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Key Stage 3 National Strategy Foundations of algebra © Crown copyright 2004 Developing algebra in Key Stage 3: overall objectives • To explore the teaching of constructing and solving linear equations by: – considering the rationale for teaching algebra – developing a range of effective teaching approaches – reflecting on research evidence about the teaching of algebra • To plan progression from Y7 to Y9 in constructing and solving linear equations Key Stage 3 National Strategy Mathematics © Crown copyright 2004 1.1 Objectives • To consider pupils’ perceptions of algebra • To consider the distinctive features of algebra in KS3 • To consider the ‘why’ and the ‘how’ of teaching algebra Key Stage 3 National Strategy Mathematics © Crown copyright 2004 1.2 As many ways as you can Perimeter = 12 + 7 + 12 + 7 Walking round the shape Perimeter = 12 + 12 + 7 + 7 Pairing up lengths and widths Perimeter = 12 × 2 + 7 × 2 Length-doubled plus width-doubled Perimeter = 2 × 12 + 2 × 7 Double-length plus double-width Perimeter = (12 + 7) × 2 Length-plus-width, doubled Perimeter = 2 × (12 + 7) Two lots of length-plus-width Key Stage 3 National Strategy Mathematics © Crown copyright 2004 1.3 An opportunity to generalise Perimeter = l + w + l + w Walking round the shape Perimeter = l + l + w + w Pairing up lengths and widths Perimeter = l × 2 + w × 2 Length-doubled plus width-doubled Perimeter = 2 × l + 2 × w Double-length plus double-width Perimeter = (l + w) × 2 Length-plus-width, doubled Perimeter = 2 × (l + w) Two lots of length-plus-width Key Stage 3 National Strategy Mathematics © Crown copyright 2004 1.4 Generating expressions and equations Key Stage 3 National Strategy Mathematics © Crown copyright 2004 1.5 Pupils’ perceptions of algebra Pupils sometimes perceive algebra as difficult. What do your pupils struggle with most? Key Stage 3 National Strategy Mathematics © Crown copyright 2004 1.6 Why do we teach algebra? Mason and Sutherland argue that: • Algebra is fundamental to our way of life – It provides a mathematical language to express economic activity – It underpins models that inform national policy decisions – It underpins databases and spreadsheets • Algebra allows individuals to discover, establish and express generality Key Stage 3 National Strategy Mathematics © Crown copyright 2004 1.7 Adding two odd numbers Key Stage 3 National Strategy Mathematics © Crown copyright 2004 1.8 Distinctive features of algebra in KS3 • Developing understanding that algebra is a way of generalising from arithmetic, from particular cases or from patterns 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 Key Stage 3 National Strategy Mathematics © Crown copyright 2004 1.9 Distinctive features of algebra at KS3 • 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 Key Stage 3 National Strategy Mathematics © Crown copyright 2004 1.10 Four-digit numbers Write down a four-digit number. Reverse the digits and add the two numbers together. Is the answer divisible by 11? Does this work for any four-digit number? Can you prove it using algebra? Key Stage 3 National Strategy Mathematics © Crown copyright 2004 1.11 Symbol sense ‘Symbol sense’ includes: • an aesthetic feel for the power of symbols – understanding how and when symbols can and should be used to display relationships, generalisations and proofs which otherwise are hidden and invisible • an ability to manipulate and to read symbolic expressions as two complementary aspects of solving algebraic problems (Adapted from Arcavi, 1994, p. 31) Key Stage 3 National Strategy Mathematics © Crown copyright 2004 1.12 Key principles in teaching algebra 1. Providing opportunities for pupils to express generality 2. Asking pupils to ‘find as many ways as you can’ Key Stage 3 National Strategy Mathematics © Crown copyright 2004 1.13 Principle 1: Providing opportunities for pupils to express generality Benefits: • Pupils appreciate the purpose of algebra • Pupils are better able to understand the meaning of expressions if they have generated some for themselves • Knowing how expressions are built up helps to clarify the process of ‘undoing’, needed when solving equations Key Stage 3 National Strategy Mathematics © Crown copyright 2004 1.14 Principle 2: Asking pupils to ‘find as many ways as you can’ Benefits: • Pupils appreciate that the same general relationship can be expressed in more than one way • They manipulate expressions to demonstrate that one expression is equivalent to another • They experience forming and transforming expressions in different ways • They have opportunities to discuss which transformations are the most efficient to use in a particular context, e.g. when solving an equation Key Stage 3 National Strategy Mathematics © Crown copyright 2004 1.15 True or false? If 6x + 8 = 4x + 11, which of the following statements are true? 14x = 15x 6x + 4x = 11 + 8 6x = 4x + 3 2x + 8 = 11 2x = 3 –3 = –2x Key Stage 3 National Strategy Mathematics © Crown copyright 2004 1.16 Crown copyright statement The content of this presentation may be reproduced free of charge by schools, local education authorities and ITT providers provided that the material is acknowledged as Crown copyright, the publication title is specified, it is reproduced accurately and not used in a misleading context. Anyone else wishing to reuse part or all of the content of this publication should apply to HMSO for a core licence. The permission to reproduce Crown copyright protected material does not extend to any material in this publication which is identified as being the copyright of a third party. Applications to reproduce the material from this publication should be addressed to: HMSO, The Licensing Division, St Clements House, 2–16 Colegate, Norwich NR3 1BQ Fax: 01603 723000 Email: [email protected] Key Stage 3 National Strategy Mathematics © Crown copyright 2004 1.17