VIEWS: 24 PAGES: 53 POSTED ON: 11/5/2012 Public Domain
NUMERACY POLICY The application of this policy and procedure ensures that no employee receives less favourable treatment on grounds of sex, trans-gender status, sexual orientation, religion or belief, marital status, civil partnership status, age, race, colour, nationality, national origin, ethnic origin, disability, part time status or trade union activities. POLICY AGREED BY __________________________________________________ DATE _____________ POLICY REVIEW DATE ________________________________________________________________________ VERSION 1 – JULY 2012 1 CONTENTS PAGE Numeracy Policy Aims 3 What is Numeracy? 4 The National Numeracy Strategy’s approach to calculation 5 Introducing and developing algebra 11 Shape, space and measures 14 Handling Data 16 Using and applying mathematics and thinking skills 18 Our approach to teaching 22 Key Objectives for different abilities 24 Key mathematical vocabulary 28 Mathematical competencies required in other subjects 37 Appendix 41 Audit 44 2 NUMERACY POLICY AIMS The aims of the numeracy policy are: To raise standards of numeracy and mathematics for all pupils throughout the school. To ensure that pupils do not experience and discontinuity from subject to subject, either in the way certain mathematical methods and strategies are taught or in relation to the level of difficulty of mathematics expected of them. To promote a positive attitude towards mathematics amongst all pupils. To promote a positive attitude towards mathematics amongst all staff and for all staff to actively promote its development across the curriculum. To ensure that all staff are aware of what numeracy is and what the National Numeracy Strategy’s approach to calculation is. To ensure that all staff, but particularly those in key departments (e.g. Science, Geography, Technology, ICT, P.E. and History), are aware of the range of mathematical skills that pupils bring to their lessons. To ensure that all staff in key departments are aware of the mathematical demands of their own subject in order that this is reflected in schemes of work. To ensure that staff in key departments provide opportunities for pupils to develop and apply their mathematical skills in their own subject. To promote a common approach to the teaching of key mathematical ideas and processes in all subjects which require them. 3 WHAT IS NUMERACY? Numeracy is a proficiency, which is developed mainly in mathematics but also in other subjects. It is more than an ability to do basic arithmetic. It involves developing confidence and competence with numbers and measures. It requires and understanding of the number system, a repertoire of mathematical techniques and an inclination and ability to solve quantitative or spatial problems in a range of contexts. Numeracy also demands practical understanding of the ways in which data are gathered by counting and measuring, and presenting in graphs, diagrams, charts and tables. As a teacher you can help children to acquire this proficiency by giving a sharp focus to the relevant aspects of the programmes of study for mathematics. The outcome should be numerate pupils who are confident enough to tackle mathematical problems without going immediately to teachers or friends for help. Your pupils should: Have a sense of the size of a number and where it fits into the number system; Know by heart number facts such as number bonds, multiplication tables, doubles and halves; Use what they know by heart to figure out answers mentally; Calculate accurately and efficiently, both mentally and with pencil and paper, drawing on a range of calculation strategies; Recognise when it is appropriate to use a calculator and be able to do so efficiently; Make sense of number problems, including non-routine problems, and recognise the operations needed to solve them; Explain their methods and reasoning using correct mathematical terms; Judge whether their answers are reasonable and have strategies for checking them where necessary; Suggest suitable units for measuring and make sensible estimates of measurements; Explain and make predictions from the numbers in graphs, diagrams, charts and tables. 4 THE NATIONAL NUMERACY STRATEGY’S APPROACH TO CALCULATION This section outlines what the strategy says about mental calculation, written calculation and the role of calculators. It explains how the four operations are taught in the mathematics department and how pupils are taught to record their calculations. It is important to stress here that pupils are not being taught to follow pencil and paper algorithms blindly and without understanding. There are various pencil and paper methods for each of the four operations; in particular it is intended to place an emphasis on the grid method for multiplication and the “chunking” method for division. Examples of the various methods are contained in the appendix. Number: From KS2 to KS3 An ability to calculate mentally lies at the heart of numeracy. As a teacher, whether of mathematics or another subject, you should stress the importance of mental calculation methods and give all pupils regular opportunities to develop the skills involved. The skills include an ability to: Remember number facts and recall them without hesitation; Use known facts to figure out new facts: for example, knowing that half of 250 is 125 can be used to work out 250 – 123; Draw on a repertoire of mental strategies to work out calculations such as 326 – 81, 223 x 4 or 2.5% of £3000, with some thinking time; Understand and use the relationships between operations to work out answers and check results: for example, 900 ÷ 15 = 60, since 6 x 150 = 900; Approximate calculations to judge whether or not an answer is about the right size: for example, recognise that ¼ of 57.9 is just under ¼ of 60, or 15; Solve problems such as: “How many CDs at £3.99 ech can I buy with “25?” or “Roughly how long will it take me to go 50 miles at 30 mph?”. An increased emphasis in KS1 and KS2 on mental calculation does not mean that written methods are not taught in primary years, but the balance between mental and written methods and the progression from one to the other become increasingly important in the later years of KS2 and in KS3. 5 CALCULATION IN KS1 AND KS2 In the early years pupils work orally with numbers. Alongside their oral work they learn to read, interpret and complete statements such as 5 + 8 = * or 13 = * + 5, and then to record the results of their mental calculations in the correct way, using a horizontal format like 43 – 8 = 35. As they progress to working with larger numbers they learn more sophisticated mental methods and tackle more complex problems. They develop some of these methods intuitively and some they are taught explicitly. Through a process of regular explanation and discussion of their own and other people’s methods they begin to acquire a repertoire of mental calculation strategies. It can be hard to hold all the intermediate steps of a calculation in the head and so informal pencil and paper notes, recording some or all the solution, become part of a mental strategy. These personal jottings may not be easy for someone else to follow but they are an important staging post to getting the right answer and acquiring fluency in mental calculation. The approach in the primary years builds on the use of the number line, first with numbers marked and then a blank line, to record steps in calculations such as 47 + 26 or 261 – 174. Jottings: Pupils make jottings to assist their mental calculations, e.g. 47 + 26 +3 +20 +3 47 50 70 73 They record steps so that they and others can see what they have done, as in: 36 + 27: 36 + 20 = 56 and then 56 + 7 = 63 Or: 30 + 20 = 50 and 6 + 7 = 13, so 50 + 13 = 63 Not everyone does a mental calculation like 81 – 26 in the same way (nor is it necessary for them to do so) but some methods are more efficient and reliable than others. Some methods only work for particular cases. Your role is not simply to accept pupils’ personal methods but to help them adopt better ones. By explaining, discussing and comparing different part-written, part-mental methods, you can guide pupils towards choosing and using methods that are quicker, can be applied more widely and are helpful for their future learning. 6 STANDARD WRITTEN METHODS Standard written methods are reliable and efficient procedures for calculating which, once grasped, can be used in many different contexts. But they are of no use to someone who applies them accurately and who cannot judge whether the answer in reasonable. For each operation, a standard written method should be taught to most pupils, then refined and practised. The method chosen should fulfil several criteria: It should not be too time-consuming to carry out; Pupils should be able to explain why it works and apply it reliably; The way it is set out should help them to avoid mistakes; It should support their further learning of mathematics. The progression towards written methods is crucial, since standard written methods are based on steps which are done mentally and which need to be secured first. For example, the calculation of 487 + 356, done by the traditional method in columns, requires the mental calculations 7 + 6 = 13, 8 + 5 + 1 = 14 and 14 + 3 + 1 = 8, while a division calculation such as 987 ÷ 23 can involve mental experiment with multiples of 23 before the correct multiple is chosen. Many countries, and in particular those which are most successful at teaching number, avoid the premature teaching of standard written methods in order not to jeopardise the development of mental calculation strategies. The bridge from recording part-written, part-mental methods to learning standard methods of written calculations begins only when pupis can add or subtract reliably any pair of two-digit numbers in their heads, usually when they are about nine years old. Standard written methods for addition and subtraction should be well established by Year 6 for nearly all pupils and will be used in KS3 with an increasing range of whole numbers and decimals. Multiplication and division methods will need to be developed further. When they transfer to KS3, some pupils may still use informal written methods to record, support and explain their multiplication and division calculations. For example, a few may use a “grid method” or Napiers’ bones (examples are in the appendix) for multiplying two or three-digit numbers, which you will need to consolidate and build on in KS3 and relate to work in algebra. For division they may use a “chunking” method; this too will need to be developed and refined to ensure efficiency. The aim is that where it is appropriate to do so all pupils use standard written methods efficiently and accurately, and with understanding. When they have reached the stage of working out more complex calculations using written methods, pupils in KS3 still need to practise and refine their mental calculation strategies. Help them to develop estimation skills in all aspects of calculation, but particularly in multiplication and division. When faced with any calculation, no matter how large or how difficult the numbers may appear to be, 7 encourage them first to ask themselves: “Can I do this in my head?”. They then need to ask: “Do I know the approximate size of the answer?” and “Does the answer make sense in the context of the question?” so that they can be reasonably sure that their calculation is right. CALCULATOR METHODS The calculator is a powerful and efficient tool in the right hands. It has an important part to play in subjects such as design and technology, geography, history and science, since it allows pupils to make use of real data from their research or experiments – often numbers with several digits. As with numeracy, the appropriate use of calculators is a whole-school matter. All subjects need to adopt a similar approach and agree when, how and for what purpose calculators are to be used. Before Year 5, the calculator’s main role in mathematics is not as a calculating tool, since pupils are still developing the mental calculation skills and written methods that they will need in later years. But, it does offer a unique way of learning about mathematical ideas throughout all key stages. For example, pupils might use a calculator to find two consecutive numbers with a given product and then discuss their methods. If pupils are to use the basic facilities of a calculator constructively and efficiently for calculating purposes, you will need to teach them in KS3 the technical skills that they will require. For example, during KS3 they need to learn: How to select from the display the number of figures appropriate to the context of the calculation; How to enter numbers and interpret the display when the numbers represent money, metric measurements, units of time or fractions; The order in which to use the keys for calculations involving more than one step; How to use facilities such as the memory, brackets, the square root and cube root keys, the sign change key, the fraction key, the constant facility, and so on. By the end of key stage 3, pupils should have the knowledge and skills to use a calculator to work out expressions such as 3250 x 1.05³ or √(7.82² - 2.91²). All pupils need to continue to learn when it is, and when it is not, appropriate to use a calculator, and their first-line strategy should involve mental calculations. They should have sufficient understanding of the calculation in front of them to be able to decide which method to use – mental, written or calculator, or a combination of these. When they do use a calculator, they should be able to draw on well- established skills of rounding numbers and calculating mentally to gain a sense of the approximate size of the answer, and have strategies to check and repeat the 8 calculation if they are not sure whether it is right. These skills do not happen by accident. They need to be taught in mathematics and applied across the curriculum whenever calculators are used. PROPORTIONAL REASONING Throughout KS3 pupils will extend their understanding of the number system to positive and negative numbers and, in particular, to fraction and their representations as terminating or recurring decimals. Fractions, decimals, percentages, ratio and proportion are different ways of expressing related ideas and relationships. The connections start to be established in KS2, particularly the equivalence between fractions, decimals and percentages. The ideas of ratio and proportion, and the relationship between them, should be a strong feature of work in KS3. By the end of the key stage, pupils should be able to solve problems involving fractions, decimals, percentages, ratio and proportion, and their interconnections. After calculation, the application of proportional reasoning is the most important aspect of elementary number. Proportionality underlies key aspects of number, algebra, shape, space and measures, and handling data. It is also central in applications of mathematics in subjects such as science, technology, geography and art. The study of proportion begins in KS2 but it is in KS3 where secure foundations need to be established. Problems involving proportion are often solved by informal methods, particularly when the numbers involved are easy to deal with mentally. But it is important to teach methods that can be applied generally. For example, the unitary method is useful for solving problems involving proportion, and multiplicative methods involving fractions or decimals are useful for solving percentage problems. When you are teaching proportional reasoning: Emphasise the language and notation of ratio and proportion, and the links to fractions, decimals and percentages; Teach pupils specific methods for solving proportion problems so that they do not remain dependent on informal approaches; Help pupils to understand what they are calculating: for example, a distance divided by a time gives a speed – an example of a rate; but a distance divided by another distance gives a scale factor or multiplier – a dimensionless number; Make explicit links between ideas of proportionality in number, algebra, shape, space and measures, and handling data. 9 In algebra, direct proportion is viewed as a linear relationship of the form y = mx. The graphical representation of this equation helps pupils to visualise ideas such as rate of change and gradient. The algebraic representation of a proportion (e.g. a:b = c:d or a/b = c/d) underpins a general method for solving problems. In shape, space and measures, proportionality arises when enlargement by different scale factors is considered. Scaling has a wide range of applications, for example, in maps, plans and scale drawings. Similar figures have sides or dimensions that are in proportion. Recognition of the similarity of all circles leads to an understanding that the circumference is directly proportional to the diameter, while awareness of the similarity of triangles with the same angles leads to an understanding of trigonometry. In statistics, proportions are often calculated when data is interpreted and inferences drawn. Proportions are also used when probabilities are estimated or calculated based on outcomes that, in theory, are equally likely. FEATURES OF NUMBER IN KEY STAGE 3 To summarise, the distinctive features of number in KS3 are: Developing understanding of the number line; Building on the approach to calculation developed in Key Stages 1 and 2, which emphasises mental methods and gradually refined written methods, extending to calculations with fractions, decimals and percentages; Developing effective use of calculators, including choosing appropriate methods for estimating, calculating and checking; Developing proportional reasoning, including making links to algebra, shape, space and measures, and handling data. 10 INTRODUCING AND DEVELOPING ALGEBRA Algebra in Key Stage 3 is generalised arithmetic. Its origins lie in arithmetic, in the art of manipulating sums, products and powers of numbers. The same rules are seen to hold true for all numbers, of whatever type, so it becomes possible to generalise the rules with letters in place of numbers. Indeed all numerical entities, coefficients as well as unknowns, can be represented by letters. This insight releases in due course the full power of algebra. The diagram below is drawn form the Royal Society and Joint Mathematical Council’s publication Teaching and learning algebra pre-19. The lines showing the links between topics are the most significant aspect. Arithmetic with Functions numbers Y = f(x) form Understandin Non-linear g equivalent functional form relationships General Relationships between Sequences terms variables Recursive Sequences functions Formulae - in words - in symbols Arithmetic with Equations Graphs symbols Inequalities Identities Coordinates Understanding Graphs of equivalent functions form - linear General terms - Quadratic Sequences Graphical Manipulation solution of of symbols equations Factorising, Relationships changing the expressed subject… graphically Algebra in Years 7 to 9 includes equations, formulae and identities, and sequences, functions and graphs. You need to stress the links between these topics and with arithmetic. Letters do not represent quantities like length or cost; they represent numbers. Pupils will have spent much time manipulating numbers in Key Stages 1 and 2, and you can build on their experience. 11 For example, in response to the question: “6 x 8 = 48. What can you deduce from this?”, a year 6 class might suggest: 8 x 6 = 48 8 = 48/6 80 x 6 = 480 16 x 6 = 96 8 x 6 + 1 = 49 6 = 48/8 80 x 60 = 4800 and so on In KS3, this can be extended into the simple manipulation of equations, referring back regularly to the number examples. For example. ab = c implies: ba = c b = c/a a = c/b 2ab = 2c 2ab + 1 = 2c + 1 You can draw other examples from pupils’ experience of multiplication based on the distributive law. By encouraging pupils to record their mental calculations in algebraic form by making use of brackets, you can help them to generalise. For example, you can draw parallels between: 6 x 42 = 6(40 + 2) = 6 x 40 + 6 x 2 = 240 + 12 = 252 and: a(b + c) = ab + ac or between: 147 – 99 = 147 – (100 – 1) = 147 – 100 + 1 = 48 and: a – (b – c) = a – b + c EQUATIONS, FORMULAE AND IDENTITIES The initial approach to manipulating number statements needs to extend into a set of rules for solving equations, under a general heading: “do the same to both sides of the equation”. Pupils’ developing skills in solving equations will be dependent on their ability to add, subtract, multiply and divide directed numbers and to simplify expressions and collect like terms. Although to begin with you will frequently refer to number examples, in time pupils will appreciate that expressions and equations can be manipulated in their own right according to given rules and conventions. The generalised algebraic use of brackets, multiplying out a single term over a bracket, and the inverse process of taking out a common factor, can also develop from examples with numbers. Multiplying out a pair of brackets can be based on the “grid method” of multiplication and developed into an algebraic process. The most able pupils can attempt the reverse process of factorisation but for most pupils this will be an activity for KS4. 12 By the end of the Key Stage, pupils need to understand that a statement such as: (x + 1)(x – 1) = x² - 1 is an identity that is true for every value of x, unlike an equation, where the purpose is to find the value(s) of x that alone satisfy the equation. SEQUENCES, FUNCTIONS AND GRAPHS Many applications of algebra involve finding a formula that generates the general term of a sequence: for example, in predicting the number of matchsticks needed for a certain pattern, or the number of paving slabs for the border of a rectangular pond. It is important for pupils to justify their formulae from physical patterns, rather than merely from number sequences, since this allows them to “prove” their solutions, not just illustrate or verify them. The apparently different but equivalent formulae that arise from alternative ways of looking at the problem can help pupils to understand equivalent algebraic expressions. Functions and graphs can be taught and learned in tandem. At KS3 the main emphasis is on linear functions and their graphs. A graphical calculator, or graph plotting software, has an important role since it helps pupils to learn from exploring problems. FEATURES OF ALGEBRA IN KEY STAGE 3 To summarise, the distinctive features of algebra in Key Stage 3 are: Developing understanding that algebra is a way of generalising from arithmetic, from particular cases or from patters and sequences; Providing regular opportunities to construct algebraic expressions and formulae and to transform one expression into another – collecting like terms, taking out common factors, working with inverses, solving linear equations; Using opportunities to represent a problem and its solution in tabular, graphical or symbolic form, using a graphical calculator or a spreadsheet where appropriate, and to relate solutions to the context of the problem; Developing algebraic reasoning, including an appreciation that while a number pattern may suggest a general result, a proof is derived from the structure of the situation being considered. 13 SHAPE, SPACE AND MEASURES Geometry in Key Stage 3 is the study of points, lines and planes and the shapes that they can make, together with a study of plane transformations. A key aspect is the use and development of deductive reasoning in geometric contexts. Geometrical activities can be linked to accurate drawing, construction and loci, and work on measures. By ensuring that pupils have a range of suitable experiences you can develop their knowledge and understanding of shape and space and their appreciation of the ways that properties of shapes enrich our culture and environment. GEOMETRICAL REASONING Pupils can be aware of and use geometrical facts or properties that they have discovered intuitively from practical work before they can prove them analytically. The aim in KS3 is for pupils to use and develop their knowledge of shapes and space to support geometrical reasoning. For example, they need to appreciate that tearing the corners off a triangle and placing them side by side at best indicates that the angle sum of a triangle is approximately 180°, and that however many particular cases they can find of triangles with an angle sum of 180°, this does not prove the general case. In Key Stage, you can build on pupils’ experience and the practical demonstrations and explanations that have sufficed in Key Stages 1 and 2. Teach them to understand and use short chains of deductive reasoning and results about alternate and corresponding angles to reach a proof. Later, pupils should be able to explain why the angle sum of any quadrilateral is 360°, and to deduce formulae for the area of a parallelogram and of a triangle from the formula for that area of a rectangle. These chains of reasoning are essential steps towards the proofs that are introduced in KS4. APPRECIATION OF SHAPE AND SPACE Geometry cannot be learned successfully solely as a series of logical results. Pupils also need opportunities to use instruments accurately, draw shapes and appreciate how they can link together, for example, in tessellations. In Key Stage 3, it is vital to distinguish between the imprecision of constructions which involve protractors and rulers, and the “exactness in principle” of standard constructions which use only compasses and a straight edge. Geometrical reasoning can show pupils why construction methods work, for example, the method to construct a perpendicular bisector of a line segment. Practical work with transformations will produce interesting problems to solve as well as helping pupils to understand the topic more fully. Urge them to visualise solutions 14 to problems such as: “When a triangle is rotated through 180° about the mid-point of one side, what shape do the original and final triangles form?” Linking geometry to subjects such as art, through symmetry or tessellations, or religious education, perhaps through a study of the properties of Islamic patterns or cathedral rose windows, offers good opportunities to develop creativity. By encouraging pupils to speculate why the properties they have found hold true you can strengthen their reasoning skills. USE OF ICT ICT offers good opportunities to develop geometrical reasoning and an appreciation of shape and space. For example, pupils can use the programming language Logo to explore properties of plane shapes, such as the exterior angles of polygons. With dynamic geometry software, such as Geometers’ Sketchpad they can use rapid geometric drawing to explore a condition such as “one pair of opposite angles of a quadrilateral is equal”, and discover the special circumstances under which condition is true. More able pupils may be able to prove their conjectures analytically, but the formal use of congruent triangles is often needed, and for most pupils this will be tackled in Key Stage 4. MEASURES Pupils in KS3 need to develop their awareness of the relative sizes of units, converting between them, and using the rough equivalence of common imperial and metric units. Towards the end of the key stage, they need to become familiar with compound measures such as speed or density. Help them to appreciate the imprecision of measurement and to recognise the accuracy to which measurements can be stated. Draw as far as possible on their practical experience of measures in other subjects, particularly design and technology, science, geography and physical education. Work on perimeter, area and volume will extend to a range of shapes, including rectangles, parallelograms, circles, cuboids and prisms. A project such as “design a swimming pool” allows pupils to exercise imagination, practise calculations of length, area and volume, and experience working with larger numbers and units. The heart of work on measure will be with triangles which, at the end of the key stage, will extend to Pythagoras’ Theorem and similarity, leading on to trigonometry. As far as possible, the relevant formulae for calculating perimeters, areas and volumes should be explored and justified logically, not simply stated as facts. The use of formulae can be linked to work in algebra, and can be enhanced by the use of spreadsheets and graphical calculators. 15 FEATURES OF SHAPE, SPACE AND MEASURES IN KS3 To summarise, the distinctive features of shape, space and measures in KS3 are: Developing geometrical reasoning and construction skills, and an appreciation of logical deduction; Developing visualisation and sketching skills, including a dynamic approach to geometry, making use of ICT and other visual aids; Developing awareness of the degree of accuracy of measurements. HANDLING DATA Handling data is described by the cycle shown in the diagram. Evaluate results Specify the problem and plan Interpret and discuss Collect data from a data variety of sources Process and represent data Data handling is best taught in a coherent way in the context of real statistical enquiries so that teaching objectives arise naturally from the whole cycle. As an enquiry develops, you will need to reinforce and develop certain skills by direct teaching of particular objectives. It is easier to make sure that problems are relevant if at least some of the enquiries are linked to other subjects. For example, a question can be formulated and data collected in geography, with mathematics lessons concentrating on processing, representing and interpreting the data. The other subject can make further interpretations and consider the implications of what has been discovered. The supplement of examples in the Framework suggests a variety of possible projects, many of them linked directly to QCA’s schemes of work for other subjects, including science, geography and physical education, with further possibilities in history, religions education, languages and personal, social and health education. 16 As pupils move through KS3, the cross-curricular aspects of data handling become more important. It is usually best for a cross-curricular enquiry to be defined in the other subject, but good preparation is needed to check that the mathematical skills, techniques and representations that pupils need to learn next are likely to arise. In Year 7, much of the work may take place in mathematics lessons, with small sets of data that pupils can generate readily from simple experiments and easily accessible secondary sources. In Year 9, pupils should engage with large sets of real data from a much wider range of sources and contexts. The experience of working with real data in KS3 is an important preparation for controlled assessments in GCSE Statistics. PRIMARY AND SECONDARY SOURCES Give pupils experience of collecting and using primary data from, for example, questionnaires or results of an experiment, and secondary data from published sources, including reference materials, ICT databases and the Internet. Real data present problems that “textbook” or contrived data can skirt around, such as the accuracy of recording or how to deal with data that are ambiguous. The sizes of numbers can be problematic, either because they are large or, in the case of a pie chart, because they are not factors of 360. The time to process and represent real data is likely to be greater thatn with textbook examples but by using them pupils will have gained useful skills that can be transferred to other investigations. PROBABILITY Pupils will have met some of the language of probability in KS2 but will have had little experience of quantitative probability. Pupils need to understand that probability is a measure of what might happen. Help them to reason out what the probability is for simple cases by considering all the possible outcomes for particular events. The data handling cycle illustrated above applies also to probability. There are two aspects to develop in tandem through this cycle: probability described by the proportion of successes in an experiment, and probability derived from theoretical considerations. As they compare their practical and theoretical results, pupils will begin to sense that as the number of trials becomes very large the proportion of successes converges to the theoretical probability. FEATURES OF HANDLING DATA IN KEY STAGE 3 To summarise, the distinctive features of handling data in KS3 are: Basing work on purposeful enquiry, using situations of interest and relevance to pupils and making appropriate links to other subjects; 17 Placing an emphasis on making inferences from data, drawing on a range of secondary sources to ensure that samples are sufficiently large; Using ICT as a powerful source of data, and as a means of processing data and simulating situations. USING AND APPLYING MATHEMATICS AND THINKING SKILLS Thinking skills underpin using and applying mathematics and the broad strands of problem solving, communication and reasoning. Well-chosen mathematical activities will develop pupils’ thinking skills. For example, you might devote occasional stand-alone lessons to an investigation of a problem. Used well, this approach can focus pupils’ attention on the “using and applying” or thinking skills that they have used so that they can apply these skills more generally in their mathematics work. But the main approach to using and applying mathematics and thinking skills is to integrate them within everyday teaching, thereby helping pupils to make the connections in learning associated with success in mathematics. You can introduce problem solving, applications of mathematics and the use of reasoning and thinking skills at many points in a unit of work. A problem can serve as an introduction, to assess pupils’ prior knowledge or to set a context for the work; it can be used to provide motivation for acquiring a skill; or it can be set as a class activity or as homework towards the end of a topic, so that pupils use and apply the mathematics they have been taught. Pupils need to be able to select the mathematics required to solve a problem and recognise that an idea that they meet in one strand of mathematics can be applied to another. A good “diet” will include: Problems and applications that extend content beyond what has just been taught; Familiar and unfamiliar problems in a range of numerical, algebraic and graphical contexts, some with a unique solution and some with several possible solutions; Activities that develop short chains of deductive reasoning and concepts of proof in algebra and geometry; Occasional opportunities to sustain thinking by tackling more complex problems. There are five categories of thinking skills in the National Curriculum: information processing skills, enquiry skills, creative thinking skills, reasoning skills and evaluation skills. The contribution of mathematics to these skills is drawn directly from using and applying mathematics. 18 INFORMATION PROCESSING SKILLS These enable pupils to find and organise relevant information, to compare and contrast it, and to identify and analyse relationships. For example, pupils acquire information processing skills when they process the information in a 2D picture of a 3D object, or when they investigate what factors influence the distribution of grass and non-grass plants on the school field. They compare and contrast when, for example, they classify quadrilaterals, order a set of decimals, or consider two different representations of the same set of data. The National Curriculum for using and applying mathematics states that pupils should: Collect data from a variety of suitable sources, including experiments and surveys, and primary and secondary sources, including the Internet; Decide what statistical analyses are needed; Process and represent the data, using ICT as appropriate, turning raw data into usable information that gives insight into the problem or enquiry; Interpret data and draw conclusions from them. ENQUIRY SKILLS Enquiry lies at the heart of mathematics. Enquiry skills enable pupils to ask questions, define questions for enquiry, plan research, predict outcomes, anticipate consequences, and draw conclusions. Central to enquiry is an ability to see connections between different aspects of mathematics and thus open up further ways of tackling a problem. Pupils develop enquiry skills when they modify a problem they have solved to create a new problem: for example, when they find the smallest number with exactly three factors, then extend this to four factors, or other numbers of factors. They use enquiry skills when they find and eliminate alternatives: for example, when they explore how many different shapes can be made with five identical squares, and decide what to do with reflections or rotations. They practise enquiry skills in cross-curricular statistical surveys: for example, by examining possible indicators for economic development and deciding which of them are likely to highlight differences between countries most clearly. 19 The National Curriculum for using and applying mathematics requires pupils to: Pose problems and plan how to solve them; Predict outcomes; Decide what data or information to collect, the degree of precision or accuracy required, and the inferences they will be able to draw; Choose the appropriate mathematics and resources to use, including ICT; Interpret and discuss representations of data and results of analyses, looking for patterns and relationships, and explaining and justifying how they reached conclusions; Identify what further information is needed to pursue a supplementary enquiry. CREATIVE THINKING SKILLS These enable pupils to generate and develop ideas, to hypothesise, to apply imagination, and to seek innovative alternatives. For example, pupils need to think creatively when they visualise the path of a moving point or how a shape might change as slices are taken from it, when they look for pairs of numbers whose sum equals their product, or when they formulate algebraic expressions to make this addition square work: ? ? ? 3a + 2b a + 4b ? 4a + 3b 2a + 5b The National Curriculum for using and applying mathematics requires pupils to: Select and combine known facts and problem-solving strategies in creative ways to solve problems, using alternative approaches to overcome difficulties; Represent problems and solutions in numerical, algebraic, geometric or graphical form, moving from one to another to gain a different perspective on the problem and to identify similarities and differences; 20 Explore connections in their mathematical work; Explore, identify, and use pattern and symmetry in mathematical contexts; Visualise and use mathematical imagery; Conjecture, hypothesise, and ask questions such as” What is...?” or “Why?”; Investigate whether particular cases can be generalised further. REASONING SKILLS These enable pupils to give reasons for opinions and actions, to infer and deduce, to make informed judgements and decisions, and to use precise language to reason. Pupils apply reasoning skills when, for example, they investigate how to draw two straight lines from the vertex of a square to divide the square into three equal parts and then prove analytically that their construction holds, or when they look for examples of pairs of unit fractions whose sum is another unit fraction. They apply reasoning when they argue, for example, that it is necessary but not sufficient that a multiple of 4 is an even number, or that a square is a trapezium is not necessarily a square. The National curriculum for using and applying mathematics states that pupils should: Use step-by-step deduction and efficient techniques for solving a problem, including breaking down complex problems into simpler steps or a series of tasks, and working systematically; Present concise, reasoned arguments, explaining and justifying inferences, deductions and conclusions, using mathematical notation, symbols and diagrams correctly and consistently; Identify exceptional cases or counter-examples that do not accord with an argument and explain why; Distinguish between practical demonstration and proof, and between conventions, definitions and derived properties. EVALUATION SKILLS These enable pupils to develop and apply evaluation criteria and to judge the value of information and ideas. For example, pupils use evaluation skills when they compare mental methods of calculation to judge which is the most efficient, or when they evaluate whether the use of a calculator is appropriate. They use evaluation skills when they generate a number sequence in a practical context 21 and then consider the best way to describe and express the general term. They might evaluate the questions used in a statistical survey to compare attitudes to fairly traded goods and decide that the questions used to explore attitudes are crude compared with the complexity of personal beliefs. The National curriculum for using and applying mathematics requires pupils to: Review progress as they work; Move from one form of representation to another to gain different perspectives on a mathematical problem; Evaluate the effectiveness of their chosen methods, techniques and problem-solving strategies, and the resources they have chosen to use. Check and evaluate their solutions, including the accuracy of their calculations, analyses and results; Consider anomalies in data or measurements and try to explain them; Examine critically, improve, then justify their choices of mathematical presentation. OUR APPROACH TO TEACHING This section provides a simple checklist of what teachers need to be aware of and what pupils need to be encouraged to do in all lessons where there is a mathematical element to the work. As teachers we strive to: Be aware of the range of mathematical attainment that pupils bring to our lessons. (Note: Just because they are in, for example, the top set for your subject doesn’t mean that they will be in the top set for mathematics. The list of key objectives, see below, will support this). Build pupils’ confidence when they are struggling with a calculation. (Note: Encourage the pupils to make sense of the numbers involved, to take a step back and consider what is needed to answer the question before diving in with a set method. Encourage short cuts based on mental strategies if possible). Encourage pupils to understand the methods that they are using. (Note: Be wary of teaching tricks which have no meaning for the pupil. For example, rules like “to multiply by 10, add a nought” or when transforming formulae, “change the side, change the sign” can be very helpful if used by 22 pupils who understand the mathematics behind these mnemonics. However, if they don’t relate to any understanding on the pupils’ part they will be half remembered and mis-used). Use mathematical language accurately and consistently within the department and across the school. (Note: The exclusive use of “times” for multiply and “take-away” for subtraction can subtly limit pupils’ understanding of important concepts. Also, if pupils encounter different words for the same ideas in different lessons this will cause confusion). Value pupils’ different methods for calculation and regularly ask “How did you do that?” and “Did anyone do that a different way?” (Note: It is important to encourage pupils to see calculation as an act of problem-solving, something that they think about and make their own decisions about regarding the most appropriate and efficient method for them to use, rather than blindly rushing to a set pencil and paper procedure). We encourage pupils to: See mental calculation as the first resort when faced with any calculation. (Note: It is uncommon for pupils to reach for the calculator or a rote learnt pencil and paper procedure for calculation that they could, with a little thought, do in their head. If all teachers are working together to discourage this whenever it arises in lessons; this will do much to encourage pupils’ numeracy skills). Explain any calculation they have done by showing all their working out. Encourage the idea that “show your working” does not necessarily mean the standard pencil and paper method. It could mean any writing or other jotting which genuinely describes how they have arrived at the answer. Estimate before a calculation is done whenever possible. Consider the reasonableness of their answers after a calculation has been done. Know how to use all the relevant buttons on their calculator efficiently and effectively when it is appropriate and to be able to interpret the display sensibly. Use appropriate mathematical language confidently. (Note: In the same way that we want pupils to “show their working” in their writing, we want them to “show their thinking” in their talk. Teachers need to create regular opportunities for pupils to explain how they make calculations that arise in their lessons in order to encourage this use of language). 23 KEY OBJECTIVES FOR DIFFERENT ABILITIES This section shows what a below average, average and above-average pupils should be able to cope with in Years 7, 8 and 9. It should be pointed out that the Years 5 and 6 objectives are included since some of Years 7, 8 and 9 may still be working at this level. This will enable colleagues in other departments to make sure that they are not making undue mathematical demands of pupils in their own subject. Year 5 Multiply and divide any positive integer up to 10000 by 10 or 100 and understand the effect. Order a given set of positive and negative integers. Use decimal notation for tenths and hundredths. Round a number with one or two decimal places to the nearest integer. Relate fractions to division and to their decimal representations. Calculate mentally a difference such as 8006 – 2993. Carry out column addition and subtraction of positive numbers less than 10000. Know by heart all multiplication facts up to 10 x 10. Carry out short multiplication and division of a three-digit by a single-digit integer. Carry out long multiplication of a two-digit by a two-digit integer. Understand area measured in square centimetres (cm²); understand and use the formula in words “length x breadth” for the area of a rectangle. Recognise parallel and perpendicular lines, and properties of rectangles. Use all four operations to solve simple word problems involving numbers and quantities, including time, explaining methods and reasoning. Year 6 Multiply and divide decimals mentally by 10 or 100, and integers by 1000, and explain the effect. Order a mixed set of numbers with up to three decimal places. Reduce a fraction to its simplest form by cancelling common factors. Use a fraction as an operator to find fraction of numbers or quantities (e.g. 5/8 of 32, 7/10 of 40, 9/100 of 400 centimetres). Understand percentage as the number of parts in every 100, and find simple percentages of small whole-number quantities. Solve simple problems involving ratio and proportion. Carry out column addition and subtraction of numbers involving decimals. Derive quickly division facts corresponding to multiplication tables up to 10 x 10. 24 Carry out short multiplication and division of numbers involving decimals. Carry out long multiplication of a three-digit by a two-digit integer. Use a protractor to measure acute and obtuse angles to the nearest degree. Calculate the perimeter and area of simple compound shapes that can be split into rectangles. Read and plot coordinates in all four quadrants. Identify and use the appropriate operations (including combinations of operations) to solve word problems involving numbers and quantities, and explain methods and reasoning. Solve a problem by extracting and interpreting information presented in tables, graphs and charts. Year 7 Simplify fractions by cancelling all common factors; identify equivalent fractions. Recognise the equivalence of percentages, fractions and decimals. Extend mental methods of calculation to include decimals, fractions and percentages. Multiply and divide three-digit by two-digit whole numbers; extend to multiplying and dividing decimals with one or two places by single-digit whole numbers. Break a complex calculation into simpler steps, choosing and using appropriate and efficient operations and methods. Check a result by considering whether it is of the right order of magnitude. Use letter symbols to represent unknown numbers or variables. Know and use the order of operations and understand that algebraic operations follow the same conventions and order as arithmetic operations. Plot the graphs of simple-linear functions. Identify parallel and perpendicular lines; know the sum of angles at a point, on a straight line and in a triangle. Convert one metric unit to another (e.g. grams to kilograms); read and interpret scales on a range of measuring instruments. Compare two simple distributions using the range and one of the mode, median or mean. Understand and use the probability scale from 0 to 1; find and justify probabilities based on equally likely outcomes in simple contexts. Solve word problems and investigate in a range of contexts, explaining and justifying methods and conclusions. Year 8 Add, subtract, multiply and divide integers Use the equivalence of fractions, decimals and percentages to compare proportions; calculate percentages and find the outcome of a given percentage increase or decrease. Divide a quantity into two or more parts in a given ratio; use the unitary method to solve simple word problems involving ratio and direct proportion. 25 Use standard column procedures for multiplication and division of integers and decimals, including by decimals such as 0.6 or 0.06; understand where to position the decimal point by considering equivalent calculations. Simplify or transform linear expressions by collecting like terms; multiply a single term over a bracket. Substitute integers into simple formulae. Plot the graphs of linear functions, where y is given explicitly in terms of x; recognise that equations of the form y = mx + c correspond to straight-line graphs. Identify alternate and corresponding angles; understand a proof that the sum of the angles of a triangle is 180° and of a quadrilateral is 360°. Enlarge 2D shapes, given a centre of enlargement and a positive whole- number scale factor. Use straight edge and compasses to do standard constructions. Deduce and use formulae for the area of a triangle and parallelogram, and the volume of a cuboid; calculate volumes and surface areas of cuboids. Construct, on paper and using ICT, a range of graphs and charts; identify which are most useful in the context of a problem. Find and record all possible mutually exclusive outcomes for single events and two successive events in a systematic way. Identify the necessary information to solve a problem; represent problems and interpret solutions in algebraic, geometric or graphical form. Use logical argument to establish the truth of a statement. Year 9 Add, subtract, multiply and divide fractions. Use proportional reasoning to solve a problem, choosing the correct numbers to take as 100%, or as a whole. Make and justify estimates and approximations of calculations. Construct and solve linear equations with integer coefficients, using an appropriate method. Generate terms of a sequence using term-to-term and position-to-term definitions of the sequence, on paper and using ICT; write an expression to describe the nth term of an arithmetic sequence. Given values for m and c, find the gradient of lines given by equations of the form y = mx + c. Construct functions arising from real-life problems and plot their corresponding graphs; interpret graphs arising from real situations. Solve geometrical problems using properties of angles, of parallel and intersecting lines, and of triangles and other polygons. Knowing that translations, rotations and reflections preserve length and angle and map objects on to congruent images. Know and use the formulae for the circumference and area of a circle. Design a survey or experiment to capture the necessary data from one or more sources; determine the sample size and degree of accuracy needed; design, trial and if necessary refine data collection sheets. 26 Communicate interpretations and results of a statistical enquiry using selected tables, graphs and diagrams in support. Know that the sum of probabilities of all mutually exclusive outcomes is 1 and use this when solving problems. Solve substantial problems by breaking them into simpler tasks, using a range of efficient techniques, methods and resources, including ICT; give solutions to an appropriate degree of accuracy. Present a concise, reasoned argument, using symbols, diagrams, graphs and related explanatory text. YEAR 9 OBJECTIVES FOR ABLE PUPILS Know and use the index laws for multiplication and division of positive integer powers. Understand and use proportionality and calculate the result of any proportional change using multiplicative methods. Square a linear expression and expand the product of two linear expressions of the form x ± n; establish identities. Solve a pair of simultaneous linear equations by eliminating one variable; link a graphical representation of an equation or a pair of equations to the algebraic solution. Change the subject of a formula. Know that if two 2D shapes are similar, corresponding angles are equal and corresponding angles are in the same ratio. Understand and apply Pythagoras’ Theorem. Know from experience of constructing them that triangles given SSS, SAS, ASA or RHS are unique, but that triangles given SSA or AAA are not; apply these conditions to establish the congruence of triangles. Use measures of speed and other compound measures to solve problems. Identify possible sources of bias in a statistical enquiry and plan how to minimise it. Examine critically the results of a statistical enquiry and justify choice of statistical representation in written presentations. Generate fuller solutions to mathematical problems. Recognise limitations on the accuracy of data and measurements. KEY MATHEMATICAL VOCABULARY This section contains lists of keywords for each of Years 7, 8 and 9, which pupils should know and be encouraged to use in lessons. For definitions of the words you will need to refer to a mathematical dictionary or to the National Curriculum glossary. The glossary can be found on the QCA website at www.qca.org.uk. Year 7 This list contains the key words used in the Year 7 teaching programme and supplement of examples. Some words will be familiar to pupils in Year 7 from earlier work. 27 APPLYING MATHEMATICS AND SOLVING PROBLEMS Answer Evidence Explain Explore Investigate method Problem Reason Results Solution (of a problem) Solve True, False NUMBERS AND THE NUMBER SYSTEM PLACE VALUE, ORDERING AND ROUNDING Approximate Approximately equal to (≈) Between Compare Decimal number Decimal place Digit Equals (=) Greater than (>), less than (<) Greatest value, least value Most/least significant digit Nearest Order Place value Round Tenth, hundredth, thousandth To one decimal place (to 1 d.p.) Value Zero place holder Integers, powers and roots Classify Common factor Consecutive Divisible, divisibility Divisor Factor Factorise Highest common factor (HCF) Integer Lowest common multiple (LCM) Multiple Negative (e.g. – 6) Plus, minus Positive (e.g. + 6) Prime Prime factor Property Sign Square number, squared Square root Fractions, decimals, percentages, ratio and proportion Cancel, cancellation Convert Decimal fraction Equivalent Fraction Lowest terms Mixed number Numerator, denominator Percentage (%) Proper/improper fraction Proportion Ratio, including notation 3 :2 Simplest form 28 Calculations Add, addition Amount Brackets Calculate, calculation Calculator: clear, display, enter Key, memory Change (money) Commutative Complements (in 10, 100) Currency Difference Discount Divide, division Double, halve Estimate Exact, exactly Exchange rate Factor Increase, decrease Inverse Multiply, multiplication Nearly Operation Order of operations Partition Product Quotient Remainder Rough, roughly Sale price Sign Subtract, subtraction Sum Total Algebra Equations, formulae and identities Algebra Brackets Commutative Equals (=) Equation Expression Evaluate Prove Simplify, simplest form Solution (of an equation) Solve (an equation) Squared Substitute Symbol Term Unknown Value Variable Verify Sequences, functions and graphs Axis, axes Consecutive Continue Coordinate pair Coordinate point Coordinates Equations (of a graph) Finite, infinite Function Function machine 29 Generate Graph Increase, decrease Input, output Mapping n th term Origin Predict Relationship, rule Sequence Straight-line graph Term X-axis, y-axis X-coordinate, y-coordinate Shape, space and measures Geometrical reasoning: lines, angles and shapes Adjacent (side) Angle: acute, obtuse, right, reflex Angles at a point Angles on a straight line Base (of plane shape or solid) Base angles Centre Circle Concave, convex Degree (°) Diagonal Diagram Edge (of solid) Equal (sides, angles) Face Horizontal, vertical Identical (shapes) Intersect, intersection Line, line segment Opposite (sides, angles) Parallel Perpendicular Plane Point Polygon: pentagon, hexagon Octagon Quadrilateral: arrowhead, delta Kite, parallelogram, rectangle Rhombus, square, trapezium Regular, irregular Shape Side (of 2D shape) Solid (3D) shape: cube, cuboid Cylinder, hemisphere, prism Pyramid, square-based pyramid Sphere, Tetrahedron Three-dimensional (3D) Triangle: equilateral, isosceles Scalene, right-angled Two-dimensional (2D) Vertex, vertices Vertically opposite angles Transformations Axis of symmetry Centre of rotation Congruent Line of symmetry Line symmetry Mirror line Object, image Order of rotational symmetry Reflect, reflection Reflection symmetry Rotate, rotation Rotational symmetry Symmetrical Transformation Translate, Translation Coordinates Axis, axes Coordinates 30 Direction Grid Intersecting, intersection Origin Position Quadrant Row, column X-axis, y-axis X-coordinate, y-coordinate Construction and loci Construct Draw Measure Net Perpendicular Protractor (angle measurer) Ruler Set square Sketch Measures Area: mm², cm², m², km² Capacity: millilitre, centilitre, litre Pint, gallon Length: millimetre, centimetre, metre, km Mile Mass: gram, kilogram, ounce Pound Time: second, minute, hour, day Week, month, year, decade Century, millennium Temperature: °C, °F Depth Distance Height, high Perimeter Surface, surface area Width Handling data Average Bar chart Bar-line graph Class interval Data, grouped data Data collection sheet Database Experiment Frequency Frequency chart Frequency diagram Interpret Interval Label Mean Median Mode, modal class/group Pie chart Questionnaire Range Represent Statistics Survey Table Tally Title Probability Certain, uncertain Chance, no chance, good chance Poor chance, 50-50 Even chance Dice Doubt 31 Equally likely Fair, unfair Likelihood Likely, unlikely Outcome Possible, impossible Probability Probability scale Probable Random Risk Spin, spinner Year 8 This list contains the new key words introduced in the Year 8 teaching programme and supplement of examples. Words from earlier years are also used. Applying mathematics and solving problems Conclude, conclusion Counter-example Deduce Exceptional case Justify Prove, proof Numbers and the number system Place value, ordering and rounding Ascending, descending Billion Index Power Integers, powers and roots Cube, cube number Cube root (e.g. 3√8) Cubed (e.g. 2³) Prime factor decomposition To the power of n (e.g. 6^4) Fractions, decimals, percentages, ratio and proportion Direct proportion Recurring decimal Terminating decimal Unit fraction Unitary method Calculations Associative Best estimate Degree of accuracy Distributive Interest Profit, loss 32 Service charge Sign change key Tax Value added tax (VAT) Algebra Equations, formulae and identities Algebraic expression Collect like terms Formula, formulae Linear equation Linear expression Multiply out (expressions) Prove, proof Transform Verify Sequences, functions and graphs Arithmetic sequence Difference pattern Flow chart General term Gradient Intercept Linear function Linear relationship Linear sequence Notation T(n) Slope Steepness Shape, space and measures Geometrical reasoning: lines, angles and shapes Alternate angles Bisect, bisector Complementary angles Congruent, congruence Corresponding angles Elevation Equidistant Exterior angle Heptagon Interior angle Isometric Mid-point Plan view Prove, proof Supplementary angles Tessellate, tessellation Triangular prism View Transformations Centre of enlargement Enlarge, enlargement Map Plan Scale, scale factor Scale drawing Constructions and loci Compasses Construction lines Locus, loci Perpendicular bisector 33 Straight edge Measures Bearing, three-figure bearing Displacement Foot, yard Hectare Tonne Volume: mm³, cm³, m³ Handling data Continuous Data log Discrete Distance-time graph Distribution Interrogate Line graph Population pyramid Primary source Sample Scatter graph Secondary source Stem-and-leaf diagram Two-way table Probability Biased Event Experimental probability Sample Sample space Theoretical probability Theory Year 9 This list contains the new key words introduced in the Year 9 teaching programme and supplement of examples. Words from earlier years are also used. Applying mathematics and solving problems Generalise Trial and improvement Numbers and the number system Place value, ordering and rounding Exponent Greater that or equal to (≥) Less than or equal to (≤) Significant figures Standard (index) form Upper bound, lower bound Integers, powers and roots Index, indices Index law 34 Index notation Fractions, decimals, percentages, ratio and proportion Proportional to Proportionality Calculations Compound interest Constant Cost price, selling price Reciprocal Algebra Equations, formulae and identities And, or Common factor Cubic equation Cubic expression Expand the product (of two linear expressions) Factorise Identically equal to Identity Index law Inequality Quadratic equation Quadratic expression Region Simultaneous equations Subject of the formula Take out common factors Sequences, functions and graphs Cubic function Curve First/second difference Identity function Inverse function Inverse mapping Quadratic function Quadratic sequence Maximum/ minimum point Maximum/ minimum value Self-inverse Shape, space and measures Geometrical reasoning: lines, angles and shapes Arc Centre (of circle) Chord Circumference Convention Cross-section Definition Derived property Diameter Hypotenuse Pi (Π) Plane Projection Pythagoras’ Theorem Similar, similarity Region Radius Section Sector Segment 35 Tangent (to a curve) Transformations Axis of rotational symmetry Plane symmetry Plane of symmetry Construction and loci Circumcentre Circumcircle Circumscribed Inscribed Measures Adjacent, opposite, hypotenuse Angle of depression Angle of elevation Density Pressure Sine (sin), cosine (cos), tangent (tan) Speed: miles per hour, metres per second Handling data Bias Census Cumulative frequency Estimate of the mean/median Interquartile range Line of best fit Quartiles Raw data Representative (sample) Probability Exhaustive Independent Limit Mutually exclusive Notation: P(n) for probability of event “n” Relative frequency Tree diagram Mathematical competencies required in other subjects 36 The link between Numeracy and Literacy The role of language is important in numeracy and there will be regular contact between the Numeracy and Literacy Co-ordinators to ensure that both are aware of developments in their respective areas. Language is an important tool for learning mathematics. Explaining to oneself, or someone else “putting it into words”, can be a powerful means of working through and clarifying ideas. Pupils should use language as a tool for reflecting on their mathematical experiences and hence for their own mathematical learning. Pupils also need to develop the skills of recording their mathematics. The first forms of recording are likely to be in everyday language or in pictures or diagrams. Gradually these representations may be shortened, leading to the need to use symbols. Pupils should develop spatial language in much the same way as they learn to talk about various animals and objects – by hearing it used appropriately by others and being encourages to use progressively more sophisticated language in describing their experiences. Links to other subjects English English lessons can help to develop and support pupils’ numeracy skills, for example, by use of mathematical vocabulary and technical terms, by asking children to read and interpret problems to identify the mathematical content, and by encouraging them to explain, argue and present their conclusions to others. Pupils use statistical analysis in the study of advertising and of the media. They learn to distinguish between fact and opinion by looking at differences that are backed up by data. Sequencing plays an important part in placing events in order when compiling a writing frame. Use of metre, syllables and rhythm in poetry and Shakespeare. 37 Science Almost every scientific investigation or experiment is likely to require one or more of the mathematical skills of classifying, counting, measuring, calculating, estimating and recording in tables and graphs. In science pupils will, for example, order numbers, including decimals, calculate simple means and percentages, use negative numbers when taking temperatures and relate these to boiling and melting points, collect, display and interpret data, decide whether it is more appropriate to use a line graph or bar chart, and plot, interpret and predict from graphs. They need to understand the scale on the axes of a graph, draw and interpret lines of best fit, use and manipulate formulae, convert units for example change minutes into seconds, use the correct units for measuring length, area, volume, mass, charge, voltage, time etc. Pupils will meet the concept of variation, squares and square roots, compound measures such as speed and density. They will need to know how to use a calculator properly and will use standard form for very large and very small numbers. Speed = distance ÷ time Force = mass x acceleration Acceleration = change in velocity ÷ time taken Density = mass ÷ volume Work done = force x distance moved in direction of force Energy transferred = work done KE = ½ mass x speed² (KE = ½ mv²) Change in potential energy = mass x gravitational field strength x change in height Weight = mass x gravitational field strength Pressure = force ÷ area Moment = force x perpendicular distance from pivot Charge = current x time Voltage = current x resistance Electrical power = voltage x current Wave speed = frequency x wavelength 38 Art and Technology Measurements are often needed in Art and Technology. Many patterns and constructions are based on spatial ideas and properties of shapes, including symmetry and tessellations. Pupils use the concepts of perspective in landscape drawing and proportion in various topics. A lot of work is also undertaken using estimation of measurements and quantities. Design may need enlarging or educing, introducing ideas of multiplication and ratio. Trigonometry and Pythagoras’ Theorem may be used in the calculation of forces, moments and stress and strain and Pi is used when there are calculations involving circles. Equations may need to be rearranged and a calculator has to be used properly. The understanding of angles and measurements is used in drawing and nets arise in surface development. When dealing with recipes and cooking pupils will carry out a great deal of measurement calculations that include working out times and calculating cost. Pupils may need to change the units of measurement from one unit to another. Surveys are carried out and questionnaire design results in the analysis of data. Pupils draw graphs and need to be able to interpret them. History, Geography and Religious Education In History and Geography, pupils will collect data by counting and measuring and make use of measurements of many kinds. In Geography pupils statistical analysis includes pie charts, bar and line graphs, Spearman’s Rank Correlation Co-efficient (beyond the scope of GCSE mathematics) and mean chloropleth maps. They look at patterns of numbers and interpret their significance, which uses the concept of proportion. The study of maps includes the use of co-ordinates and ideas of angle, direction, position, scale and ratio. In History, the study of maps and distances includes the effect on people and events. In particular they look at Roman miles when considering distance to put events into perspective. Pupils look at the percentage of people surviving a battle and the percentage of trade across the Roman Empire. Roman numerals are studied in History and pupils examine how shape affects the strength of a castle. The pattern of the days of the week, the calendar and recurring annual festivals all have a mathematical basis and pupils look at the lunar calendar and Ramadan in RE. Historical ideas require understanding of the passage of time, which can be illustrated on a time line, similar to the number line that they already know. Symmetry is important in RE, especially when looking at Islamic design and pupils consider the importance of symbols such as the Star of David. Pupils use measurements when they study Noah’s Ark and they interpret statistics and graphs in the topic of abortion. They look at mathematics in the workplace and study poverty and wealth, using payment and economics in speculating, for example world wide depression. ICT Pupils will apply and use mathematics in a variety of ways when they solve problems using ICT. For example, they will collect and classify data, enter it into data handling software, produce graphs and tables, and interpret and explain their results. Pupils enter formulae into spreadsheets using mathematical operators, but 39 substitute * for x and type = at the beginning of the formula. Their work in control includes the measurement of distance and angle, using uniform non-standard then standard measures. When they use computer models and simulations they will draw on their abilities to manipulate numbers and identify patterns and relationships. Modern Foreign Languages Pupils study foreign currency and numbers within a country. They perform simple addition, subtraction and multiplication. They tell the time in a foreign language and have to be able to use analogue and 24 hour notation. Percentages occur when they analyse surveys and graphs and the terms “more than”, “less than” and “the most popular answer was…” are used. Physical Education and Music Athletic activities require measurement of height, distance, angle and time, while ideas of counting, time, symmetry, movement, position and direction are used extensively in music, dance, gymnastics and ball games. 40 Appendix This appendix contains different approaches to calculation that teachers may be unfamiliar with. Often pupils will partition numbers to make the numbers easier to handle. The following are examples where the numbers have been partitioned into hundreds, tens and units. 345 + 47 = (300 + 40 + 5) + (40 + 7) = 300 + 80 + 12 = 392 53 – 27 = 53 – 20 – 7 = 33 – 7 = 26 Half of 238 = half of (200 + 30 + 8) = 100 + 15 + 4 = 119 This is the grid method for multiplication: 242 x 3 x 200 40 2 3 600 120 6 = 726 36 x 48 x 30 6 40 1200 240 = 1440 + 8 240 48 = 288 1728 The following are examples of counting on in multiples of 100, 10 or 1 (counting on from the smaller number to the larger number). 86 + 57 = 86 + 50 + 7 = 136 + 7 41 +50 +4 +3 86 136 140 143 84 – 56 This problem is changed to “How much do i add to get from 56 to 84?” 56 + 4 + 20 + 4 = 84 +4 +20 +4 56 60 80 84 Change from £5 after spending £2.47 Count on from £2.47: 3p + 50p + £2 = £2.53 Compensation 34 + 19 = 34 + 20 – 1 = 54 – 1 = 53 98 + 32 = 100 + 32 – 2 = 132 – 2 = 130 56 – 38 = 56 – 40 + 2 = 18 £3.98 x 5 = (£4 x 5) – (2p x 5) = £20 – 10p = £19.90 13 x 21 = (13 x 20) + 13 = 260 + 13 = 273 Chunking Considering division as repeated subtraction. Estimation skills are needed for division in order for students to make a sensible choice of multiple to subtract. 72 ÷ 5 72 - 50 (10 X 5) 22 - 20 (4 X 5) 2 Answer: 14 remainder 2 42 There is a problem that pupils encounter with this method and that iis the subtraction, hence the following is preferred: In the following 72 ÷ 5 the 72 is seen as the target number and it is made up by adding multiples of 5. 50 10 x 5 20 4x5 70 14 x 5 We need 2 to make 72, thus the answer is 14 remainder 2. The standard method for 72 ÷ 5 is: 14 remainder 2 5 72 Napier’s bones or Gelosia Napier’s bones method for multiplying three digits by two digits: 3 2 7 1 0 2 2 8 8 4 2 1 5 4 6 6 8 1 5 6 9 6 327 x 48 = 15626 Explanation: The numbers outside the grid are multiplied and the answer goes inside one of the cells with the tens digit above and the units digit below. 7 2 8 4 This cell shows how 7 x 4 = 28 is entered. When all of the cells have been filled, the numbers are then added along the diagonals, starting from the right hand side. The first number is 6, since it is the only number in that diagonal. To make the answer in the next diagonal we do 6 + 5 + 1 = 12. 2 is written down and the 1 from the tens is carried to the next diagonal. 43 This method is continued until the end. The standard method for 327 x 48 is long multiplication: 327 x48 2616 (327 x 8) 13080 (327 x 40) 15696 Mathematical Association Audit Reproduced here is the Maths Association audit. It has been modified to fit with the Key Stage 3 Curriculum – adaptations could be made for KS4. Mathematical Skills N.C. NNS Year Year 7 Year 8 Year 9 Other Level Number and Algebra Understand place value and order numbers up to 100 2 2 Know when to add or subtract when solving problems 2 2, 3, 4 Recognise odd and even number 2 2 Read and write numbers in figures and words, up to a million 4, 5 Round numbers to the nearest 10, 100 or 1000 3 3, 4, 5 Use <, >, <, > and = symbols 4, 5 Mental addition and subtraction of 2 digit numbers 3 4 Written addition and subtraction of three digit numbers 3 4 Multiplication tables – 2, 3, 4, 5 and 10 and associated division 3 4 facts Use simple fractions and recognise when two simple 3,4 4 44 fractions are equivalent Use decimal notation – money 3 4 Recognise and use negative 3 4 numbers in context Multiplication tables up to 10 X 10 4 5 and associated division facts Multiply and divide whole 4 4,5,6 numbers by 10 and 100 N.C. NNS Year Year 7 Year 8 Year 9 Other Level Written addition and subtraction 4 5 of whole numbers Short multiplication and division of 4 5 whole numbers Add and subtract decimals to 4 5,6 two decimal places Order decimals to three decimal 4 6 places Recognise and use simple 4 5 percentages Use simple formulae expressed in 4 4,5,6 words Use co-ordinates in the first 4 4,5 quadrant Multiply and divide decimals by 5 10, 100, 1000 Order, add and subtract negative 5 5,6 numbers in context +, -, X, ÷ with up to 2 decimal 5 5, 6 places Calculate fractions of quantities 5 5, 6 Use simple ratio and proportion 5 5, 6 Long multiplication 5 5, 6 Long division 5 45 Check answers using inverse 5 4, 5, 6 operations Estimate and check answers using 5 4, 5, 6 approximations Construct simple formulae 5 5, 6 Use co-ordinates in four 5 6 quadrants Calculate a number as a fraction 6 4, 5, 6 of another N.C. NNS Year Year 7 Year 8 Year 9 Other Level Calculate a number as a 6 6 percentage of another Understand and use fraction, 6 5, 6 decimal and percentage equivalence Add and subtract fractions 6 7 Calculate using ratios 6 7 Solve linear equations 6 7 Use co-ordinates for 6 7 geographical representation Calculate percentages of 5 5, 6 quantities Use simple formulae 5 5, 6 Use significant figures 7 9 Multiply and divide fractions and 7 9 decimals Use a calculator efficiently and 7 9 appropriately Use proportional change 7 9 Solve simultaneous linear 7 9 equations graphically Solve simultaneous linear 7 9 equations algebraically 46 Solve simple inequalities 7 9 Calculate power and roots 8 Use standard form 8 Use formulae involving fractions, 8 decimals or negative numbers Calculate the original quantity 8 given the result of proportional change Solve problems involving 8 repeated proportional change N.C. NNS Year Year 7 Year 8 Year 9 Other Level Interpret graphs modelling real life 8 situations Draw graphs modelling real life 8 situations Transform formulae 8 Solve inequalities in two variables 8 Determine the bounds of intervals EP Find formulae that approximately EP connect data and express general laws in symbolic form Use direct proportion EP Use indirect proportion EP Use the rules of indices EP Solve problems using intersections EP or gradients of graphs 47 Mathematical Skills N.C. Level NNS Year Year 7 Year 8 Year 9 Other Shape, Space and Measures Mathematical names for 2-D 2 4, 5, 6 and 3-D shapes Understand angle as a 2 4, 5, 6 measure of turn Recognise right angles 2 4 Understand reflective 3 4, 5, 6 symmetry Use metric units of length, 3 4, 5, 6 capacity, mass and time Solve problems involving 4, 5 time or timetables Make simple 3-D models 4 4, 5, 6 from nets Draw 2-D shapes in different 4 orientations Understand rotational 4 4, 5, 6 symmetry Reflect simple shapes in a 4 4, 5, 6 mirror line Measure and read scales 4 4, 5, 6 using appropriate units and accuracy Find perimeters of simple 4 4, 5, 6 shapes Find areas by counting 4 4, 5 squares Measure and draw angles 5 5, 6 Know the angle sum of a 5 6 triangle Know the sum of angles at a 5 6 point Identify all the symmetries of 5 4, 5, 6 2-D shapes Convert one metric unit to 5 5, 6 48 another Know rough metric/imperial 5 6 equivalence of common units Estimate measures 5 4, 5, 6 N.C. Level NNS Year Year 7 Year 8 Year 9 Other Know and use the formula 5 5 for the area of a rectangle Draw and interpret simple scale drawings Recognise 2-D 6 4, 5, 6 representations of 3-D shapes Know and use properties of 6 7 quadrilaterals Use angle and symmetry 6 7 properties of polygons Use angle properties of 6 8 intersecting and parallel lines Devise instructions for a 6 7,8,9 computer to generate and transform shapes Use the formula for the 6 9 circumference of a circle Use the formula for the area 6 9 of a circle Find the areas of plane 6 7 rectilinear figures Use the formulae for the 6 8 volume of a cuboid Enlarge shapes by a positive 6 8 whole number scale factor Use Pythagoras' Theorem in 7 9 2D Calculate lengths and areas 7 in plane shapes Calculate volumes of prisms 7 9 49 Enlarge shapes by a 9 fractional scale factor Determine the locus of a 7 8,9 moving object Understand the limitations of 7 accuracy of measurements Other Year 9 N.C. Level NNS Year Year 7 Year 8 Understand and use 7 compound measures Understand and use similarity 8 and congruence Use trigonometry in 2-D 8 Distinguish between EP formulae for perimeter, area and volume by considering dimensions Use Pythagoras' Theorem in EP 3D Use trigonometry in 3-D EP Calculate lengths of circular EP arcs Calculate areas of sectors EP Calculate surface area of EP Cylinders Calculate volume of cones EP and spheres 50 Other Mathematical Skills N.C. Level NNS Year Year 7 Year 8 Year 9 Handling data Sort and classify objects by 2 4 more than one criterion Record results in simple lists, 2 2 tables and block graphs Interpret simple tables and 3 3 lists Interpret pictograms 3 3, 4 Draw pictograms 3 3, 4 Interpret bar graphs 3 4, 5, 6 Draw bar graphs 3 4, 5, 6 Collect data and record 4 4, 5, 6 them using frequency tables Understand and use the 4 4, 5 mode, the median and the range of a set of data Group collected data into 4 equal class intervals Draw frequency diagrams 4 6 using grouped data Interpret line graphs 4 5, 6 Select and use appropriate scales for axes Draw line graphs 4 5, 6 Understand and use the 5 6 mean of a set of data Use averages and ranges to 5 7 compare two sets of data Interpret pie charts 5 6 Understand and use the 5 7 probability scale from 0 to 1 Find probabilities using 5 7 equally likely outcomes or experiment N.C. Level NNS Year Year 7 Year 8 Year 9 Other Create frequency tables with 6 8 equal class intervals to record continuous data Interpret frequency diagrams 6 8 Draw frequency diagrams 6 8 Draw pie charts 6 8 Draw scatter diagrams 6 8 Understand simple 6 9 correlation Use two-way tables to record 6 8 all the possible outcomes of two events Use the fact that the total 6 8 probability of all mutually exclusive outcomes of an experiment is 1 Specify and test hypotheses 7 9 using appropriate methods and taking account of variability and bias Find modal class of grouped 7 9 data Estimate the mean, median 7 9 and range of grouped data Use averages and ranges 7 9 and frequency polygons to compare two sets of data Draw a line of best fit on a 7 9 scatter diagram Use relative frequency to 7 9 estimate probability Interpret cumulative 8 frequency tables and diagrams Construct cumulative 8 frequency tables and diagrams 52 Estimate the median, 8 quartiles and inter-quartile range from a cumulative frequency diagram Interpret histograms with EP unequal class intervals N.C. Level NNS Year Year 7 Year 8 Year 9 Other Understand and use EP sampling Draw histograms with EP unequal class intervals Use Spearman's coefficient EP of correlation 53