pria

te to

th

e co

ntex

t or p

robl

em; e

valu

ate

the

effici

ency

of

alte

rnat

ive

stra

tegi

es a

nd a

ppro

ache

s

Just

ify th

e m

athe

mat

ical

feat

ures

dra

wn

from

a

cont

ext a

nd th

e ch

oice

of a

ppro

ach;

gen

erat

e fu

ller s

olut

ions

by

pres

entin

g a

conc

ise,

re

ason

ed a

rgum

ent u

sing

sym

bols

, dia

gram

s,

grap

hs a

nd re

late

d ex

plan

atio

ns

Com

mun

icat

ing

and

refle

ctin

gCo

mm

unic

ate

own

findi

ngs e

ffect

ivel

y, o

rally

and

in

writ

ing,

and

dis

cuss

and

com

pare

app

roac

hes

and

resu

lts w

ith o

ther

s; re

cogn

ise

equi

vale

nt

appr

oach

es

Refin

e ow

n fin

ding

s and

app

roac

hes o

n th

e ba

sis

of d

iscu

ssio

ns w

ith o

ther

s; re

cogn

ise

effici

ency

in

an a

ppro

ach;

rela

te th

e cu

rren

t pro

blem

and

st

ruct

ure

to p

revi

ous s

ituat

ions

Revi

ew a

nd re

fine

own

findi

ngs a

nd a

ppro

ache

s on

the

basi

s of d

iscu

ssio

ns w

ith o

ther

s; lo

ok fo

r an

d re

flect

on

othe

r app

roac

hes a

nd b

uild

on

prev

ious

exp

erie

nce

of si

mila

r situ

atio

ns a

nd

outc

omes



Year

7, 8

and

9 a

lgeb

ra o

bjec

tive

s

Alg

ebra

Year

7Ye

ar 8

Year

9

Equa

tion

s,

form

ulae

, ex

pres

sion

s an

d id

enti

ties

Use

lett

er s

ymbo

ls to

repr

esen

t unk

now

n nu

mbe

rs o

r var

iabl

es; k

now

the

mea

ning

s of t

he

wor

ds te

rm, e

xpre

ssio

n an

d eq

uatio

n

Und

erst

and

that

alg

ebra

ic o

pera

tions

follo

w th

e ru

les o

f arit

hmet

ic

Sim

plify

line

ar a

lgeb

raic

exp

ress

ions

by

colle

ctin

g lik

e te

rms;

mul

tiply

a si

ngle

term

ove

r a

brac

ket (

inte

ger c

oeffi

cien

ts)

Reco

gnis

e th

at le

tter

sym

bols

pla

y di

ffere

nt ro

les

in e

quat

ions

, for

mul

ae a

nd fu

nctio

ns; k

now

the

mea

ning

s of t

he w

ords

form

ula

and

func

tion

Und

erst

and

that

alg

ebra

ic o

pera

tions

, inc

ludi

ng

the

use

of b

rack

ets,

follo

w th

e ru

les o

f arit

hmet

ic;

use

inde

x no

tatio

n fo

r sm

all p

ositi

ve in

tege

r po

wer

s

Sim

plify

or t

rans

form

line

ar e

xpre

ssio

ns b

y co

llect

ing

like

term

s; m

ultip

ly a

sing

le te

rm o

ver a

br

acke

t

Dis

tingu

ish

the

diffe

rent

role

s pla

yed

by le

tter

sy

mbo

ls in

equ

atio

ns, i

dent

ities

, for

mul

ae a

nd

func

tions

Use

inde

x no

tatio

n fo

r int

eger

pow

ers a

nd si

mpl

e in

stan

ces o

f the

inde

x la

ws

Sim

plify

or t

rans

form

alg

ebra

ic e

xpre

ssio

ns b

y ta

king

ou

t sin

gle-

term

com

mon

fact

ors;

add

sim

ple

alge

brai

c fr

actio

ns

Cons

truc

t and

solv

e si

mpl

e lin

ear e

quat

ions

with

in

tege

r coe

ffici

ents

(unk

now

n on

one

side

onl

y)

usin

g an

app

ropr

iate

met

hod

(e.g

. inv

erse

op

erat

ions

)

Cons

truc

t and

solv

e lin

ear e

quat

ions

with

inte

ger

coeffi

cien

ts (u

nkno

wn

on e

ither

or b

oth

side

s,

with

out a

nd w

ith b

rack

ets)

usi

ng a

ppro

pria

te

met

hods

(e.g

. inv

erse

ope

ratio

ns, t

rans

form

ing

both

side

s in

sam

e w

ay)

Use

gra

phs a

nd se

t up

equa

tions

to so

lve

sim

ple

prob

lem

s inv

olvi

ng d

irect

pro

port

ion

Cons

truc

t and

solv

e lin

ear e

quat

ions

with

inte

ger

coeffi

cien

ts (w

ith a

nd w

ithou

t bra

cket

s, n

egat

ive

sign

s any

whe

re in

the

equa

tion,

pos

itive

or n

egat

ive

solu

tion)

Use

sys

tem

atic

tria

l and

impr

ovem

ent m

etho

ds a

nd

ICT

tool

s to

find

appr

oxim

ate

solu

tions

to e

quat

ions

su

ch a

s 2

20

xx

+=

Use

alg

ebra

ic m

etho

ds to

solv

e pr

oble

ms i

nvol

ving

di

rect

pro

port

ion;

rela

te a

lgeb

raic

solu

tions

to

grap

hs o

f the

equ

atio

ns; u

se IC

T as

app

ropr

iate

Expl

ore

way

s of c

onst

ruct

ing

mod

els o

f rea

l-life

sit

uatio

ns b

y dr

awin

g gr

aphs

and

cons

truc

ting

alge

brai

c equ

atio

ns a

nd in

equa

litie

s

Use

sim

ple

form

ulae

from

mat

hem

atic

s and

ot

her s

ubje

cts;

subs

titut

e po

sitiv

e in

tege

rs in

to

linea

r exp

ress

ions

and

form

ulae

and

, in

sim

ple

case

s, d

eriv

e a

form

ula

Use

form

ulae

from

mat

hem

atic

s and

oth

er

subj

ects

; sub

stitu

te in

tege

rs in

to si

mpl

e fo

rmul

ae, i

nclu

ding

exa

mpl

es th

at le

ad to

an

equa

tion

to so

lve;

subs

titut

e po

sitiv

e in

tege

rs

into

exp

ress

ions

invo

lvin

g sm

all p

ower

s, e

.g.

2

34

x+

or

32

x ; d

eriv

e si

mpl

e fo

rmul

ae

Use

form

ulae

from

mat

hem

atic

s and

oth

er su

bjec

ts;

subs

titut

e nu

mbe

rs in

to e

xpre

ssio

ns a

nd fo

rmul

ae;

deriv

e a

form

ula

and,

in si

mpl

e ca

ses,

cha

nge

its

subj

ect



Teaching and learning approaches

Guidance on teaching and learning approaches is presented in three sections:

’Some principles for effective learning’, based on the 2008 programme of study in mathematics ●

’Some principles for effective teaching’, based on research over many years into the teaching of ●

mathematics

’Further support to develop pedagogy and practice’, which references existing Strategy ●

guidance on lesson design, teaching repertoire, etc.

Some principles for effective learningThis section is informed by the curriculum aims of the 2008 programme of study. By synthesising and interpreting the aims, key processes, key concepts and curriculum opportunities the intention is to provide a supportive reference paper which the whole department can use to reflect on priorities for development in teaching and learning and so phase the implementation of the new programme of study.

Pupils learn about and learn through the key mathematical processesKey processes need to be experienced as components of a whole cycle and this can be reflected within a single lesson as well as through a unit of work. Investigative and problem-solving opportunities should be designed so that pupils cycle through the processes several times and also move backwards and forwards between the stages as ideas mature, modify and change. In this way the notion of a cycle provides a helpful structure but does not become restrictive.

Representing

InterpretingEvaluating

Use proceduresAnalysing

Use reasoning

CommunicatingReflectingCONTEXT MATHEMATICS



The diagram represents the dual nature of mathematics, both as a tool for solving problems in a wide range of contexts and as a discipline with a distinctive and rigorous structure. So pupils become successful learners by developing competence in applying mathematics effectively in a range of contexts, including those from within mathematics itself. There are two ways of thinking about pupils’ experience of the key mathematical processes that lie at the heart of the revised programme of study.

They need opportunities to learn about the mathematical processes and to reflect on how they are improving in these skills. This could include designing lessons or units where there is no new content and the focus is on improving the process skills.

They need opportunities to learn through using the mathematical processes. As pupils gain confidence in the skills of applying these processes they can use them to develop their understanding of topics within the range and content of the curriculum.

Pupils work collaboratively and engage in mathematical talkIt is through paired and group work that pupils gain confidence in their ability to communicate mathematics effectively. Choosing a rich task will usually provide pupils with the chance to explain and justify, question and disagree. Over time the level of dialogue in the classroom becomes more mathematically rich as pupils pose questions to each other and develop more convincing arguments orally. As this kind of dialogue becomes a regular part of their work on mathematics pupils are forced to think in this way, preparing questions for one another and rehearsing arguments. We could describe this as developing a habit of ‘self-talk’; that is they are naturally developing the thinking which will support more independent work.

Pupils work on sequences of tasks Within the planning and teaching of units of work there need to be sequences of lessons which do not move too quickly from one topic to another or from one task to another. Instead, pupils need to be provided with a sequence of learning which is planned to become more challenging within a phase of a unit. One way of doing this is to select a task or sequence of related tasks which develop over a number of lessons. This has the advantage of reducing the burden of producing and introducing different tasks in each lesson. It means that more of a lesson is dedicated to pupils actively doing mathematics rather than listening to instructions for new and different topics and tasks. Most importantly a sequence of tasks, involving the same mathematics in increasingly difficult or unfamiliar contexts, or increasingly demanding mathematics in similar contexts makes mathematical progression more explicit to the pupils. In this way pupils develop the competence to apply suitable mathematics accurately within the classroom and beyond.

Pupils select the mathematics to use Pupils can begin to see the power and purpose of their mathematical learning when they are given the opportunity to make decisions about the mathematical tools (including ICT) to help them to solve a problem or investigate from a given stimulus. As pupils use existing mathematical knowledge to investigate or create solutions to unfamiliar problems their confidence increases and they come to see that doing mathematics is an interesting and enjoyable activity. They are then more likely to apply mathematical skills in life effectively, in their wider studies and ultimately in employment. Unit plans adapted to meet the new curriculum should build in a variety of open and closed tasks, ensuring that the contexts for some task are real and others are abstract. In this way pupils come to appreciate mathematics for itself as well as understanding that it is used as a tool in a wide range of contexts.



Pupils tackle relevant contexts beyond the mathematics classroom In order for pupils to be functional in mathematics and motivated to take their learning further they need to hone their knowledge, skills and understanding by applying suitable mathematics accurately within the classroom and beyond. This means planning units where pupils are not learning new content but are working on problems that arise in other subjects and in contexts beyond the school, such as architecture or engineering. In many cases a solution will involve using mathematics as a model to interpret or represent situations. Applications involving modelling changes in society and the environment or managing risk (for example, insurance and investments) could be used to stimulate discussion about important issues such as financial capability or environmental dilemmas. The assumptions and simplifications involved in the process of modelling a real context should be made explicit so that pupils come to realise that mathematics itself is essentially abstract and that a model or representation has limitations to its scope.

Pupils are exposed to the historical and cultural roots of mathematics If they are given the chance to learn about problems from the past that led to the development of particular areas of mathematics, pupils can begin to appreciate that people of all cultures use mathematics to make sense of the world around them. They may be fascinated to find out that pure mathematical findings sometimes precede practical applications, and their curiosity may be aroused to think that mathematics continues to develop and evolve. This will engage and motivate pupils to become more aware of the nature of mathematics and of the mathematics around them.

Some principles for effective teachingResearch shows that the following principles underlie effective teaching. They are based on those included in Improving learning in mathematics: challenges and strategies, by Malcolm Swan University of Nottingham, in the ‘Standards Unit box’ (Improving learning in mathematics, The Standards Unit, DfES1599-2005DOC-EN). The list is provided to support evaluation of current teaching approaches and to stimulate departmental discussions about improving the effectiveness of current teaching.

Build on the knowledge pupils bring to a sequence of lessons Design activities which uncover prior learning and offer pupils opportunities to express their understanding. For example:

pose a problem to the whole class to stimulate paired discussion and to set the agenda for the ●

next few lessons

set up pairs or groups to draw and share a concept map or equivalent diagram showing the ●

interconnections of existing understanding. Revisit the ‘maps’ and add to them as the learning emerges throughout the unit.

The following Strategy resources in the Secondary mathematics planning toolkit may help with this approach (Rich tasks folder):

Leading in learning ● (KS3 and KS4 training materials and exemplification in mathematics)

Bridging plans: from KS3 to KS4 ●

Interacting with mathematics in Key Stage 3 – proportional reasoning ●

Y – ear 7 Fractions and ratio minipack and resources, especially the key lesson



Expose and discuss common misconceptionsPupils make mistakes for a variety of reasons. Some are due to lapses in concentration, hasty reasoning, memory overload or a failure to notice important features of a problem. Others, however, are symptoms of more profound mathematical difficulties. Where mistakes are the result of consistent, alternative interpretations of mathematical ideas we refer to them as misconceptions. These should not be dismissed as ‘wrong thinking’ as they may be necessary stages of conceptual development. Design activities so that misconceptions are systematically exposed by allowing time in the lessons for pupils to reflect and discuss these conflicts. For example:

ask pupils to complete a task, using more than one method, and then to resolve conflicting ●

answers

present statements to be classified and justified as always true, sometimes true or never true. ●

The following Strategy resources in the Secondary mathematics planning toolkit may help with this approach:

Misconceptions in mathematics ● (Pedagogy folder, Improving subject knowledge sub-folder))

Teaching mental mathematics from level 5 ●

Shape and space – (Rich tasks folder)

You might also find useful, as a separately available CD-ROM including video:

Mathematics: developing dialogue and reasoning ● (DfES 00243-2006CDO-EN)

Develop effective questioningAim to invite a range of responses to your questions by asking more open and probing questions which promote higher-level reflective thinking. Allow time for pupils to think before offering help or moving on to ask someone else and allow time for yourself so that you think about your response. For example:

establish a routine through which pupils share their answers in pairs before you take whole-class ●

feedback. This ‘pair/share’ not only builds confidence, it also increases the number of pupils who feel that their response has been heard.

be explicit about types of questions you use and encourage the pupils to use the same types of ●

question. A display of question stems can be helpful for you and the pupils:

What if…?

Why do you think…?

When would it not work…?

How do you know…?

The following Strategy resources in the Secondary mathematics planning toolkit may help with this approach (Pedagogy folder):

Standards Unit ● Improving learning in mathematics (Pedagogy folder)

Pedagogy and practice: Teaching and learning in secondary schools ●

Unit 7 questioning – (Pedagogy folder)



Also, as separately available CD-ROMs:

Assessing pupils’ progress in mathematics at Key Stage 3 ● , probing questions (DfES 00007-2007CDO-EN)

Mathematics: developing dialogue and reasoning ● (DfES 00243-2006CDO-EN)

Use cooperative small-group work Ensure that everyone is confident and benefits from participating in discussions by designing tasks which require collaboration in pairs or small groups and establish this as a regular feature of mathematics lessons. For example:

use classification activities with only one set of objects per pair or group so that joint decisions ●

have to be made

ask pupils to create a spider diagram of connections with one large sheet of paper for three ●

pupils.

The following Strategy resources in the Secondary mathematics planning toolkit may help with this approach:

Pedagogy and practice: Teaching and learning in secondary schools ●

Unit 10 group work – (Pedagogy folder)

Interacting with mathematics in Key Stage 3 – handling data ●

Y8 handling data minipack – (Rich tasks folder)

Emphasise methods rather than answers Focus on pupils developing their repertoire of appropriate methods rather than on obtaining correct answers to a long list of similar problems. This is likely to involve aiming to work on fewer problems than would appear in a typical textbook exercise. For example:

direct pupils to tackle the same problem, using more than one method, and work in pairs to ●

compare solutions and evaluate their efficiency

ask pupils to redesign a problem so that it is more challenging or simpler and give it to their ●

group to solve.

The following Strategy resources in the Secondary mathematics planning toolkit may help with this approach.

Interacting with mathematics in Key Stage 3 ● (Rich tasks folder)

Handling data – (Wise words and other tasks)

Proportional reasoning – (Year 8 multiplicative relationships, Year 9 proportional reasoning, Enhancing PR in Year 8 and Year 9)

Use rich collaborative tasksThink about how to design tasks that motivate a need to learn and encourage the pupils to be creative, curious and reflective. Pupils’ mathematical thinking will be improved if they have to make decisions and ask questions. The learning is made memorable when pupils enjoy the tasks and are surprised by outcomes. Richer tasks allow all learners to find something challenging and at an appropriate level to work on. Examples of accessible and extendable tasks can be developed from ‘routine’ tasks by changing the initial stimulus and the questions asked. For example:

help pupils to consolidate their understanding of algebraic factorisation, expansion and ●

simplification by working with jigsaw or domino cards showing matching expressions. Extend this to include their own design of such cards



ask two groups of pupils to debate opposing arguments which support or refute a hypothesis ●

where data is supplied in a spreadsheet.

The following resources in the Secondary mathematics planning toolkit may help with this approach (Rich tasks folder):


Standards Unit ● Improving learning in mathematics

Create connections between mathematical topics Design activities for existing units which make explicit connections within and across mathematical topics. For example:

matching tasks which require pupils to recognise different representations of the same ●

mathematical idea.

You may also plan to include more cross-strand units to develop stronger connections. For example:

a functional mathematics unit presenting a real context requiring exploration or investigation. In ●

such units pupils could work on a range of mathematical connections needed to reach a resolution.

The following resources in the Secondary mathematics planning toolkit may help with this approach (Rich tasks folder):

Standards Unit Improving learning in mathematics ●


Measures and mensuration booklets –

Interacting with mathematics in KS3 – proportional reasoning ●

Year 9 proportional reasoning –

Use technology in appropriate ways Present mathematical concepts in dynamic, visually exciting ways that engage and motivate learners. Introduce, explore and represent concepts, structures and processes in new and revealing ways. Often dynamic images will permit insights and understandings which are difficult to convey in other ways. For example:

Display an equation of the form ● y mx c= + on the same screen as the associated table and graph in order to explore the relationship between them

Explore a dynamic diagram showing how the angle formed between two straight lines changes ●

as the lines move. Extend to parallel lines and an intercept.

The following resources from the Secondary mathematics planning toolkit may help with this approach (Rich tasks folder):

Interacting with mathematics in Key Stage 3 – proportional reasoning ●

Year 7 fractions and ratio – , interactive teaching programmes

ICT in mathematics ● , ICT lesson plans



Further support to develop pedagogy and practice The 2001 Framework for teaching mathematics: Years 7, 8 and 9 established several principles which teachers found useful in guiding their planning. They included dimensions of a teaching repertoire such as modelling, questioning and explaining and aspects of lesson design such as structuring learning and starters and plenaries.

As further support for these developments the Pedagogy and Practice materials were published a few years later.

Pedagogy and Practice: Teaching and learning in secondary schools is often referred to as ‘the ped pack’. It is a suite of study guides created to support the professional development of teachers across all subjects in secondary schools. They provide guidance on the relationship between pedagogic approaches (teaching models), teaching strategies, techniques and methods of creating the conditions for learning in order to inform lesson design. The techniques suggested in each study guide are tried and tested and draw on both academic research and the experience of practising teachers.

Many teachers, who began to work on new teaching strategies from the initial, brief guidance in the Framework, moved on to more detailed developments through the ‘ped pack’ guidance. For example, the structuring learning booklet elaborates the original Framework guidance on structured lessons. It describes dividing lessons into a series of episodes, choosing from a range of strategies and techniques to motivate pupils and examines three pedagogic approaches – direct interactive, inductive and exploratory – to show how they can help pupils develop tools for learning, such as inductive thinking or enquiry skills.

The full list of booklets is given below, but you are unlikely to require the entire set at any one time. Instead, think about what support you need and consider downloading one or two booklets; most are only 24 pages and can be accessed at www.standards.dcsf.gov.uk/secondary/keystage3/all/respub/sec ppt10.



Designing lessonsUnit 1 Structuring learning

Unit 2 Teaching models

Unit 3 Lesson design for lower attainers

Unit 4 Lesson design for inclusion

Unit 5 Starters and plenaries

Teaching repertoireUnit 6 Modelling

Unit 7 Questioning

Unit 8 Explaining

Unit 9 Guided learning

Unit 10 Group work

Unit 11 Active engagement techniques

Creating effective learnersUnit 12 Assessment for learning

Unit 13 Developing reading

Unit 14 Developing writing

Unit 15 Using ICT to enhance learning

Unit 16 Developing effective learners

Creating conditions for learningUnit 17 Improving the climate for learning

Unit 18 Learning styles

Unit 19 Classroom management



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ther

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ns.

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hist

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t abo

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e w

ays t

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mat

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ontin

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velo

p.



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e as

pect

s of

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fect

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hing

Sum

mar

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plan

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ore

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each

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on th

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ring

to

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quen

ce o

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nits

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g ac

tiviti

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hich

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over

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r lea

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g an

d off

er p

upils

opp

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nitie

s to

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unde

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Expo

sing

and

dis

cuss

ing

com

mon

mis

conc

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nsLe

sson

s or u

nits

incl

udin

g ac

tiviti

es so

that

mis

conc

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re sy

stem

atic

ally

exp

osed

, allo

win

g fo

r tim

e in

the

less

ons

for p

upils

to re

flect

and

dis

cuss

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e co

nflic

ts.

Dev

elop

ing

effec

tive

ques

tioni

ngTe

ache

rs in

vitin

g a

rang

e of

resp

onse

s to

open

and

/or p

robi

ng q

uest

ions

, pro

mot

ing

high

er-le

vel r

eflec

tive

thin

king

, al

low

ing

time

for p

upils

to th

ink

befo

re o

fferin

g he

lp o

r mov

ing

on a

nd a

llow

ing

time

for t

hem

selv

es to

thin

k ab

out

thei

r res

pons

e.

Usi

ng c

oope

rativ

e sm

all-

grou

p w

ork

Less

ons o

r uni

ts in

volv

ing

task

s whi

ch re

quire

col

labo

ratio

n in

pai

rs o

r sm

all g

roup

s, p

upils

show

ing

confi

denc

e an

d be

nefit

ing

from

par

ticip

atin

g in

dis

cuss

ions

.

Emph

asis

ing

met

hods

ra

ther

than

ans

wer

sPu

pils

wor

king

on

thei

r rep

erto

ire o

f app

ropr

iate

met

hods

rath

er th

an o

n ob

tain

ing

corr

ect a

nsw

ers t

o a

long

list

of

sim

ilar p

robl

ems.

Thi

s is l

ikel

y to

invo

lve

wor

king

in d

epth

on

few

er p

robl

ems i

n ea

ch le

sson

, aim

ing

to d

evel

op m

ore

pow

erfu

l and

gen

eral

isab

le m

etho

ds.

Usi

ng ri

ch c

olla

bora

tive

task

sPu

pils

wor

king

on

enjo

yabl

e an

d ex

tend

able

task

s whi

ch e

ncou

rage

mat

hem

atic

al ta

lk a

nd th

inki

ng. T

here

is

som

ethi

ng a

ppro

pria

tely

cha

lleng

ing

for a

ll pu

pils

, inv

olvi

ng th

em in

mor

e co

mpl

ex ta

sks w

hich

mot

ivat

e a

need

to

lear

n.

Crea

ting

conn

ectio

ns

betw

een

mat

hem

atic

al

topi

cs

Pupi

ls w

orki

ng o

n ta

sks w

hich

exp

licitl

y us

e kn

owle

dge

and

conn

ect c

lose

ly re

late

d co

ncep

ts a

nd n

otat

ions

. The

se

idea

s may

be

abst

ract

and

exp

lore

d w

ithin

mat

hem

atic

s or b

roug

ht to

geth

er in

app

licat

ion,

solv

ing

a pr

oble

m in

a re

al

cont

ext.

Usi

ng te

chno

logy

in

appr

opria

te w

ays

Mat

hem

atic

al c

once

pts p

rese

nted

in d

ynam

ic, v

isua

lly e

xciti

ng w

ays,

intr

oduc

ing,

exp

lorin

g an

d re

pres

entin

g co

ncep

ts,

stru

ctur

es a

nd p

roce

sses

in n

ew a

nd re

veal

ing

way

s. U

sing

dyn

amic

imag

es to

gai

n in

sigh

ts a

nd u

nder

stan

ding

s.



Tea

chin

g a

nd le

arn

ing

revi

ew te

mp

late

: pup

ils' v

iew

s

This

is a

n ad

apta

ble

tem

plat

e. It

shou

ld b

e us

ed to

supp

ort y

our i

mpl

emen

tatio

n of

the

new

pro

gram

me

of st

udy

and

help

you

wor

k to

geth

er a

s a

depa

rtm

ent o

n yo

ur c

hose

n pr

iorit

ies.

It is

inte

nded

to b

e us

ed to

pro

mpt

focu

sed

disc

ussi

ons w

ith sm

all g

roup

s of p

upils

, per

haps

five

or s

ix. T

he d

iscu

ssio

ns sh

ould

be

used

as a

maj

or so

urce

of

evid

ence

that

agr

eed

deve

lopm

ents

in te

achi

ng a

nd le

arni

ng a

re h

avin

g an

impa

ct o

n th

e pu

pils

.

The

first

step

is to

com

plet

e th

e te

mpl

ate

by u

sing

the

thre

e 'co

py a

nd p

aste

' she

ets t

o dr

op (a

nd a

dapt

) que

stio

ns w

hich

rela

te to

the

prio

ritie

s for

revi

ew.

To k

eep

the

disc

ussi

on a

cces

sibl

e, g

ener

al q

uest

ions

hav

e be

en su

gges

ted

for t

he 'k

ey p

roce

ss' s

ectio

n. Y

ou m

ay w

ish

to e

labo

rate

on

thes

e qu

estio

ns w

ith

part

icul

ar e

xam

ples

of a

spec

ts o

f key

pro

cess

es w

hich

hav

e be

en a

rece

nt fo

cus f

or d

evel

opm

ent.

Ther

e ar

e on

ly tw

o ce

lls b

esid

e ea

ch p

riorit

y to

enc

oura

ge y

ou to

focu

s clo

sely

. You

may

wis

h to

adj

ust t

his n

umbe

r bea

ring

in m

ind

that

refle

ctin

g on

a

smal

ler n

umbe

r of d

evel

opm

ent p

oint

s can

hav

e a

mor

e si

gnifi

cant

impa

ct.



Prom

pt s

heet

for d

iscu

ssio

n w

ith

smal

l gro

ups

of p

upils

In th

is d

iscu

ssio

n w

e ar

e tr

ying

to fi

nd o

ut

……

abou

t the

dev

elop

men

t of t

hese

cho

sen

prio

ritie

sN

otes

– re

spon

ses

…pu

pils

' aw

aren

ess o

f the

key

pro

cess

esIn

you

r rec

ent l

esso

ns o

n (in

sert

topi

c an

d ta

sk) y

ou

wer

e w

orki

ng o

n (in

sert

spec

ific

key

proc

ess)

– c

an y

ou

desc

ribe

this?

Use

exa

mpl

es o

f wha

t you

did

in th

e le

sson

s.

Tell

me

abou

t whe

re y

ou h

ave

wor

ked

on (i

nser

t sp

ecifi

c ke

y pr

oces

s) b

efor

e?

Did

the

less

ons a

nd ta

sks h

elp

you

get b

ette

r at (

topi

c)?

Did

you

get

bet

ter a

t (in

sert

spec

ific

key

proc

ess)?

…ho

w o

ften

pup

ils e

xper

ienc

e th

ese

aspe

cts

of le

arni

ng

…an

d if

it is

hel

ping

them

to m

ake

prog

ress

…if

pupi

ls a

re a

war

e of

an

incr

ease

d fo

cus o

n th

ese

aspe

cts o

f tea

chin

g

…an

d if

it is

hel

ping

them

to m

ake

prog

ress



Som

e as

pect

s of

eff

ecti

ve

lear

ning

Sum

mar

y ex

plan

atio

n

– fo

r mor

e de

tail

see

'Teac

hing

and

lear

ning

app

roac

hes'

Ada

pt th

ese

ques

tion

s/pr

ompt

s to

suit

the

stag

e of

impl

emen

tati

on

and

the

grou

ps y

ou a

re c

onve

ning

Pupi

ls le

arni

ng

abou

t the

key

pr

oces

ses

Lear

ning

abo

ut th

e m

athe

mat

ical

pro

cess

es a

nd re

flect

ing

on

how

they

are

impr

ovin

g in

thes

e sk

ills.

Eng

agin

g in

less

ons o

r uni

ts

with

no

new

con

tent

and

focu

sing

on

impr

ovin

g th

e pr

oces

s ski

lls.

Tell

me

abou

t how

you

r rec

ent m

athe

mat

ics l

esso

ns h

ave

help

ed y

ou to

get

bet

ter a

t (in

sert

sp

ecifi

c ke

y pr

oces

s).

Whe

n an

d w

here

do

you

thin

k you

will

use

this

skill

? In

mat

hem

atic

s, in

oth

er su

bjec

ts, in

dai

ly lif

e?

Pupi

ls le

arni

ng

thro

ugh

the

key

proc

esse

s

Lear

ning

thro

ugh

usin

g th

e m

athe

mat

ical

pro

cess

es. G

aini

ng

confi

denc

e in

the

skill

s of a

pply

ing

thes

e pr

oces

ses a

nd u

sing

th

em to

dev

elop

thei

r und

erst

andi

ng o

f top

ics w

ithin

the

rang

e an

d co

nten

t of t

he c

urric

ulum

.

We

have

rece

ntly

bee

n w

orki

ng o

n (in

sert

spec

ific

key

proc

ess)

, tel

l me

abou

t how

this

has

he

lped

you

to m

ake

prog

ress

in (i

nser

t top

ic).

Pupi

ls w

orki

ng

colla

bora

tivel

y an

d en

gagi

ng in

m

athe

mat

ical

talk

Wor

king

in p

airs

or g

roup

s on

a ric

h ta

sk, e

xpla

inin

g an

d ju

stify

ing,

qu

estio

ning

and

dis

agre

eing

, dev

elop

ing

the

thin

king

whi

ch w

ill

supp

ort m

ore

inde

pend

ent w

ork.

Tell

me

abou

t som

e m

athe

mat

ical

task

s whi

ch m

eant

you

nee

ded

to w

ork

toge

ther

in p

airs

or

gro

ups.

W

hat w

as it

abo

ut th

e ta

sk w

hich

mad

e yo

u w

ork

in a

gro

up o

r pai

r?

Whe

n yo

u w

ork

as p

art o

f a g

roup

or i

n a

pair:

How

do

you

wor

k on

the

task

toge

ther

, e.g

. do

you

each

do

diffe

rent

par

ts?

●

Wha

t typ

es o

f dec

isio

ns d

o yo

u m

ake?

●

Are

you

abl

e to

lear

n fr

om o

ther

s in

the

grou

p. H

ow? G

ive

an e

xam

ple.

● D

oes t

he c

hanc

e to

talk

and

thin

k to

geth

er h

elp

you

to le

arn?

Giv

e ex

ampl

es.

Pupi

ls w

orki

ng o

n se

quen

ces o

f tas

ksW

orki

ng o

n a

sequ

ence

of l

earn

ing

thro

ugh

an e

xten

ded

task

or

sequ

ence

of c

lose

ly-r

elat

ed ta

sks o

ver a

few

less

ons,

eith

er

invo

lvin

g th

e sa

me

mat

hem

atic

s in

incr

easi

ngly

diffi

cult

or

unfa

mili

ar c

onte

xts,

or i

ncre

asin

gly

dem

andi

ng m

athe

mat

ics i

n si

mila

r con

text

s.

Tell

me

abou

t a se

t of l

esso

ns w

hich

wer

e lin

ked

in so

me

way

, e.g

. the

y w

ere

on th

e sa

me

topi

c or

you

wer

e w

orki

ng o

n a

task

whi

ch e

xten

ded

acro

ss a

few

less

ons.

How

did

eac

h le

sson

pro

gres

s fro

m th

e on

e be

fore

?

●

How

did

the

sequ

ence

of l

esso

ns e

nd?

●

Do

you

know

whe

n yo

u m

ight

use

wha

t you

hav

e le

arne

d th

roug

h th

e se

quen

ce o

f les

sons

?

●

Pupi

ls se

lect

ing

the

mat

hem

atic

s to

use

Mak

ing

deci

sion

s abo

ut th

e m

athe

mat

ical

tool

s (in

clud

ing

ICT)

, us

ing

exis

ting

mat

hem

atic

al k

now

ledg

e to

inve

stig

ate

or c

reat

e so

lutio

ns to

unf

amili

ar p

robl

ems,

enj

oyin

g m

athe

mat

ics f

rom

real

an

d ab

stra

ct c

onte

xts.

Tell

me

abou

t a ta

sk w

here

you

had

to m

ake

deci

sion

s abo

ut h

ow to

star

t it.

Tell

me

abou

t an

occa

sion

whe

n yo

u ha

d to

cho

ose

the

met

hods

or t

ools

to u

se. G

ive

an

exam

ple

of c

hoos

ing

to u

se IC

T fo

r a m

athe

mat

ical

task

. Te

ll m

e ab

out a

tim

e w

hen

you

wer

e se

t a c

halle

nge

that

mea

nt y

ou h

ad to

pul

l tog

ethe

r kn

owle

dge

from

diff

eren

t par

ts o

f mat

hem

atic

s.

Pupi

ls ta

cklin

g re

leva

nt c

onte

xts

from

bey

ond

the

mat

hem

atic

s cl

assr

oom

App

lyin

g su

itabl

e m

athe

mat

ics a

ccur

atel

y w

ithin

the

clas

sroo

m

and

beyo

nd, w

orki

ng o

n pr

oble

ms t

hat a

rise

in o

ther

subj

ects

and

in

con

text

s bey

ond

the

scho

ol, u

sing

mat

hem

atic

s as a

mod

el to

in

terp

ret o

r rep

rese

nt si

tuat

ions

.

Tell

me

abou

t tas

ks/a

ctiv

ities

that

you

hav

e do

ne re

cent

ly w

here

you

use

d m

athe

mat

ics i

n ot

her s

ubje

cts o

r at h

ome,

e.g

. in

club

s, in

you

r hob

by, a

s par

t of a

job.

H

ave

you

used

you

r und

erst

andi

ng o

f mat

hem

atic

s to

expl

ain

som

ethi

ng to

any

one

rece

ntly

? G

ive

me

an e

xam

ple

of w

hen

you

unde

rsto

od so

met

hing

out

side

you

r mat

hem

atic

s les

son

beca

use

of y

our l

earn

ing

in m

athe

mat

ics.

Pupi

ls e

ngag

ing

with

the

hist

oric

al

and

cultu

ral r

oots

of

mat

hem

atic

s

Find

ing

out a

bout

the

mat

hem

atic

s of o

ther

cul

ture

s, le

arni

ng

abou

t pro

blem

s fro

m th

e pa

st, fi

ndin

g ou

t abo

ut th

e w

ays t

hat

mat

hem

atic

s con

tinue

s to

deve

lop.

Do

you

know

whe

re a

ny o

f the

mat

hem

atic

s you

hav

e le

arnt

cam

e fro

m o

r why

it w

as d

evel

oped

? Ca

n yo

u na

me

any

fam

ous m

athe

mat

icia

ns fr

om th

e pr

esen

t or t

he p

ast?

Why

are

they

fam

ous?

W

hat d

o yo

u kn

ow a

bout

mat

hem

atic

s fro

m d

iffer

ent c

ultu

res?

– W

ho? W

hich

? Wha

t?

Do

you

thin

k m

athe

mat

ics i

s stil

l dev

elop

ing?

How

? Why

?



Som

e as

pect

s of

eff

ecti

ve

teac

hing

Sum

mar

y ex

plan

atio

n

– fo

r mor

e de

tail

see

'Tea

chin

g an

d le

arni

ng a

ppro

ache

s'

Ada

pt th

ese

ques

tion

s to

suit

the

stag

e of

impl

emen

tati

on a

nd th

e gr

oups

you

are

con

veni

ng

Build

ing

on th

e kn

owle

dge

pupi

ls

brin

g to

a se

quen

ce

of le

sson

s

Less

ons a

nd u

nits

incl

udin

g ac

tiviti

es w

hich

unc

over

prio

r lea

rnin

g an

d off

er p

upils

opp

ortu

nitie

s to

expr

ess t

heir

unde

rsta

ndin

g.Ar

e yo

ur m

athe

mat

ics l

esso

ns li

nked

to so

met

hing

you

've

done

bef

ore?

– D

escr

ibe

an e

xam

ple.

W

hen

and

how

do

you

get t

he o

ppor

tuni

ty to

show

wha

t you

kno

w a

bout

a to

pic?

H

ow c

ould

you

show

som

eone

wha

t you

alre

ady

know

abo

ut th

e m

athe

mat

ics y

ou a

re

lear

ning

? Doe

s thi

s hap

pen

in y

our l

esso

ns?

Expo

sing

and

di

scus

sing

co

mm

on

mis

conc

eptio

ns

Less

ons o

r uni

ts in

clud

ing

activ

ities

so th

at m

isco

ncep

tions

are

sy

stem

atic

ally

exp

osed

, allo

win

g tim

e in

the

less

ons f

or p

upils

to

refle

ct a

nd d

iscu

ss th

ese

confl

icts

.

Wha

t do

you

do w

hen

you

notic

e th

at y

ou, o

r the

per

son

you

are

wor

king

with

, has

m

isun

ders

tood

the

topi

c/pr

oble

m o

r has

the

wro

ng id

ea?

Do

you

get o

ppor

tuni

ties t

o ex

plai

n w

hy a

pie

ce o

f mat

hem

atic

s is w

rong

in le

sson

s? W

hy

is th

is im

port

ant?

Te

ll m

e ab

out a

tim

e w

hen

you

thou

ght y

ou u

nder

stoo

d so

met

hing

but

then

you

wor

ked

on a

task

or h

ad a

dis

cuss

ion

that

exp

osed

a g

ap o

r flaw

in y

our u

nder

stan

ding

.

Dev

elop

ing

effec

tive

ques

tioni

ng

Teac

hers

invi

ting

a ra

nge

of re

spon

ses t

o op

en a

nd p

robi

ng

ques

tions

, pro

mot

ing

high

er-le

vel r

eflec

tive

thin

king

, allo

win

g tim

e fo

r pup

ils to

thin

k be

fore

offe

ring

help

or m

ovin

g on

and

al

low

ing

time

for t

hem

selv

es to

thin

k ab

out t

heir

resp

onse

.

Tell

me

abou

t an

inte

rest

ing

ques

tion

you

have

bee

n as

ked

in a

mat

hem

atic

s les

son.

Te

ll m

e ab

out t

he k

inds

of q

uest

ion

that

mak

e yo

u th

ink

a lo

t. Ca

n yo

u re

mem

ber a

n ex

ampl

e?

Whe

n a

ques

tion

is d

ifficu

lt, w

hat d

o yo

u do

? W

hat h

appe

ns in

the

clas

s whe

n so

meo

ne is

find

ing

a qu

estio

n ha

rd?

How

long

do

you

get t

o th

ink

abou

t you

r ans

wer

s? D

oes t

his v

ary?

D

oes j

ust o

ne p

erso

n gi

ve a

n an

swer

to a

que

stio

n or

are

a ra

nge

of a

nsw

ers t

aken

?

Usi

ng c

oope

rativ

e sm

all-g

roup

wor

kLe

sson

s or u

nits

invo

lvin

g ta

sks w

hich

requ

ire c

olla

bora

tion

in

pairs

or s

mal

l gro

ups,

pup

ils sh

owin

g co

nfide

nce

and

bene

fitin

g fr

om p

artic

ipat

ing

in d

iscu

ssio

ns.

Tell m

e ab

out s

ome

mat

hem

atic

al ta

sks w

hich

mea

nt yo

u ne

eded

to w

ork t

oget

her i

n pa

irs o

r gro

ups?

W

hat w

as it

abo

ut th

e ta

sk w

hich

mad

e yo

u w

ork

in a

gro

up o

r pai

r?

Whe

n yo

u w

ork

as p

art o

f a g

roup

or i

n a

pair:

How

do

you

wor

k on

the

task

toge

ther

, e.g

. do

you

each

do

diffe

rent

par

ts?

●

Wha

t typ

es o

f dec

isio

ns d

o yo

u m

ake?

●

Are

you

abl

e to

lear

n fr

om o

ther

s in

the

grou

p –

how

/giv

e an

exa

mpl

e?

● Doe

s the

cha

nce

to ta

lk a

nd th

ink

toge

ther

hel

p yo

u to

lear

n? G

ive

exam

ples

Emph

asis

ing

met

hods

rath

er

than

ans

wer

s

Pupi

ls w

orki

ng o

n th

eir r

eper

toire

of a

ppro

pria

te m

etho

ds ra

ther

th

an o

n ob

tain

ing

corr

ect a

nsw

ers t

o a

long

list

of s

imila

r pr

oble

ms.

Thi

s is l

ikel

y to

invo

lve

wor

king

in d

epth

on

few

er

prob

lem

s in

each

less

on, a

imin

g to

dev

elop

mor

e po

wer

ful a

nd

gene

ralis

able

met

hods

.

Do

you

get o

ppor

tuni

ties t

o he

ar h

ow o

ther

s in

your

cla

ss w

orke

d ou

t a p

robl

em o

r re

ache

d a

solu

tion?

H

ow d

oes t

his h

elp

you

if yo

u've

alre

ady

got t

he p

robl

em ri

ght?

If

you

get a

n an

swer

to a

mat

hem

atic

s que

stio

n w

rong

, do

you

get t

he c

hanc

e to

show

yo

ur te

ache

r or p

artn

er h

ow y

ou w

orke

d it

out?

W

hat i

s mor

e im

port

ant,

the

met

hod

or th

e an

swer

?



Som

e as

pect

s of

eff

ecti

ve

teac

hing

Sum

mar

y ex

plan

atio

n

– fo

r mor

e de

tail

see

'Tea

chin

g an

d le

arni

ng a

ppro

ache

s'

Ada

pt th

ese

ques

tion

s to

suit

the

stag

e of

impl

emen

tati

on a

nd th

e gr

oups

you

are

con

veni

ng

Usi

ng ri

ch

colla

bora

tive

task

sPu

pils

wor

king

on

enjo

yabl

e an

d ex

tend

able

task

s whi

ch

enco

urag

e m

athe

mat

ical

talk

and

thin

king

. The

re is

som

ethi

ng

appr

opria

tely

cha

lleng

ing

for a

ll pu

pils

invo

lvin

g th

em in

mor

e co

mpl

ex ta

sks w

hich

mot

ivat

e a

need

to le

arn.

Tell

me

abou

t a ta

sk w

hich

you

wor

ked

on th

at w

as re

ally

inte

rest

ing.

Why

did

you

enj

oy w

orki

ng o

n it?

●

Wha

t was

it a

bout

the

task

that

was

inte

rest

ing?

●

Did

this

task

mak

e yo

u w

ant t

o le

arn?

● Do

you

find

it he

lps t

o ta

lk in

a g

roup

whe

n yo

u ar

e fa

ced

with

a re

ally

har

d m

athe

mat

ics

prob

lem

? Doe

s it h

elp

you

lear

n m

ore?

Crea

ting

conn

ectio

ns

betw

een

mat

hem

atic

al

topi

cs

Pupi

ls w

orki

ng o

n ta

sks w

hich

exp

licitl

y us

e kn

owle

dge

and

conn

ect c

lose

ly-r

elat

ed c

once

pts a

nd n

otat

ions

. The

se id

eas m

ay

be a

bstr

act a

nd e

xplo

red

with

in m

athe

mat

ics o

r bro

ught

toge

ther

in

app

licat

ion,

solv

ing

a pr

oble

m in

a re

al c

onte

xt.

Giv

e m

e an

exa

mpl

e of

how

diff

eren

t top

ics i

n m

athe

mat

ics l

ink

toge

ther

. Te

ll m

e ab

out a

pro

blem

or i

nves

tigat

ion

whe

re y

ou h

ad to

con

nect

you

r kno

wle

dge

of

diffe

rent

par

ts o

f mat

hem

atic

s in

orde

r to

reac

h a

solu

tion.

Usi

ng te

chno

logy

in

app

ropr

iate

w

ays

Mat

hem

atic

al c

once

pts p

rese

nted

in d

ynam

ic, v

isua

lly e

xciti

ng

way

s, in

trod

ucin

g, e

xplo

ring

and

repr

esen

ting

conc

epts

, st

ruct

ures

and

pro

cess

es in

new

and

reve

alin

g w

ays.

Usi

ng

dyna

mic

imag

es to

gai

n in

sigh

ts a

nd u

nder

stan

ding

s.

Tell

me

abou

t the

last

tim

e yo

u us

ed IC

T in

a m

athe

mat

ics l

esso

n. W

as it

you

r cho

ice?

G

ive

an e

xam

ple

whe

re u

sing

ICT

in m

athe

mat

ics r

eally

hel

ped

you

unde

rsta

nd th

e to

pic

or re

ach

a so

lutio

n. C

an y

ou e

xpla

in w

hy?



Key processes classifying task

Representing

Analysing – use mathematical reasoning

Analysing – use appropriate mathematical procedures


Communicating and reflecting

Process statements

identify the mathematical aspects of a situation or problem

manipulate numbers, algebraic expressions and equations and apply routine algorithms

choose between representations use accurate notation, including correct syntax when using ICT

simplify the situation or problem in order to represent it mathematically, using appropriate variables, symbols, diagrams and models

record methods, solutions and conclusions

select mathematical information, methods and tools to use

estimate, approximate and check working



make connections within mathematics form convincing arguments based on findings and make general statements

use knowledge of related problems consider the assumptions made and the appropriateness and accuracy of results and conclusions

visualise and work with dynamic images

be aware of the strength of empirical evidence and appreciate the difference between evidence and proof

identify and classify patterns look at data to find patterns and exceptions

make and begin to justify conjectures and generalisations, considering special cases and counter-examples

relate findings to the original context, identifying whether they support or refute conjectures

explore the effects of varying values and look for invariance and covariance

engage with someone else’s mathematical reasoning in the context of a problem or particular situation

take account of feedback and learn from mistakes

consider the effectiveness of alternative strategies

work logically towards results and solutions, recognising the impact of constraints and assumptions

communicate findings effectively

appreciate that there are a number of different techniques that can be used to analyse a situation

engage in mathematical discussion of results



reason inductively and deduce consider the elegance and efficiency of alternative solutions

make accurate mathematical diagrams, graphs and constructions on paper and on screen

look for equivalence in relation to both the different approaches to the problem and different problems with similar structures

calculate accurately, selecting mental methods or calculating devices as appropriate

make connections between the current situation and outcomes, and situations and outcomes they have already encountered



‘Hundred square’ prompt sheet

This is a rich task which could be used as an early experience of algebra so that pupils can see how it helps them to generalise and explain. Think about a Year 7 class, perhaps starting along the following lines.

Tell me what you know about patterns of numbers in the 100 square…

24

33 34 35

44

The cross-shaped ‘window’ can be moved to different positions on the grid, to reveal different sets of five numbers. Place the window anywhere you like.

Add the left- and right-hand numbers in the window and compare with the middle number. Do the same for the top and bottom numbers. What do you notice?

Try different positions of the window. What do you notice?

Will it always work? Can you explain why? (Pupils are encouraged to give verbal explanations.)

Let’s see if we can use algebra to make the explanation simpler and clearer…

You told me that lots of numbers could be in the middle position. We could give the middle number a symbol, let’s call it n.

If we have chosen this symbol what can we say about the left-hand number? … and the right-hand number? How can we write this in symbols? (Pupils may use words and then try symbols.)

So the sum can be written as…

(n – 1) + (n + 1) = n + n −1 + 1 = n + n = n × 2 = 2n

(Explain the notation as necessary.)

Can you write down an expression for the sum of the top and bottom numbers?

(n – 10) + (n + 10) = 2n

Can this help us to explain why the two totals are the same for any position of the window?

…

By extending the task, pupils could work more independently to try other ideas such as starting by giving a symbol to the top number or exploring sums and differences of various numbers in the window. The shape of the window could also be changed.



Line

s of

pro

gre

ssio

n in

ma

the

ma

tics

Ove

rvie

w o

f str

ands

Stra

nds

Sub-

stra

nds

1 M

athe

mat

ical

pro

cess

es a

nd a

pplic

atio

ns

1.1

Repr

esen

ting

1.2

Ana

lysi

ng –

use

reas

onin

g

1.3

Ana

lysi

ng –

use

pro

cedu

res

1.4

Inte

rpre

ting

and

eval

uatin

g

1.5

Com

mun

icat

ing

and

refle

ctin

g

2 N

umbe

r

2.1

Plac

e va

lue,

ord

erin

g an

d ro

undi

ng

2.2

Inte

gers

, pow

ers a

nd ro

ots

2.3

Frac

tions

, dec

imal

s, p

erce

ntag

es, r

atio

and

pro

port

ion

2.4

Num

ber o

pera

tions

2.5

Men

tal c

alcu

latio

n m

etho

ds

2.6

Writ

ten

calc

ulat

ion

met

hods

2.7

Calc

ulat

or m

etho

ds

2.8

Chec

king

resu

lts



3 A

lgeb

ra

3.1

Equa

tions

, for

mul

ae, e

xpre

ssio

ns a

nd id

entit

ies

3.2

Sequ

ence

s, fu

nctio

ns a

nd g

raph

s

4 G

eom

etry

and

mea

sure

s

4.1

Geo

met

rical

reas

onin

g

4.2

Tran

sfor

mat

ions

and

coo

rdin

ates

4.3

Cons

truc

tion

and

loci

4.4

Mea

sure

s and

men

sura

tion

5 St

atis

tics

5.1

Spec

ifyin

g a

prob

lem

, pla

nnin

g an

d co

llect

ing

data

5.2

Proc

essi

ng a

nd re

pres

entin

g da

ta

5.3

Inte

rpre

ting

and

disc

ussi

ng re

sults

5.4

Prob

abili

ty



Lear

ning

obj

ecti

ves

1 M

athe

mat

ical

pro

cess

es a

nd a

pplic

atio

nsSo

lve

prob

lem

s, e

xplo

re a

nd in

vest

igat

e in

a ra

nge

of c

onte

xts

Incr

ease

the

chal

leng

e an

d bu

ild p

rogr

essi

on a

cros

s the

key

stag

e, a

nd fo

r gro

ups o

f pup

ils b

y:

incr

easi

ng th

e

●co

mpl

exit

y of

the

appl

icat

ion,

e.g

. non

-rou

tine,

mul

ti-st

ep p

robl

ems,

ext

ende

d en

quiri

es

redu

cing

the

●

fam

iliar

ity

of th

e co

ntex

t, e.

g. n

ew c

onte

xts i

n m

athe

mat

ics,

con

text

s dra

wn

from

oth

er su

bjec

ts, o

ther

asp

ects

of p

upils

’ liv

es

incr

easi

ng th

e

●te

chni

cal d

eman

d of

the

mat

hem

atic

s req

uire

d, e

.g. m

ore

adva

nced

con

cept

s, m

ore

diffi

cult

proc

edur

es

incr

easi

ng th

e de

gree

of

●

inde

pend

ence

and

aut

onom

y in

pro

blem

-sol

ving

and

inve

stig

atio

n

1.1

Repr

esen

ting

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

iden

tify

the

●

nece

ssar

y in

form

atio

n to

un

ders

tand

or

sim

plify

a c

onte

xt

or p

robl

em;

repr

esen

t pr

oble

ms,

mak

ing

corr

ect u

se o

f sy

mbo

ls, w

ords

, di

agra

ms,

tabl

es

and

grap

hs; u

se

appr

opria

te

proc

edur

es a

nd

tool

s, in

clud

ing

ICT

iden

tify

the

●

mat

hem

atic

al

feat

ures

of a

co

ntex

t or

prob

lem

; try

out

an

d co

mpa

re

mat

hem

atic

al

repr

esen

tatio

ns;

sele

ct a

ppro

pria

te

proc

edur

es a

nd

tool

s, in

clud

ing

ICT

brea

k do

wn

●

subs

tant

ial t

asks

to

mak

e th

em m

ore

man

agea

ble;

re

pres

ent p

robl

ems

and

synt

hesi

se

info

rmat

ion

in

alge

brai

c,

geom

etric

al o

r gr

aphi

cal f

orm

; m

ove

from

one

fo

rm to

ano

ther

to

gain

a d

iffer

ent

pers

pect

ive

on th

e pr

oble

m

com

pare

and

●

eval

uate

re

pres

enta

tions

; ex

plai

n th

e fe

atur

es

sele

cted

and

just

ify

the

choi

ce o

f re

pres

enta

tion

in

rela

tion

to th

e co

ntex

t

choo

se a

nd

●

com

bine

re

pres

enta

tions

fr

om a

rang

e of

pe

rspe

ctiv

es;

intr

oduc

e an

d us

e a

rang

e of

m

athe

mat

ical

te

chni

ques

, the

m

ost e

ffici

ent f

or

anal

ysis

and

mos

t eff

ectiv

e fo

r co

mm

unic

atio

n

syst

emat

ical

ly

●

mod

el c

onte

xts o

r pr

oble

ms t

hrou

gh

prec

ise

and

cons

iste

nt u

se o

f sy

mbo

ls a

nd

repr

esen

tatio

ns,

and

sust

ain

this

th

roug

hout

the

wor

k



1.2

Ana

lysi

ng –

use

reas

onin

g

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

clas

sify

and

●

visu

alis

e pr

oper

ties

and

patt

erns

; ge

nera

lise

in

sim

ple

case

s by

wor

king

logi

cally

; dr

aw si

mpl

e co

nclu

sion

s and

ex

plai

n re

ason

ing;

un

ders

tand

the

sign

ifica

nce

of a

co

unte

r-ex

ampl

e;

take

acc

ount

of

feed

back

and

lear

n fr

om m

ista

kes

visu

alis

e an

d

●

man

ipul

ate

dyna

mic

imag

es;

conj

ectu

re a

nd

gene

ralis

e; m

ove

betw

een

the

gene

ral a

nd th

e pa

rtic

ular

to te

st

the

logi

c of

an

argu

men

t; id

entif

y ex

cept

iona

l cas

es

or c

ount

er-

exam

ples

; mak

e co

nnec

tions

with

re

late

d co

ntex

ts

use

conn

ectio

ns

●

with

rela

ted

cont

exts

to

impr

ove

the

anal

ysis

of a

si

tuat

ion

or

prob

lem

; pos

e qu

estio

ns a

nd

mak

e co

nvin

cing

ar

gum

ents

to

just

ify

gene

ralis

atio

ns o

r so

lutio

ns;

reco

gnis

e th

e im

pact

of

cons

trai

nts o

r as

sum

ptio

ns

iden

tify

a ra

nge

of

●

stra

tegi

es a

nd

appr

ecia

te th

at

mor

e th

an o

ne

appr

oach

may

be

nece

ssar

y; e

xplo

re

the

effec

ts o

f va

ryin

g va

lues

and

lo

ok fo

r inv

aria

nce

and

cova

rianc

e in

m

odel

s and

re

pres

enta

tions

; ex

amin

e an

d re

fine

argu

men

ts,

conc

lusi

ons a

nd

gene

ralis

atio

ns;

prod

uce

sim

ple

proo

fs

mak

e pr

ogre

ss b

y

●

expl

orin

g m

athe

mat

ical

ta

sks,

dev

elop

ing

and

follo

win

g al

tern

ativ

e ap

proa

ches

; ex

amin

e an

d ex

tend

ge

nera

lisat

ions

; su

ppor

t as

sum

ptio

ns b

y cl

ear a

rgum

ent

and

follo

w th

roug

h a

sust

aine

d ch

ain

of re

ason

ing,

in

clud

ing

proo

f

pres

ent r

igor

ous

●

and

sust

aine

d ar

gum

ents

; rea

son

indu

ctiv

ely,

ded

uce

and

prov

e; e

xpla

in

and

just

ify

assu

mpt

ions

and

co

nstr

aint

s

1.3

Ana

lysi

ng –

use

pro

cedu

res

With

in th

e ap

prop

riate

rang

e an

d co

nten

t:

mak

e ac

cura

te m

athe

mat

ical

dia

gram

s, g

raph

s and

con

stru

ctio

ns o

n pa

per a

nd o

n sc

reen

; cal

cula

te a

ccur

atel

y, se

lect

ing

men

tal m

etho

ds o

r cal

cula

ting

devi

ces a

s app

ropr

iate

; man

ipul

ate

num

bers

, alg

ebra

ic e

xpre

ssio

ns a

nd e

quat

ions

, and

app

ly ro

utin

e al

gorit

hms;

use

accu

rate

not

atio

n, in

clud

ing

corr

ect

synt

ax w

hen

usin

g IC

T; re

cord

met

hods

, sol

utio

ns a

nd c

oncl

usio

ns; e

stim

ate,

app

roxi

mat

e an

d ch

eck

wor

king



1.4

Inte

rpre

ting

and

eva

luat

ing

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

inte

rpre

t

●

info

rmat

ion

from

a

mat

hem

atic

al

repr

esen

tatio

n or

co

ntex

t; re

late

fin

ding

s to

the

orig

inal

con

text

; ch

eck

the

accu

racy

of

the

solu

tion;

ex

plai

n an

d ju

stify

m

etho

ds a

nd

conc

lusi

ons;

com

pare

and

ev

alua

te

appr

oach

es

use

logi

cal

●

argu

men

t to

inte

rpre

t the

m

athe

mat

ics i

n a

give

n co

ntex

t or t

o es

tabl

ish

the

trut

h of

a st

atem

ent;

give

ac

cura

te so

lutio

ns

appr

opria

te to

the

cont

ext o

r pr

oble

m; e

valu

ate

the

effici

ency

of

alte

rnat

ive

stra

tegi

es a

nd

appr

oach

es

just

ify th

e

●

mat

hem

atic

al

feat

ures

dra

wn

from

a c

onte

xt a

nd

the

choi

ce o

f ap

proa

ch; g

ener

ate

fulle

r sol

utio

ns b

y pr

esen

ting

a co

ncis

e, re

ason

ed

argu

men

t usi

ng

sym

bols

, dia

gram

s,

grap

hs a

nd re

late

d ex

plan

atio

ns

mak

e se

nse

of, a

nd

●

judg

e th

e va

lue

of,

own

findi

ngs a

nd

thos

e pr

esen

ted

by

othe

rs; j

udge

the

stre

ngth

of

empi

rical

evi

denc

e an

d di

stin

guis

h be

twee

n ev

iden

ce

and

proo

f; ju

stify

ge

nera

lisat

ions

, ar

gum

ents

or

solu

tions

show

insi

ght i

nto

●

the

mat

hem

atic

al

conn

ectio

ns in

the

cont

ext o

r pr

oble

m; c

ritic

ally

ex

amin

e st

rate

gies

ad

opte

d an

d ar

gum

ents

pr

esen

ted;

con

side

r th

e as

sum

ptio

ns in

th

e m

odel

and

re

cogn

ise

limita

tions

in th

e ac

cura

cy o

f res

ults

an

d co

nclu

sion

s

just

ify a

nd e

xpla

in

●

solu

tions

to

prob

lem

s inv

olvi

ng

an u

nfam

iliar

co

ntex

t or a

nu

mbe

r of f

eatu

res

or v

aria

bles

; co

mm

ent

cons

truc

tivel

y on

re

ason

ing,

logi

c,

proc

ess,

resu

lts a

nd

conc

lusi

ons



1.5

Com

mun

icat

ing

and

refle

ctin

g

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

com

mun

icat

e ow

n

●

findi

ngs e

ffect

ivel

y,

oral

ly a

nd in

w

ritin

g, a

nd d

iscu

ss

and

com

pare

ap

proa

ches

and

re

sults

with

oth

ers;

reco

gnis

e eq

uiva

lent

ap

proa

ches

refin

e ow

n fin

ding

s

●

and

appr

oach

es o

n th

e ba

sis o

f di

scus

sion

s with

ot

hers

; rec

ogni

se

effici

ency

in a

n ap

proa

ch; r

elat

e th

e cu

rren

t pr

oble

m a

nd

stru

ctur

e to

pr

evio

us si

tuat

ions

revi

ew a

nd re

fine

●

own

findi

ngs a

nd

appr

oach

es o

n th

e ba

sis o

f dis

cuss

ions

w

ith o

ther

s; lo

ok

for a

nd re

flect

on

othe

r app

roac

hes

and

build

on

prev

ious

ex

perie

nce

of

sim

ilar s

ituat

ions

an

d ou

tcom

es

use

a ra

nge

of

●

form

s to

com

mun

icat

e fin

ding

s effe

ctiv

ely

to d

iffer

ent

audi

ence

s; re

view

fin

ding

s and

look

fo

r equ

ival

ence

to

diffe

rent

pro

blem

s w

ith si

mila

r st

ruct

ure

rout

inel

y re

view

●

and

refin

e fin

ding

s an

d ap

proa

ches

; id

entif

y ho

w o

ther

co

ntex

ts w

ere

diffe

rent

from

, or

sim

ilar t

o, th

e cu

rren

t situ

atio

n an

d ex

plai

n ho

w

and

why

the

sam

e or

diff

eren

t st

rate

gies

wer

e us

ed

use

mat

hem

atic

al

●

lang

uage

and

sy

mbo

ls e

ffect

ivel

y in

pre

sent

ing

conv

inci

ng

conc

lusi

ons o

r fin

ding

s; cr

itica

lly

refle

ct o

n ow

n lin

es

of e

nqui

ry w

hen

expl

orin

g; se

arch

fo

r and

app

reci

ate

mor

e el

egan

t for

ms

of c

omm

unic

atin

g ap

proa

ches

and

so

lutio

ns; c

onsi

der

the

effici

ency

of

alte

rnat

ive

lines

of

enqu

iry o

r pr

oced

ures



2 N

umbe

r2.

1 Pl

ace

valu

e, o

rder

ing

and

roun

ding

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

unde

rsta

nd a

nd

●

use

deci

mal

no

tatio

n an

d pl

ace

valu

e; m

ultip

ly a

nd

divi

de in

tege

rs a

nd

deci

mal

s by

10, 1

00,

1000

, and

exp

lain

th

e eff

ect

com

pare

and

ord

er

●

deci

mal

s in

diffe

rent

con

text

s; kn

ow th

at w

hen

com

parin

g m

easu

rem

ents

the

units

mus

t be

the

sam

e

read

and

writ

e

●

posi

tive

inte

ger

pow

ers o

f 10;

m

ultip

ly a

nd d

ivid

e in

tege

rs a

nd

deci

mal

s by

0.1,

0.

01

orde

r dec

imal

s

●

exte

nd k

now

ledg

e

●

of in

tege

r pow

ers

of 1

0; re

cogn

ise

the

equi

vale

nce

of 0

.1,

1 ⁄10 a

nd 1

0–1;

mul

tiply

and

div

ide

by a

ny in

tege

r po

wer

of 1

0

expr

ess n

umbe

rs in

●

stan

dard

inde

x fo

rm, b

oth

in

conv

entio

nal

nota

tion

and

on a

ca

lcul

ator

dis

play

conv

ert b

etw

een

●

ordi

nary

and

st

anda

rd in

dex

form

re

pres

enta

tions

use

stan

dard

inde

x

●

form

to m

ake

sens

ible

est

imat

es

for c

alcu

latio

ns

invo

lvin

g m

ultip

licat

ion

and/

or d

ivis

ion

roun

d po

sitiv

e

●

who

le n

umbe

rs to

th

e ne

ares

t 10,

100

or

100

0, a

nd

deci

mal

s to

the

near

est w

hole

nu

mbe

r or o

ne

deci

mal

pla

ce

roun

d po

sitiv

e

●

num

bers

to a

ny

give

n po

wer

of 1

0;

roun

d de

cim

als t

o th

e ne

ares

t who

le

num

ber o

r to

one

or tw

o de

cim

al

plac

es

use

roun

ding

to

●

mak

e es

timat

es

and

to g

ive

solu

tions

to

prob

lem

s to

an

appr

opria

te d

egre

e of

acc

urac

y

roun

d to

a g

iven

●

num

ber o

f si

gnifi

cant

figu

res;

use

sign

ifica

nt

figur

es to

ap

prox

imat

e an

swer

s whe

n m

ultip

lyin

g or

di

vidi

ng la

rge

num

bers

unde

rsta

nd h

ow

●

erro

rs c

an b

e co

mpo

unde

d in

ca

lcul

atio

ns

unde

rsta

nd u

pper

●

and

low

er b

ound

s



2.2

Inte

gers

, pow

ers

and

root

s

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

unde

rsta

nd

●

nega

tive

num

bers

as

pos

ition

s on

a nu

mbe

r lin

e; o

rder

, ad

d an

d su

btra

ct

inte

gers

in c

onte

xt

reco

gnis

e an

d us

e

●

mul

tiple

s, fa

ctor

s,

prim

es (l

ess t

han

100)

, com

mon

fa

ctor

s, h

ighe

st

com

mon

fact

ors

and

low

est

com

mon

mul

tiple

s in

sim

ple

case

s; us

e si

mpl

e te

sts o

f di

visi

bilit

y

reco

gnis

e th

e fir

st

●

few

tria

ngul

ar

num

bers

; rec

ogni

se

the

squa

res o

f nu

mbe

rs to

at l

east

12

× 1

2 an

d th

e co

rres

pond

ing

root

s

add,

subt

ract

,

●

mul

tiply

and

div

ide

inte

gers

use

mul

tiple

s,

●

fact

ors,

com

mon

fa

ctor

s, h

ighe

st

com

mon

fact

ors,

lo

wes

t com

mon

m

ultip

les a

nd

prim

es; fi

nd th

e pr

ime

fact

or

deco

mpo

sitio

n of

a

num

ber,

e.g.

800

0 =

26 × 5

3

use

squa

res,

●

posi

tive

and

nega

tive

squa

re

root

s, c

ubes

and

cu

be ro

ots,

and

in

dex

nota

tion

for

smal

l pos

itive

in

tege

r pow

ers

use

the

prim

e

●

fact

or

deco

mpo

sitio

n of

a

num

ber

use

ICT

to e

stim

ate

●

squa

re ro

ots a

nd

cube

root

s

use

inde

x no

tatio

n

●

for i

nteg

er p

ower

s; kn

ow a

nd u

se th

e in

dex

law

s for

m

ultip

licat

ion

and

divi

sion

of p

ositi

ve

inte

ger p

ower

s

use

inde

x no

tatio

n

●

with

neg

ativ

e an

d fr

actio

nal p

ower

s,

reco

gnis

ing

that

th

e in

dex

law

s can

be

app

lied

to th

ese

as w

ell

know

that

●

1 2n

=

n a

nd

1 3n

= f

or

any

posi

tive

num

ber n

use

inve

rse

●

oper

atio

ns,

unde

rsta

ndin

g th

at

the

inve

rse

oper

atio

n of

rais

ing

a po

sitiv

e nu

mbe

r to

pow

er n

is

rais

ing

the

resu

lt of

th

is o

pera

tion

to

pow

er 1 n

unde

rsta

nd a

nd

●

use

ratio

nal a

nd

irrat

iona

l num

bers



2.3

Frac

tion

s, d

ecim

als,

per

cent

ages

, rat

io a

nd p

ropo

rtio

n

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

expr

ess a

smal

ler

●

who

le n

umbe

r as a

fr

actio

n of

a la

rger

on

e; si

mpl

ify

frac

tions

by

canc

ellin

g al

l co

mm

on fa

ctor

s an

d id

entif

y eq

uiva

lent

fr

actio

ns; c

onve

rt

term

inat

ing

deci

mal

s to

frac

tions

, e.g

. 0.2

3 =

23

100

; use

dia

gram

s to

com

pare

two

or

mor

e si

mpl

e fr

actio

ns

reco

gnis

e th

at a

●

recu

rrin

g de

cim

al is

a

frac

tion;

use

di

visi

on to

con

vert

a

frac

tion

to a

de

cim

al; o

rder

fr

actio

ns b

y w

ritin

g th

em w

ith a

co

mm

on

deno

min

ator

or b

y co

nver

ting

them

to

deci

mal

s

unde

rsta

nd th

e

●

equi

vale

nce

of

sim

ple

alge

brai

c fr

actio

ns; k

now

th

at a

recu

rrin

g de

cim

al is

an

exac

t fr

actio

n

dist

ingu

ish

●

betw

een

frac

tions

w

ith d

enom

inat

ors

that

hav

e on

ly

prim

e fa

ctor

s 2 o

r 5

(term

inat

ing

deci

mal

s), a

nd

othe

r fra

ctio

ns

(recu

rrin

g de

cim

als)

use

an a

lgeb

raic

●

met

hod

to c

onve

rt

a re

curr

ing

deci

mal

to

a fr

actio

n

add

and

subt

ract

●

sim

ple

frac

tions

an

d th

ose

with

co

mm

on

deno

min

ator

s; ca

lcul

ate

sim

ple

frac

tions

of

quan

titie

s and

m

easu

rem

ents

(w

hole

-num

ber

answ

ers)

; mul

tiply

a

frac

tion

by a

n in

tege

r

add

and

subt

ract

●

frac

tions

by

writ

ing

them

with

a

com

mon

de

nom

inat

or;

calc

ulat

e fr

actio

ns

of q

uant

ities

(fr

actio

n an

swer

s);

mul

tiply

and

div

ide

an in

tege

r by

a fr

actio

n

use

effici

ent

●

met

hods

to a

dd,

subt

ract

, mul

tiply

an

d di

vide

fr

actio

ns,

inte

rpre

ting

divi

sion

as a

m

ultip

licat

ive

inve

rse;

can

cel

com

mon

fact

ors

befo

re m

ultip

lyin

g or

div

idin

g

unde

rsta

nd a

nd

●

appl

y effi

cien

t m

etho

ds to

add

, su

btra

ct, m

ultip

ly

and

divi

de

frac

tions

, in

terp

retin

g di

visi

on a

s a

mul

tiplic

ativ

e in

vers

e



unde

rsta

nd

●

perc

enta

ge a

s the

‘n

umbe

r of p

arts

pe

r 100

’; cal

cula

te

sim

ple

perc

enta

ges

and

use

perc

enta

ges t

o co

mpa

re si

mpl

e pr

opor

tions

reco

gnis

e th

e

●

equi

vale

nce

of

perc

enta

ges,

fr

actio

ns a

nd

deci

mal

s

unde

rsta

nd th

e

●

rela

tions

hip

betw

een

ratio

and

pr

opor

tion;

use

di

rect

pro

port

ion

in si

mpl

e co

ntex

ts;

use

ratio

not

atio

n,

sim

plify

ratio

s and

di

vide

a q

uant

ity

into

two

part

s in

a gi

ven

ratio

; sol

ve

sim

ple

prob

lem

s in

volv

ing

ratio

and

pr

opor

tion

usin

g in

form

al st

rate

gies

inte

rpre

t

●

perc

enta

ge a

s the

op

erat

or ‘s

o m

any

hund

redt

hs o

f’ an

d ex

pres

s one

giv

en

num

ber a

s a

perc

enta

ge o

f an

othe

r; ca

lcul

ate

perc

enta

ges a

nd

find

the

outc

ome

of

a gi

ven

perc

enta

ge

incr

ease

or

decr

ease

use

the

equi

vale

nce

●

of fr

actio

ns,

deci

mal

s and

pe

rcen

tage

s to

com

pare

pr

opor

tions

appl

y

●

unde

rsta

ndin

g of

th

e re

latio

nshi

p be

twee

n ra

tio a

nd

prop

ortio

n; si

mpl

ify

ratio

s, in

clud

ing

thos

e ex

pres

sed

in

diffe

rent

uni

ts,

reco

gnisi

ng li

nks

with

frac

tion

nota

tion;

div

ide

a qu

antit

y in

to tw

o or

m

ore

part

s in

a gi

ven

ratio

; use

the

unita

ry m

etho

d to

so

lve

sim

ple

prob

lem

s inv

olvi

ng

ratio

and

dire

ct

prop

ortio

n

reco

gnis

e w

hen

●

frac

tions

or

perc

enta

ges a

re

need

ed to

com

pare

pr

opor

tions

; sol

ve

prob

lem

s inv

olvi

ng

perc

enta

ge

chan

ges

use

prop

ortio

nal

●

reas

onin

g to

solv

e pr

oble

ms,

ch

oosi

ng th

e co

rrec

t num

bers

to

take

as 1

00%

, or a

s a

who

le; c

ompa

re

two

ratio

s; in

terp

ret

and

use

ratio

in a

ra

nge

of c

onte

xts

unde

rsta

nd a

nd

●

use

prop

ortio

nalit

y an

d ca

lcul

ate

the

resu

lt of

any

pr

opor

tiona

l ch

ange

usi

ng

mul

tiplic

ativ

e m

etho

ds

calc

ulat

e an

●

orig

inal

am

ount

w

hen

give

n th

e tr

ansf

orm

ed

amou

nt a

fter

a

perc

enta

ge

chan

ge; u

se

calc

ulat

ors f

or

reve

rse

perc

enta

ge

calc

ulat

ions

by

doin

g an

ap

prop

riate

di

visi

on

calc

ulat

e an

●

unkn

own

quan

tity

from

qua

ntiti

es

that

var

y in

dire

ct

prop

ortio

n us

ing

alge

brai

c m

etho

ds

whe

re a

ppro

pria

te

unde

rsta

nd a

nd

●

use

dire

ct a

nd

inve

rse

prop

ortio

n;

solv

e pr

oble

ms

invo

lvin

g in

vers

e pr

opor

tion

(incl

udin

g in

vers

e sq

uare

s) u

sing

al

gebr

aic

met

hods



2.4

Num

ber o

pera

tion

s

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

unde

rsta

nd a

nd

●

use

the

rule

s of

arith

met

ic a

nd

inve

rse

oper

atio

ns

in th

e co

ntex

t of

posi

tive

inte

gers

an

d de

cim

als

use

the

orde

r of

●

oper

atio

ns,

incl

udin

g br

acke

ts

unde

rsta

nd a

nd

●

use

the

rule

s of

arith

met

ic a

nd

inve

rse

oper

atio

ns

in th

e co

ntex

t of

inte

gers

and

fr

actio

ns

use

the

orde

r of

●

oper

atio

ns,

incl

udin

g br

acke

ts,

with

mor

e co

mpl

ex

calc

ulat

ions

unde

rsta

nd th

e

●

effec

ts o

f m

ultip

lyin

g an

d di

vidi

ng b

y nu

mbe

rs b

etw

een

0 an

d 1;

con

solid

ate

use

of th

e ru

les o

f ar

ithm

etic

and

in

vers

e op

erat

ions

unde

rsta

nd th

e

●

orde

r of

prec

eden

ce o

f op

erat

ions

, in

clud

ing

pow

ers

reco

gnis

e an

d us

e

●

reci

proc

als;

unde

rsta

nd

‘reci

proc

al’ a

s a

mul

tiplic

ativ

e in

vers

e; k

now

that

an

y nu

mbe

r m

ultip

lied

by it

s re

cipr

ocal

is 1

, and

th

at z

ero

has n

o re

cipr

ocal

bec

ause

di

visi

on b

y ze

ro is

no

t defi

ned

use

a m

ultip

lier

●

rais

ed to

a p

ower

to

repr

esen

t and

solv

e pr

oble

ms i

nvol

ving

re

peat

ed

prop

ortio

nal

chan

ge, e

.g.

com

poun

d in

tere

st



2.5

Men

tal c

alcu

lati

on m

etho

ds

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

reca

ll nu

mbe

r fac

ts,

●

incl

udin

g po

sitiv

e in

tege

r co

mpl

emen

ts to

10

0 an

d m

ultip

licat

ion

fact

s to

10

× 10

, and

qu

ickl

y de

rive

asso

ciat

ed d

ivis

ion

fact

s

reca

ll eq

uiva

lent

●

frac

tions

, dec

imal

s an

d pe

rcen

tage

s; us

e kn

own

fact

s to

deriv

e un

know

n fa

cts,

incl

udin

g pr

oduc

ts in

volv

ing

num

bers

such

as

0.7

and

6, a

nd 0

.03

and

8

use

know

n fa

cts t

o

●

deriv

e un

know

n fa

cts;

exte

nd

men

tal m

etho

ds o

f ca

lcul

atio

n,

wor

king

with

de

cim

als,

frac

tions

, pe

rcen

tage

s,

fact

ors,

pow

ers a

nd

root

s; so

lve

prob

lem

s men

tally

use

surd

s and

●�

in

exac

t cal

cula

tions

, w

ithou

t a

calc

ulat

or;

ratio

nalis

e a

deno

min

ator

such

as

1

3 =

3 3

stre

ngth

en a

nd

●

exte

nd m

enta

l m

etho

ds o

f ca

lcul

atio

n to

in

clud

e de

cim

als,

fr

actio

ns a

nd

perc

enta

ges,

ac

com

pani

ed

whe

re a

ppro

pria

te

by su

itabl

e jo

ttin

gs;

solv

e si

mpl

e pr

oble

ms m

enta

lly

stre

ngth

en a

nd

●

exte

nd m

enta

l m

etho

ds o

f ca

lcul

atio

n,

wor

king

with

de

cim

als,

frac

tions

, pe

rcen

tage

s,

squa

res a

nd sq

uare

ro

ots,

and

cub

es

and

cube

root

s; so

lve

prob

lem

s m

enta

lly

mak

e an

d ju

stify

●

estim

ates

and

ap

prox

imat

ions

of

calc

ulat

ions

mak

e an

d ju

stify

●

estim

ates

and

ap

prox

imat

ions

of

calc

ulat

ions

mak

e an

d ju

stify

●

estim

ates

and

ap

prox

imat

ions

of

calc

ulat

ions

mak

e an

d ju

stify

●

estim

ates

and

ap

prox

imat

ions

of

calc

ulat

ions

by

roun

ding

num

bers

to

one

sign

ifica

nt

figur

e an

d m

ultip

lyin

g or

di

vidi

ng m

enta

lly



2.6

Wri

tten

cal

cula

tion

met

hods

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

use

effici

ent

●

writ

ten

met

hods

to

add

and

subt

ract

w

hole

num

bers

an

d de

cim

als w

ith

up to

two

plac

es

mul

tiply

and

div

ide

●

thre

e-di

git b

y tw

o-di

git w

hole

nu

mbe

rs; e

xten

d to

m

ultip

lyin

g an

d di

vidi

ng d

ecim

als

with

one

or t

wo

plac

es b

y si

ngle

-di

git w

hole

nu

mbe

rs

use

effici

ent

●

writ

ten

met

hods

to

add

and

subt

ract

in

tege

rs a

nd

deci

mal

s of a

ny

size

, inc

ludi

ng

num

bers

with

di

fferin

g nu

mbe

rs

of d

ecim

al p

lace

s

use

effici

ent

●

writ

ten

met

hods

fo

r mul

tiplic

atio

n an

d di

visi

on o

f in

tege

rs a

nd

deci

mal

s, in

clud

ing

by d

ecim

als s

uch

as

0.6

or 0

.06;

un

ders

tand

whe

re

to p

ositi

on th

e de

cim

al p

oint

by

cons

ider

ing

equi

vale

nt

calc

ulat

ions

use

effici

ent

●

writ

ten

met

hods

to

add

and

subt

ract

in

tege

rs a

nd

deci

mal

s of a

ny

size

; mul

tiply

by

deci

mal

s; di

vide

by

deci

mal

s by

tran

sfor

min

g to

di

visi

on b

y an

in

tege

r



2.7

Calc

ulat

or m

etho

ds

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

carr

y ou

t

●

calc

ulat

ions

with

m

ore

than

one

step

us

ing

brac

kets

and

th

e m

emor

y; u

se

the

squa

re ro

ot a

nd

sign

cha

nge

keys

ente

r num

bers

and

●

inte

rpre

t the

di

spla

y in

diff

eren

t co

ntex

ts (d

ecim

als,

pe

rcen

tage

s,

mon

ey, m

etric

m

easu

res)

carr

y ou

t mor

e

●

diffi

cult

calc

ulat

ions

eff

ectiv

ely

and

effici

ently

usi

ng th

e fu

nctio

n ke

ys fo

r si

gn c

hang

e,

pow

ers,

root

s and

fr

actio

ns; u

se

brac

kets

and

the

mem

ory

ente

r num

bers

and

●

inte

rpre

t the

di

spla

y in

diff

eren

t co

ntex

ts (e

xten

d to

ne

gativ

e nu

mbe

rs,

frac

tions

, tim

e)

use

a ca

lcul

ator

●

effici

ently

and

ap

prop

riate

ly to

pe

rfor

m c

ompl

ex

calc

ulat

ions

with

nu

mbe

rs o

f any

si

ze, k

now

ing

not

to ro

und

durin

g in

term

edia

te st

eps

of a

cal

cula

tion;

use

th

e co

nsta

nt, �

and

si

gn c

hang

e ke

ys;

use

the

func

tion

keys

for p

ower

s,

root

s and

frac

tions

; us

e br

acke

ts a

nd

the

mem

ory

use

an e

xten

ded

●

rang

e of

func

tion

keys

, inc

ludi

ng th

e re

cipr

ocal

and

tr

igon

omet

ric

func

tions

use

stan

dard

inde

x

●

form

, exp

ress

ed in

co

nven

tiona

l no

tatio

n an

d on

a

calc

ulat

or d

ispl

ay;

know

how

to e

nter

nu

mbe

rs in

st

anda

rd in

dex

form

use

calc

ulat

ors t

o

●

expl

ore

expo

nent

ial g

row

th

and

deca

y, u

sing

a

mul

tiplie

r and

the

pow

er k

ey

calc

ulat

e w

ith

●

stan

dard

inde

x fo

rm, u

sing

a

calc

ulat

or a

s ap

prop

riate

use

calc

ulat

ors,

or

●

writ

ten

met

hods

, to

cal

cula

te th

e up

per a

nd lo

wer

bo

unds

of

calc

ulat

ions

in a

ra

nge

of c

onte

xts,

pa

rtic

ular

ly w

hen

wor

king

with

m

easu

rem

ents

2.8

Chec

king

resu

lts

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

chec

k re

sults

by

●

cons

ider

ing

whe

ther

they

are

of

the

right

ord

er o

f m

agni

tude

and

by

wor

king

pro

blem

s ba

ckw

ards

sele

ct fr

om a

rang

e

●

of c

heck

ing

met

hods

, inc

ludi

ng

estim

atin

g in

co

ntex

t and

usi

ng

inve

rse

oper

atio

ns

chec

k re

sults

usi

ng

●

appr

opria

te

met

hods

chec

k re

sults

usi

ng

●

appr

opria

te

met

hods

chec

k re

sults

usi

ng

●

appr

opria

te

met

hods

chec

k re

sults

usi

ng

●

appr

opria

te

met

hods



3 A

lgeb

ra3.

1 Eq

uati

ons,

form

ulae

, exp

ress

ions

and

iden

titi

es

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

use

lett

er sy

mbo

ls

●

to re

pres

ent

unkn

own

num

bers

or

var

iabl

es; k

now

th

e m

eani

ngs o

f th

e w

ords

term

, ex

pres

sion

and

equa

tion

unde

rsta

nd th

at

●

alge

brai

c op

erat

ions

follo

w

the

rule

s of

arith

met

ic

reco

gnis

e th

at

●

lett

er sy

mbo

ls p

lay

diffe

rent

role

s in

equa

tions

, fo

rmul

ae a

nd

func

tions

; kno

w

the

mea

ning

s of

the

wor

ds fo

rmul

a an

d fu

nctio

n

unde

rsta

nd th

at

●

alge

brai

c op

erat

ions

, in

clud

ing

the

use

of b

rack

ets,

follo

w

the

rule

s of

arith

met

ic; u

se

inde

x no

tatio

n fo

r sm

all p

ositi

ve

inte

ger p

ower

s

dist

ingu

ish

the

●

diffe

rent

role

s pl

ayed

by

lett

er

sym

bols

in

equa

tions

, id

entit

ies,

form

ulae

an

d fu

nctio

ns

use

inde

x no

tatio

n

●

for i

nteg

er p

ower

s an

d si

mpl

e in

stan

ces o

f the

in

dex

law

s

know

and

use

the

●

inde

x la

ws i

n ge

nera

lised

form

fo

r mul

tiplic

atio

n an

d di

visi

on o

f in

tege

r pow

ers

squa

re a

line

ar

●

expr

essi

on; e

xpan

d th

e pr

oduc

t of t

wo

linea

r exp

ress

ions

of

the

form

xn

±

and

sim

plify

the

corr

espo

ndin

g qu

adra

tic

expr

essi

on;

esta

blis

h id

entit

ies

such

as

22

()(

)a

ba

ba

b−

=+

−

fact

oris

e qu

adra

tic

●

expr

essi

ons,

in

clud

ing

the

diffe

renc

e of

two

squa

res,

e.g

. 2

9(

3)(

3)x

xx

−=

+−

ca

ncel

com

mon

fa

ctor

s in

ratio

nal

expr

essi

ons,

e.

g.

22(

1)(

1)x x

+ +

sim

plify

sim

ple

●

alge

brai

c fr

actio

ns

to p

rodu

ce li

near

ex

pres

sion

s; us

e fa

ctor

isat

ion

to

sim

plify

com

poun

d al

gebr

aic

frac

tions



sim

plify

line

ar

●

alge

brai

c ex

pres

sion

s by

colle

ctin

g lik

e te

rms;

mul

tiply

a

sing

le te

rm o

ver a

br

acke

t (in

tege

r co

effici

ents

)

cons

truc

t and

solv

e

●

sim

ple

linea

r eq

uatio

ns w

ith

inte

ger c

oeffi

cien

ts

(unk

now

n on

one

si

de o

nly)

usi

ng a

n ap

prop

riate

m

etho

d (e

.g.

inve

rse

oper

atio

ns)

sim

plify

or

●

tran

sfor

m li

near

ex

pres

sion

s by

colle

ctin

g lik

e te

rms;

mul

tiply

a

sing

le te

rm o

ver a

br

acke

t

cons

truc

t and

solv

e

●

linea

r equ

atio

ns

with

inte

ger

coeffi

cien

ts

(unk

now

n on

eith

er

or b

oth

side

s,

with

out a

nd w

ith

brac

kets

) usi

ng

appr

opria

te

met

hods

(e.g

. in

vers

e op

erat

ions

, tr

ansf

orm

ing

both

si

des i

n sa

me

way

)

use

grap

hs a

nd se

t

●

up e

quat

ions

to

solv

e si

mpl

e pr

oble

ms i

nvol

ving

di

rect

pro

port

ion

sim

plify

or

●

tran

sfor

m a

lgeb

raic

ex

pres

sion

s by

taki

ng o

ut si

ngle

-te

rm c

omm

on

fact

ors;

add

sim

ple

alge

brai

c fr

actio

ns

cons

truc

t and

solv

e

●

linea

r equ

atio

ns

with

inte

ger

coeffi

cien

ts (w

ith

and

with

out

brac

kets

, neg

ativ

e si

gns a

nyw

here

in

the

equa

tion,

po

sitiv

e or

neg

ativ

e so

lutio

n)

use

syst

emat

ic tr

ial

●

and

impr

ovem

ent

met

hods

and

ICT

tool

s to

find

appr

oxim

ate

solu

tions

to

equa

tions

such

as

220

xx

+=

solv

e lin

ear

●

equa

tions

in o

ne

unkn

own

with

in

tege

r and

fr

actio

nal

coeffi

cien

ts; s

olve

lin

ear e

quat

ions

th

at re

quire

prio

r si

mpl

ifica

tion

of

brac

kets

, inc

ludi

ng

thos

e w

ith

nega

tive

sign

s an

ywhe

re in

the

equa

tion

solv

e lin

ear

●

ineq

ualit

ies i

n on

e va

riabl

e; re

pres

ent

the

solu

tion

set o

n a

num

ber l

ine

solv

e a

pair

of

●

sim

ulta

neou

s lin

ear

equa

tions

by

elim

inat

ing

one

varia

ble;

link

a

grap

h of

an

equa

tion

or a

pai

r of

equ

atio

ns to

the

alge

brai

c so

lutio

n;

cons

ider

cas

es th

at

have

no

solu

tion

or

an in

finite

num

ber

of so

lutio

ns

solv

e eq

uatio

ns

●

invo

lvin

g al

gebr

aic

frac

tions

with

co

mpo

und

expr

essi

ons a

s the

nu

mer

ator

s and

/or

deno

min

ator

s

solv

e lin

ear

●

ineq

ualit

ies i

n on

e an

d tw

o va

riabl

es;

find

and

repr

esen

t th

e so

lutio

n se

t

expl

ore

‘opt

imum

’

●

met

hods

of s

olvi

ng

sim

ulta

neou

s eq

uatio

ns in

di

ffere

nt fo

rms

solv

e qu

adra

tic

●

equa

tions

by

fact

oris

atio

n

solv

e ex

actly

, by

●

elim

inat

ion

of a

n un

know

n, tw

o si

mul

tane

ous

equa

tions

in tw

o un

know

ns, w

here

on

e is

line

ar in

eac

h un

know

n an

d th

e ot

her i

s lin

ear i

n on

e un

know

n an

d qu

adra

tic in

the

othe

r or o

f the

form

2

22

xy

r+

=

solv

e qu

adra

tic

●

equa

tions

by

fact

oris

atio

n,

com

plet

ing

the

squa

re a

nd u

sing

th

e qu

adra

tic

form

ula,

incl

udin

g th

ose

in w

hich

the

coeffi

cien

t of t

he

quad

ratic

term

is

grea

ter t

han

1



use

alge

brai

c

●

met

hods

to so

lve

prob

lem

s inv

olvi

ng

dire

ct p

ropo

rtio

n;

rela

te a

lgeb

raic

so

lutio

ns to

gra

phs

of th

e eq

uatio

ns;

use

ICT

as

appr

opria

te

Y9 e

xten

sion

ob

ject

ive

expl

ore

way

s of

●

cons

truc

ting

mod

els

of re

al-li

fe si

tuat

ions

by

dra

win

g gr

aphs

an

d co

nstr

uctin

g al

gebr

aic e

quat

ions

an

d in

equa

litie

s

use

sim

ple

●

form

ulae

from

m

athe

mat

ics a

nd

othe

r sub

ject

s; su

bstit

ute

posi

tive

inte

gers

into

line

ar

expr

essi

ons a

nd

form

ulae

and

, in

sim

ple

case

s,

deriv

e a

form

ula

use

form

ulae

from

●

mat

hem

atic

s and

ot

her s

ubje

cts;

subs

titut

e in

tege

rs

into

sim

ple

form

ulae

, inc

ludi

ng

exam

ples

that

lead

to

an

equa

tion

to

solv

e; su

bstit

ute

posi

tive

inte

gers

in

to e

xpre

ssio

ns

invo

lvin

g sm

all

pow

ers,

e.g

.

23

4x

+ o

r ;

32

x

deriv

e si

mpl

e fo

rmul

ae

use

form

ulae

from

●

mat

hem

atic

s and

ot

her s

ubje

cts;

subs

titut

e nu

mbe

rs

into

exp

ress

ions

an

d fo

rmul

ae;

deriv

e a

form

ula

and,

in si

mpl

e ca

ses,

cha

nge

its

subj

ect

deriv

e an

d us

e

●

mor

e co

mpl

ex

form

ulae

; cha

nge

the

subj

ect o

f a

form

ula,

incl

udin

g ca

ses w

here

a

pow

er o

f the

su

bjec

t app

ears

in

the

ques

tion

or

solu

tion,

e.g

. find

gi

ven

that

2

Ar

=

deriv

e an

d us

e

●

mor

e co

mpl

ex

form

ulae

; cha

nge

the

subj

ect o

f a

form

ula,

incl

udin

g ca

ses w

here

the

subj

ect o

ccur

s tw

ice

deriv

e re

latio

nshi

ps

●

betw

een

diffe

rent

fo

rmul

ae th

at

prod

uce

equa

l or

rela

ted

resu

lts

r



3.2

Sequ

ence

s, fu

ncti

ons

and

grap

hs

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

desc

ribe

inte

ger

●

sequ

ence

s; ge

nera

te te

rms o

f a

sim

ple

sequ

ence

, gi

ven

a ru

le (e

.g.

findi

ng a

term

from

th

e pr

evio

us te

rm,

findi

ng a

term

gi

ven

its p

ositi

on in

th

e se

quen

ce)

gene

rate

●

sequ

ence

s fro

m

patt

erns

or

prac

tical

con

text

s an

d de

scrib

e th

e ge

nera

l ter

m in

si

mpl

e ca

ses

gene

rate

term

s of a

●

linea

r seq

uenc

e us

ing

term

-to-

term

an

d po

sitio

n-to

-te

rm ru

les,

on

pape

r and

usi

ng a

sp

read

shee

t or

grap

hics

cal

cula

tor

use

linea

r

●

expr

essi

ons t

o de

scrib

e th

e nt

h te

rm o

f a si

mpl

e ar

ithm

etic

se

quen

ce,

just

ifyin

g its

form

by

refe

rrin

g to

the

activ

ity o

r pra

ctic

al

cont

ext f

rom

whi

ch

it w

as g

ener

ated

gene

rate

term

s of a

●

sequ

ence

usi

ng

term

-to-

term

and

po

sitio

n-to

-ter

m

rule

s, o

n pa

per a

nd

usin

g IC

T

gene

rate

●

sequ

ence

s fro

m

prac

tical

con

text

s an

d w

rite

and

just

ify a

n ex

pres

sion

to

desc

ribe

the

th

n

term

of a

n ar

ithm

etic

se

quen

ce

find

the

next

term

●

and

the

nth

term

of

quad

ratic

se

quen

ces a

nd

expl

ore

thei

r pr

oper

ties;

dedu

ce

prop

ertie

s of t

he

sequ

ence

s of

tria

ngul

ar a

nd

squa

re n

umbe

rs

from

spat

ial

patt

erns



expr

ess s

impl

e

●

func

tions

in w

ords

, th

en u

sing

sy

mbo

ls; r

epre

sent

th

em in

map

ping

s

expr

ess s

impl

e

●

func

tions

al

gebr

aica

lly a

nd

repr

esen

t the

m in

m

appi

ngs o

r on

a sp

read

shee

t

find

the

inve

rse

of a

●

linea

r fun

ctio

n

gene

rate

poi

nts

●

and

plot

gra

phs o

f lin

ear f

unct

ions

, w

here

y is

giv

en

impl

icitl

y in

term

s of

x (e

.g.

0a

ybx

+=

, 0

ybx

c+

+=

), o

n pa

per a

nd u

sing

IC

T; fi

nd th

e gr

adie

nt o

f lin

es

give

n by

equ

atio

ns

of th

e fo

rm , g

iven

va

lues

for m

and

c

plot

the

grap

h of

●

the

inve

rse

of a

lin

ear f

unct

ion

unde

rsta

nd th

at

●

equa

tions

in th

e fo

rm y

mx

c=

+re

pres

ent a

stra

ight

lin

e an

d th

at m

is

the

grad

ient

and

c is

the

valu

e of

the

y

-inte

rcep

t; in

vest

igat

e th

e gr

adie

nts o

f pa

ralle

l lin

es a

nd

lines

per

pend

icul

ar

to th

ese

lines

iden

tify

the

●

equa

tions

of

stra

ight

-line

gra

phs

that

are

par

alle

l; fin

d th

e gr

adie

nt

and

equa

tion

of

a st

raig

ht-li

ne

grap

h th

at is

pe

rpen

dicu

lar

to a

giv

en li

ne

plot

gra

phs o

f

●

mor

e co

mpl

ex

quad

ratic

and

cu

bic

func

tions

; es

timat

e va

lues

at

spec

ific

poin

ts,

incl

udin

g m

axim

a an

d m

inim

a

know

and

●

unde

rsta

nd th

at

the

inte

rsec

tion

poin

ts o

f the

gr

aphs

of a

line

ar

and

quad

ratic

fu

nctio

n ar

e th

e ap

prox

imat

e so

lutio

ns to

the

corr

espo

ndin

g si

mul

tane

ous

equa

tions

cons

truc

t the

●

grap

hs o

f sim

ple

loci

, inc

ludi

ng th

e ci

rcle

2

22

xy

r+

= ;

find

grap

hica

lly th

e in

ters

ectio

n po

ints

of

a g

iven

stra

ight

lin

e w

ith th

is c

ircle

an

d kn

ow th

is

repr

esen

ts th

e so

lutio

n to

the

corr

espo

ndin

g tw

o si

mul

tane

ous

equa

tions



gene

rate

●

coor

dina

te p

airs

th

at sa

tisfy

a si

mpl

e lin

ear r

ule;

plo

t the

gr

aphs

of s

impl

e lin

ear f

unct

ions

, w

here

y is

giv

en

expl

icitl

y in

term

s of

x, o

n pa

per a

nd

usin

g IC

T; re

cogn

ise

stra

ight

-line

gra

phs

para

llel t

o th

e x

-axi

s or

y - a

xis

plot

and

inte

rpre

t

●

the

grap

hs o

f si

mpl

e lin

ear

func

tions

aris

ing

from

real

-life

si

tuat

ions

, e.g

. co

nver

sion

gra

phs

gene

rate

poi

nts i

n

●

all f

our q

uadr

ants

an

d pl

ot th

e gr

aphs

of

line

ar fu

nctio

ns,

whe

re y

is g

iven

ex

plic

itly

in te

rms

of x

, on

pape

r and

us

ing

ICT;

reco

gnis

e th

at e

quat

ions

of

the

form

y

mx

c=

+

corr

espo

nd to

st

raig

ht-li

ne g

raph

s

cons

truc

t lin

ear

●

func

tions

aris

ing

from

real

-life

pr

oble

ms a

nd p

lot

thei

r co

rres

pond

ing

grap

hs; d

iscu

ss a

nd

inte

rpre

t gra

phs

aris

ing

from

real

si

tuat

ions

, e.g

. di

stan

ce–t

ime

grap

hs

cons

truc

t fun

ctio

ns

●

aris

ing

from

re

al-li

fe p

robl

ems

and

plot

thei

r co

rres

pond

ing

grap

hs; i

nter

pret

gr

aphs

aris

ing

from

re

al si

tuat

ions

, e.g

. tim

e se

ries g

raph

s

expl

ore

sim

ple

●

prop

ertie

s of

quad

ratic

fu

nctio

ns; p

lot

grap

hs o

f sim

ple

quad

ratic

and

cub

ic

func

tions

, e.g

. 2

yx

= ,

23

4y

x=

+,

3

yx

=

unde

rsta

nd th

at

●

the

poin

t of

inte

rsec

tion

of tw

o di

ffere

nt li

nes i

n th

e sa

me

two

varia

bles

that

si

mul

tane

ousl

y de

scrib

e a

real

si

tuat

ion

is th

e so

lutio

n to

the

sim

ulta

neou

s eq

uatio

ns

repr

esen

ted

by th

e lin

es

find

appr

oxim

ate

●

solu

tions

of a

qu

adra

tic e

quat

ion

from

the

grap

h of

th

e co

rres

pond

ing

quad

ratic

func

tion

iden

tify

and

sket

ch

●

grap

hs o

f lin

ear

and

sim

ple

quad

ratic

and

cub

ic

func

tions

; un

ders

tand

the

effec

t on

the

grap

h of

add

ition

of (

or

mul

tiplic

atio

n by

) a

cons

tant

plot

and

reco

gnis

e

●

the

char

acte

ristic

sh

apes

of g

raph

s of

sim

ple

cubi

c fu

nctio

ns (e

.g.

3

yx

= ),

reci

proc

al

func

tions

(e.g

. 1

yx

=,

0x≠

), ex

pone

ntia

l fu

nctio

ns ( y

kx=

fo

r int

eger

val

ues

of x

and

sim

ple

posi

tive

valu

es o

f k)

and

trig

onom

etric

fu

nctio

ns, o

n pa

per

and

usin

g IC

T

appl

y to

the

grap

h

●

f(

)y

x=

the

tran

sfor

mat

ions

f()

yx

a=

+,

f(

)y

ax

=,

f(

)y

xa

=+

and

f()

ya

x=

for

line

ar,

quad

ratic

, sin

e an

d co

sine

func

tions



Y9 e

xten

sion

ob

ject

ives

Use

ICT

to e

xplo

re

●

the

grap

hica

l re

pres

enta

tion

of

alge

brai

c equ

atio

ns

and

inte

rpre

t how

pr

oper

ties o

f the

gr

aph

are

rela

ted

to

feat

ures

of t

he

equa

tion,

e.g

. pa

ralle

l and

pe

rpen

dicu

lar l

ines

inte

rpre

t the

●

mea

ning

of v

ario

us

poin

ts a

nd se

ctio

ns

of st

raig

ht-li

ne

grap

hs, i

nclu

ding

in

terc

epts

and

in

ters

ectio

n, e

.g.

solv

ing

simul

tane

ous l

inea

r eq

uatio

ns



4 G

eom

etry

and

mea

sure

s4.

1 G

eom

etri

cal r

easo

ning

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

use

corr

ectly

the

●

voca

bula

ry,

nota

tion

and

labe

lling

co

nven

tions

for

lines

, ang

les a

nd

shap

es

iden

tify

para

llel

●

and

perp

endi

cula

r lin

es; k

now

the

sum

of

ang

les a

t a p

oint

, on

a st

raig

ht li

ne

and

in a

tria

ngle

; re

cogn

ise

vert

ical

ly

oppo

site

ang

les

iden

tify

alte

rnat

e

●

angl

es a

nd

corr

espo

ndin

g an

gles

; und

erst

and

a pr

oof t

hat:

– th

e an

gle

sum

of

a tr

iang

le is

180

° an

d of

a

quad

rilat

eral

is 3

60°

– th

e ex

terio

r ang

le

of a

tria

ngle

is

equa

l to

the

sum

of

the

two

inte

rior

oppo

site

ang

les

dist

ingu

ish

●

betw

een

conv

entio

ns,

defin

ition

s and

de

rived

pro

pert

ies

expl

ain

how

to fi

nd,

●

calc

ulat

e an

d us

e:

– th

e su

ms o

f the

in

terio

r and

ex

terio

r ang

les o

f qu

adril

ater

als,

pe

ntag

ons a

nd

hexa

gons

–

the

inte

rior a

nd

exte

rior a

ngle

s of

regu

lar p

olyg

ons

dist

ingu

ish

●

betw

een

prac

tical

de

mon

stra

tion

and

proo

f in

a ge

omet

rical

co

ntex

t

solv

e m

ulti-

step

●

prob

lem

s usi

ng

prop

ertie

s of

angl

es, o

f par

alle

l lin

es, a

nd o

f tr

iang

les a

nd o

ther

po

lygo

ns, j

ustif

ying

in

fere

nces

and

ex

plai

ning

re

ason

ing

with

di

agra

ms a

nd te

xt

show

step

-by-

step

●

dedu

ctio

n in

so

lvin

g m

ore

com

plex

ge

omet

rical

pr

oble

ms

unde

rsta

nd th

e

●

nece

ssar

y an

d su

ffici

ent

cond

ition

s und

er

whi

ch

gene

ralis

atio

ns,

infe

renc

es a

nd

solu

tions

to

geom

etric

al

prob

lem

s rem

ain

valid



iden

tify

and

use

●

angl

e, si

de a

nd

sym

met

ry

prop

ertie

s of

tria

ngle

s and

qu

adril

ater

als;

expl

ore

geom

etric

al

prob

lem

s inv

olvi

ng

thes

e pr

oper

ties,

ex

plai

ning

re

ason

ing

oral

ly,

usin

g st

ep-b

y-st

ep

dedu

ctio

n su

ppor

ted

by

diag

ram

s

solv

e ge

omet

rical

●

prob

lem

s usi

ng

side

and

ang

le

prop

ertie

s of

equi

late

ral,

isos

cele

s and

rig

ht-a

ngle

d tr

iang

les a

nd

spec

ial

quad

rilat

eral

s,

expl

aini

ng

reas

onin

g w

ith

diag

ram

s and

text

; cl

assi

fy

quad

rilat

eral

s by

thei

r geo

met

rical

pr

oper

ties

know

the

defin

ition

●

of a

circ

le a

nd th

e na

mes

of i

ts p

arts

; ex

plai

n w

hy

insc

ribed

regu

lar

poly

gons

can

be

cons

truc

ted

by

equa

l div

isio

ns o

f a

circ

le

solv

e pr

oble

ms

●

usin

g pr

oper

ties o

f an

gles

, of p

aral

lel

and

inte

rsec

ting

lines

, and

of

tria

ngle

s and

oth

er

poly

gons

, jus

tifyi

ng

infe

renc

es a

nd

expl

aini

ng

reas

onin

g w

ith

diag

ram

s and

text

know

that

the

●

tang

ent a

t any

po

int o

n a

circ

le is

pe

rpen

dicu

lar t

o th

e ra

dius

at t

hat

poin

t; ex

plai

n w

hy

the

perp

endi

cula

r fr

om th

e ce

ntre

to

the

chor

d bi

sect

s th

e ch

ord

prov

e an

d us

e th

e

●

fact

s tha

t: –

the

angl

e su

bten

ded

by a

n ar

c at

the

cent

re o

f a

circ

le is

twic

e th

e an

gle

subt

ende

d at

an

y po

int o

n th

e ci

rcum

fere

nce

– th

e an

gle

subt

ende

d at

the

circ

umfe

renc

e by

a

sem

icirc

le is

a ri

ght

angl

e –

angl

es in

the

sam

e se

gmen

t are

eq

ual

– op

posi

te a

ngle

s on

a c

yclic

qu

adril

ater

al su

m

to 1

80°

prov

e an

d us

e th

e

●

alte

rnat

e se

gmen

t th

eore

m



know

that

if tw

o

●

2-D

shap

es a

re

cong

ruen

t, co

rres

pond

ing

side

s and

ang

les

are

equa

l

unde

rsta

nd

●

cong

ruen

ce a

nd

expl

ore

sim

ilarit

y

know

that

if tw

o

●

2-D

shap

es a

re

sim

ilar,

corr

espo

ndin

g an

gles

are

equ

al

and

corr

espo

ndin

g si

des a

re in

the

sam

e ra

tio;

unde

rsta

nd fr

om

this

that

any

two

circ

les a

nd a

ny tw

o sq

uare

s are

m

athe

mat

ical

ly

sim

ilar w

hile

in

gene

ral a

ny tw

o re

ctan

gles

are

not

prov

e th

e

●

cong

ruen

ce o

f tr

iang

les a

nd v

erify

st

anda

rd ru

ler a

nd

com

pass

co

nstr

uctio

ns

usin

g fo

rmal

ar

gum

ents

Y9 e

xten

sion

ob

ject

ive

inve

stig

ate

●

Pyth

agor

as’

theo

rem

, usin

g a

varie

ty o

f med

ia,

thro

ugh

its h

istor

ical

an

d cu

ltura

l roo

ts

incl

udin

g ‘p

ictu

re’

proo

fs



use

2-D

●

repr

esen

tatio

ns to

vi

sual

ise

3-D

sh

apes

and

ded

uce

som

e of

thei

r pr

oper

ties

visu

alis

e 3-

D

●

shap

es fr

om th

eir

nets

; use

geo

met

ric

prop

ertie

s of

cubo

ids a

nd sh

apes

m

ade

from

cu

boid

s; us

e si

mpl

e pl

ans a

nd

elev

atio

ns

visu

alis

e an

d us

e

●

2-D

repr

esen

tatio

ns

of 3

-D o

bjec

ts;

anal

yse

3-D

shap

es

thro

ugh

2-D

pr

ojec

tions

, in

clud

ing

plan

s and

el

evat

ions

unde

rsta

nd a

nd

●

appl

y Py

thag

oras

’ th

eore

m w

hen

solv

ing

prob

lem

s in

2-D

and

sim

ple

prob

lem

s in

3-D

unde

rsta

nd a

nd

●

use

trig

onom

etric

re

latio

nshi

ps in

rig

ht-a

ngle

d tr

iang

les,

and

use

th

ese

to so

lve

prob

lem

s,

incl

udin

g th

ose

invo

lvin

g be

arin

gs

unde

rsta

nd a

nd

●

use

Pyth

agor

as’

theo

rem

to so

lve

3-D

pro

blem

s

use

trig

onom

etric

●

rela

tions

hips

in

right

-ang

led

tria

ngle

s to

solv

e 3-

D p

robl

ems,

in

clud

ing

findi

ng

the

angl

es b

etw

een

a lin

e an

d a

plan

e

calc

ulat

e th

e ar

ea

●

of a

tria

ngle

usi

ng

the

form

ula

1 2si

na

bC

draw

, ske

tch

and

●

desc

ribe

the

grap

hs o

f tr

igon

omet

ric

func

tions

for

angl

es o

f any

size

, in

clud

ing

tran

sfor

mat

ions

in

volv

ing

scal

ings

in

eith

er o

r bot

h of

th

e x a

nd y

di

rect

ions

use

the

sine

and

●

cosi

ne ru

les t

o so

lve

2-D

and

3-D

pr

oble

ms



4.2

Tran

sfor

mat

ions

and

coo

rdin

ates

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

unde

rsta

nd a

nd

●

use

the

lang

uage

an

d no

tatio

n as

soci

ated

with

re

flect

ions

, tr

ansl

atio

ns a

nd

rota

tions

iden

tify

all t

he

●

sym

met

ries o

f 2-D

sh

apes

iden

tify

refle

ctio

n

●

sym

met

ry in

3-D

sh

apes

reco

gnis

e an

d

●

visu

alis

e th

e sy

mm

etrie

s of a

2-

D sh

ape

tran

sfor

m 2

-D

●

shap

es b

y:

– re

flect

ing

in g

iven

m

irror

line

s –

rota

ting

abou

t a

give

n po

int

– tr

ansl

atin

g

tran

sfor

m 2

-D

●

shap

es b

y ro

tatio

n,

refle

ctio

n an

d tr

ansl

atio

n, o

n pa

per a

nd u

sing

IC

T

reco

gnis

e th

at

●

tran

slat

ions

, ro

tatio

ns a

nd

refle

ctio

ns p

rese

rve

leng

th a

nd a

ngle

, an

d m

ap o

bjec

ts

on to

con

grue

nt

imag

es

tran

sfor

m 2

-D

●

shap

es b

y co

mbi

natio

ns o

f tr

ansl

atio

ns,

rota

tions

and

re

flect

ions

, on

pape

r and

usi

ng

ICT;

use

co

ngru

ence

to

show

that

tr

ansl

atio

ns,

rota

tions

and

re

flect

ions

pre

serv

e le

ngth

and

ang

le



expl

ore

thes

e

●

tran

sfor

mat

ions

an

d sy

mm

etrie

s us

ing

ICT

try

out

●

mat

hem

atic

al

repr

esen

tatio

ns o

f si

mpl

e co

mbi

natio

ns o

f th

ese

tran

sfor

mat

ions

devi

se in

stru

ctio

ns

●

for a

com

pute

r to

gene

rate

and

tr

ansf

orm

shap

es

use

any

poin

t as

●

the

cent

re o

f ro

tatio

n; m

easu

re

the

angl

e of

ro

tatio

n, u

sing

fr

actio

ns o

f a tu

rn

or d

egre

es;

unde

rsta

nd th

at

tran

slat

ions

are

sp

ecifi

ed b

y a

vect

or

expl

ore

and

●

com

pare

m

athe

mat

ical

re

pres

enta

tions

of

com

bina

tions

of

tran

slat

ions

, ro

tatio

ns a

nd

refle

ctio

ns o

f 2-D

sh

apes

, on

pape

r an

d us

ing

ICT



unde

rsta

nd a

nd

●

use

the

lang

uage

an

d no

tatio

n as

soci

ated

with

en

larg

emen

t; en

larg

e 2-

D sh

apes

, gi

ven

a ce

ntre

of

enla

rgem

ent a

nd a

po

sitiv

e in

tege

r sc

ale

fact

or;

expl

ore

enla

rgem

ent u

sing

IC

T

enla

rge

2-D

shap

es,

●

give

n a

cent

re o

f en

larg

emen

t and

a

posi

tive

inte

ger

scal

e fa

ctor

, on

pape

r and

usi

ng

ICT;

iden

tify

the

scal

e fa

ctor

of a

n en

larg

emen

t as t

he

ratio

of t

he le

ngth

s of

any

two

corr

espo

ndin

g lin

e se

gmen

ts;

reco

gnis

e th

at

enla

rgem

ents

pr

eser

ve a

ngle

but

no

t len

gth,

and

un

ders

tand

the

impl

icat

ions

of

enla

rgem

ent f

or

perim

eter

enla

rge

2-D

shap

es

●

usin

g po

sitiv

e,

frac

tiona

l and

ne

gativ

e sc

ale

fact

ors,

on

pape

r an

d us

ing

ICT;

re

cogn

ise

the

sim

ilarit

y of

the

resu

lting

shap

es;

unde

rsta

nd a

nd

use

the

effec

ts o

f en

larg

emen

t on

perim

eter

unde

rsta

nd a

nd

●

use

the

effec

ts o

f en

larg

emen

t on

area

s and

vol

umes

of

shap

es a

nd

solid

s

mak

e sc

ale

●

draw

ings

use

and

inte

rpre

t

●

map

s and

scal

e dr

awin

gs in

the

cont

ext o

f m

athe

mat

ics a

nd

othe

r sub

ject

s



use

conv

entio

ns

●

and

nota

tion

for

2-D

coo

rdin

ates

in

all f

our q

uadr

ants

; fin

d co

ordi

nate

s of

poin

ts d

eter

min

ed

by g

eom

etric

in

form

atio

n

find

the

mid

poin

t

●

of th

e lin

e se

gmen

t A

B, g

iven

the

coor

dina

tes o

f po

ints

A a

nd B

use

the

coor

dina

te

●

grid

to so

lve

prob

lem

s inv

olvi

ng

tran

slat

ions

, ro

tatio

ns,

refle

ctio

ns a

nd

enla

rgem

ents

find

the

poin

ts th

at

●

divi

de a

line

in a

gi

ven

ratio

, usi

ng

the

prop

ertie

s of

sim

ilar t

riang

les;

calc

ulat

e th

e le

ngth

of

AB,

giv

en th

e co

ordi

nate

s of

poin

ts A

and

B

unde

rsta

nd a

nd

●

use

vect

or n

otat

ion

to d

escr

ibe

tran

sfor

mat

ion

of

2-D

shap

es b

y co

mbi

natio

ns o

f tr

ansl

atio

ns;

calc

ulat

e an

d re

pres

ent

grap

hica

lly th

e su

m

of tw

o ve

ctor

s

calc

ulat

e an

d

●

repr

esen

t gr

aphi

cally

the

sum

of

two

vect

ors,

the

diffe

renc

e of

two

vect

ors a

nd a

scal

ar

mul

tiple

of a

ve

ctor

; cal

cula

te

the

resu

ltant

of t

wo

vect

ors

unde

rsta

nd a

nd

●

use

the

com

mut

ativ

e an

d as

soci

ativ

e pr

oper

ties o

f ve

ctor

add

ition

solv

e si

mpl

e

●

geom

etric

al

prob

lem

s in

2-D

us

ing

vect

ors



4.3

Cons

truc

tion

and

loci

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

use

a ru

ler a

nd

●

prot

ract

or to

: –

mea

sure

and

dr

aw li

nes t

o th

e ne

ares

t mill

imet

re

and

angl

es,

incl

udin

g re

flex

angl

es, t

o th

e ne

ares

t deg

ree

– co

nstr

uct a

tr

iang

le, g

iven

two

side

s and

the

incl

uded

ang

le

(SA

S) o

r tw

o an

gles

an

d th

e in

clud

ed

side

(ASA

)

use

ICT

to e

xplo

re

●

cons

truc

tions

use

rule

r and

●

prot

ract

or to

co

nstr

uct s

impl

e ne

ts o

f 3-D

shap

es,

e.g.

cub

oid,

regu

lar

tetr

ahed

ron,

sq

uare

-bas

ed

pyra

mid

, tria

ngul

ar

pris

m

use

stra

ight

edg

e

●

and

com

pass

es to

co

nstr

uct:

– th

e m

idpo

int a

nd

perp

endi

cula

r bi

sect

or o

f a li

ne

segm

ent

– th

e bi

sect

or o

f an

angl

e –

the

perp

endi

cula

r fr

om a

poi

nt to

a

line

– th

e pe

rpen

dicu

lar

from

a p

oint

on

a lin

e –

a tr

iang

le, g

iven

th

ree

side

s (SS

S)

use

ICT

to e

xplo

re

●

thes

e co

nstr

uctio

ns

find

sim

ple

loci

,

●

both

by

reas

onin

g an

d by

usi

ng IC

T, to

pr

oduc

e sh

apes

an

d pa

ths,

e.g

. an

equi

late

ral t

riang

le

use

stra

ight

edg

e

●

and

com

pass

es to

co

nstr

uct t

riang

les,

gi

ven

right

ang

le,

hypo

tenu

se a

nd

side

(RH

S)

use

ICT

to e

xplo

re

●

cons

truc

tions

of

tria

ngle

s and

oth

er

2-D

shap

es

find

the

locu

s of a

●

poin

t tha

t mov

es

acco

rdin

g to

a

sim

ple

rule

, bot

h by

re

ason

ing

and

by

usin

g IC

T

unde

rsta

nd fr

om

●

expe

rienc

e of

co

nstr

uctin

g th

em

that

tria

ngle

s giv

en

SSS,

SA

S, A

SA o

r RH

S ar

e un

ique

, but

th

at tr

iang

les g

iven

SS

A o

r AA

A a

re n

ot

find

the

locu

s of a

●

poin

t tha

t mov

es

acco

rdin

g to

a

mor

e co

mpl

ex ru

le,

both

by

reas

onin

g an

d by

usi

ng IC

T



4.4

Mea

sure

s an

d m

ensu

rati

on

Year

7Ye

ar 8

Year

9Ye

ar10

Year

11

Exte

nsio

n

choo

se a

nd u

se u

nits

●

of m

easu

rem

ent t

o m

easu

re, e

stim

ate,

ca

lcul

ate

and

solv

e pr

oble

ms i

n ev

eryd

ay co

ntex

ts;

conv

ert o

ne m

etric

un

it to

ano

ther

, e.g

. gr

ams t

o ki

logr

ams;

read

and

inte

rpre

t sc

ales

on

a ra

nge

of

mea

surin

g in

stru

men

ts

dist

ingu

ish b

etw

een

●

and

estim

ate

the

size

of a

cute

, obt

use

and

refle

x an

gles

choo

se a

nd u

se

●

units

of

mea

sure

men

t to

mea

sure

, est

imat

e,

calc

ulat

e an

d so

lve

prob

lem

s in

a ra

nge

of c

onte

xts;

know

roug

h m

etric

eq

uiva

lent

s of

impe

rial m

easu

res

in c

omm

on u

se,

such

as m

iles,

po

unds

(lb)

and

pi

nts

use

bear

ings

to

●

spec

ify d

irect

ion

solv

e pr

oble

ms

●

invo

lvin

g m

easu

rem

ents

in a

va

riety

of c

onte

xts;

conv

ert b

etw

een

area

mea

sure

s (e.

g.

mm

2 to c

m2 , c

m2 to

m

2 , and

vic

e ve

rsa)

an

d be

twee

n vo

lum

e m

easu

res

(e.g

. mm

3 to c

m3 ,

cm3 to

m3 , a

nd v

ice

vers

a)

unde

rsta

nd a

nd

●

use

mea

sure

s of

spee

d (a

nd o

ther

co

mpo

und

mea

sure

s suc

h as

de

nsity

or

pres

sure

); so

lve

prob

lem

s inv

olvi

ng

cons

tant

or a

vera

ge

rate

s of c

hang

e

appl

y kn

owle

dge

●

that

mea

sure

men

ts

give

n to

the

near

est

who

le u

nit m

ay b

e in

accu

rate

by

up to

on

e ha

lf of

the

unit

in e

ither

dire

ctio

n an

d us

e th

is to

un

ders

tand

how

er

rors

can

be

com

poun

ded

in

calc

ulat

ions

reco

gnis

e

●

limita

tions

in th

e ac

cura

cy o

f m

easu

rem

ents

an

d ju

dge

the

prop

ortio

nal e

ffect

on

solu

tions

Y9 e

xten

sion

ob

ject

ive

inte

rpre

t and

exp

lore

●

com

bini

ng m

easu

res

into

rate

s of c

hang

e in

ever

yday

cont

exts

(e

.g. k

m p

er h

our,

penc

e per

met

re);

use

com

poun

d m

easu

res

to co

mpa

re in

re

al-li

fe co

ntex

ts (e

.g.

trave

l gra

phs a

nd

valu

e for

mon

ey),

usin

g IC

T as

ap

prop

riate



know

and

use

the

●

form

ula

for t

he a

rea

of a

rect

angl

e;

calc

ulat

e th

e pe

rimet

er a

nd a

rea

of sh

apes

mad

e fr

om re

ctan

gles

calc

ulat

e th

e

●

surf

ace

area

of

cube

s and

cub

oids

deriv

e an

d us

e

●

form

ulae

for t

he

area

of a

tria

ngle

, pa

ralle

logr

am a

nd

trap

eziu

m;

calc

ulat

e ar

eas o

f co

mpo

und

shap

es

know

and

use

the

●

form

ula

for t

he

volu

me

of a

cub

oid;

ca

lcul

ate

volu

mes

an

d su

rfac

e ar

eas o

f cu

boid

s and

shap

es

mad

e fr

om c

uboi

ds

know

and

use

the

●

form

ulae

for t

he

circ

umfe

renc

e an

d ar

ea o

f a c

ircle

calc

ulat

e th

e

●

surf

ace

area

and

vo

lum

e of

righ

t pr

ism

s

solv

e pr

oble

ms

●

invo

lvin

g le

ngth

s of

circ

ular

arc

s and

ar

eas o

f sec

tors

solv

e pr

oble

ms

●

invo

lvin

g su

rfac

e ar

eas a

nd v

olum

es

of c

ylin

ders

solv

e pr

oble

ms

●

invo

lvin

g su

rfac

e ar

eas a

nd v

olum

es

of c

ylin

ders

, py

ram

ids,

con

es

and

sphe

res

unde

rsta

nd a

nd

●

use

the

form

ulae

fo

r the

leng

th o

f a

circ

ular

arc

and

are

a an

d pe

rimet

er o

f a

sect

or

cons

ider

the

●

dim

ensi

ons o

f a

form

ula

and

begi

n to

reco

gnis

e th

e di

ffere

nce

betw

een

form

ulae

for

perim

eter

, are

a an

d vo

lum

e in

sim

ple

cont

exts

solv

e pr

oble

ms

●

invo

lvin

g m

ore

com

plex

shap

es

and

solid

s,

incl

udin

g se

gmen

ts o

f circ

les

and

frus

tum

s of

cone

s

unde

rsta

nd th

e

●

diffe

renc

e be

twee

n fo

rmul

ae

for p

erim

eter

, are

a an

d vo

lum

e by

co

nsid

erin

g di

men

sion

s



5 St

atis

tics

5.1

Spec

ifyin

g a

prob

lem

, pla

nnin

g an

d co

llect

ing

data

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

sugg

est p

ossi

ble

●

answ

ers,

giv

en a

qu

estio

n th

at c

an

be a

ddre

ssed

by

stat

istic

al m

etho

ds

deci

de w

hich

dat

a

●

wou

ld b

e re

leva

nt

to a

n en

quiry

and

po

ssib

le so

urce

s

disc

uss a

pro

blem

●

that

can

be

addr

esse

d by

st

atis

tical

met

hods

an

d id

entif

y re

late

d qu

estio

ns to

ex

plor

e

deci

de w

hich

dat

a

●

to c

olle

ct to

ans

wer

a

ques

tion,

and

the

degr

ee o

f acc

urac

y ne

eded

; ide

ntify

po

ssib

le so

urce

s; co

nsid

er

appr

opria

te sa

mpl

e si

ze

sugg

est a

pro

blem

●

to e

xplo

re u

sing

st

atis

tical

met

hods

, fr

ame

ques

tions

an

d ra

ise

conj

ectu

res

disc

uss h

ow

●

diffe

rent

sets

of

data

rela

te to

the

prob

lem

; ide

ntify

po

ssib

le p

rimar

y or

se

cond

ary

sour

ces;

dete

rmin

e th

e sa

mpl

e si

ze a

nd

mos

t app

ropr

iate

de

gree

of a

ccur

acy

inde

pend

ently

●

devi

se a

suita

ble

plan

for a

su

bsta

ntia

l st

atis

tical

pro

ject

an

d ju

stify

the

deci

sion

s mad

e

iden

tify

poss

ible

●

sour

ces o

f bia

s and

pl

an h

ow to

m

inim

ise

it

brea

k a

task

dow

n

●

into

an

appr

opria

te

serie

s of k

ey

stat

emen

ts

(hyp

othe

ses)

, and

de

cide

upo

n th

e be

st m

etho

ds fo

r te

stin

g th

ese

cons

ider

pos

sibl

e

●

diffi

culti

es w

ith

plan

ned

appr

oach

es,

incl

udin

g pr

actic

al

prob

lem

s; ad

just

th

e pr

ojec

t pla

n ac

cord

ingl

y

deal

with

pra

ctic

al

●

prob

lem

s suc

h as

no

n-re

spon

se o

r m

issi

ng d

ata

iden

tify

wha

t ext

ra

●

info

rmat

ion

may

be

requ

ired

to p

ursu

e a

furt

her l

ine

of

enqu

iry

sele

ct a

nd ju

stify

a

●

sam

plin

g sc

hem

e an

d a

met

hod

to

inve

stig

ate

a po

pula

tion,

in

clud

ing

rand

om

and

stra

tified

sa

mpl

ing

unde

rsta

nd h

ow

●

diffe

rent

met

hods

of

sam

plin

g an

d di

ffere

nt sa

mpl

e si

zes m

ay a

ffect

th

e re

liabi

lity

of

conc

lusi

ons d

raw

n



plan

how

to c

olle

ct

●

and

orga

nise

smal

l se

ts o

f dat

a fr

om

surv

eys a

nd

expe

rimen

ts:

– de

sign

dat

a co

llect

ion

shee

ts o

r qu

estio

nnai

res t

o us

e in

a si

mpl

e su

rvey

–

cons

truc

t fr

eque

ncy

tabl

es

for g

athe

ring

disc

rete

dat

a,

grou

ped

whe

re

appr

opria

te in

eq

ual c

lass

inte

rval

s

colle

ct sm

all s

ets o

f

●

data

from

surv

eys

and

expe

rimen

ts,

as p

lann

ed

plan

how

to c

olle

ct

●

the

data

; con

stru

ct

freq

uenc

y ta

bles

w

ith e

qual

cla

ss

inte

rval

s for

ga

ther

ing

cont

inuo

us d

ata

and

two-

way

tabl

es

for r

ecor

ding

di

scre

te d

ata

colle

ct d

ata

usin

g a

●

suita

ble

met

hod

(e.g

. obs

erva

tion,

co

ntro

lled

expe

rimen

t, da

ta

logg

ing

usin

g IC

T)

desi

gn a

surv

ey o

r

●

expe

rimen

t to

capt

ure

the

nece

ssar

y da

ta

from

one

or m

ore

sour

ces;

desi

gn,

tria

l and

if

nece

ssar

y re

fine

data

col

lect

ion

shee

ts; c

onst

ruct

ta

bles

for g

athe

ring

larg

e di

scre

te a

nd

cont

inuo

us se

ts o

f ra

w d

ata,

cho

osin

g su

itabl

e cl

ass

inte

rval

s; de

sign

an

d us

e tw

o-w

ay

tabl

es

gath

er d

ata

from

●

spec

ified

se

cond

ary

sour

ces,

in

clud

ing

prin

ted

tabl

es a

nd li

sts,

and

IC

T-ba

sed

sour

ces,

in

clud

ing

the

inte

rnet

gath

er d

ata

from

●

prim

ary

and

seco

ndar

y so

urce

s,

usin

g IC

T an

d ot

her

met

hods

, inc

ludi

ng

data

from

ob

serv

atio

n,

cont

rolle

d ex

perim

ent,

data

lo

ggin

g, p

rinte

d ta

bles

and

list

s



5.2

Proc

essi

ng a

nd re

pres

enti

ng d

ata

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

calc

ulat

e st

atis

tics

●

for s

mal

l set

s of

disc

rete

dat

a:

– fin

d th

e m

ode,

m

edia

n an

d ra

nge,

an

d th

e m

odal

cla

ss

for g

roup

ed d

ata

– ca

lcul

ate

the

mea

n, in

clud

ing

from

a si

mpl

e fr

eque

ncy

tabl

e,

usin

g a

calc

ulat

or

for a

larg

er n

umbe

r of

item

s

calc

ulat

e st

atis

tics

●

for s

ets o

f dis

cret

e an

d co

ntin

uous

da

ta, i

nclu

ding

w

ith a

cal

cula

tor

and

spre

adsh

eet;

reco

gnis

e w

hen

it is

ap

prop

riate

to u

se

the

rang

e, m

ean,

m

edia

n an

d m

ode

and,

for g

roup

ed

data

, the

mod

al

clas

s

calc

ulat

e st

atis

tics

●

and

sele

ct th

ose

mos

t app

ropr

iate

to

the

prob

lem

or

whi

ch a

ddre

ss th

e qu

estio

ns p

osed

use

an a

ppro

pria

te

●

rang

e of

stat

istic

al

met

hods

to e

xplo

re

and

sum

mar

ise

data

; inc

ludi

ng

estim

atin

g an

d fin

ding

the

mea

n,

med

ian,

qua

rtile

s an

d in

terq

uart

ile

rang

e fo

r lar

ge d

ata

sets

(by

calc

ulat

ion

or u

sing

a

cum

ulat

ive

freq

uenc

y di

agra

m)

use

an a

ppro

pria

te

●

rang

e of

stat

istic

al

met

hods

to e

xplo

re

and

sum

mar

ise

data

; inc

ludi

ng

calc

ulat

ing

an

appr

opria

te

mov

ing

aver

age

for

a tim

e se

ries

use

a m

ovin

g

●

aver

age

to id

entif

y se

ason

ality

and

tr

ends

in ti

me

serie

s dat

a, u

sing

th

em to

mak

e pr

edic

tions



cons

truc

t, on

pap

er

●

and

usin

g IC

T,

grap

hs a

nd

diag

ram

s to

repr

esen

t dat

a,

incl

udin

g:

– ba

r-lin

e gr

aphs

–

freq

uenc

y di

agra

ms f

or

grou

ped

disc

rete

da

ta

– si

mpl

e pi

e ch

arts

cons

truc

t gra

phic

al

●

repr

esen

tatio

ns, o

n pa

per a

nd u

sing

IC

T, a

nd id

entif

y w

hich

are

mos

t us

eful

in th

e co

ntex

t of t

he

prob

lem

. Inc

lude

: –

pie

char

ts fo

r ca

tego

rical

dat

a –

bar c

hart

s and

fr

eque

ncy

diag

ram

s for

di

scre

te a

nd

cont

inuo

us d

ata

– si

mpl

e lin

e gr

aphs

for t

ime

serie

s –

sim

ple

scat

ter

grap

hs

– st

em-a

nd-le

af

diag

ram

s

sele

ct, c

onst

ruct

●

and

mod

ify, o

n pa

per a

nd u

sing

IC

T, su

itabl

e gr

aphi

cal

repr

esen

tatio

ns to

pr

ogre

ss a

n en

quiry

and

id

entif

y ke

y fe

atur

es p

rese

nt in

th

e da

ta. I

nclu

de:

– lin

e gr

aphs

for

time

serie

s –

scat

ter g

raph

s to

deve

lop

furt

her

unde

rsta

ndin

g of

co

rrel

atio

n

sele

ct, c

onst

ruct

●

and

mod

ify, o

n pa

per a

nd u

sing

IC

T, su

itabl

e gr

aphi

cal

repr

esen

tatio

n to

pr

ogre

ss a

n en

quiry

and

id

entif

y ke

y fe

atur

es p

rese

nt in

th

e da

ta. I

nclu

de:

– cu

mul

ativ

e fr

eque

ncy

tabl

es

and

diag

ram

s –

box

plot

s –

scat

ter g

raph

s an

d lin

es o

f bes

t fit

(by

eye)

sele

ct, c

onst

ruct

●

and

mod

ify, o

n pa

per a

nd u

sing

IC

T, su

itabl

e gr

aphi

cal

repr

esen

tatio

n to

pr

ogre

ss a

n en

quiry

, inc

ludi

ng

hist

ogra

ms f

or

grou

ped

cont

inuo

us d

ata

with

equ

al c

lass

in

terv

als

cons

truc

t

●

hist

ogra

ms,

in

clud

ing

thos

e w

ith u

nequ

al c

lass

in

terv

als

Y9 e

xten

sion

ob

ject

ive

wor

k th

roug

h th

e

●

entir

e ha

ndlin

g da

ta

cycl

e to

exp

lore

re

latio

nshi

ps w

ithin

bi

varia

te d

ata,

in

clud

ing

appl

icat

ions

to

glob

al ci

tizen

ship

, e.

g. h

ow fa

ir is

our

soci

ety?



5.3

Inte

rpre

ting

and

dis

cuss

ing

resu

lts

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

inte

rpre

t dia

gram

s

●

and

grap

hs

(incl

udin

g pi

e ch

arts

), an

d dr

aw

sim

ple

conc

lusi

ons

base

d on

the

shap

e of

gra

phs a

nd

sim

ple

stat

istic

s for

a

sing

le d

istr

ibut

ion

com

pare

two

●

sim

ple

dist

ribut

ions

us

ing

the

rang

e an

d on

e of

the

mod

e, m

edia

n or

m

ean

writ

e a

shor

t rep

ort

●

of a

stat

istic

al

enqu

iry, i

nclu

ding

ap

prop

riate

di

agra

ms,

gra

phs

and

char

ts, u

sing

IC

T as

app

ropr

iate

; ju

stify

the

choi

ce o

f pr

esen

tatio

n

inte

rpre

t tab

les,

●

grap

hs a

nd

diag

ram

s for

di

scre

te a

nd

cont

inuo

us d

ata,

re

latin

g su

mm

ary

stat

istic

s and

fin

ding

s to

the

ques

tions

bei

ng

expl

ored

com

pare

two

●

dist

ribut

ions

usi

ng

the

rang

e an

d on

e or

mor

e of

the

mod

e, m

edia

n an

d m

ean

writ

e ab

out a

nd

●

disc

uss t

he re

sults

of

a st

atis

tical

en

quiry

usi

ng IC

T as

app

ropr

iate

; ju

stify

the

met

hods

us

ed

inte

rpre

t gra

phs

●

and

diag

ram

s and

m

ake

infe

renc

es to

su

ppor

t or c

ast

doub

t on

initi

al

conj

ectu

res;

have

a

basi

c un

ders

tand

ing

of

corr

elat

ion

com

pare

two

or

●

mor

e di

strib

utio

ns

and

mak

e in

fere

nces

, usi

ng

the

shap

e of

the

dist

ribut

ions

and

ap

prop

riate

st

atis

tics

revi

ew

●

inte

rpre

tatio

ns a

nd

resu

lts o

f a

stat

istic

al e

nqui

ry

on th

e ba

sis o

f di

scus

sion

s; co

mm

unic

ate

thes

e in

terp

reta

tions

and

re

sults

usi

ng

sele

cted

tabl

es,

grap

hs a

nd

diag

ram

s

anal

yse

data

to fi

nd

●

patt

erns

and

ex

cept

ions

, and

try

to e

xpla

in

anom

alie

s; in

clud

e so

cial

stat

istic

s su

ch a

s ind

ex

num

bers

, tim

e se

ries a

nd su

rvey

da

ta

appr

ecia

te th

at

●

corr

elat

ion

is a

m

easu

re o

f the

st

reng

th o

f as

soci

atio

n be

twee

n tw

o va

riabl

es;

dist

ingu

ish

betw

een

posi

tive,

ne

gativ

e an

d ze

ro

corr

elat

ion,

usi

ng

lines

of b

est fi

t; ap

prec

iate

that

ze

ro c

orre

latio

n do

es n

ot

nece

ssar

ily im

ply

‘no

rela

tions

hip’

bu

t mer

ely

‘no

linea

r rel

atio

nshi

p’

inte

rpre

t and

use

●

cum

ulat

ive

freq

uenc

y di

agra

ms t

o so

lve

prob

lem

s

reco

gnis

e th

e

●

limita

tions

of a

ny

assu

mpt

ions

and

th

e eff

ects

that

va

ryin

g th

e as

sum

ptio

ns c

ould

ha

ve o

n th

e co

nclu

sion

s dra

wn

from

dat

a an

alys

is

com

pare

two

or

●

mor

e di

strib

utio

ns

and

mak

e in

fere

nces

, usi

ng

the

shap

e of

the

dist

ribut

ions

and

m

easu

res o

f av

erag

e an

d sp

read

, inc

ludi

ng

med

ian

and

quar

tiles

use,

inte

rpre

t and

●

com

pare

hi

stog

ram

s,

incl

udin

g th

ose

with

une

qual

cla

ss

inte

rval

s



exam

ine

criti

cally

●

the

resu

lts o

f a

stat

istic

al e

nqui

ry;

just

ify c

hoic

e of

st

atis

tical

re

pres

enta

tions

an

d re

late

su

mm

aris

ed d

ata

to th

e qu

estio

ns

bein

g ex

plor

ed



5.4

Prob

abili

ty

Year

7Ye

ar 8

Year

9Ye

ar 1

0Ye

ar 1

1Ex

tens

ion

use

voca

bula

ry a

nd

●

idea

s of p

roba

bilit

y,

draw

ing

on

expe

rienc

e

unde

rsta

nd a

nd

●

use

the

prob

abili

ty

scal

e fr

om 0

to 1

; fin

d an

d ju

stify

pr

obab

ilitie

s bas

ed

on e

qual

ly li

kely

ou

tcom

es in

sim

ple

cont

exts

; ide

ntify

al

l the

pos

sibl

e m

utua

lly e

xclu

sive

ou

tcom

es o

f a

sing

le e

vent

inte

rpre

t the

resu

lts

●

of a

n ex

perim

ent

usin

g th

e la

ngua

ge

of p

roba

bilit

y;

appr

ecia

te th

at

rand

om p

roce

sses

ar

e un

pred

icta

ble

know

that

if th

e

●

prob

abili

ty o

f an

even

t occ

urrin

g is

p ,

then

the

prob

abili

ty o

f it n

ot

occu

rrin

g is

1p

− ;

use

diag

ram

s and

ta

bles

to re

cord

in a

sy

stem

atic

way

all

poss

ible

mut

ually

ex

clus

ive

outc

omes

fo

r sin

gle

even

ts

and

for t

wo

succ

essi

ve e

vent

s

inte

rpre

t res

ults

●

invo

lvin

g un

cert

aint

y an

d pr

edic

tion

iden

tify

all t

he

●

mut

ually

exc

lusi

ve

outc

omes

of a

n ex

perim

ent;

know

th

at th

e su

m o

f pr

obab

ilitie

s of a

ll m

utua

lly e

xclu

sive

ou

tcom

es is

1 a

nd

use

this

whe

n so

lvin

g pr

oble

ms

use

tree

dia

gram

s

●

to re

pres

ent

outc

omes

of t

wo

or

mor

e ev

ents

and

to

calc

ulat

e pr

obab

ilitie

s of

com

bina

tions

of

inde

pend

ent

even

ts

know

whe

n to

add

●

or m

ultip

ly tw

o pr

obab

ilitie

s: if

A

and

B ar

e m

utua

lly

excl

usiv

e, th

en th

e pr

obab

ility

of A

or

B oc

curr

ing

is P

(A) +

P(

B), w

here

as if

A

and

B ar

e in

depe

nden

t ev

ents

, the

pr

obab

ility

of A

an

d B

occu

rrin

g is

P(

A) ×

P(B

)

use

tree

dia

gram

s

●

to re

pres

ent

outc

omes

of

com

poun

d ev

ents

, re

cogn

isin

g w

hen

even

ts a

re

inde

pend

ent a

nd

dist

ingu

ishi

ng

betw

een

cont

exts

in

volv

ing

sele

ctio

n bo

th w

ith a

nd

with

out

repl

acem

ent

reco

gnis

e w

hen

●

and

how

to w

ork

with

pro

babi

litie

s as

soci

ated

with

in

depe

nden

t and

m

utua

lly e

xclu

sive

ev

ents

whe

n in

terp

retin

g da

ta



estim

ate

●

prob

abili

ties b

y co

llect

ing

data

fr

om a

sim

ple

expe

rimen

t and

re

cord

ing

it in

a

freq

uenc

y ta

ble;

co

mpa

re

expe

rimen

tal a

nd

theo

retic

al

prob

abili

ties i

n si

mpl

e co

ntex

ts

com

pare

est

imat

ed

●

expe

rimen

tal

prob

abili

ties w

ith

theo

retic

al

prob

abili

ties,

re

cogn

isin

g th

at:

– if

an e

xper

imen

t is

repe

ated

the

outc

ome

may

, and

us

ually

will

, be

diffe

rent

–

incr

easi

ng th

e nu

mbe

r of t

imes

an

expe

rimen

t is

repe

ated

gen

eral

ly

lead

s to

bett

er

estim

ates

of

prob

abili

ty

com

pare

●

expe

rimen

tal a

nd

theo

retic

al

prob

abili

ties i

n a

rang

e of

con

text

s; ap

prec

iate

the

diffe

renc

e be

twee

n m

athe

mat

ical

ex

plan

atio

n an

d ex

perim

enta

l ev

iden

ce

unde

rsta

nd re

lativ

e

●

freq

uenc

y as

an

estim

ate

of

prob

abili

ty a

nd u

se

this

to c

ompa

re

outc

omes

of

expe

rimen

ts

unde

rsta

nd th

at if

●

an e

xper

imen

t is

repe

ated

, the

ou

tcom

e m

ay –

and

us

ually

will

– b

e di

ffere

nt, a

nd th

at

incr

easi

ng th

e sa

mpl

e si

ze

gene

rally

lead

s to

bett

er e

stim

ates

of

prob

abili

ty a

nd

popu

latio

n pa

ram

eter

s

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PHOTO REDACTED DUE TO THIRD PARTY RIGHTS OR OTHER …Year 10 2 tier UAM in exam 2 tier UAM in exam 2...

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